2.1 Physical domain

Following Watson (Reference Watson1964), four distinct regions of flow are identified for a slipping fluid, with smooth passage from one region to the next: a stagnation flow region (i), a developing boundary-layer region (ii) and a fully viscous thin-film region (iii). Under some flow conditions, a hydraulic jump may form in region (iv). In the vicinity of the stagnation point, in region (i), $x=O(1)$ . The speed outside the boundary layer rises rapidly from 0 at the stagnation point to the impingement velocity in the inviscid far region. For an adhering as well as a slipping jet, the impinging inviscid flow is not affected by slip, and the boundary-layer thickness remains negligibly small. Obviously, this would be the case for a jet at relatively large Reynolds number, especially for a slipping fluid. Ideally, the flow at the boundary-layer edge should correspond to the inviscid flow near the stagnating jet, with the boundary-layer leading edge coinciding with the stagnation point (Liu, Gabour & Lienhard Reference Liu, Gabour and Lienhard1993). However, the assumption of uniform horizontal flow near the wall and outside the boundary layer (as illustrated in figure 1) is not unreasonable. For an adhering jet, the flow acquires a similar character; the position or effect of the leading edge is irrelevant. This is not the case for a slipping jet, where, as we shall see, a non-similarity solution is sought subject to initial conditions at the leading edge. However, as argued above, the dominance of inertia and slip should make the assumption of uniform flow and a delayed leading edge plausible.

In region (ii), the boundary layer is expected to grow like $\sqrt{x/Re}$ , at a rate that diminishes with $S$ (see Khayat Reference Khayat2016). The elongation rate or normal stress weakens with distance, but the flow remains two-dimensional as a result of non-negligible surface curvature. The speed outside the boundary layer remains almost constant, equal to 1 (in units of $V_{\text{0}}$ ), as the fluid here is unaffected by the viscous stresses. For $x\gg 1$ , the flow field in region (ii) is not affected by the stagnation flow of region (i). The region $1\ll x<x_{\text{0}}$ will be referred to as the developing boundary-layer region, with boundary-layer thickness $\unicode[STIX]{x1D6FF}(x)$ , outside which the flow is inviscid and constant. Here, $x_{\text{0}}$ is the location of the transition point at which viscous stresses become appreciable right up to the free surface, where the whole flow is of the boundary-layer type. At this point, for an adhering jet, the velocity profile changes from the Blasius type to the similarity profile. In contrast, a similarity profile does not exist for a slipping jet. The flow in region (iii), $x>x_{\text{0}}$ , which will be referred to as the fully developed boundary-layer region, is bounded by the flat plate and the free surface $y=h(x)$ .

Finally, the hydraulic jump in region (iv) occurs at a location $x=x_{J}$ , which can be larger or smaller than $x_{\text{0}}$ since the jump may occur at any point in the flow. The jump height $H(x)$ is generally not constant but different from the height immediately downstream of the jump, $H_{J}=H(x=x_{J})$ . In practice, the fluid is drained at the end $x=x_{\infty }$ (trailing edge) of the plate to maintain steady flow, with the film thickness denoted by $H_{\infty }$ . Although it is common practice to assume the jump height to remain equal to $H_{J}$ , this assumption is only valid for fluids of low viscosity. In this study, we consider both an inviscid layer and a viscous layer downstream of the jump.

2.2 Governing equations and boundary conditions

Unless otherwise specified, $Re$ will be assumed to be moderately large. Thus, for a thin jet, the conservation equations reduce to

The gravitational effect will only be considered downstream of the jump. Here, a prime denotes total differentiation (with respect to $x$ in this case) and a subscript with respect to $x$ or $y$ denotes partial differentiation. At the plate, the (partial) slip and no-penetration conditions are assumed to hold, so that

The flow field is sought separately in the developing boundary-layer region $(x<x_{0})$ , fully developed viscous boundary-layer region $(x>x_{0})$ and hydraulic-jump region. Consequently, the additional boundary conditions are as follows. In region (ii), or the developing boundary-layer region ( $1\ll x<x_{\text{0}}$ ), the flow is assumed to be sufficiently inertial for inviscid flow to prevail between the boundary-layer limit and the free surface (see figure 1). In this case, the following condition at the outer edge of the boundary layer must hold:

Additionally, a boundary condition is needed at the origin ( $x=0$ ), namely

which reflects the uniformity of the flow at the leading edge of the boundary layer and will be referred to as the initial condition. For region (iii), we let $U(x)\equiv u(x>x_{0},y=h)$ and $V(x)\equiv v(x>x_{0},y=h)$ denote the velocity components at the free surface. In the absence of surface tension, the kinematic and dynamic conditions can be written respectively as

Here again, given the non-similar character of the flow, an initial condition is needed for the solution in region (iii), which is provided by matching the flow at the transition point:

Although the Reynolds number and slip parameter appear explicitly in problem (2.1)–(2.7), it is possible to deduce universal behaviour for the flow with respect to these two dimensionless groups. It is not difficult to show that, upon substituting $x$ by $x/Re$ and $v$ by $Rev$ , the entire problem can be treated independently of the Reynolds number. Less obvious is the universality with respect to $S$ , which will be discussed in the next sections.

