3.2.1 Regular perturbations

Hereafter, we confidently ignore any disturbances of the fully developed flow resorting to jet impact as these do not affect the subsequent analysis crucially. As a consequence, potential eigensolutions of the linearised operator in (3.8) can be safely ignored. This justifies the form of the expansion (3.1) for regular perturbations, varying in the streamwise direction only with $x$ , i.e. on the global scale, in the limits considered.

As seen from (2.2a ), those deviations of the shallow-water limit then arise by the pressure and normal-stress gradients in terms of the inhomogeneities $-\unicode[STIX]{x1D716}^{2}\bar{h}^{3}(\bar{p}_{x}-\bar{\unicode[STIX]{x1D713}}_{zxx})$ and $-\unicode[STIX]{x1D716}^{2}\bar{g}\bar{h}^{3}\bar{h}^{\prime }$ added to the right-hand side of the expanded form of (3.8a ). In turn, $h_{\unicode[STIX]{x1D716},g}$ are found by virtue of the linearised kinematic BCs (2.2c ) and free-slip condition in (2.2d )

(3.11a ) $$\begin{eqnarray}\displaystyle & \displaystyle [\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D716},g},\unicode[STIX]{x2202}_{z}\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D716},g}](x,0)=[0,0],\quad \unicode[STIX]{x1D713}_{\unicode[STIX]{x1D716},g}(x,\bar{h})=-h_{\unicode[STIX]{x1D716},g}\bar{\unicode[STIX]{x1D713}}_{z}(x,\bar{h}), & \displaystyle\end{eqnarray}$$

(3.11b ) $$\begin{eqnarray}\displaystyle & \displaystyle z=\bar{h}:[\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D716}},\unicode[STIX]{x1D713}_{g}]_{zz}=-h_{\unicode[STIX]{x1D716},g}\bar{\unicode[STIX]{x1D713}}_{zzz}[1,1]+[\bar{\unicode[STIX]{x1D713}}_{xx}+2\bar{h}^{\prime }\bar{\unicode[STIX]{x1D713}}_{zx},0]. & \displaystyle\end{eqnarray}$$

We first consider those essentially accounting for momentum transfer and normal stresses in the vertical direction. The representations of $\bar{\unicode[STIX]{x1D713}}$ in (3.9) and $\bar{p}$ in (3.10) and (3.8) suggest we write $[\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D716}},h_{\unicode[STIX]{x1D716}}/h_{W}]=[f_{\unicode[STIX]{x1D716}}(\unicode[STIX]{x1D702}),H_{\unicode[STIX]{x1D716}}]$ with the latter quantities determined by the boundary-value problem

(3.12a-c ) $$\begin{eqnarray}\displaystyle 2a_{W}f_{W}^{\prime }f_{\unicode[STIX]{x1D716}}^{\prime }+f_{\unicode[STIX]{x1D716}}^{\prime \prime \prime }=-(\unicode[STIX]{x1D702}^{3}f_{W}^{\prime })^{\prime \prime }/\unicode[STIX]{x1D702},\quad f_{\unicode[STIX]{x1D716}}(0)=f_{\unicode[STIX]{x1D716}}^{\prime }(0)=0,\quad (f_{\unicode[STIX]{x1D716}}^{\prime \prime }/f_{\unicode[STIX]{x1D716}})(1)=(f_{W}^{\prime \prime \prime }/f_{W}^{\prime })(1) & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

and $H_{\unicode[STIX]{x1D716}}=-(f_{\unicode[STIX]{x1D716}}/f_{W}^{\prime })(1)$ . Here the expressions of the inhomogeneities ensue after some lengthy rearrangements; the second contribution to $f_{\unicode[STIX]{x1D716}}^{\prime \prime }(1)$ as given by (3.11) is found to vanish. Standard methods allow for an analytical albeit (due to the inhomogeneous BCs) implicit representation of the unique solution to (3.12). We therefore solved (3.12) numerically in a straightforward fashion. With $f_{\unicode[STIX]{x1D716}}\geqslant 0$ (figure 2 a), those perturbations are of an expansive (accelerating) nature: they increase the wall shear stress ( $f_{\unicode[STIX]{x1D716}}^{\prime \prime }(0)\simeq 15.0837$ ) and decrease the film height accordingly ( $H_{\unicode[STIX]{x1D716}}\simeq -3.0338$ ).

Figure 2.

(a) Shape functions $f_{\unicode[STIX]{x1D716}}^{\prime }(\unicode[STIX]{x1D702})/4$ (dashed), $f_{g}^{\prime }(\unicode[STIX]{x1D702})$ (solid) referring to cross-stream and hydrostatic perturbations of Watson’s flow profile $f_{W}^{\prime }(\unicode[STIX]{x1D702})$ (dotted); (b) detached flow, normalised by $\bar{h}_{0}$ : flow profiles $f_{\unicode[STIX]{x1D702}}\bar{h}_{0}/\bar{h}$ (solid) versus $\unicode[STIX]{x1D702}$ for $x=0$ (Watson’s profile), $x/\bar{h}_{0}=0.005i$ ( $i=1,2,\ldots ,20$ ),  $x/\bar{h}_{0}=0.999$ , note $f_{\unicode[STIX]{x1D702}x}(x,0)>0$ , and $\bar{h}/\bar{h}_{0}$ versus $x/\bar{h}_{0}$ (dashed), asymptotes of $\bar{h}/\bar{h}_{0}$ for $x\rightarrow 0$ , $x\rightarrow \infty$ considered in §§ 3.3.3, 3.3.2 (dotted).

The subsequently more relevant hydrostatically induced regular disturbances (§ 4.1) add the term $-ga_{W}h_{W}^{3}$ to the right-hand side of the correspondingly linearised version of (3.8a ). We in turn arrive at $[\unicode[STIX]{x1D713}_{g},h_{g}/h_{W}]=(1+x)^{3}[f_{g}(\unicode[STIX]{x1D702}),H_{g}]$ and conveniently set $f_{g}=f_{W}^{\prime }G_{g}$ . With the aid of (3.9) and (3.11), $G_{g}$ and $H_{g}$ then satisfy

(3.13a-c ) $$\begin{eqnarray}4a_{W}f_{W}^{\prime 4}G_{g}^{\prime }=(f_{W}^{\prime 3}G_{g}^{\prime \prime })^{\prime }-a_{W}^{4}f_{W}^{\prime 2},\quad G_{g}(0)=G_{g}^{\prime \prime }(1)=0,\quad H_{g}=-G_{g}(1).\end{eqnarray}$$

The regularity condition $G_{g}^{\prime \prime }(0)=a_{W}^{4}/(3\unicode[STIX]{x1D706}_{W})$ completes (3.13) to form a boundary-value problem. A numerical investigation shows that the associated self-adjoint homogeneous problem has no solution (and thus the absence of the aforementioned eigensolutions or free perturbations for the specific streamwise variation considered). Hence, (3.13) uniquely determines $G_{g}^{\prime }$ and $G_{g}$ , $H_{g}$ by subsequent integration. Here the straightforward numerical solution of (3.13) predicts $f_{g}\leqslant 0$ (figure 2 a), $f_{g}^{\prime \prime }(0)=2\unicode[STIX]{x1D706}_{W}[G_{g}^{\prime }(0)+H_{g}]\simeq -2.6272$ and $H_{g}\simeq 0.9907$ . Contrasting sharply with the irregular short-scale perturbations studied below and in § 4.1, the adverse hydrostatic pressure gradient first yields a weakly compressive (retarded) flow.

3.2.2 Irregular perturbations

Bowles & Smith (Reference Bowles and Smith1992) showed how weak perturbations of rapid downstream growth can exist in very supercritical developed shallow-water flow so as to provoke an accordingly strong hydraulic jump via local free viscous–inviscid interaction of double-layer type. For finite values of $x+1$ , that interaction process would occur along a horizontal length of $O(g^{3})$ in the current specification of the flow. The similar scenario for disturbances due to cross-stream effects yields an interaction length of $\unicode[STIX]{x1D716}^{6/7}$ as derived originally by Smith (Reference Smith1977) and Smith & Duck (Reference Smith and Duck1977). Raising the magnitude of $g$ to $O(\unicode[STIX]{x1D716}^{2/7})$ establishes the distinguished limit Bowles & Smith (Reference Bowles and Smith1992) considered. Complying with Higuera’s (Reference Higuera1994) analysis, however, here the presumed existence of the trailing edge, so introducing the basic streamwise scale, avoids that jump in the limit $g\rightarrow 0$ to keep the flow correspondingly close to supercritical conditions. Hence, the present embedding keeps the magnitude of those rapidly growing perturbations from attaining a magnitude that triggers that strong interaction sufficiently far upstream from the plate edge. On the other hand, Brotherthon-Ratcliffe & Smith (Reference Brotherton-Ratcliffe and Smith1989) identified a type of Korteweg–de-Vries wave interacting with a Stokes layer that perturbs a developed layer described by the hydrostatic extension of the shallow-water approximation, with the basic phase speed given by the unperturbed flow, on a relatively large longitudinal scale measured by $\unicode[STIX]{x1D716}^{-1/5}$ , and slowly modulated on an associated time scale measured by $\unicode[STIX]{x1D716}^{-3/5}$ . Following their analysis (also Balmforth Reference Balmforth1999), the so-manifested upstream influence and possible destabilisation of the hydrostatic extension of the shallow-water approximation vanish only for $g=0$ : this then precisely coins the notion of supercritical developed flow.

