2.2.1. Main deck

Since the MD describes a predominantly inviscid flow in the long-wave limit, the central local expansion reads as

(2.11)\begin{align} \left[\psi,h,h_-,h_+\right] &\sim \left[\psi_0(z),h_0,0,h_0\right]+ \epsilon^{2/7}m\left[A(X)\,\psi_0'(z),-A(X),H_-(X),H_+(X)\right] \nonumber\\ &\quad + O(\epsilon^{4/7}),\quad H_+:= H_--A,\quad m:=(M/\lambda^{4})^{1/7},\quad z:= y-h_-(x), \end{align}

and ${p=O(\epsilon ^{4/7})}$. The local streamwise variable ${X=O(1)}$ is defined in (2.13a–c) below. The expansion (2.11) induces the following hierarchy of equations resulting from the Euler operator in (2.3a,b). The dominant viscous displacement exerted by the LD, ${-A(X)}$, generates typically the dominant perturbation of $\psi$ about $\psi _0$ in terms of the pressure-free eigensolution of the linearised streamwise momentum equation (2.3a), where we have conveniently introduced the Prandtl transposition. Entering (2.3b), this $O(\epsilon ^{2/7})$-contribution to $\psi$ governs streamline curvature and, by virtue of integration with respect to $y$, supplements the hydrostatic portion of $p$ with the convective one, also of $O(\epsilon ^{4/7})$. The disturbances described so far account for the role of the MD for the interactive mechanism. The $O(\epsilon ^{4/7})$-contributions to $p$ and to $\psi$, the latter induced subsequently by the streamwise pressure gradient, are specified in Reference Scheichl, Bowles and PasiasSBP18.

2.2.2. Lower deck

In the LD the expansion

(2.12)\begin{equation} [\psi,p]/\epsilon^{4/7}\sim\left[(M^{2}/\lambda)^{1/7}\,\varPsi(X,Z),(M^{2}\lambda^{6})^{1/7}P(X)\right]+ O(\epsilon^{6/7}) \end{equation}

employs the stretched coordinates

(2.13a–c)\begin{equation} X:= x l/\epsilon^{6/7},\quad (Y,Z):=(y,z)/(\epsilon^{2/7}m),\quad l:=(\lambda^{5}/M^{3})^{1/7}. \end{equation}

To describe the flow up- and downstream of the plate edge, the variable $Z$ is preferred over $Y$ in the slender LD. In turn, (2.3a,b) reduce locally to the boundary layer equation

(2.14a)\begin{equation} \varPsi_Z\varPsi_{ZX}-\varPsi_X\varPsi_{ZZ}={-}P'+\varPsi_{ZZZ}, \end{equation}

and (2.3c,d) to the mixed BCs expressing the downstream passage from no- to free-slip along

(2.14b)\begin{equation} Z=0\colon\quad \varPsi=\varPsi_Z\theta({-}X)=\varPsi_{ZZ}\theta(X)=0. \end{equation}

To match (2.12) and (2.11) subject to (2.5a,b), we require that for

(2.14c)\begin{equation} Z{\rightarrow\infty}\colon\quad \varPsi\sim[Z+A(X)]^{2}/2+[P(X)-G+\text{TST}]. \end{equation}

The rightmost bracketed contribution herein is a consequence of (2.14a) and that the interactive flow branches off the unperturbed state given by ${[\varPsi ,P]\equiv [Z^{2}/2,G]}$ infinitely far upstream; TST means transcendentally small terms.

Relating the displacement function $A$ to $P$ closes the interactive feedback loop and the weakly elliptic free-interaction problem. For ${X<0}$, that relationship is given by the jet-type interaction law ${P-G={\textrm {sgn}}(T-1)(A''-H_-'')}$, typically provoked by the streamline curvature in the MD (as introduced by Smith (Reference Smith1977), Smith & Duck (Reference Smith and Duck1977) and, for an unconfined wall jet passing an abrupt edge, Smith Reference Smith1978) and the (counteracting) capillary pressure jump across the uppermost streamline. For ${X>0}$, one eliminates $H_-$ from the interaction law via the representation of $P$ in terms of the pressure jump across the lowermost streamline to which (2.3e) reduces,

(2.14d)\begin{equation} {\rm \Delta} P\theta(X)=TH_-''/|T-1|,\quad {\rm \Delta} P:= P-P_-. \end{equation}

(in Reference Scheichl, Bowles and PasiasSBP18 only the case ${P_- =0}$ was considered). We thus arrive at the $P$/$A$ law in the form

(2.14e)\begin{gather} {\rm \Delta} P=C(T)(G+SA''-P_-),\quad S:={\textrm{sgn}}(T-1), \end{gather}

(2.14f)\begin{gather}C(T):=\left\{\begin{array}{@{}cl} 1 & (X\leq~0), \\ T/(2T-1) & (X>0). \end{array}\right. \end{gather}

We furthermore introduce ${D(T)=1-C(T)}$. The upstream case (${X\leq ~0}$) is included in this interaction law for the sake of completeness and clarity. Downstream of the edge, it accounts for a subtle interplay of capillarity with inertia; the functions C and D plotted in figure 4 are consulted tacitly from here on. The pole of $C$ points to an interesting local increase of the capillary action for ${T\sim 1/2}$. The passage of $T$ over this threshold (where surface tension exactly compensates the streamwise momentum of the pressure-free base flow) is associated with an unbounded increase of $P$ and $H$ over $A$ and implies the onset of condensed interaction, which causes a breakdown of the existing flow description for the free jet. This requires the introduction of a streamwise scale relatively short as compared with the stretched interactive one and can be interpreted as choking of a capillary wave. A second critical value ${T=1}$ (${S=0}$) describes the cancelling of the counteracting effects of streamline curvature and capillarity on the transverse momentum transfer. Both are subsumed by $A''$ and, thus, actually originate in the viscous forcing of the LD. The absence of their net influence hampers the interaction pressure from becoming effective, where $H_-$ remains unspecified according to (2.14d), unless $A$ grows significantly to allow for a proper regularisation over a suitably shortened scale. Both exceptional situations are skated over below (§ 2.2.3) and still the subject of ongoing investigations.

Figure 4.

Plots of $C(T)$ (solid) and $D(T)$ (dashed) by (2.14f) (${X>0}$) with their asymptote and poles (all dotted), fixed point and zeros (all as circles).

The rescaled shear stress exerted at the plate, ${\varLambda (X):=\varPsi _{ZZ}(X,0)}$, plays a crucial role for the (unambiguous) formulation of the initial conditions (ICs) imposed at the plate edge ${X=0}$ by Reference Scheichl, Bowles and PasiasSBP18 for the detached flow, controlling its upstream influence on the plate-bounded flow in a unique manner. The detailed rationale underlying these deserves to be clarified in terms of the following three steps.

1. (i) The two original demands on the interaction mechanism were the simultaneous continuous approach of the overall pressure jump across the layer towards $-P_-$ and of $\varLambda$ towards zero in the limit ${X {\rightarrow 0}-}$, but only the first of these typical edge conditions can be met.

2. (ii) If

   (2.14g)\begin{equation} \epsilon^{12/7}\ll T<1\quad (S={-}1,\ T\neq~1/2), \end{equation}
   which is the case pursued here, the conditions the flow has to meet at the edge can then be formulated without resorting to the analysis of smaller regions enclosing the edge.

3. (iii) Then a least-degenerate flow description that allows for a smooth gradual transition from attachment to detachment of the flow quantities on smaller streamwise scales requires continuity of $\varPsi$ and $A'$ above the edge.

The sought quantities $\varPsi$ and $P$ satisfy the, with respect to $X$, first- and second-order equations (2.14a) and (2.14e). In turn, three ICs are required to continue marching over the edge,

(2.14h)\begin{equation} \varPsi_0:=\varPsi(0+,Z)=\varPsi(0-,Z)\enspace (Z>0),\quad T[A'(0+)-A'(0-)]=0,\quad P(0)=P_- \end{equation}

(or, equivalently, ${A''(0)=-SG}$). These complete the interaction problem (2.14) for the free jet. Here the flow profile at detachment $\varPsi (0-,Z)$ and $A'(0-)$ are taken as obtained by the preceding sweep of numerical marching towards the edge. It is stressed that $\varPsi$, $P$ behave regularly as ${X {\rightarrow 0}-}$. Moreover, these quantities are continuous across the edge except for the shear stress $\varPsi _{ZZ}$ on ${Z=0}$, owing to (2.14b).

We also recall the behaviour, inferred from (2.14a,b), for

(2.15)\begin{equation} Z{\rightarrow 0}\colon \varPsi\sim\left\{\begin{array}{@{}lc} \varLambda(X)Z^{2}/2+P'(X)Z^{3}/6+O(Z^{5}) & (X\leq~0), \\ {U_{s}}(X)Z+[P'+{U_{s}}{U_{s}}'](X)Z^{3}/6+O(Z^{5}) & (X>0). \end{array}\right. \end{equation}

Hence, the finite slip emerging along the lower free streamline, ${U_{s}}$, supersedes the finite plate stress $\varLambda$ upstream of the edge. We note that (2.15) first implies that

(2.16a,b)\begin{equation} \varPsi_0\sim\varLambda_0 Z^{2}/2+P'(0-)Z^{3}/6+O(Z^{5})\quad (Z{\rightarrow 0}),\quad \varLambda_0(G,T):=\varLambda(0-). \end{equation}

The apparent non-uniformity of (2.16a,b) for ${X=0+}$ is the topic of § 3.2 below. The parameters $G$ and $P_-$, representing the freely chosen support pressure, enter the solution of the interaction problem only via (2.14h), i.e. $G$ in terms of the imposed momentum flux, and subsequent integration of $P'(X)$ found in the course of the marching procedure. The decoupled calculation of $H_-$ is finally provided by (2.14d). Eliminating $P$ with the aid of (2.14e) gives the alternative relation

(2.17)\begin{equation} H_-(X)=D(T)\left[A(X)-A(0-)-A'(0-)X+(G-P_-)SX^{2}/2\right], \end{equation}

i.e. ${H_-(0)=H_-'(0)=0}$. Evidently, the support pressure behaves as a body force counteracting gravity.