2.3 Overall solution strategy

We use different solution strategies in each flow region. We propose two approaches in region (ii) to examine the boundary-layer flow, namely a rigorous series or asymptotic approach for large and small slip lengths, and a K–P approximation, the latter being most convenient to estimate the boundary-layer height and validate the series solution. A numerical approach is also implemented and discussed. However, as expected, the numerical solution performs poorly near the leading edge due to the flow singularity. The flow in the entire boundary-layer domain is obtained upon matching the flows between the developing and fully viscous boundary-layer regions. A series solution is also used in region (iii), this time taking advantage of the small film thickness. In region (iv) downstream of the hydraulic jump, inertia will be neglected, allowing an analytical solution in the presence of gravity.

3.1 Flow behaviour for large slip length

A series solution is sought by expanding the streamfunction and velocity in suitable powers of $\unicode[STIX]{x1D709}$ . For the slip velocity $u(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702}=0)$ to remain finite and non-zero, the slip condition in (3.4) suggests that we must have $f_{\unicode[STIX]{x1D702}\unicode[STIX]{x1D702}}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702}=0)\sim \sqrt{\unicode[STIX]{x1D709}}$ to leading order. Similarly, for $v(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})$ to remain finite near the origin, equation (3.2) suggests that $f(\unicode[STIX]{x1D709}\rightarrow 0,\unicode[STIX]{x1D702})\sim \sqrt{\unicode[STIX]{x1D709}}$ . Thus, we set

Expansions (3.6) satisfy the initial conditions (3.5a,b ), and are expected to be valid (converge) for large and/or small $\unicode[STIX]{x1D709}$ . The extent of the developing boundary-layer region will indeed turn out to be small for moderately large Reynolds number, allowing relatively smaller slip lengths to be considered. Upon substituting (3.6) into (3.3), we get

In this case, it is not difficult to show that the recurrence relation takes the form

The corresponding boundary conditions read

We list the first two modes for reference, namely

where the following constants are introduced for simplicity: $a=0.009$ , $b=2.134$ and $r=1.281$ .

Figure 2. Streamfunction profiles at different positions from the leading edge of the boundary layer.

The non-similar character of the flow is perhaps best illustrated by examining the behaviour of $f(x,\unicode[STIX]{x1D702})$ . Figure 2 shows the variation of $f$ with $\unicode[STIX]{x1D702}$ at various $x$ positions, with an increment of $\unicode[STIX]{x1D6E5}=0.01/ReS^{2}$ over the range $0\leqslant x/ReS^{2}\leqslant 0.1$ . The figure illustrates clearly the dependence of $f$ on $x$ , and suggests that saturation occurs as the distance grows from the leading edge of the boundary layer. The saturation is a reflection of the growing viscous effect as the fluid advances, slowing the flow. More importantly, the slowing rate can be determined accurately as function of $Re$ and $S$ . The overall behaviour appears to be close to linear, with a slope diminishing with distance, confirming the weakening of the flow across the boundary layer, and not just at the plate.

Figure 3. Streamwise (a) and transverse (b) velocity profiles at different distances from the leading edge in the developing boundary-layer region (ii), based on series solution. The inset in (a) shows the slip velocity against the distance along the plate.

The flow in the boundary layer is illustrated in figure 3, which displays the profiles of the streamwise (figure 3 a) and transverse (figure 3 b) velocity components over the range $0.01<x/S^{2}Re<0.1$ . As for an adhering jet, $u$ and $v$ increase monotonically with height. While these observations essentially hold for both adhering and slipping jets, there is a dramatic departure in behaviour for a slipping jet.

The contrast to an adhering jet becomes particularly evident when examining the flow at the plate. For this, we have included the slip velocity dependence on distance, reported in the inset of figure 3(a). The velocity is readily available from (3.6) as $u(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702}=0)=1+\sum _{n=1}F_{n}^{\prime }(\unicode[STIX]{x1D702}=0)(\sqrt{\unicode[STIX]{x1D709}}/S)^{n}$ . It should be recalled that, for an adhering jet, there is a sudden drop in velocity from 1 to 0 at the leading edge of the boundary layer, accompanied by a sudden jump in the shear stress, causing in fact the emergence of the boundary layer at the leading edge. For a slipping jet, the transition is continuous for both the velocity and the shear stress, with the latter behaving like $\sqrt{\unicode[STIX]{x1D709}}$ and the former remaining close to 1 near the leading edge (see the inset in figure 3 a). Consequently, it is the discontinuity in the normal stress $u_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702}=0)$ , which behaves like $-1/\sqrt{\unicode[STIX]{x1D709}}$ near the origin, that causes the formation of the boundary layer. It should be recalled that the normal stress remains continuously zero for an adhering jet. Moreover, unlike an adhering jet, the $v$ profiles in figure 3(b) display a non-zero slope at the plate. For $S=0$ , equation (3.2) shows that the transverse velocity component reduces to $2\sqrt{Re\unicode[STIX]{x1D709}}v(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})=\unicode[STIX]{x1D702}f_{\unicode[STIX]{x1D702}}(\unicode[STIX]{x1D702})-f(\unicode[STIX]{x1D702})$ , leading to $v_{\unicode[STIX]{x1D702}}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702}=0)=0$ . More generally, for $S\geqslant 0$ , equation (3.2) leads to

which confirms the singular character of an adhering fluid.

