2.1 Non-dimensional key groups and orders of magnitude

We conveniently describe the flow in an inertial frame of reference, using polar coordinates $r$ , $z$ made non-dimensional with $\tilde{a}$ in directions respectively radial from the stagnation point ${\mathcal{S}}$ , i.e. the centre of rotation, and normal from the wetted face of the disc. Accordingly, let $u$ , $v$ , $w$ denote the velocity components, non-dimensional with $\tilde{U}$ , in the radial, azimuthal and normal directions respectively, and $p$ the pressure in the flow, non-dimensional with $\tilde{\unicode[STIX]{x1D70C}}\tilde{U}^{2}$ . Our focus lies on the flow downstream of jet impingement. Henceforth, $z=h(r)$ rather than its inverse $r=a(z)$ preferably describes the position of the free surface $\unicode[STIX]{x1D6F4}$ such that $h$ denotes the local height of the liquid layer and $a$ its local radial extent (double valued due to radial contraction of the freely falling jet).

Dimensional analysis shows that the sought flow quantities $[u,v,w,p](r,z)$ and $h(r)$ are parametrised by the complete set of non-dimensional key groups equivalent to the above physical input quantities which characterise the flow:

(2.1a-c ) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}:=\tilde{d}/\tilde{a},\quad r_{e}:=\tilde{r}_{e}/\tilde{a},\quad h_{e}:=\tilde{h}_{e}/\tilde{a},\end{eqnarray}$$

(2.1d-g ) $$\begin{eqnarray}\displaystyle Re_{a}:=\frac{\tilde{U}\tilde{a}}{\tilde{\unicode[STIX]{x1D708}}}=\frac{\tilde{Q}}{\unicode[STIX]{x03C0}\tilde{a}\tilde{\unicode[STIX]{x1D708}}},\quad Ro_{a}:=\frac{\tilde{U}}{\tilde{a}\tilde{\unicode[STIX]{x1D6FA}}}=\frac{\tilde{Q}}{\unicode[STIX]{x03C0}\tilde{a}^{3}\tilde{\unicode[STIX]{x1D6FA}}},\quad Fr_{a}:=\frac{\tilde{U}}{\sqrt{\tilde{g}\tilde{a}}},\quad We_{a}:=\frac{\tilde{\unicode[STIX]{x1D70C}}\tilde{U}^{2}\tilde{a}}{\tilde{\unicode[STIX]{x1D70E}}}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

In (2.1d–g ), $Re_{a}$ is the Reynolds number formed with the typical jet radius $\tilde{a}$ and characteristic of the internal flow upstream of the nozzle, $Ro_{a}$ the Rossby number $Ro_{a}$ measuring the ratio of inertial to centrifugal forces, $Fr_{a}$ the Froude and $We_{a}$ the Weber number, all defined consistently. The associated Bond and capillary numbers $Bo_{a}=We_{a}/Fr_{a}^{2}$ and $Ca_{a}=We_{a}/Re_{a}$ provide useful combinations.

The vast majority of applications advocate the usual assumption of a slender free jet and radially spreading film. Moreover, it is realistically accompanied by that of large values of $Re_{a}$ :

(2.2a-c ) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\gg 1,\quad r_{e}\gg 1,\quad Re_{a}\gg 1.\end{eqnarray}$$

Simultaneously, $Re_{a}$ is taken as sufficiently small so that laminar–turbulent transition of the flow is expected to occur downstream of any region considered subsequently (a realistic scenario as pointed out by Higuera (Reference Higuera1994)). We add the important restriction

(2.3) $$\begin{eqnarray}Re_{a}/\unicode[STIX]{x1D6FF}\gg 1\end{eqnarray}$$

as this furnishes an appealing possibility to control the behaviour of the jet and, in turn, the film flow (in terms of its purging/etching performances, for instance). The asymptotic analysis will make clear that particular order-of-magnitude requirements for $Ro_{a}$ , $Fr_{a}$ , $We_{a}$ are not as crucial (and can be relaxed) as the fundamental ones in (2.2). Although justified for the majority of potential applications, largeness of $We_{a}$ , say, does not substantially simplify the treatment of jet deflection where capillary effects are thus kept for the sake of generality (§ 3.2). Most important, assuming formal largeness of $Ro_{a}^{2}$ is backed by the scaling of the thin wall-bounded film, associated with a least-degenerate flow description (§ 3.3.1).

The data of the input quantities listed in table 1 refer to an example typical of genuine application: a liquid/environment constellation water/air at standard conditions. Table 2 displays the values of the corresponding non-dimensional parameters. It is noteworthy that a wide range of species and flow configurations of practical interest give $Bo_{a}=O(1)$ and $Ca_{a}\ll 1$ . In the authors’ opinion, the data in table 1 represent a reliable input for the vast amount of engineering problems. Subsequently, any range of values discussed refers essentially to the variation of $\tilde{\unicode[STIX]{x1D6FA}}$ in table 1, viz. in table 2 that of $Ro_{a}^{2}$ as a key quantity subsequently.

Table 1. Typical input values (water jet at $20^{\circ }$ , $\tilde{U}$ calculated, rounding due to the usual measurements).

Table 2. Values of dimensionless groups according to table 1 (suitably rounded).

It is instructive to first pose the flow problem in full, irrespective of approximations ensuing from the preceding order-of-magnitude estimates.

2.2 Governing equations

The computational domain is fixed by $0\leqslant r\leqslant r_{e}$ , $0\leqslant z\leqslant h\leqslant \unicode[STIX]{x1D6FF}$ . As usual, subscripts unambiguously indicate partial derivatives.

The equations of motion comprise the continuity equation

(2.4) $$\begin{eqnarray}(ru)_{r}+rw_{z}=0,\end{eqnarray}$$

conveniently satisfied identically through a streamfunction $\unicode[STIX]{x1D713}(r,z)$ defined by

(2.5a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D713}_{z}=ru,\quad \unicode[STIX]{x1D713}_{r}=-rw,\end{eqnarray}$$

and the NSE for, respectively, the radial, azimuthal and vertical directions

(2.5c ) $$\begin{eqnarray}uu_{r}+wu_{z}-v^{2}/r=Re_{a}^{-1}\{[(ru)_{r}/r]_{r}+u_{zz}\}-p_{r},\end{eqnarray}$$

(2.5d ) $$\begin{eqnarray}uv_{r}+wv_{z}+uv/r=Re_{a}^{-1}\{[(rv)_{r}/r]_{r}+v_{zz}\},\end{eqnarray}$$

(2.5e ) $$\begin{eqnarray}uw_{r}+ww_{z}=Re_{a}^{-1}\{(rw_{r})_{r}/r+w_{zz}\}-p_{z}-Fr_{a}^{-2}.\end{eqnarray}$$

The kinematic BCs

(2.5f-i ) $$\begin{eqnarray}\unicode[STIX]{x1D713}(0,z)=w_{r}(0,z)=0,\quad \unicode[STIX]{x1D713}(r,0)=u(r,0)=0,\quad v(r,0)=rRo_{a}^{-1},\quad \unicode[STIX]{x1D713}(r,h(r))={\textstyle \frac{1}{2}}\end{eqnarray}$$

account for axial symmetry, no penetration and no slip at the disc, and conservation of the volumetric flow rate. Finite disc rotation yields a non-trivial azimuthal flow component $v$ above the disc by viscous diffusion, so that (2.5d ) involving the Coriolis acceleration is non-degenerate. The resultant centrifugal body force showing up in (2.5c ) and the mutual coupling of (2.5d ) with (2.5c ) are of specific interest in this study.

In order to formulate the dynamic BCs, required to determine $h(r)$ , we recall the well-known Bonnet expression for the mean curvature

(2.5j ) $$\begin{eqnarray}\unicode[STIX]{x1D705}(r)=-\frac{\text{d}[rh^{\prime }(1+h^{\prime 2})^{-1/2}]/\text{d}r}{2r}=-\frac{h^{\prime }}{2r(1+h^{\prime 2})^{1/2}}-\frac{h^{\prime \prime }}{2(1+h^{\prime 2})^{3/2}}\end{eqnarray}$$

of $\unicode[STIX]{x1D6F4}$ and decompose the viscous surface stress, non-dimensional with $\tilde{\unicode[STIX]{x1D70C}}\tilde{U}^{2}$ , into its tangential components $\unicode[STIX]{x1D70F}_{m}$ and $\unicode[STIX]{x1D70F}_{a}$ , referring to the median and the azimuthal direction respectively, and its normal component $\unicode[STIX]{x1D70F}_{n}$ . This gives

(2.5k ) $$\begin{eqnarray}Re_{a}(1+h^{\prime 2})\unicode[STIX]{x1D70F}_{m}\equiv [2h^{\prime }(w_{z}-u_{r})+(1-h^{\prime 2})(u_{z}+w_{r})]_{z=h}=0,\end{eqnarray}$$

(2.5l ) $$\begin{eqnarray}Re_{a}(1+h^{\prime 2})^{1/2}\unicode[STIX]{x1D70F}_{a}\equiv [v_{z}-h^{\prime }(v_{r}-v/r)]_{z=h}=0,\end{eqnarray}$$

(2.5m ) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{n}\equiv \frac{2[h^{\prime 2}u_{r}-h^{\prime }(u_{z}+w_{r})+w_{z}]_{z=h}}{Re_{a}(1+h^{\prime 2})}=p(r,h(r))-\frac{2\unicode[STIX]{x1D705}(r)}{We_{a}}.\end{eqnarray}$$

The following excursus elucidates these zero-stress simplifications, resorting to the estimates stated at the beginning of § 2. The liquid and ambient gaseous phase share the flow speed and the shear stress on their interface $\unicode[STIX]{x1D6F4}$ . Then the inertia terms and, in turn, the balanced shear stress gradients in the adjacent liquid shear layer and the relatively thin induced gaseous drag layer are of comparable magnitude. Accordingly, $\sqrt{\tilde{\unicode[STIX]{x1D708}}/\tilde{\unicode[STIX]{x1D708}}_{e}}$ measures the ratio of their penetration depths normal to $\unicode[STIX]{x1D6F4}$ , thus $(\tilde{\unicode[STIX]{x1D707}}/\tilde{\unicode[STIX]{x1D707}}_{e})\sqrt{\tilde{\unicode[STIX]{x1D708}}_{e}/\tilde{\unicode[STIX]{x1D708}}}=\sqrt{(\tilde{\unicode[STIX]{x1D70C}}/\tilde{\unicode[STIX]{x1D70C}}_{e})(\tilde{\unicode[STIX]{x1D707}}/\tilde{\unicode[STIX]{x1D707}}_{e})}\gg 1$ that of the shear stresses inside those layers. Equating these when evaluated on $\unicode[STIX]{x1D6F4}$ implies vanishing non-dimensional shear stress as expressed by (2.5k ) and (2.5l ); extending this consideration to the normal-force balance on $\unicode[STIX]{x1D6F4}$ leads to (2.5m ).

The problem is closed once appropriate upstream and downstream conditions are posed. It is reasonable to assume a more-or-less fully developed Hagen–Poiseuille flow in a cylindrical tube ending concentrically in the relatively short, slender nozzle (both having circular cross-sections). It is controlled by the supply and the discharge pressure, the latter given by $We_{a}^{-1}$ according to (2.5j ), (2.5m ). By the largeness of $Re_{a}$ , the flow undergoes a predominantly inviscid modification inside the nozzle. This raises an idea which might be of interest in engineering applications but, to the authors’ knowledge, is actually not realised so far: as noted above, controlling $w$ and hence the behaviour of the thin radial wall flow can be accomplished via a properly designed axial shape of the nozzle. Its suitably tapered and smooth (polished) inside wall maintains stable laminar flow (cf. McCarthy & Molloy Reference McCarthy and Molloy1974). Specifically, for the sake of simplicity, let its contour exhibit a vertical tangent at the orifice. Then $u$ can be considered as negligibly small there, which motivates the model initial conditions (ICs)

(2.5n ) $$\begin{eqnarray}[u,v,w,p](r,\unicode[STIX]{x1D6FF})=[0,0,w_{o}(r),We_{a}^{-1}]\quad (0\leqslant r\leqslant 1),\quad h(1)=\unicode[STIX]{x1D6FF}.\end{eqnarray}$$

Here it is sensible to describe the velocity profile $w_{o}(r)$ by a superposition of a typical parabolic profile and a uniform one, accounting for the targeted distortion of the former:

(2.5o ) $$\begin{eqnarray}w_{o}(r)=2(1-\unicode[STIX]{x1D707})r^{2}-2+\unicode[STIX]{x1D707}\quad ({\leqslant}0,0\leqslant \unicode[STIX]{x1D707}\leqslant 1).\end{eqnarray}$$

The parameter $\unicode[STIX]{x1D707}$ specifies variations of the nozzle shape and controls the vorticity introduced under the last condition in (2.5f–i ). The limits $\unicode[STIX]{x1D707}=0$ and $\unicode[STIX]{x1D707}=1$ refer to the full Hagen–Poiseuille and a parallel flow respectively. Notably, the irrotational contribution violates the no-slip condition at the nozzle outlet. However, this is met correctly by a rather steep but smoothing gradient of $w_{o}$ as $r\rightarrow 1$ in a BL where $1-r=O(Re_{a}^{-1/2})$ originating inside the nozzle. This renders the proposed flow model admissible within our requirements of asymptotic accuracy, given the primarily inviscid flow inside of the nozzle suggested to modify the flow in a supply pipe further upstream. Furthermore, the current deliberate simplification of a one-parametric in lieu of a more involved model of $w_{o}(r)$ is acceptable given the possibility of further analytical progress, and the focus of our study. Finally, specifying

(2.5p ) $$\begin{eqnarray}h(r_{e})=h_{e},\end{eqnarray}$$

accommodates the ellipticity of the flow problem. More advanced and physically more meaningful replacements of (2.5p ) are conceivable in the shallow-water limit (cf. Higuera (Reference Higuera1994) and the appendix A).

Also, it proves expedient to identify the components $-v_{z}$ , $u_{z}-w_{r}$ , $(rv)_{r}/r$ of the vorticity for the $r$ -, azimuthal, and $z$ -direction respectively.

Any attempt to solve the full flow problem governing $[\unicode[STIX]{x1D713},v](r,z)$ and the position $z=h(r)$ of the free surface $\unicode[STIX]{x1D6F4}$ as formulated by (2.5p ) numerically requires a major subtleness; at least to the extent of the desired grade of resolution. Specifically the better the restriction (2.3) is satisfied, the more calculating the position of $\unicode[STIX]{x1D6F4}$ with satisfactory accuracy is compromised. This becomes obvious from the slenderness of both the jet and the resultant radially spreading film; the first associated with viscous forces acting in the aforementioned rather slowly growing thin sublayers along $\unicode[STIX]{x1D6F4}$ , the second with a complex interplay of viscous and centrifugal forces. This without doubt renders the problem an intriguing one. Consequently, resolving the richness of interesting flow details is not routine. In fact, it has not been accomplished so far. These assessments strongly substantiate the need for asymptotic techniques. A systematic reduction process then obviates the need of a multiparametric, full numerical study of (2.5p ). The so obtained much more feasible one highlights all the features of the film flow on the conditions (2.2) and (2.3) at drastically reduced computational costs and under variation of only a few similarity parameters (§ 4). Most important, the achieved parabolicity of the leading-order problem makes (2.5p ) ultimately obsolete by suppressing the difficile upstream influence by $r_{e}$ , $h_{e}$ in (2.1a–c ).

3.1 Jet regime: $r=O(1)$ , $h\gg 1$

We first make reference to the earlier discussion of the radial distribution of $w$ and an associated thin edge layer. Initially, $w\sim w_{o}$ given by (2.5o ) holds in the bulk of the just forming jet where $a-r$ is not too small; in the second viscosity regularises (for any value of $Re_{a}$ ) the jump at the orifice from a no-slip towards a free-slip condition (or $w=O(a-r)$ to $w_{r}=O(a-r)$ as $r\rightarrow a$ ). The latter follows from (2.5k ) in the ‘jet limit’ $h^{\prime }\rightarrow -\infty$ .