Both contexts suggest the existence of longitudinal short-scale disturbances in the present situation. We study these in the necessary breadth and depth as their strengths and variations through the flow are found to both complete the parametrisation of the strongly interactive near-edge flow in § 4.1 and to identify the full extent of the upstream influence.

The effects of gravity and of streamline curvature including the capillary influence generate a family of perturbations characterised by locally rapid variation but of exponentially small amplitude. These promote further contributions to (3.1) for $\unicode[STIX]{x1D716}\rightarrow 0$ and $g\rightarrow 0$ where the distinguished limit (3.4) can be relaxed. Due to the scale reduction, they represent homogeneous disturbances about the shallow-water limit above their generating viscous near-plate region where $N:=\unicode[STIX]{x1D702}/\unicode[STIX]{x1D6E5}=O(1)$ , say; the asymptotically small gauge function $\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D716};x)$ is positive downstream of the virtual origin (for $x>-1$ ). Since a nonlinear distortion of the unperturbed shear flow in this viscous sublayer (SL), represented by $f_{W}^{\prime }\sim \unicode[STIX]{x1D706}_{W}\unicode[STIX]{x1D702}$ , would typically require $1+x=O(\unicode[STIX]{x1D6E5}^{1/3})$ along the plate, it is ruled out. This renders a linearisation of (2.2) across the entire layer sufficient. The viscous displacement of the streamlines in the SL provokes likewise rapid disturbances in the core layer, hence satisfying the inviscid-flow equations to leading order. For larger values of $\bar{g}$ , the purely hydrostatically caused disturbances were first shown by Bowles (Reference Bowles1990) to originate from the singular point $x=-1$ , associated with their exponentially fast streamwise growth; cf. Scheichl & Bowles (Reference Scheichl and Bowles2017) and Scheichl & Kluwick (Reference Scheichl and Kluwick2017). Here we revise and extend this analysis of the associated singularly perturbed eigenproblems.

It is useful at this stage to first demonstrate why the thereby introduced shortest streamwise scales are, however, sufficiently long to avoid the formation of a Rayleigh stage and thus a corresponding finite streamwise slip at its base. Its inevitable breakdown close to the latter would either be of inviscid and nonlinear character or directly controlled by viscous shear. Together with the no-slip condition holding at the plate, the first type would yield a further viscous slip flow adjacent to it, driven by an imposed pressure gradient. However, this impedes a strong interaction mechanism (of the above kind) required to control the flow around the edge as outlined further below. The second would imply an (interactive) two-tiered structure, crucially tied in with the displacement effect mentioned before. But in conflict with the assumption of a Rayleigh limit, then the pressure disturbances in the core tier are typically controlled by their cross-stream variation (meaning a hierarchical slender-layer approximation).

We bear these arguments in mind when it comes to the rationale underlying the interactive-flow structure. In the present context of linearising (2.2a,b ) for $\unicode[STIX]{x1D702}=O(1)$ , they justify us taking the irregular pressure perturbations balancing streamline curvature (and the gradients of the normal stresses), see (2.2b ), as insignificantly small in the expanded form of (2.2a ). By the smallness of $g$ , this also holds for their hydrostatically induced counterparts. We then linearise (2.2) by ignoring the BCs for $y=0$ and noting $p_{W}(x,\unicode[STIX]{x1D702})$ as given by (3.10). This yields $\unicode[STIX]{x1D713}\sim f_{W}$ perturbed by the aforesaid local eigenfunction at negligibly small pressure disturbances ensuing from (2.2b ) and the last of the BCs (2.2d ) with the aid of (3.1) (subject to $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D702}}p_{W}(x,1)=0$ ):

(3.14a ) $$\begin{eqnarray}\displaystyle \displaystyle [\unicode[STIX]{x1D713},h/h_{W}(x)] & {\sim} & \displaystyle [f_{W}(\unicode[STIX]{x1D702}),1]+\cdots +\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D716};x)[f_{W}^{\prime }(\unicode[STIX]{x1D702}),-1]+\text{c.c.}+o(\unicode[STIX]{x1D6FE}),\end{eqnarray}$$

(3.14b ) $$\begin{eqnarray}\displaystyle \displaystyle p/\unicode[STIX]{x1D716}^{2} & {\sim} & \displaystyle p_{W}(x,\unicode[STIX]{x1D702})+\cdots +\unicode[STIX]{x1D6FE}_{xx}P_{\unicode[STIX]{x1D716}}(x,\unicode[STIX]{x1D702})-\unicode[STIX]{x1D6FE}\bar{g}h_{W}(x)+\text{c.c.}+o(\unicode[STIX]{x1D6FE}_{xx},\unicode[STIX]{x1D6FE}\bar{g}),\qquad \quad\end{eqnarray}$$

(3.14c ) $$\begin{eqnarray}\displaystyle & \displaystyle P_{\unicode[STIX]{x1D716}}(x,\unicode[STIX]{x1D702}):=\unicode[STIX]{x1D70F}h_{W}(x)-\int _{\unicode[STIX]{x1D702}}^{1}f_{W}^{\prime 2}(t)\,\text{d}t=\unicode[STIX]{x1D70F}h_{W}-\frac{f_{W}^{\prime \prime }}{a_{W}}.\qquad \quad & \displaystyle\end{eqnarray}$$

$\unicode[STIX]{x1D6FE}$

$x$

$\unicode[STIX]{x1D716}\rightarrow 0$

$\unicode[STIX]{x1D6FE}=o(\unicode[STIX]{x1D6FE}_{x})$

$\unicode[STIX]{x1D6FE}_{x}=o(\unicode[STIX]{x1D6FE}_{xx})$

$\unicode[STIX]{x1D6FE}_{xx}=o(\unicode[STIX]{x1D6FE}_{xxx})$

(3.15) $$\begin{eqnarray}\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D6FE}_{xxx}=o(\unicode[STIX]{x1D6FE}_{xx}).\end{eqnarray}$$

This prompts us to disregard the viscous stresses when expanding (2.2b,d ) and express $P_{\unicode[STIX]{x1D716}}$ in (3.14c ) through streamline curvature solely, with the help of (3.9).

From (3.14a )–(3.15), we deduce the following estimates holding in the SL: with $p_{W}(x,0)\equiv 0$ by (3.10), irregular disturbances of $O(\unicode[STIX]{x1D6FE}_{xx})$ contribute to the accordingly reduced pressure $p/\unicode[STIX]{x1D716}^{2}+(\unicode[STIX]{x1D6FE}-1)\bar{g}h_{W}(x)$ ; therefore, the magnitudes of its $x$ - and $y$ -gradients (of $O(\unicode[STIX]{x1D6FE}_{xxx})$ and $O(\unicode[STIX]{x1D6FE}_{xx}/\unicode[STIX]{x1D6E5})$ respectively) are much larger than those of the respective perturbations of the normal-stress gradients in (2.2a ) and (2.2b ) (of $O(\unicode[STIX]{x1D6FE}_{xx})$ and $O(\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D6E5}\unicode[STIX]{x1D6FE}_{xxx})$ respectively). Hence, for $N=O(1)$ we expand

(3.16a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D713}-\unicode[STIX]{x1D706}_{W}\unicode[STIX]{x1D702}^{2}/2\sim \cdots +\unicode[STIX]{x1D6E5}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D706}_{W}F(N)+\text{c.c.}+o(\unicode[STIX]{x1D6E5}\unicode[STIX]{x1D6FE}), & \displaystyle\end{eqnarray}$$

(3.16b ) $$\begin{eqnarray}\displaystyle & \displaystyle p/\unicode[STIX]{x1D716}^{2}+(\unicode[STIX]{x1D6FE}-1)\bar{g}h_{W}(x)\sim \cdots +\unicode[STIX]{x1D6FE}_{xx}P_{p}(x)+\text{c.c.}+o(\unicode[STIX]{x1D6E5}^{2}\unicode[STIX]{x1D6FE}_{xx},\unicode[STIX]{x1D6FE}\bar{g}), & \displaystyle\end{eqnarray}$$

