2.1 The physical domain

Following the treatment of Watson (Reference Watson1964), we identify four distinct flow regions for the jet over a circular disk, with smooth passage from one region to the next (see figure 1): a stagnation flow region (i), a developing boundary-layer region (ii), where the boundary layer grows until it reaches the film surface at the transition location $r=r_{0}$ and a fully viscous thin-film region (iii). A hydraulic jump emerges in region (iv), located at a radius $r=r_{J}$ . We observe that $r=O(1)$ near the stagnation point in region (i). The velocity outside the boundary layer rises rapidly from 0 at the stagnation point to the impingement velocity in the inviscid far region. In region (ii), and as we shall confirm, the boundary layer is not expected to grow like $\sqrt{r/Re}$ in the presence of gravity. The speed outside the boundary layer remains almost constant, as the fluid here is unaffected by the viscous stresses. For $r\gg 1$ , the flow field in region (ii) is not significantly affected by the stagnation flow of region (i). The region $1\ll r<r_{0}$ will be referred to as the developing boundary-layer region, with boundary-layer thickness $\unicode[STIX]{x1D6FF}(r)$ , outside which the flow is inviscid and uniform. Here $r_{0}$ is the location of the transition point at which the viscous stresses become appreciable right up to the free surface, where the whole flow is of the boundary-layer type. At this point, in the absence of gravity, the velocity profile changes from the Blasius type to the self-similar profile. In contrast, in the presence of gravity, a similarity profile does not exist. The flow in region (iii), $r>r_{0}$ , which will be referred to as the fully developed viscous region, is bounded by the disk and the free surface $z=h(r)$ .

Finally, the hydraulic jump in region (iv) occurs at a location $r=r_{J}$ , which is larger than $r_{0}$ since the jump typically occurs downstream of the transition point. Referring to figure 1, we conveniently introduce the supercritical film thickness (upstream of the jump) as $h(r)=h(r<r_{J})$ , and the subcritical thickness (downstream of the jump) as $H(r)\equiv h(r>r_{J})$ . The height immediately upstream of the jump is denoted by $h_{J}\equiv h(r=r_{J})$ , and the height immediately downstream of the jump is denoted by $H_{J}\equiv H(r=r_{J})$ . The subcritical height $H(r)$ is generally not constant and is different from the height $H_{J}$ . In this study, the fluid is assumed to be drained at the edge of the disk, at $r=r_{\infty }$ , and the flow remains steady, with the film thickness denoted by $H_{\infty }=H(r=r_{\infty })$ . Although it is common practice to assume the jump height to remain equal to $H_{J}$ , this assumption is valid for fluids of low viscosity. The edge thickness is not expected to depend heavily on the flow rate (Rojas et al. Reference Rojas, Argentina and Tirapegui2013; Mohajer & Li Reference Mohajer and Li2015).

2.2 Governing equations and boundary conditions

Unless otherwise specified, the Reynolds number is assumed to be moderately large so that our analysis is confined to the laminar regime. Consequently, for steady axisymmetric thin-film flow, in the presence of gravity, the mass and momentum conservation equations are formulated using a thin-film or Prandtl boundary-layer approach, which amounts to assuming that the radial flow varies much slower than the vertical flow (Schlichtling Reference Schlichtling2000). By letting a subscript with respect to $r$ or $z$ denote partial differentiation, the reduced dimensionless conservation equations become

(2.1a ) $$\begin{eqnarray}\displaystyle & \displaystyle u_{r}+\frac{u}{r}+w_{z}=0, & \displaystyle\end{eqnarray}$$

(2.1b ) $$\begin{eqnarray}\displaystyle & \displaystyle Re(uu_{r}+wu_{z})=-\frac{Re}{Fr^{2}}p_{r}+u_{zz}, & \displaystyle\end{eqnarray}$$

(2.1c ) $$\begin{eqnarray}\displaystyle & \displaystyle p_{z}=-1. & \displaystyle\end{eqnarray}$$

$p$

These are the thin-film equations commonly used to model the spreading liquid flow (Tani Reference Tani1949; Bohr et al. Reference Bohr, Dimon and Putzkaradze1993, Reference Bohr, Ellegaard, Hansen and Haaning1996; Kasimov Reference Kasimov2008). We observe that the pressure for a thin film is hydrostatic as a result of its vanishing at the film surface (in the absence of surface tension) and the small thickness of the film. In addition, upstream of the jump, the variation of the film thickness with the radius is expected to be smooth and gradual. In this case, the radial variation of the hydrostatic pressure is also small. Unlike the case of liquids of low viscosity, gravity cannot be neglected in the supercritical range. At the disk, the no-slip and no-penetration conditions are assumed to hold for any $r$ . In this case:

(2.2) $$\begin{eqnarray}\displaystyle u(r,z=0)=w(r,z=0)=0. & & \displaystyle\end{eqnarray}$$

At the free surface $z=h(r<r_{J})$ or $z=H(r>r_{J})$ , the kinematic and dynamic conditions for steady flow take the form

(2.3a,b ) $$\begin{eqnarray}\displaystyle w(r,z=h)=u(r,z=h)h^{\prime }(r),\quad u_{z}(r,z=h)=p(r,z=h)=0. & & \displaystyle\end{eqnarray}$$

Here a prime denotes total differentiation. The flow field is sought separately in the developing boundary-layer region (ii) for $0<r<r_{0}$ , the fully developed viscous boundary-layer region (iii) for $r_{0}<r<r_{J}$ and the hydraulic jump region (iv) for $r_{J}<r<r_{\infty }$ . We observe that region (i) is neglected and will not be considered here. In this case, the leading edge of the boundary layer in region (ii) is taken to coincide with the disk centre. Consequently, the additional boundary conditions are as follows. In region (ii), the flow is assumed to be sufficiently inertial for inviscid flow to prevail between the boundary-layer outer edge and the free surface (see figure 1). In this case, the following conditions at the outer edge of the boundary layer and beyond must hold:

(2.4a,b ) $$\begin{eqnarray}\displaystyle u_{z}(r<r_{0},z=\unicode[STIX]{x1D6FF})=0,\quad u(r<r_{0},\unicode[STIX]{x1D6FF}\leqslant z<h)=1. & & \displaystyle\end{eqnarray}$$

Integrating (2.1c ) subject to condition (2.3b ), the pressure becomes $p(r,z)=h(r)-z$ , which is then eliminated so that (2.1b ) reduces to

(2.5) $$\begin{eqnarray}\displaystyle Re(uu_{r}+wu_{z})=-\frac{Re}{Fr^{2}}h^{\prime }+u_{zz}. & & \displaystyle\end{eqnarray}$$

Finally, the conservation of mass at any location upstream and downstream of the jump yields the following relation in dimensionless form:

(2.6) $$\begin{eqnarray}\displaystyle \frac{1}{2r}=\int _{0}^{h(r)}u(r<r_{j},z)\,\text{d}z=\int _{0}^{H(r)}u(r>r_{j},z)\,\text{d}z. & & \displaystyle\end{eqnarray}$$

The presence of gravity causes the flow to be non-self-similar in character. Therefore, in the present study, approximate solutions are sought in each region. An integral approach of the Kármán–Pohlhausen (KP) type (Schlichtling Reference Schlichtling2000) is adopted upstream of the jump, similar to the formulation of Prince et al. (Reference Prince, Maynes and Crockett2012) for a jet impinging on a disk with slip. The cubic profile is used for the velocity, which is considered to be the leading-order solution in a comprehensive spectral approach when inertia is included (Khayat & Kim Reference Khayat and Kim2006; Rojas et al. Reference Rojas, Argentina and Tirapegui2010). The cubic profile seems to be amply adequate as it leads to close agreement with Watson’s (Reference Watson1964) similarity solution for a jet impinging on a stationary disk (Prince et al. Reference Prince, Maynes and Crockett2012). The cubic profile was also assessed by Khayat (Reference Khayat2016) for a planar jet impinging on a surface with slip, and was found to yield a good agreement against his numerical solution. See also Rao & Arakeri (Reference Rao and Arakeri1998) for an integral analysis of a rotating film. Higher-order polynomial velocity profiles were also used. In their study on flow separation and wave breaking, Bohr et al. (Reference Bohr, Ellegaard, Hansen and Haaning1996) used a quartic profile to illustrate the emergence of a singularity at the separation point for a thin film. The coefficients in the polynomial expansion were not obtained explicitly. A cubic velocity profile was later adopted by Bohr et al. (Reference Bohr, Putkaradze and Watanabe1997), accounting for regions of separation. The cubic profile was also adopted in our recent study for a jet impinging on a rotating disk (Wang & Khayat Reference Wang and Khayat2018), and was found to yield close agreement with experiment.