For any howsoever small initial surface slip $\unicode[STIX]{x1D707}>0$ , Khayat (Reference Khayat2016) aids in describing the non-trivial local modification of the aforementioned shear layer. If $\unicode[STIX]{x1D707}=0$ , the findings by Khayat (Reference Khayat2017) and the regularisation of the Goldstein singularity by localised viscous–inviscid interaction of the type put forward by Scheichl, Bowles & Pasias (Reference Scheichl, Bowles and Pasias2018) apply. Since vertical convection balances viscous diffusion in that edge layer, inspection analysis of (2.5e ) yields the order-of-magnitude estimates $w\sim a-r$ , by (2.5o ), and $w^{2}/(\unicode[STIX]{x1D6FF}-z)\sim w/[Re_{a}(a-r)^{2}]$ there. One then typically locates its formation immediately downstream of interaction for

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D707}=0:\quad (a-r)^{3}\sim (1-\breve{z})\unicode[STIX]{x1D6FF}/Re_{a}\ll 1,\quad \breve{z}:=z/\unicode[STIX]{x1D6FF}.\end{eqnarray}$$

Slenderness of the bulk region of the jet implies $u=O(\unicode[STIX]{x1D6FF}^{-1})$ via (2.4) and (2.5o ) and $p\sim (We_{a}\,a)^{-1}+O(\unicode[STIX]{x1D6FF}^{-2})$ according to (2.5m ) and (2.5c ). To confirm the previous and subsequent scalings as adequate for the targeted flow control, let us first elucidate the flow in a least-degenerate fashion. To this end, we relax (2.3) and take $Re_{a}/\unicode[STIX]{x1D6FF}$ but also $We_{a}$ as of $O(1)$ . This fastens the merging of the edge BL and the bulk layer by viscous diffusion. There the one-term expansions $\unicode[STIX]{x1D713}\sim \breve{\unicode[STIX]{x1D713}}(r,\breve{z})$ , $w\sim \breve{w}=-\breve{\unicode[STIX]{x1D713}}_{r}/r$ , $a\sim \breve{a}(\breve{z})$ ( $\breve{z}=O(1)$ , $\unicode[STIX]{x1D6FF}\gg 1$ ) and (2.5a,b ) yield the shear-layer approximation

(3.2a ) $$\begin{eqnarray}\breve{\unicode[STIX]{x1D713}}_{r}\breve{\unicode[STIX]{x1D713}}_{r\breve{z}}-\breve{\unicode[STIX]{x1D713}}_{\breve{z}}\breve{\unicode[STIX]{x1D713}}_{rr}+\breve{\unicode[STIX]{x1D713}}_{\breve{z}}\breve{\unicode[STIX]{x1D713}}_{r}/r=-r^{2}\breve{p}^{\prime }(\breve{z})-(\unicode[STIX]{x1D6FF}/Re_{a})r[r(\breve{\unicode[STIX]{x1D713}}_{r}/r)_{r}]_{r}\end{eqnarray}$$

of (2.5e ) with the suitably reduced pressure, zero at the nozzle exit ( $\breve{z}=1$ ),

(3.2b ) $$\begin{eqnarray}\breve{p}(\breve{z}):=Fr_{d}^{-2}(\breve{z}-1)+We_{a}^{-1}(\breve{a}^{-1}-1),\quad Fr_{d}:=Fr_{a}/\sqrt{\unicode[STIX]{x1D6FF}}.\end{eqnarray}$$

For the resulting typical value of the Froude number $Fr_{d}$ formed with the height of fall $\tilde{d}$ see table 2. Equation (3.2a ) is then subject to the ICs and kinematic BCs

(3.2c,d ) $$\begin{eqnarray}[(\breve{\unicode[STIX]{x1D713}}_{r}/r)(r,1),\breve{a}(1)]=[-w_{o}(r),1],\quad [\breve{\unicode[STIX]{x1D713}},(\breve{\unicode[STIX]{x1D713}}_{r}/r)_{r}](0,\breve{z})=[0,0],\breve{\unicode[STIX]{x1D713}}(\breve{a},\breve{z})={\textstyle \frac{1}{2}},\end{eqnarray}$$

cf. (2.5n ), (2.5o ) and (2.5f–i ) respectively, and the free-slip condition

(3.2e ) $$\begin{eqnarray}(\breve{\unicode[STIX]{x1D713}}_{r}/r)_{r}(\breve{a},\breve{z})=0\end{eqnarray}$$

provided by (2.5k ). Owing to the lack of a symmetry-breaking no-slip condition, the BL problem (3.2) governing $\breve{\unicode[STIX]{x1D713}}$ and $\breve{a}$ , as entering the sought pressure contribution, for $0\leqslant r\leqslant \breve{a}$ , $0<\breve{z}\leqslant 1$ is of parabolic type. That is, the jet does not ‘feel’ a feedback of its impact in the leading approximation: consequently, stopping the (numerical) downstream integration of (3.2) for $\breve{z}=0$ yields the terminal jet contraction $a_{0}:=\breve{a}(0)$ and $\breve{w}$ -profile $w_{0}(r):=\breve{w}(r,0)$ . For more interesting details see Tillett (Reference Tillett1968).

The assumption (2.3) ensures a predominantly inviscid and yet swirl-free ( $v\sim 0$ ) $90^{\circ }$ -deflection of the flow where $r,z=O(a_{0})$ and the jet approximation breaks down. By the largeness of $Re_{a}$ , the spin of the disc is indeed noticeable only via (2.5f–i ) in an adjacent BL, typically having a vertical extent not larger than of $O(Re_{a}^{-1/2})$ even for arbitrarily small values of $Ro_{a}$ . This scenario is investigated in § 3.3. It alleviates drastically the prediction of the modification of the initial $w$ -profile in the nozzle and thus the control of the $u$ -profile in the developed film (for etching/cleaning purposes). Assuming a fully viscous falling jet, however, requires the numerical solution of (3.2) and a systematic variation of $\unicode[STIX]{x1D6FF}/Re_{a}$ , $Fr_{d}$ , $We_{a}$ and $\unicode[STIX]{x1D707}$ . That is, as long as (2.3) is not met, a desirable simplification arises only if $Fr_{d}$ is small, i.e. $\tilde{d}$ chosen as correspondingly large, since the viscous term in (3.2a ) is predominant only for $\breve{z}=O(Fr_{d}^{2})$ . For larger values of $\breve{z}$ , the increasing action of gravity and the absence of external shear forces accelerate the jet whereas vorticity is no longer diffused, and inspection of (3.2) recovers Toricelli’s law for uniform flow and the associated jet thinning:

(3.3a,b ) $$\begin{eqnarray}\breve{w}\sim -\sqrt{2(1-\breve{z})}/Fr_{d},\quad \breve{a}\sim 1/\sqrt{-\breve{w}}.\end{eqnarray}$$

(This scenario applies even for only moderately large values of $Re_{a}$ : then the viscous and, by (3.2c,d ), also the convective terms dominate (3.2a ) already for smaller values of $\breve{z}$ before gravity becomes important, which finally confirms the above results further downstream.)

The marked increase of the irrotational contribution to $w$ by (3.3) also hampers severely the desired control of the vorticity. Hence, the demand for a viable control strategy underpins the requirement (2.3), i.e. of a predominantly inviscid fall of the jet already for $\breve{z}=O(1)$ . Then the viscous term in (3.2a ) remains small, thus the edge BL slender and of a thickness $\unicode[STIX]{x1D6E5}:=\sqrt{\unicode[STIX]{x1D6FF}/Re_{a}}$ , see figure 1. In this inviscid-flow limit, the expensive systematic numerical investigation of the full problem (3.2) is superseded by the analytical evaluation of its first integral. This expresses Bernoulli’s theorem in von Mises form: $\breve{w}=-\sqrt{w_{o}^{2}-2\breve{p}}$ is a function of $\breve{\unicode[STIX]{x1D713}}$ and $\breve{z}$ . Integrating $r\breve{w}=-\breve{\unicode[STIX]{x1D713}}_{r}$ for $\breve{z}=1$ reveals $w_{o}=-\sqrt{(2-\unicode[STIX]{x1D707})^{2}-8(1-\unicode[STIX]{x1D707})\breve{\unicode[STIX]{x1D713}}}$ , according to (3.2c,d ) and (2.5o ), and once again for $\breve{z}\leqslant 1$ subject to (3.2c,d ) preservation of vorticity along streamlines:

(3.4a ) $$\begin{eqnarray}\displaystyle & \breve{\unicode[STIX]{x1D713}}=(\unicode[STIX]{x1D707}-1)r^{4}/2-\breve{w}_{c}(\breve{z})r^{2}/2,\quad \breve{w}_{c}(\breve{z}):=-\sqrt{(2-\unicode[STIX]{x1D707})^{2}-2\breve{p}(\breve{z})}, & \displaystyle\end{eqnarray}$$

(3.4b ) $$\begin{eqnarray}\displaystyle & \breve{w}=-\sqrt{(2-\unicode[STIX]{x1D707})^{2}-2\breve{p}-8(1-\unicode[STIX]{x1D707})\breve{\unicode[STIX]{x1D713}}}=2(1-\unicode[STIX]{x1D707})r^{2}+\breve{w}_{c}(\breve{z})\quad (0\leqslant r\leqslant \breve{a}).\qquad & \displaystyle\end{eqnarray}$$

$\breve{w}_{c}$

$We_{a}<\infty$

$We_{a}=\infty$

$\breve{a}$

(3.5) $$\begin{eqnarray}\unicode[STIX]{x1D6EC}(\breve{z}):=Fr_{d}^{-2}(1-\breve{z})+We_{a}^{-1}=(We_{a}\breve{a})^{-1}-\breve{p}(\breve{z}),\end{eqnarray}$$

$We_{a}$ (cf. Gurevich Reference Gurevich1961), and $1-\unicode[STIX]{x1D707}$ as a measure of the vorticity. It is advantageously written as

(3.6) $$\begin{eqnarray}[\breve{a}^{-2}+(1-\unicode[STIX]{x1D707})\breve{a}^{2}]^{2}=\breve{w}_{c}^{2}=2\unicode[STIX]{x1D6EC}-2(We_{a}\breve{a})^{-1}+(2-\unicode[STIX]{x1D707})^{2}.\end{eqnarray}$$

This finally yields the flow factors $a_{0}$ and $w_{0,c}:=\breve{w}_{c}(0)$ by (3.4a ) and (3.5), providing the match with the flows in the jet and the region of its marked deflection.

For the distributions of $\breve{a}$ and $\breve{w}_{c}$ see figure 2. Here varying $\unicode[STIX]{x1D707}$ has only little effect; according to table 3, with

(3.7) $$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{0}:=\unicode[STIX]{x1D6EC}(0)=Fr_{d}^{-2}+We_{a}^{-1}\end{eqnarray}$$

and the typical data for these two addends, a rather moderate end contraction rate $a_{0}$ ensues. In general, $0<\breve{a}\leqslant 1$ holds, and $\breve{a}$ decreases as $\breve{z}$ , $Fr_{d}^{-1}$ , $We_{a}^{-1}$ , $\unicode[STIX]{x1D707}$ decrease. Thus, $a_{0}$ becomes quite small for a very slender jet, elongated by gravity, eventually in the asymptotic limit (3.3) implied by (3.4b ), (3.6) and (3.2b ). The behaviour of $\breve{w}_{c}$ turns out to be strictly reciprocal. We note that jet formation at the nozzle exit is governed by the limit $\unicode[STIX]{x1D706}:=\unicode[STIX]{x1D6EC}-We_{a}^{-1}\rightarrow 0_{+}$ . In turn, setting $\unicode[STIX]{x1D6FE}:=1-\breve{a}$ gives $\breve{p}\sim -\unicode[STIX]{x1D706}+We_{a}^{-1}[\unicode[STIX]{x1D6FE}+\unicode[STIX]{x1D6FE}^{2}+O(\unicode[STIX]{x1D6FE}^{3})]$ where $\unicode[STIX]{x1D6FE}$ expands in three different forms: $\unicode[STIX]{x1D6FE}\sim [We_{a}^{-1}+2\unicode[STIX]{x1D707}(2-\unicode[STIX]{x1D707})]\unicode[STIX]{x1D706}+O(\unicode[STIX]{x1D706}^{2})$ ( $\unicode[STIX]{x1D707}>0$ ); $\unicode[STIX]{x1D6FE}\sim \sqrt{\unicode[STIX]{x1D706}/8}+O(\unicode[STIX]{x1D706})$ ( $\unicode[STIX]{x1D707}=We_{a}^{-1}=0$ ); $\unicode[STIX]{x1D6FE}\sim We_{a}\unicode[STIX]{x1D706}-We_{a}^{2}(8We_{a}^{-1}+1)\unicode[STIX]{x1D706}^{2}+O(\unicode[STIX]{x1D706}^{3})$ ( $\unicode[STIX]{x1D707}=0$ , $We_{a}<\infty$ ). Here the first two imply $\breve{w}_{c}\sim -(2-\unicode[STIX]{x1D707})+O(\unicode[STIX]{x1D706})$ , the second an irregular sudden expansion of the jet (smoothed out by surface tension, $We_{a}\gg 1$ ) and the last $\breve{w}_{c}\sim -(2-\unicode[STIX]{x1D707})-16(We_{a}\unicode[STIX]{x1D706})^{2}/(2-\unicode[STIX]{x1D707})+O(\unicode[STIX]{x1D706}^{3})$ . As for what follows, even more important, we also find the behaviour

(3.8) $$\begin{eqnarray}[\breve{a},\breve{w}_{c}]\sim [(2\unicode[STIX]{x1D6EC})^{-1/4},\,-(2\unicode[STIX]{x1D6EC})^{1/2}]+O([\unicode[STIX]{x1D6EC}^{-9/4},\unicode[STIX]{x1D6EC}^{-1/2}][1+We_{a}^{-1}\unicode[STIX]{x1D6EC}^{1/4}])\quad (\unicode[STIX]{x1D6EC}\gg 1).\end{eqnarray}$$

Figure 2. Jet contraction $\breve{a}$ (a) and centre speed $\breve{w}_{c}$ (b) by varying $\unicode[STIX]{x1D6EC}$ , $We_{a}$ , $\unicode[STIX]{x1D707}$ (same legend).

Table 3. Characteristic values governing jet deflection and the thin film according to table 2.

The parabolic contribution to $\breve{w}$ scales with $\breve{a}^{2}$ but the irrotational one $\breve{w}_{c}$ with $\breve{a}^{-2}$ . We hence consider the characteristic flow speed $\tilde{U}_{0}:=\tilde{U}/a_{0}^{2}$ and jet radius $\tilde{a}_{0}:=a_{0}\tilde{a}$ in the region of pronounced jet deflection $z=O(a_{0})$ , with (3.3) providing a lower bound for $a_{0}$ . In turn, adequately defined Froude, Reynolds and Weber numbers are

(3.9a-c ) $$\begin{eqnarray}Fr_{0}:=\tilde{U}_{0}/\sqrt{\tilde{g}\tilde{a}_{0}}=Fr_{a}/a_{0}^{5/2},\quad Re_{0}:=Re_{a}/a_{0},\quad We_{0}:=We_{a}/a_{0}^{3}.\end{eqnarray}$$

The premise (2.3) renders (3.3) valid for $Fr_{d}\ll 1$ in (3.2b ), which implies $\unicode[STIX]{x1D6EC}\sim Fr_{d}^{-2}$ and $a_{0}\sim Fr_{d}^{1/2}/2^{1/4}$ per (3.5) and (3.8), hence $Fr_{0}/Fr_{a}\sim 2^{5/8}/Fr_{d}^{5/4}=(2\unicode[STIX]{x1D6FF})^{5/8}/Fr_{a}^{5/4}$ . The values in table 2 also imply $Fr_{0}\gg 1$ , but rather due to the largeness of $Fr_{a}$ than the smallness of $Fr_{d}$ . The latter is argued by Higuera (Reference Higuera1994, p. 72, bottom) when he claims that ‘ $\tilde{U}_{0}^{2}/(2\tilde{g})$ is the head of the fluid in the jet, typically large compared with its half-width $a_{0}$ ’ (our notations used) – which means $Fr_{0}\gg 1$ finally. The present view agrees on neglecting the effect of gravity on jet bending but sees it only moderately increasing $\breve{w}_{c}$ and decreasing $a_{0}$ as $Fr_{d}$ is considered to take on rather not so small corresponding values. In the irrotational-flow limit $\unicode[STIX]{x1D707}\rightarrow 1_{-}$ , one also confirms $a_{0}=O(1)$ for $\breve{w}\sim \breve{w}_{c}$ and again $\breve{a}\sim 1/\sqrt{-\breve{w}}=1/[1+2\unicode[STIX]{x1D6EC}-2/(We_{a}\breve{a})]^{1/4}$ (cf. figure 2), but now holding for the more realistic, moderate values of $Fr_{d}$ . However, in the following $Fr_{d}$ and thus $a_{0}$ are allowed to become very small. Therefore, taking $\tilde{a}_{0}$ as the new length scale representative of jet impingement includes this case conveniently.

With (3.2a ), (3.2c,d ) and (3.2e ), we now may also be more specific as to the viscous perturbations of $\breve{\unicode[STIX]{x1D713}}$ and hence the surface slip $W(\breve{z}):=-\breve{w}(\breve{a},\breve{z})\;({>}0)$ by the edge shear layer. There $\unicode[STIX]{x1D713}\sim 1/2-\unicode[STIX]{x1D6E5}W+\unicode[STIX]{x1D6E5}^{2}\unicode[STIX]{x1D6F9}(\breve{R},\breve{z})$ where $\breve{R}:=[\breve{a}(\breve{z})-r]/\unicode[STIX]{x1D6E5}=O(1)$ so that the rescaled streamfunction $\unicode[STIX]{x1D6F9}$ satisfies the locally linearised form of (3.2),

(3.10a ) $$\begin{eqnarray}\displaystyle & (\breve{a}W)^{\prime }(\unicode[STIX]{x1D6F9}_{\breve{R}}-\breve{R}\,\unicode[STIX]{x1D6F9}_{\breve{R}\breve{R}})-2\breve{a}^{\prime }W\unicode[STIX]{x1D6F9}_{\breve{R}}+\breve{a}W\unicode[STIX]{x1D6F9}_{\breve{R}\breve{z}}=\breve{a}\,\unicode[STIX]{x1D6F9}_{\breve{R}\breve{R}\breve{R}}, & \displaystyle\end{eqnarray}$$

(3.10b ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D6F9}(0,z)=\unicode[STIX]{x1D6F9}_{\breve{R}}(0,\breve{z})=0,\quad \unicode[STIX]{x1D6F9}_{\breve{R}\breve{R}}(\infty ,\breve{z})=\breve{\unicode[STIX]{x1D713}}_{rr}(0,\breve{z}). & \displaystyle\end{eqnarray}$$

$W\rightarrow \unicode[STIX]{x1D707}$

$\breve{z}\rightarrow 1$

$\unicode[STIX]{x1D707}$

$\unicode[STIX]{x1D6E5}$

$1-\breve{z}=O(\unicode[STIX]{x1D6E5})$

3.2 Jet-bending regime:  $r=O(a_{0})$ , $h=O(a_{0})$

As stated above, all the assumptions (2.2) and (2.3) are of paramount importance for enabling a desirable simplified analysis of both the jet and its massive deflection. Concerning its bulk regime, we have just sketched a flow picture consistent with the scalings proposed in § 2.1. According to the definitions (3.9), here jet bending is categorised most precisely by the validity of the following order-of-magnitude estimates:

1. (i) $Re_{0}\gg 1$ , $Fr_{0}\gg 1$ ; in this respect,

2. (ii) $We_{0}^{-1}$ assumes values ranging from of $o(1)$ to $O(1)$ ( $\simeq 0.0210$ here); also

3. (iii) $a_{0}$ assumes values ranging from of $o(1)$ to $O(1)$ (meaning a very large dimensionless fall $\unicode[STIX]{x1D6FF}$ and a correspondingly small relevant Bond number $Bo_{0}:=We_{0}/Fr_{0}^{2}=Bo_{a}a_{0}^{2}$ ).

