Developed liquid film passing a trailing edge: small-scale analysis and the `teapot effect' at large Reynolds numbers

Recently, the authors considered a thin steady developed viscous liquid wall jet passing the sharp trailing edge of a horizontally aligned flat plate under surface tension and the weak action of gravity acting vertically in the asymptotic slender-layer limit (J. Fluid Mech. 850, pp. 924--953, 2018). We revisit the capillarity-driven short-scale viscous--inviscid interaction, on account of the inherent upstream influence, immediately downstream of the edge and scrutinise flow detachment on all smaller scales. We adhere to the assumption of a Froude number so large that choking at the plate edge is insignificant but envisage the variation of the relevant Weber number of O(1). The aspect in the main focus, tackled essentially analytically, is the continuation of the structure of the flow towards scales much smaller then the interactive ones and where it no longer can be treated as slender. As a remarkable phenomenon, this analysis predicts harmonic capillary ripples of Rayleigh type, prevalent on the free surface upstream of the trailing edge. They exhibit an increase of both the wavelength and amplitude as the characteristic Weber number decreases. Finally, the theory clarifies the actual detachment process, unprecedented in the rational theories of flow separation. At this stage, the wetting properties of the fluid and the microscopically wedge-shaped edge, viewed as infinitely thin on the larger scales, come into play. As this geometry typically models the exit of a spout, the predicted wetting of the wedge is related to what in the literature is referred to as the so-called teapot effect.


Introduction
We continue to analyse a flow problem of fundamental importance as started in our forerunner study (Scheichl, Bowles & Pasias 2018, hereafter referenced as SBP18). Let a nominally steady and two-dimensional, developed, slender stream of a Newtonian liquid having uniform properties and at constant flow rate in an inertial frame of reference detach from a horizontal, solid, impenetrable, perfectly smooth plate with a trailing edge that is initially considered as abrupt and sharp. Downstream, the resulting fluid jet divides its gaseous environment, fully at rest and under constant pressures, into two parts. Here this picture is relaxed insofar as the upper one still defines the zero pressure level but we allow for a nonzero, constant support pressure prescribed at the downside of the detached layer. The body and interface forces crucially at play are the constant gravitational acceleration acting vertically towards the wetted side of the plate and surface tension. Based on the principle of least degeneration, our rigorous theoretical description of the detaching thin film under the assumption of very supercritical flow adopts a specific distinguished limit where the relevant Reynolds and Froude numbers are taken as asymptotically large but the corresponding Weber number as of O(1). Hence, the details accompanying the detachment process are governed by a strong viscous-inviscid, shortened-scale interaction at the outset of our present study.
Subsequently, we refer to the sketch in figure 1 throughout, illustrating the different flow regions considered when viewed on the global vertical scale defined by the height of the detaching layer. Specific interest is aroused by the so-called "teapot effect", here observed in the flow in the immediate vicinity of the trailing edge and thus strongly affected by its microscopic geometrical resolution. As a start, we critically review the prevailing, rather phenomenological view on this effect and its previous modelling.
1.1. The teapot effect: a digression The frequently observed, at a first glance spontaneous (and often undesired) tendency of a liquid pouring from a spout to instead stick to its underside was originally reported by Reiner (1956, also see the references therein) and later its current abstraction for a planar, horizontal high-momentum jet in fact passing a rounded wedge of angle α, detailing the flow around the trailing edge in figure 1, typical no slip on the plate and free slip along the free streamlines, blue: free and internal streamlines and detachment point, red: plate and original (virtual) tip in figure 1.
by Watson (1984): see figure 2(a). More precisely, Reiner coined the notion "teapot effect" for pouring liquid along a rigid convex wall with a marked corner and adjoining to another (even liquid) fluid. He untangled the riddle of its occurrence experimentally: his observations ruled out the hitherto widely held belief that the wetting properties in terms of short-range inter-molecular adhesion forces, promoted by wetting agents, are its essential cause. However, his various experiments demonstrated that "adhesion" as the reaction force on the fluid flowing over a solid phase as well as surface tension at its common interface with the surrounding fluid play a decisive role. A recent survey of the various treatments of this scenario presented by Jambon-Puillet et al. (2019, see the references therein) spans the rigorous approach within the framework of classical fluid mechanics, outlined below, to the nowadays more common but less stringent approach. This proposes that the pivotal cause for the fluid sticking lies in the hydrophilic tendency of the liquid/wall pairing rather than the mechanisms of the pouring. The latter authors provide new insight by coupling these ideas with classical arguments resorting to the first principles of continuum mechanics. Notably, Duez et al. (2010) indicate a significant reduction of the effect via the application of superhydrophobic substrates. We advocate continuum mechanics for providing a satisfactory, rational unravelling of the effect. In agreement with the above mentioned early observations, we interpret it as a subtle interplay of inertia, capillarity and gravity in a twodimensional setting. This is crucially tied in with the breakdown of viscousinviscid interaction and thus the slender-layer approximation made on larger scales due to the assumed largeness of the globally defined Reynolds number of the oncoming attached flow. The significance of capillarity and inertia lies also in its proper adjustment immediately upstream of detachment. Our asymptotic theory proposes a fully rational account of the onset of this phenomenon in the realistic situation of a developed incident flow. As a specific ingredient, the trailing edge is replaced by a tip, i.e. a wedge formed by an acute cut-back angle or lip: this "attracts" the liquid film such that it clings to it before the liquid sheet breaks away from it as a whole from its underside. This phenomenon of free rather than forced gross separation from a convex rigid surface, consequently referred to as the teapot effect from here onwards, does not yet have a satisfactorily rigorous and complete description. Duez et al. (2010) previously considered this "inertialcapillary" mechanism, investigated here in depth and breadth, as a crucial step towards a breakthrough in the explanation of the effect.
An initial self-consistent clarification of the effect benefitted from the quite restrictive assumption of irrotational free-surface flow of a weightless ideal fluid and the neglect of surface tension past a horizontal plate, terminated by the aforementioned lip: remaining firmly attached both with the neglect of gravity (Keller 1957) and under gravity (Vanden-Broeck & Keller 1986); detaching grossly from the underside at zero gravity (Vanden-Broeck & Keller 1989). In these investigations, the flow is stipulated to cling to the wall and, due to the absence of viscosity, the position of detachment is also prescribed (Vanden-Broeck & Keller 1989). However, the well-known Brioullin-Villat condition, met for vanishingly small effects of capillarity (and viscosity), fixes the physically admissible detachment point.
Rather little is known when it comes to the rigorous inclusion of viscosity in this flow picture. At least, the passage of a layer over an asymptotically small convex wall corner (and in related situations) considered by Gajjar (1987) (and the refined numerical results by Yapalparvi 2012) is relevant. Specifically, there the unperturbed oncoming flow is fully developed (so as to model a real situation), as being already inclined towards gravity, and viscous-inviscid interaction of the double-layer structure in the high-Reynolds-number limit, adopted here, negotiates the slender obstacle which the corner forms. However, the counteracting impact of surface tension in the resulting combined hypersonic-and wall-jet-type interaction law (cf. Bowles & Smith 1992) is ignored in the analysis although mentioned. Although the interactive flow considered by Gajjar (1987) is assumed to remain grossly attached, it is certainly interesting that the numerical solutions predict a closed separation bubble beyond the mild wedge for both sufficiently large turning angles and Froude numbers.
A seminal reference for the teapot effect in a realistic, i.e. developed, flow is the numerical and partially analytical investigation of the full Navier-Stokes problem by Kistler & Scriven (1994). They unambiguously highlighted its viscous and capillary, i.e. hydrodynamic, nature as underpinned by experimental evidence. This prompted them to conclude that "the teapot effect is more than merely an issue of wetting". Most remarkably, they pointed out how the restrictions of the microscopic wedge-type geometry of what is on larger scales viewed as an "infinitely sharp" edge implies a contact-angle hysteresis, associated with nonunique flow states, but the point of flow detachment becomes the apex of the wedge when the jet Reynolds number, i.e. the momentum it carries, becomes sufficiently large. The present asymptotic analysis corroborates this finding, where we deal with a horizontal oncoming flow past a wedge originally represented by a cut-back angle α (0 < α < π), using equal horizontal and vertical scales. However, here the wedge is no longer necessarily sharp as as we allow for its tip being realistically rounded: see the sketch in figure 2(b).