Figure 4. Dependence of the slip velocity on the position in the developing boundary-layer region (ii). The profiles based on the series solution with 1, 2 and 3 modes, the Karman–Pohlhausen approximation and the numerical solution are shown.

Figure 4 shows a comparison between the numerical and series solutions for the evolution of the slip velocity with distance. We also show the influence of the number of modes on the series solution, which reflects rapid convergence when only three modes are included. As expected, the need for higher-order modes increases as the discrepancy grows with distance. However, as we shall see below, the transition to fully developed flow occurs at a relatively short distance from the leading edge (origin), and three modes turn out to be amply sufficient for our purposes. Comparison with the numerical solution indicates an overall agreement of 2–4 %. However, there is a significant qualitative disagreement near the origin; the stress singularity at the origin is not captured by the numerical solution. Ideally, some local (asymptotic) treatment is required that would provide a suitable initial condition for the numerical integration to start at some point downstream of the origin (see Khayat Reference Khayat2016). There are additional challenges facing the implementation of a numerical approach for the present problem, particularly when matching the flow at the transition location. Further details and discussion are given in appendix A. We shall not pursue the numerical approach beyond this point.

Figure 4 also shows the slip velocity based on the K–P parabolic approximation, which will be discussed below and given in (3.22). There is good agreement with the series solution near the origin, reflecting the capability of the K–P solution to capture the singular character at the leading edge. However, it tends to underestimate the level of slip velocity overall by a few per cent. A velocity profile different from the parabolic profile (3.22) may yield closer agreement. However, in addition to the expected cumbersome manipulation of a more complicated profile, there is no physical justification or systematic approach for the generation of such a profile (Prince et al. Reference Prince, Maynes and Crockett2012).

Finally, it should be noted that the series solution (3.6) is valid for finite $S$ only; in fact, the solution is singular for $S=0$ . Thus, for the solution to apply close to the leading edge, $S$ must be relatively large. We propose next a series solution that is valid for small $S$ , including the $S=0$ limit.

3.2 Flow behaviour for small slip length

If $S$ is small, it can be adopted as the small parameter for the problem, and we can seek a perturbation expansion of the streamfunction in powers of $S$ . In this case, the similarity solution $f_{0}(\unicode[STIX]{x1D702})$ , corresponding to an adhering jet, is taken as the leading-order term. Clearly, this term should not be confused with the one in (A 3a,b ). Thus, the solution of (3.3) can be written as

It should be recalled that the leading-order function satisfies the following similarity problem (Schlichting Reference Schlichting2000):

Upon substituting (3.12) into (3.3), and using (3.13), we obtain the following recurrence equations for the perturbation coefficients, namely, for $n\geqslant 1$ :

By inspecting (3.14) and corresponding boundary conditions, it is not difficult to see that the solution is of the form $f_{n}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})=H_{n}(\unicode[STIX]{x1D702})\unicode[STIX]{x1D709}^{-n/2}$ , where the coefficients $H_{n}$ are governed by

We observe that for small $S$ , $S/\sqrt{\unicode[STIX]{x1D709}}$ must remain small for expansion (3.12) to be valid, whereas for finite $S$ , it is $\sqrt{\unicode[STIX]{x1D709}}/S$ that must be small for expansion (3.6) to hold. In other words, expansion (3.12) is valid far from the leading edge, whereas (3.6) is applicable for small distance.

The flow is then given by

Figure 5. Influence of the slip length on the velocity profile for small $S$ , plotted at various positions along the plate in the developing boundary-layer region.

The slip velocity is given by $u(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702}=0)=0.332S/\sqrt{\unicode[STIX]{x1D709}}-0.0805S^{2}/\unicode[STIX]{x1D709}+O(S^{3}/\unicode[STIX]{x1D709}^{3/2})$ , where we recall that $f_{0}^{\prime \prime }(\unicode[STIX]{x1D702}=0)\approx 0.332$ (Schlichting Reference Schlichting2000). As an illustration, figure 5 depicts the influence of slip length on the velocity profile for small $S$ , to $O(S^{3}/\unicode[STIX]{x1D709}^{3/2})$ . In particular, one should note the gradual departure from the similarity profile $(S^{2}Re/x=0)$ , and the contrast to figure 3(a).