4.1 The equation for the film thickness

In region (iii), the potential flow in the radial direction ceases to exist, with the velocity $U(r)=u(r,z=h)$ at the free surface becoming dependent on $r$ . We again assume a cubic velocity profile subject to conditions (2.3a ) and (2.3b ). In this case, the radial velocity profile is given as function of the surface velocity $U(r)$ as

(4.1a,b ) $$\begin{eqnarray}\displaystyle u(r_{0}<r<r_{J},z)=U(r)f(\unicode[STIX]{x1D702}),\quad \unicode[STIX]{x1D702}=\frac{z}{h(r)}. & & \displaystyle\end{eqnarray}$$

Here, we observe that $f(\unicode[STIX]{x1D702})$ is still given by (3.4). Using the mass conservation (2.6) yields the following relation:

(4.2) $$\begin{eqnarray}\displaystyle U(r_{0}<r<r_{J})=\frac{4}{5hr}. & & \displaystyle\end{eqnarray}$$

This equation agrees with equation (15) of Prince et al. (Reference Prince, Maynes and Crockett2012) when setting their slip parameter equal to zero.

Similar to (3.2), the integral form of the momentum equation reads:

(4.3) $$\begin{eqnarray}\displaystyle \frac{Re}{r}\frac{\text{d}}{\text{d}r}\int _{0}^{h}ru^{2}\,\text{d}z=-\frac{Re}{Fr^{2}}hh^{\prime }-u_{z}(r,z=0). & & \displaystyle\end{eqnarray}$$

Substituting (4.1) into (4.3) and using (4.2) to eliminate $U$ , we obtain the equation for the film thickness in the fully viscous region:

(4.4) $$\begin{eqnarray}\displaystyle Re\left(\frac{1}{Fr^{2}}-\frac{272}{875r^{2}h^{3}}\right)h^{\prime }=\frac{4}{5rh^{2}}\left(\frac{68Re}{175}\frac{1}{r^{2}}-\frac{3}{2h}\right), & & \displaystyle\end{eqnarray}$$

which is solved for $r\geqslant r_{0}$ subject to $h(r=r_{0})=\unicode[STIX]{x1D6FF}(r_{0})$ . We observe that the flow in the absence of gravity is recovered in the limit $Fr\rightarrow \infty$ . In this case, the problem reduces to the following equation and boundary condition:

(4.5a,b ) $$\begin{eqnarray}\displaystyle \frac{\text{d}h}{\text{d}r}=-\frac{h}{r}+\frac{525}{136}\frac{r}{Re},\quad h(r_{0})=2\sqrt{\frac{70}{39}\frac{r_{0}}{Re}}, & & \displaystyle\end{eqnarray}$$

which admits

(4.6) $$\begin{eqnarray}\displaystyle h(r>r_{0})=\frac{175}{136Rer}(r^{3}-r_{0}^{3})+2\frac{r_{0}}{r}\sqrt{\frac{70}{39}\frac{r_{0}}{Re}}=\frac{175}{136}\frac{r^{2}}{Re}+\frac{233}{340}\frac{1}{r}, & & \displaystyle\end{eqnarray}$$

as solution, where we recall $r_{0}=((78/875)Re)^{1/3}$ . For comparison, Watson’s expression is reproduced here in dimensionless form:

(4.7) $$\begin{eqnarray}\displaystyle h(r>r_{0})=\frac{2\unicode[STIX]{x03C0}}{3\sqrt{3}}\frac{r^{2}}{Re}+\frac{3c(3\sqrt{3}c-\unicode[STIX]{x03C0})}{8\unicode[STIX]{x03C0}}\frac{1}{r}. & & \displaystyle\end{eqnarray}$$

Thus, we have $h\approx 1.21(r^{2}/Re)+0.685(1/r)$ from (4.6) compared to Watson’s $h\approx 1.28(r^{2}/Re)+0.69(1/r)$ from (4.7), showing a surprisingly close agreement, and validity of the cubic profile.

4.2 The supercritical flow and the location of the hydraulic jump

Equation (4.4) indicates that a singularity exists, occurring at some distance where the slope of the free surface becomes infinite. An equation similar to (4.4) was obtained by Bohr et al. (Reference Bohr, Dimon and Putzkaradze1993) by approximating the mean of the derivative of $u^{2}$ in the averaged momentum equation (4.3) in terms of the derivative of the mean square. They showed that the singularity is not an artefact of the averaging process, but is inherent to the thin-film equations. Of closer relevance is the equation obtained by Kasimov (Reference Kasimov2008) using a parabolic velocity profile, incorporating the shape of a finite disk with a sudden falling edge (see below).

We conjecture that the location of the singularity coincides with the radius of the jump. Consequently, we now have a relation between the jump radius $r_{J}$ and the film height $h_{J}$ immediately upstream of the jump:

(4.8) $$\begin{eqnarray}\displaystyle Fr^{-2}=\frac{272}{875}\frac{1}{r_{J}^{2}h_{J}^{3}}. & & \displaystyle\end{eqnarray}$$

We therefore identify the jump location or radius to occur when the slope of the free surface upstream of the jump becomes infinite, that is $h^{\prime }(r=r_{J})\rightarrow \infty$ , which coincides with the occurrence of the singularity of equation (4.4). At this location the relation between the jump radius and height is given by (4.8). Obviously, this claim is bold and needs to be validated, which we shall do shortly. The jump location is found by simply integrating (4.4) numerically subject to $h(r_{0})=\unicode[STIX]{x1D6FF}(r_{0})$ from $r_{0}$ to a distance $r_{J}$ until (4.8) is satisfied to within a certain tolerance. More details on the numerical treatment are given below.

Before comparing the predicted jump radius with existing measurements, it is helpful to explore the general supercritical flow behaviour (upstream of the jump). Once again, the flow becomes governed by a one-parameter problem when transformation (3.7) is used. In this case, (4.4) reduces to

(4.9) $$\begin{eqnarray}\displaystyle \left(\frac{1}{\unicode[STIX]{x1D6FC}}-\frac{272}{875}\frac{1}{\bar{r}^{2}\bar{h}^{3}}\right)\bar{h}^{\prime }=\frac{1}{\bar{r}\bar{h}^{2}}\left(\frac{272}{875}\frac{1}{\bar{r}^{2}}-\frac{6}{5\bar{h}}\right), & & \displaystyle\end{eqnarray}$$

where we recall $\unicode[STIX]{x1D6FC}$ from (3.9). In this case, relation (4.8) becomes

(4.10) $$\begin{eqnarray}\displaystyle \bar{r}_{J}^{2}\bar{h}_{J}^{3}={\textstyle \frac{272}{875}}\unicode[STIX]{x1D6FC}. & & \displaystyle\end{eqnarray}$$

We observe that the velocity at the free surface remains invariant under transformation (3.7). Equation (4.9) is integrated numerically subject to the condition $\bar{h}(\bar{r}=\bar{r}_{0})=\bar{\unicode[STIX]{x1D6FF}}(\bar{r}_{0})$ at the transition point, using MATLAB sixth-order Runge–Kutta scheme. The integration is carried out at equal steps in the distance taken equal to $10^{-3}$ , until the turning point is reached at the singularity. The program is terminated when the slope $h^{\prime }>10^{2}$ , giving an accuracy in the jump location $r_{J}$ to the third decimal. The pre-jump height $h_{J}$ is then deduced from (4.10).