2.2.3. Some important aspects

To achieve the last requirement in (2.14h), the interaction is initiated in the limit ${X {\rightarrow -\infty }}$ by a controlled branching from the oncoming base flow, here maintained as the trivial solution ${\varPsi \equiv Z}$ for ${X\leq ~0}$ if ${G=P_-\geq ~0}$. Hence, the case ${G>P_-}$ requires branching of expansive type as scrutinised by Reference Scheichl, Bowles and PasiasSBP18 (where ${P_-=0}$ throughout) and the opposite one ${0\leq G< P_-}$ compressive branching (unconsidered so far). However, since $A''(X)$ is the streamline curvature in the interactive limit, it becomes evident from (2.14e) that the interactive feedback loop triggers stationary capillary waves iff ${SC>0}$. Here this implies ${0< T<1/2}$ or ${T>1}$; see the preceding studies by Bowles & Smith (Reference Bowles and Smith1992) and Reference Scheichl, Bowles and PasiasSBP18 and the preliminary presentation of these interactive undulations by Scheichl, Bowles & Pasias (Reference Scheichl, Bowles and Pasias2019). Their revealing linkage to unsteady linear capillary waves is given in Appendix B.

Moreover, Reference Scheichl, Bowles and PasiasSBP18 demonstrated how the phenomenon of stationary waves up- and downstream of the edge for ${T>1}$ is associated with pre-detachment and severely violates the considerations (i)–(iii) and the notion of expansive branching. They finally disclosed non-uniqueness of the solutions due to an arbitrary phase shift far upstream, presumed fixed by an as yet missing further downstream condition. We are therefore still left with the two constraints (2.14g) in our consistent description of the flow continued downstream of the edge by dint of (2.14). The first states that not only $A(X)$ but also $A'(X)$ is continuous at ${X=0}$, so that we henceforth omit the signs in the arguments $0-$ and $0+$ of $A$, expressing one-sided limits. The second guarantees strictly forward interacting flow upstream of the edge, thus, ${\varLambda _0>0}$ in (2.16a,b). Since realistic values of $\tau$ and $J$ by (2.8) yields ${T\lesssim 10}$, assuming ${T<1}$ seems acceptable; see table 1 and the last comment in Appendix A.

However, $A$ becomes discontinuous at the edge in the limit ${T {\rightarrow 0}}$ in (2.14e) and (2.14h), implying the absence of interaction (${P'\equiv 0}$) for ${X>0}$. Here the possibility of free interaction exists but the conditions at ${X=0}$ do not provoke it even upstream of the edge in the formal limit ${G-P_-=T=0}$. Then the classical Goldstein wake (Goldstein Reference Goldstein1930) is recovered immediately downstream as the trivial solution ${[\varPsi ,P]\equiv [Z^{2}/2,G]}$, representing the oncoming base flow, applies upstream of it.

3.3.1. Preliminaries

Introductory considerations lay the foundation for the outer and inner mechanisms for the further regularisation of the HRW, as follows.

1. (a) The interactive $u$- and $p$-variations, on account of streamline curvature via the vertical pressure variation in (2.3b), are of respectively $O(\epsilon ^{2/7}X^{8/3})$ and $O(\epsilon ^{4/7}X^{2/3})$ as ${X {\rightarrow 0}+}$. They and the non-interactive $u$-perturbation in (2.11), provoked by the streamwise pressure variation through (2.3a), all become of $O(\epsilon ^{2/3})$ in the outer RS (§ 3.3.2) where, cf. (2.13a–c),

   (3.6)\begin{equation} \bar{X}:= x/\epsilon=X/(l\epsilon^{1/7})=O(1). \end{equation}

2. (b) Conversely, $v$ of $O(\epsilon ^{5/7}X^{-1/3})$ grows significantly to become comparable in size to the $u$-perturbation of ${\epsilon ^{2/7}X^{2/3}}$ across most of the LD for ${X=O(\epsilon ^{3/7})}$, i.e. in the inner RS (§ 3.3.5) where

   (3.7)\begin{equation} \hat{X}:= x/(m\epsilon^{9/7})=X/(lm\,\epsilon^{3/7})=O(1). \end{equation}
   However, as $p$ and $\psi$ of $O(\epsilon ^{2/3})$ at its base and downstream of the edge are still prescribed by the HRW, the inner RS cannot regularise the associated singularity expressed by (3.2) and (3.5). Therefore, the analysis of the inner RS is of only subordinate importance compared with that of the outer one.

3. (c) A quick justification of the expansions of the flow quantities below for both square regions relies on the relevant inviscid-flow approximation of the elliptic vorticity transport equation, obtained from elimination of the pressure in (2.3a,b),

   (3.8)\begin{equation} \psi_{yy}+\epsilon^{2}\psi_{xx}\sim{-}\varOmega(\psi):=\psi_0''\left(\psi_0^{{-}1}(\psi)\right). \end{equation}
   To express $\varOmega$ as the vorticity conserved along the streamlines, we use $\psi _0^{-1}$ to symbolise the inversion of the corresponding leading-order relationship ${\psi \sim \psi _0(y)}$. As a consequence, the contributions to those expansions are triggered by the vorticity imposed by the surrounding interactive flow and, in addition, the vorticity produced by the HRW and entering via non-trivial matching or BCs. These are provided by (3.5) with (3.2) for ${Z {\rightarrow \infty }}$ at the base of the outer RS and on top of the inner RS and by matching (3.5) for ${Z {\rightarrow 0}}$ and (3.1a,b) at the base of the latter. Consequently, eigensolutions of the linearised operator in (3.8) are absent.

It is noteworthy that Stewartson (Reference Stewartson1968) discovered the generic advent of a linearised Euler or Rayleigh stage when he solved the (non-rigorous) Oseen approximation of the NS problem governing the unconfined flow in a small region around a trailing edge, and prior to the far-reaching rigorous appreciation of viscous–inviscid interaction on larger scales (Stewartson Reference Stewartson1969; Messiter Reference Messiter1970).

3.3.2. Outer Rayleigh stage: main deck

In the outer square region, $p$ is, as in the surrounding MD, of $O(\epsilon ^{4/7})$, and the viscous terms in (2.3a,b) become formally of $O(\epsilon )$ as all remaining ones can be scaled to $O(1)$. Following the comments (a) and (c) above, substitution of (3.3) into (2.11) suggests, in this domain, the expansion ${\psi \sim \psi _0(y)+\epsilon ^{2/7}\psi _1(y)+\epsilon ^{3/7}\psi _2(y)+\epsilon ^{4/7}\psi _3(y)+O(\epsilon ^{4/6})}$. The sought functions $\psi _{1,2,3}$ satisfy the hierarchy of Rayleigh equations

(3.9a,b)\begin{equation} ({\partial}_{yy}+{\partial}_{\bar{X}\bar{X}}-\psi_0'''/\psi_0')\psi_{1,2}=0,\quad ({\partial}_{yy}+{\partial}_{\bar{X}\bar{X}}-\psi_0'''/\psi_0')\psi_3=\psi_1^{2}(\psi_0'''/\psi_0')'/(2\psi_0') \end{equation}

resulting from expanding (3.8). According to the considerations following (3.8) and the regularity of (2.11) upstream of the trailing edge, $\psi _{1,2}$ consist just of the pressure-free disturbances given by the Taylor series of $A(X)$ up to second order, where ${A''(0-)=G-P_-}$ from (2.14e) subject to (2.14h). This and the inhomogeneity in the last equation in (3.9a,b), caused by the inertia-based nonlinearities, require an additional $y$-dependent component of $\psi _3$.

Specifying these findings gives

(3.10)\begin{align} \left[\psi,h_+\right] &\sim \left[\psi_0(y),h_0\right] \nonumber\\ &\quad + m\left[\epsilon^{2/7}A(0)+\epsilon^{3/7}A'(0)l\bar{X}+\epsilon^{4/7}(G-P_-)(l\bar{X})^{2}/2\right] \left[\psi_0'(y),-1\right] \nonumber\\ &\quad + \epsilon^{4/7}m\left[\psi_\ast(y),-\psi_\ast(h_0)/u_0^{+}\right]+ \epsilon^{4/6}\left[\bar{\varPsi}(\bar{X},y),\bar{H}(\bar{X})\right]+O(\epsilon^{5/7}). \end{align}

Hence, $\psi _\ast$ denotes the limiting value of the corresponding $O(\epsilon ^{4/7})$-contribution to the expansion (2.11) of $\psi$ in ${X=0}$. That quantity satisfies $\psi _\ast ''-(\psi _0'''/\psi _0')\psi _\ast =(\psi _0'''/\psi _0')'A(0)^{2}/2-A''(0-)\psi _0 '$, where the last inhomogeneity reflects the action of the streamwise pressure gradient. We furthermore expand

(3.11)\begin{equation} p\sim\epsilon^{4/7}m l^{2}\left[(G-P_-)\int_0^{y}\psi_0'^{2}(t)\,\text{d} t- M\left(\frac{Gy}{h_0}-P_-\right)\right] +\epsilon^{4/6}\bar{P}(\bar{X},y)+O(\epsilon^{5/7}). \end{equation}

The $X$-independent leading-order term in (3.11) is again just the dominant contribution to $p$ in the MD up- and downstream of the trailing edge (cf. Reference Scheichl, Bowles and PasiasSBP18) evaluated at ${X=0}$ and rewritten with the aid of (2.9a–c). Here the irregular terms in (3.3) play no role. It follows from inserting (3.10) into (2.3b) and integrating its thereby reduced form ${p_y\sim \epsilon ^{4/7}m l^{2}\psi _0'^{2} -g}$ subject to ${p\sim p_-}$ as ${y {\rightarrow 0}}$, to match $p$ in the LD. Moreover, (3.10) fulfils (2.3c) supplemented with (2.6a,b) and, together with (3.11), complies with the capillary pressure jump at ${y=h_+}$ in (2.3e) up to $O(\epsilon ^{4/7})$ for ${\bar {X}=O(1)}$. The $O(\epsilon ^{4/6})$-contributions to (3.10), (3.11) serve to regularise the flow quantities in the MD. As the subsequent analysis of $\bar {\varPsi }$, $\bar {H}$, $\bar {P}$ makes clear, those expansions do not contain lower-order eigenfunctions having sufficiently strong decay for ${|\bar {X}| {\rightarrow \infty }}$, consistent with (2.11).