(3.16c ) $$\begin{eqnarray}\displaystyle & \displaystyle P_{p}(x):=P_{\unicode[STIX]{x1D716}}(x,0)=a_{W}\unicode[STIX]{x1D706}[T(x+1)-1],\quad T:=\unicode[STIX]{x1D70F}/J_{0}, & \displaystyle\end{eqnarray}$$

$T$

$P_{p}$

$\unicode[STIX]{x1D6E5}$

$\unicode[STIX]{x1D6FE}$

$F$

(3.17) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{W}^{2}h_{W}\unicode[STIX]{x1D6E5}^{3}\unicode[STIX]{x1D6FE}_{x}(NF^{\prime }-F)\sim h_{W}^{3}(\unicode[STIX]{x1D716}\unicode[STIX]{x1D6E5})^{2}[\unicode[STIX]{x1D6FE}_{x}\bar{g}h_{W}-\unicode[STIX]{x1D6FE}_{xxx}P_{p}(x)]+\unicode[STIX]{x1D706}_{W}\unicode[STIX]{x1D6FE}F^{\prime \prime \prime }.\end{eqnarray}$$

One finds $\unicode[STIX]{x1D6FE}$ varying much more rapidly with $x$ than $\unicode[STIX]{x1D6E5}$ . Differentiation of (3.17) with respect to $N$ suggests to write $\unicode[STIX]{x1D70E}NF^{\prime \prime }=F^{(4)}$ subject to $F(0)=F^{\prime }(0)=0$ and $F^{\prime }(\infty )=1$ and with the (over $x$ piecewise) constant $\unicode[STIX]{x1D70E}:=\unicode[STIX]{x1D706}_{W}h_{W}\lim _{\unicode[STIX]{x1D716}\rightarrow 0}(\unicode[STIX]{x1D6E5}^{3}\unicode[STIX]{x1D6FE}_{x}/\unicode[STIX]{x1D6FE})$ by consistency with (3.17). In turn,

(3.18a,b ) $$\begin{eqnarray}F(N)=3\unicode[STIX]{x1D70E}^{1/3}\int _{0}^{N}(N-t)\text{Ai}(\unicode[STIX]{x1D70E}^{1/3}t)\,\text{d}t,\quad \frac{\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D6FE}_{0}(\unicode[STIX]{x1D716})}\sim \exp \int _{0}^{x}\frac{\unicode[STIX]{x1D70E}+o(h_{W}\unicode[STIX]{x1D6E5}^{3})}{\unicode[STIX]{x1D706}_{W}h_{W}(s)\unicode[STIX]{x1D6E5}^{3}(\unicode[STIX]{x1D716};s)}\,\text{d}s\end{eqnarray}$$

for $-1<x<0$ and $-\unicode[STIX]{x03C0}<\arg \unicode[STIX]{x1D70E}<\unicode[STIX]{x03C0}$ . As $\unicode[STIX]{x1D6E5}/\unicode[STIX]{x1D70E}^{1/3}$ proves invariant against a variation in the definition of $\unicode[STIX]{x1D6E5}$ , this is seen to absorb any potential $x$ -dependence of $\unicode[STIX]{x1D70E}$ and thus of $F$ . We then preferably choose $|\unicode[STIX]{x1D70E}|=1$ , thus

(3.19) $$\begin{eqnarray}\unicode[STIX]{x1D70E}=1\end{eqnarray}$$

as long as the streamwise variation of $\unicode[STIX]{x1D6FE}$ stays real, and are aware that a potential sudden emergence of a non-zero $\arg \unicode[STIX]{x1D70E}$ is tied in with a corresponding shortening of the streamwise scale, then requiring a proper smoothing. The, at first arbitrary, (complex) quantity $\unicode[STIX]{x1D6FE}_{0}$ in (3.18) parametrises the strength of the irregular disturbances once they have evolved towards a terminal state asymptotically close to the edge. The herewith introduced upstream influence not only accounts for the singularity of the integrand in (3.18) at the virtual origin but will become fixed in § 4.1. In combination with (3.18), the premise $\unicode[STIX]{x1D6FE}=o(\unicode[STIX]{x1D6E5})$ made implicitly in (3.14a ) yields $(\unicode[STIX]{x1D6E5}^{3})_{x}=o(h_{W}^{-1})$ , which will turn out to hold even for $x\rightarrow -1$ .

Consistency with (3.17) also requires, after some algebra and with the help of (3.16c ),

(3.20a ) $$\begin{eqnarray}\displaystyle & \displaystyle I^{7}-G_{\unicode[STIX]{x1D716}}(1+x)^{3}I^{6}=1-T(1+x), & \displaystyle\end{eqnarray}$$

(3.20b ) $$\begin{eqnarray}\displaystyle & \displaystyle I:=\frac{3^{2/21}\unicode[STIX]{x1D706}^{3/7}a_{W}}{\unicode[STIX]{x0393}(1/3)^{1/7}}\frac{\unicode[STIX]{x1D6E5}}{\unicode[STIX]{x1D70E}^{1/3}\unicode[STIX]{x1D716}^{2/7}},\quad G_{\unicode[STIX]{x1D716}}:=\frac{\unicode[STIX]{x0393}(1/3)^{6/7}}{3^{4/7}\unicode[STIX]{x1D706}^{11/7}}\frac{g}{\unicode[STIX]{x1D716}^{2/7}}. & \displaystyle\end{eqnarray}$$

$\unicode[STIX]{x1D6FE}_{xxx}/\unicode[STIX]{x1D6FE}\sim (\unicode[STIX]{x1D6FE}_{x}/\unicode[STIX]{x1D6FE})^{3}$

$I$

$\unicode[STIX]{x1D6E5}/\unicode[STIX]{x1D70E}^{1/3}$

$G_{\unicode[STIX]{x1D716}}$

$T$

$G_{\unicode[STIX]{x1D716}}$

$g=O(\unicode[STIX]{x1D716}^{2/7})$

$G_{\unicode[STIX]{x1D716}}\gg 1$

$I\sim (1+x)^{3}G_{\unicode[STIX]{x1D716}}$

$g=O(1)$

$x$

$-1$

$I-I^{-6}\sim G_{\unicode[STIX]{x1D716}}(1+x)^{3}=O(1)$

$G_{\unicode[STIX]{x1D716}}$

$\unicode[STIX]{x1D6E5}=O(\unicode[STIX]{x1D716}^{2/7})$

$-1\leqslant x<0$

$x\rightarrow -1$

$\text{Re}(\unicode[STIX]{x1D70E}^{1/3})>0$

$\text{Re}\,I>0$

$\text{Re}\,\unicode[STIX]{x1D70E}>0$

$I\rightarrow 1$

$x\rightarrow -1$

For the large Froude numbers considered here, consistency with § 4, below, which takes $g=O(\unicode[STIX]{x1D716}^{4/7})$ , requires us to concentrate here on the case $G_{\unicode[STIX]{x1D716}}=0$ , i.e. to neglect the gravitational influence in (3.20a ). We include it, however, for completeness and for consideration in our subsequent work examining smaller values of the Froude number, as outlined in § 5. Some of the impact of finite but small values of $G_{\unicode[STIX]{x1D716}}$ is taken up in appendix A.

For the current negligibly small values of $G_{\unicode[STIX]{x1D716}}$ and also $T$ , then $I\sim 1$ throughout and (3.18) signifies a strong algebraic downstream growth, which close to the plate edge is transformed into an exponential one:

(3.21a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{0}\underset{x=O(1)}{{\sim}}(1+x)^{E}\underset{x\rightarrow 0-}{{\sim}}\exp (Ex),\quad E:=(B\unicode[STIX]{x1D716})^{-6/7}\rightarrow \infty , & \displaystyle\end{eqnarray}$$

(3.21b ) $$\begin{eqnarray}\displaystyle & \displaystyle B:=\frac{\unicode[STIX]{x0393}(1/3)^{3/7}}{(3\unicode[STIX]{x1D706})^{2/7}}=\left(\frac{4\unicode[STIX]{x03C0}^{3}\unicode[STIX]{x0393}(5/6)^{6}}{3\unicode[STIX]{x0393}(1/3)^{3}}\right)^{1/7}\simeq 1.2376, & \displaystyle\end{eqnarray}$$

$G_{\unicode[STIX]{x1D716}}$

$T$

$x$

$h$

$x$

$x+1\rightarrow 0+$

$\text{Re}\,\unicode[STIX]{x1D70E}>0$

(3.22) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{0}\sim \exp [Ex/I_{0}^{3}+o(x)],\quad x/\unicode[STIX]{x1D716}^{6/7}\rightarrow 0-.\end{eqnarray}$$

Herein, $I_{0}$ denotes a permitted root of (3.20a ) for $x=0$ . The exponentially growing amplitude of the perturbations has attained its final order of magnitude for $x=O(\unicode[STIX]{x1D716}^{6/7})$ upstream of the plate edge.