By (i), jet impingement is seen to be a predominantly inviscid process. The lower bound in prerequisite (ii) asserts that jet bending might be significantly affected by capillary effects. Given the original presumption $Bo_{a}=O(1)$ and the findings $Fr_{a}\gg 1$ and issue (iii) of § 3.1, all justified by the associated data, this represents a relaxation we allow for hereafter as it is readily included in the subsequent asymptotic analysis of the flow properties. In this most general setting, jet inflexion is then asymptotically described by (i) and the match with the developed jet flow where $We_{a}$ , $Fr_{d}$ and thus $a_{0}$ are taken as of $O(1)$ .

Before we proceed, a few words should be said on the concept of a steady and sufficiently smooth jet under the capillary impact accounted for by the tacitly made assumptions $We_{a}=O(1)$ (above) and $We_{a}<We_{0}=O(1)$ (below). As was pointed out kindly by a referee, the capillary hoop stress induces the well-known Rayleigh–Plateau instability if, in our setting, the value of $\unicode[STIX]{x1D6FF}$ exceeds the critical wavelength, $\unicode[STIX]{x1D6FF}_{c}\simeq 2.74\times \sqrt{We_{a}}$ (see e.g. Chandrasakhar Reference Chandrasakhar1981, p. 537 ff.). The practically ubiquitous values summarised in tables 1 and 2, consistent with the essential scaling of the flow in this and related studies, predict $\unicode[STIX]{x1D6FF}_{c}\simeq 13.6$ and thus indeed supercritical conditions. Conversely, one then would like to have $We_{a}$ being sufficiently larger than $(\unicode[STIX]{x1D6FF}/2.74)^{2}$ , i.e. $(20/2.74)^{2}\simeq 53$ here, or even

(3.11) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}^{2}\ll We_{a}\end{eqnarray}$$

to safely avoid any danger of a dripping rather than a contiguous jet. However, the proposed jet flow can be easily reproduced by a ‘classroom experiment’ with a tap in a kitchen or bath sink, where an only slightly wavy jet surface confirms that instability visually but dispels any concern regarding a markedly wavy jet or even its pinch-off. This observation is supported by experimental evidence from other studies dealing with jet impingement on rotating/non-rotating discs for jet geometries and kinematic viscosities comparable with those in table 1, where no signs of significant instabilities were reported: cf. Charwat et al. (Reference Charwat, Kelly and Gazley1972), Astarita & Cardone (Reference Astarita and Cardone2008), Mohajer & Li (Reference Mohajer and Li2015) (for an even more elongated jet) and Bhagat et al. (Reference Bhagat, Jha, Linden and Wilson2018) (specifically, we refer to figure 2 in the last study). It might be traced back to weak viscous damping in the dispersion relation entered by $Ca_{a}$ , which here is small. Therefore, assuming (3.11) proves a sufficient but not necessary criterion to avoid that instability. Admittedly, since $\unicode[STIX]{x1D6FF}_{c}$ decreases with $a$ , this might be of concern if capillarity considerably influences jet bending for $a_{0}\ll 1$ , i.e. a relatively large fall, entailing $We_{a}=We_{0}\,a_{0}^{3}\ll 1$ by (3.9). Nonetheless, the following analysis deliberately includes capillarity as a dominant effect (at a negligible small gravitational one) for the sake of generality. In other words, our analysis not only seems to confidently model reality but remains valid (and becomes accordingly simplified) if one takes $We_{a}$ as consistently large to avoid safely a marked waviness of the jet or even its breakage into droplets.

In leading order, (2.5c )–(2.5e ) reduce to the Euler equations by virtue of the expansions

(3.12a,b ) $$\begin{eqnarray}\{[\unicode[STIX]{x1D713},a_{0}^{4}\,p],v,h/a_{0}\}\sim \{[\bar{\unicode[STIX]{x1D713}},\bar{p}](\bar{r},\bar{z}),0,\bar{h}(\bar{r})\},\quad a_{0}^{2}[u,w]\sim [\bar{u},\bar{w}]=[\bar{\unicode[STIX]{x1D713}}_{\bar{z}}/\bar{r},-\bar{\unicode[STIX]{x1D713}}_{\bar{r}}/\bar{r}],\quad\end{eqnarray}$$

(3.12c ) $$\begin{eqnarray}[\bar{r},\bar{z}]:=[r,z]/a_{0}.\end{eqnarray}$$

Via the match with the jet flow, $[\bar{\unicode[STIX]{x1D713}},\bar{p}]\rightarrow [\breve{\unicode[STIX]{x1D713}}(a_{0}\bar{r},0),We_{0}^{-1}]$ ( $\bar{z}\rightarrow \infty$ ) with (3.4) and (3.5) for $\breve{z}=0$ , Bernoulli’s law expressed with the appropriately rescaled quantities in (3.12) becomes

(3.13) $$\begin{eqnarray}(\bar{u}^{2}+\bar{w}^{2})/2+\bar{p}=B(\bar{\unicode[STIX]{x1D713}}):=a_{0}^{4}w_{0}^{2}/2+We_{0}^{-1}=a_{0}^{4}[(2-\unicode[STIX]{x1D707})^{2}/2+\unicode[STIX]{x1D6EC}_{0}-4(1-\unicode[STIX]{x1D707})\bar{\unicode[STIX]{x1D713}}]\end{eqnarray}$$

with $B\sim 1/2$ , $w_{0}\sim w_{0,c}\sim -1/a_{0}^{2}$ for $a_{0}\ll 1$ by (3.8). Helmholtz’s transport equation for the vorticity $\bar{\unicode[STIX]{x1D714}}:=\bar{u}_{\bar{z}}-\bar{w}_{\bar{r}}$ then expresses the invariance of $\bar{\unicode[STIX]{x1D714}}/\bar{r}$ along streamlines in the form

(3.14a ) $$\begin{eqnarray}\bar{\unicode[STIX]{x1D714}}/\bar{r}\equiv \bar{\unicode[STIX]{x1D713}}_{\bar{\unicode[STIX]{x1D70C}}\bar{\unicode[STIX]{x1D70C}}}+\bar{\unicode[STIX]{x1D713}}_{\bar{z}\bar{z}}/(2\bar{\unicode[STIX]{x1D70C}})=B^{\prime }(\bar{\unicode[STIX]{x1D713}}),\quad \bar{\unicode[STIX]{x1D70C}}:=\bar{r}^{2}/2.\end{eqnarray}$$

This equation governs $\bar{\unicode[STIX]{x1D713}}$ subject to the kinematic BCs as hitherto and the dynamic one ensuing from (3.13) and (2.5m ). With $a_{0}\unicode[STIX]{x1D705}\sim \bar{\unicode[STIX]{x1D705}}(\bar{r})$ defining $\bar{\unicode[STIX]{x1D705}}$ , see (2.5j ), these now read

(3.14b,c ) $$\begin{eqnarray}\bar{\unicode[STIX]{x1D713}}(0,\bar{z})=\bar{\unicode[STIX]{x1D713}}(\bar{r},0)=\bar{\unicode[STIX]{x1D713}}(\bar{r},\bar{h}(\bar{r}))-{\textstyle \frac{1}{2}}=0,\quad (\bar{u}^{2}+\bar{w}^{2})(\bar{r},\bar{h}(\bar{r}))+4We_{0}^{-1}\bar{\unicode[STIX]{x1D705}}=2B({\textstyle \frac{1}{2}}).\quad\end{eqnarray}$$

Once the Bernoulli function $B$ is specified, the nonlinear free-surface problem (3.14), effectively parametrised by $\unicode[STIX]{x1D6EC}_{0}$ and $We_{0}$ where gravity plays a key role only far upstream but surface tension also locally, is already completed by the in- and outflux conditions

(3.14d,e ) $$\begin{eqnarray}\bar{h}\rightarrow \infty \quad (\bar{r}\rightarrow 1,\bar{z}\rightarrow \infty ),\quad \bar{h}\rightarrow 0\quad (\bar{r}\rightarrow \infty ).\end{eqnarray}$$

Together with (3.14a ) and the suppression of super-algebraically growing eigensolutions (or, equivalently, strict forward flow), these requirements anticipate the match of the jet bending with both the developed-jet and stratified-film regions. Flow reversal at $\unicode[STIX]{x1D6F4}$ is possible sufficiently upstream for $\unicode[STIX]{x1D705}>0$ but only once ( $\text{d}\unicode[STIX]{x1D705}/\text{d}r<0$ ) and is hence unacceptable. The first of the conditions (3.14d,e ) implies the above ones of matching as $\bar{u}\rightarrow 0$ , $\bar{\unicode[STIX]{x1D705}}\rightarrow 1/2$ ( $\bar{h}^{\prime }\rightarrow -\infty$ ) and involves a double valued $\bar{h}(\bar{r})$ in case the jet radius undergoes a minimum. The second complies with the ellipticity of (3.14a ) and states that $\bar{\unicode[STIX]{x1D713}}=O(1)$ and $\bar{u}=O(1)$ as $\bar{w}\rightarrow 0$ according to the last kinematic and the dynamic BC in (3.14b,c ) respectively.

As (3.14d,e ) entails $\bar{w}\rightarrow 0$ , one infers directly from (3.13), specifying $B$ via the incidental flow, that

(3.15) $$\begin{eqnarray}[\bar{\unicode[STIX]{x1D713}},\bar{u},\bar{p},\bar{h},\bar{\unicode[STIX]{x1D705}}\,]\sim \left[\frac{f_{0}(\unicode[STIX]{x1D702})}{2},\frac{f_{0}^{\prime }(\unicode[STIX]{x1D702})}{m_{0}},\frac{2\bar{\unicode[STIX]{x1D705}}}{We_{0}},\frac{m_{0}}{2\bar{r}},-\frac{m_{0}}{4\bar{r}^{3}}\right]\quad \left(\unicode[STIX]{x1D702}:=\frac{\bar{z}}{\bar{h}}=O(1),\bar{r}\rightarrow \infty \right),\end{eqnarray}$$

(3.16) $$\begin{eqnarray}f_{0}^{\prime }(\unicode[STIX]{x1D702})=m_{0}\sqrt{2B(f_{0}/2)},\quad f_{0}(0)=0,\quad f_{0}(1)=1,\end{eqnarray}$$

and $\bar{w}\sim -\unicode[STIX]{x1D702}f_{0}^{\prime }/(2\bar{r}^{2})$ accordingly. The relations (3.15) embody the conservation of the volumetric flow rate and of the linear momentum such that  $\bar{u}$ remains of $O(1)$ . Specifically, evaluation of (3.14b,c ) yields the asymptote of $\bar{p}$ and the problem (3.16) governing $f_{0}$ and the affine parameter $m_{0}$ that quantifies the radial flattening of $\unicode[STIX]{x1D6F4}$ . This completes the sought modification of the prescribed (parabolic) profile $w_{o}$ of nozzle outflow, see (2.5o ), via $w_{0}$ , see (3.4b ) and (3.13), towards the linear radial profile $f_{0}^{\prime }/m_{0}$ . This represents a most viable paradigm for the inviscid distortion of a developed jet by its deflection. It initiates the thin-film flow with the upstream history controlled by $Fr_{d}$ , $We_{a}$ , $\unicode[STIX]{x1D707}$ still in a quite general fashion. Recalling $-\bar{w}=\bar{\unicode[STIX]{x1D713}}_{\bar{\unicode[STIX]{x1D70C}}}\sim \sqrt{2B(\bar{\unicode[STIX]{x1D713}})-2We_{0}^{-1}}=a_{0}^{2}w_{0}$ ( $\bar{z}\rightarrow \infty$ ) expresses the acceleration of the fluid due to the pressure drop $We_{0}^{-1}$ accompanying jet bending. In view of the zoomed-in detail in figure 1 and (3.16), we set $u_{0}(\unicode[STIX]{x1D702}):=f_{0}^{\prime }(\unicode[STIX]{x1D702})/(a_{0}^{2}m_{0})$ , $u_{0,0}:=u_{0}(0)=\sqrt{w_{0,c}^{2}+2/(We_{a}a_{0})}$ . Knowing the function $B$ puts $f_{0}$ in closed form in two cases: firstly, an insignificant pressure recovery ( $We_{0}\gg 1$ ) gives $f_{0}(\unicode[STIX]{x1D702})=2\breve{\unicode[STIX]{x1D713}}(a_{0}\sqrt{m_{0}\unicode[STIX]{x1D702}},0)$ as noticed by Rubel (Reference Rubel1980); secondly, if $B(\bar{\unicode[STIX]{x1D713}})$ varies not stronger than quadratically, as is the case here, where $\breve{\unicode[STIX]{x1D713}}(r,0)$ and thus $w_{0}(r)$ can again be expressed explicitly.

Specifying $B$ by the (right-hand side of) (3.13) gives

(3.17) $$\begin{eqnarray}B^{\prime }(\bar{\unicode[STIX]{x1D713}})=-4a_{0}^{4}(1-\unicode[STIX]{x1D707})\end{eqnarray}$$

in (3.14a ) and yields with the aid of (3.4a ) and (3.5) for the radial slip exerted on the disc in the original scaling

(3.18) $$\begin{eqnarray}u_{0,0}=\sqrt{(2-\unicode[STIX]{x1D707})^{2}+2\unicode[STIX]{x1D6EC}_{0}}\end{eqnarray}$$

to leading order. We furthermore deduce from (3.16) (after some tedious manipulations)

(3.19a ) $$\begin{eqnarray}\displaystyle & f_{0}(\unicode[STIX]{x1D702})=\unicode[STIX]{x1D702}-\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D702}-\unicode[STIX]{x1D702}^{2})/2,\quad \unicode[STIX]{x1D70E}:=2a_{0}^{4}m_{0}^{2}(\unicode[STIX]{x1D707}-1)\quad (-2\leqslant \unicode[STIX]{x1D70E}\leqslant 0), & \displaystyle\end{eqnarray}$$

(3.19b ) $$\begin{eqnarray}\displaystyle & m_{0}=\frac{\displaystyle \sqrt{(2-\unicode[STIX]{x1D707})^{2}+2\unicode[STIX]{x1D6EC}_{0}}-\sqrt{\unicode[STIX]{x1D707}^{2}+2\unicode[STIX]{x1D6EC}_{0}}}{\displaystyle \sqrt{(2-\unicode[STIX]{x1D707})^{2}+2\unicode[STIX]{x1D6EC}_{0}-2/(We_{a}a_{0})}-\displaystyle \sqrt{\unicode[STIX]{x1D707}^{2}+2\unicode[STIX]{x1D6EC}_{0}-2/(We_{a}a_{0})}}\;({\leqslant}1),\quad \qquad & \displaystyle\end{eqnarray}$$

(3.19c ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D70E}=\left(\sqrt{(2-\unicode[STIX]{x1D707})^{2}+2\unicode[STIX]{x1D6EC}_{0}}-\sqrt{\unicode[STIX]{x1D707}^{2}+2\unicode[STIX]{x1D6EC}_{0}}\right)^{2}(2\unicode[STIX]{x1D707}-2), & \displaystyle\end{eqnarray}$$

(3.19d ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D70E}/(a_{0}^{2}m_{0})=\sqrt{\unicode[STIX]{x1D707}^{2}+2\unicode[STIX]{x1D6EC}_{0}}-\sqrt{(2-\unicode[STIX]{x1D707})^{2}+2\unicode[STIX]{x1D6EC}_{0}}. & \displaystyle\end{eqnarray}$$

$We_{a}$

$a_{0}$

$m_{0}$

$\unicode[STIX]{x1D6EC}_{0}$

$\unicode[STIX]{x1D707}$

$We_{a}$

$\unicode[STIX]{x1D6F4}$

$\unicode[STIX]{x1D70E}$

$\unicode[STIX]{x1D6EC}_{0}$

$\unicode[STIX]{x1D707}$

$u_{0}^{\prime }(\unicode[STIX]{x1D702})\equiv \unicode[STIX]{x1D70E}/(a_{0}^{2}m_{0})$

$f_{0}^{\prime \prime }(\unicode[STIX]{x1D702})\equiv \unicode[STIX]{x1D70E}$

$f_{0}^{\prime }(\unicode[STIX]{x1D702})$

$f^{\prime }(0)=1-\unicode[STIX]{x1D70E}/2$

$f_{0}^{\prime }(1)=1+\unicode[STIX]{x1D70E}/2$

Figure 3. Flattening factor $m_{0}$ (a,b) and shear rate $\unicode[STIX]{x1D70E}$ (c,d) under variation of $Fr_{d}$ , $We_{a}$ , $\unicode[STIX]{x1D707}$ : (a) legend see figure 2; (b) $We_{a}=1$ , $\unicode[STIX]{x1D6EC}_{0}=1.1,1.2,1.3,1.4,1.5$ (bottom to top), ▫ indicate extrema.