We observe that equation (4.9), similar to equation (33) of Bohr et al. (Reference Bohr, Dimon and Putzkaradze1993) and equation (3.1)) of Kasimov (Reference Kasimov2008) for a flat disk, has only one critical point $\bar{h}_{c}=(6\unicode[STIX]{x1D6FC}/5)^{1/4},\bar{r}_{c}=\sqrt{272/875}(5/6)^{3/8}\unicode[STIX]{x1D6FC}^{1/8}$ , which is the root of the system $1/\unicode[STIX]{x1D6FC}-(272/875)(1/\bar{r}^{2}\bar{h}^{3})=(272/875)(1/\bar{r}^{2})-6/5\bar{h}=0$ , corresponding to a pair of complex conjugate eigenvalues of the Jacobian of the linearized two-dimensional dynamical system (Kasimov Reference Kasimov2008). The real part is positive, indicating that the critical point is an unstable spiral, which in turn indicates that the solution cannot pass through the critical point. Bohr et al. (Reference Bohr, Dimon and Putzkaradze1993) estimated that the jump is located close to the critical point. They computed the flow and the free surface by choosing the pre-jump (inner) branch to correspond to a constant average velocity, and chose the post-jump (outer) branch that emanates from the point of singularity. The two branches are then joined by the shock when they reach the same radial position, at a point that is identified as the jump radius (see their figure 3). This method led them to deduce the scaling for the jump radius to be $R_{J}\approx Q^{5/8}\unicode[STIX]{x1D708}^{-3/8}g^{-1/8}$ . In fact, if we assume the jump to occur at or near the critical point and recall (3.9), we obtain $\bar{r}_{J}\approx \bar{r}_{c}=\sqrt{272/875}(5/6)^{3/8}(Re^{3}Fr^{2})^{1/8}$ , which is the dimensionless form of the scaling deduced by Bohr et al. (Reference Bohr, Dimon and Putzkaradze1993).

Later, Kasimov (Reference Kasimov2008) derived an equation of closer similarity to (4.9) but introduced the shape of a flat disk with a cutoff at the edge. See his equation (3.1) and figure 2. The addition of the variable disk shape led to the existence of a critical saddle point near the disk edge, in addition to the spiral critical point. Kasimov determined the flow and the surface height on the two sides of the jump. The upstream branch is sought by solving his equation (3.1) subject to an initial condition corresponding the location where the jet velocity at impact equals the free-surface velocity. The downstream branch is sought by integrating (3.1) inward toward the jump starting at the far critical saddle point through which the solution effectively must pass. The integration is terminated on each side at the turning points, corresponding to an infinite slope in the surface height or the singularity in (3.1). The two heights computed on either side are subsequently used to determine the location of the jump by applying the discretized momentum equation.

Figure 4 gives an overview of the influence of gravity on the film thickness distribution up to the jump location. The film thickness exhibits a minimum typically downstream of the transition location. In general, gravity does not seem to have much of an effect until the jump is reached. In particular, and as expected by inspecting (2.5), near the minimum, gravity has a negligible effect. The jump occurs sooner and closer to transition point for a flow with stronger gravity.

Figure 4. Influence of gravity on the developing boundary-layer height and film thickness for supercritical flow. Also indicated in vertical lines are the locations of the hydraulic jump location $Re^{-1/3}r_{J}$ for different $\unicode[STIX]{x1D6FC}$ values.

The influence of gravity on the corresponding free-surface velocity profiles is depicted in figure 5. Here the velocity in the developing boundary-layer region (ii) outside the boundary layer is equal to 1 (the uniform jet velocity), which then decreases monotonically with distance downstream of the transition location. In the absence of gravity $(\unicode[STIX]{x1D6FC}\rightarrow \infty )$ , the surface velocity decreases rapidly. This behaviour is easily deduced from (4.2) by substituting (4.6) to obtain

(4.11) $$\begin{eqnarray}\displaystyle \bar{U}(\bar{r}_{0}<\bar{r}<\bar{r}_{J})\sim {\textstyle \frac{4}{5}}\left({\textstyle \frac{175}{136}}(\bar{r}^{3}-\bar{r}_{0}^{3})+2\bar{r}_{0}\sqrt{{\textstyle \frac{70}{39}}\bar{r}_{0}}\right)^{-1},\quad \text{as }\unicode[STIX]{x1D6FC}\rightarrow \infty . & & \displaystyle\end{eqnarray}$$

In this case, $U$ decreases like $r^{-3}$ at large distance. The figure indicates that gravity tends to enhance the radial flow near the transition point similar to the effects of disk rotation (Wang & Khayat Reference Wang and Khayat2018) and slip (Prince et al. Reference Prince, Maynes and Crockett2012; Khayat Reference Khayat2016). The rate at which the surface velocity decays with radial distance is also enhanced by gravity. However, further downstream, gravity inhibits flow movement as the film thickens ahead of the jump.

Figure 5. Influence of gravity on the free-surface velocity for supercritical flow.

Figure 6 illustrates the development of the dimensionless wall shear stress at the disk (skin friction) for the same gravity levels as in figures 4 and 5. The figure shows that the wall shear stress is always larger for higher gravity except near the jump. This larger shear stress, which reflects a larger shear rate at the disk, is the result of a thinner film thickness and a greater free-surface velocity caused by a higher gravity effect as already reported in figures 4 and 5. The shear stress decays rapidly with radial distance in the developing boundary-layer region. In the absence of gravity,

(4.12) $$\begin{eqnarray}\displaystyle \bar{\unicode[STIX]{x1D70F}}_{w}(\bar{r}_{0}<\bar{r}<\bar{r}_{J})\sim {\textstyle \frac{6}{5}}\bar{r}\left[{\textstyle \frac{175}{136}}(\bar{r}^{3}-\bar{r}_{0}^{3})+2\bar{r}_{0}\sqrt{{\textstyle \frac{70}{39}}\bar{r}_{0}}\right]^{-2},\quad \text{as }\unicode[STIX]{x1D6FC}\rightarrow \infty . & & \displaystyle\end{eqnarray}$$

For any gravity level, after the rapid drop, $\unicode[STIX]{x1D70F}_{w}$ exhibits a maximum before decaying monotonically. At large radial distance, the shear stress decays like $r^{-5}$ in the absence of gravity effect. The strength of the maximum is essentially uninfluenced by gravity, but tends to occur further downstream with increasing gravity effect.

Although both the film height in figure 4 and the free-surface velocity in figure 5 are not significantly influenced by gravity, the location of the jump reflects a significant influence. This is depicted in figure 7 where the jump radius and corresponding film thickness immediately upstream of the jump are plotted against $\unicode[STIX]{x1D6FC}$ . The growth of the jump radius and the height closely follows the overall behaviour $\bar{r}_{J}\approx \unicode[STIX]{x1D6FC}^{1/6}$ and $\bar{h}_{J}\approx \unicode[STIX]{x1D6FC}^{2/9}$ , yielding $\bar{r}_{J}^{2}\bar{h}_{J}^{3}\approx \unicode[STIX]{x1D6FC}$ , which agrees with the original expression (4.10). The fractional power growth is also reflected from the position of the vertical lines in figure 4 as well.

Figure 6. Influence of gravity on the wall shear stress for supercritical flow.

Figure 7. Dependence on the jump location and height immediately upstream of the jump on $\unicode[STIX]{x1D6FC}$ .

4.3 Comparison with experiment

Figure 8 shows the dependence of the dimensional jump location on the flow rate, where comparison is carried out with the measurements of Hansen et al. (Reference Hansen, Horluck, Zauner, Dimon, Ellegaard and Creagh1997) as well as the numerical predictions of Rojas et al. (Reference Rojas, Argentina and Tirapegui2010) for silicon oils of two different viscosities. The same experimental data were also used by Rojas et al. (Reference Rojas, Argentina and Tirapegui2010) when they validated their spectral solution. We have included our results using the same log–log ranges used by Rojas et al. (Reference Rojas, Argentina and Tirapegui2010) in their figure 2. Our predictions are in excellent agreement with their numerical results. The qualitative and quantitative agreement for the highest-viscosity case $\unicode[STIX]{x1D708}=95~\text{cS}$ is especially encouraging given the simplicity of the present approach compared to their spectral approach. In particular, and in contrast to the numerical approach, the present formulation does not require imposing a boundary condition downstream of the jump. The agreement with experiment appears to suggest that the location of the jump can be determined without knowledge of downstream conditions such as the disk diameter or the thickness at the edge of the disk. This observation corroborates well the experimental findings of Brechet & Neda (Reference Brechet and Neda1999). We recall that Rojas et al. (Reference Rojas, Argentina and Tirapegui2010) had to impose the thickness at the edge of the disk as measured by Hansen et al. (Reference Hansen, Horluck, Zauner, Dimon, Ellegaard and Creagh1997). Both the present theoretical and existing numerical predictions tend to overestimate slightly the jump radius compared to experiment. The discrepancy appears to be higher for low flow rates, for a given liquid. A plausible explanation for the discrepancy is the difficulty to accurately locate the jump radius in reality. Moreover, it is likely that the jump for $\unicode[STIX]{x1D708}=95~\text{cSt}$ is of type II, with two vortex rolls downstream of the jump.