As a result, we first trace a single branch of positive roots of (3.20a ) in the downstream direction, starting with $I=1$ , along with the setting (3.19). For $T<1$ , all involved perturbation quantities stay real up to the plate edge. If $T>1$ , however, we encounter a sudden modification of the shape function $F$ in (3.18) (on the original streamwise $O(1)$ -scale) as

(3.23a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D70E}=\exp (-3\text{i}\unicode[STIX]{x03C0}/7),\quad x_{T}:=1/T-1<x<0.\end{eqnarray}$$

Here $x=x_{T}$ indicates a so-called ‘capillary threshold’ where $P_{p}(x)$ in (3.16c ) changes sign as its capillary portion overcompensates that by vertical momentum transfer for $x>x_{T}$ . With the aid of (3.20a ) and (3.22), we finally track the logarithmic attenuation $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{0}$ of the capillarity-controlled disturbances from the edge in upstream direction in the form

(3.24a,b ) $$\begin{eqnarray}B\unicode[STIX]{x1D716}^{6/7}\ln \left(\frac{\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D6FE}_{0}}\right)\sim \unicode[STIX]{x1D6FA}(x;T):=\int _{0}^{x}\frac{\text{d}s}{(1+s)[1-T(1+s)]^{3/7}},\quad I_{0}=\frac{1}{(1-T)^{1/7}},\end{eqnarray}$$

where we note (3.21) arising in the limit $T\rightarrow 0$ . Figure 3 illustrates the numerical evaluation of the scaled exponent $\unicode[STIX]{x1D6FA}$ in (3.18): one clearly identifies the logarithmic singularity of  $\unicode[STIX]{x1D6FA}$ and the associated algebraic one of $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{0}$ at $x=-1$ , included in (3.21); its integrable counterpart of $\unicode[STIX]{x1D6FA}$ at $x=x_{T}$ , accompanying the abrupt switch from (3.19) to (3.23) in the case $T>1$ ; the exponential steepening of $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{0}$ upstream of the plate edge, the more extreme the smaller $\unicode[STIX]{x1D716}$ is. Most important, for $T>1$ we find $\arg I=\unicode[STIX]{x03C0}/7$ (0) for $x>x_{T}$ ( $x<x_{T}$ ) such that $\text{Im}\,\unicode[STIX]{x1D6FA}$ increases from zero for $x=0$ in the upstream direction towards its final value assumed for $x\leqslant x_{T}$ , where $\arg \unicode[STIX]{x1D6FE}=\arg \unicode[STIX]{x1D6FE}_{0}+\arg \unicode[STIX]{x1D6FA}$ . This expresses a subtle cumulative upstream influence. As a physically intriguing phenomenon, stationary capillarity-driven undulations of the flow originate from the capillary threshold to grow exponentially towards the plate edge. Both their frequency and the slope of their envelope increase for decreasing values of $\unicode[STIX]{x1D716}$ . For figure 3(c–d), we have used $\unicode[STIX]{x1D6FE}/|\unicode[STIX]{x1D6FE}_{0}|\sim \exp [\unicode[STIX]{x1D6FA}/(B\unicode[STIX]{x1D716}^{6/7})+\text{i}\arg \unicode[STIX]{x1D6FE}_{0}]$ . Hence, considering just the real part by varying $\arg \unicode[STIX]{x1D6FE}$ from 0 to $\unicode[STIX]{x03C0}$ surveys the potential variety of the streamwise oscillations.

Figure 3.

$\unicode[STIX]{x1D6FA}$ and $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{0}$ versus $x$ given by (3.24) (cubic-spline interpolation of computed data points); (a)  $\unicode[STIX]{x1D6FA}$  for $T=0$ (dotted), 0.2 (dot-dot-dashed), 0.5 (dot-dashed), 0.7 (dashed), 0.9 (solid); (b)  $\text{Re}\,\unicode[STIX]{x1D6FA}$ (solid, right ordinate), $\text{Im}\,\unicode[STIX]{x1D6FA}$ (dashed, left ordinate) for $T=4/3$ ( $x_{T}=-0.25$ , $\text{Im}\,\unicode[STIX]{x1D6FA}\lesssim 0.8185$ ),  $T=2$ ( $-0.5$ , ${\lesssim}1.3099$ ),  $T=4$ ( $-0.75$ , ${\lesssim}1.8404$ ); (c)  $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{0}$ parametrised as in (a) for $\unicode[STIX]{x1D716}=0.3$ (upper family of curves), 0.03 (lower family); (d)  $\text{Re}\,\unicode[STIX]{x1D6FE}/|\unicode[STIX]{x1D6FE}_{0}|$ parametrised as in (b) for $\arg \unicode[STIX]{x1D6FE}_{0}=0$ and $\unicode[STIX]{x1D716}=0.3$ (dashed), $\unicode[STIX]{x1D716}=0.03$ (solid); (e) as in (d) but for $\arg \unicode[STIX]{x1D6FE}_{0}=\unicode[STIX]{x03C0}/4$ ; (f) as in (d) but for $\arg \unicode[STIX]{x1D6FE}_{0}=\unicode[STIX]{x03C0}/2$ .

The importance of distinguishing between $T>1$ and $T<1$ as in (3.14c ), here due to the zero in (3.16c ), will be met again in the course of the analysis (§ 3.3.3, § 4). The following notes complement the analysis of the irregular perturbations.

1. (i) We first consider the weak singularity $\unicode[STIX]{x1D6FA}$ exhibits for $T>1$ at the capillary threshold, see (3.24). There the sublayer thickness $\unicode[STIX]{x1D6E5}$ vanishes like $|x-x_{T}|^{1/7}$ and $\unicode[STIX]{x1D70E}$ has a discontinuity as $P_{p}$ is zero, see (3.16c ). The latter neutralises the associated increase of the pressure disturbance in (3.14b ) in the sublayer, described by (3.16b ). For $x-x_{T}=O(\unicode[STIX]{x1D716}^{2/13})$ , we find $(x-x_{T})\unicode[STIX]{x1D6FE}=O(\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D6FE}_{xx})$ such that the irregular correction of the velocity profile produced by the inviscid-flow balance in (2.2a ) affects the streamline curvature in a way that the zero of $P_{p}$ is shifted downstream accordingly. Since $P_{\unicode[STIX]{x1D716}}=O(y^{3})$ at $x=x_{T}$ , see (3.14c ), the associated relative pressure correction of $O(x-x_{T})$ , acting on the sublayer, induces a passive buffer layer where $y=O(\unicode[STIX]{x1D716}^{2/39})$ . Simultaneously, the thickness of the sublayer $\unicode[STIX]{x1D6E5}$ has decreased to $O(\unicode[STIX]{x1D716}^{4/13})$ . However, this new structure just delays the singularity. Its regularisation is finally accomplished by a Rayleigh stage, namely when the magnitude of the pressure perturbation has reached that of the streamwise velocity. According to (3.14c ), then $\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D6FE}_{x}x=O(\unicode[STIX]{x1D6FE})$ , giving $x-x_{T}=O(\unicode[STIX]{x1D716}^{1/3})$ . Gravity plays no active role in this process. However, examining its details is left to smoothing of the stronger form of this singularity for larger values of $G_{\unicode[STIX]{x1D716}}$ , cf. appendix A.

2. (ii) Determining the gauge functions $\unicode[STIX]{x1D6E5}$ and $\unicode[STIX]{x1D6FE}$ , for some given $\unicode[STIX]{x1D6FE}_{0}$ , requires the simultaneous investigation of the core region and the SL. This reveals an already interactive character of the yet very small disturbances along the entire adjustment of the flow towards the plate edge, tied in with their upstream influence.

3.3.1 Full wake

We obtained the corresponding numerical solution of (3.8) by adopting a backward-differentiation Keller–Box scheme (method of lines) with automated adaptive remeshing of the $\unicode[STIX]{x1D702}$ -grid ( ${\approx}2000$ points) and a rather high absolute accuracy in the $x$ -direction of approximately $10^{-8}$ (using the routine d03prc of the NAG C Library 2017). If not stated otherwise, in the remainder of this paper, the output of this solver is considered at specific locations of the downstream and the vertical coordinate where an associated interpolation routine is adopted for the latter; visualisation is then accomplished by cubic-spline interpolation. The solution plotted in figure 2(b) discloses a quite rapid approach of a uniform parallel-flow far downstream through the action of viscous diffusion.