3.3 Boundary-layer height

A more physically insightful and reasonably accurate approach is to determine the boundary-layer height $\unicode[STIX]{x1D6FF}(x)$ using the K–P integral method. Considering the mass and force balance over the boundary-layer region, the integral form of (2.1) leads to

As in the case of an adhering jet, the balance of terms in (2.1b ) suggests that $\unicode[STIX]{x1D6FF}\simeq \sqrt{x/Re}$ for a slipping jet as well. In this case, letting

and recalling the slip condition in (2.2), equation (3.18) reduces to

We adopt a parabolic profile for the velocity, subject to the following conditions:

In this case,

This profile yields an error of a few per cent for $S=0$ , relative to the estimate of the boundary-layer thickness based on the similarity solution White (Reference White2015). As reported in figure 4, the same level of accuracy holds for a slipping jet as well. The slip velocity becomes

which reflects similar qualitative flow features, distinct to a slipping jet, to the series solution (3.6). In particular, the continuity of the flow velocity and discontinuity of normal stress are ensured for a slipping jet. Thus, for $S>0$ , near the origin, $u\sim 1$ and $u_{\unicode[STIX]{x1D709}}\sim -1/\sqrt{\unicode[STIX]{x1D709}}$ . This explains the agreement between the K–P and series solutions depicted in figure 4 for small distance, and establishes the range of validity of the K–P approach. Despite its limitation, the profile (3.22) has the significant advantage of being uniformly valid for any $S$ value. In contrast, the asymptotic solutions in §§ 3.1 and 3.2 are limited to the large and small $S$ values respectively.

Inserting (3.22) into (3.20), we obtain the following equation:

Assuming that $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D709}=0)=0$ , we finally obtain a transcendental relation for the boundary-layer height, namely

Equation (3.24) is the counterpart of equation (11) of Prince et al. (Reference Prince, Maynes and Crockett2012) for a circular jet.

Figure 6. Influence of slip on the boundary-layer thickness (a) and slip velocity (b). All curves collapse onto the $S=1$ curve when $\unicode[STIX]{x1D6FF}/S$ is plotted against $x/ReS^{2}$ . The large- $S$ limit is also added as a solid (lowest) curve.

Figure 6 depicts the influence of slip on the profiles of the boundary layer (figure 6 a) and slip velocity (figure 6 b), including the limit of an adhering film $(S=0)$ . While the boundary-layer thickness grows indefinitely with distance, the slip velocity eventually vanishes regardless of the slip length, as also suggested by (3.23). As expected, the more slipping jet tends to exhibit a thinner boundary layer, moving faster at the plate. The drop in $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D709})$ and the jump in $u(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702}=0)$ , relative to an adhering jet, are significant for a weakly slipping jet. For small $S$ and a position not too close to the leading edge $(S\prec \sqrt{\unicode[STIX]{x1D709}})$ , equation (3.25) suggests that $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D709})\sim \sqrt{30\unicode[STIX]{x1D709}}-2S$ . In the limit $S=0$ , one recovers the boundary-layer height for an adhering jet, namely $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D709})=\sqrt{30\unicode[STIX]{x1D709}}$ (White Reference White2015). For small slip, (3.23) suggests that the slip velocity behaves like $u(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702}=0)\sim S/\sqrt{30\unicode[STIX]{x1D709}}$ . In other words, both the drop in the boundary-layer height and the jump in the slip velocity are linear with respect to the slip length for weakly slipping flow. For larger slip length, $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D709})$ displays saturation and a non-zero limit, whereas the slip velocity reaches the free-stream value. This saturation suggests insensitivity of the flow to the effect of slip length for strongly slipping liquids, and is also evident from figure 3 of Prince et al. (Reference Prince, Maynes and Crockett2012) for a circular jet. Equations (3.23) and (3.25) indicate that the curves in figure 6 are collapsible onto one curve when $\unicode[STIX]{x1D6FF}/S$ and the slip velocity are plotted against $x/ReS^{2}$ . In fact, these curves correspond to $S=1$ in figures 6(a) and 6(b) respectively. This same universality was also exhibited by the series solution in figures 3 and 4.

These predictions corroborate closely the observations and measurements of Dressaire et al. (Reference Dressaire, Courbin, Crest and Stone2009, Reference Dressaire, Courbin, Crest and Stone2010), who examined the flow of a circular jet impinging on smooth and microdecorated substrate surfaces, the latter consisting of a regular array of micron-size posts. They rationalized their observations by considering a leakage flow (through the posts) coupled to an effective slip length. Dressaire et al. (Reference Dressaire, Courbin, Crest and Stone2010) observed that the thickness of the boundary layer decreases as the slip coefficient and the leakage flow in the porous layer increase. In their experiment, the $y=0$ plane corresponds to the top of the posts. Thus, as the slip length corresponds to the distance below the actual surface at which the velocity is equal to zero, they interpreted the reduced thickness as the development of the boundary layer from a plane below the $y=0$ plane; see below.

Figure 7. Dependence of the boundary-layer thickness (a) and slip velocity (b) on the slip parameter at different positions $(0.1\leqslant x/Re\leqslant 1)$ . It should be noted that $\unicode[STIX]{x1D6FF}(S;x=0)=0$ and $u(y=0,S;x=0)=1$ .