Figure 8. Dependence of the hydraulic jump radius on the flow rate. The figure shows the comparison between the present theoretical predictions and the measurements of Hansen et al. (Reference Hansen, Horluck, Zauner, Dimon, Ellegaard and Creagh1997) for two silicon oils of viscosities $\unicode[STIX]{x1D708}=15~\text{cSt}$ and 95 cSt. The numerical predictions of Rojas et al. (Reference Rojas, Argentina and Tirapegui2010) are also included.

We further assess the validity of our approach by comparison against the scaling law proposed by Rojas et al. (Reference Rojas, Argentina and Tirapegui2013), which relates the radius of the jump, in particular, to the height downstream of the jump (see their relation (15)). In the absence of surface tension, the relation, written here as $R_{J}\approx ((9/70)(Q^{3}/\unicode[STIX]{x03C0}^{3}\unicode[STIX]{x1D708}g\tilde{H}_{\infty }))^{1/4}$ , becomes based on their spectral approach for inertial lubrication flow (Rojas et al. Reference Rojas, Argentina and Tirapegui2010) and the inviscid Belanger equation (White Reference White2006). Figure 9 shows the comparison between our predictions and the scaling law.

Figure 9. Dependence of the hydraulic jump radius on the flow rate. The figure shows the comparison between the present theoretical predictions (solid lines) and the ones based on the scaling law (dashed lines) of Rojas et al. (Reference Rojas, Argentina and Tirapegui2013) for two silicon oils of viscosities $\unicode[STIX]{x1D708}=15~\text{cSt}$ and 95 cSt.

Finally, we observe that our earlier approach (Wang & Khayat Reference Wang and Khayat2018), where gravity is neglected in the supercritical regime, could not capture the location of the jump in comparison to experiment for heavily viscous liquids. On the other hand, the case of water is not considered here, not just because of the low viscosity but also due to the high surface tension. We considered the case of water in our recent work dealing with high surface tension liquids along with comparisons with experiment (Wang & Khayat Reference Wang and Khayat2018). In the present work, we neglect surface tension upstream in order to investigate the role of gravity on the hydraulic jump. Our objective is to also confirm or dismiss the claim and observation of Duchesne et al. (Reference Duchesne, Lebon and Limat2014) for highly viscous liquids. Further comparison with other existing measurements, including those of Duchesne et al. (Reference Duchesne, Lebon and Limat2014), will be carried out in § 5.

4.4 Further validation

As further general assessment of the validity of equation (4.9), we examine its solution against that of the shallow-water equations for weak gravity. Thus, we set $\unicode[STIX]{x1D700}\equiv \unicode[STIX]{x1D6FC}^{-1}$ as the small parameter, and expand the thickness as $\bar{h}(\bar{r})=\sum _{m=0}\unicode[STIX]{x1D700}^{m}\bar{h}_{m}(\bar{r})$ . To leading order, (4.9) yields the following equation for $\bar{h}_{0}$ :

(4.13) $$\begin{eqnarray}\displaystyle (\bar{r}\bar{h}_{0})^{\prime }={\textstyle \frac{525}{136}}\bar{r}^{2}, & & \displaystyle\end{eqnarray}$$

which corresponds to the thickness in the absence of gravity. The solution of this equation was already given earlier and is equivalent to (4.6). To next order:

(4.14) $$\begin{eqnarray}\displaystyle (\bar{r}\bar{h}_{1})^{\prime }={\textstyle \frac{875}{272}}\bar{r}^{3}\bar{h}_{0}^{3}\bar{h}_{0}^{\prime }. & & \displaystyle\end{eqnarray}$$

We next examine the corresponding solution of the shallow-water equations, which are first rescaled to involve the only parameter $\unicode[STIX]{x1D6FC}=Re^{1/3}Fr^{2}$ by recalling (3.7) and introducing all the barred variables as

(4.15a-e ) $$\begin{eqnarray}\displaystyle r=Re^{1/3}\bar{r},\quad h=Re^{-1/3}\bar{h},\quad z=Re^{-1/3}\bar{z},\quad u=\bar{u},\quad w=Re^{-2/3}\bar{w}. & & \displaystyle\end{eqnarray}$$

In this case, equations (2.1a ), (2.5) and (2.6) become

(4.16a-c ) $$\begin{eqnarray}\displaystyle \bar{u}_{\bar{r}}+\frac{\bar{u}}{\bar{r}}+\bar{w}_{\bar{z}}=0,\quad \bar{u}\bar{u}_{\bar{r}}+\bar{w}\bar{u}_{\bar{z}}=-\unicode[STIX]{x1D6FC}^{-1}\bar{h}^{\prime }+\bar{u}_{\bar{z}\bar{z}},\quad \int _{0}^{\bar{h}}\bar{u}\,\text{d}\bar{z}=\frac{1}{2\bar{r}}. & & \displaystyle\end{eqnarray}$$

In the presence of gravity, a similarity solution is possible only under some conditions.

Starting with the mapping $\bar{\unicode[STIX]{x1D709}}(\bar{r},\bar{z})=\bar{r},\bar{\unicode[STIX]{x1D702}}(\bar{r},\bar{z})=\bar{z}/\bar{h}(\bar{r})$ , and taking $\bar{u}(\bar{r},\bar{z})=\bar{U}(\bar{\unicode[STIX]{x1D709}})g(\bar{\unicode[STIX]{x1D702}})$ , (4.16a ) becomes

(4.17) $$\begin{eqnarray}\displaystyle \bar{U}^{\prime }g-\frac{\bar{h}^{\prime }}{\bar{h}}\bar{\unicode[STIX]{x1D702}}\bar{U}g_{\bar{\unicode[STIX]{x1D702}}}+\frac{\bar{U}g}{\bar{\unicode[STIX]{x1D709}}}+\frac{1}{\bar{h}}\bar{w}_{\bar{\unicode[STIX]{x1D702}}}=0. & & \displaystyle\end{eqnarray}$$

Isolating $\bar{w}_{\bar{\unicode[STIX]{x1D702}}}$ and rearranging terms:

(4.18) $$\begin{eqnarray}\displaystyle \bar{w}_{\bar{\unicode[STIX]{x1D702}}}=\bar{\unicode[STIX]{x1D709}}^{-1}[-(\bar{\unicode[STIX]{x1D709}}\bar{h}\bar{U})^{\prime }g+\bar{\unicode[STIX]{x1D709}}\bar{h}^{\prime }\bar{U}(\bar{\unicode[STIX]{x1D702}}g)_{\bar{\unicode[STIX]{x1D702}}}]. & & \displaystyle\end{eqnarray}$$

Now, from conservation of mass or (4.16c ), we have

(4.19) $$\begin{eqnarray}\displaystyle \bar{U}\bar{h}\bar{r}=\frac{1}{2\displaystyle \int _{0}^{1}g\,\text{d}\bar{\unicode[STIX]{x1D702}}}=\text{Const., yielding}\quad (\bar{\unicode[STIX]{x1D709}}\bar{h}\bar{U})^{\prime }=0. & & \displaystyle\end{eqnarray}$$

Consequently, (4.18) reduces to $\bar{w}_{\bar{\unicode[STIX]{x1D702}}}=\bar{h}^{\prime }\bar{U}(\bar{\unicode[STIX]{x1D702}}g)_{\bar{\unicode[STIX]{x1D702}}}$ . Integrating and recalling that $\bar{w}(\bar{\unicode[STIX]{x1D709}},\bar{\unicode[STIX]{x1D702}}=0)=0$ , we get $\bar{w}=\bar{h}^{\prime }\bar{U}\bar{\unicode[STIX]{x1D702}}g(\bar{\unicode[STIX]{x1D702}})$ .