Studied phenomena and open questions
Our current concern is with the analytical/numerical challenges arising in the analysis of the free jet with particular emphasis placed on the description of its detachment at the abrupt plate edge on the smallest scales and the freely interacting flow immediately downstream of the trailing edge. As a key observation in SBP18, the free layer is strongly dependent on its history and therefore, of the no-slip condition satisfied upstream of its detachment. Since the interaction mechanism is not alone capable of smoothing the flow quantities at the sharp edge, coping with this demand addresses the flow on still smaller and down to the smallest scales discernable and eventually the wetting properties of the plate as well as the detailed geometry forming its edge. The threefold conclusions drawn from such an analysis attempt to shed light on some unsettled questions of fundamental interest: (i) As a first cornerstone, it reveals the existence of (stationary) undamped capillary Rayleigh modes upstream of its break-away from the plate.
(ii) The multi-layer slenderness of the flow, given the largeness of the Reynolds number, prevents its separation upstream of the trailing edge, which confirms the initially made assumption of detachment "at the edge" considered on larger scales.
(iii) As a second highlight, the implied wetting of the edge suggests a novel, rational explanation of the teapot effect observed in a high-momentum liquid layer when a convex corner provides -in a most simple but nevertheless sufficiently complex manner -the non-degenerate geometry modelling the plate edge.
1.3. Organisation of the paper, used notation and numerical software The process of asymptotic scale separation, starting with the largest global scale down to the smallest ones where the teapot effect is at play, guides the structure of our study. Visualising this in figure 3 serves to illustrate and accompany the subsequent analysis of the individual flow regimes governed by those spatial scales. Hence, figure 3(f ) recovers the linkage to the teapot effect as in figure 2.
The paper is organised as follows. § 2: We first pose the problem based on first principles in full. Our basic scaling arguments ( § 2.1 and appendix A) justify the use of asymptotic analysis as the means of choice to study the flow, initiated by completing the formulation of the interaction problem, originally posed in SBP18. It then governs the continuation of the freely interacting jet downstream of the edge in a rigorous manner as long as the value of the appropriately rescaled Weber number does not fall below a certain threshold, so avoiding the onset of nonlinear stationary capillary waves even above the plate ( § 2.2 and appendix B). Over the interactive streamwise scale, this brings into play the splitting of the film into the main deck (MD) and the lower deck (LD), this initiated by a viscous sublayer (VSL) adjacent to the plate.
§ 3: A multi-structured small-scale flow, essentially controlled by capillarity only ( § 3.1), supersedes locally the two-tiered interactive one. The Hakkinen-Rott-type near wake (HRW) forming at the base of the LD just downstream  of the plate ( § 3.2) is central for understanding the multi-structured small-scale flow locally superseding the two-tiered interactive one. Its thorough investigation reveals two nested square Euler regions ( § 3.3). These outer and inner Rayleigh stages (RSs) govern weak perturbations around the flow at detachment. The exterior one extends vertically across most of the layer and is the source of phenomenon (i) above on the top free surface. Simultaneously, a viscous (passive) slip layer (SL) forms at the base of the predominantly inviscid flow.
§ 4: This essentially inviscid description of flow detachment paves the way for a full Navier-Stokes (NS) regime detected on even smaller streamwise and vertical scales, where the flow structure of § 3 collapses. ( § 4.1). Its analytical study leads to the implication (ii) above ( § 4.2). As a pivotal finding, achievement (iii), we also identify one or two interlaced Stokes regions resolving the smallest scales and the actual wedge-type resolution of the plate end ( § 4.3), until now seen as infinitely thin. Consequently, it is this flow regime where the break-away of the film, interacting with the larger-scale flow through the NS region, is finally controlled by both the effective edge geometry and the static wetting angle. Thereby, an awareness of the close relationship of this situation to to the teapot effect is gained.
§ 5: Surveying the current results and anticipating the inclusion of e.g. unsteadiness and the aforementioned capillary undulations in our ongoing research completes the study.
So as not to distract attention away from the main arguments and their physical impact, the detailed steps of the asymptotic analysis, together with further technical side aspects, potentially of interest for the more mathematically orientated readership, are put forward as the accompanying "Other supplementary material". It consists of the individual Supplements A-E. Cross-references between these, its numbered subsections and the main document are conveniently employed. We add citations exclusively in the supplement to the list of references. In addition to the usual conventions for mathematical expressions, we adopt the following usage of accents and sub-and superscripts (cf. figure  § 1). Indices typically indicate orders in asymptotic expansions and partial derivatives unambiguously, and lowered "−" and "+" refer to respectively the lower and upper boundaries of the liquid layer (e.g. h − and h + ) or the states of the flow infinitely far upstream ("−") and downstream ("+"). We endow dimensional quantities with tildes. Furthermore, we attempt a systematic as possible denotation of the dependent and independent O(1)-variables characteristic of the individual regimes: lowercase for the MD (e.g. x), capitalised for the LD (X), capitalised with overbars for the outer RS (X), capitalised with hats for the inner RS (X), lowercase with overbars for the full NS region (x), lowercase with hats for the Stokes regions (x).
All our numerical calculations used the widely-used, proprietary programming language and numerical-computing environment MATLAB (2020), supplemented with The NAG Toolbox (2020). In particular, the computations benefit from its convenient handling of complex arithmetic and the, in principle, built-in arbitrarily high accuracy and precision.

Statement of the extended problem
It proves expedient to first reappraise the fundamental assumptions and the problem in full before revisiting the interactive limit.

Non-dimensional groups and governing equations
The problem has the following central ingredients. The slender layer of densitỹ ρ and kinematic viscosityν and experiencing a tensile surface stressτ and gravitational accelerationg carries a volume flow rate per lateral unit widthQ. It adjusts to a developed state over some sufficiently large distanceL, serving as the basic length scale and measured along the plate from its trailing edge in the upstream direction. Simultaneously,L is required to be so short that the vertical layer height has not grown sufficiently to allow for a significant impact of the hydrostatic pressure on streamwise convection. Then a layer heightH =Lν/Q and flow speedŨ =Q 2 /(νL) representative of this near-supercritical film follow from conservation of the flow rate and the streamwise momentum, here expressed by the balance between convection and the shear stress gradient, respectivelỹ (2.1) In many applications, the vertical height and, accordingly, the speed of the layer have respectively increased and decreased so markedly overL that it has almost attained its well-known perfectly supercritical, fully developed or self-preserving state discovered by Watson (1964): for related discussions see Bowles & Smith (1992), Higuera (1994) and, in the context of an axisymmetric and rotatory layer generated by vertical jet impingement, Scheichl & Kluwick (2019).
The flow is then controlled by the slenderness parameter or reciprocal Reynolds number and corresponding reciprocal Froude and Weber numbers g and τ : : (2.2a) Regarding the distinguished limit involving g, locally strong viscous-inviscid interaction describes the abrupt transformation of the wall-bounded flow on crossing the lip towards the free liquid jet in a least-degenerate, self-consistent and sufficiently smooth manner. We remark that the conventionally defined capillary number or the alternative Ohnesorge number, here / √ τ 1, provide different albeit less preferable measures of the surface tension for a layer of slenderness expressed by : since the streamline curvature scales withH/L 2 = 2 /H, the ratio of the viscous (deviatoric) stress, normal to a free surface and scaling withρνŨ /L =ρŨ 2 2 , to the capillary hoop pressure measured byτH/L 2 = τ 2ρŨ 2 is expressed by the augmented capillary number Ca/ = 1/τ = O(1), taking into account the aspect ratio of the flow. This indicates that in the limit provided by (2.2a) the surface jump of the total normal stress is fully retained in the dynamic boundary conditions (BCs) below.
Order-of-magnitude arguments considering realistic flow situations support the above asymptotic scaling and demonstrate its applicability to the teapot phenomenon in typical settings: see appendix A.
We introduce Cartesian coordinates x and y pointing respectively horizontally from the trailing edge and vertically towards the flow, the streamfunction ψ and the pressure p, non-dimensional withL,H,Q andρŨ 2 respectively. Then u := ψ y is the horizontal and v := − ψ x the vertical flow component made dimensionless withŨ . These O(1)-quantities satisfy the NS equations in the form (2.3b) Here and hereafter, the subscripts − and + indicate the evaluation along the lower-and the uppermost free streamline respectively. Accordingly, y = h − (x) (≡ 0 for x 0) and y = h + (x) denote their positions, hence h(x) := h + − h − the vertical film thickness and p ± the given pressure levels along the free streamlines. Adopting the Heaviside step function θ then gives the kinematic boundary conditions including the conventional requirements of no slip at and no penetration through the plate as follows.
The dynamic BCs express vanishing tangential stresses and total normal stresses equal to the capillary pressure jumps on the free surfaces of curvatures κ ± (x) and subject to the Young-Laplace equilibrium. Therefore, at This completes the problem (2.3) as proper up-and downstream conditions will be condensed into requirements of continuity holding at the trailing edge x = 0.
2.2. Free interaction across the trailing edge The governing equations (2.3) and (2.2a) immediately give rise to regular expansions valid for the flow above the plate on the original large streamwise scale, i.e. for 1 + In the leading order of this non-interactive limit, the classical parabolic shallowwater approximation of (2.3) is recovered, predicting a pressure-free base flow described by ψ 0 and h 0 . These quantities approach regularly some values ψ 0 and h 0 at the trailing edge. The higher-order contributions in (2.4a) control the modification by the hydrostatic pressure distributions and non-parallel-flow effects, the latter predominantly due to streamline curvature, capillary action and the viscous normal stresses ± 2 ψ yx , in the following iterative manner. At each level of improvement, the obtained approximation for ψ feeds into (2.3b) subject to (2.3e). The resulting pressure correction then forces a problem that emerges from expanding (2.3a) subject to (2.3c) and (2.3d) and governs a further correction for ψ, and so on. Following SBP18, this hierarchy is singularly perturbed by weak irregular disturbances exhibiting exponential growth over a short streamwise scale measured by 6/7 . Thus, they are active in the VSL adjacent to the plate. Hence, subject to free viscous-inviscid interaction governed by streamline curvature, not accounted for in the classical shallow-water limit, they describe the intrinsic upstream influence in the film caused by both gravity and capillarity. Finally, the growth of these two effects renders the above hierarchy invalid around the trailing edge where x = O( 6/7 ) and they provoke a locally strong interaction over that scale in the limits (2.2a). This typically involves a nonlinear distortion of the strongly viscosity-affected slow flow in the LD, here originating from the VSL, adjacent to the lowermost streamline where y = O( 2/7 ). The latter exerts a linear response in the MD that comprises the bulk of the layer, beneath the upper free streamline.
The background flow enters the interactive scalings at leading order solely through two quantities condensing its upstream history: the momentum flux J at the trailing edge and the shear stress λ exerted on it, J := The coeffcient ω is only relevant in the small-scale analysis of § 3.3.2. We also note (2.3c) and the free-slip condition resulting from (2.3d): Usually,H is definitely larger than the height of the film immediately downstream of its origin (as given by jet impingement) and where the flow starts to become developed: see table 1 in appendix A. This prompts us to assume that the base flow is already described by Watson's (1964) self-similar solution and so to neglect the small deviations from this due to the flow history, as in SBP18 and without any substantial loss of generality. In this idealisation, h 0 = π(x − x v )/ √ 3 provided some x = x v < 0 indicates the virtual origin of the fully developed flow and ψ 0 is a universal function of y/h 0 . At x = 0, ψ 0 then satisfies and has an exact representation given by Scheichl & Kluwick (2019): writing u + 0 := ψ 0 (h 0 ) from here on, this implies the important canonical results The interaction process itself is parametrised by suitably redefined reciprocal Froude and Weber numbers G and T and the rescaled support pressure P − , all of O(1). Specifically, T is formed with the local momentum flux and thus measures the influence of capillarity relative to fluid inertia. We thus introduce The above propositions enable us to reconsider the interaction problem, at first under the assumption that T is not too close to unity. For the details of its numerical treatment by specifying ψ 0 as Watson's flow profile and marching downstream we refer to SBP18.
The given adjustment lengthL serves to defineH andŨ via (2.1). Hence, for a given flow, we note the invariance of (2.1) and thus of , ψ, (2.3) and G, T , P − under the affine transformation L ,H,Ũ , x, y, h ± , p, g, τ, J, λ → aL, aH,Ũ a , x a , y a , h ± a , a 2 p, a 3 g, aτ, aJ, a 2 λ (2.10) with a > 0 being an arbitrary scaling factor. This confirms the independence tõ H of the canonical formulation of the interaction problem below and thus on the specific choice of the streamwise length scaleL (for a sufficiently small =H/L). In particular, its solution downstream of the edge does not depend on the scaling of the attached flow and, specifically, the position of the aforementioned virtual origin. For any subsequent numerical evaluation involving ψ 0 and h 0 , however, we not only assume the flow as being fully developed but also adopt the natural standardisation x v = −1 from here on, i.e. we specifyL to be the full development length.