Invoking the inverse Prandtl shift in (3.10) gives ${\psi _0(y)\sim \psi _0(z)+h_-\psi _0'(z)}$ for ${h_-=O(\epsilon ^{2/3})}$, see (3.4), and brings to mind matching $\psi$ up to $O(\epsilon ^{2/3})$ in (2.11) and also in the LD, according to (3.5) and (3.2). Furthermore, $\bar {\varPsi }$, $\bar {P}$ are seen to satisfy the linearised Euler equations

(3.12a,b)\begin{equation} \psi_0''\bar{\varPsi}_{\bar{X}}-\psi_0'\bar{\varPsi}_{y\bar{X}}=\bar{P}_{\bar{X}},\quad \psi_0'\bar{\varPsi}_{\bar{X}\bar{X}}=\bar{P}_{y}. \end{equation}

To separate the influence of the shear stress at detachment, $\varLambda _0$, effective in the LD and of a potential $\bar {X}$-independent contribution to $\bar {\varPsi }$ arising from integration of (3.12a,b) (i.e. no $O(\epsilon ^{2/3})$-contribution to $\varOmega$, cf. (3.8), in the surrounding MD), we advantageously consider the scaled vertical flow perturbation

(3.13a,b)\begin{equation} \bar{V}:={-}\bar{\varPsi}_{\bar{X}}/\bar{\varLambda},\quad \bar{\varLambda}:=~2\,{p_{{HR}}}\,\lambda^{1/3}\varLambda_0^{4/3}/3. \end{equation}

Equations (3.12a,b) yield the Rayleigh equation governing $\bar {V}$ in accordance with (3.9a,b), i.e.

(3.14a)\begin{equation} ({\partial}_{yy}+{\partial}_{\bar{X}\bar{X}}-\psi_0'''/\psi_0')\bar{V}=0. \end{equation}

Matching $\psi$ and $p$ in the outer RS and the LD with the support of (2.12) and $m$, $l$ given by (2.11), (2.13a–c) requires for

(3.14b)\begin{equation} y=0\colon\ \bar{V}={-}\theta(\bar{X})\bar{X}^{{-}1/3}.\end{equation}

Furthermore, expanding (2.3c) and (2.3e) gives

(3.14c)\begin{equation} \bar{H}={-}\bar{\varPsi}(\bar{X},h_0)/u_0^{+} \end{equation}

and ${\bar {P}(\bar {X},h_0)=-\tau \bar {H}''(\bar {X})}$, respectively. By the same token, inspection of (3.12a,b) with the help of (2.6a,b), (2.9a–c) and (3.14c) gives for

(3.14d)\begin{equation} y=h_0\colon\ u_0^{+}{}^{2}\,\bar{V}_{y}={-}TJ\bar{V}_{\bar{X}\bar{X}}, \end{equation}

i.e. the explicit dependence of $\bar {V}$ on $T$.

Also, matching (3.10), (3.11) with (2.11) and $p$ in the MD subject to (3.3) and (3.2) yields ${\bar {\varPsi } {\rightarrow 0}}$ and ${\bar {P} {\rightarrow 0}}$, thus, ${\bar {V} {\rightarrow 0}}$ and ${\bar {H} {\rightarrow 0}}$ by (3.14c), as ${\bar {X} {\rightarrow -\infty }}$. In contrast, ${\epsilon ^{2/3}\bar {\varPsi }}$ for ${\bar {X}\gg 1}$ must match the dominant singular behaviour of ${\psi -\psi _0\sim \epsilon ^{2/7}m(A-H_-)(X)\psi _0'(y)}$ for ${X\ll 1}$ as inferred from (2.11). The expansions (3.1a,b), (3.3) and ${\psi -\psi _0\sim -(9\,{p_{{HR}}}/40)\epsilon ^{2/3}m\varLambda _0^{4/3}(l\bar {X})^{8/3}}$ imply ${\bar {V}/\psi _0'(y)\sim 9\lambda \bar {X}^{5/3}/(10M)+O(\bar {X}^{-1/3})}$ (${\bar {X} {\rightarrow \infty }}$). Likewise, (3.13a,b) and (3.14c) give ${\bar {H}/\bar {\varLambda }\sim 27\lambda \bar {X}^{8/3}/(80M)+O(\bar {X}^{2/3})}$. This proves consistent with the interplay of the two free surfaces in § 3.2.

It is illuminating to demonstrate that the up- and downstream asymptotes are already intrinsic to the problem (3.14) governing $\bar {\varPsi }$ and $\bar {H}$. To this end, we consider the weakest admissible, i.e. first algebraic, decays of $\bar {V}$ for ${\bar {X}\to \pm \infty }$ with unknown dominant corresponding rates $\bar {a}_\pm (\bar {X})$, say. We obtain from (3.14a,b), using ${( {\partial }_{yy}-\psi _0'''/\psi _0')\bar {V}\equiv (\psi _0'\bar {V}_{y}-\psi _0''\bar {V})_y/\psi _0'}$ and standard methods and (2.5a,b), the long-wave approximation of $\bar {V}$,

(3.15)\begin{align} \frac{\bar{V}}{\psi_0'(y)} &\sim\bar{a}_\pm{+}\bar{a}_\pm'' \left[\bar{b}_\pm{-}\int_0^{y}\frac{\text{d} t}{\psi_0'^{2}(t)}\int_0^{t}\psi_0'^{2}(s)\,\text{d} s\right] \nonumber\\ &\quad -\frac{\lambda\theta(\bar{X})}{\bar{X}^{1/3}}\int_y^{h_0}\frac{\text{d} t}{\psi_0'^{2}(t)}+ O(\bar{a}_\pm'''',\bar{X}^{{-}7/3}), \end{align}

where $\bar {a}_\pm$ and the constants $\bar {b}_\pm$ are determined by solvability conditions of the inhomogeneous problems governing the $O(\bar {a}_\pm '')$-term and the $O(\bar {a}_\pm '''')$-term, respectively. The small-$y$ behaviour of $\psi _0$ in (2.5a,b) grants a corresponding regularity of the right-hand side of (3.15). Substitution of (3.15) into (3.14d) using (2.5a,b) and (2.6a,b) gives, after division by $u_0^{+}$, the solvability relation ${\bar {a}_\pm ''J-\lambda \theta (\bar {X})\bar {X}^{-1/3}\sim \bar {a}_\pm ''\tau }$. In the upstream case this statement can only be met in the limit ${T\to 1-}$; cf. (2.9a–c). Consequently, ${\bar {a}_-\equiv 0}$, ${\bar {b}_-=0}$, and the upstream decay is indeed exponential, although the limit of an undamped (neutral or harmonic) oscillation may also be taken into consideration and an unbounded increase of $\bar {V}$ is expected for ${T\to 1-}$. In contrast,

(3.16)\begin{equation} \bar{a}_+{=}9\lambda\bar{X}^{5/3}/[10 J(1-T)] \end{equation}

confirms the aforementioned leading-order asymptote involving $M$ defined in (2.9a–c). This shows that matching (3.10) and (2.11) requires ${T<1}$.

As a further result, (3.12a,b) yields

(3.17)\begin{equation} \bar{P}=\psi_0''\bar{\varPsi}-\psi_0'\bar{\varPsi}_{y}, \end{equation}

and ${\bar {P}\sim 3\lambda \bar {\varLambda }\bar {X}^{2/3}/2}$ (${\bar {X} {\rightarrow \infty }}$) provides the match of $p$ in the MD, according to (3.2), (3.5) and (3.14b). This and ${\bar {P}(\bar {X},0)=3\lambda \bar {\varLambda }\,\theta (\bar {X})\bar {X}^{2/3}/2}$ make evident how $\bar {\varPsi }$ and $\bar {P}$ resort to these behaviours originating in the HRW and why the inner RS is required to complete the regularisation closer to the trailing edge. Since the coefficient $\psi _0'''/\psi _0'$ in (3.14a) becomes, from (2.5a,b), ${\omega y}$ for ${y\ll 1}$, (3.14b) allows $\bar {V}$ to attain an undesired potential-flow pole in the origin, as described by the singular eigensolutions of the Laplacian ${r^{-N}\sin (N\vartheta )}$, where

(3.18a,b)\begin{equation} r:=\sqrt{\smash[b]{\bar{X}^{2}+y^{2}}}{\rightarrow 0},\quad 0\leq\vartheta:=\arctan(y/\bar{X})\leq{\rm \pi} \end{equation}

and ${N>0}$ is some integer (cf. Scheichl Reference Scheichl2014). Its occurrence has to be avoided in the further treatment of (3.14). Rather, (3.14b) and the vorticity term provoke a weaker singularity as one readily finds that

(3.19a,b)\begin{equation} \bar{V}\sim\bar{V}_{0}+\bar{c}_1 y+\bar{c}_2 xy+O(r^{8/3})\quad (r{\rightarrow 0}),\quad \bar{V}_{0}:=~2 r^{{-}1/3}\sin({\rm \pi}/3-\vartheta/3)/\sqrt{\smash[b]{3}}, \end{equation}

and (3.17) recovers the pressure induced by the HRW as ${\bar {P}=O(r^{2/3})}$. The first three contributions to $\bar {V}$ in (3.19a,b) are of potential-flow type, and the coefficients $\bar {c}_{1,2}$ of the homogeneous ones are determined by the overall solution for $\bar {V}$. The (lengthy expression of the) $O(r^{8/3})$-term in (3.19a,b) solves the Poisson problem to which (3.14a) reduces, with ${\psi _0'''\bar {V}/\psi _0'\sim \omega y\bar {V}_{0}}$ forming the inhomogeneity. The singularity described by $\bar {V}_{0}$ is pivotal in § 3.3.5 where it comes to its regularisation by the inner RS.