With $\bar{h}_{+}=\bar{h}+\bar{h}_{-}$ , the auxiliary condition (3.7) decouples from (3.2) as a consequence of the Prandtl shift. Integrating the first twice subject to $\bar{h}_{-}(0)=0$ yields for $x\geqslant 0$ :

(3.25) $$\begin{eqnarray}(1-2T)\bar{h}_{-}(x)=T(\bar{h}-h_{W})(x)+\frac{K_{W}-K(x)}{J_{0}}-\frac{\bar{g}}{J_{0}}\int _{0}^{x}(x-s)\bar{h}(s)\,\text{d}s.\end{eqnarray}$$

Hence, conservation of the angular momentum determines the vertical shift $h_{-}(x)$ of the wake parametrised by $\unicode[STIX]{x1D70F}$ and $\bar{g}$ . Here $K(x)$ , $\bar{h}(x)$ result from the solution of (3.8). In turn, $\bar{p}(x,z)$ for $x>0$ ensues from (3.3). The constant of the first integration gives a contribution to $(1-2T)\bar{h}_{-}$ varying linearly in $x$ . We choose this as $-Ta_{W}x$ , which produces the first term on the right-hand side of (3.25). Thereby, $\bar{h}_{-}^{\prime }(x)$ is continuous if Watson’s attached-flow solution ( $\bar{h}_{-}\equiv 0$ , $\bar{h}\equiv h_{W}$ , $K\equiv K_{W}$ ) is continued over the edge. The following subtle rationale for that specific choice anticipates findings relevant for the behaviour of the regular contribution to $\bar{h}_{+}$ for $|x|$ being much smaller, elucidated later (§ 4.1).

It proves useful at this stage to not necessarily resort to a fully developed (Watson’s) flow and the associated value of $\unicode[STIX]{x1D706}$ . Then the forms of $\bar{\unicode[STIX]{x1D713}}$ , $\bar{h}$ for $x\rightarrow 0+$ consist of the Maclaurin expansion

(3.26a,b ) $$\begin{eqnarray}\displaystyle [\bar{\unicode[STIX]{x1D713}}-\bar{\unicode[STIX]{x1D713}}_{0},\bar{h}-\bar{h}_{0}]\sim x[\bar{\unicode[STIX]{x1D713}}_{1}(z),\bar{h}_{1}],\quad [\bar{\unicode[STIX]{x1D713}}_{1}(z),\bar{h}_{1}]:=\left[\bar{\unicode[STIX]{x1D713}}_{0}^{\prime }(z)\int _{0}^{y}\frac{\bar{\unicode[STIX]{x1D713}}_{0}^{\prime \prime \prime }(t)}{\bar{\unicode[STIX]{x1D713}}_{0}^{\prime 2}(t)}\,\text{d}t,-\frac{\bar{\unicode[STIX]{x1D713}}_{1}(\bar{h}_{0})}{\bar{\unicode[STIX]{x1D713}}_{0}^{\prime }(\bar{h}_{0})}\right], & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

for $x\rightarrow 0-$ readily derived from (3.2) to this leading approximation, but they are also singularly perturbed by an additive irregular contribution arising by the loss of the no-slip condition, cf. (3.2b ). The latter is smoothed on shorter scales where the hierarchical slender-layer approximation ceases to be valid (§ 4.1). However, the flow has to pass the edge in an entirely regular fashion at smaller $x$ -scales according to (2.2a,b ). This finally rules out an initial arbitrariness in the non-trivial regular continuation of $\bar{h}_{-}$ .

The accurate numerical evaluation of (3.25) for several values of $T$ and $\bar{g}$ is displayed in figure 4. We note that the first term on the right-hand side of (3.25) is negative and the second positive throughout. Remarkably, $\bar{h}_{-}$ changes abruptly sign as $T\rightarrow 1/2$ , accompanied by an unbounded deflection of the layer. Therefore, this special limit deserves a separate attention (beyond the scope of the current investigation).

Figure 4.

Distributions of $\bar{h}_{-}$ versus $x/\bar{h}_{0}$ by (3.25); (a)  $\bar{g}=0$ (solid, left ordinate), $\bar{g}=0.25$ (dashed, left ordinate), $\bar{g}=1$ (dot-dashed, right ordinate), each set for $T=0.2,0,1.2$ (bottom to top for $x/\bar{h}_{0}=1$ ); (b) zoomed-in detail of (a); note that abscissae have a common origin.

In the context of (3.26), the solution of (3.2) has the following notable properties: the regular expansion

(3.27) $$\begin{eqnarray}\bar{\unicode[STIX]{x1D713}}_{0}\sim \unicode[STIX]{x1D706}z^{2}/2+\unicode[STIX]{x1D6FD}z^{5}+O(z^{8}),\quad z\rightarrow 0,\end{eqnarray}$$

with some constant $\unicode[STIX]{x1D6FD}$ (equal to $-\unicode[STIX]{x1D706}^{2}/60$ for Watson’s flow); the BCs imposed at $z=\bar{h}$ in (3.2b ) imply also $\bar{\unicode[STIX]{x1D713}}_{zzzz}(x,\bar{h})=0$ .

3.3.2 Far wake, downfall and ‘capillary ascent’

For $x\rightarrow \infty$ , the normalised streamwise-flow component $f_{\unicode[STIX]{x1D702}}\bar{h}_{0}/\bar{h}$ tends towards $\bar{h}_{0}/h_{\infty }$ as $h\sim \bar{h}$ attains some terminal value $h_{\infty }$ . In the $(x,z)$ -plane, it becomes horizontally aligned. Preservation of its streamwise momentum expressed by (3.5) confirms a contraction of the detached layer towards that uniform free jet: $h_{\infty }/\bar{h}_{0}\sim 1/(J_{0}\bar{h}_{0})<1$ . This constraint follows from the Cauchy–Schwarz or Jensen’s inequality and the conserved volumetric flow rate. Specifically, Watson’s flow passing the edge predicts $\bar{h}_{0}/h_{\infty }=\bar{h}_{0}J_{0}=a_{W}\unicode[STIX]{x1D706}\simeq 1.2570$ , $h_{\infty }/\bar{h}_{0}\simeq 0.7956$ . Linearising (3.2) about that ultimate flow state shows exponentially decaying and vertically oscillatory eigenmodes during the final stage of viscous diffusion:

(3.28) $$\begin{eqnarray}\left[\bar{\unicode[STIX]{x1D713}}-\frac{z}{h_{\infty }}-\mathop{\sum }_{n=1}^{\infty }a_{n}\text{e}^{-(n\unicode[STIX]{x03C0})^{2}x/h_{\infty }}\sin \left(\frac{n\unicode[STIX]{x03C0}z}{h_{\infty }}\right),\bar{h}-h_{\infty }\right]=O(\text{e}^{-2\unicode[STIX]{x03C0}^{2}x/h_{\infty }}),\end{eqnarray}$$

with the constants  $a_{n}$ determined by the full solution to (3.8). The remainder term in (3.28) indicates that the higher-order eigenfunctions are dominated by the omitted contribution provoked by the nonlinear self-interaction of the dominant one.

For $g=0$ , (3.25) predicts $\bar{h}_{-}\sim a_{W}Tx/(2T-1)+O(1)$ ( $x\rightarrow \infty$ ), where the specific form of this linear growth is due to the continuity of $\bar{h}_{-}$ for $T=0$ at the plate edge. For any howsoever small value of $g$ , this behaviour is superseded by $\bar{h}_{-}\sim -\bar{g}x^{2}/[2J_{0}(1-2T)]+O(\bar{g}x,Tx)$ for $\bar{g}x^{2}$ becoming large. This reveals the classical downfall parabola which the essentially inviscid free layer assumes under sufficiently small capillary action. Increasing $T$ reinforces the vertical pressure gradient and thus the downward bending of the flow, cf. (3.3). Finally, as the value of $T$ exceeds $1/2$ , this feed-forward is replaced by an intriguing counteracting one: the flow curves more and more upwards in order to sustain a corresponding negative vertical pressure gradient increasingly dominated by surface tension. We thus consider the present results based upon (3.8) and (3.25) as the ‘viscous extension’ of the early ones by Keller & Weitz (Reference Keller and Weitz1957). Accordingly, boundedness of $\bar{h}_{-}$ for $x\rightarrow \infty$ occurs for $g=T=0$ solely: with $h_{\infty }=1/J_{0}$ and $K(\infty )=1/2$ , (3.25) yields in this limit when body and surface forces are ignored

(3.29) $$\begin{eqnarray}\frac{\bar{h}_{+}^{2}-\bar{h}_{-}^{2}}{2h_{\infty }}\sim \frac{\bar{h}_{\pm }}{h_{\infty }}\mp \frac{1}{2}\sim K_{W}+\frac{2}{\unicode[STIX]{x03C0}}\mathop{\sum }_{n=1}^{\infty }a_{n}\text{e}^{-(n\unicode[STIX]{x03C0})^{2}x/h_{\infty }}\frac{1-(-1)^{n}}{n}+O(\text{e}^{-2\unicode[STIX]{x03C0}^{2}x/h_{\infty }}).\end{eqnarray}$$

The left-most term in (3.29) obviously expresses the angular momentum around the plate edge exerted by the flow infinitely far downstream. We find $\bar{h}_{-}(\infty )\simeq 0.3157$ and $\bar{h}_{+}(\infty )\simeq 1.1121$ describing the terminal upwards shift of the detached layer.