Main deck
Since the MD describes a predominantly inviscid flow in the long-wave limit, the central local expansion reads (2.11) and p = O( 4/7 ). The local streamwise variable X = O(1) is defined in (2.13) 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 ψ about ψ 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( 2/7 )-contribution to ψ 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( 4/7 ). The disturbances described so far account for the role of the MD for the interactive mechanism. The O( 4/7 )contributions to p and to ψ, the latter induced subsequently by the streamwise pressure gradient, are specified in SBP18.

Lower deck
In the LD, the expansion 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 and (2.3c,d ) to the mixed BCs expressing the downstream passage from no-to free-slip along (2.14b) To match (2.12) and (2.11) subject to (2.5), we require that for (2.14c) The rightmost bracketed contribution herein is a consequence of (2.14a) and that the interactive flow branches off the unperturbed state given by [Ψ, P ] ≡ [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 = sgn(T − 1)(A − H − ), typically provoked by the streamline curvature in the MD (as introduced by Smith 1977; Smith & Duck 1977 and, for an unconfined wall jet passing an abrupt edge, Smith 1978) 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: (in SBP18 only the case P − = 0 was considered). We thus arrive at the P /A law in the form (2.14f ) We furthermore introduce D(T ) = 1 − C(T ). The upstream case (X 0) is included in this interaction law for the sake of completeness and clarity. Downstream of the edge, it accounts for a subtle interplay are referred to tacitly from here on. The pole of C points to an interesting local increase of the capillary action for T ∼ 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 to 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 subject of ongoing investigations. The rescaled shear stress exerted at the plate, Λ(X) := Ψ ZZ (X, 0), plays a crucial role for the (unambiguous) formulation of the initial conditions (ICs) imposed at the plate edge X = 0 by SBP18 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.
(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 Λ towards zero in the limit X → 0−, but only the first of these typical edge conditions can be met.
(II) If 12/7 T < 1 (S = −1, T = 1/2), (2.14g) 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.
(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 Ψ and A above the edge.
The sought quantities Ψ and P satisfy the, with respect to X, first-and secondorder equations (2.14a) and (2.14e). In turn, three ICs are required to continue marching over the edge: (2.14h) (or, equivalently, A (0) = −SG). These complete the interaction problem (2.14) for the free jet. Here the flow profile at detachment Ψ (0−, Z) and A (0−) are taken as obtained by the preceding sweep of numerical marching towards the edge. It is stressed that Ψ , P behave regularly as X → 0−. Moreover, these quantities are continuous across the edge except for the shear stress Ψ ZZ on Z = 0, owing to (2.14b).
We also recall the behaviour, inferred from (2.14a,b), for Hence, the finite slip emerging along the lower free streamline, U s , supersedes the finite plate stress Λ upstream of the edge. We note that (2.15) first implies The apparent non-uniformity of (2.16) 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 Evidently, the support pressure behaves as a body force counteracting gravity.

Some important aspects
To achieve the last requirement in (2.14h), the interaction is initiated in the limit X → −∞ by a controlled branching from the oncoming base flow, here maintained as the trivial solution Ψ ≡ Z for X 0 if G = P − 0. Hence, the case G > P − requires branching of expansive type as scrutinised by SBP18 (where P − = 0 throughout) and the opposite one 0 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 (1992) and SBP18 and the preliminary presentation of these interactive undulations by Scheichl, Bowles & Pasias (2019). Their revealing linkage to unsteady linear capillary waves is given in appendix B. Moreover, SBP18 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 Λ 0 > 0 in (2.16). Since realistic values of τ and J by (2.8) yields T 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 → 0 in (2.14e) and (2.14h), implying the absence of interaction (P ≡ 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 1930) is recovered immediately downstream as the trivial solution [Ψ, P ] ≡ [Z 2 /2, G], representing the oncoming base flow, applies upstream of it.
3. Inviscid detachment at smaller scales As emphasised in more detail below, the interactive flow structure leaves us with a still singular transition from no-to free slip. It therefore initiates its own breakdown on scales much smaller than the interactive ones. The bottom line of the subsequent analysis is that of demonstrating self-consistency of the interaction theory and a required smooth behaviour of all flow quantities at the edge demands a thorough analysis of the smaller scales (figures 3b-d). This will also highlight the strikingly different characteristics of the gross break-away of the film, i.e. the formation of a free streamline at the solid wall, in the present situation and (well-understood) steady internal separation. In the first, the flow quantities appear to undergo weak algebraic singularities, whereas in the second their behaviour is well-known to be regular at separation (Goldstein 1930).

The influence of capillarity
To advance further in completing the description of flow detachment, it proves useful to first summarise the analysis in SBP18 of the interplay of surface tension and the Goldstein wake in the non-interactive limit x → 0+.
Here the latter exerts a displacement −ax 1/3 with some constant a > 0 (a 1.0079 if ψ 0 is given by Watson's profile on top of the wake), so that ψ ∼ ψ 0 (z) + ax 1/3 ψ 0 (z) + O(x 2/3 ). Accordingly, (2.3c), (2.6) and the . By integration across the unperturbed layer, from y = 0 to y = h 0 , one finally obtains from (2.3e) the limiting overall capillary pressure jump in the form (a − a − ) (2.14f). One draws the important conclusion that h − (x) is required to be regularised on the interactive and again on smaller scales even for T 0, whereas h + (x) (> 0) remains continuous at x = 0 for T = 0 as the inverse Prandtl shift produces additional irregular terms in the core region for x → 0+ and a cuspidal distortion of h + (x) exists for T > 0 only. Even then, however, the complete regularisation of h + (x) is left to higher orders over the interactive x-scale, where it is accomplished by the introduction of a thin shear layer adjacent to the upper free surface in order to satisfy (2.3d) (cf. SBP18, § 3.3.4).
It is noteworthy to highlight the difference to the related classical situation of the gravity-and capillarity-free axisymmetric flow exiting a pipe (Tillett 1968). There symmetry cancels the leading-order displacement in the core region but the vorticity gradient of the Hagen-Poiseuille profile (as opposed to streamline curvature) provokes an higher-order displacement and vertical pressure, requiring a regularisation similar to that discussed below.
Keeping in mind the above preliminary considerations operating for arbitrarily small values of T , we consider the precise regularisation of h ± for finite values of T . To this end, we first reappraise the interaction under the first of the restrictions (2.14g). The details of the detached flow in the close vicinity of the edge as reported by SBP18 provide an insight into how the full interactive structure is recovered for 9/14 X = O(T 3/8 ). In general, the so-called near-near wake, replacing the pressure-free Goldstein near-wake, emerges as a subregion split off the main portion of the LD to absorb the nonlinearity of the interaction immediately downstream of the trailing edge. Most importantly, it dictates the onset of free slip according to (2.14b).