For what follows, we introduce the Fourier transform of a function $f(\bar {X},y)$ for complex wavenumbers $k$,

(3.20)\begin{equation} \phi\{\,f\}(k,y)=\frac{1}{2{\rm \pi}}\int_{-\infty}^{\infty} f(\bar{X},y)\,{\textrm{e}}^{-\text{i} k\bar{X}}\,\text{d}\bar{X}. \end{equation}

We first assume that $\bar {V}$ decays exponentially far upstream. Since it grows with $O(\bar {X}^{5/3})$ as $\bar {X}$ becomes large, (3.20) defines $\phi \{\bar {V}\}$ first in the open strip ${-u_1(T)< \mbox{Im}\,k<0}$, where ${-u_1}$ denotes the imaginary coordinate of the pole in the lower half-plane ${\mbox{Im}\,k\leq ~0}$ lying closest to the real axis. The analytic continuation of $\bar {V}$ into the entire $k$-plane excluding the locations of singularities is provided by the convenient decomposition

(3.21a,b)\begin{equation} \phi\{\bar{V}\}(k,y)=\mathcal{B}(k)\mathcal{V}(k,y),\quad \mathcal{B}(k):=\phi\left\{\theta(\bar{X})\bar{X}^{{-}1/3}\right\}(k)= 1/\left[\sqrt{\smash[b]{3}}\,\mathrm{\Gamma}\left({\tfrac{1}{3}}\right)(\text{i} k)^{2/3}\right]. \end{equation}

The last expression is understood in connection with a branch cut along the positive imaginary $k$-axis. Absorbing (3.14b) and accommodating the non-integer growth with $\bar {X}$ in (3.16), it captures the influence of the HRW and gives a non-trivial $\bar {V}$. Poles of $\mathcal {V}$ on the real $k$-axis allow for relaxing the original assumption of exponential decay by the inclusion of harmonic modes surviving far upstream. From (3.14) we deduce the Rayleigh equation

(3.22a)\begin{equation} ({\partial}_{yy}-k^{2}-\psi_0'''/\psi_0')\mathcal{V}=0 \end{equation}

subject to the then inhomogeneous lower and the homogeneous upper BC,

(3.22b)\begin{gather} y=0\colon\enspace \mathcal{V}={-}1, \end{gather}

(3.22c)\begin{gather}y=h_0\colon\enspace \psi_0'^{2}\mathcal{V}_y=T J k^{2}\mathcal{V}; \end{gather}

cf. (2.5a,b). The solution of the two-point boundary value problem (3.22), parametrised by $k$, facilitates the semi-analytical inversion of (3.20) so as to determine $\bar {V}$, parametrised by $\psi _0(y)$ and $T$, in an elegant manner, avoiding the abovementioned Laplacian eigensolutions; all the more, as our focus lies on $\bar {H}(\bar {X})$ given by (3.14c). For the numerical implementation of (3.22), we recall that $\psi _0$ is typically specified by Watson's (Reference Watson1964) flow profile. In turn, the properties (2.6a,b), (2.7) and the closed form of $\psi _0$ in Scheichl & Kluwick (Reference Scheichl and Kluwick2019) and the values for ${J=\lambda }$ and $u_0^{+}$ given by (2.8) are employed. Detailing the properties of (3.22), especially the behaviours of $\mathcal {V}$ for ${k {\rightarrow 0}}$ and ${|\mbox{Re}\,k| {\rightarrow \infty }}$ and the analysis of its poles, which select the discrete spectrum of $\bar {V}$ out of the continuous one (and where (3.22) does not have a solution but its homogeneous form does), is relegated to supplement B. These findings enable the representation of $\bar {V}$ in a most efficient manner as envisaged next.

The poles of $\mathcal {V}$ lie symmetrically with respect to both the real and the imaginary $k$-axes. There are a double pole at ${k=0}$, exactly two real simple poles where ${k=\pm k_u(T)}$ with ${k_u>0}$ (§ B.1) and an infinite number of simple poles lying on ${k=\pm \text {i}u_i(T)}$ (${i=1,2,\ldots }$) with ${u_i>0}$ (§ B.3). Since ${\mathcal{V}(-k,y)\equiv\mathcal {V}(k,y)}$, ${{\textrm {Res}}_{k=-k_u}(\mathcal{V})=-{\textrm {Res}}_{k=k_u}(\mathcal{V})}$ and real, and ${{\textrm {Res}}_{k=-\text {i}u_i}(\mathcal{V})={\textrm {Res}}_{k=\text {i}u_i}(\mathcal{V})}$ and imaginary. We then have

(3.23)\begin{equation} \left[\bar{V},\frac{\bar{\varPsi}}{\bar{\varLambda}}\right](\bar{X},y)=\int_\mathcal{C} \mathcal{B}(k)\mathcal{V}(k,y)\,{\textrm{e}}^{\text{i} k\bar{X}} \left[1,\frac{\text{i}}{k}\right]\text{d} k, \end{equation}

where all possible paths of integration $\mathcal {C}$ stretch from ${\mbox{Re}\,k=-\infty }$ to ${\mbox{Re}\,k=+\infty }$ and originate from one another through a continuous deformation as they divide the $k$-plane in two portions: the origin and all poles ${k=\text {i}u_i(T)}$ lie in the upper and all poles ${k=-\text {i}u_i(T)}$ in the lower part. We furthermore anticipate that both real poles are located either in the upper or the lower part to guarantee $\bar {V}$ being real. Indeed, as will be argued below to render $\bar {V}$ unique, $\mathcal {C}$ must bypass both real poles such that they lie in the lower part. This situation is sketched in figure 5 with the path $\mathcal {C}$ specified for the numerical calculation of $\bar {H}(\bar {X})$ by means of (3.23) and (3.14c) for ${\bar {X}\geq ~0}$. There the branch cut prevents a more efficient treatment of (3.23) using Cauchy's residue formula: to avoid accuracy issues associated with complex integration, we specified $\mathcal {C}$ to follow the real axis apart from small squares of lengths $2\varepsilon$ with the midpoints ${k=\pm k_u}$ and of length $\varepsilon$ with the midpoint in the origin. Consistency of the results is confirmed for values of $\varepsilon$ ranging from 0.1 to 0.3. On the other hand, applying Cauchy's residue theorem to (3.23) yields with (3.21a,b), the fact that ${{\textrm {Res}}_{k=-k_u}(\mathcal{V})=-{\textrm {Res}}_{k=k_u}(\mathcal{V})}$ and Euler's reflection formula after some algebra

(3.24)\begin{equation} \frac{\bar{\varPsi}}{\mathrm{\Gamma}\left(\tfrac{2}{3}\right)\bar{\varLambda}}=2{\textrm{Res}}_{k=k_u}(\mathcal{V}) \frac{\cos(k_u\bar{X}-{\rm \pi}/3)}{k_u^{5/3}}+ \text{i}\sum_{i=1}^{\infty}{\textrm{Res}}_{k={-}\text{i}\mu_i}(\mathcal{V}) \frac{\exp(\mu_i\bar{X})}{\mu_i^{5/3}}\quad (\bar{X}\leq~0) \end{equation}

(cf. Tillett Reference Tillett1968). This series of residues converges (uniformly) for any ${\bar {X}<0}$. The full evaluation of (3.23) and smoothness of $\bar {\varPsi }$ for ${y>0}$ in ${\bar {X}=0}$ confirms that (3.24) holds even there although the decay of the exponentials has disappeared.

Figure 5.

Sketch of $k$-plane: double-symmetric singular points (circles), actual path $\mathcal {C}$ and direction of integration.

Finally, $\bar {H}/\bar {\varLambda }$ for ${\bar {X}\leq ~0}$ follows from (3.14c) and directly from (3.24) in a convenient manner. This approach allows us to check the accuracy of the full integration according to (3.23). It is definitely preferred for resolving most accurately the novel discrete undamped capillary Rayleigh modes, forming a wave crest upstream of the edge. These are revealed, as arising from the real poles, with wavenumbers ${k=k_u}$, found to strictly increase as $T$ decreases. Here we point to the classical dispersion relation of small-amplitude capillary waves in a finite-depth layer of uniform parallel flow with uniform speed scaled to unity over a flat bed (see Drazin & Reid Reference Drazin and Reid2004, p. 30 and Vanden-Broeck Reference Vanden-Broeck2010, § 2.4.2 therein). We can infer it directly from that of symmetric Squire modes (Squire Reference Squire1953) as discussed in Appendix B, note that $k/2$ therein is replaced by $k$ here. Such stationary modes then exist for the two wavenumbers ${k=\pm k_u}$ satisfying ${1=T k_u\tanh {k_u}}$. In the current setting we extract from (3.24) the neutral amplitude normalised with $\bar {\varLambda }$,

(3.25)\begin{equation} \bar{a}_u:=~2\mathrm{\Gamma}\left({\tfrac{2}{3}}\right){\textrm{Res}}_{k=k_u}[\mathcal{V}(k,h_0)]/ \left(u_0^{+} k_u^{5/3}\right). \end{equation}

A linchpin of the analysis in supplement B is the asymptotic representation of $k_u$ and ${\textrm {Res}}_{k=k_u}(\mathcal {V})$ as $k_u$ vanishes and $\bar {a}_u$ diverges for ${T\to 1-}$ (§ B.1) as well as the qualitatively reciprocal behaviour for ${T {\rightarrow 0}}$ (§ B.2). In combination with (2.8) (for ${x_v=-1}$), this boils down to the following, numerically valuable, finite limits obtained with high accuracy:

(3.26a)\begin{gather} k_u/\sqrt{\smash[b]{1-T}}\simeq 1.8046,\quad \bar{a}_u(1-T)^{7/3}\simeq~0.2805 \quad (T\to 1-), \end{gather}