We thus have so far

(4.20a-c ) $$\begin{eqnarray}\displaystyle \bar{u}=\bar{U}g(\bar{\unicode[STIX]{x1D702}}),\quad \bar{w}=\bar{h}^{\prime }\bar{U}\bar{\unicode[STIX]{x1D702}}g(\bar{\unicode[STIX]{x1D702}}),\quad \bar{r}\bar{U}\bar{h}=C\equiv \left(2\int _{0}^{1}g(\unicode[STIX]{x1D702})\,\text{d}\unicode[STIX]{x1D702}\right)^{-1}, & & \displaystyle\end{eqnarray}$$

where (4.20c ) is deduced from (4.16c ). Substituting the velocity components from (4.20a,b ) into equation (4.16b ), and eliminating $\bar{U}$ using (4.20c ), yields the following problem for $g$ :

(4.21a-c ) $$\begin{eqnarray}\displaystyle C^{2}(\bar{r}\bar{h})^{\prime }g^{2}-\unicode[STIX]{x1D700}\bar{h}^{3}\bar{r}^{3}\bar{h}^{\prime }+Cr^{2}g_{\bar{\unicode[STIX]{x1D702}}\bar{\unicode[STIX]{x1D702}}}=0,\quad g(0)=g_{\bar{\unicode[STIX]{x1D702}}}(1)=0,\quad g(1)=1. & & \displaystyle\end{eqnarray}$$

We again seek the solution by expanding the thickness as $\bar{h}(\bar{r})=\sum _{m=0}\unicode[STIX]{x1D700}^{m}\bar{h}_{m}(\bar{r})$ . To leading order, we recover the classical equation of Watson (Reference Watson1964):

(4.22) $$\begin{eqnarray}\displaystyle C(\bar{r}\bar{h}_{0})^{\prime }g^{2}+\bar{r}^{2}g_{\bar{\unicode[STIX]{x1D702}}\bar{\unicode[STIX]{x1D702}}}=0, & & \displaystyle\end{eqnarray}$$

which suggests that $\bar{r}^{-2}(\bar{r}\bar{h}_{0})^{\prime }$ must be constant. Multiplying (4.22) by $g_{\bar{\unicode[STIX]{x1D702}}}$ , and integrating using the conditions in (4.21), yields the following equation:

(4.23) $$\begin{eqnarray}\displaystyle (rh_{0})^{\prime }=\frac{3c^{2}}{2C}r^{2},\quad \text{where }c=\int _{0}^{1}\frac{\text{d}g}{\sqrt{1-g^{3}}}=1.402. & & \displaystyle\end{eqnarray}$$

The value of $C$ is determined by noting that $c=g_{\bar{\unicode[STIX]{x1D702}}}/\sqrt{1-g^{3}}$ , yielding $\int _{0}^{1}g\,\text{d}g=\int _{0}^{1}g\,\text{d}g/\sqrt{1-g^{3}}=0.615$ , so that $C=0.813$ . To the next order in $\unicode[STIX]{x1D700}$ , (4.21) gives

(4.24) $$\begin{eqnarray}\displaystyle (\bar{r}\bar{h}_{1})^{\prime }=\frac{3c}{2C^{2}}\bar{r}^{3}\bar{h}_{0}^{3}\bar{h}_{0}^{\prime }, & & \displaystyle\end{eqnarray}$$

where $c\int _{0}^{1}g^{2}\,\text{d}\unicode[STIX]{x1D702}=(1/3)\int _{0}^{1}\text{d}g^{3}/\sqrt{1-g^{3}}=2/3$ was used. Comparison between the numerical coefficients of (4.14) and (4.24) indicates a discrepancy of 6 %. The discrepancy for the first-order contribution is 1 % when (4.15) is compared with (4.24).

Finally, it is worth observing that Bohr et al. (Reference Bohr, Dimon and Putzkaradze1993) and Rojas et al. (Reference Rojas, Argentina and Tirapegui2013) found that the jump radius scales roughly as $R_{J}\approx Q^{5/8}\unicode[STIX]{x1D708}^{-3/8}g^{-1/8}$ . Using the current scaling, the dimensionless form of the estimate of Bohr et al. can be written as $r_{J}\approx Re^{3/8}Fr^{1/4}$ . Avedisian & Zhao (Reference Avedisian and Zhao2000) investigated the circular hydraulic jump experimentally for normal and reduced gravity conditions. They measured the jump diameter and shape at the free liquid surface for an impinging jet on a stationary disk. Based on the reported two values of the flow rate and two gravitational acceleration data provided, we find that the location of the jump behaves close to $g^{-1/9}$ , roughly confirming the scaling of Bohr et al. (Reference Bohr, Dimon and Putzkaradze1993) for low gravity. We can also estimate the behaviour of the jump radius from (4.8) by assuming the thickness from (4.6) or (4.7) for large distance or $h_{J}\approx r_{J}^{2}/Re$ . When inserted in (4.8), we obtain $r_{J}\approx Re^{3/8}Fr^{1/4}$ , which is precisely the scaling suggested by Bohr et al. (Reference Bohr, Dimon and Putzkaradze1993) cast in dimensionless form. Interestingly, this scaling law can also be expressed in terms of only one parameter as $\bar{r}_{J}\approx \unicode[STIX]{x1D6FC}^{1/8}$ .

5.1 Conservation of momentum across the jump

We first recall the integral form (4.3) of the momentum conservation equation, which holds for any position $r>r_{0}$ in the super- and subcritical regions. Across the jump, equation (4.3) is applied for a control volume of width $\unicode[STIX]{x0394}r$ in the radial direction, taking the following discretized form:

(5.1) $$\begin{eqnarray}\displaystyle Re\unicode[STIX]{x0394}\int _{0}^{h}u^{2}\,\text{d}z=-\frac{Re}{Fr^{2}}\frac{\unicode[STIX]{x0394}h^{2}}{2}-\unicode[STIX]{x0394}ru_{z}(r_{J},z=0). & & \displaystyle\end{eqnarray}$$

Since the width of the jump $\unicode[STIX]{x0394}r$ is assumed to be small, equation (5.1) reduces to

(5.2) $$\begin{eqnarray}\displaystyle \frac{Fr^{-2}}{2}(H_{J}^{2}-h_{J}^{2})=\int _{0}^{h_{J}}u^{2}(r_{J-},z)\,\text{d}z-\int _{0}^{H_{J}}u^{2}(r_{J+},z)\,\text{d}z. & & \displaystyle\end{eqnarray}$$

We observe that the supercritical velocity is already available from (4.1) and (4.2), yielding $u(r_{J-},z)=(4/5r_{J}h_{J})f(\unicode[STIX]{x1D702})$ , where $\unicode[STIX]{x1D702}=z/h_{J}$ and $f(\unicode[STIX]{x1D702})$ is given in (3.4). We also use relation (4.8) to eliminate $h_{J}$ , In this case, (5.2) becomes

(5.3) $$\begin{eqnarray}\displaystyle H_{J}^{2}-3\left(\frac{272}{875}\frac{Fr^{2}}{r_{J}^{2}}\right)^{2/3}+2Fr^{2}\int _{0}^{H_{J}}u^{2}(r_{J+},z)\,\text{d}z=0. & & \displaystyle\end{eqnarray}$$

Thus, the jump height is completely determined as a function of the Froude and the Reynolds numbers once the subcritical velocity profile $u(r_{J+},z)$ is imposed. Various assumptions have been adopted in the literature, ranging from inviscid to fully viscous flows. Both regimes will be explored next.