Extended Hakkinen-Rott wake
As the second of the ICs (2.14h) requires A − A(0) = O(X) (X → 0), the nearnear wake must suppress any larger contribution to A, hence transferred passively through the core of the LD. As a consequence of this leading-order analysis, this wake itself then provides an example of condensed interaction through an interesting, capillarity-controlled specification of the pressure-driven Hakkinen-Rott wake (HRW, Hakkinen & Rott 1965): P vanishes as X → 0 in an irregular manner such that the wake exerts zero displacement. Since the canonical pressure gradient in the HRW turns out to be adverse, the capillary pressure jump (2.14d) enforces the lower free streamline to be convex immediately downstream of detachment in X = 0 (where it is curvature-free). It thus bends vertically upwards as X grows. The strong pressure rise provokes an enhanced streamline curvature, and this in turn the aforementioned breakdown and required smoothing of the interaction theory for sufficiently small values of X, as already indicated in figure 1. In the LD, this behaviour may be fully understood if one considers only the behaviour of the leading-order quantities Ψ and P , i.e. under the neglect of the vertical pressure variations.
The flow profile in the HRW matches that at detachment at its upper extent in its limiting form given by (2.16). As a result, the self-preserving flow in the HRW discerned for X → 0+ resolves the non-uniformity of (2.16). It is expressed as the inner limit (0) and the matching condition f HR ∼ η + TST as η → ∞ is recalled. The absence of a constant displacement term determines the eigenvalue p HR and prevents A from being of O(X 1/3 ) as X → 0+ and enforces continuity of A as required by (2.14h). Our refined numerical study yields p HR 0.61334 and a rescaled free slip f HR (0) 0.89915 obtained with max(η) = 50 (cf. Hakkinen & Rott 1965, SBP18). This gives U s ∼ f HR (0)X 1/3 (X → 0+) in (2.15) when rewritten in the limit η → 0.
Next, we propose the regular/singular upstream/downstream behaviour including higher orders with the logarithmic variations and the constants c 1 , c 2 to be determined through a higher-order analysis of the HRW. Accordingly, from (2.14e-g) or (2.17), Our expectation of a more nonlinear theory superseding the current one when T crosses 1/2, at the pole of C(T ), complies with the sign change of the singular contribution to A provided by the HRW. That weak downstream irregularity is also transferred to H + , cf. (2.11), as (3.4) By the expansive type of interaction for S = −1, A(X) bends convexly but P (X) concavely throughout (SBP18). That is, we can expect here A(0) > 0, A (0) > 0, but P (0−) < 0. One infers from (2.14c) that the i-th (i = 1, 2, . . .) contribution to the expansion for Ψ − Ψ 0 as X → 0 attains the form d i (X)Z + e i (X) + TST as Z → ∞ where the series of gauge functions d i and e i are determined by the expansions (3.2) and (3.3) and add up to respectively A(X) − A(0) and [A(X) 2 − A(0) 2 ]/2 + P (X). Typically, e i (X)Ψ 0 (Z) are the eigensolutions of the linearised convective operator in (2.14a). By matching Ψ in the LD and the MD, the solution of the inviscid version of (2.14a) indeed yields the accordingly refined form of the expansion for Ψ given by SBP18 (as (3.2), correctly including the logarithmic terms). So, with ∆P expanded as in (3.2), we have for (3.5) A detailed higher-order analysis of the HRW demonstrates self-consistency of the interactive asymptotic structure for X → 0. Amongst other aspects, it fixes the dependence of the coefficients c 1 , c 2 in (3.2) on the parameters characterising the LD flow in the limit X → 0−. Here we refer the interested reader to Supplement A.
The breakdown and so a required regularisation of the interactive flow structure for sufficiently small values of X is due to an unbounded vertical flow component and vertical pressure gradient evoked by the O(X 2/3 )-term in (3.2) and (3.5) and the associated O(X 3/8 )-term in (3.3). As a crucial observation, even then the pressure gradient in the HRW stays imposed by the flow on its top and must vary such that a potential singular displacement varying with X 1/3 is suppressed. Since the self-similar structure of the HRW already absorbs this type of condensed interaction and is recovered at its origin closer to the trailing edge, (3.1) prevails even over an x-scale much smaller than the interactive one. As a result, h − is still given by (3.1) in § 3.3 below.

Outer and inner Euler regions
We here consider the two nested square outer and inner vortical-flow regions (when measured by the equally scaled global horizontal and vertical coordinates x and y) that supersede locally the MD (outer) and the LD (inner) but where ψ ∼ ψ 0 and Ψ ∼ Ψ 0 still govern the flow at leading order. The associated linearised Euler stages (outer and inner RS) account for the small-scale upstream influence, within that on the interactive scale, and serve to regularise the singular behaviour predicted in § 3.2; most importantly, h + (x) by virtue of H + (outer). It is furthermore noted that the aforementioned large-Z representation of the expansion (3.5) accompanies a passive re-ordering of its hierarchy, so as to match the small-X limit of (2.11) provided by (3.3). Accordingly, the last expansion enforces a contribution of O(X 3/8 ) to (3.5) and this in turn a pressure-driven one of O(X 2/3 ) to the non-interactive disturbance of O( 4/7 ) in (2.11).