At first, $m_{0}$ ( $\unicode[STIX]{x1D70E}$ ) increases if $\unicode[STIX]{x1D6EC}_{0}$ ( $\unicode[STIX]{x1D6EC}_{0}$ or $\unicode[STIX]{x1D707}$ ) increases towards the asymptotes $m_{0}\sim 1+O(We_{a}^{-1}a_{0}^{-1})$ ( $We_{a}a_{0}\gg 1$ ), $[m_{0},\unicode[STIX]{x1D70E}]\sim [1+O(\unicode[STIX]{x1D6EC}_{0}^{-1}),(\unicode[STIX]{x1D707}-1)/\unicode[STIX]{x1D6EC}_{0}+O(\unicode[STIX]{x1D6EC}_{0}^{-2})]$ ( $\unicode[STIX]{x1D6EC}_{0}\gg 1$ ), $[m_{0},\unicode[STIX]{x1D70E}]\sim [\sqrt{1-2/[We_{a}a_{0}(1+2\unicode[STIX]{x1D6EC}_{0})]},\,-2\unicode[STIX]{x1D712}/(1+2\unicode[STIX]{x1D6EC}_{0})]+O(\unicode[STIX]{x1D712}^{2})$ ( $\unicode[STIX]{x1D712}:=1-\unicode[STIX]{x1D707}\rightarrow 0_{+}$ ). The variation of $m_{0}$ with respect to $\unicode[STIX]{x1D707}$ exhibits a second extremum (local minimum) for $We_{a}<\infty$ where the values of $\unicode[STIX]{x1D6EC}_{0}$ occupy a rather small range (figure 3 b). Also, the important assumptions $Fr_{d}\gg 1$ (negligibly small drop height of the jet), $We_{a}\gg 1$ imply

(3.20) $$\begin{eqnarray}m_{0}\sim 1,\quad \unicode[STIX]{x1D70E}\sim 2\unicode[STIX]{x1D707}-2+\unicode[STIX]{x1D6EC}_{0}(4-4\unicode[STIX]{x1D707})/(2\unicode[STIX]{x1D707}-\unicode[STIX]{x1D707}^{2})+O(\unicode[STIX]{x1D6EC}_{0}^{2})\quad (\unicode[STIX]{x1D6EC}_{0}\rightarrow 0),\end{eqnarray}$$

see figure 3(a,c). On the other hand, by (3.19c ) the limit $\unicode[STIX]{x1D70E}\rightarrow -2$ (negligibly small surface speed, maximum disc slip) means $\unicode[STIX]{x1D6EC}_{0}\rightarrow 0$ for $\unicode[STIX]{x1D707}=0$ (maximum vorticity).

The dynamic BCs in (3.14b,c ) confirm the results (3.19) and determine unambiguously the higher-order extension of (3.15), based upon (3.19a ). Investigation of (3.14a ) supplemented with (3.17) and the no-slip condition gives

(3.21) $$\begin{eqnarray}2[\bar{\unicode[STIX]{x1D713}},\bar{r}\bar{h}/m_{0}]\sim [f_{0}(\unicode[STIX]{x1D702}),1]+[f_{1}(\unicode[STIX]{x1D702}),m_{1}]/(We_{0}\bar{r}^{3})+O(\bar{r}^{-4})\quad (\bar{r}\rightarrow \infty ).\end{eqnarray}$$

Hence, the dominant correction to purely inviscid flattening of the jet is dictated by surface tension, but the next higher one of $O(\bar{r}^{-4})$ solely by convection. Taking into account that the capillary correction leaves the volume flux unchanged, we obtain $f_{1}^{\prime }(1)=-f_{1}^{\prime }(0)=m_{1}\unicode[STIX]{x1D70E}$ by

(3.22a,b ) $$\begin{eqnarray}f_{1}(\unicode[STIX]{x1D702})=2m_{1}[f_{0}(\unicode[STIX]{x1D702})-\unicode[STIX]{x1D702}],\quad m_{1}=2m_{0}^{3}/(\unicode[STIX]{x1D70E}^{2}-4)\;\;({<}0).\end{eqnarray}$$

The downstream tail of the locally strongly concave surface $\unicode[STIX]{x1D6F4}$ causes a pressure drop increasing $u\sim f_{0}^{\prime }/m_{0}$ by the amount $(m_{1}/m_{0})[f_{0}^{\prime }(\unicode[STIX]{x1D702})-2]/(We_{0}\bar{r}^{3})$ , stratifying the layer.

The flow speed $\bar{u}_{0}(\bar{r}):=\bar{u}(\bar{r},0)$ along the disc driving the BL and the associated penetration depth of fluid rotation deserve some comments regarding the full numerical solution of (3.14). In the authors’ opinion, here the most relevant candidates in the vast quantity of quotable studies involve various specifications of $B(\bar{\unicode[STIX]{x1D713}})$ or inflow profiles and semi-analytical techniques: see the overviews by Lienhard (Reference Lienhard V and Tien1995) and Webb & Ma (Reference Webb, Ma, Sparrow, Abraham and Gorham1995), the classical solutions of the potential-flow case ( $B^{\prime }\equiv 0$ ) via classical Green’s function approaches (Trefftz Reference Trefftz1916; Birkhoff & Zarantonello Reference Birkhoff and Zarantonello1957; Gurevich Reference Gurevich1966), later rectified by more tenable series expansions (cf. Liu, Gabour & Lienhard Reference Liu, Gabour and Lienhard V1993), and the likewise recent extension to the more intriguing and interesting case of a curved influx profile, albeit ignoring capillary effects ( $B^{\prime }\!\leqslant 0$ ; Phares, Smedley & Flagan Reference Phares, Smedley and Flagan2000a ). See also the overview on adopted approaches given by Deshpande & Vaishnav (Reference Deshpande and Vaishnav1982). This improves the original treatment, which was approximate, by resorting to slender-flow assumptions (Bradbury Reference Bradbury1972; Rubel Reference Rubel1980, Reference Rubel1983). The last two pioneering investigations merit specific mention. Most important, Rubel (Reference Rubel1983) demonstrated that a (closed) separated-flow region can only emerge adjacent to the disc and only if $B^{\prime }(\bar{\unicode[STIX]{x1D713}})$ takes on sufficiently large positive values owing to a centred dent in the influx profile $w_{o}(r)$ . Let us finally confirm this analytically for an eddy that is typically convex at the reattachment point $(\bar{r},\bar{z})=(\bar{r}_{r},0)$ with some $\bar{r}_{r}>0$ .

The branched portions of the streamline $\bar{\unicode[STIX]{x1D713}}=0$ confine that bubble. Since (3.14a ) expresses preservation of $\bar{\unicode[STIX]{x1D714}}/\bar{r}$ along the streamlines $\bar{\unicode[STIX]{x1D713}}=\text{const.}$ , we introduce the value $\bar{\unicode[STIX]{x1D714}}_{R}=\bar{r}_{r}B^{\prime }(0)$ of $\bar{\unicode[STIX]{x1D714}}$ carried by the forward flow. Then (3.14a ) together with the slip condition $\bar{\unicode[STIX]{x1D713}}(\bar{r},0)=0$ yields the quadratic approximation $\bar{\unicode[STIX]{x1D713}}\sim B^{\prime }(0)(\bar{r}_{r}\bar{z})^{2}/2+O((\bar{r}-\bar{r}_{r})\bar{z})$ for $(\bar{r}-\bar{r}_{r},\bar{z})\rightarrow (0_{+},0_{+})$ . Convexity of the reattaching part of the streamline $\bar{\unicode[STIX]{x1D713}}=0$ requires $\bar{u}\sim B^{\prime }(0)\,\bar{r}_{r}\bar{z}>0$ for sufficiently small values of $\bar{z}$ and thus $B^{\prime }(0)>0$ . Evaluation of (3.14a ) in the limit $\bar{z}\rightarrow \infty$ gives $B^{\prime }(\bar{\unicode[STIX]{x1D713}})=-a_{0}^{4}w_{0}^{\prime }(r)/r$ according to (3.12) and thereby $\lim _{r\rightarrow 0}w_{0}^{\prime }(r)/r=w_{0}^{\prime \prime }(0)<0$ , i.e. a ‘dented’ $w_{0}(r)$ -distribution. The analysis of § 3.1 indicates that this originates in the aforementioned depression of the $w_{o}(r)$ -profile. (Applying this rationale to the symmetric bending of a planar jet yields an analogous conclusion.)

Also, a description of closed streamlines of a steady and inviscid flow (considered as an asymptotic limit) remains intrinsically incomplete and must thus be accepted with some caution. Here, the modified Prandtl–Batchelor theorem holding for axisymmetric flow with a zero azimuthal component aids by predicting a constant value of $\bar{\unicode[STIX]{x1D714}}/\bar{r}$ within the toroidal recirculating-flow region. (The uncertainty in determining the recirculating flow was kindly pointed to us by one of the referees.)

3.3 Rotatory boundary layer

In all the cases considered here, (3.17) yields strict forward flow throughout the region of inviscid jet bending. Moreover, the radial convergence of the streamlines according to (3.14a ) suggests strictly radially accelerating flow adjacent to the disc: $\bar{u}_{0}^{\prime }(\bar{r})>0$ . Hence, we are concerned with a BL under a favourable pressure gradient. This exhibits a quite rich variety of flow patterns, unappreciated so far both experimentally and theoretically, which depends on the relative magnitude of the centrifugal force. It can be understood in considerable depth even though $\bar{u}_{0}(r)$ is available only qualitatively. Two important approximations are specified rigorously: the Blasius and the von Kármán BL.

3.3.1 General scaling of fully viscous flow

In a first step, we determine the onset of the evolution of a radially thinning wall jet as expressed by (3.15). More precisely, the flow regime governing jet deflection is superseded by a new one once viscous diffusion, initially concentrated in the rapidly growing aforementioned BL, has spread vertically across the entire layer.

We first supplement (3.12) with $\bar{v}:=a_{0}^{2}v$ and (3.9) with the Rossby and Ekman numbers respectively

(3.23a,b ) $$\begin{eqnarray}Ro_{0}:=Ro_{a}/a_{0}^{3},\quad Ek_{0}:=Ro_{0}/Re_{0}=\tilde{\unicode[STIX]{x1D708}}/(\tilde{\unicode[STIX]{x1D6FA}}\tilde{a}_{0}^{2}).\end{eqnarray}$$

Formed with $\tilde{U}_{0}$ and $\tilde{a}_{0}$ , these absorb the contraction of the jet in the full governing equations including BCs, (2.4)–(2.5f–i ), as these prove invariant under the variable transformation $q\mapsto \bar{q}$ ( $q=r,z,\unicode[STIX]{x1D713},u,v,w,p$ ) and $(Re_{a},Ro_{a},Fr_{a})\mapsto (Re_{0},Ro_{0},Fr_{0})$ . Inspection analysis of (2.4), (2.5c )–(2.5f-i ) and the downstream tail of an inviscid jet bending expressed by (3.15) then launches the following sets (i), (ii) and (A)–(C) of apparent order-of-magnitude balances for $Re_{0}\gg 1$ .

1. (i) As long as disc rotation is so weak that $\bar{u}\sim 1$ (by inviscid jet bending) even in the BL, this merges into the entire layer to form a developed thin film for $\bar{r}\sim \bar{r}_{v}:=Re_{0}^{1/3}$ (from conservation of volume or $\bar{u}\bar{h}\bar{r}\sim 1$ and the radial inertial–viscous balance $\bar{u}^{2}/\bar{r}\sim Re_{0}^{-1}\bar{u}/\bar{h}^{2}$ );

2. (ii) in the BL, the centrifugal force induces a radial velocity component ${\sim}1$ for $\bar{r}\sim \bar{r}_{c}:=Ro_{0}$ (from $\bar{v}\sim \bar{r}/Ro_{0}$ ).

Then the effect of rotation extends to the full film for $\bar{u}\sim \bar{v}\sim \bar{r}/Ro_{0}$ or, with the last balances in (i),

(3.24) $$\begin{eqnarray}\bar{r}\sim \bar{r}_{r}:=(Re_{0}Ro_{0})^{1/4}\end{eqnarray}$$

to yield the subsequent basic case analysis the more complex classification precipitated at the end of § 3.3.1 pivots around.

1. (A) $Ro_{0}\gg Re_{0}^{1/3}:$ the magnitude of rotation matches up to that of the radial flow in (2.5c )–(2.5e ) relatively far downstream of the formation of a developed film, i.e. for $(\bar{r}_{c}\gg )\;\bar{r}_{r}\gg \bar{r}_{v}$ ;

2. (B) $Ro_{0}\sim Re_{0}^{1/3}:$ generic case, rotation becomes important where the developed film emerges, i.e. for $\bar{r}_{c}\sim \bar{r}_{r}\sim \bar{r}_{v}$ ;

3. (C) $Ro_{0}\ll Re_{0}^{1/3}:$ centrifugal forces come into play when the BL is still much thinner than the whole film, implying a (generalised) von Kármán BL where $\bar{u}\sim \bar{v}\sim \bar{r}/Ro_{0}$ , i.e. the whole film is entrained into this for $\bar{r}_{c}\ll \bar{r}_{r}\;(\ll \bar{r}_{v})$ .

As a consequence of these preliminary findings, the BL is discerned for $0\leqslant \bar{r}\ll Re_{0}^{1/3}$ , where the latter bound defines its downstream end in the cases (A) and (B). At the same time, $\bar{u}_{0}(\bar{r})$ grows monotonically from $\bar{u}_{0}=O(\bar{r})$ for $\bar{r}\ll 1$ , given the locally stagnant Euler flow, but saturates as $\bar{u}_{0}(\infty )=\bar{u}_{0,\infty }:=a_{0}^{2}u_{0,0}=f_{0}^{\prime }(0)/m_{0}$ , hence defining the downstream tail of the BL with a radial stretch depending on the magnitude of $Ro_{0}$ .

By the typical expansions holding within a distance ${\sim}1/\sqrt{Re_{0}}$ from the disc

(3.25a-d ) $$\begin{eqnarray}\bar{p}\sim B(0)-\frac{\bar{u}_{0}^{2}(\bar{r})}{2},\quad \bar{\unicode[STIX]{x1D713}}\sim \frac{\bar{r}\bar{u}_{0}(\bar{r})F(\bar{r},Z)}{\sqrt{Re_{0}}},\quad \bar{v}\sim \frac{\bar{r}G(\bar{r},Z)}{Ro_{0}},\quad Z:=\bar{z}\sqrt{Re_{0}}=O(1),\end{eqnarray}$$

equation (2.5c )–(2.5e ) assume their BL approximations in the form

(3.26a ) $$\begin{eqnarray}\displaystyle & \bar{u}_{0}^{\prime }(F_{Z}^{2}-1)-(\bar{r}\bar{u}_{0})^{\prime }FF_{ZZ}/\bar{r}+\bar{u}_{0}(F_{Z}F_{\bar{r}Z}-F_{\bar{r}}F_{ZZ})-\bar{r}G^{2}/(Ro_{0}^{2}\bar{u}_{0})=F_{ZZZ}, & \displaystyle\end{eqnarray}$$

(3.26b ) $$\begin{eqnarray}\displaystyle & [2\bar{u}_{0}F_{Z}G-(\bar{r}\bar{u}_{0})^{\prime }FG_{Z}]/\bar{r}+\bar{u}_{0}(F_{Z}G_{\bar{r}}-F_{\bar{r}}G_{Z})=G_{ZZ}. & \displaystyle\end{eqnarray}$$

Supplemented with the BCs (2.5f–i ) and matching conditions,

(3.26c,d ) $$\begin{eqnarray}Z=0:\quad F=F_{Z}=0,G=1,\quad Z=\infty :\quad F_{Z}=1,G=0,\end{eqnarray}$$

they determine the functions $F$ and $G$ . The potential flow induced by their displacement and their perturbations caused by its feedback are not of concern here. For $\bar{u}_{0}(\bar{r})$ prescribed, downstream integration of (3.26) starts at stagnation, given by $\bar{r}=0$ . Since we have $\bar{u}_{0}\sim b\bar{r}$ ( $\bar{r}\rightarrow 0$ ) with a positive initial slope $b$ of $\bar{u}$ , the Hannah flow (Hannah Reference Hannah1947; Tifford & Chu Reference Tifford and Chu1952), here parametrised by $Ro_{0}$ , is recovered. A Wentzel–Kramers–Brillouin (WKB) ansatz inserted into (3.26a ) and (3.26b ) yields

(3.27a ) $$\begin{eqnarray}[F-Z_{A},\,G]\sim [\text{e}^{\unicode[STIX]{x1D711}},\,\text{e}^{\unicode[STIX]{x1D711}}]+O(\text{e}^{2\unicode[STIX]{x1D711}})\quad (Z_{A}:=Z+A\rightarrow \infty )\end{eqnarray}$$

with the here unknown displacement function $-A(\bar{r};Ro_{0})$ and the exponent

(3.27b ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D711}\sim -\unicode[STIX]{x1D718}(\bar{r})Z_{A}^{2}-3\ln Z_{A}+\unicode[STIX]{x1D6FA}(\bar{r})+o(1), & \displaystyle\end{eqnarray}$$

(3.27c ) $$\begin{eqnarray}\displaystyle & [\unicode[STIX]{x1D718},\unicode[STIX]{x1D6FA}]:=\left[\bar{r}^{2}\bar{u}_{0}^{2}(\bar{r})\!\left/\!\left(4\displaystyle \int _{0}^{\bar{r}}t^{2}\bar{u}_{0}(t)\,\text{d}t\right)\right.,\unicode[STIX]{x1D6FA}_{0}+12\displaystyle \int _{0}^{\bar{r}}\!\left(\displaystyle \frac{\unicode[STIX]{x1D718}(t)}{\bar{u}_{0}(t)}-\displaystyle \frac{1}{t}\right)\text{d}t+\ln \left(\displaystyle \frac{\unicode[STIX]{x1D718}^{1/2}\bar{r}^{7}}{\bar{u}_{0}^{7}}\right)\!\right] & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

and some constant $\unicode[STIX]{x1D6FA}_{0}$ . This representation of the transition towards the external flow, used below, satisfies ICs found for $\bar{r}\rightarrow 0$ (Hannah flow; $\unicode[STIX]{x1D718}\sim b$ , $\unicode[STIX]{x1D6FA}\sim \unicode[STIX]{x1D6FA}_{0}$ ) and predicts $[\unicode[STIX]{x1D718},\unicode[STIX]{x1D6FA}]\sim [3\bar{u}_{0,\infty }/(4\bar{r}),(7/2)\ln \bar{r}]$ for $\bar{r}\gg 1$ . A key quantity potentially of interest in applications is the shear stress dragging the disc against rotation, here written as