The asymptotic behaviour depicted in figure 6 for large $S$ becomes more apparent from figure 7, where the boundary-layer height and slip velocity are plotted against $S$ at different streamwise distances in figures 7(a) and 7(b) respectively. This behaviour was also reported by Dressaire et al. (Reference Dressaire, Courbin, Crest and Stone2010) for any leakage flow rate, as shown in their figure 7(b). Interestingly, the large- $S$ behaviour appears to be insensitive to the leakage flow rate, as the curves converge together for large slip length. For a planar jet with large slip length, (3.27) suggests an asymptotic height $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D709})\sim \sqrt{6\unicode[STIX]{x1D709}}$ , signalling a 55 % drop from the $\sqrt{30\unicode[STIX]{x1D709}}$ level for an adhering jet. Comparatively, for a circular jet, figure 7(b) of Dressaire et al. (Reference Dressaire, Courbin, Crest and Stone2010) indicates a 77 % drop, suggesting that drag is easier to reduce for a circular jet than a liquid sheet. The linear behaviour against $S$ for small slip length, discussed earlier, is evident from the current figures 7(a) and 7(b).

3.4 Transition location

As the boundary layer grows with distance, it eventually invades the entire jet width, merging with the jet free surface at $x=x_{0}$ . For $x<x_{0}$ and above the boundary-layer outer edge, at some $y=h(x)>\unicode[STIX]{x1D6FF}(x)$ , lies the free surface. The height of the free surface is determined from volume conservation inside and outside the boundary layer, namely

Upon using (3.22), we deduce that

For an adhering jet, $h(\unicode[STIX]{x1D709}<\unicode[STIX]{x1D709}_{0})=1+1/3\sqrt{30\unicode[STIX]{x1D709}}$ . More generally, for small $S$ , we recall that $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D709})\sim \sqrt{30\unicode[STIX]{x1D709}}-2S$ , so that $h(\unicode[STIX]{x1D709}<\unicode[STIX]{x1D709}_{0})\sim 1+1/3(\sqrt{30\unicode[STIX]{x1D709}}-2S)^{2}/\sqrt{30\unicode[STIX]{x1D709}}\simeq 1+1/3\sqrt{30\unicode[STIX]{x1D709}}(1-4S/\sqrt{30\unicode[STIX]{x1D709}})$ . Hence, both the boundary-layer and the film thickness in region (ii) tend to decrease linearly with the slip length. On the other hand, for large $S$ , $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D709})\sim \sqrt{6\unicode[STIX]{x1D709}}$ , so that $h(\unicode[STIX]{x1D709}<\unicode[STIX]{x1D709}_{0})\sim 1+2(\unicode[STIX]{x1D709}/\sqrt{6\unicode[STIX]{x1D709}}+2S)$ .

The transverse velocity component at the outer edge of the boundary layer is obtained upon using (2.5) and (3.27), namely

This expression illustrates clearly the character of the flow at the leading edge of the boundary layer. For an adhering jet, $V$ is infinite at $x=0$ as it behaves like $1/\sqrt{x}$ . For a slipping jet, the jump is finite with $V(\unicode[STIX]{x1D709}=0)=1/2S$ . The flow structure is further discussed once the solution in the fully viscous layer is determined in the next section.

Figure 8. Influence of slip on the transition location and free-surface height, $h(x_{0})=\unicode[STIX]{x1D6FF}(x_{0})$ , at the transition point. The large- $S$ asymptote is also shown for $\unicode[STIX]{x1D6FF}$ .

The transition point is reached when the boundary layer absorbs all of the flow. Consequently, $\unicode[STIX]{x1D709}_{0}$ and the corresponding jet thickness are determined upon setting $h(\unicode[STIX]{x1D709}_{0})=\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D709}_{0})$ in (3.27) and using (3.25), yielding

Figure 8 displays the influence of slip on the transition location and corresponding jet thickness. Overall, the transition location increases with slip length, and the jet becomes thinner. In the limit of an adhering jet, one recovers $h(\unicode[STIX]{x1D709}_{0})\sim 3/2$ and $\unicode[STIX]{x1D709}_{0}\sim 3/40$ . For an infinitely slipping jet, (3.29) and (3.30) suggest that $h(\unicode[STIX]{x1D709}_{0})\sim 1+3/16S$ and $\unicode[STIX]{x1D709}_{0}\sim 1/6(1+3/8S)$ . The asymptote for the height is also shown in the figure.

4.1 Series expansion in powers of $\unicode[STIX]{x1D701}$

Before seeking a series solution, it is helpful to eliminate the integral from equation (4.4) by differentiating equation (4.4) with respect to $\unicode[STIX]{x1D701}$ and reusing (4.4) to obtain

It is also convenient to make the transformation $\unicode[STIX]{x1D701}\rightarrow 1-\unicode[STIX]{x1D701}$ . In this case, equation (4.8) remains of the same form, subject to the following conditions, recast from (4.5)–(4.7) as

The solution of (4.8) is symmetric with respect to $\unicode[STIX]{x1D701}=0$ ; that is, $g(\unicode[STIX]{x1D709},\unicode[STIX]{x1D701})=g(\unicode[STIX]{x1D709},-\unicode[STIX]{x1D701})$ . Therefore, we let

which satisfy the latter two conditions in (4.9). Upon substituting (4.11) into (4.8), one arrives at the following relations for the first two coefficients:

and the recurrence relation for the remaining $G_{m\geqslant 3}$ coefficients:

Additionally, the slip condition in (4.9) and the conservation of mass relation in (4.10) lead respectively to

Finally, the following initial conditions ensue from the matching of the flow at the transition point:

This completes the set of equations needed to determine the unknown variables $U(\unicode[STIX]{x1D709})$ , $h(\unicode[STIX]{x1D709})$ and $G_{m\geqslant 1}(\unicode[STIX]{x1D709})$ . It should be noted that, despite the apparent decoupling process, the problem (4.12)–(4.16) remains highly nonlinear and constitutes a coupled algebraic–differential system. The results discussed next are obtained with an error of the order of $10^{-5}$ .