Preliminaries
Introductory considerations lay the foundation for the outer and the inner mechanism for the further regularisation of the HRW, as follows.
( 3.7) However, as p and ψ of O( 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 inner RS is of only subordinate importance compared to that of the outer one.
(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): To express Ω as the vorticity conserved along the streamlines, we use ψ −1 0 to symbolise the inversion of the corresponding leading-order relationship ψ ∼ ψ 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 → ∞ at the base of the outer RS and on top of the inner RS and by matching (3.5) for Z → 0 and (3.1) at the base of the latter. Consequently, eigensolutions of the linearised operator in (3.8) are absent.
It is noteworthy that Stewartson (1968) 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 1969;Messiter 1970).
It is illuminating to demonstrate that the up-and downstream asymptotes are already intrinsic to the problem (3.14) governingΨ andH. To this end, we consider the weakest admissible, i.e. first algebraic, decays ofV forX → ±∞ with unknown dominant corresponding ratesā ± (X), say. We obtain from (3.14a,b), using (∂ yy − ψ 0 /ψ 0 )V ≡ (ψ 0V y − ψ 0V ) y /ψ 0 and standard methods and (2.5), the long-wave approximation ofV (3.15) whereā ± and the constantsb ± are determined by solvability conditions of the inhomogeneous problems governing the O(ā ± )-and the O(ā ± )-term respectively. The small-y behaviour of ψ 0 in (2.5) grants a corresponding regularity of the right-hand side of (3.15). Substitution of (3.15) into (3.14d) using (2.5) and (2.6) gives, after division by u + 0 , the solvability relationā ± J − λθ(X)X −1/3 ∼ā ± τ . In the upstream case, this statement can only be met in the limit T → 1−, cf. (2.9). Consequently,ā − ≡ 0,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 ofV is expected for T → 1−. In contrast,ā ( 3.16) confirms the aforementioned leading-order asymptote involving M defined in (2.9). This shows that matching (3.10) and (2.11) requires T < 1. As a further result, (3.12) yields andP ∼ 3λΛX 2/3 /2 (X → ∞) provides the match of p in the MD, according to (3.2), (3.5) and (3.14b). This andP (X, 0) = 3λΛ θ(X)X 2/3 /2 make evident how Ψ andP 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 ψ 0 /ψ 0 in (3.14a) becomes, from (2.5), ωy for y 1, (3.14b) allows V to attain an undesired potential-flow pole in the origin, as described by the singular eigensolutions of the Laplacian r −N sin(N ϑ) where and N > 0 is some integer (cf. Scheichl 2014). 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 and ( We first assume thatV decays exponentially far upstream. Since it grows with O(X 5/3 ) asX becomes large, (3.20) defines φ{V } first in the open strip −µ 1 (T ) < Im k < 0 where −µ 1 denotes the imaginary coordinate of the pole in the lower half-plane Im k 0 lying closest to the real axis. The analytic continuation ofV into the entire k-plane excluding the locations of singularities is provided by the convenient decomposition (3.21) 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 withX in (3.16), it captures the influence of the HRW and gives a non-trivialV . Poles of 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 subject to the then inhomogeneous lower and the homogeneous upper BC, cf. (2.5). 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 determineV , parametrised by ψ 0 (y) and T , in an elegant manner, avoiding the above-mentioned Laplacian eigensolutions; all the more, as our focus lies on H(X) given by (3.14c). For the numerical implementation of (3.22), we recall that ψ 0 is typically specified by Watson's (1964) flow profile. In turn, the properties (2.6), (2.7) and the closed form of ψ 0 in Scheichl & Kluwick (2019) and the values for J = λ and u + 0 given by (2.8) are employed. Detailing the properties of (3.22), especially the behaviours of V for k → 0 and |Re k| → ∞ and the analysis of its poles, which select the discrete spectrum ofV 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 ofV in most efficient manner as envisaged next.
The poles of 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 = ±k u (T ) with k u > 0 ( § B.1) and an infinite number of simple poles lying on k = ±iµ i (T ) (i = 1, 2, . . .) with µ i > 0 ( § B.3). Since V(−k, y) ≡ V(k, y), Res k=−ku (V) = − Res k=ku (V) and real, and Res k=−iµi (V) = Res k=iµi (V) and imaginary. We then have where all possible paths of integration C stretch from Re k = −∞ to Re k = +∞ and originate from one another through a continuous deformation as they divide branch cut the k-plane in two portions: the origin and all poles k = iµ i (T ) lie in the upper and all poles k = −iµ i (T ) in the lower part. We furthermore anticipate that both real poles are located either in the upper or the lower part to guaranteeV being real. Indeed, as will be argued below to renderV unique, C must bypass both real poles such that they lie in the lower part. This situation is sketched in figure 5 with the path C specified for the numerical calculation ofH(X) by means of (3.23) and (3.14c) forX 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 C to follow the real axis apart from small squares of lengths 2ε with the midpoints k = ±k u and of length ε with the midpoint in the origin. Consistency of the results is confirmed for values of ε ranging from 0.1 to 0.3. On the other hand, applying Cauchy's residue theorem to (3.23) yields with (3.21), the fact that Res k=−ku (V) = −Res k=ku (V) and Euler's reflection formula after some algebrā Tillett 1968). This series of residues converges (uniformly) for anyX < 0. The full evaluation of (3.23) and smoothness ofΨ for y > 0 inX = 0 confirms that (3.24) holds even there although the decay of the exponentials has disappeared. Finally,H/Λ forX 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 2004, p. 30;Vanden-Broeck 2010, § 2.4.2). We can infer it directly from that of symmetric Squire modes (Squire 1953) as discussed in appendix B, hence with k/2 therein replaced by k here. Such stationary modes then exist for the two wavenumbers k = ±k u satisfying 1 = T k u tanh k u . In the current setting, we extract from (3.24) the neutral amplitude normalised withΛ (3.25) A linchpin of the analysis in Supplement B is the asymptotic representation of k u and Res k=ku (V) as k u vanishes andā u diverges for T → 1− ( § B.1) and the the qualitatively reciprocal behaviour for T → 0 ( § B.1). In combination with (2.8) (for x v = −1), this boils down to the following, numerically valuable, finite limits obtained with high accuracy: To compute (3.23) (for y = h 0 ), we restrict the numerical integration to the interval |Re k| 20, which in view of the exponential large-k tails of V ( § B.1) gives satisfactorily accurate results. Specifically, we find V(k, h 0 ) = O(exp[−|k|h 0 ]/k). The evaluation of the integrand employs a cubic-spline interpolation of the solution 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 V(k, h 0 ) ( § B.1) and finally adding their inverse Fourier transform, which results in the reciprocal large-X representation ofV andH 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 ofH by integration. To evaluate (3.24) (for y = h 0 ) and discreteX-values, the poles of V are detected as the roots k = k p , say, of V −1 (k, y). Since V ∼ Res k=kp (V)/(k − k p ) as k → k p , the according residuals (given by a homogeneous solution to (3.22), see above) are computed as 1/[∂ k V −1 (k p , y)] (y = h 0 ). For i > 7 andX 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 resulting plots in figures 6 and 7 also employ 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 ofX and for small values of T than those found by the direct evaluation of (3.23). Figure 7(b) indicates that excellent agreement with the asymptotes found analytically can be achieved. It is seen thatH undergoes a trough immediately downstream of the edge before it recovers to rapidly assume the algebraic fardownstream 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 → 0. Even the maximum value of k u shown lies on the part of C considered for the numerical integration, but the suppression of exponentially growing terms in the calculation of V and the residuals becomes a numerically delicate task when |k| becomes sufficiently large. In the long-wave limit k u → 0 as T → 1−,Ψ diverges both immediately upstream of the trailing edge, asā u grows like k −14/3 u , and for constant but sufficiently large positive values ofX. Also these findings compare favourably with the curves in figure 7. The intriguing further implications of the long-wave limit are addressed in § 5. We complete the numerical analysis by demonstrating the excellent agreement between the computed wavelengths 2π/k u , see figure 7(a), and their leading-order asymptotes: for T = 0.95 and T = 0.8, (3.26a) predicts those as about 15.570 and 7.785 respectively; for T = 0.1, (3.26b) gives a wavelength of about 0.542. In addition,ā u 8.654 × 10 −10 . The details of this case displayed in figure 8 shows that our numerical method resolves even the rapid oscillations of exponentially small amplitude for rather small T -values with surprisingly high accuracy.

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 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 ( 2/3 ) in (3.10) morphs into the pressure-free one 2/3Ψ y (X, 0)Ψ 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 viscous sublayer or SL (figure 3b) where Z = O( 1/21 ) is required on account of the no-slip BC. For very negative values ofX, this shear layer is of the type provoked by the rapid small-scale disturbances considered in SBP18. ForX 1/7 (X 1), it becomes absorbed into the HRW, there serving as the viscous correction of the LD; for larger values ofX, an additional perturbation in the expansion of − 2/3Ψ y (X, 0)U s (X) of h − serves to satisfy the free-slip condition ψ zz ∼ 0 on z = 0 to which (2.3d) 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 (1992) 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( 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 (1979). The last authors also point to the rigorous justification of the above observation by solving the signalling problem, following DePrima & Wu (1957). 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 ψ 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 1990, § 3.4). This plausibility argument serves to single out the mode upstream as the physically meaningful alternative.

Diffusive overlayer
Expansion (3.10) accounts for the second dynamic BC (2.3d), requiring vanishing shear stress on the top free surface, up to O( 3/7 ), i.e. as long as (2.3d) reduces to ψ yy ∼ 0 on y = h + . Moreover, it was indicated in SBP18 how (2.3d) alters the highest-order contribution of O( 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( 3/7 ). Since the flow therein itself becomes inviscid over the shortened Rayleigh scale, a further diffusion layer of reduced vertical depth arises wherē X and ξ := (y − h + )/ 1/2 are of O(1) and (2.3d) is formally retained in full. A comprehensive completion of the present self-consistent theory requires a brief examination of this overlayer meeting (2.3d): see Supplement C.

Inner Rayleigh stage: lower deck
Following the outline (b) at the end of § 3.3, the inner square region regularises P by taking into account the transverse variation of p, which becomes of O( 6/7 ) according to (3.19) and ( where we advantageously revert to the inverse Prandtl transposition in (2.13). Again, the quantitiesΨ ,P describe a linearised Euler flow, now with Ψ 0 providing the base profile. Therefore,V (X, Y ) := −ΨX 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 2/7 ) = (X 2 + Y 2 ) 1/2 , where the displacement of the HRW controlsV by virtue of a R −1/3 -variation matching (3.19). 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 (3.19) recovers the far-field singularity also for R → 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 slipΨ y (X, 0) exerted by the outer RS becomes of O(X 1/3 ) as X → 0−, the vertical extent of the associated SL introduced in § 3.3.3 shrinks typically to Y = O( 1/21X 1/3 ). It is continued as a sublayer covering the inner region where Y = O( 1/7 ) (figure 3c). There the driving slip is replaced bŷ Ψ Y (X, 0), which again attains aX 1/3 -behaviour asX 1/3 → 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.

Full Navier-Stokes and Stokes regions
As the conditions (2.14h) take into account the detachment of the lowermost streamline but not the edge as a geometric restriction or even its micro-geometry on the length scales considered so far, the prior analysis does not determine whether detachment occurs actually at the edge or further upstream. Therefore, this question is taken up first through an examination of even smaller scales, governing first a full NS regime. This ensues from a breakdown of (3.27) initiated by the unresolved singularity ofP , just discussed, and the associated unbounded growth of the vertical flow component, −ΨX. The associated growth of v evaluated in the HRW shows the emergence of the NS region. We will see that it in turn contains at least one Stokes region around detachment so that the flow can accommodate the wetting properties controlling the emerging meniscus and defined by the thermodynamic three-phase equilibrium holding in the detachment point.
(4.3e) The smallness of the remainder term imposed onψ provides the required second kinematic far-field BC. Since we are dealing with the full NS equations, (4.3) already captures the inner Rayleigh region and its subregions both upstream (SL) and downstream (HRW,ȳ ∼rϑ = O(r 1/3 ) there) of detachment; cf. figure 3(d ). That is, (4.3e) already implies (ψ,p) = O(r 2/3 ) andh = O(r 8/3 ) at the onset of the HRW. The BCs for ϑ = 0 in (4.3c) describe zero tangential stress along the detached streamline and the net normal-stress jump across it. Eventually, eliminatingp from (4.3a,b) yields the vorticity transport equation to be solved subject to the first two BCs in (4.3c) and (4.3d,e). Hence,ψ is solely induced and parametrised by the externally exerted shear rate Λ 0 . We recall that this is determined by the solution of the viscous-inviscid interaction problem on a larger scale and accounts for the upstream momentum flux, gravity and capillarity. The variation ofh withr is then found from integrating the capillary normalstress jump in (4.3c) and, given the identical match ofh and H − according to (3.1), two ICs to be imposed asr → 0. Before tackling their determination, we first identify the flow topology near detachment, solely based on the information extracted from the NS problem posed above in the limitr → 0. The importance of this insight by far outweighs the perspective of obtaining the full numerical solution. Therefore, we have refrained from tackling this considerable challenge. (The considerations below suggest spectral collocation in the ϑ-direction as the method of choice.)