(3.28) $$\begin{eqnarray}\bar{v}_{\bar{z}}(\bar{r},0)\sim \bar{r}G_{Z}(\bar{r},0)\sqrt{Re_{0}}/Ro_{0}\;({<}0).\end{eqnarray}$$

In (3.26a ), the centrifugal-force term in (2.5c ) becomes of $O(\bar{r}/Ro_{0}^{2})$ and involves the local ratio

(3.29) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}:=Ro_{0}\,\bar{u}_{0}/\bar{r}\end{eqnarray}$$

of the two local reference speeds $\tilde{U}_{0}\bar{u}_{0}$ and $\tilde{\unicode[STIX]{x1D6FA}}\tilde{a}_{0}\bar{r}$ at play. It grows drastically with $\bar{r}$ when the BL approaches its downstream end. Once the fully viscous film has started to evolve, in its bulk the quantity $\bar{r}\bar{h}^{2}Re_{0}/Ro_{0}^{2}$ expresses the order of magnitude of the centrifugal relative to the viscous term. According to the above estimates, this is equivalent to $(Re_{0}^{1/3}/Ro_{0})^{2}$ or $1/\unicode[STIX]{x1D6FD}^{2}$ . However, this also provides a confident upper bound for the ratio of those terms in the whole BL, where the viscous one is consistently dominant. Anticipating the analysis of the developed film, we conveniently measure this bound by a novel parameter $\unicode[STIX]{x1D6FC}$ . This is the squared reciprocal of a Rossby number characteristic of the film flow, i.e. $Ro_{0}$ reduced by a small parameter $\unicode[STIX]{x1D716}$ assessing the radial stretch of the BL relative to that of the film:

(3.30a-c ) $$\begin{eqnarray}\unicode[STIX]{x1D716}:=\left(\frac{4}{Re_{0}}\right)^{1/3},\quad Ro:=\unicode[STIX]{x1D716}Ro_{0}=\left(\frac{4}{a_{0}^{8}Re_{a}}\right)^{1/3}Ro_{a}=(4Ek_{0})^{1/3}Ro_{0}^{2/3},\quad \unicode[STIX]{x1D6FC}:=\frac{1}{Ro^{2}}.\end{eqnarray}$$

We have $Ro=\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D716}\bar{r}/\bar{u}_{0}$ with $\unicode[STIX]{x1D716}\bar{r}\ll 1$ in the BL. Typical values of $\unicode[STIX]{x1D716}$ , the ‘viscous’ radius

(3.31) $$\begin{eqnarray}r_{\unicode[STIX]{x1D708}}:=a_{0}/\unicode[STIX]{x1D716}\end{eqnarray}$$

characteristic of the developed flow in units of the nozzle radius $\tilde{a}$ , $Ro$ , $\unicode[STIX]{x1D6FC}$ and $Ek_{0}$ formed with quantities estimated earlier complete table 3. Hence, full adjustment to the film is expected to be accomplished quite close to jet impingement. The coupling parameter $\unicode[STIX]{x1D6FC}$ properly describes a series of distinguished limits for $Re_{0}\gg 1$ involving the viscous and centrifugal forces. So measuring the spin of the flow in both the BL region and the thin-film region further away from the disc centre by $\unicode[STIX]{x1D6FC}$ requires us to distinguish between the following cases, resorting to these two main flow regimes.

1. (a) $\unicode[STIX]{x1D6FC}\ll 1$ : slowly rotating flow in both flow regions, associated with a regular perturbation about the limit $\unicode[STIX]{x1D6FC}=0$ of purely radial flow in the BL but a singular one in the thin-film region;

2. (b) $\unicode[STIX]{x1D6FC}=O(1)$ : generic case of a modestly rotating film and thus a slowly rotating BL;

3. (c) $\unicode[STIX]{x1D6FC}=O(Re_{0}^{2/3})$ or $Ro_{0}=O(1)$ : modestly rotating BL and thus strongly rotating film, raises a singular perturbation problem in the region of the latter;

4. (d) $\unicode[STIX]{x1D6FC}\gg Re_{0}^{2/3}$ or $Ro_{0}\ll 1$ : strongly rotating BL and thus very strongly rotating film, raises a series of singular perturbation problems in either region.

In view of (3.24), the situations (a) and (b) refer to the original ones (A) and (B) respectively, (c) and (d) to the scenario (C) above. Concerning the BL flow, we next subsume the cases (a) and (b) in § 3.3.2, then (c) and (d) in § 3.3.3. Our main concern is with the behaviour of the BL relatively far downstream, i.e. in the intermediate region $1\ll \bar{r}\ll 1/\unicode[STIX]{x1D716}$ providing the match with the film flow.

3.3.2 Slowly rotating boundary layer: $\unicode[STIX]{x1D6FC}\ll 1$ , $\unicode[STIX]{x1D6FC}=O(1)$

In this scenario with $\unicode[STIX]{x1D6FD}\gg 1$ , the approximation $[F,G]\sim [F_{0},G_{0}]$ ( $Ro_{0}\gg 1$ ), with $F_{0}$ describing the BL above a disc at rest and $G_{0}$ the one-sidedly coupled associated azimuthal flow, is uniformly valid within the entire BL. Near stagnation, the Hannah flow is then approximated by the classical Homann flow (Homann Reference Homann1936). In the intermediate region, the BL thickens at a rate ${\sim}\sqrt{\bar{r}}$ , defining its absorption by the developed film flow, and thus assumes Blasius form: $[F_{0},G_{0}]\sim [\sqrt{\bar{r}/c}\,F_{B}(\unicode[STIX]{x1D701}),\,G_{B}(\unicode[STIX]{x1D701})]$ , $\unicode[STIX]{x1D701}:=Z\sqrt{c/\bar{r}}$ , $c:=3\bar{u}_{0,\infty }$ , with $F_{B}$ and $G_{B}$ satisfying the reduced form of (3.26)

(3.32a,b ) $$\begin{eqnarray}\displaystyle -F_{B}F_{B}^{\prime \prime }=2F_{B}^{\prime \prime \prime },\quad 4F_{B}^{\prime }G_{B}/3-F_{B}G_{B}^{\prime }=2G_{B}^{\prime \prime }, & & \displaystyle\end{eqnarray}$$

(3.32c,d ) $$\begin{eqnarray}\displaystyle F_{B}(0)=F_{B}^{\prime }(0)=G_{B}(\infty )=0,\quad F_{B}^{\prime }(\infty )=G_{B}(0)=1. & & \displaystyle\end{eqnarray}$$

The (straightforward) numerical solution of (3.32) is shown in figure 4(a). According to (3.27) in the limit of large $\bar{r}$ , we recover the behaviours $A\sim A_{\infty }\sqrt{\bar{r}/c}$ with $A_{\infty }\simeq -1.7208$ and $[F_{B}-\unicode[STIX]{x1D701}_{A},G_{B}]\sim [\text{e}^{\unicode[STIX]{x1D6F7}},\text{e}^{\unicode[STIX]{x1D6F7}}]$ with $\unicode[STIX]{x1D701}_{A}:=\unicode[STIX]{x1D701}-A_{\infty }$ and $\unicode[STIX]{x1D6F7}\sim -\unicode[STIX]{x1D701}_{A}^{2}/4-2\ln \unicode[STIX]{x1D701}_{A}+O(1)$ . Its independence of the upstream history conforms to a subtle breakdown of (3.27c ) for $\unicode[STIX]{x1D701}_{A}=O(1)$ . We also find $\bar{r}G_{Z}(\bar{r},0)\sim \sqrt{c\bar{r}}\,G_{B}^{\prime }(0)$ with $G_{B}^{\prime }(0)\simeq -0.48982$ in view of (3.28).

Figure 4. Self-preserving BL flow of Blasius type (a), von Kármán type (b), flow profiles along abscissae, $F_{K}^{\prime }$ augmented accordingly.

For semi-empirical methods to obtain approximate solutions of (3.26a ) for $Ro_{0}^{-1}=0$ and good agreement of calculated flow data with accurately measured ones we refer to Phares, Smedley & Flagan (Reference Phares, Smedley and Flagan2000b ); see also the references therein. These authors calculated $\bar{u}_{0}(\bar{r})$ with the method devised in their aforementioned study (Phares et al. Reference Phares, Smedley and Flagan2000a ), where they adopted Bickley’s or Schlichting’s well-known far-field expressions for the axial velocity of an either fully developed laminar free jet or its fully turbulent counterpart, see Batchelor (Reference Batchelor1970, pp. 344–345) and Schlichting & Gersten (Reference Schlichting and Gersten2017), as upstream input for inviscid jet bending (contrasting with the present flow configuration). Most important, their data confirm both the trends in the radial development of $\bar{r}_{0}$ and the wall shear rate $F_{ZZ}(\bar{r},0)$ , predicted here by asymptotic analysis: the latter initially increases at a rate ${\sim}\bar{r}$ , provoked by the accelerating Hannah flow, then reaches a maximum and eventually decreases, at a rate ${\sim}1/\sqrt{\bar{r}}$ .

3.3.3 Modestly to rapidly rotating boundary layer: $\unicode[STIX]{x1D6FC}\gg 1$

In this situation, $Ro_{0}$ is small enough to render the centrifugal-force term in (3.26a ) dominant for $\bar{r}\ll 1/\unicode[STIX]{x1D716}$ . Then (3.26) has to be solved in full for a prescribed value of $Ro$ where downstream integration is initiated by Hannah’s solution. For increasing disc rotation, that term and, correspondingly, $F_{Z}$ and thus $F$ increase close to the disc. Simultaneously, the second contribution to $F_{Z}^{2}-1$ in (3.26a ) feeling the imposed flow speed $\bar{u}_{0}$ becomes more and more insignificant. Its truncation renders the BL problem (3.26) singular: in case (c) above in the intermediate region; in case (d) within the whole streamwise extent of the BL. In either case, a reformation of the effective BL, defined by the predominance of both convective and centripetal acceleration of the fluid, takes place. Most important, this implies incomplete similarity of $F$ and $G$ . We adequately rescale $Z$ and $F$ by setting

(3.33a,b ) $$\begin{eqnarray}[\bar{F},\bar{G}](\bar{r},\bar{Z}):=[\unicode[STIX]{x1D6FD}/\sqrt{Ro_{0}}F,G](\bar{r},Z),\quad \bar{Z}:=Z/\sqrt{Ro_{0}}=\bar{z}/\sqrt{Ek_{0}}\end{eqnarray}$$

in the regions of non-uniformity. For $\bar{r}=O(1)$ , (3.29) implies a speed ratio $\unicode[STIX]{x1D6FD}$ of $O(1)$ in case (c) and $\unicode[STIX]{x1D6FD}\ll 1$ in case (d). Nevertheless, in both cases $\unicode[STIX]{x1D6FD}$ and thus the scaling factors of $F$ in (3.33) are asymptotically small in those regions. In turn, there the expansion

(3.34) $$\begin{eqnarray}[\bar{F},\bar{G}]\sim [F_{K},G_{K}](\bar{Z})+\unicode[STIX]{x1D6FD}[F_{1},G_{1}](\bar{r},\bar{Z})+\unicode[STIX]{x1D6FD}^{2}[F_{2},G_{2}](\bar{r},\bar{Z})+O(\unicode[STIX]{x1D6FD}^{3})\quad (\unicode[STIX]{x1D6FD}\rightarrow 0)\end{eqnarray}$$

holds. Our interest is with the bracketed functions. Re-expanding the azimuthal shear rate accordingly turns (3.28) into

(3.35) $$\begin{eqnarray}\bar{v}_{\bar{z}}(\bar{r},0)\sim [\bar{r}G_{K}^{\prime }(0)/Ro_{0}+\bar{u}_{0}{G_{1}}_{\bar{Z}}(\bar{r},0)]/\sqrt{Ek_{0}}.\end{eqnarray}$$

To leading order, equation (3.26) then restates the classical problem of von Kármán (Reference von Kármán1921), see also Cochran (Reference Cochran1934), as

(3.36a,b ) $$\begin{eqnarray}\displaystyle F_{K}^{\prime 2}-2F_{K}F_{K}^{\prime \prime }-G_{K}^{2}=F_{K}^{\prime \prime \prime },\quad 2(F_{K}^{\prime }G_{K}-F_{K}G_{K}^{\prime })=G_{K}^{\prime \prime }, & & \displaystyle\end{eqnarray}$$

(3.36c,d ) $$\begin{eqnarray}\displaystyle F_{K}(0)=F_{K}^{\prime }(0)=F_{K}^{\prime }(\infty )=0,\quad G_{K}(0)=1. & & \displaystyle\end{eqnarray}$$

Figure 4(b) displays our numerical solution of (3.36). The following properties of $F_{K}$ and $G_{K}$ are essential. As only strict forward flow is admissible and possible, $F_{K}^{\prime }(\bar{z})$ describes an overshooting streamwise flow component or wall jet, triggered by the strong centripetal acceleration given by $-G_{K}^{2}$ in (3.36a,b ) and thus barely affected by the relatively slow motion on its top. It sounds counterintuitive at first, however, that increasing disc rotation progressively impedes the penetration of the azimuthal flow component into the bulk of the flow. In case (c), the effective or von Kármán layer just suppresses the Blasius-type growth of the BL; in case (d), it is globally squeezed towards the disc by the factor $\sqrt{Ro_{0}}$ such that (3.36) describes the BL adjacent to the disc in canonical manner. One finds $G_{K}^{\prime }(0)\simeq -0.61592$ entering (3.35). The desired condition $G_{K}(\infty )=0$ is implied by $F_{K}^{\prime }(\infty )=0$ rather than to be stated as a second BC (as done usually). Then $F_{K}$ cannot grow stronger than with $\bar{Z}$ raised to some non-negative power smaller than unity for $\bar{Z}\rightarrow \infty$ ; examining (3.36a,b ) asymptotically shows that this exponent is zero and $F_{\infty }:=F_{K}(\infty )\simeq 0.44115$ . In turn, (3.25a–d ) and (3.33) predict the strong rate of entrainment $-\bar{w}\sim 2F_{\infty }/Ro$ for the von Kármán layer. Let $C$ and $D$ denote positive constants being part of the full solution, we finally arrive at the well-established exponential decay

(3.37a,b ) $$\begin{eqnarray}[F_{K}^{\prime },G_{K}]\sim -[C,D]E+O(E^{2}),\quad E:=\exp (-2F_{\infty }\bar{Z})\quad (\bar{Z}\rightarrow \infty ).\end{eqnarray}$$

The next higher approximation in (3.34) gives rise to the homogeneous linear equations

(3.38a ) $$\begin{eqnarray}\displaystyle & [F_{K}^{\prime }(\bar{r}\bar{u}_{0}{F_{1}}_{\bar{Z}})_{\bar{r}}-F_{K}^{\prime \prime }(\bar{r}\bar{u}_{0}F_{1})_{\bar{r}}]/\bar{u}_{0}-2(F_{K}{F_{1}}_{\bar{Z}\bar{Z}}+G_{K}G_{1})={F_{1}}_{\bar{Z}\bar{Z}\bar{Z}}, & \displaystyle\end{eqnarray}$$

(3.38b ) $$\begin{eqnarray}\displaystyle & [F_{K}^{\prime }(\bar{r}\bar{u}_{0}G_{1})_{\bar{r}}-G_{K}^{\prime }(\bar{r}\bar{u}_{0}F_{1})_{\bar{r}}]/\bar{u}_{0}-2(F_{K}{G_{1}}_{\bar{Z}}-G_{K}{F_{1}}_{\bar{Z}})={G_{1}}_{\bar{Z}\bar{Z}} & \displaystyle\end{eqnarray}$$

controlling the perturbation functions $F_{1}$ and $G_{1}$ . By the initial matching condition $F_{Z}\sim 1$ ( $Z\rightarrow \infty$ ), they are supposed to be non-trivial. We have in turn

(3.38c ) $$\begin{eqnarray}\bar{Z}=0:\quad F_{1}={F_{1}}_{\bar{Z}}=G_{1}=0,\quad \bar{Z}=\infty :\quad {F_{1}}_{\bar{Z}}=1,\quad G_{1}=0.\end{eqnarray}$$

As a consequence of (3.37), the three linearly independent contributions to $F_{1}$ behave for $\bar{Z}\rightarrow \infty$ to leading order like $\bar{Z}+O(\bar{Z}^{2}E)$ , $1+O(\bar{Z}E)$ and $E$ ; correspondingly, the two ones to $G_{1}$ like $1$ , except for terms of $O(\bar{Z}^{2}E)$ and $O(\bar{Z}E)$ arising from $F_{1}$ , and $E$ . For evaluating the solvability of (3.38), let us consider (3.38b ) an inhomogeneous equation governing $G_{1}$ where $F_{1}$ already satisfies (3.38a ) and (3.38c ). By varying ${G_{1}}_{\bar{Z}}(\bar{r},0)$ , we can then construct a contribution to $G_{1}$ satisfying the truncated homogeneous equation and the first of the BCs (3.38c ) such that $G_{1}$ vanishes for $\bar{Z}\rightarrow \infty$ (and $F_{1}$ is modified accordingly). As a result of these considerations, both $F_{1}$ and $G_{1}$ are uniquely determined (unless the homogeneous form of (3.38b ) subject to (3.38c ) allows for a non-trivial solution at exceptional values of $\bar{r}$ ). The linear growth of $F_{1}$ reminds us of the external flow ( $\unicode[STIX]{x1D6FD}\neq 0$ ). Since $\bar{u}_{0}^{\prime }({F_{1}}_{\bar{Z}}^{2}-1)$ , see (3.26a ), represents an inhomogeneity in the equation governing $F_{2}$ , this quantity grows just like $\bar{Z}^{2}E$ and not stronger than $F_{1}$ . This grants consistency of the expansion (3.34) with the flow on top of the BL.