We first consider the flow to be inviscid downstream of the jump. Although the present work is focused on heavily viscous liquids, the inviscid subcritical flow would correspond to a jump that is relatively high. A boundary layer always exists, but it is relatively thin in this case. Consequently, the velocity right downstream of the jump would be essentially uniform. This is an assumption that has been widely adopted in the literature in various contexts (see, for instance, Watson Reference Watson1964; Bush & Aristoff Reference Bush and Aristoff2003; Dressaire et al. Reference Dressaire, Courbin, Crest and Stone2010; Prince et al. Reference Prince, Maynes and Crockett2012). At the very least, the inclusion of the uniform subcritical flow is helpful as a reference limit. The assumption of uniform flow does not necessarily have to hold over the entire domain downstream of the jump. It is not unreasonable to assume that the velocity is uniform right downstream of the jump, which allows the jump height to be readily determined from (5.3). Further downstream, the flow may be considered fully viscous, and can be determined by taking the jump height and velocity as initial conditions.

Thus, assuming uniform flow downstream of the jump, and using the mass conservation equation (2.6), equation (5.3) reduces to

(5.4) $$\begin{eqnarray}\displaystyle H_{J}^{2}-3\left(\frac{272}{875}\frac{Fr^{2}}{r_{J}^{2}}\right)^{2/3}+\frac{Fr^{2}}{2r_{J}^{2}H_{J}}=0. & & \displaystyle\end{eqnarray}$$

Equation (5.4) takes an interesting form when cast in terms of the Froude number based on the jump radius and height. In this regard, there is a close connection with the recent experimental findings and claim of Duchesne et al. (Reference Duchesne, Lebon and Limat2014), which we will now explore.

Duchesne et al. (Reference Duchesne, Lebon and Limat2014) introduced the Froude number defined in terms of the jump height and the average velocity immediately after the jump, namely $Fr_{J}=Fr/2r_{J}H_{J}^{3/2}$ in our notations. Their measurements suggest that $Fr_{J}$ remains sensibly independent of the flow rate (constant with respect to $Fr$ ). However, they could not explain or theoretically support this observation, which, in turn, begs the question whether the constancy of $Fr_{J}$ has any theoretical basis. This turns out to be indeed the case as we shall now demonstrate.

It is easy to see that (5.4) yields the following equation for $Fr_{J}$ :

(5.5) $$\begin{eqnarray}\displaystyle Fr_{J}^{2}-{\textstyle \frac{24}{25}}({\textstyle \frac{17}{7}})^{2/3}Fr_{J}^{4/3}+{\textstyle \frac{1}{2}}=0. & & \displaystyle\end{eqnarray}$$

This equation indicates that $Fr_{J}$ is indeed a constant that is independent of the $Fr$ or, equivalently, of the flow rate, confirming the observation of Duchesne et al. (Reference Duchesne, Lebon and Limat2014). This is a cubic equation in $Fr_{J}^{2/3}$ , admitting $Fr_{J}=0.58$ as a solution. Thus, we have established the constancy of $Fr_{J}$ when the subcritical flow to be inviscid.

We next address the question whether $Fr_{J}$ remains actually independent of the flow rate if the subcritical flow is assumed to be viscous, and see whether this yields a closer constant $Fr_{J}$ value to the one measured by Duchesne et al. (Reference Duchesne, Lebon and Limat2014). We follow Duchesne et al. (Reference Duchesne, Lebon and Limat2014), and adopt a lubrication flow approach. In this case, a differential equation for $H$ can be obtained by neglecting the inertial terms in equation (2.5), yielding the following profile for the radial velocity:

(5.6) $$\begin{eqnarray}\displaystyle u(r>r_{J},z)=\frac{Re}{Fr^{2}}\frac{\text{d}H}{\text{d}r}\left(\frac{z^{2}}{2}-Hz\right). & & \displaystyle\end{eqnarray}$$

Inserting $u$ into the mass conservation (2.6) and integrating, the equation governing the film thickness downstream of the jump becomes

(5.7) $$\begin{eqnarray}\displaystyle \frac{\text{d}H}{\text{d}r}=-\frac{3}{2}\frac{Fr^{2}}{Re}\frac{H^{-3}}{r}, & & \displaystyle\end{eqnarray}$$

which leads to the velocity profile just downstream of the jump as

(5.8) $$\begin{eqnarray}\displaystyle u(r_{J+},z)=-\frac{3}{2r_{J}H_{J}^{3}}\left(\frac{z^{2}}{2}-H_{J}z\right). & & \displaystyle\end{eqnarray}$$

Finally, inserting (5.8) into (5.3), we obtain the following equation for $Fr_{J}$ :

(5.9) $$\begin{eqnarray}\displaystyle Fr_{J}^{2}-{\textstyle \frac{4}{5}}({\textstyle \frac{17}{7}})^{2/3}Fr_{J}^{4/3}+{\textstyle \frac{5}{12}}=0. & & \displaystyle\end{eqnarray}$$

Similar to (5.5), equation (5.9) also confirms that $Fr_{J}$ is independent of $Fr$ (flow rate), with $Fr_{J}=0.71$ .

What we have established so far, based on the discretized mass and momentum equations across the jump, is that $Fr_{J}$ is indeed constant (independent of the flow rate) as Duchesne et al. (Reference Duchesne, Lebon and Limat2014) claim from their measurements. Surprisingly, this is the case whether the subcritical flow is assumed to be inviscid or viscous obeying the lubrication regime, thus covering a wide range of viscosity and flow rate. The value of $Fr_{J}$ is found to be slightly lower for inviscid compared to viscous subcritical flow. However, both values remain higher than the measured value by Duchesne et al. (Reference Duchesne, Lebon and Limat2014): $Fr_{J}\approx 0.35{-}0.40$ . It is important to observe that the values of $Fr_{J}$ can be found theoretically without the knowledge of downstream conditions of the jump such as the disk radius or the thickness at the edge of the disk. Such conditions are not needed when the discretized conservation equations are invoked. Another important observation to make is whether the discretized (5.2) itself is valid. It is expected that (5.2) remains reasonably valid for low-viscosity liquids or at high flow rate since the jump is of negligible thickness and its location is well defined. However, for high-viscosity liquids such as the silicon oils used by Duchesne et al. (Reference Duchesne, Lebon and Limat2014), the jump is expected to be wide, and (5.2) cannot be entirely valid. This brings us to the second alternative when seeking the subcritical flow.

5.2 The influence of disk radius and edge thickness

We proceed by examining the flow in the subcritical range, downstream of the jump, without invoking (5.3). In this case, an approximate or asymptotic solution of (4.4) can be found by keeping the three dominant terms for large distance, reducing it to a lubrication-like equation for $H$ :

(5.10) $$\begin{eqnarray}\displaystyle \frac{\text{d}H}{\text{d}r}=-\frac{6}{5}\frac{Fr^{2}}{Re}\frac{H^{-3}}{r}. & & \displaystyle\end{eqnarray}$$

Subject to $H(r=r_{\infty })=H_{\infty }$ , (5.10) can be integrated analytically to give

(5.11) $$\begin{eqnarray}\displaystyle H(r)=\left(H_{\infty }^{4}+\frac{24}{5}\frac{Fr^{2}}{Re}\ln \left(\frac{r_{\infty }}{r}\right)\right)^{1/4}. & & \displaystyle\end{eqnarray}$$

The prediction of the edge thickness was considered in our recent study for the flow on a rotating disk (Wang & Khayat Reference Wang and Khayat2018). The case of a stationary disk was also examined. Both static and dynamic contributions were considered, which yielded an accurate prediction established by comparing against experiment for the edge thickness. Direct measurements by Duchesne et al. (Reference Duchesne, Lebon and Limat2014) of the edge thickness, performed at nearly 5 mm of the disk perimeter in their experiment, give a nearly constant value of 1.5 mm with a weak power-law variation with the flow rate, not exceeding a few per cent. This constant thickness value, for a liquid of surface tension $\unicode[STIX]{x1D6FE}$ , is very close to the capillary length $\sqrt{\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D70C}g}$ of the fluid, which results from the balance of forces between the hydrostatic pressure and the surface tension at the disk perimeter. This value is also consistent with the measurements of Dressaire et al. (Reference Dressaire, Courbin, Crest and Stone2010). Consequently, we assumed that the film thickness at the edge of the disk is essentially equal to the film thickness the liquid exhibits under static conditions. Lubarda & Talke (Reference Lubarda and Talke2011) proposed an expression for this static thickness as $h_{s}=2\sqrt{\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D70C}g}\sin (\unicode[STIX]{x1D703}_{Y}/2)$ , based on the minimum free energy principle. Here $\unicode[STIX]{x1D703}_{Y}$ is the contact angle, which depends on both the liquid and the solid, and may then be deduced from experiment. For water, we achieved an excellent agreement with the measurements of Dressaire et al. (Reference Dressaire, Courbin, Crest and Stone2010) for the jump radius by taking $\unicode[STIX]{x1D703}_{Y}=90^{\circ }$ . This value is within the range of values measured for water on polydimethylsiloxane (Diversified Enterprises 2009), which is the material used by Dressaire et al. (Reference Dressaire, Courbin, Crest and Stone2010) for their disk. For water flowing on glass, $\unicode[STIX]{x1D703}_{Y}=35^{\circ }$ (Vicente, Andre & Ferreira Reference Vicente, Andre and Ferreira2012), a value used recently in our comparison with the measurements of Hansen et al. (Reference Hansen, Horluck, Zauner, Dimon, Ellegaard and Creagh1997) for water (Wang & Khayat Reference Wang and Khayat2018).