Our careful numerical evaluations perfectly confirm the above findings: see figures 2(b) and 4.

3.3.3 Goldstein’s near wake and consequences

The analysis in this section substantially resorts to (3.27), but again without the need to specify $\bar{\unicode[STIX]{x1D713}}_{0}$ to represent Watson’s flow.

The sudden loss of the no-slip BC at the edge for $x\rightarrow 0+$ typically provokes a Goldstein-type near wake, with its vertical extent controlled by the local viscous length scale $x^{1/3}$ :

(3.30a,b ) $$\begin{eqnarray}\bar{\unicode[STIX]{x1D713}}/(\unicode[STIX]{x1D706}x^{2})^{1/3}\sim f_{G}(\unicode[STIX]{x1D701})+x^{\unicode[STIX]{x1D6FC}}f_{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D701})+O(x^{2\unicode[STIX]{x1D6FC}}),\quad \unicode[STIX]{x1D701}:=z(\unicode[STIX]{x1D706}/x)^{1/3}=O(1),\quad x\rightarrow 0+,\end{eqnarray}$$

$\unicode[STIX]{x1D6FC}$ is some positive constant. The function $f_{G}$ is governed by the arising well-known leading-order problem $f_{G}^{\prime 2}-2f_{G}f_{G}^{\prime \prime }=3f_{G}^{\prime \prime \prime }$ , $f_{G}(0)=f_{G}^{\prime \prime }(0)=0$ and fixed uniquely by setting $f_{G}^{\prime \prime }(\infty )$ equal to $1$ by matching. This yields $f_{G}^{\prime }\sim \bar{\unicode[STIX]{x1D701}}+\,\text{transcendentally small terms}$ (TST) for $\bar{\unicode[STIX]{x1D701}}:=\unicode[STIX]{x1D701}+a_{G}\rightarrow \infty$ ; our numerical computation of $f_{G}(\unicode[STIX]{x1D701})$ gives $a_{G}\simeq 0.8920$ . Hence,

(3.31a-c ) $$\begin{eqnarray}(2+3\unicode[STIX]{x1D6FC})(f_{G}^{\prime }f_{\unicode[STIX]{x1D6FC}}^{\prime }-f_{G}^{\prime \prime }f_{\unicode[STIX]{x1D6FC}})-2f_{G}f_{\unicode[STIX]{x1D6FC}}^{\prime \prime }=f_{\unicode[STIX]{x1D6FC}}^{\prime \prime \prime },\quad f_{\unicode[STIX]{x1D6FC}}(0)=f_{\unicode[STIX]{x1D6FC}}^{\prime \prime }(0)=0,\quad f_{\unicode[STIX]{x1D6FC}}^{\prime }(0)\neq 0,\end{eqnarray}$$

and $f_{\unicode[STIX]{x1D6FC}}=f_{G}^{\prime }$ satisfies only the first of these relationships. Thus, $f_{\unicode[STIX]{x1D6FC}}\sim \unicode[STIX]{x1D6FD}_{1}\bar{\unicode[STIX]{x1D701}}^{3\unicode[STIX]{x1D6FC}+2}+\unicode[STIX]{x1D6FD}_{2}\bar{\unicode[STIX]{x1D701}}+\text{TST}$ with some constants $\unicode[STIX]{x1D6FD}_{1,2}$ holds for $\bar{\unicode[STIX]{x1D701}}\rightarrow \infty$ . Rewriting (3.30) in terms of $x$ , $z$ yields $\unicode[STIX]{x1D6FC}=1$ , $\unicode[STIX]{x1D6FD}_{1}=\unicode[STIX]{x1D6FD}/\unicode[STIX]{x1D706}^{2}$ by (3.27) ( $\unicode[STIX]{x1D6FD}_{1}=-1/60$ for Watson’s flow) via matching with $\bar{\unicode[STIX]{x1D713}}\sim \bar{\unicode[STIX]{x1D713}}_{0}$ ( $z\gg x^{1/3}$ ), and with (2.2d ) finally the sought irregular extension of the expansion (3.26) holding for $x\rightarrow 0+$ as $z=O(1)$ is found:

(3.32) $$\begin{eqnarray}\displaystyle [\bar{\unicode[STIX]{x1D713}}-\bar{\unicode[STIX]{x1D713}}_{0},\bar{h}-\bar{h}_{0}] & {\sim} & \displaystyle ax^{1/3}[\bar{\unicode[STIX]{x1D713}}_{0}^{\prime },-1]+a^{2}x^{2/3}[\bar{\unicode[STIX]{x1D713}}_{0}^{\prime \prime }/2,0]+x[a^{3}\bar{\unicode[STIX]{x1D713}}_{0}^{\prime \prime \prime }/6+\bar{\unicode[STIX]{x1D713}}_{1}(z),\bar{h}_{1}]\nonumber\\ \displaystyle & & \displaystyle +\,O(x^{4/3}),\quad a:=a_{G}/\unicode[STIX]{x1D706}^{1/3}\simeq 1.0079,\end{eqnarray}$$

with that numerical value valid for Watson’s flow. The detaching flow stays intact immediately downstream of the edge apart from being slightly perturbed by a local eigensolution $x^{1/3}\bar{\unicode[STIX]{x1D713}}^{\prime }$ of (3.2a ) as in (3.14a ); this inviscid-flow disturbance is triggered by the displacement of Goldstein’s wake forming underneath.

From (3.30) and (3.32), the dominant contribution to $z=\bar{h}_{-}(x)$ in (3.25) as $x\rightarrow 0+$ originates in the core layer $z=O(1)$ . The layer responds to the capillary jumps of the pressure and its cross-flow variation by streamline curvature in terms of a cuspidal bending, confirmed by the numerical results shown in figure 4(b) and visualised in figure 5(a):

(3.33) $$\begin{eqnarray}[\bar{h}_{-},\bar{h}_{+}]\sim [1-T,T]ax^{1/3}/(1-2T)+O(x^{2/3}),\end{eqnarray}$$

with $T$ defined in (3.16c ). In addition to the case $T=1/2$ , we also discard the further physically exceptional, degenerate one $T=1$ , which marks the onset of capillary undulations above the plate (§ 3.2.2). The layer contracts, i.e. accelerates, almost independently of $T$ in agreement with (3.32) and the numerical results (figure 2 b) as free slip supersedes no slip at its base. For $T>0$ , the inverse Prandtl shift produces additional irregular terms in the core region for $x\rightarrow 0+$ , defining a cuspidal distortion of $h_{+}$ originating from the near wake. The hydrostatically caused $O(x^{2})$ -contribution to $\bar{h}_{-}$ in (3.3) arises from both $f_{G}$ for $z=O(x^{1/3})$ and the regular part of (3.32) for $z=O(1)$ .

Figure 5.

Goldstein’s wake; (a) local distributions of $\bar{h}_{-}^{\prime }/(ax^{1/3})$ (solid), $\bar{h}_{+}/(ax^{1/3})$ (dashed) versus $T$ with zeros and asymptotes (dotted) as given by (3.33); (b)  $F_{G}^{\prime }(\unicode[STIX]{x1D701})$ (solid), $f_{G}^{\prime }(\unicode[STIX]{x1D701})$ (dashed), median as asymptote $F_{G}^{\prime }\sim \unicode[STIX]{x1D701}$ (dotted).

Exploiting (3.3) and (3.33) gives for $x\rightarrow 0+$

(3.34a,b ) $$\begin{eqnarray}\bar{p}-\bar{p}(0,z)\sim \frac{T}{2T-1}\left(\frac{J(z)}{J(0)}-T\right)\frac{2a}{9}x^{-5/3}+O(x^{-4/3}),\quad J(z):=\int _{z}^{\bar{h}_{0}}\bar{\unicode[STIX]{x1D713}}_{0}^{\prime 2}(y)\,\text{d}t,\end{eqnarray}$$

so henceforth, $J(0)=J_{0}$ . In order to estimate $\bar{p}$ in the near wake, the integral in (3.3) has to be split asymptotically. With the aid of (3.7) and with $(\ast )$ abbreviating the integrand in (3.3) of $O(x^{-1})$ , one obtains $\bar{p}-\unicode[STIX]{x1D70F}\bar{h}_{-}^{\prime \prime }\sim \int _{0}^{z}(\ast )\,\text{d}t-\bar{\unicode[STIX]{x1D713}}_{zx}-\bar{\unicode[STIX]{x1D713}}_{zx,-}+\bar{\unicode[STIX]{x1D713}}_{zz}\bar{h}_{-}^{\prime \prime }=O(x^{-2/3})$ for $z=O(x^{1/3})$ . As Watson’s flow having $\text{d}\bar{p}_{p}/\text{d}x\equiv 0$ provides the regular extension of (3.26), regular pressure terms contribute to (3.30) and in turn induce contributions to (3.32) at higher orders only.