4.2 Slip and transition flow

The influence of slip on the flow through the transition region is discussed now that the solutions have been established in the developing and fully viscous boundary layers. Figure 9 illustrates the influence of slip on the film free-surface height (figure 9 a), streamwise velocity component at the free surface (figure 9 b) and slip velocity at the plate (figure 9 c). Both regions (ii) and (iii) are covered for coherence and completeness. Figure 9(a) depicts a relatively sharp increase in jet thickness above the boundary layer near the leading edge $(x<x_{0})$ in region (ii) and a milder essentially linear growth beyond the transition point $(x>x_{0})$ in region (iii). The linear growth was also predicted by Watson (Reference Watson1964) for an adhering jet. For a more slipping jet, the film is thinner and the growth remains practically linear throughout. The velocity at the surface in figure 9(b) exhibits essentially a linear decrease with distance, at a rate diminishing with slip.

Figure 9. Influence of slip on the jet free-surface height (a), surface velocity (b) and slip velocity (c) across the developing and transition regions (ii) and (iii). The locus of transition points is also shown in (a) and (c). The limit of an adhering jet is shown in bold solid lines in all three panels.

Figure 9(c) should be contrasted to figure 6(b) where the slip velocity was only shown over the developing boundary-layer region (ii). The rather sharp drop in region (ii), as a result of the sharp increase in boundary-layer height (see figures 6 a and 9(a)), is tempered significantly in the viscous boundary-layer region (iii) as a result of the slower growth of film thickness (figure 9 a) compared with the boundary-layer height (figure 6 a).

Figure 10. Influence of slip on the jet and boundary-layer thicknesses.

This difference in growth rate between the boundary-layer height and the film thickness is evident from figure 10, where the boundary-layer profiles are shown near the leading edge. The locus of the transition points is also shown in figures 9(a) and 9(c), which indicates that the transition becomes relatively insensitive to slip for large $S$ . This saturation is also reflected in the profiles of figure 9(b) and 10, and, to a lesser extent, in the slip velocity profiles of figure 9(c). Finally, despite the difference in solution methodologies and levels of accuracy in the fully developed and viscous boundary-layer regions, the free-surface slope in figure 9(a) as well as the slopes of the slip velocity in figure 9(c) remain reasonably smooth at the transition point, $x=x_{\text{0}}$ .

5.1 The case of an inviscid fluid

This case mimics the flow of a film of low viscosity, draining at a relatively high rate. Hence, the influence of slip originates from the upstream side of the jump. Following Watson (Reference Watson1964), the position of the hydraulic jump, $x=x_{J}$ , is determined by equating the rate of change of momentum to the force associated with the difference in pressure that arises from the elevation change. This assumption is legitimate provided that the width (measured in the streamwise direction) of the jump is small so that skin friction can be ignored. In this case, the height remains constant, with $H(x\geqslant x_{J})=H_{J}$ and $u(x>x_{J},y)=1/H$ . In this case, the integral form for the condition of momentum (2.1) is expressed as

The momentum outside the wave, which is $O(h/H)$ compared with that inside, will be included only approximately since it is assumed that the speed of flow immediately outside the jump is uniform. In this case, (5.1) becomes

The integral on the right-hand side of (5.2) must be evaluated separately for $x_{J}>x_{0}$ and $x_{J}<x_{0}$ , since the jump may occur at any point in the development of the boundary layer.

When $x_{J}\leqslant x_{0}$ , the flow upstream of the jump consists of the developing boundary layer and external inviscid flow. In this case,

where $\unicode[STIX]{x1D6FF}_{J}=\unicode[STIX]{x1D6FF}(x=x_{J})$ and $\unicode[STIX]{x1D709}_{J}=x_{J}/Re$ . Although an accurate expression such as (3.9) can be used to evaluate the integral in (5.3), the approximate profile (3.22) is perhaps more physically revealing, and certainly much more convenient to use. In this case,

It should be noted that the upstream film height is determined from (3.27).

When $x_{J}\geqslant x_{o}$ , the substitution of the velocity profile from (4.11) leads to

where $U_{J}=U(x=x_{J})$ is the velocity at the surface just upstream of the jump. Since, from (4.15a,b ), $\sum _{m=1}G_{m}/2m+1=1/hU-1$ , (5.2) becomes, for $\unicode[STIX]{x1D709}_{J}\geqslant \unicode[STIX]{x1D709}_{o}$ ,

In this case, $h_{J}$ and $U_{J}$ are obtained from (4.15a,b ).

Figure 11. Influence of slip on the dependence of the hydraulic-jump height on the jump position for an inviscid film. The locus of transition locations is also shown.