Then the according expansions of $F^{\prime }$ and $G$ break down passively in a layer set apart from the disc around $Z=-\sqrt{Ro_{0}}\ln \unicode[STIX]{x1D6FD}/(2F_{\infty })$ and having a width again of $O(\sqrt{Ro_{0}})$ . The final question is how to unravel the gradual modification of the exponential towards the Gaussian tail in accordance with (3.27), triggered by the radial flow on top of the BL, for larger values of $Z$ . It must be answered separately for each of the cases (c) and (d).

Case (c). Here we have $\bar{u}_{0}\sim \bar{u}_{0,\infty }$ and take $\bar{r}$ large but not $Ro_{0}$ in (3.29), (3.33) and (3.34). In turn, this expansion describes incomplete similarity of the BL flow far downstream such that $F_{1}$ and $G_{1}$ are functions of $\bar{Z}$ solely. Then (3.38) reduces to

(3.39a ) $$\begin{eqnarray}\displaystyle & F_{K}^{\prime }F_{1}^{\prime }-F_{K}^{\prime \prime }F_{1}-2(F_{K}F_{1}^{\prime \prime }+G_{K}G_{1})=F^{\prime \prime \prime }, & \displaystyle\end{eqnarray}$$

(3.39b ) $$\begin{eqnarray}\displaystyle & F_{K}^{\prime }G_{1}-G_{K}^{\prime }F_{1}-2(F_{K}G_{1}^{\prime }-G_{K}F_{1}^{\prime })=G_{1}^{\prime \prime } & \displaystyle\end{eqnarray}$$

subject to (3.38c ). Figure 4(b) shows also the numerical solution of this problem. It gives $G_{1}^{\prime }(0)\simeq -0.21073$ in (3.35). The slightly lowered decay for $\bar{Z}$ being large is clearly visible.

Case (d). The aforementioned transition already indicated for situation (c) above now takes place for $\bar{r}=O(1)$ . Its detailed analysis is beyond the scope of the present study. Here we only note that the structure of the Hannah flow for $Ro_{0}\rightarrow 0$ just represents a special case.

Our asymptotic analysis of the BL discloses the origin of a radial wall jet, slightly perturbed by the radial flow on top. It complements the theoretical investigation by Liu et al. (Reference Liu, Gabour and Lienhard V1993) and the extensive experimental one by Broderson, Metzger & Fernando (Reference Broderson, Metzger and Fernando1996a ,Reference Broderson, Metzger and Fernando b ).

3.4 Thin-film regime: $r\gg a_{0}$ , $h\ll a_{0}$

The preliminary analysis of the thin film in § 3.3.1 lays the basis for the formulation of the problem governing the developed film flow in the limit of a small stretching parameter $\unicode[STIX]{x1D716}$ .

The earlier scalings invoke the expansion for $\unicode[STIX]{x1D716}\rightarrow 0$

(3.40) $$\begin{eqnarray}[\bar{\unicode[STIX]{x1D713}},\bar{v},\bar{u},\bar{w},\bar{h}\,]\sim \left[\frac{f(R,\unicode[STIX]{x1D702})}{2},\frac{Rg(R,\unicode[STIX]{x1D702})}{Ro},U(R,\unicode[STIX]{x1D702}),\unicode[STIX]{x1D716}^{2}W(R,\unicode[STIX]{x1D702}),\frac{\unicode[STIX]{x1D716}H(R)}{2}\right],\quad R:=\unicode[STIX]{x1D716}\bar{r}.\end{eqnarray}$$

It is valid in the thin-film region where the compressed radial coordinate $R$ is of $O(1)$ or greater and the associated stretched vertical coordinate $2\bar{z}/\unicode[STIX]{x1D716}$ , the rescaled film height $H$ , the film coordinate $\unicode[STIX]{x1D702}\sim 2\bar{z}/(\unicode[STIX]{x1D716}H)$ already defined by (3.15) and the newly introduced functions $f$ , $g$ , $U$ , $W$ are of $O(1)$ . This casts (3.28) into

(3.41) $$\begin{eqnarray}\bar{v}_{\bar{z}}(\bar{r},0)\sim 2Rg_{\unicode[STIX]{x1D702}}(R,0)/(\unicode[STIX]{x1D716}H).\end{eqnarray}$$

Furthermore, we write

(3.42a,b ) $$\begin{eqnarray}U=f_{\unicode[STIX]{x1D702}}/m,\quad 2W=\unicode[STIX]{x1D702}H^{\prime }f_{\unicode[STIX]{x1D702}}/m-f_{R}/R,\end{eqnarray}$$

where we have conveniently introduced

(3.43a ) $$\begin{eqnarray}m(R):=RH.\end{eqnarray}$$

Then plugging (3.40) into (2.5c )–(2.5e ) reduces these equations to their parabolic shallow-water forms respectively

(3.43b ) $$\begin{eqnarray}\displaystyle & m(f_{\unicode[STIX]{x1D702}}f_{R\unicode[STIX]{x1D702}}-f_{R}f_{\unicode[STIX]{x1D702}\unicode[STIX]{x1D702}})-m^{\prime }f_{\unicode[STIX]{x1D702}}^{2}-\unicode[STIX]{x1D6FC}R\,m^{3}g^{2}=R^{2}f_{\unicode[STIX]{x1D702}\unicode[STIX]{x1D702}\unicode[STIX]{x1D702}}, & \displaystyle\end{eqnarray}$$

(3.43c ) $$\begin{eqnarray}\displaystyle & m(f_{\unicode[STIX]{x1D702}}g_{R}-f_{R}g_{\unicode[STIX]{x1D702}}+2f_{\unicode[STIX]{x1D702}}g/R)=R^{2}g_{\unicode[STIX]{x1D702}\unicode[STIX]{x1D702}}. & \displaystyle\end{eqnarray}$$

Supplemented with (2.5f–i ) and the free-slip conditions resulting from (2.5k ) and (2.5l ) and ICs,

(3.43d ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D702}=0:\quad f=f_{\unicode[STIX]{x1D702}}=0,\quad g=1,\quad \unicode[STIX]{x1D702}=1:\quad f=1,\quad f_{\unicode[STIX]{x1D702}\unicode[STIX]{x1D702}}=g_{\unicode[STIX]{x1D702}}=0, & \displaystyle\end{eqnarray}$$

(3.43e ) $$\begin{eqnarray}\displaystyle & R=0:\quad [f,f_{\unicode[STIX]{x1D702}},g,m]=[f_{0}(\unicode[STIX]{x1D702}),f_{0}^{\prime }(\unicode[STIX]{x1D702})\text{sgn}(\unicode[STIX]{x1D702}),1-\text{sgn}(\unicode[STIX]{x1D702}),m_{0}], & \displaystyle\end{eqnarray}$$

they govern the quantities $f$ , $g$ , $m$ . These are parametrised by $\unicode[STIX]{x1D6FC}$ taken as of $O(1)$ and $\unicode[STIX]{x1D6EC}_{0}$ , $We_{a}$ , $\unicode[STIX]{x1D707}$ . The latter enter the ICs and reflect the match with the predominantly inviscid process of jet bending. The jumps of $f^{\prime }$ , $g$ at the disc surface introduced in (3.43e ) account for the no-slip condition in (3.43d ) and thus render the resulting parabolic problem (3.43) singular. This triggers the evolution of the associated BL for small values of $R$ , whereas that along $\unicode[STIX]{x1D6F4}$ emerges naturally as we have $f_{0}^{\prime \prime }\equiv 0$ ( $0<\unicode[STIX]{x1D702}\leqslant 1$ ).

Specifically, equation (3.43b ) yields together with the rescaling in (3.40), (3.25a–d ), (3.15) and § 3.3.2

(3.44) $$\begin{eqnarray}[f,g]\sim [\sqrt{\bar{u}_{0,\infty }R^{3}/3}F_{B}(\unicode[STIX]{x1D701}),G_{B}(\unicode[STIX]{x1D701})]\quad (\unicode[STIX]{x1D701}=m_{0}\unicode[STIX]{x1D702}\sqrt{3\bar{u}_{0,\infty }/R^{3}}=O(1),\quad R\rightarrow 0).\end{eqnarray}$$

That is, the solutions to (3.43) are inherently tied in with the excitation of a Blasius BL for any value of $\unicode[STIX]{x1D6FC}$ , howsoever large. They naturally include its modification associated with the generation of a wall jet for $R=O(\unicode[STIX]{x1D6FC}^{-1/2})$ and $\unicode[STIX]{x1D6FC}\gg 1$ , pointing to the predominance of the centrifugal-force term in (3.43b ) in the BL limit and the cases (c) and (d) in § 3.3.1.

The subsequent section, forming the core of our study, is devoted to the leading-order analysis of the flow regime governed by (3.43) in the whole radial domain $0\leqslant R<\infty$ . Perturbations caused by the higher-order terms in (3.21) as well as by gravity and capillarity are at first discarded as they are recognised to be weak and to bear no decisive physical relevance. Inspection of (2.5b,d,f,g ) shows upon substitution of (3.9), (3.12), (3.15) and (3.40) that

(3.45) $$\begin{eqnarray}\bar{p}\sim \unicode[STIX]{x1D716}(1-\unicode[STIX]{x1D702})H/(2Fr_{0}^{2})-\unicode[STIX]{x1D716}^{3}(RH^{\prime })^{\prime }/(2We_{0}R)+O(\unicode[STIX]{x1D716}^{4})\end{eqnarray}$$

holds for $R=O(1)$ . Thus, there the film pressure is indeed negligibly small.

4.1 Disc at rest revisited: $\unicode[STIX]{x1D6FC}=0$ (benchmark case)

The Mangler–Stepanov transformation (see e.g. Schlichting & Gersten Reference Schlichting and Gersten2017, p. 323)

(4.1a-c ) $$\begin{eqnarray}[x,y]:=[R^{3}/3,\,m(R)\unicode[STIX]{x1D702}],\quad \hat{f}(x,\unicode[STIX]{x1D702}):=f(R,\unicode[STIX]{x1D702}),\quad \hat{h}(x):=m(R)\end{eqnarray}$$

casts (3.43b ) into an advantageous autonomous form, governing the planar counterpart of the flow exhibiting the same streamwise velocity component: let $x$ and $y$ denote the streamwise and disc-normal coordinates respectively and $\hat{f}$ and $\hat{h}$ the corresponding streamfunction and film height. With $\unicode[STIX]{x1D702}\sim y/\hat{h}$ , these quantities then satisfy

(4.2) $$\begin{eqnarray}\hat{h}(\hat{f}_{\unicode[STIX]{x1D702}}\hat{f}_{x\unicode[STIX]{x1D702}}-\hat{f}_{x}\hat{f}_{\unicode[STIX]{x1D702}\unicode[STIX]{x1D702}})-\hat{h}^{\prime }\hat{f}_{\unicode[STIX]{x1D702}}^{2}-\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6EF}=\hat{f}_{\unicode[STIX]{x1D702}\unicode[STIX]{x1D702}\unicode[STIX]{x1D702}},\quad \unicode[STIX]{x1D6EF}:=\hat{h}^{3}g^{2}/R.\end{eqnarray}$$

We take $\unicode[STIX]{x1D6FC}=0$ first to reconsider the flow over a disc at rest. It was studied first exhaustively by Watson (Reference Watson1964); the axisymmetric case was taken up by Bowles & Smith (Reference Bowles and Smith1992), the planar counterpart revisited by Higuera (Reference Higuera1994). Here we focus on large values of $x$ in more depth so as to conveniently gain the results for large values of $R$ with conforming asymptotic accuracy. These provide a benchmark for the precision of our numerical predictions if we assume an irrotational jet and neglect surface tension: equations (3.19a,b ) yield $m_{0}=\unicode[STIX]{x1D707}=1$ and $\unicode[STIX]{x1D70E}=0$ . Hence, downstream integration is initiated by $[f,m]\sim [\unicode[STIX]{x1D702},1]$ ( $R\rightarrow 0$ ) (equivalent to $[\hat{f},\hat{h}]\sim [\unicode[STIX]{x1D702},1]$ $(x\rightarrow 0)$ as conservation of the flow rate yields $\hat{h}\sim 1$ initially for $\bar{u}\sim 1$ , see Higuera Reference Higuera1994). The numerical solution of the so resulting initial-value problem was evaluated at three rather moderate end values $R_{e}:=1,2,3$ of $R$ ; $f$ and $m$ were found to adjust quite rapidly to their far-field values of $O(1)$ computed via (4.1). Comparing key figures with those obtained originally by Watson (Reference Watson1964) and Bowles & Smith (Reference Bowles and Smith1992) convincingly confirm the high reliability and downstream stability of the present numerical schemes. The results are visualised together with those for $\unicode[STIX]{x1D6FC}>0$ in § 4.3.

Watson employed a simplified von Kármán–Pohlhausen method interpolating between the small- and large- $R$ form of $f$ he analysed first; Bowles & Smith chose $R$ , $y$ as independent variables, which is equivalent to setting $m\equiv 1$ and identifying $\unicode[STIX]{x1D702}$ with $y$ in (3.43b ). They adopted a Crank–Nicholson discretisation. Although this approach copes perfectly with the initial development of the flow (yet exhibiting an almost inviscid bulk), it does less adequately than our scheme with the rapidly increasing growth of  $H$ .

Equation (4.2) implies $\hat{f}\sim f_{W}(\unicode[STIX]{x1D702})+[x^{-\unicode[STIX]{x1D714}}f_{W}^{\prime }(\unicode[STIX]{x1D702})F_{\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D702})+\text{c.c.}]/2+o(x^{-\unicode[STIX]{x1D714}})$ associated with a nearly linear thickening of the layer: $\hat{h}-d^{3}\sim h_{W}x+[h_{\unicode[STIX]{x1D714}}x^{1-\unicode[STIX]{x1D714}}+\text{c.c.}]/2+o(x^{1-\unicode[STIX]{x1D714}})$ and $\bar{u}\sim f_{W}^{\prime }/(h_{W}x)$ for $x\rightarrow \infty$ and some constants $\unicode[STIX]{x1D714}$ ( $\text{Re}\,\unicode[STIX]{x1D714}>0$ , $\unicode[STIX]{x1D714}\neq 1$ ), $h_{W}>0$ , $h_{\unicode[STIX]{x1D714}}$ and $d$ (real and due to integration). The last two reflect the flow history specified by the ICs (3.43e ). This behaviour is equivalent to $m\sim h_{W}(R^{3}+d^{3})/3$ ( $R\rightarrow \infty$ ) or $H$ increasing with $R^{2}$ accordingly (Watson Reference Watson1964). In accordance with (4.1) and (4.2), $H$ and $f$ ( $\hat{h}$ ) can be viewed as symmetric in $R$ (antisymmetric in $x$ ), which confirms the form above expansions (which amounts to $\unicode[STIX]{x1D714}$ being even). Moreover, extending the next-order terms towards the limits $\unicode[STIX]{x1D714}\rightarrow 0_{+}$ and $\unicode[STIX]{x1D714}\rightarrow \infty$ includes the possibility of a respectively sub- and super-algebraic variation in $x$ . The associated functions $f_{W}$ , $F_{\unicode[STIX]{x1D714}}$ then satisfy

(4.3) $$\begin{eqnarray}\displaystyle & -h_{W}f_{W}^{\prime 2}=f_{W}^{\prime \prime \prime },\quad f_{W}(0)=f_{W}^{\prime }(0)=f_{W}^{\prime \prime }(1)=0,\quad f_{W}(1)=1, & \displaystyle\end{eqnarray}$$

(4.4) $$\begin{eqnarray}\displaystyle & (1-\unicode[STIX]{x1D714})f_{W}^{\prime 4}(h_{W}F_{\unicode[STIX]{x1D714}}^{\prime }-h_{\unicode[STIX]{x1D714}})=(f_{W}^{\prime 3}F_{\unicode[STIX]{x1D714}}^{\prime \prime })^{\prime },\quad F_{\unicode[STIX]{x1D714}}(0)=F_{\unicode[STIX]{x1D714}}(1)=F_{\unicode[STIX]{x1D714}}^{\prime \prime }(1)=0. & \displaystyle\end{eqnarray}$$

Problem (4.3) is one of the few of Falkner–Skan-type where a closed-form solution is available. Let us abbreviate $\unicode[STIX]{x1D6E4}:=\unicode[STIX]{x0393}[1/3]/\unicode[STIX]{x0393}[5/6]$ . By integration, the BCs fix the rescaled surface slip $f_{W}^{\prime }(1)$ as $\unicode[STIX]{x1D6E4}^{2}/(2\sqrt{3})\simeq 1.6260$ , the respective shear rate at the disc $f_{W}^{\prime \prime }(0)$ as $\sqrt{\unicode[STIX]{x03C0}/3}\,\unicode[STIX]{x1D6E4}^{3}/6\simeq 2.2799$ and the eigenvalue $h_{W}$ as $\unicode[STIX]{x03C0}/\sqrt{3}\simeq 1.8138$ . One in turn finds $f_{W}^{\prime }(\unicode[STIX]{x1D702})/f_{W}^{\prime }(1)=1-\sqrt{3}\tan ^{2}(\unicode[STIX]{x1D719}/2)$ where $\unicode[STIX]{x1D719}$ is the Jacobi amplitude $\text{am}(c(1-\unicode[STIX]{x1D702});k)$ with $c=\sqrt{\unicode[STIX]{x03C0}}\,\unicode[STIX]{x1D6E4}/3^{3/4}\simeq 1.8454$ and the elliptic modulus $k=(\sqrt{3}/4+1/2)^{1/2}\simeq 0.96593$ . This solution of (3.43) describing self-similar flow for $\unicode[STIX]{x1D6FC}=0$ was originally put forward by Watson (Reference Watson1964) but with an ambiguous (inaccurate) value of $k$ . We detected non-zero residua $f_{\unicode[STIX]{x1D702}}(R_{e},1)-\hat{f}^{\prime }(1)$ , $f_{\unicode[STIX]{x1D702}\unicode[STIX]{x1D702}}(R_{e},0)-\hat{f}^{\prime \prime }(0)$ , $h^{\prime }(R_{e})-h_{W}$ in the third digits. Also, Watson (Reference Watson1964) estimated $d\simeq 2.29$ by his approximate method, whereas we extracted $d\simeq 2.33$ from our as did Bowles & Smith (Reference Bowles and Smith1992) from their data.