Flow close to detachment
Asψ must satisfy four BCs in (4.3c,d ), the viscous terms are retained in the limiting forms of (4.3a,b) asr → 0 and ϑ ∈ [0, π]. Requiring strict forward flow in the immediate vicinity of detachment, is initially seen as a natural additional constraint. It is supported by the extensive numerical investigation by Kistler & Scriven (1994) of the full NS problem for a flow passing a wedge-shaped lip, see figures 2(b) and 3(f ): this predicts an eddy at its underside in some situations associated with rather low to moderate Reynolds numbers but strictly forward flow detaching at its tip in the present high-Reynolds-number limit. The analysis below, however, demonstrates that (4.5) is only met in the least singular situation chosen from the initial alternatives.

The full inertial-viscous limit
The convective-viscous balance in (4.4) is restored in full ifψ varies essentially with lnr: We are thus concerned with a spiralling extension of a special type of a radial Jeffery-Hamel (JH) flow described byḡ(ϑ) (see Fraenkel 1962), exhibiting the vorticity −∆ψ = −ḡ /r 2 and an outwards flow speedḡ (ϑ)/r as collapsing in a line source of strengthḡ (ϑ), due to a superimposed potential vortex of some strength Γ . Here the homogeneous BCsḡ(0) =ḡ (0) =ḡ(π) =ḡ (π) = 0 originating in (4.3c,d ) require Γ = 0, andḡ represents an eigensolution of the full NS problem. Nevertheless, the case Γ = 0 andḡ ≡ 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 (4.6),ḡ(ϑ) varies sinusoidally in general but linearly in the case of a potential flow.
An analytical-numerical study shows that there exist two eigensolutionsḡ. Each describes a distinctly different canonical flow topology as both exhibit a dividing streamlineḡ = 0 for ϑ = ϑ 0 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 (ḡ < 0 for ϑ 0 < ϑ < π) or fully detached as bounded by the free streamline (ḡ < 0 for 0 < ϑ < ϑ 0 ): see figure 9(a). In the first case, the flow undergoes pre-separation to reattach in the originr = 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. 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.

An extended Stokes limit as the alternative
Discarding the possibility of a full inner NS problem, (4.6), leaves us with the degenerate situation of the dominant Stokes balances 0 ∼∆ 2ψ ,pr ∼∆ψ ϑ /r,p ϑ ∼ −r(∆ψ)r (4.7) andψ → 0 as the originr = 0 is approached along any path from within the flow. We then expandψ into the eigensolutionsψ i of the biharmonic operator in (4.7) when supplemented with the homogeneous BCs in (4.3c,d ) found by separation in the polar variables, following Moffatt (1964) 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 forr → 0 towards a separating flow (ψȳȳ = 0): The constant a 5 and the offset pressurep 0 are part of the solution to the full NS problem, in turn, forced by the value of of Λ 0 . Let us first indicate how to fix the unknown coefficientsh(0) andh (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,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 π that it is approximated by π − 3/2h (0). This determines a positive value ofh (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 constrainth > 0 forx > 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, thenh (0) remains undetermined as long as its microscopic shape remains unresolved.
The outcome of these considerations is threefold. Firstly, we expect bothh(0) andh (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 third, that new length scale must resolve the microscopic contour of the plate with sufficient accuracy.

Distinguished Stokes limits and wetting properties
Although possibly not satisfied in a particular realisation of the flow, let us treat the surface of the plate as locally chemically heterogeneous and ignore distributed roughness on all scales for the sake of clarity. Then the so-called quasi-static apparent contact angle, β, is observed between the wetted plate and the tangent to the free streamline at its point of detachment and formation, where three phases (locally) at rest meet under the Young-Dupré equilibrium: for its precise conceptual foundation we refer to Teletzke, Davis & Scriven (1988), Kistler & Scriven (1994), Whyman, Bormashenko & Stein (2008) and Bonn et al. (2009). Since this macroscopic contact angle summarises all related submicroscopic phenomena (see Kistler & Scriven 1994, and references therein) and shall apply even to the smallest scales identified in the flow, we have consistently used the notion "microscopic" in the context of the resolved geometry of the trailing edge.
The case of a perfectly flat surface associated withĥ 0 ≡ 0 and the trivial solutionψ = −4a 5ŷ 3 of (4.12) recovers the dominant Stokes limit of the full NS solution forr → 0 and the aforementioned pathological case of fully hydrophobic dewetting with both x d and 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 . 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 β (cf. Duez et al. 2010). 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 α and with an apex lying at the coordinate origin. Then the curvature radius typical of the rounded nose conveniently defines ; the degenerate situation of a wedge still found sharp when viewed on the scale 2 is assumed in the limit ∆ → 0. The case of specific relevance α = 0 can be interpreted as a plate-type thin tip formed by a semi-circle and of local thickness 2 (figure 10b).
Assuming ∆ = 1 in (4.11) andH = 1 mm (table 1 in appendix A) typically gives a quite small physical scale H 0.01−0.04 µm. 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 β as the wedge geometry imposes a closure condition on (4.12). This fixes the location of D on Σ : dŷ/dx ∼ tan β + o( 3/2 ). (4.15) This describes the general, non-degenerate case whereh(0) is found in virtue of (4.10). Evidently, then alsoĥ (0) = 0 as the linear follow-up problem to (4.12) governing disturbances of O( / 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 (4.15). Proceeding in this manner determines successively the two initial conditions that each term arising in the expansion of h − in (4.1) has to meet asr → 0. This consideration confirms selfconsistency of the proposed theoretical framework. As a crucial result, the flow wets the underside of the wedge asĥ 0 represents a (strictly) monotonic function of β, which decreases from 0 as β decreases from π. This justifies our reference to the teapot effect. The pathological limit β → π− or (x d ,ĥ 0 ) → (0, 0), however, leads to a non-trivial value ofĥ (0). Here we only note that the above analysis by inspection gives = O( ) in the degenerate caseh(0) = 0,h (0) > 0. On the other extreme, D has reached the point on the nose where its curvature vanishes once β has become as small as α. All together, we arrive at the geometrical constraint π − 3/2h (0) β α. (4.16) The variation ofĥ 0 with β is more and more squeezed towards the edge as this gets sharpened. Finally, D is seen as pinned to the edge as (4.16) is interpreted as the well-known Gibbs inequality: see Oliver, Huh & Mason (1977); Dyson (1988); Kistler & Scriven (1994). In accordance with the last authors, we find that the distance of D from the apex decreases with both increasing values of β and the Reynolds number. The formidable task of solving the Stokes problem (4.12), parametrised by α and β, 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 π > β > 0 of physical significance. If, however, β α, determining the actual position of D requires the introduction of a further, inner Stokes region, as indicated in figure 10(a). Contrasting with its counterpart (4.12), there the governing problem is of 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 D to be found the further away from the apex the smaller is β, with its position fixed by a constraint arising of the interplay of these nested Stokes regions. This is a topic of our future activities.
As the final step, we focus on the flow properties in the immediate vicinity of detachment, specified on condition (4.16). Here we again follow Moffatt (1964) in his analysis of local eigensolutions of (4.12) 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 (1964, § 3.2) also showed that a related class of eigensolutions controls the behaviour ofψ at small distanceŝ d = [(x −x d ) 2 + (ŷ −ĥ 0 ) 2 ] 1/2 from the detachment point: using (4.12a,c,d ) and reusing the azimuthal angle, ϑ := arctan[(ŷ −ĥ 0 )/(x −x d )], yields for 0 ϑ β and (4.17) The constantâ is determined by the full solution to (4.12), and σ appears to be a (complex) eigenvalue related to β by (σ − 1) sin(2β) = sin[2β(σ − 1)] (σ = 2), tan(2β) = 2β (σ = 2). (4.18) A continuous relationship requires β = 3π/4 for σ = 2. One readily confirms that the eigensolutions considered in § 4.2.1 are recovered in the limit β → π. Equation normal stress (the pressure), both of O(d σ−2 ), on Σ being integrable asd → 0 (and not to compromise the validity of the Young-Dupré equilibrium). Thus only values of σ having Re σ > 1 are permitted, anticipated by the requirementψ = 0 at detachment in (4.12d). The plot of the real branches of (4.18) in figure 11 illustrates the infinite multiplicity of σ, not considered by Moffatt (1964), the asymptotes β → π/2, π as Re σ → ∞ and the local extrema of β. There (4.18) is continued to complex values of σ, via (4.17) associated with Moffat's (1964) prominent and exceptional infinite sequence of eddies. Hence, our flow model does not predict a single eddy as do the calculations by Kistler & Scriven (1994) for moderately large Reynolds numbers but this series of eddies if the value of β falls below its absolute minimum. Moffatt (1964) predicted this well-established threshold as 78 • ; here we recompute it as 79.557 • for σ 3.7818. A more elaborate discussion of these details and their consequences requires the yet pending full numerical solution of (4.12).