(3.26b)\begin{gather}k_u T\simeq~1.1596,\quad \bar{a}_u k_u^{2/3}\exp(k_u h_0)\simeq~6.0422 \quad (T{\rightarrow 0}). \end{gather}

To compute (3.23) (for ${y=h_0}$), we restrict the numerical integration to the interval ${|\mbox{Re}\,k|\leq ~20}$, which in view of the exponential large-$k$ tails of $\mathcal {V}$ (§ B.2) gives satisfactorily accurate results. Specifically, we find that ${\mathcal{V}(k,h_0)=O(\exp [-|k| h_0]/k)}$. The evaluation of the integrand employs a cubic-spline interpolation of the solution $\mathcal {V}$ of the Rayleigh problem (3.22) for discrete values of $k$. We advantageously mitigated the singularity at ${k=0}$, circumvented at a small distance (see figure 5), by splitting off the first two terms in the small-$k$ expansion of $\mathcal {V}(k,h_0)$ (§ B.1) and finally adding their inverse Fourier transform, which results in the reciprocal large-$\bar {X}$ representation of $\bar {V}$ and $\bar {H}$ via (3.23). We skip the details of this alternative derivation of (3.15) in its more complete form, supplemented with (3.16), also yielding the corresponding asymptote of $\bar {H}$ by integration. To evaluate (3.24) (for ${y=h_0}$) and discrete $\bar {X}$-values, the poles of $\mathcal {V}$ are detected as the roots ${k=k_p}$, say, of $\mathcal {V}^{-1}(k,y)$. Since ${\mathcal{V}\sim {\textrm {Res}}_{k=k_p}(\mathcal{V})/(k-k_p)}$ as ${k\to k_p}$, the according residuals (given by a homogeneous solution to (3.22), see above) are computed as ${1/[ {\partial }_k \mathcal{V}^{-1}(k_p,y)]}$ (${y=h_0}$). For ${i>7}$ and $\bar {X}$ lying not too close to zero, the values of the exponentials in (3.24) have already fallen below the round-off error; a few more modes calculated using the asymptotic behaviour of the residuals (§ B.3) were, however, added.

The plots in figures 6 and 7 are also constructed by cubic-spline interpolation of the pointwise data sets. Figure 7(a) displays the results obtained by summation of residuals. As one expects, these are slightly more accurate for very negative values of $\bar {X}$ and for small values of $T$ than those found by the direct evaluation of (3.23). The contributions of the residua of imaginary poles in (3.24) become only marked when $\bar {H}$ starts to set off from its oscillating behaviour further upstream. Figure 7(b) indicates that excellent agreement with the asymptotes found analytically can be achieved. It is seen that $\bar {H}$ undergoes a trough immediately downstream of the edge before it recovers to rapidly assume the algebraic far-downstream growth governed by (3.15), (3.16) (see also § B.1). The second result in (3.26b) corroborates the extremely rapid upstream decay of the Rayleigh modes found numerically as ${T {\rightarrow 0}}$. Even the maximum value of $k_u$ shown lies on the part of $\mathcal {C}$ considered for the numerical integration, but the suppression of exponentially growing terms in the calculation of $\mathcal {V}$ and the residuals becomes a numerically delicate task when $|k|$ becomes sufficiently large. In the long-wave limit ${k_u {\rightarrow 0}}$ as ${T\to 1-}$, $\bar {\varPsi }$ diverges both immediately upstream of the trailing edge, as $\bar {a}_u$ grows like $k_u^{-14/3}$, and for constant but sufficiently large positive values of $\bar {X}$. Also these findings compare favourably with the curves in figure 7. The intriguing further implications of the long-wave limit are addressed in § 5.

Figure 6.

Wavenumber $k_u$ (solid) and amplitude $\bar {a}_u$ (dashed), see (3.25), of the neutral capillary mode vs inverse Weber number $T$, asymptotes for ${T\to 1}$ (dotted) and ${T {\rightarrow 0}}$ (dash–dotted) from (3.26).

Figure 7.

Plots of $\bar {H}$ vs $\bar {X}$ and $T$ (a) upstream, (b) downstream of trailing edge: labels indicate $T$-values; multiples ${\neq ~1}$ of $\bar {H}$ (in parentheses) shown for enhanced visibility; plot resolution of strongly augmented oscillations for ${T=0.1}$ discerned in (a); two-terms downstream asymptotes (dashed) from (3.15) with (3.16).

We complete the numerical analysis by pointing to the promising agreement between the computed wavelengths ${\lambda _u:=~2{\rm \pi} /k_u}$ and oscillation amplitudes, see figure 7(a), and those found from the leading-order asymptotes

(3.27)\begin{equation} \bar{H}/\bar{\varLambda}\sim{-}(\bar{a}_u/u_0^{+})\cos(k_u\bar{X}-{\rm \pi}/3)\quad (\bar{X}\ll{-}1), \end{equation}

following from (3.24) and (3.14c). For ${T=0.95}$, (3.26a) predicts ${\lambda _u\simeq ~15.570}$ and ${\bar {a}_u\simeq ~339.741}$; for ${T=0.8}$, (3.26a) still predicts the reasonably good approximation ${\lambda _u\simeq ~7.785}$. For ${T=0.1}$, (3.26b) gives ${k_u\simeq ~11.5960}$ or ${\lambda _u\simeq ~0.5418}$ and $\bar {a}_u/u_0^{+}\simeq ~9.655\times 10^{-10}$ (cf. § B.2), whereas ${k_u\simeq ~11.5570}$ or a slightly larger wavelength ${\lambda _u\simeq ~0.5437}$ is extracted from the numerical data. The details of this case displayed in figure 8 give evidence of the capability of our numerical method to resolve even the rapid oscillations of exponentially small amplitude for small $T$-values with surprisingly high accuracy. In figure 8(b) the difference between the asymptotically and numerically found wavenumbers explains a rather small cumulative phase shift of about $0.3895$ between the curve found numerically and its harmonic approximation provided by (3.27).

Figure 8.

Plots of $\bar {H}$ vs $\bar {X}$ for ${T=0.1}$ far upstream over (a) several wavelengths, (b) approximately a single wavelength: data points (circles) interpolated by cubic splines (solid) vs harmonic asymptote (dashed) in (b).

3.3.3. Why capillary undulations exist only upstream of the trailing edge

In fact, the decision whether the oscillatory capillary modes occur either up- or downstream of the trailing edge, which depends on whether the real poles are within the lower or upper part of the $k$-plane divided by $\mathcal {C}$, cannot be left to the present steady-flow analysis. In both cases, these small-scale Rayleigh waves are also manifest above the MD of the interactive flow, modulating their amplitude over the interactive streamwise length scale. We now return to a convincing (although not rigorous) argument restricting their presence to upstream of the edge, as already anticipated in figure 1.

As inferred from the long-wave limit of (3.14a), the Rayleigh-type perturbation of the streamfunction $(\epsilon ^{2/3})$ in (3.10) morphs into the pressure-free one ${\epsilon ^{2/3}\bar {\varPsi }_{y}(\bar {X},0)\varPsi _0'(Z)}$ in the LD. It exhibits a rapid (harmonic) streamwise variation, either far up- or far downstream. Inspection of (2.3a) shows that typically a further VSL or SL (figure 3b) where ${Z=O(\epsilon ^{1/21})}$ is required on account of the no-slip BC. For very negative values of $\bar {X}$, this shear layer is of the type provoked by the rapid small-scale disturbances considered in Reference Scheichl, Bowles and PasiasSBP18. For ${\bar {X}\ll \epsilon ^{1/7}}$ (${X\ll 1}$), it becomes absorbed into the HRW, there serving as the viscous correction of the LD; for larger values of $\bar {X}$, an additional perturbation in the expansion of ${-\epsilon ^{2/3}\bar {\varPsi }_{y}(\bar {X},0){U_{s}}(X)}$ of $h_-$ serves to satisfy the free-slip condition ${\psi _{zz}\sim 0}$ on ${z=0}$ to which (2.3e) reduces; see (2.15).

These observations allow for the existence of the undular modes up- or downstream of the edge, i.e. without preferring one of these alternatives. That is, a steady-flow analysis cannot rule out one of these two possibilities. We therefore justify our choice by making a recourse to the detection of capillary modes exclusively upstream of a wall-mounted obstacle, serving as a compact forcing, by Bowles & Smith (Reference Bowles and Smith1992) and Rayleigh's celebrated radiation principle, which exploits the anomalous dispersion relation for small-amplitude capillarity waves. Acknowledging their essentially inviscid nature in both situations (despite their amplitude of $O(\epsilon ^{2/3})$ here), we consider this analogy as reasonable.

As a serious objection, however, we have to admit that this principle applies strictly only to a uniform (potential) background flow, where it was adopted by Cumberbatch & Norbury (Reference Cumberbatch and Norbury1979). The last authors also point to the rigorous justification of the above observation by solving the signalling problem, following DePrima & Wu (Reference DePrima and Wu1957). When applied to the current situation (in a separate study), this demands the solution of the unsteady extension of (3.14a) subject to an artificial, spontaneous introduction of the trailing edge in the unperturbed flow described by $\psi _0$. That is, one expects a pertinent neutral mode for zero frequency in the long-time response in the spectrum, to occur upstream rather than downstream of the edge. An easier modification of this ideal, rigorous approach is the introduction of artificial viscosity and tracing that particular wavenumber in the $k$-plane when the then complex frequency tends to zero (cf. Huerre & Monkewitz (Reference Huerre and Monkewitz1990), § 3.4 therein). This plausibility argument serves to single out the mode upstream as the physically meaningful alternative.