Problem (4.4) governs the perturbation quantities $F_{\unicode[STIX]{x1D714}}$ , $\unicode[STIX]{x1D714}$ , $h_{\unicode[STIX]{x1D714}}$ . They are worth considering in more depth and breadth with respect to the case of finite disc rotation. Let $F_{\unicode[STIX]{x1D714}}^{h}$ denote a homogeneous solution, i.e. obtained by setting $h_{\unicode[STIX]{x1D714}}=0$ . Then (4.4) represents a generalised singular self-adjoint Sturm–Liouville eigenvalue problem fixing the eigenvalues of $\unicode[STIX]{x1D714}$ and $F_{\unicode[STIX]{x1D714}}^{h\prime }$ , the latter subject to $F_{\unicode[STIX]{x1D714}}^{h\prime \prime }(1)=0$ and sorting out an $\unicode[STIX]{x1D702}^{-2}$ -singularity for $\unicode[STIX]{x1D702}\rightarrow 0$ . Thus $\unicode[STIX]{x1D714}$ and all other quantities involved are real. Writing $F_{\unicode[STIX]{x1D714}}^{h}(\unicode[STIX]{x1D702})\sim \unicode[STIX]{x1D702}+O(\unicode[STIX]{x1D702}^{4})$ in this limit uniquely defines $F_{\unicode[STIX]{x1D714}}^{h}(\unicode[STIX]{x1D702})$ for each admissible value of $\unicode[STIX]{x1D714}$ . Hence, the full solution reads $F_{\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D702})=C_{\unicode[STIX]{x1D714}}F_{\unicode[STIX]{x1D714}}^{h}(\unicode[STIX]{x1D702})+h_{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D702}/h_{W}$ with some constant $C_{\unicode[STIX]{x1D714}}$ fixed by the upstream history of the flow and $h_{\unicode[STIX]{x1D714}}/C_{\unicode[STIX]{x1D714}}$ given by $-h_{W}F_{\unicode[STIX]{x1D714}}^{h}(1)$ . A numerical investigation then shows that $F_{\unicode[STIX]{x1D714}}^{\prime \prime }(1)$ vanishes for a discrete set of eigenvalues where $F_{\unicode[STIX]{x1D714}}^{h}(\unicode[STIX]{x1D702})$ becomes more and more oscillatory the larger $\unicode[STIX]{x1D714}$ is. (Finally, a sublayer forms where $\unicode[STIX]{x1D702}$ is of $O(\unicode[STIX]{x1D714}^{-1/3})$ as the problem becomes singular, and the algebraic variations in $x$ even merge with exponentially weak ones once $\unicode[STIX]{x1D714}$ has become of $O(x/\ln x)$ for some sufficiently large $x$ .). Quite interestingly, it seems that $\unicode[STIX]{x1D714}^{\ast }$ exactly equal to $14$ (and even, as proposed above) represents the smallest of those eigenvalues ( $F_{\unicode[STIX]{x1D714}^{\ast }}^{h\,\prime }(\unicode[STIX]{x1D702})$ changes sign once). This (surprisingly large) value explains the very fast approach of $f$ towards Watson’s solution $f_{W}$ observed via full downstream integration. For the sake of completeness, we note that $F_{\unicode[STIX]{x1D714}^{\ast }}^{h}(1)$ seemingly also equals a rational number, namely 0.48.

4.2 Slowly rotating film: $\unicode[STIX]{x1D6FC}\ll 1$

For $\unicode[STIX]{x1D6FC}\rightarrow 0$ , (3.43b ) decouples from (3.43c ). According to the study of the BL in § 3.3.2, for not too large values of $R$ the power series

(4.5) $$\begin{eqnarray}[f,g,m]\sim [\hat{f}(x,\unicode[STIX]{x1D702}),\hat{g}(x,\unicode[STIX]{x1D702}),\hat{h}(x)]+\unicode[STIX]{x1D6FC}[f^{1}(R,\unicode[STIX]{x1D702}),g^{1}(R,\unicode[STIX]{x1D702}),m^{1}(R)]+O(\unicode[STIX]{x1D6FC}^{2})\end{eqnarray}$$

holds. With $\hat{f}$ known and satisfying (3.43b ) in the form (4.2) for $\unicode[STIX]{x1D6FC}=0$ , a one-sided coupling between (3.43b ) and (3.43c ) and a hierarchy of linear problems is generated determining the sequence $\hat{g},f^{1},g^{1},\ldots$ : the second equation, together with the (jump) condition for $g$ in (3.43e ) and the inhomogeneous BC $\hat{g}(x,0)=1$ , determines $\hat{g}$ and the centripetal acceleration $-\unicode[STIX]{x1D6EF}$ in (4.2); this forces its linearised version governing $f^{1}$ , feeding into the corresponding equation for $g^{1}$ ; reiteration of this scheme yields higher-order corrections. Specifically, $\hat{g}$ satisfies $\hat{h}[\hat{f}_{\unicode[STIX]{x1D702}}\,\hat{g}_{x}-\hat{f}_{x}\,\hat{g}_{\unicode[STIX]{x1D702}}+2\hat{f}\hat{g}/(3x)]=\hat{g}_{\unicode[STIX]{x1D702}\unicode[STIX]{x1D702}}$ .

For $R\gg 1$ , weak disc rotation extends Watson’s solution: $\hat{g}\sim g_{W}(\unicode[STIX]{x1D702})$ with $g_{W}$ governed by

(4.6) $$\begin{eqnarray}2h_{W}f_{W}^{\prime }g_{W}=3g_{W}^{\prime \prime },\quad g_{W}(0)=1,g_{W}^{\prime }(1)=0.\end{eqnarray}$$

From the straightforward numerical solution of (4.6), $g_{W}^{\prime }(0)\simeq -0.76126$ is adopted in (3.41). The corresponding effect of disc spin felt by the axial flow is readily at hand by studying the linearised full form of (4.2) for large $R$ , viz. $\unicode[STIX]{x1D6EF}\sim h_{W}^{3}g_{W}^{2}R^{8}$ providing a forcing term. The analysis of § 4.1 subsequent to (4.1) immediately predicts

(4.7a,b ) $$\begin{eqnarray}f^{1}/f_{W}^{\prime }(\unicode[STIX]{x1D702})\sim R^{8}F^{1}(\unicode[STIX]{x1D702})+R^{-3\unicode[STIX]{x1D714}^{\ast }}F_{\unicode[STIX]{x1D714}^{\ast }}(\unicode[STIX]{x1D702}),\quad m^{1}\sim h^{1}(R/3^{1/3})^{11}+h_{\unicode[STIX]{x1D714}^{\ast }}(R/3^{1/3})^{3-3\unicode[STIX]{x1D714}^{\ast }}\end{eqnarray}$$

and $g^{1}=O(R^{8})$ ( $R\rightarrow \infty$ ). Herein, $\unicode[STIX]{x1D714}^{\ast }$ designates the unique value of $\unicode[STIX]{x1D714}$ found before, and the function $F^{1}$ and the constant $h^{1}$ are uniquely determined by the accordingly inhomogeneous counterpart to (4.4) with $\unicode[STIX]{x1D714}=-8/3$ :

(4.8) $$\begin{eqnarray}(11/3)f_{W}^{\prime 4}(h_{W}{F^{1}}^{\prime }-h^{1})-(h_{W}^{3}/3^{1/3})(f_{W}^{\prime }g_{W})^{2}=(f_{W}^{\prime 3}{F^{1}}^{\prime \prime })^{\prime }\end{eqnarray}$$

subject to identical homogeneous BCs. With regard to the dominant homogeneous contributions to $m^{1}$ and $f^{1}$ in (4.7), the constants $h_{\unicode[STIX]{x1D714}^{\ast }}$ and thus $C_{\unicode[STIX]{x1D714}^{\ast }}$ , i.e. the strength of the perturbation $F_{\unicode[STIX]{x1D714}^{\ast }}$ , are expected to differ from $h_{\unicode[STIX]{x1D714}}$ and $C_{\unicode[STIX]{x1D714}}$ fixing the large- $x$ form of $\hat{f}$ above when evaluated for $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}^{\ast }$ .

For $R$ becoming large, the rapid approach towards Watson’s solution for $\unicode[STIX]{x1D6FC}=0$ is seen to be attenuated likewise by the advent of a finite disc speed and the rapidly increasing magnitude of $\unicode[STIX]{x1D6EF}$ in (4.2). In turn, the expansions (4.5) and (4.7) trigger a region $R=O(\unicode[STIX]{x1D6FC}^{-1/8})$ of non-uniformity. Therein, (3.43b ) or (4.2) are re-established in full. Simultaneously, in this leading approximation the flow is no longer reminiscent of any jet characteristics. We beneficially introduce rescaled $O(1)$ -quantities by

(4.9a,b ) $$\begin{eqnarray}[f,g](R,\unicode[STIX]{x1D702})\sim [\bar{f},\bar{g}](\bar{R},\unicode[STIX]{x1D702}),\quad [H,m](R)\sim [\unicode[STIX]{x1D6FC}^{-1/4}\bar{H},\unicode[STIX]{x1D6FC}^{-3/8}\bar{m}](\bar{R}),\quad \bar{R}:=\unicode[STIX]{x1D6FC}^{1/8}R.\end{eqnarray}$$

The underlying affine transformation identifies the region $\bar{R}=O(1)$ where the disc speed is representative of both relevant velocity components: $\bar{m}\bar{u}$ and $\bar{v}$ are of $O(\unicode[STIX]{x1D6FC}^{3/8})$ by (3.40). Moreover, it recasts (3.43b )–(3.43e ) into the canonical form of the momentum equations,

(4.10a ) $$\begin{eqnarray}\displaystyle & \bar{m}(\bar{f}_{\unicode[STIX]{x1D702}}\bar{f}_{\bar{R}\unicode[STIX]{x1D702}}-\bar{f}_{\bar{R}}\bar{f}_{\unicode[STIX]{x1D702}\unicode[STIX]{x1D702}})-\bar{m}^{\prime }\bar{f}_{\unicode[STIX]{x1D702}}^{2}-\bar{R}\,\bar{m}^{3}\bar{g}^{2}=\bar{R}^{2}\bar{f}_{\unicode[STIX]{x1D702}\unicode[STIX]{x1D702}\unicode[STIX]{x1D702}}, & \displaystyle\end{eqnarray}$$

(4.10b ) $$\begin{eqnarray}\displaystyle & \bar{m}(\bar{f}_{\unicode[STIX]{x1D702}}\,\bar{g}_{\bar{R}}-\bar{f}_{\bar{R}}\,\bar{g}_{\unicode[STIX]{x1D702}}+2\bar{f}_{\unicode[STIX]{x1D702}}\,\bar{g}/\bar{R})=\bar{R}^{2}\bar{g}_{\unicode[STIX]{x1D702}\unicode[STIX]{x1D702}}, & \displaystyle\end{eqnarray}$$

(4.10c ) $$\begin{eqnarray}\unicode[STIX]{x1D702}=0:\quad \bar{f}=\bar{f}_{\unicode[STIX]{x1D702}}=0,\quad \bar{g}=1,\quad \unicode[STIX]{x1D702}=1:\quad \bar{f}=1,\quad \bar{f}_{\unicode[STIX]{x1D702}\unicode[STIX]{x1D702}}=\bar{g}_{\unicode[STIX]{x1D702}}=0,\end{eqnarray}$$

and ICs (3.43e ) apart from $\bar{m}\sim \unicode[STIX]{x1D6FC}^{3/8}m_{0}$ ( $\bar{R}\rightarrow 0$ ). By (4.10a ), the perturbations of $O(R^{3})$ about $\bar{f}\sim f_{0}$ then account for viscous effects for any howsoever small finite value of $\unicode[STIX]{x1D6FC}$ . Consequently, the problem is singular in the limit $\unicode[STIX]{x1D6FC}\rightarrow 0$ as the above structure of the flow captures its transition for $R\ll \unicode[STIX]{x1D6FC}^{-1/8}$ towards a state close to Watson’s self-preserving one. To leading order, the ICs have to be replaced by

(4.10d,e ) $$\begin{eqnarray}[\bar{f},\bar{g}](\bar{R},\unicode[STIX]{x1D702})\rightarrow [f_{W},g_{W}](\unicode[STIX]{x1D702}),\quad \bar{m}\sim h_{W}\bar{R}^{3}/3\quad (\bar{R}\rightarrow 0).\end{eqnarray}$$

Herein the remainder terms of $O(\bar{R}^{8})$ satisfy the hierarchy of inhomogeneous equations arising from (4.10a ) and (4.10b ): Watson’s solution is attracting, and no branching eigensolutions arise by linearisation about it in the limit $\bar{R}\rightarrow 0$ (confirmed by extending the analysis of § 4.1 to the case $x\rightarrow 0$ , $\unicode[STIX]{x1D714}<0$ ). This slow, regular departure from it due to disc rotation complies with matching the expansions (4.9) and (4.5). However, plugging (4.10d,e ) into (4.10a ) renders the problem (4.10) a singular one. In applying the numerical schemes (I) and (II), one combines (4.10d,e ) in the first $\bar{R}$ -step with the more relaxed condition $\bar{m}(0)=0$ . For $\bar{R}=0$ , Watson’s solution, obtained earlier, is interpolated at the points of the $\unicode[STIX]{x1D702}$ -grid.

Figure 5. Canonical representation of film flow including leading-order asymptotes (dotted).

Figure 6. Flow profiles: canonical limits in radial (a) and azimuthal (b) directions.

Figure 5 shows the key quantities resolved by the numerical solution of (4.10). These consist of $H$ , the surface speed $\bar{U}_{s}(\bar{R}):=\bar{f}_{\unicode[STIX]{x1D702}}(\bar{R},1)/\bar{m}(\bar{R})$ and the radial and circumferential components of the shear at the disc here represented by $[\bar{\unicode[STIX]{x1D70E}},\bar{\unicode[STIX]{x1D70F}}]:=[\bar{f}_{\unicode[STIX]{x1D702}\unicode[STIX]{x1D702}},-\bar{g}_{\unicode[STIX]{x1D702}}](\bar{R},0)$ . They inherit all the exciting features of the film flow originating in its departure from Watson’s solution, essentially characterised by the initial growth of $\bar{H}$ with $\bar{R}^{2}$ . However, the radially increasing strength of the centrifugal and thus the shear forces finally amplifies the radial flow and so decreases the film height. This implies a local maximum of the latter for $[\bar{R},\bar{H}]=[\bar{R}^{\ast },\bar{H}^{\ast }]\simeq [1.7000,1.1906]$ , a striking feature of the flow observed in some previous computations (cf. Prieling & Steiner Reference Prieling and Steiner2013a ,Reference Prieling and Steiner b ). Eventually, the flow becomes fully developed as the fluid particles assume a solid-body rotation, $\bar{g}\sim 1$ , and their centripetal acceleration balancing the shear stress gradient predominates in (4.10a ). One spots the behaviour

(4.11a ) $$\begin{eqnarray}\displaystyle & [\bar{f},\bar{R}^{8/3}(\bar{g}-1),\bar{R}^{2/3}\bar{H}]\sim [f_{\infty }(\unicode[STIX]{x1D702}),g_{\infty }(\unicode[STIX]{x1D702}),3^{1/3}]+O(\bar{R}^{-8/3})\quad (\bar{R}\rightarrow \infty ), & \displaystyle\end{eqnarray}$$

(4.11b ) $$\begin{eqnarray}\displaystyle & [f_{\infty },g_{\infty }](\unicode[STIX]{x1D702})=[(3\unicode[STIX]{x1D702}^{2}-\unicode[STIX]{x1D702}^{3})/2,-3^{1/3}(2\unicode[STIX]{x1D702}-\unicode[STIX]{x1D702}^{3}+\unicode[STIX]{x1D702}^{4}/4)]. & \displaystyle\end{eqnarray}$$

$\bar{R}\rightarrow 0$

$\bar{R}\rightarrow \infty$

$\bar{U}_{s}\sim (9/\bar{R})^{1/3}/2$

$\bar{f}_{\unicode[STIX]{x1D702}}$

$\bar{g}$

$g_{W}(1)\simeq 0.53718$

$\bar{f}_{\unicode[STIX]{x1D702}}$

$\bar{R}$

The following observation deserves mentioning. The novel scaling (4.9) of the radius and the streamwise flow velocity describes a deviation from Watson’s flow state still far from a fully developed one, expected to hold further downstream. In both of these limiting cases, however, the flow quantities obey laws of full similarity, whereas their $\bar{R}$ -dependence represents incomplete similarity according to the presence of the two disparate scalings expressed by (4.10d,e ) and (4.11).