It is seen that gravity has no essential impact on the behaviour of the just detaching film given the magnitude of the Froude numbers considered. However, we find the following two sources of non-uniformity of the above expansions.

1. (I) In the case $\unicode[STIX]{x1D70F}=0$ , we conclude that in the parabolic slender-flow limit Watson’s classical flow is abruptly superseded by Goldstein’s one at the trailing edge. The irregular shape of the detaching streamline reflects the absence of any upstream influence. However, in the region $x=O(\unicode[STIX]{x1D716}^{3/2})$ , $y=O(\unicode[STIX]{x1D716}^{1/2})$ encompassing the edge and where $[\unicode[STIX]{x1D713},p,h_{-}]=O(\unicode[STIX]{x1D716})$ the slender-layer approximation anticipated in (2.2c,d ) ceases to be valid, and all terms in (2.2), apart from $-g$ in (2.2b ), are retained to leading order. Most important, the solution to this full NS problem is expected to facilitate a smooth transition from no-slip to free-slip, according to (2.2c ). This then completes the current flow description (§ 4.1).

2. (II) For any howsoever small values of $\unicode[STIX]{x1D70F}$ , even the upper streamline exhibits an unacceptable cusp for $x\rightarrow 0+$ , which cannot be smoothed out by the aforementioned NS region. We observe that $\bar{p}=O(\unicode[STIX]{x1D70F}x^{-5/3})$ in the near wake for $x\ll \unicode[STIX]{x1D70F}$ and the magnitude of $\bar{p}_{p}$ immediately upstream of the edge can only be increased accordingly by the action of viscous forces. Consequently, the required smoothing of the flow quantities in the vicinities of both the edge as well the upper free streamline is typically accomplished by viscous–inviscid interaction due to streamline curvature, as identified in § 3.2.2: on a reduced streamwise length scale of $O(\unicode[STIX]{x1D716}^{6/7})$ between the flow in the lower deck (LD), continued as the near wake further downstream, where $z=O(\unicode[STIX]{x1D716}^{2/7})$ and the predominantly inviscid one in the main deck (MD) comprising the core region. As shown next, this loss of parabolicity, with the slender-layer approximation staying intact first, completes the flow picture for $p$ – and thus $g\bar{h}$ – becoming of $O(\unicode[STIX]{x1D716}^{4/7})$ . So $g$ is increased above $O(\unicode[STIX]{x1D716}^{2})$ and the distinguished limit (3.4) relaxed accordingly.

4.2.1 Solution strategy

Downstream integration was performed with the same method, routine and precision as in § 3.3. To this end, (4.4a ) together with (4.8) are formulated as a system of first-order equations with respect to $X_{s}$ . We prescribed the second of the conditions (4.4c ) at a sufficiently large maximum value of $Z$ . A sufficiently small (and for $T<1$ negative) initial value $P^{i}$ , say, of the difference pressure $P-G$ triggers the exponential departure, where $X_{s}$ takes on some correspondingly large negative value $X_{s}^{i}$ , say. Noting that (4.10) implies $-3^{2/3}/[\unicode[STIX]{x1D707}^{1/3}\unicode[STIX]{x0393}(1/3)]=S\unicode[STIX]{x1D707}^{2}$ and taking the explicit expression for $P$ in (4.9) as real gives in combination with (4.1) and (4.13) $\exp (\unicode[STIX]{x1D707}X_{s}^{i}+\text{i}\unicode[STIX]{x1D719})=P^{i}/(2S\unicode[STIX]{x1D707}^{2})$ . Then the initial kick for $\unicode[STIX]{x1D6F9}$ is readily calculated upon substitution of the exponential and integration. Moreover, this representation of $P^{i}$ yields

(4.19a,b ) $$\begin{eqnarray}X_{s}^{i}=\ln |P^{i}/(2\unicode[STIX]{x1D707}^{2})|/\text{Re}\unicode[STIX]{x1D707},\quad \unicode[STIX]{x1D719}=\arg (P^{i}/S)-2\arg \unicode[STIX]{x1D707}-X_{s}^{i}\text{Im}\,\unicode[STIX]{x1D707}\quad \text{mod}\;2\unicode[STIX]{x03C0}\end{eqnarray}$$

to determine $\unicode[STIX]{x1D719}$ as a function of $P^{i}$ in the oscillatory case $T>1$ . Here implementing the classical FLARE technique (Reyhner & Flügge-Lotz Reference Reyhner and Flügge-Lotz1968) helped to stabilise downstream integration when it comes to flow reversal.

The remaining details of the adopted discretisation differ in the two distinct cases and are therefore specified below. The thereby obtained results are not significantly altered by any substantial refinement of the resolution, as became evident through rather elaborate numerical tests. Consequently, those were accepted as essentially resolution-independent and sufficiently accurate. The solution is considered on consecutive $X_{s}$ -steps of 0.05.

4.2.2 Canonical expansive case: $T<1$

As outlined before, in this gravity-dominated case only the expansive branching (4.9) enables the local acceleration of the flow towards the edge required by $P=0$ there. According to (4.13), this together with $\unicode[STIX]{x1D6E4}_{0}$ is found by a single numerical downstream sweep over $X_{s}>-\infty$ . It yields $P-G<0$ as a strictly monotonically decreasing function of $X_{s}$ solely. In turn, the terminal value of $X_{s}$ indicating the plate edge ( $X=0$ ) and, by (4.12), $\unicode[STIX]{x1D6E4}_{0}$ are strictly monotonically increasing functions of $G$ . We then write $\unicode[STIX]{x1D6E4}_{0}(G)$ with $\unicode[STIX]{x1D6E4}_{0}(0)=0$ , tied in with the trivial solution $\unicode[STIX]{x1D6F9}-Z\equiv P\equiv H_{+}\equiv 0$ , and expect some maximum value $\unicode[STIX]{x1D6E4}_{0}(\infty )=\exp (\unicode[STIX]{x1D707}X_{s}^{t})$ where the position of the intrinsic expansive breakdown described by (4.14) has approached $X=0$ from above. Only values of $\unicode[STIX]{x1D6E4}_{0}$ less than this prove meaningful within the framework of the current theory.

It proved sufficient to choose $\max (Z)=10$ and an initial uniform step size in the $Z$ -direction of $0.05$ . With the satisfactorily small value $10^{-4}$ of $-P^{i}$ , substituting (3.24) into (4.19) yields $X_{s}^{i}=\ln [-P^{i}/(2\unicode[STIX]{x1D707}^{2})]/\unicode[STIX]{x1D707}\simeq -10.7960$ . The key results are displayed in figure 6. By (4.12), only the portions of the plots left of $X_{s}<\ln \unicode[STIX]{x1D6E4}_{0}/\unicode[STIX]{x1D707}$ , where $P=0$ , are relevant and thus fix $\unicode[STIX]{x1D6E4}_{0}$ for a given value of $G$ . The near collapse of the data in figure 6(b) supports the numerical attainment of (4.14) with highly satisfactory accuracy. Linear interpolation (by a least-squares fit) of a few last reliable data points allows for their extrapolation. This yields estimates for the difference between the very last reliable value of $X_{s}\simeq 2.1765$ and $X_{s}^{t}$ : $0.5973$ ( $\unicode[STIX]{x1D6F4}$ ) $0.6085$ ( $G-P$ ) $0.6057$ ( $A$ ). Adopting their arithmetic mean $\simeq 0.6038$ gives $X_{s}^{t}\simeq 2.008$ as our most confident estimate and $\unicode[STIX]{x1D6E4}_{0}(\infty )\simeq 6.06$ with some reservation, in view of the obvious numerical uncertainties.

Figure 6.

Case $T<1$ ; (a)  $\unicode[STIX]{x1D6F4}$ (solid), $G-P$ (dashed), $A$ (dot-dashed) versus $X_{s}$ ; (b)  $-\unicode[STIX]{x0394}X$ calculated from (4.14) versus $\unicode[STIX]{x0394}X$ by extrapolation: data points for $\unicode[STIX]{x1D6F4}$ (▫), $G-P$ (○), $A$  (▵) connected by straight lines (dotted), equality of the two forms for $\unicode[STIX]{x0394}X$ (solid).