Figure 11 shows the influence of slip on the variation of the hydraulic-jump height with the position of the jump, incorporating the influence of the Froude number. The term $h_{J}^{2}/2Fr^{2}$ , which is the pressure thrust on the inward side of the wave jump, is only $O(h^{2}/H^{2})$ compared with the thrust on the outward side, and is assumed to be negligible. The locus of the transition locations is also included. It is shown that, as for an adhering jet, the hydraulic jump occurs earlier when the film depth downstream of the jump is larger. The figure illustrates that greater slip leads to farther location of the jump, as a result of reduced friction at the plate. In this case, the jet spreads more easily. These observations are in agreement with those of Prince et al. (Reference Prince, Maynes and Crockett2012) for a circular jet (see their figure 10). The influence of slip is very much like the effect of shear thinning (see figure 19(a) of Zhao & Khayat (Reference Zhao and Khayat2008)), where the level of viscosity, and not necessarily the degree of shear thinning, dictates the position of the hydraulic jump.

Figure 12. Influence of slip on the jump position for $Fr=2.75$ . The jump height is kept the same at $H=2.7$ for all slip lengths. The locus of transition locations is also shown.

Finally, figure 12 displays the influence of slip on the location of the hydraulic jump while the jump height is kept the same ( $H=2.7$ ) for any slip length. This corresponds to a value of $(H^{3}+2Fr^{2})/2Fr^{2}H=0.85$ in figure 11, with $Fr=2.75$ . The locus of the transition locations is also shown, which indicates that the jump occurs in the fully developed boundary layer for $S<0.5$ and in the fully viscous layer for the more slipping fluid.

5.2 The case of a viscous fluid

Following Duchesne et al. (Reference Duchesne, Lebon and Limat2014), we determine the thickness distribution by adopting a thin-film approach downstream of the jump. The formulation is valid at large distance $x$ compared with the jump height. As for the ideal case, the jump height and position cannot be determined uniquely. However, the measurements of Duchesne et al. for a circular jet provide an additional relation for the unique determination of $H_{J}$ and $x_{J}$ . The local Reynolds number and the local slope are both assumed to be small; they are of the order of $10^{\text{-2}}$ in the experiment of Duchesne et al. (Reference Duchesne, Lebon and Limat2014).

Since experimental data are typically reported in terms of the flow rate $Q$ (Dressaire et al. Reference Dressaire, Courbin, Crest and Stone2009, Reference Dressaire, Courbin, Crest and Stone2010; Duchesne et al. Reference Duchesne, Lebon and Limat2014), we shall discuss our results in terms of $Fr$ and $G\equiv Re/Fr$ (rather than $Fr$ and $Re$ ) given that $G$ is independent of the flow rate. Neglecting inertia, the balance between hydrostatic pressure and viscous friction in (2.1), and the conditions of slip at the substrate (2.2) and free slip at the free surface (2.5) yield the following expressions for the velocity components and the pressure:

In addition, conservation of mass leads to

Integration between the positions of the jump $x_{J}$ and the trailing edge of the plate $x_{\infty }$ gives

The measurements of Duchesne et al. (Reference Duchesne, Lebon and Limat2014) for an adhering circular jet indicate that their experimental Froude number, $Fr_{J}=Q/2\unicode[STIX]{x03C0}\bar{R}_{J}\sqrt{g}\bar{H}_{J}^{3/2}$ , defined in terms of the (dimensional) jump height $\bar{H}_{J}$ and radius $\bar{R}_{J}$ , remains constant when the flow rate $Q$ is varied. Equivalently, for the current planar jet, and in terms of our (dimensionless) notations, we may then assume that $Fr_{J}=Fr/x_{J}H_{J}^{3/2}$ remains constant (independent of the Froude number, $Fr$ ). This assumption is not unreasonable since planar flow is the limit of circular flow with infinite radius of curvature, and also since there does not seem to be any restriction on the jump radius in the experiment. Moreover, the insensitivity of the experimental Froude number, $Fr_{J}$ , to variation of the flow rate was verified for both low- and high-viscosity liquids. It should be noted that $Fr_{J}$ is independent of the viscosity. In this case, it is plausible to assume that the constancy of $Fr_{J}$ would also hold for a slipping jet. Consequently, we assume that the value of $Fr_{J}$ is the same for any $S$ . This yields the following additional relations:

which, together with (5.9), determine the jump height and position uniquely.

Admittedly, the assumptions we make here are speculative, awaiting experimental verification for an impinging liquid sheet. However, they seem to yield a realistic physical picture of the flow in the jump region. For instance, the first relation in (5.11), suggesting a decrease in the jump position with the jump height for a fixed Froude number, agrees well with the measurements and scaling arguments of Bohr et al. (Reference Bohr, Dimon and Putzkaradze1993) for the jump radius versus height at a fixed flow rate, as reported in their figure 5. Finally, the flow structure is expected to change as the jump location decreases with increasing depth (Bush & Aristoff Reference Bush and Aristoff2003). Such a change, however, cannot be captured by the current thin-film formulation.

Figure 13. Jump height (a) and position (b) behaviour with the Froude number (flow rate) for various slip lengths. Here, $G=0.2$ , $Fr_{J}=1$ and $x_{\infty }=2$ .