The aforementioned leading-order equilibrium expressed by $\hat{f}\sim \hat{f}_{\infty }$ has attracted attention previously. For instance, Aroesty et al. (Reference Aroesty, Gross, Illickal and Maloney1967) and Brauer (Reference Brauer1971, p. 323 ff.) considered it as a model for spraying along a rotating disc. Specifically, perturbations of the fully developed state die out with integer powers of $\bar{R}^{-8/3}$ . This has provided the starting point when it comes to the rigorous higher-order analysis of spinning-disc flow available in the literature; see the note on Rauscher et al. (Reference Rauscher, Kelly and Cole1973) in § 1. Further, Needham & Merkin (Reference Needham and Merkin1987) studied the stability and the development of surface waves by imposing localised disturbances and deriving a simple kinematic wave equation. In general, all the virtually self-preserving or universal solutions to (3.43) we dealt with above might provide base states in forthcoming studies on the spatial/temporal stability of the axisymmetric film.

4.3 General case of modest to rapid film rotation: $\unicode[STIX]{x1D6FC}>0$

In general, equation (3.43) must be solved for some given value of $\unicode[STIX]{x1D6FC}$ and (3.43e ) specified. Fortunately, numerical experiments suggest a very weak influence of the choice of $m_{0}$ for a given value of $\unicode[STIX]{x1D70E}$ , see (3.19a ) and (3.19b ), on the film flow. Therefore, it is essentially parametrised by the latter parameter apart from $\unicode[STIX]{x1D6FC}$ . Accordingly, the downstream influence of the jet is just felt via $\unicode[STIX]{x1D6EC}_{0}$ and $\unicode[STIX]{x1D707}$ . This observation prompts us to focus on the systematic variation of $\sqrt{\unicode[STIX]{x1D6FC}}=1/Ro$ and $\unicode[STIX]{x1D70E}\;(=0,-2/3,-2)$ with $m_{0}$ fixed by unity. For $\unicode[STIX]{x1D6FC}$ increasing and ${\gtrsim}1000$ , stable downstream integration requires a progressively refined resolution of the $\unicode[STIX]{x1D702}$ -direction. It is quite amazing however, that it was found possible to achieve converged solutions for values of $\unicode[STIX]{x1D6FC}$ up to the magnitude of $10^{4}$ . This refers to a regime for $\unicode[STIX]{x1D6FC}\gg 1$ where (4.11) and the transformation underlying (4.9) allow for the asymptotic representation

(4.12) $$\begin{eqnarray}[U,W,g-1,H]\sim \left[\left(\frac{\unicode[STIX]{x1D6FC}}{3R}\right)^{1/3}f_{\infty }^{\prime },-\frac{\unicode[STIX]{x1D702}f_{\infty }^{\prime }}{3R^{2}},\frac{g_{\infty }}{(\unicode[STIX]{x1D6FC}R^{8})^{1/3}},\left(\frac{3}{\unicode[STIX]{x1D6FC}R^{2}}\right)^{1/3}\right]\quad (\bar{R}\rightarrow \infty ).\end{eqnarray}$$

For sufficiently large distances from the jet, the key properties of the universal solution to (3.43) discussed above are recovered for any non-zero (and not too large) value of $\unicode[STIX]{x1D6FC}$ . However, the deviation from Watson’s solution and, consequently, the maximum film height and hence the state of the flow close to a fully developed one are shifted upstream as $\unicode[STIX]{x1D6FC}$ increases. The rich variety of flow details, in dependence of $R$ and $\unicode[STIX]{x1D6FC}$ , is essentially reflected by the distributions of $m$ , the radial flow component $U(R,\unicode[STIX]{x1D702})$ and the associated speed $U_{s}(R):=U(R,1)=\unicode[STIX]{x1D6FC}^{3/8}\bar{U}_{s}(\bar{R})$ at the surface $\unicode[STIX]{x1D6F4}$ . These features are exemplified first for the case where the inviscid upstream flow is free of vorticity.

Figure 7. Solutions of (3.43) for $\unicode[STIX]{x1D70E}=0$ and $\unicode[STIX]{x1D6FC}=0$ (dashed), $\sqrt{\unicode[STIX]{x1D6FC}}=0.01$ , 0.02, 0.05, 0.2, 0.723, 2, 10, 100; asymptotes A1–A4 (dotted) discussed in body text: $\unicode[STIX]{x1D6FC}$ increases from (a) top to bottom with $\circ$ indicating maxima, (b) bottom to top with right ordinate referring to $\unicode[STIX]{x1D6FC}=100$ and $10^{4}$ .

4.3.1 Irrotational jet: $\unicode[STIX]{x1D70E}=0$ , $m_{0}=1$

We refer to figure 7 with the asymptotes A1–A4 defined in the following. Figure 7(a) illustrates, amongst others, the rather fast transition between Watson’s solution (A1, see § 4.1) and the canonical state even for not so small values of $\unicode[STIX]{x1D6FC}$ . It is concentrated increasingly close to the disc centre for $\unicode[STIX]{x1D6FC}$ becoming larger. Applying the similarity law $H/\bar{H}^{\ast }\sim (R/\bar{R}^{\ast })^{2}$  (A2) ensuing from (4.9) to the position of the maximum film height $H$ serves to assess the deviation from that limiting behaviour. If $\unicode[STIX]{x1D6FC}$ increases, Watson’s solution becomes less attracting as that local maximum of $H$ becomes less pronounced. It finally collapses with the minimum located by forming a flat point upstream as early as for $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}^{\ast }$ with $\unicode[STIX]{x1D6FC}^{\ast }\simeq 0.723^{2}\simeq 0.5227$ ; for larger values of $\unicode[STIX]{x1D6FC}$ , $H(R)$ decreases strictly monotonically. The surface speed shown in figure 7(b) exhibits a local maximum for all values of $\unicode[STIX]{x1D6FC}$ and a minimum upstream for not too large ones, as does $\bar{U}_{s}(\bar{r})$ . For $\unicode[STIX]{x1D6FC}\gg 1$ , $H$ and $U_{s}$ initially follow closely the inviscid-flow asymptotes $[H,U_{s}]\sim [1/R,1]$  (A3) before they undergo a rather instantaneous turn to already assume their terminal states given by (4.12) with $U_{s}\sim (9\unicode[STIX]{x1D6FC}/R)^{1/3}/2$ (A4: $\unicode[STIX]{x1D6FC}=100$ , $10^{4}$ ) for moderate values of $R$ .

Figure 8. Solutions of (3.43) for $\unicode[STIX]{x1D70E}=0$ and $\unicode[STIX]{x1D6FC}=36$ : (a) $R=0.02$ , $0.721$ , $0.046+0.05i\leqslant 0.746$ ( $i=0,1,\ldots ,14$ ); (b) $R=0.746+0.05i\leqslant 1.096$ ( $i=0,1,\ldots ,7$ , solid), $1.121+0.05i\leqslant 1.821$ ( $i=0,1,\ldots ,14$ , dashed), 7 (dash–dotted); (c) as (a) and (b) combined but with $2i$ replacing $i$ ; (d) $R=0.02$ , $0.196$ , $1.221$ , $7$ (dash-dotted); note asymptotes for $R\rightarrow 0$ and $R\rightarrow \infty$ (dotted).

Figure 8 displays the velocity components, see (3.40) and (3.42), for the case $\unicode[STIX]{x1D6FC}=36$ and values of $R$ ranging from 0.02 up to 7. Here $\unicode[STIX]{x1D6FC}$ is large enough that a wall jet forms but still so small that the evolution of the BL towards the developed film and the overshooting are recognised in great detail. Moreover, the salient properties of the flow are qualitatively representative also for other values of $\unicode[STIX]{x1D6FC}$ . The modification of the flow with $R$ is inferred from that of $U_{s}$ (see upper abscissae in $U$ -plots in figure 8 a,b) as follows. For sufficiently small and increasing values of $R$ , the velocity deficit $1-U_{s}$ is positive and increases monotonically; it is exponentially small as long as a BL can be discerned. Eventually, the $U$ -profile in disc-normal direction forms a minimum and a maximum. The latter lies more close to the disc and initiates the internal radial jet. The minimum vanishes approximately for those radial positions where $U_{s}$ undergoes its local minimum and the maximum value of $U$ again reaches unity: $R\simeq 0.721$ . Further downstream, the flow accelerates rapidly as the maximum exceeds unity and the jet becomes more pronounced. However, it dies out for $R\simeq 1.096$ , so $U$ is again strictly convex in $\unicode[STIX]{x1D702}$ for $R$ being larger. For $R\simeq 1.821$ , $U_{s}$ attains its maximum. More outwards, the flow becomes rapidly fully developed. The corresponding distributions of $g$ in dependence of $\unicode[STIX]{x1D702}$ in figure 8(c) are shifted monotonically towards $g\sim 1$ as $R$ increases. Specifically, figure 8(d) shows selected profiles of $R^{2}W$ versus $\unicode[STIX]{x1D702}$ : for $R=0.02$ , with their limiting value $-m_{0}\,\unicode[STIX]{x1D702}f_{0}^{\prime }/2$ for $R\rightarrow 0$ as lying very close, for $R=0.196$ and $R=1.221$ where they now assume a maximum and minimum respectively at the free surface, and $R=7$ . The flow is towards the disc as $w$ is negative throughout the film. The same is true for $w_{z}$ , so $ru$ increases with the radius according to (2.4). Finally, one cannot distinguish visually the results from their asymptotic representations in (4.12) as early as for the terminal $R$ -value. This is demonstrated for good measure in figure 8(c) where the plot of $(\unicode[STIX]{x1D6FC}R^{8})^{1/3}[1-g(R,\unicode[STIX]{x1D702})]/2$ for $R=7$ coincides with that for $R=\infty$ as given by the asymptote $g_{a}(\unicode[STIX]{x1D702}):=-g_{\infty }(\unicode[STIX]{x1D702})/2$ .

4.3.2 Influence of nozzle shape:  $-2\leqslant \unicode[STIX]{x1D70E}\leqslant 0$ , $m_{0}=1$

Sufficient vorticity at the nozzle outlet and thus the inlet of the film region, here (3.20) gives $\unicode[STIX]{x1D70E}=2\unicode[STIX]{x1D707}-2$ , provokes the formation of an internal wall jet by viscous diffusion. As mentioned earlier, this consolidates the possibility of controlling the strength of the radial flow even for moderate values of $\unicode[STIX]{x1D6FC}$ . More precisely, a targeted enhancement of the associated shear rates adjacent to the disc and, consequently, the diffusive–convective mass transfer of some specific constituent of the fluid (cf. Prieling & Steiner Reference Prieling and Steiner2013a ) becomes attainable. Maintaining an axisymmetric wall jet in some annular region without the need to keep the rotational speed over some threshold, seen critical for other practical reasons, could provide an efficient strategy to homogenise cleansing and etching the surface of e.g. a silicium wafer.

Choosing $\unicode[STIX]{x1D6FC}$ as 1, 100 and 400 results in figure 9 (cf. figure 3 d), allowing for the following discussion. The individual $U$ -profiles are identified advantageously by their radial positions $R=R_{i}$ ( $i=0,1,\ldots ,21$ ) defined by $R_{0}=0.01$ , $R_{i}=0.011+0.1(i-1)\leqslant 1.911$ ( $i=1,\ldots ,20$ ), $R_{21}=2$ . For the sake of clarity, the labels $i=2$ , 3, 4, 5 are omitted in figure 9(b) for $\unicode[STIX]{x1D70E}=-2/3$ but can be assigned readily to the curves lying very close together, given the evident monotonic dependence on $i$ maintained for different values of $\unicode[STIX]{x1D70E}$ . Although the curves for $i=0$ and $i=1$ are hardly distinguishable, a pronounced wall jet, already emerging rapidly for small values of $R$ and rather low levels of inlet vorticity, here represented by $\unicode[STIX]{x1D70E}=-2/3$ , is clearly discernible. The curves for the extreme case $\unicode[STIX]{x1D70E}=-2$ of vanishing $U_{s}(0)$ refer to the same $i$ -values and give qualitatively the same picture as those for $\unicode[STIX]{x1D70E}=-2/3$ and, therefore, are not indexed at all. The numerical runs predict the local extrema of $U_{s}(R)$ to occur at radial positions largely independent of $\unicode[STIX]{x1D70E}$ . For $\unicode[STIX]{x1D6FC}=1$ , the current resolution predicts the local minimum at $R\simeq R_{17}=1.611$ . In accordance with our previous findings, cf. figure 7(b), both the minimum and maximum are shifted upstream when $\unicode[STIX]{x1D6FC}$ is increased; a minimum is yet barely recognised for $\unicode[STIX]{x1D6FC}=100$ and $\unicode[STIX]{x1D70E}=0$ at $R\simeq R_{6}=0.511$ besides the maximum, detected for all three cases at $R\simeq R_{16}=1.511$ . For $\unicode[STIX]{x1D6FC}$ equal to 1 and 100, the plots for $\unicode[STIX]{x1D70E}=-2/3$ and $\unicode[STIX]{x1D70E}=-2$ show qualitatively very similar results by resorting to identical values of $i$ . The minimum has disappeared once $\unicode[STIX]{x1D6FC}$ has been increased up to 400. For this value, only increasing instances of $U_{s}$ are plotted: $i=0,\ldots ,10$ and also $R=1$ for $\unicode[STIX]{x1D70E}=0$ and $\unicode[STIX]{x1D70E}=-2/3$ ; $i=0$ , 2, 4, 6, 8, 10, 12, 13 for $\unicode[STIX]{x1D70E}=-2$ . In the last case, $U_{s}$ assumes its maximum for $R\simeq R_{13}=1.211$ . As an overall tendency, a sufficient increase of the imposed vorticity upstream suppresses the advent of a minimum as do larger values of $\unicode[STIX]{x1D6FC}$ . In any case, however, the jet dies out at $R\lesssim 1$ , which complies with the results for $\unicode[STIX]{x1D70E}=0$ shown in figure 8(b), and the flow is essentially independent of its history further downstream as quickly morphed into its terminal asymptotic state. Hence, a sufficiently large disc spin fortifies the generation of a wall jet induced by upstream vorticity, but the associated flattening of the film limits its persistence, turned off at a distance almost independent of the flow parameters.

Figure 9. Solutions of (3.43) for $\unicode[STIX]{x1D6FC}=1$ (a), $\unicode[STIX]{x1D6FC}=100$ (b), $\unicode[STIX]{x1D6FC}=400$ (c): left-/rightmost abscissae values slightly right-/left-shifted, discussion in body text.

4.4 Rapidly rotating film: $\unicode[STIX]{x1D6FC}\gg 1$

It is noted that (4.11a ) invokes integer powers of $(Ro/R^{4})^{2/3}$ as gauge functions governing perturbations about the fully developed state (4.12) for $Ro\rightarrow 0$ and $R$ kept fixed, conflicting with those proposed by Rauscher et al. (Reference Rauscher, Kelly and Cole1973). However, that state inevitably represents the far-downstream form of one governed by (3.43b ) and (3.43c ) restored in full. We here recognise the significance of the canonical transformation tied in with (4.9), but with $\bar{R}$ now being a likewise stretched radial coordinate and $\bar{f}$ , $\bar{g}$ , $\bar{m}$ , $\bar{H}$ redefined accordingly in the compressed region $\bar{R}=O(1)$ . There again (4.10a )–(4.10c ) and (4.11) hold.

However, the match with the predominantly inviscid flow further upstream is a non-trivial matter, as inferred from the mismatch of $m\sim m_{0}$ and $m\sim \unicode[STIX]{x1D6FC}^{-3/8}\bar{m}(\bar{R})$ . Here a more severe form of non-uniformity of the expansion (4.9) applies given the horizontal two-layer splitting of the flow upstream analysed in § 3.3.3, promoting the formation of a wall jet. Most important, the exponentially growing BL displacement provokes a ‘frozen’, still essentially inviscid, state of the flow in a further, relatively small region around a small value of $\bar{R}$ . This explains the observed abrupt departure from the inviscid-flow asymptotes in figure 7, shifted closer to jet impingement for larger values of $\unicode[STIX]{x1D6FC}$ , before transition towards the fully developed flow sets in but shall be specified in a separate study. Putting previous results in the affine scaling provided by (4.9) indicates both the attraction to that similarity for small but growing values of $R$ and a fast approach of the anticipated master function $[\bar{U}_{s},\bar{H}\,](\bar{R})$ towards $[(9/\bar{R})^{1/3}/2,\,3^{1/3}/\bar{R}^{2/3}]$ : see figure 10.

Figure 10. Solutions of (3.43) ( $\bar{H}$ : dashed) including asymptotes given by (4.11) (dotted).

We are now able to precisely assess the validity of the current flow structure, which might also be of practical value. At first, the shallow-water regime remains intact as long as $h\ll r$ holds. Given the scaling (3.40) and the quadratic growth rate of $H$ predicted by Watson’s self-similar state of the flow in § 4.1, this requires $R\ll \unicode[STIX]{x1D716}^{-2}$ for $\unicode[STIX]{x1D6FC}=0$ . However, it is maintained for $R<\infty$ for quite slow but still sufficiently strong disc rotation expressed by $\unicode[STIX]{x1D6FC}\gg \unicode[STIX]{x1D716}^{1/4}$ according to (4.9). Here the current inspection of very rapid rotation yields the second constraint, namely the region having $\bar{R}=O(1)$ must not collapse with the square region of inviscid jet impingement having $\bar{r}=O(1)$ . This requires $Ro_{0}\gg Re_{0}^{1/12}$ and, in combination with the first bound,

(4.13) $$\begin{eqnarray}\unicode[STIX]{x1D716}^{1/4}\ll \unicode[STIX]{x1D6FC}\ll \unicode[STIX]{x1D716}^{-1/8}.\end{eqnarray}$$

As indicated by the values in table 3 ( $a_{0}\simeq 0.8$ ), the latter estimate might be violated in a realistic scenario. However, this paves the way to a new flow structure in the immediate vicinity of jet impingement. Not covered at all by the present analysis, this flow regime is defined by ultimately strong rotation of the film, a notion coined to complete the categorisation of the flow ending § 3.3.1 and referenced in future efforts.