In addition to the static contribution, and in order to explore the small variation of edge thickness with flow rate as observed by Duchesne et al. (Reference Duchesne, Lebon and Limat2014), we resorted to a minimum mechanical energy principle (Yang & Chen Reference Yang and Chen1992; Yang, Chen & Hsu Reference Yang, Chen and Hsu1997), which states that a fluid flowing over the edge of a disk under the influence of a hydrostatic pressure gradient will adjust itself so that the mechanical energy within the fluid will be minimum with respect to the film thickness at the disk edge. Consequently, the contribution to the thickness at the edge of the disk is determined by setting the derivative of the mechanical energy with respect to the film thickness equal to zero. In dimensionless form, the total energy is $E=\int _{0}^{H}(Fr^{2}(u^{2}/2)+z+p)\,\text{d}z$ . Since the pressure is hydrostatic: $p=-z+H$ , yielding $E=\int _{0}^{H}(Fr^{2}(u^{2}/2)+H)\,\text{d}z$ . Consequently, we set $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}H\int _{0}^{H}(Fr^{2}(u^{2}/2)+H)\,\text{d}z=0$ at the edge of the disk. Using (5.8), we obtain the dimensionless film thickness at the edge as

(5.12) $$\begin{eqnarray}\displaystyle H_{\infty }=2\sqrt{\frac{1}{Bo}}\sin \left(\frac{\unicode[STIX]{x1D703}_{Y}}{2}\right)+\left(\frac{3}{40}\right)^{1/3}\left(\frac{Fr}{r_{\infty }}\right)^{2/3}. & & \displaystyle\end{eqnarray}$$

Here the Bond number is given by $Bo=\unicode[STIX]{x1D70C}ga^{2}/\unicode[STIX]{x1D6FE}$ . Clearly, in the presence of relatively strong gravity or surface tension and large disk radius, the second term tends to be dominated by the static contribution. As we shall see next, even the static contribution will turn out to be uninfluential for the heavily viscous liquids considered in the present work.

Indeed, once the thickness $H(r=r_{\infty })=H_{\infty }$ at the edge of the disk is determined as per (5.12), we obtain the film thickness distribution downstream of the jump from (5.10). In particular, and given that the jump location has already been determined, the jump height is now obtained through

(5.13) $$\begin{eqnarray}\displaystyle H_{J}=\left(H_{\infty }^{4}+\frac{24}{5}\frac{Fr^{2}}{Re}\ln \left(\frac{r_{\infty }}{r_{J}}\right)\right)^{1/4}. & & \displaystyle\end{eqnarray}$$

In this case, it is not difficult to confirm that, for a large disk and relatively small Reynolds number, the logarithmic term dominates on the right-hand side of (5.13). This is obviously the case of very viscous liquids. As a comparison, the Reynolds number for the flow of silicon oils (Duchesne et al. Reference Duchesne, Lebon and Limat2014) is of the order of $10^{2}$ , whereas for water (Dressaire et al. Reference Dressaire, Courbin, Crest and Stone2010) its order is closer to $10^{4}$ . It is important to note that the range of Froude number in both sets of measurements is essentially the same. Based on the range of Froude numbers in figure 10 below, it is not difficult to deduce that the static contribution to the edge thickness is also dominated. In fact, experiment (Duchesne et al. Reference Duchesne, Lebon and Limat2014) indicate that $H_{\infty }=O(1)$ at most, and $H_{\infty }\approx 0$ for a liquid of high viscosity.

We now turn, once again, to examining conditions where $Fr_{J}$ may remain independent of the flow rate. This time, (5.13) is used instead of (5.3). In this case, since the pre-jump height $h_{J}$ and the jump radius $r_{J}$ must be computed numerically, it is not possible to derive a closed form equation similar to (5.5) or (5.9), confirming that $Fr_{J}$ is constant. In fact, (5.13) suggests that $Fr_{J}$ is not independent of $Fr$ . However, the dependence on $Fr$ turns out to be weak, as figure 10 suggests. The figure shows the variation of $Fr_{J}$ against $Fr$ for the silicon oil of viscosity 20 cSt used by Duchesne et al. (Reference Duchesne, Lebon and Limat2014) in their measurements, which are also shown in the figure. The experimental data are reproduced in dimensionless form. Surprisingly, the figure indicates that $Fr_{J}$ not only is indeed sensibly constant but agrees closely with experiment. Some discrepancy is, however, noted for low flow rates, which is not surprising given the difficulty in measuring accurately the jump radius and height.

Figure 10. Dependence of $Fr_{J}$ on the flow rate (Froude number). The solid line correspond to predictions based on relations (4.4) and (5.13). The experimental data corresponding to silicon oil are from Duchesne et al. (Reference Duchesne, Lebon and Limat2014). Also added as dashed and dash-dotted lines the result based on (5.5) and (5.9), respectively.

Now that the solution is available in the supercritical and subcritical regions, we are in a position to validate our model over the entire domain, against existing numerical results and experiment. We also take the opportunity to assess the validity of the parabolic profile. For the supercritical range, an equation similar to (4.4) is obtained when using the parabolic profile: $f(\unicode[STIX]{x1D702})=2\unicode[STIX]{x1D702}-\unicode[STIX]{x1D702}^{2}$ , namely

(5.14) $$\begin{eqnarray}\displaystyle Re\left(\frac{1}{Fr^{2}}-\frac{3}{10r^{2}h^{3}}\right)h^{\prime }=\frac{3}{10rh^{2}}\left(\frac{Re}{r^{2}}-\frac{5}{h}\right). & & \displaystyle\end{eqnarray}$$

We observe that the asymptotic form of (5.14) for large $r$ is precisely the lubrication (5.7), which when integrated yields

(5.15) $$\begin{eqnarray}\displaystyle H(r)=\left(H_{\infty }^{4}+6\frac{Fr^{2}}{Re}\ln \left(\frac{r_{\infty }}{r}\right)\right)^{1/4}. & & \displaystyle\end{eqnarray}$$

Figure 11 shows the comparison between the present theory and the numerical results of Rojas & Tirapegui (Reference Rojas and Tirapegui2015) as well as the measurements of Ellegaard et al. (Reference Ellegaard, Hansen, Haaning, Hansen and Bohr1996) for ethylene glycol. Both formulations based on the parabolic and cubic profiles are represented. The results for the free-surface velocity are reported in dimensionless form, with corresponding parameters being $Re=334$ , $Fr=14.4$ and $Bo=1.21$ . The contact angle used to determine the edge thickness is $\unicode[STIX]{x1D703}_{Y}=70^{\circ }$ , yielding a total edge thickness of $H_{\infty }=1.7$ , including the height of the vertical edge. While the position of the jump is well reproduced by the present theory based on the cubic profile and the numerical method, figure 11 shows that the theory tends to slightly underestimate the level of the surface velocity in both the super- and subcritical ranges. Figure 11 indicates that the numerical approach of Rojas et al. tends to agree slightly better with experiment than the present theory. The figure also indicates a larger discrepancy when the parabolic profile is used.