Conclusions and further perspectives
As an unexpected extension of the interactive flow structure around flow detachment at the free plate edge, we report neutral capillary Rayleigh modes on the upper free surface solely and immediately upstream of the edge. Demonstrating this confidently calls for solving a signalling problem where typically a compact forcing dividing the flow into an upstream and an downstream part. Here this is provided by a delta functional describing the transition of the vertical flow component over the geometric discontinuity formed by the trailing edge but tied in with an additional non-compact excitation by the displacement of the Hakkinen-Rott wake, necessary to provoke the non-trivial Rayleigh state.
The small-scale/small-amplitude ripples differ markedly in their origin from those already predicted by Bowles & Smith (1992) upstream of a wall-mounted obstacle over the interactive length scale. Accordingly, they are separated a streamwise extent of O( ) from those of much larger wavelength found in the solutions of the interaction problem and set off by that wake in the downstream direction on both free surfaces in phase for 0 < T < 1/2 ). On the other hand, since these rather long waves on the upper free surface are observed even upstream of the trailing edge for T > 1 (Scheichl et al. 2018), they collapse there with the short Rayleigh modes when T ∼ 1, as the longwave limit of the latter indicates ( figure 6). This heralds how the introduction of a reduced streamwise length scale paves the way for an Euler stage to regularise the breakdown of viscous-inviscid interaction in a more general setting when the measure T − 1 of the typical counteracting dispersive effects, namely capillary versus convective streamline curvature, of classical Korteweg-de-Vriestype changes sign. Although already identified in related studies (Gajjar 1987;Bowles & Smith 1992;Kluwick et al. 2010), this has not been investigated in due detail so far. Having in mind the anomalous dispersion for classical linear capillary waves, we find it appropriate to speak of "choking" when both the wavelength and the amplitude of the capillary ripples, controlled by the dominant eigensolution ψ 0 (y) of the Rayleigh operator and triggered by the displacement of the Hakkinen-Rott wake, diverge for T → 1−. This consideration highlights the identical nature of the threshold T = 1 in this long-wave limit as for the interactive flow. For the current state of our research on the interactive stationary waves we refer to  and appendix B.
Neither the onset of the interactive, long waves above the plate for T → 1+ nor the formulation of additional conditions imposed at the plate edge to account correctly for the upstream influence that render them unique have yet been clarified satisfactorily (cf. Scheichl et al. 2018). This and other exciting related phenomena attributed to the solution of the interaction problem downstream of the edge, such as its sound regularisation when T − 1/2 changes sign and attracting attention through (2.14d -f ) and (2.17), are topics of our current research. A stability analysis of the detached interactive flow, where unsteadiness of the streamwise momentum balance becomes typically explicit in the lower deck, should clarify the analogy of the capillary waves with the classical linear Squire modes (Squire 1953).
As a major conclusion of our analysis, the layer undergoes its break-away from the trailing edge at its underside when this is geometrically resolved in a leastdegenerate but most simple manner as a (cut-back) wedge having a rounded nose. As a rule of thumb, the higher the wettability, the more the fluid "sticks" on the underside and the more the point of three-phase contact or detachment is remote from the plane wall on top. In the authors' mind, the present analysis rationalises for the first time how a high-Reynolds-number flow negotiates the formation of free streamline with due rigour. As the vital idea, any physically viable flow always "feels" a small reference length (the nose radius ) that resolves the abstraction of geometrical perfection (the sharp trailing edge). This then defines the smallest scales at play and hence controls the thereby arising Stokes limits and local dewetting or film rupture. As an interesting aspect, the convective influence and thus the flow profile stretching towards the upper free surface is only felt through a single coefficient of the dominant eigensolution of the Stokes operator. Pinpointing the flow on those smallest, geometrically induced length scales provides a self-consistent and qualitatively reasonable explanation of the teapot effect observed in the detachment of a high-momentum liquid layer. The underlying continuum hypothesis is admissible as long as the smallest scales are so large that the liquid/gaseous interface can be taken as infinitely thin. We hope that this appealing and promising approach stimulates future research in this direction.
The Stokes problems governing the steady, capillarity-dominated free-surface flow on the smallest scales constitute the central building block for completing the rigorous examination of the teapot effect. This appears as an essentially hydrodynamic phenomenon, but the adjustment of the flow to the three-phase equilibrium defining the wetting properties in terms of the apparent contact angle represents the most salient ingredient. More will be able to be said and further progress sparked once the inner Stokes problem is established and the outstanding solutions of these core problems are available.
Last although not least, we feel an urgent need for careful and systematic laboratory experiments, with the ultimate goal to corroborate the theoretical findings on all scales. Here the values in the tables 1 and 2 in appendix A might be helpful.
Acknowledgements. The authors express their thanks to Dr. S. N. Timoshin (Department of Mathematics, UCL) and the referees for fruitful discussions and helpful comments.
Funding. Financial support from the Austrian Research Promotion Agency (FFG,grant no.: 849109,COMET K2 program: XTribology) and from a UCL Mathematics Teaching Assistantship is greatly acknowledged.
Declaration of interest. The authors report no conflict of interests.
Author contributions. All authors contributed equally to this paper.

Appendix A. Orders of magnitudes and their physical relevance
Even though left unspecified here, a horizontal nozzle or the impingement of a vertical jet represent the most likely methods of generating a flow configuration of engineering concern and of the type considered here. The work then is certainly relevant for a variety of physical scenarios. However, one might question the validity of the order-of-magnitude requirements made in (2.2a) in a conceivable situation of industrial importance or even of observations in daily life -such as the falling jet generated by wielding a teapot. Such settings are characterised by feasible geometrical and flow conditions and an aqueous, viscous fluid under the action of gravity and surface tension. Indeed, the chosen largeness of the Froude number at a moderate Weber number deserves some comment. The following arguments yield the values, presented in table 2, of the non-dimensional groups in (2.2) and (2.2b), relevant to a film of pure water under standard conditions and based on the reference values of the input quantities as well asH andŨ , following from (2.1), presented in table 1.
Withτ 100 mN/m throughout (water as a polar liquid, and of potential interest, has a comparatively high surface tension), an adequately small Bond number g/τ =gρH 2 /τ (allowing the neglect of gravity over surface tension) is, however, definitely not smaller than 10 5 m −2 ×H 2 . This requiresH ≈ 1 mm; for much smaller film heights, effects originating in technically unavoidable geometric imperfections of the plate surface might no longer be negligible (but worthy of study). Likewise, g 1 (allowing the neglect of gravity over inertia) is achieved if U 0.1 m/s. Given their rather narrow range of physically acceptable values and prediction of an extremely thin and fast film, these estimates have admittedly to be adopted with some caution. As an essential finding, the Reynolds number −1 proves to be indeed large but not to the extent that laminar-turbulent transition becomes an issue. However, the accompanying rather large value of ρ (kg/m 3 )ν (mm 2 /s)τ (mN/m)g (m/s 2 )Q (l/min)L (mm)H (mm)Ũ (m/s) 998.20 1.00 72.75 9.81 6 ≈ 50−60 ≈ 1 > 0.1 τ alleviates these doubts as it points to a numerically rather high sensitivity of the key parameters to slight variations of the input data. Also, the requirement = O(g 7/4 ) for e.g. g = 0.1 implies a reference or effective plate lengthL of about 5 to 6 cm, which seems sensible, and ≈ 0.018. We may check the reliability of the last estimate on the basis of the second relationship defining in (2.2a): the above estimate forŨ predicts values for barely smaller than 0.01. Given the potential variety of the input data, we achieve a satisfactorily good agreement. Our prerequisites, summarised in (2.2a), can then be considered as self-consistent.
Most critically, the validity and sensitiveness of the scalings originate in a sufficiently small typical film heightH rather than in the values of the remaining physical parameters. Nonetheless, the subsequent asymptotic analysis of (2.3) in the distinguished limits provided by (2.2a) remains valuable even if the underlying order-of-magnitude estimates should be interpreted with a greater flexibility. In particular, the actual value of τ is taken as definitely smaller than its upper bound stated in table 2.
Appendix B. Small-amplitude waves For the following instructive analogy to (unconditionally stable) linear Squire modes, perturbing weakly a planar, uniform jet having constant speed in the xdirection and two free surfaces y = h − = 0 and y = h + = h 0 , we refer the reader to Squire (1953), Drazin & Reid (2004, p. 30) and Villermaux (2020, § A.4).
Let k denote their wavenumber, non-dimensional withH, and c the ratio of their phase speed relative to the unperturbed jet speed. Using the definition of J in (2.5) yields the classical anomalous dispersion relation in the form (c − 1) 2 = T k × coth(k/2) (skew-symmetric modes), tanh(k/2) (symmetric modes). (B 1) Here the symmetry refers unambiguously to the u-perturbation with respect to the centreline y = h 0 /2. Hence, the antisymmetric modes give the picture of a sinusoidally meandering or flapping jet as h + ∼ h 0 + h − and h − are in phase. On the contrary, they are in antiphase as h + ∼ h 0 − h − for the symmetric modes, producing a "varicose" or symmetrically looking jet. These latter modes appear visually as the classical axisymmetric Rayleigh-Plateau modes, thus forming their counterpart on a circular jet (see Drazin & Reid 2004, p. 22 ff.; Villermaux 2020, § A.5). There exists a single stationary, choked mode (c = 0) for each value of T in the symmetric case but only for T < 1/2 in the antisymmetric one, where indeed T → 1/2− in the long-wave limit k → 0, resembling the interactive limit. Moreover, our first numerical solutions of (2.14) predict a sinusoidal modulation only of the detached jet if 0 < T < 1/2 and of varicose kind in the yet poorlyunderstood case T > 1, where the onset of the waviness of the upper free surface approaches the edge from upstream as T tends to 1 from above (see SBP18). These results allow for the following interpretation. The undulations for 0 < T < 1/2 represent a nonlinear, viscosity-affected variation of their classical counterpart, also strongly impacted by the background vorticity or the reduced fluid velocity at the lower interface. Like the classical ones, these vanish only for vanishing capillarity. For T sufficiently exceeding 1, the predominance of capillarity over both vorticity and the symmetry-breaking displacement effect implements a nonlinear modification of steady varicose modes. This analogy becomes evident from inspection of (2.11), (2.17) and figure 4: for sufficiently large |A|, we have H + ∼ (D − 1)H − ; thus sgn(H + ) = sgn(H − ) for T < 1/2 and sgn(H + ) = − sgn(H − ) for T > 1/2, where the symmetry of the varicose waves downstream of the plate allows also for their emergence above the plate; their failure occurring for T → 1+ is again associated with an unbounded LD displacement −A.