3.3.4. Diffusive overlayer

Expansion (3.10) accounts for the second dynamic BC (2.3e), requiring vanishing shear stress on the top free surface, up to $O(\epsilon ^{3/7})$, i.e. as long as (2.3e) reduces to ${\psi _{yy}\sim 0}$ on ${y=h_+}$. Moreover, it was indicated in Reference Scheichl, Bowles and PasiasSBP18 how (2.3e) alters the highest-order contribution of $O(\epsilon ^{4/7})$ to the inviscid flow described by (2.11) in a thin layer adjacent to the upper free streamline accounting for viscous diffusion of weak perturbations around the base flow. From inspection of (2.3a), it penetrates to values of ${h_+ -y}$ measured by the square root of its horizontal extent and thus of $O(\epsilon ^{3/7})$. Since the flow therein itself becomes inviscid over the shortened Rayleigh scale, a further diffusion layer of reduced vertical depth arises where $\bar {X}$ and ${\xi :=(y-h_+)/\epsilon ^{1/2}}$ are of $O(1)$ and (2.3e) is formally retained in full. A comprehensive completion of the present self-consistent theory requires a brief examination of this overlayer meeting (2.3e); see supplement C.

3.3.5. Inner Rayleigh stage: lower deck

Following the outline (b) in the preliminaries, see § 3.3.1, the inner square region regularises $P$ by taking into account the transverse variation of $p$, which becomes of $O(\epsilon ^{6/7})$ according to (3.19a,b) and (3.18a,b) with (3.7). Then (3.10), (3.11) yield the relevant expansion

(3.28)\begin{equation} [\psi,p-p_-]\sim\epsilon^{4/7}[(M^{2}/\lambda)^{1/7}\varPsi_0(Y),0]+\epsilon^{6/7}[\hat{\varPsi},\hat{P}](\hat{X},Y)+ O(\epsilon^{20/21}), \end{equation}

where we advantageously revert to the inverse Prandtl transposition in (2.13a–c). Again, the quantities $\hat {\varPsi }$, $\hat {P}$ describe a linearised Euler flow, now with $\varPsi _0$ providing the base profile. Therefore, ${\hat {V}(\hat {X},Y):= -\hat {\varPsi }_{\bar {X}}}$ satisfies a Rayleigh problem of the type (3.22) except for (3.14c), (3.14d) being replaced by the required decay for large values of ${R:= r/(m\,\epsilon ^{2/7})=(\hat {X}^{2}+Y^{2})^{1/2}}$, where the displacement of the HRW controls $\hat {V}$ by virtue of a $R^{-1/3}$-variation matching (3.19a,b). Since the absence of a free surface at play renders the Rayleigh operator here self-adjoint, all poles lie on the imaginary axis of the corresponding wavenumber plane, which suppresses oscillations of wavenumbers much smaller than those detected in § 3.3.2. Moreover, following the analysis leading to (3.19a,b) recovers the far-field singularity also for ${R {\rightarrow 0}}$.

This shows that the inner RS is unable to fulfil its original task of regularising the pressure provoked by the HRW in the outer RS across the LD, and the associated Rayleigh problem does not therefore merit a more detailed analysis as it proves physically insignificant.

Since the scaled slip $\bar {\varPsi }_{y}(\bar {X},0)$ exerted by the outer RS becomes of $O(\bar {X}^{1/3})$ as ${\bar {X} {\rightarrow 0}-}$, the vertical extent of the associated SL introduced in § 3.3.3 shrinks typically to ${Y=O(\epsilon ^{1/21}\bar {X}^{1/3})}$. It is continued as a sublayer covering the inner region where ${Y=O(\epsilon ^{1/7})}$ (figure 3c). There the driving slip is replaced by $\hat {\varPsi }_{Y}(\hat {X},0)$, which again attains an $\hat {X}^{1/3}$-behaviour as ${\hat {X}^{1/3} {\rightarrow 0}-}$. We are therefore driven to consider a collapse of the inner RS, the SL and the HRW into a single region (figure 3d), addressed next.

4.2.1. The full inertial–viscous limit

The convective–viscous balance in (4.4) is restored in full if $\bar {\psi }$ varies essentially with $\ln {\bar {r}}$,

(4.6a,b)\begin{equation} \bar{\psi}\sim\bar{g}(\vartheta)-\varGamma\ln{\bar{r}}/(2{\rm \pi})\quad (\bar{r}{\rightarrow 0}),\quad \varGamma\bar{g}'''/(2{\rm \pi})-2\bar{g}'\bar{g}''=(4\bar{g}+\bar{g}'')''. \end{equation}

We are thus concerned with a spiralling extension of a special type of a radial Jeffery–Hamel (JH) flow described by $\bar {g}(\vartheta )$ (see Fraenkel Reference Fraenkel1962), exhibiting the vorticity ${-\bar {{\rm \Delta} }\bar {\psi }=-\bar {g}''/\bar {r}^{2}}$ and an outwards flow speed $\bar {g}'(\vartheta )/\bar {r}$ as collapsing in a line source of strength $\bar {g}'(\vartheta )$, due to a superimposed potential vortex of some strength $\varGamma$. Here the homogeneous BCs ${\bar {g}(0)=\bar {g}''(0)=\bar {g}({\rm \pi} )=\bar {g}'({\rm \pi} )=0}$ originating in (4.3c,d) require ${\varGamma =0}$, and $\bar {g}$ represents an eigensolution of the full NS problem. Nevertheless, the case ${\varGamma \neq ~0}$ and ${\bar {g}''\not \equiv 0}$, apparently unconsidered before now, might be of interest in a different context. We also remark that for an inviscid flow, removing the Stokes operator in (4.6a,b), $\bar {g}(\vartheta )$ varies sinusoidally in general but linearly in the case of a potential flow.

An analytical–numerical study shows that there exist two eigensolutions $\bar {g}$. Each describes a distinctly different canonical flow topology as both exhibit a dividing streamline ${\bar {g}=0}$ for ${\vartheta =\vartheta _0\simeq ~1.12777}$ and, thus, violate the premise (4.5) and point to the existence of a closed reversed-flow eddy. This is located either adjacent to the plate (${\bar {g}<0}$ for ${\vartheta _0<\vartheta <{\rm \pi} }$) or fully detached as bounded by the free streamline (${\bar {g}<0}$ for ${0<\vartheta <\vartheta _0}$); see figure 9(a). In the first case, the flow undergoes pre-separation to reattach in the origin ${\bar {r}=0}$; in the second, the free streamline attaches rather than detaches there from the plate. These flow pictures are the immediate consequence of including azimuthal higher-order corrections to the purely radial JH flow and extending the streamline pattern over the full NS scales; see figure 9(b). However, our scrutiny of the related literature does not inform about what, at first sight, is a rather pathological situation. In particular, the conception of a detached eddy with a stagnation point forming at the free and material streamline, to which the fluid particles stay attached, raises serious concerns.

Figure 9.

(a) Eigensolutions of (4.4) referring to a JH flow given by (4.6a,b); (b) sketched flow patterns for the two cases in (a): reversed-flow bubble upstream of detachment or dictating attachment of free streamline.

We therefore rule out the JH solution as the local limit of the full NS solution. Notwithstanding its apparent shortcoming, however, we refer the interested reader to the higher-order corrections and some of the further impact of this limit in supplement D. These findings are not required for the core arguments at present but are potentially of interest for pursuing the study of this flow structure in a related context.

4.2.2. An extended Stokes limit as the alternative

Discarding the possibility of a full inner NS problem, (4.6a,b), leaves us with the degenerate situation of the dominant Stokes balances

(4.7a,b)\begin{equation} 0\sim\bar{{\rm \Delta}}^{2}\bar{\psi},\quad \bar{p}_{\bar{r}}\sim\bar{{\rm \Delta}}\bar{\psi}_{\vartheta}/\bar{r},\quad \bar{p}_{\vartheta}\sim{-}\bar{r}(\bar{{\rm \Delta}}\bar{\psi})_{\bar{r}} \end{equation}

and ${\bar {\psi } {\rightarrow 0}}$ as the origin ${\bar {r}=0}$ is approached along any path from within the flow. We then expand $\bar {\psi }$ into the eigensolutions $\bar {\psi }_i$ of the biharmonic operator in (4.7a,b) when supplemented with the homogeneous BCs in (4.3c,d) found by separation in the polar variables, following Moffatt (Reference Moffatt1964) and the references therein. However, here the subordinate convective terms in (4.4) control their admissibility and, thus, the form of the dominant eigensolution. This straightforward but long-winded selection process is detailed in supplement E. As the most significant result, it predicts regular behaviours for ${\bar {r} {\rightarrow 0}}$ towards a separating flow (${\bar {\psi }_{\bar {y}\bar {y}}=0}$),

(4.8a)\begin{gather} \bar{\psi}\sim{-}4a_5\bar{y}^{3}+o(\bar{r}^{3}),\quad \bar{p}-\bar{p}_0\sim{-}24a_5\bar{x}+o(\bar{r})\quad (a_5<0,\ \bar{r}{\rightarrow 0}), \end{gather}

(4.8b)\begin{gather}\bar{h}\sim\bar{h}(0)+\bar{h}'(0)\bar{x}+\bar{p}_0\bar{x}^{2}/(2\tau)-4 a_5\bar{x}^{3}/\tau+o(\bar{x}^{3})\quad (\bar{x}{\rightarrow 0}+). \end{gather}

The constant $a_5$ and the offset pressure $\bar {p}_0$ are part of the solution to the full NS problem, in turn, forced by the value of $\varLambda _0$. Let us first indicate how to fix the unknown coefficients $\bar {h}(0)$ and $\bar {h}'(0)$, governing the local elevation of the just detached streamline, and complete our analysis at this stage, i.e. without taking into consideration any smaller length scale.