Figure 11. Free-surface velocity in the supercritical and subcritical domains. Comparison between the present theory (solid line), the numerical results of Rojas & Tirapegui (Reference Rojas and Tirapegui2015), as well as the experimental data of Ellegaard et al. (Reference Ellegaard, Hansen, Haaning, Hansen and Bohr1996). Also added is the velocity distribution based on the parabolic profile.

We next examine the shape of the entire film under general flow conditions. For this, we rescale (5.11) using (3.7) to reduce the problem in terms of the parameter $\unicode[STIX]{x1D6FC}$ :

(5.16) $$\begin{eqnarray}\displaystyle \bar{H}(\bar{r})=\left(\bar{H}_{\infty }^{4}+\frac{24}{5}\unicode[STIX]{x1D6FC}\ln \left(\frac{\bar{r}_{\infty }}{\bar{r}}\right)\right)^{1/4}. & & \displaystyle\end{eqnarray}$$

Here $\bar{H}_{\infty }=2\sqrt{Re^{2/3}/Bo}\sin (\unicode[STIX]{x1D703}_{Y}/2)+((3/40)\unicode[STIX]{x1D6FC}/\bar{r}_{\infty }^{2})^{1/3}$ from (5.12). The overall effect of gravity is illustrated in figure 12, which depicts the film thickness over the entire disk. The jump height decreases with increasing gravity, simultaneously as the jump location is pushed upstream toward the stagnation point and away from the edge of the disk. We can also observe the logarithmic increase in height reported earlier in figure 7.

Figure 12. Influence of gravity on the film thickness plotted against radial distance in regions (ii), (iii) and (iv).

While the use of (4.4) or (4.9) is imperative in the supercritical range ahead of the jump, allowing us to locate the jump, it is not necessarily so for the subcritical flow, where we have the choice to use different approximations. For instance, as the flow slows down downstream of the shock, we saw that the lubrication assumption holds well between the jump and the edge of the disk, yielding a good agreement with experiment (see figures 10 and 11). Our calculations of the subcritical flow so far are based on the asymptotic form (5.11) of (4.4) or (5.16) of (4.9) for large $r$ as it is convenient to use, given the analytical distribution of the thickness (and velocity) downstream of the jump and its direct relation to the edge thickness and the disk radius. Alternatively, we now consider using (4.4), and apply it directly to capture the subcritical flow, which should allow us to assess the validity of (5.11). Simultaneously, we examine the effect of the disk radius. The comparison is reported in figure 13 for the film thickness distribution with distance in the super- and subcritical ranges for three different values of the disk radius. We take $Re=628$ , $Fr=63$ , $Bo=1.1$ and $\unicode[STIX]{x1D6E9}_{Y}=55^{\circ }$ . In this case, the values of the thickness at the edge of the disk are determined from (5.12) are $H_{\infty }=1.24$ , 1.2 and 1.18, corresponding to $r_{\infty }=70$ , 80 and 90, respectively.

Several observations are worth making here. Figure 13 shows that the subcritical branches exhibit a turning point corresponding to the singularity of (4.4), which occurs slightly upstream of the jump location. As expected, the profile of the asymptotic solution (5.11) collapses onto the profile based on (4.4) at a distance not too far from the jump. This distance, nevertheless, increases with the disk radius. The asymptotic solution yields a jump height $H_{J}$ that is slightly above the one based on the exact solution of (4.4). Finally, despite the important spread in the values of the disk radius, the location of the singularity reached by the subcritical branches is essentially the same, as reflected in the saturation near the turning point. This seems to suggest that the location of the jump in reality, if it were to fall half-way, say, between the two locations of the singularity, is independent of downstream conditions. We seem to reach this observation regardless of which branch, super- or subcritical, we are referring to.

Figure 13. Influence of the disk radius on the film thickness plotted against the radial distance in regions (ii), (iii) and (iv). Solution in solid line based on equations (4.4) and (5.11). Dashed line shows the subcritical profile based on (4.4).

Next, we pursue our assessment of the validity of (4.4) and the asymptotic form (5.11) against experiment. The comparison is reported in figure 14 for the film thickness distribution with distance in the super- and subcritical regions against the measurements of Duchesne et al. (Reference Duchesne, Lebon and Limat2014) for silicon oil (20 cSt). The data are reproduced here in dimensionless form from their figure 2, corresponding to $Re=169$ , $Fr=14.88$ , $Bo=1.19$ , $\unicode[STIX]{x1D6E9}_{Y}=55^{\circ }$ and $\bar{r}_{\infty }=94$ . In this case, the value of the thickness at the edge of the disk is determined from (5.12) as $\bar{H}_{\infty }=0.95$ . Several observations are worth making here. Figure 14 shows that the theoretical predictions, based on the solution of (4.4), are generally in good agreement with the experiment of Duchesne et al. (Reference Duchesne, Lebon and Limat2014), slightly underestimating their measurements. The location of the jump is predicted to be close to the level of the turning point or the singularity of the supercritical branch upstream of the jump. The subcritical branch also exhibits a turning point corresponding to the singularity of (4.4), occurring slightly downstream of the jump location (see figure 13). The behaviour of the two branches is in close (qualitative) agreement with the theoretical predictions of Kasimov (Reference Kasimov2008) who incorporated the shape of the bottom (flat disk with a sharp cut off at the edge). The reader is particularly referred to figure 3(a) from Kasimov (Reference Kasimov2008). The asymptotic solution cannot mimic the downward turning trend observed in the experiment, yielding a jump height HJ that is slightly above the one based on the exact solution of (4.4). Interestingly, there is no need here to integrate from a critical point coinciding with at the edge of the disk to obtain the subcritical branch as Kasimov (Reference Kasimov2008) did. Kasimov estimated the location of the jump to be somewhere between the upstream and downstream singularities, which seems to be case here. However, this may not always be the case. Based on the agreement between theory and experiment in previous figures, we saw that the location of the jump coincides rather with the upstream singularity of the averaged momentum equation.

Figure 14. Free-surface profile. Comparison between theoretical predictions and the measurements of Duchesne et al. (Reference Duchesne, Lebon and Limat2014) for silicon oil (20 cSt). Results plotted in dimensionless form with $Re=169$ , $Fr=14.88$ , $Bo=1.19$ , $\unicode[STIX]{x1D6E9}_{Y}=55^{\circ }$ and $\bar{r}_{\infty }=94$ . Theoretical profiles based on current theory or (4.4) (solid lines) and asymptotic subcritical (5.11) (dashed line).

Finally, an interesting observation can be made regarding the (constant) value of $Fr_{J}$ and its independence of $Fr$ (or the flow rate). The measurements of the film profile for heavily viscous liquids seem to give a rough estimate of the height $H_{J}$ immediately downstream of the relative the height $h_{J}$ upstream of the jump. More precisely, experiment suggests that $H_{J}\approx 2h_{J}$ (see, for instance, the measurements of Ellegaard et al. (Reference Ellegaard, Hansen, Haaning, Hansen and Bohr1996), and those in figure 2 of Duchesne et al. (Reference Duchesne, Lebon and Limat2014). By substituting $h_{J}=H_{J}/2$ in (4.8) and recalling that $Fr_{J}=Fr/2r_{J}H_{J}^{3/2}$ , we deduce that

(5.17) $$\begin{eqnarray}\displaystyle Fr_{J}=\frac{Fr}{2^{5/2}r_{J}h_{J}^{3/2}}=\frac{1}{16}\sqrt{\frac{875}{34}}\simeq 0.32, & & \displaystyle\end{eqnarray}$$

which is very close to 0.33, the value measured by Duchesne et al. (Reference Duchesne, Lebon and Limat2014). Another way of obtaining the same result is to assume that the singularity yielding (4.8) occurs at a height half-way along the jump, where we expect the slope to be largest (infinite). In this case, one would replace $h_{J}$ in (4.8) by the average height, which, in turn, can be approximated as $(H_{J}+h_{J})/2\approx H_{J}/2$ if one assumes that $h_{J}\ll H_{J}$ (Watson Reference Watson1964), and obtain (5.17). The estimated value in (5.17) confirms the important observation in this study that the jump characteristics appear to be dictated only by the supercritical flow and upstream conditions.