4.2.3 An example case of stationary capillary waves: $T>1$

In this capillarity-controlled case, the aforementioned initial pressure kick is associated with a potentially non-zero $\arg \unicode[STIX]{x1D6E4}_{0}=\arg \unicode[STIX]{x1D6FE}_{0}$ according to (4.11). The determination of the appropriate value of this phase poses a still-open question: so far, it is neither answered by the generation of the small-amplitude precursor waves at the upstream bound $x=x_{T}$ of their longitudinal adjustment over a length of $O(1)$ , addressed in § 3.2.2, (i), nor, on the other hand, fixed by the upstream influence on the nonlinear regime, provided by the downstream condition $P(0)=0$ . For both the pressure kick $P^{i}$ and the phase shift $\unicode[STIX]{x1D719}$ prescribed, here (4.9), (4.10) and (4.13) yield the actual value of $X_{s}^{i}$ as the most negative root of $\exp (X_{s}^{i}\text{Re}\,\unicode[STIX]{x1D707})\cos (X_{s}^{i}\text{Im}\,\unicode[STIX]{x1D707}+\unicode[STIX]{x03C0}/7+\unicode[STIX]{x1D719})=-P^{i}/(2|\unicode[STIX]{x1D707}|^{2})$ . Accordingly, for a given $\arg \unicode[STIX]{x1D6E4}_{0}$ , $|\unicode[STIX]{x1D6E4}_{0}|$ is, in principle and in contrast to the strictly monotonic case $T<1$ , expected to be fixed by an iterative procedure. Its three basic steps are as follows. For some educated guess of $|\unicode[STIX]{x1D6E4}_{0}|$ and hence of $\unicode[STIX]{x1D719}$ , compute an estimate for $X_{s}^{i}$ ; solve the marching problem; apply the edge condition to the so-obtained values of $P-G$ to find an updated estimate for $|\unicode[STIX]{x1D6E4}_{0}|$ via (4.12).

Figure 7.

Case $T>1$ ‘kicked’ by (4.19); (a)  $\unicode[STIX]{x1D6F4}$ (solid), $P-G$ (dashed), $A$ (dot-dashed) versus $X_{s}$ ; (b) streamlines with $\unicode[STIX]{x1D6F9}$ -values (B-spline approximation of computed data points, separation bubble: $\min (\unicode[STIX]{x1D6F9})\simeq \unicode[STIX]{x1D6F9}(2.35,1.6)\simeq -0.067835$ ); (c) velocity profiles for $X_{s}\simeq -60.00$ (first $X_{s}$ -step, dot-dot-dot-dashed), $0.05$ , (pre-separation, dot-dot-dashed), $2.40$ (minimum velocity in closed separation bubble, dot-dashed), $3.85$ (reattachment, dashed), $11.50$ (terminal profile, solid); (d) as in (c) for massive separation, $X_{s}\simeq 7.20$ (separation, dot-dot-dashed), $X_{s}\simeq 7.50+0.5i$ ( $i=0,1,\ldots ,8$ ).

Given our, for the time being, incomplete understanding of which effect determines $\arg \unicode[STIX]{x1D6E4}_{0}$ , we used (4.19) to showcase satisfactorily accurate results, which exemplify the essential qualitative traits of the solutions: figure 7. The onset of the oscillations far upstream requires a much smaller value of $-P^{i}$ than for $T<1$ to achieve their satisfactory mesh-independence. Hence, we chose $P^{i}=-10^{-5}$ so that (4.19) yields $X_{s}^{i}\simeq -60.0498$ and, with $\arg (P^{i}/S)=-\unicode[STIX]{x03C0}$ , $\unicode[STIX]{x1D719}\simeq -2.4444$ . Further, preliminary, numerical evidence demonstrates a strong quantitative but comparatively weak qualitative dependence of the solution on $\unicode[STIX]{x1D719}$ insofar as this is reproduced well in the nonlinear interactive regime. This illustrates the oscillation of the flow quantities increasing in the streamwise direction and a maintained phase lock between $A$ and $P$ , finally leading to a closed separation bubble and massive re-separation. However, the flow does not recover from the latter but forms the already mentioned negative plateau of the excess pressure $P-G$ . The computations then inevitably terminate (here at $X_{s}\simeq 11.50$ ), however, as $A\sim (P-G)X_{s}^{2}/2$ with $X_{s}$ becoming arbitrarily large in this regime. Expectations are high that a, numerically costly, mapping of $Z$ (and $X$ ) onto finite computational domains promote further advancement with required accuracy into the grossly separated flow in a future study. To deal with separation here, an initially non-uniform grid, with a densification proportional to $Z^{2}$ as $Z$ becomes sufficiently small, was adopted. The regular behaviour $\unicode[STIX]{x1D6F9}\sim \unicode[STIX]{x1D6F4}Z^{2}/2+P^{\prime }Z^{3}/6+O(\unicode[STIX]{x1D6F4}^{2}Z^{5})$ ( $Z\ll 1$ ) stays intact where $\unicode[STIX]{x1D6F4}$ changes sign.

In order to cope with the unbounded decrease of $A$ and the attainment of the terminal reverse-flow region in the LD, the choice $\max (Z)=80$ proved sufficient to generate an essentially discretisation-independent solution. The associated far-field structure of the LD has not been considered in detail so far as the second separation is merely considered for the sake of completeness in the present study: potential values of $X_{s}$ determining the plate edge via (4.12) not only require $P=0$ but simultaneously unseparated flow there (we have tacitly stipulated $\unicode[STIX]{x1D6EC}>0$ above). By these requirements, multi-valued results ensue, potentially existing only for a restricted range of values of $G$ (cf. figure 7 a). As a most exciting feature, however, this implies the possibility of pre-separation upstream of the edge.

The possibility of describing reattachment by a free-interaction problem deserves some additional discussion; all the more, as here the final word has not been spoken yet. As long as the weak ellipticity is introduced by a local and linear interaction law, a feed-forward process decreases the wall shear further. In addition, marching downstream is typically terminated (by an intrinsic ill-posedness and instantaneous branching) once the wall shear rate exceeds some sufficiently negative value. The associated sudden generation of a family of eigensolutions of the locally linearised marching operator is usually interpreted as an insurmountable upstream influence, commonly suppressed by a proper truncation of the inertia terms in numerical upwind schemes, as we have adopted here. Noteworthy exceptions reported are local interactions laws nonlinear in $P$ that allow for a non-trivial saturation far downstream. Here the asymptotically weak hydraulic jump in an underdeveloped single-layer flow, hence described by transcritical tripe-deck interaction (Kluwick et al. Reference Kluwick, Cox, Exner and Grinschgl2010), represents the pioneering and, in the present context, most prominent such example. Avoiding spurious reattachment is definitely of importance as the oscillations in $P$ subject to $P(0)=0$ entail a complex non-uniqueness in the determination of  $\unicode[STIX]{x1D6E4}_{0}$ .

4.3.1 Weak surface tension

By (4.18), $H_{+}=-A(0)+H_{1}X-GX^{2}/2$ characterises the singular case $T=0$ of a non-interactive downstream part of the problem ( $P=0$ ). Here the replacements $[x,z,\unicode[STIX]{x1D706},\bar{\unicode[STIX]{x1D713}},\bar{h}]\mapsto [X,Z,\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D6F9},H]$ (subscripts and the accordingly altered value of $a_{G}$ omitted) recover (3.30), (3.32), (3.33) and thus the aforementioned Goldstein-type ‘near-near’ wake; note (4.3). Far downstream, (4.4) reproduces the near wake in § 3.3.3: $[\unicode[STIX]{x1D6F9},A]\sim [X^{2/3}f(\unicode[STIX]{x1D701}),a_{G}X^{1/3}]$ , here $\unicode[STIX]{x1D701}=Z/X^{1/3}$ . Their collapse in the trivial case $G=T=0$ , thus $\unicode[STIX]{x1D6F9}-Z=P=0$ ( $X\leqslant 0$ ), $H_{+}\equiv 0$ , and finally $A=a_{G}X^{1/3}\unicode[STIX]{x1D703}(X)\equiv H_{-}$ confirms the flow structure outlined in § 3.3.3, (I). Inspection of (4.18) and (3.33) and also $\bar{P}$ in (4.7) for $T>0$ and (3.34) indicates the overlap of the interactive- and the non-interactive-flow regimes, with the downfall parabola predominating in the expressions for $h_{\pm }(x)$ as $G=O(1)$ ; see (3.25). These considerations complete the theory for $T\lesssim \unicode[STIX]{x1D716}^{12/7}$ , where the NS region covers the intermediate domain (4.16).