As an obvious geometrical requirement, ${\bar {h}(0)=0}$ then. In full agreement with the current status of the theory, the position of flow detachment not only defines the origin ${x=y=0}$ but an arbitrary point of the upper side of the plate rather than necessarily coinciding with the trailing edge, as a genuine geometrical constraint. Detachment further upstream then requires the actual static wetting or contact angle, observed in the NS region, as an input quantity being so close to ${\rm \pi}$ that it is approximated by ${{\rm \pi} -\epsilon ^{3/2}\bar {h}'(0)}$. This determines a positive value of $\bar {h}'(0)$. However, and as an immediate consequence of the slenderness of the lower free streamline, this thereby resulting distinguished limit refers to the quite exceptional break-away of an almost perfectly hydrophobic liquid. Additionally, such a scenario demands for the geometrical constraint ${\bar {h}>0}$ for ${\bar {x}>0}$, which admittedly cannot be guaranteed as long as the numerical solution of the above NS problem is not available. It is also not likely to occur in reality, where unavoidable (though here neglected) surface imperfections already affect the flow described on the vertical NS scale. It is a natural step, therefore, to identify the location of flow detachment indeed at the trailing edge. However, then $\bar {h}'(0)$ remains undetermined as long as its microscopic shape remains unresolved.

The outcome of these considerations is threefold. Firstly, we expect both $\bar {h}(0)$ and $\bar {h}'(0)$ to be fixed by conditions of matching the full NS and a Stokes flow in a hidden region of an extent much smaller than that of the encompassing NS region. Secondly, as we raised in the introduction to § 4, the description of that creeping flow must take into account the meniscus formed by the actual slope of the free streamline at its detachment point of three-phase contact as a hitherto unconsidered physical input. And thirdly, that a new length scale must resolve the microscopic contour of the plate with sufficient accuracy.

4.3.1. Nested Stokes problems

To progress further, we introduce the new length scale ${\ell \ll \epsilon ^{1/2}}$, non-dimensional with the nominal film thickness $\tilde {H}$. In the new flow region, ${[\hat {x},\hat {y}]:=[x/(\epsilon \ell ),y/\ell ]}$ and ${\hat {r}:= r/\ell =(\hat {x}^{2}+\hat {y}^{2})^{1/2}}$, see (3.18a,b), are of $O(1)$ as ${\ell {\rightarrow 0}}$. Hence, supplementing (4.1) with (4.8), the associated increase of ${\bar {p}-\bar {p}_0}$ and (4.8b) suggests the two-parameter expansion

(4.9)\begin{equation} \left[\frac{\psi}{\ell^{3}\,\epsilon^{{-}1/2}}, \frac{p-p_-{+}gy-\epsilon\bar{p}_0}{\ell\,\epsilon^{1/2}},\frac{h_-}{\ell}\right]\sim \left[\hat{\psi}(\hat{x},\hat{y}),\hat{p}(\hat{x},\hat{y}),\hat{h}_0(\hat{x})+\frac{\ell}{\epsilon^{1/2}}\hat{h}_1(\hat{x})\right]. \end{equation}

The $O(1)$-quantities $\hat {\psi }$, $\hat {p}$, $\hat {h}_0$ and the only first-order correction of interest $\hat {h}_1$ are to be found. The scaled elevation ${\hat {h}(\hat {x}):= h_-/\ell }$ of the detaching streamline remains to be determined by the capillary normal-stress jump in (2.3e). Since ${\hat {p}\sim \epsilon \bar {p}_0+O(\ell \epsilon ^{1/2})}$ and all viscous terms on its left side are found to be of $O(\ell \epsilon ^{1/2})$, that becomes of ${\epsilon \bar {p}_0+O(\ell \epsilon ^{1/2})}$ in the limit (4.9). However,

(4.10)\begin{equation} \varkappa_-{\sim}\ell^{{-}1}\hat{h}''/(1+\hat{h}'^{2})^{3/2}, \end{equation}

stating that the capillary number at play, $\ell ^{2}\epsilon ^{1/2}/\tau$, is small. This is also inferred from reducing ${\textit {Ca}}$ in (2.2b) by the small relative velocity scale $\ell ^{2}/\epsilon ^{1/2}$. In turn, ${\hat {h}''\equiv 0}$, and matching (4.9) and (4.8b) shows that the lower free streamline remains horizontally inclined under an angle no larger than of $O(\epsilon ^{3/2})$. Accordingly,

(4.11a,b)\begin{equation} \hat{h}_0=\bar{h}(0)/\varDelta,\quad \hat{h}_1=\bar{h}'(0)\,(\hat{x}-\hat{x}_d)/\varDelta. \end{equation}

Here the parameter $\varDelta$ measures the strength of the required distinguished limit,

(4.12a,b)\begin{equation} \ell=\varDelta\epsilon^{2},\quad 0<\varDelta=O(1), \end{equation}

and ${(\hat {x},\hat {y})=(\hat {x}_d,\hat {h}_0)}$ denote the position of the actual detachment point, $\mathcal {D}$, taken initially to be known.

Let $\varSigma$ denote the resolved surface of the plate. Inspection of (2.3) and the behaviour (4.11a,b) confirm that the leading-order quantities $\hat {\psi }$, $\hat {p}$ satisfy

(4.13a)\begin{equation} \hat{{\rm \Delta}}^{2}\hat{\psi}=0,\quad \hat{p}_{\bar{x}}=\hat{{\rm \Delta}}\hat{\psi}_{\bar{y}},\quad \hat{p}_{\hat{y}}={-}\hat{{\rm \Delta}}\hat{\psi}_{\hat{x}},\quad \hat{{\rm \Delta}}:={\partial}_{\hat{x}\hat{x}}+{\partial}_{\hat{y}\hat{y}}, \end{equation}

subject to mixed boundary conditions in the limit of zero capillary number as

(4.13b)\begin{gather} \hat{r}{\rightarrow\infty}\colon\enspace \hat{\psi}\sim{-}4a_5\hat{y}^{3}+o(\hat{r}^{5/2}), \end{gather}

(4.13c)\begin{gather}\text{on}\enspace \varSigma\ (\hat{x}<\hat{x}_d)\colon\enspace \hat{\psi}=\hat{\psi}_{\hat{y}}=0, \end{gather}

(4.13d)\begin{gather}\hat{y}=\hat{h}_0\enspace (\hat{x}\geq\hat{x}_d)\colon\enspace \hat{\psi}=\hat{\psi}_{\hat{y}\hat{y}}=0. \end{gather}

Once $\hat {\psi }$ is found, one can calculate $\hat {p}$ by integration, giving ${\hat {p}\sim -24 a_5\hat {x}+O(1)}$. This matches identically the small-$\bar {r}$ form of $p$ in (4.8a) as the remainder term negotiates a constant of integration found from the $O(\epsilon ^{3/2})$-contribution to (4.1). In accordance with the above results and likewise, the neglected remainder term in (4.13b) expresses the second necessary far-field condition and the absence of an eigensolution of the NS problem of $O(\epsilon ^{3/4})$ that would enter the right-hand side of (4.1). This seems to be a natural choice, as (3.28) would require it to die out for large values of $\hat {r}$. Rather, (4.13d) enforces an $O(1)$-correction $a_5\hat {g}(\vartheta )$, say, in the large-$\hat {r}$ form of $\hat {\psi }$. The function $\hat {g}$ is then governed by

(4.14a–d)\begin{align} (4\hat{g}+\hat{g}'')''=0,\quad \hat{g}(0)={-}4a_5\hat{h}_0^{3},\quad \hat{g}''(0)={-}24a_5\hat{h}_0,\quad \hat{g}({\rm \pi})=\hat{g}'({\rm \pi})=0; \end{align}

cf. (4.6a,b). Eventually,

(4.15a,b)\begin{align} \frac{\hat{\psi}}{a_5}\sim{-}4\hat{y}^{3}\cdots+\hat{g}(\vartheta)+o(1),\quad \hat{g}={-}6\hat{h}_0(\sin{\vartheta})^{2}-4\hat{h}_0^{3}\left[1+\frac{\sin(2\vartheta)}{2{\rm \pi}}-\frac{\vartheta}{\rm \pi}\right], \end{align}

where the dots indicate potential eigensolutions of $o(\hat {r}^{2})$. The behaviours in (4.15a,b) provide the match of (4.9) with (4.1) supplemented with an $O(\epsilon ^{9/2})$-contribution, hence, also excited by the displacement (4.2) of the interface. While that of $O(\epsilon ^{3/2})$ is controlled by the linearisation of $\bar {\psi }$ as ${\bar {y} {\rightarrow 0}}$, this is due to the corresponding third-order terms. As these dominate as ${\bar {r} {\rightarrow 0}}$ where ${\bar {\psi }\sim 4 a_5\bar {y}^{3}}$, evaluating $\hat {g}$ for ${\vartheta {\rightarrow 0}}$ describes the feedback of the displacement on the flow near detachment.

We note that the weak curvature of the detaching streamline is determined by the higher-order approximation of (2.3e), following from (4.10) as

(4.16)\begin{equation} \tau\hat{h}''\sim\epsilon\ell\bar{p}_0+2\epsilon^{1/2}\ell^{2}\hat{\psi}_{\hat{x}\hat{y}}\quad (\hat{y}=\hat{h}_0). \end{equation}

The case of a perfectly flat surface associated with ${\hat {h}_0\equiv 0}$ and the trivial solution ${\hat {\psi }=-4 a_5\hat {y}^{3}}$ of (4.13) recovers the dominant Stokes limit of the full NS solution for ${\bar {r} {\rightarrow 0}}$ and the aforementioned pathological case of fully hydrophobic dewetting with both $x_d$ and $\ell$ then remaining unspecified. This situation is therefore ruled out, and we are indeed left with flow detachment in a vicinity of the originally sharp plate edge covered by the Stokes region, where the microscopic resolution of the edge dictates the definition of $\ell$. We henceforth refer to the sketch of the flow around the resolved edge in figure 10, detailing figures 2(b) and 3( f) on the new scale for various values of $\beta$ (cf. Duez et al. Reference Duez, Ybert, Clanet and Bocquet2010). As previously discussed, the edge is, without substantial loss of generality, assumed to be given by a smoothed but at first ideal wedge of cut-back angle $\alpha$ and with an apex lying at the coordinate origin. Then the curvature radius typical of the rounded nose conveniently defines $\ell$; the degenerate situation of a wedge still found sharp when viewed on the scale $\epsilon ^{2}$ is assumed in the limit ${\varDelta {\rightarrow 0}}$. The case of specific relevance ${\alpha =0}$ can be interpreted as a plate-type thin tip formed by a semi-circle and of local thickness $2\ell$ (figure 10b). Although the film generated by Duez et al. (Reference Duez, Ybert, Clanet and Bocquet2010) is much thinner and thus the scales different, we find the comparison of their experimental with our theoretical prediction noteworthy (figure 10c).