Figure 13 illustrates the dependence of the jump height and the position on $Fr$ (flow rate) for various $S$ values. Here, $G=0.2$ , $Fr_{J}=1$ and $x_{\infty }=2$ . Both the jump height (figure 13 a) and the position (figure 13 b) increase with the Froude number, reflecting the tendency of the film to accumulate further downstream at higher flow rate. While the jump height decreases with slip, its position increases. This behaviour is typical of jets with lower viscosity, where, as in the case of slipping jets, the reduced friction enables the liquid to flow farther with diminished accumulation. The jump height becomes increasingly less sensitive to the variation of $Fr$ with slip (figure 13 a); the film behaves increasingly like an inviscid fluid, adopting a thickness closer to the film at the trailing edge. This insensitivity is contrasted by the sharper growth of the jump location with $Fr$ . In fact, in the limit of infinite slip, (5.9) and (5.11) indicate that

suggesting a linear growth of jump position with flow rate, and no change in height.

The trend in figure 13(b) against $Fr$ agrees with the measurements of Dressaire et al. (Reference Dressaire, Courbin, Crest and Stone2010) which indicate that the (average) jump radius for a circular jet also increases with the (total) flow rate. See the data reported in their figure 16(a). This observation holds for both smooth and rough substrates. The influence of slip reported in figure 13(b) is also in qualitative agreement with existing predictions, such as the K–P approach of Prince et al. (Reference Prince, Maynes and Crockett2012) and the similarity approach of Dressaire et al. (Reference Dressaire, Courbin, Crest and Stone2010). The latter assumed a slip length inversely proportional to the film thickness to secure a similarity solution. However, in contrast to the trend in figure 13(b), the measurements of Dressaire et al. (Reference Dressaire, Courbin, Crest and Stone2009, Reference Dressaire, Courbin, Crest and Stone2010), when conducted to examine the influence of topographical parameters of textured substrates on the jump radius and its dependence on flow rate, indicate that the jump radius actually decreases when roughness is added, regardless of the microtexture examined. Four square lattices were considered, with fixed post radius and different lattice spacings and post heights. In particular, they found that the jump radius decreases with increasing post height. Since the slip length is directly related to the post height, the experimental findings contradict existing theoretical predictions, including the behaviour reported in our figure 13(b) above. Dressaire et al. (Reference Dressaire, Courbin, Crest and Stone2009, Reference Dressaire, Courbin, Crest and Stone2010) attributed the systematic reduction in jump radius to a reduced flow rate. They regarded the (total) flow rate as divided into a leakage flow rate through the microtexture and the flow rate of a film above the posts.

The predictions in figure 13 corroborate well observations and findings of existing studies for an adhering jet. Bohr et al. (Reference Bohr, Dimon and Putzkaradze1993) and Rojas et al. (Reference Rojas, Argentina and Tirapegui2013) found the (dimensional) jump radius to scale roughly as $\bar{R}_{J}\sim Q^{5/8}\unicode[STIX]{x1D708}^{-3/8}g^{-1/8}$ for a circular jet. More recently, Duchesne et al. (Reference Duchesne, Lebon and Limat2014) derived a more accurate estimate involving a logarithmic correction based on the thin-film approach. In the current problem, the same scaling holds for an adhering planar jet. When cast in dimensionless form, we find that the jump location behaves like $x_{J}\sim Fr_{J}^{-1}Fr^{5/8}G^{-3/8}$ . This behaviour corresponds to the $S=0$ curve in figure 13(b).

The film shape and flow downstream of the jump are now examined by first integrating (5.8) to yield the following equation for $H(x)$ :

The velocity at the surface is determined from (5.7). In particular,

For an adhering jet, one has

which is equivalent to equation (3) of Duchesne et al. (Reference Duchesne, Lebon and Limat2014) for a circular jump.

Figure 14. Film thickness (a) and surface velocity (b) evolution with position downstream of the jump for the range $x_{J}\leqslant x\leqslant x_{\infty }$ , for different slip parameters. Here, $x_{\infty }=2$ and $G=0.2$ . The jump position for each $S$ value is also indicated by a vertical line.

Figure 14 depicts the evolution of the film height (figure 14 a) and the surface velocity (figure 14 b) with position downstream of the jump. Vertical lines indicate the jump position for each slip level. The decrease in jump height with position is obviously accompanied by an increase in surface velocity that is particularly steep near the plate trailing edge. The height profile is similar to the one measured by Duchesne et al. (Reference Duchesne, Lebon and Limat2014) for axisymmetric flow (see their figure 2a). While the jump height decreases gradually with slip, the velocity remains relatively insensitive to the variation of $S$ , except near the trailing edge of the plate. This may be surprising at first, but upon closer examination of figure 14, we observe the relatively large values of the jump height, which make the effect of slip relatively uninfluential at the surface, except near the edge where the film thickness narrows considerably. Finally, figure 14 shows the tendency of the film to become increasingly uniform with slip. In the limit of infinite slip, (5.7) and (5.8) yield $H(x)\sim H_{\infty }$ and $u(x>x_{J},y)\sim 1/H_{\infty }$ , in correlation with (5.11a,b ).