Figure 10.

Stokes flow around resolved smoothed trailing edge: (a) wedge-type (${\alpha >0}$), inner region emerging for ${\beta <\alpha }$ (green); (b) plate-type and semi-circular (${\alpha =0}$), no inner region; (c) cut of liquid interfaces in experiment, described in and reprinted with permission from Duez et al. (Reference Duez, Ybert, Clanet and Bocquet2010) (© the American Physical Society).

Assuming ${\varDelta =1}$ in (4.12a,b) and ${\tilde {H}=1\ \text {mm}}$ (table 1 in Appendix A) typically gives a quite small physical scale ${\ell \tilde{H}\simeq 0.01 - 0.04\ \mathrm {\mu }\text {m}}$. However, it is large enough to consider the asymptotic theory applicable to curvature radii achieved in manufacturing practice.

Completing our flow model at this stage is indeed possible for a non-degenerate, smoothed wedge tip and a sufficiently large apparent wetting angle $\beta$ as the wedge geometry imposes a closure condition on (4.13). This fixes the location of $\mathcal {D}$

(4.17)\begin{equation} \text{on}\ \varSigma\colon\ \text{d}\hat{y}/\text{d}\hat{x}\sim\tan{\beta}+o(\epsilon^{3/2}). \end{equation}

This describes the general, non-degenerate case where $\bar {h}(0)$ is found in virtue of (4.11a,b). Evidently, then also ${\hat {h}'(0)=0}$ as the linear follow-up problem to (4.13) governing disturbances of $O(\ell /\epsilon ^{1/2})$ in (4.9) has the zero solution. Higher-order perturbations, already affected by the curvature of the detached streamline, control the (physically insignificant) remainder term in (4.17). Proceeding in this manner determines successively the two ICs that each term arising in the expansion of $h_-$ in (4.1) has to meet as ${\bar {r} {\rightarrow 0}}$. This consideration confirms self-consistency of the proposed theoretical framework. As a crucial result, the flow wets the underside of the wedge as $\hat {h}_0$ represents a (strictly) monotonic function of $\beta$, which decreases from $0$ as $\beta$ decreases from ${\rm \pi}$. This justifies our reference to the teapot effect. The pathological limit ${\beta \to {\rm \pi}-}$ or ${(\hat {x}_d,\hat {h}_0)\to (0,0)}$, however, leads to a non-trivial value of $\hat {h}'(0)$. Here we only note that the above analysis by inspection gives ${\ell =O(\epsilon )}$ in the degenerate case ${\bar {h}(0)=0}$, ${\bar {h}'(0)>0}$. On the other extreme, $\mathcal {D}$ has reached the point on the nose where its curvature vanishes once $\beta$ has become as small as $\alpha$. All together, we arrive at the geometrical constraint

(4.18)\begin{equation} {\rm \pi}-\epsilon^{3/2}\bar{h}'(0)\geq\beta\geq\alpha. \end{equation}

The variation of $\hat {h}_0$ with $\beta$ is more and more squeezed towards the edge as this gets sharpened. Finally, $\mathcal {D}$ is seen as pinned to the edge as (4.18) is interpreted as the well-known Gibbs inequality; see Oliver, Huh & Mason (Reference Oliver, Huh and Mason1977), Dyson (Reference Dyson1988) and Kistler & Scriven (Reference Kistler and Scriven1994). In accordance with the last authors, we find that the distance of $\mathcal {D}$ from the apex decreases with both increasing values of $\beta$ and the Reynolds number.

The formidable task of solving the Stokes problem (4.13), parametrised by $\alpha$ and $\beta$, has not yet been accomplished. Most importantly, in the situation sketched in figure 10(b), mastering this challenge will establish a comprehensive flow description in the entire range ${{\rm \pi} >\beta >0}$ of physical significance. If, however, ${\beta \geq \alpha }$, determining the actual position of $\mathcal {D}$ requires the introduction of a further, inner Stokes region, as indicated in figure 10(a). Contrasting with its counterpart (4.13), there the governing problem is of a non-degenerate free-surface type, thus controlled by a capillary number of $O(1)$, to accommodate to the necessary local bending of the detaching streamline. We expect $\mathcal {D}$ to be found the further away from the apex the smaller is the value of $\beta$, with its position fixed by a constraint arising of the interplay of these nested Stokes regions. This is a topic of our upcoming activities. Adhering to the scalings of the outer region, however, and using (4.10) gives as a first result for the approach to $\mathcal {D}$,

(4.19a,b)\begin{equation} \hat{h}'\to\tan(\alpha-\beta),\quad \varkappa_-{\sim}\ell^{{-}1}[\cos(\alpha-\beta)]^{3}\hat{h}''\quad (\hat{x}\to\hat{x}_d). \end{equation}

4.3.2. Static wetting angle

As the final step, we focus on the flow properties in the immediate vicinity of detachment, specified either on conditions (4.18) or (4.19a,b). Here we again follow Moffatt (Reference Moffatt1964) in his analysis of local eigensolutions of (4.13) varying algebraically with distance from a singular point at a rigid wall. These suggest that the streamlines are locally pushed away from the nose. Moffatt (Reference Moffatt1964, § 3.2) also showed that a related class of eigensolutions controls the behaviour of $\hat {\psi }$ at small distances ${\hat {d}=[(\hat {x}-\hat {x}_d)^{2}+(\hat {y}-\hat {h}_0)^{2}]^{1/2}}$ from the detachment point: using (4.13a,c,d) and reusing the azimuthal angle, ${\vartheta :=\arctan [(\hat {y}-\hat {h}_0)/(\hat {x}-\hat {x}_d)]}$, yields first for (4.18) and ${0\leq \vartheta \leq \beta }$ in the limit

(4.20)\begin{align} \hat{d}{\rightarrow 0}\colon\enspace \frac{\hat{\psi}}{\hat{a}\hat{d}^{\,\sigma}}\sim\left\{\begin{array}{@{}ll} \sin(\sigma\vartheta)\sin[(\sigma-2)\beta]-\sin(\sigma\beta)\sin[(\sigma-2)\vartheta]+{\rm c.c.} & (\sigma\neq~2), \\ \sin(2\vartheta)-\vartheta/\beta & (\sigma=2). \end{array}\right. \end{align}

The constant $\hat {a}$ is determined by the full solution to (4.13), and $\sigma$ appears to be a (complex) eigenvalue related to $\beta$ by

(4.21a,b)\begin{equation} (\sigma-1)\sin(2\beta)=\sin[2\beta(\sigma-1)]\enspace (\sigma\neq~2),\quad \tan(2\beta)=2\beta\enspace (\sigma=2). \end{equation}

Thus, a continuous relationship requires ${\beta \simeq ~128.727^{\circ }}$ for ${\sigma =2}$. One readily confirms that the eigensolutions considered in § 4.2.2 are recovered in the limit ${\beta \to {\rm \pi}}$. Equation (4.21a,b) is symmetric with respect to ${\mbox{Re}\,\sigma -1}$. However, physically admissible solutions require the flow speed, of $O(\hat {d}^{\,\sigma -1})$, to vanish and the shear and the normal stress (the pressure), both of $O(\hat {d}^{\,\sigma -2})$, on $\varSigma$ being integrable as ${\hat {d} {\rightarrow 0}}$ (and not to compromise the validity of the Young–Dupré equilibrium). According to (4.16), $\hat {h}''$ varies at the same rate. Therefore, only values of $\sigma$ having ${\mbox{Re}\,\sigma >1}$ are permitted, anticipated by the requirement ${\hat {\psi }=0}$ and a finite slope $\hat {h}'$ at detachment in (4.13d) and, thus, the existence of a static contact angle $\beta$. The plot of the real branches of (4.21a,b) in figure 11 illustrates the infinite multiplicity of $\sigma$, not considered by Moffatt (Reference Moffatt1964), the asymptotes ${\beta \to {\rm \pi}/2,\,{\rm \pi} }$ as ${\mbox{Re}\,\sigma {\rightarrow \infty }}$ and the local extrema of $\beta$. There (4.21a,b) is continued to complex values of $\sigma$, via (4.20) associated with Moffatt's (Reference Moffatt1964) prominent and exceptional infinite sequence of eddies. Hence, our flow model does not predict a single eddy as do the calculations by Kistler & Scriven (Reference Kistler and Scriven1994) for moderately large Reynolds numbers. Rather, it predicts this series of eddies if the value of $\beta$ falls below its absolute minimum. Moffatt (Reference Moffatt1964) predicted this well-established threshold as $\simeq 78^{\circ }$; here we recompute it as $\simeq 79.557^{\circ }$ for ${\sigma \simeq ~3.7818}$.

Figure 11.

Allowed contact angle $\beta$ vs real $\sigma$, from (4.21a,b); existent for $\sigma >3/2$, smooth for $\sigma =2$ (full circle) and having a local absolute minimum (empty circle).

A more elaborate discussion of these details and their consequences requires the yet pending full numerical solution of (4.13). Since (4.21a,b) is independent of the choice of the coordinate system used, however, the above results remain valid in the vicinity of the apex $\mathcal {D}$ of the wedge-shaped inner Stokes region if (4.18) is violated (figure 10b). According to (4.19a,b), $\hat {h}''$ is then again integrable. As a crucial finding, these local properties characterise in general the solution of a Stokes problem governing flow detachment at some finite capillary number.

Moreover, as these results rely solely on the leading-order term of the Stokes expansion, they even hold for a finite-Reynolds-number flow around a resolved top; cf. Silliman & Scriven (Reference Silliman and Scriven1980) and Kistler & Scriven (Reference Kistler and Scriven1994).