Other supplementary material
In this supplement (Supplements A-E ), we present the technical details required to reproduce the details of our analysis.
Supplement A. Hakkinen-Rott wake: higher-order scheme To elucidate the structure of (3.2) and (3.5) in more detail, we have to match the latter in the limit Z → 0 to the appropriate expansion holding for η = O(1) as X → 0+ and with F 1,2 to be found. The logarithmic contributions to (3.2)-(3.4) and (A 1), by anticipating c 1,2 = 0, comply with the representation A 2) of (3.5). This is found with the help of (2.16), in the overlap 1 Z X 1/3 or X −1/3 η 1, see (3.1), with the expansion (A 1). There they cause a reordering of its terms arising for large values of η, as seen from expressing Z in (A 2) in terms of η: As it is the HRW where the inertia terms in (2.14a) are fully restored, the Z-independent contribution to Ψ in (A 2) is again in agreement with (2.14c). Thus, matching the X-independent terms in (A 1) and (A 3) confirms that f HR − η 2 /2 ∼ p HR + TST (η → ∞); matching the higher-order terms requires the successive emergence of the logarithmic terms.
Supplement B. Outer Rayleigh problem: wavenumber space We investigate the Rayleigh problem (3.22) in the distinguished limit T → 1− and k → 0, for |Re k| → ∞ and the eigenspace (poles) of V in detail.
B.1. Singular limits k → 0, T → 1− The analysis of the long-wave limit k → 0 is closely related to the discussion of (3.15) with (3.16). We substitute the expansion V ∼ ∞ i=0 κ i (k)V i (y) (κ i+1 /κ i → 0) with, at first, unknown gauge functions κ i and O(1)-functions V i into (3.22). At leading order, the problem (3.22) then only permits the homogeneous solution parametrised by κ 0 , thus V 0 = ψ 0 . To allow for deviations from this, the analysis of V 1 requires κ 1 = O(k 2 κ 0 ). Specifying κ 1 := k 2 κ 0 yields the inhomogeneous problem where we have anticipated that κ 1 = O(1), including the alternative κ 1 → ∞ as k → 0 as a limiting case. By using the first BC here and after some rearrangements, The initially arbitrary constant α 1 indicates again a homogeneous solution and the first BC in (B 1) is met in the limit y → 0. The second BC in (B 1) represents a solvability condition for (B 1) as it gives to fix κ 0 = κ 1 /k 2 with κ 1 = λ/[(T − 1)J]. This analysis reveals a double pole of V at k = 0 the strength of which becomes unbounded as T → 1. It refers to an apparent solution |X|ψ 0 (y) (X = 0) to the homogeneous problem formed by (3.14a,c). Since k 2 enters (3.22) linearly, the above expansion is now specified as and breaks down passively where y = O(k 2 ). We solve the resulting hierarchy of the inhomogeneous problems using the approach that leads to (B 2): where the last BC in (B 5) fixes the constant α i−1 as a function of T in terms of a solvability condition for the problem governing V i−1 . We finally write this Developed liquid film past a trailing edge: 'teapot effect' S 3 constraint after some manipulations as the recursive relationship Also, V i (i > 0) is inversely proportional to 1 − T . Inspection of (3.14) reveals immediately the expansionV ∼ā , which completes (3.15) subject to (3.16) as the reciprocal form to (B 4). From this, one infers thatb + = −α 1 .
The above analysis ceases to be valid when 1 − T is so small that κ −1 1 = O(k 2 ) and thus no longer enters (B 1) but, instead, the problem governing V 2 . However, the asymptotic series (B 4) captures this shift of the lower BC formally when we introduce the (positive) parameter to quantify the consequent least-degenerate distinguished limit. Then κ −1 1 is replaced by 0 and T by 1 in (B 1), (B 2) such that (B 3) is satisfied identically. Most importantly, the BCs in (B 5) are modified to The special form of the dynamic BC in (B 9) determines the value of the constant α i−2 for i > 2. It is sufficient for our purposes to concentrate on this BC for i = 2. As V 1 is given by (B 6) for i = 1, this solvability condition for (B 10) yields with V 0 specified in (B 4) and the definitions of I 1 in (B 7) and T in (B 8), and after some rearrangements, an expression for κ 2 and thus κ 0 , independent of the value of α 1 : (B 11) We find that the distinguished limit (B 8) is rich enough to disclose the behaviour of V near the critical point k = 0 and T = 1 (but the last situation is excluded in this study). First, one readily finds that V admits a regular expansion in T − 1 as T → 1 for k = 0 or T → 0. It is seen that V attains a fourth-order pole in k = 0, T = 1 which morphs into a double pole for T < 1, here recovered in the limit T → ∞ where the two forms of (B 4) considered match. This behavior is associated with the bifurcation of a simple pole for k becoming positive, whose location we trace in the (T, k)-plane as k = k u (T ), (B 12) indicating the existence of undamped capillary oscillations, where T ∼ I 1 or As suggested by these asymptotic findings, our numerical study predicts exactly one value of k 2 = k 2 u for each value of T in the relevant interval 0 < T < 1. We also infer from V ∼ Res k=ku (V)/(k − k u ) (k → k u ), (B 4), (B 11) and (B 13) that The amplitudeā u , see (3.25), varies predominantly with k −14/3 u or (1 − T ) −7/3 in this limit. To exploit the above results numerically, we specify ψ 0 by Watson's flow, using (2.7) with x v = −1. Then I 1 can be transformed into a single integral, and we add The relationships (B 13)-(B 15) set the basis for (3.26a).
B.2. Singular short-wave limit |Re k| → ∞ For |Re k| → ∞, V obviously varies exponentially weakly with k. We first identify a viscous sublayer ζ := ky = O(1) where we take V as a function of ζ and k. There (3.22a) and (2.5) give ∂ ζζ V − V ∼ ωζV/k 3 + O(k −6 ). In turn, where e is some function of k satisfying e(−k) ≡ −e(k) − 1 as (3.22) enforces symmetry of V with respect to k. For y = O(1), the exponential variation in (B 16) is morphed into a rapid one, typically captured by a Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) ansatz. Inserting this into (3.22a,c) yields after some manipulations and exploiting the above symmetry property intrinsic to (3.22)  Expanding K for y → 0 with the help of (2.5) confirms the match of (B 16) and (B 17), up to contributions of respectively O(k −3 ) and O(k −1 ) in the brackets. This first yields two relationships involving e and E: e ∼ E e kh0 where we abbreviate the algebraic variations with k in terms of The first of the relations (B 19) verifies that V = O(1) in (B 16) and the second the exponential smallness of V for y 1/|k|. From (B 17), there where the exponentially weak remainder term is of O e ∓k(y+2h0) . Therefore, with the aid of (B 18). As seen from (B 19) and (B 20), the O(k −2 )-terms in (B 21) and (B 22) are correct as long as E ∼ − e −kh0 /A(k) or |A(k)| | e −2kh0 | for Re k > 0 and E ∼ e kh0 /A(−k) or |A(−k)| | e 2kh0 | for Re k < 0. However, they increase up to O(k −1 ) when these constraints are violated, that is, when both contributions to E −1 in (B 19) become of the same order of magnitude or k ∼ ±K 0 /2 + O(T ).
(B 23) The expression for K 0 in (B 18) heralds this possibility when T is so small that T k ∼ ±u + 0 2 /J + O(T 2 ). Moreover, E −1 might change sign then, which reveals the emergence of a real pole of V that represents the small-T asymptote of (B 12). Hence, the trace of the pole is known in a first approximation as weak deviations must account for its weak straining due to the higher-order corrections in (B 21): In turn, and as also obtained directly from (B 22), These delicate consequences of matching exponentially varying terms verify a-posteriori the inclusion of the algebraically varying ones. The asymptotic behaviours (B 24) and (B 26) are finally condensed into (3.26b).

B.3. Eigenspace and poles
As an important aspect shown next, for any T 0, the homogeneous version of (3.22) is solvable only for a countable, infinite set of real eigenvalues of k 2 bounded from above.
We letV k (y) symbolise the space of eigenfunctions, V k = (k 2 + ψ 0 /ψ 0 )V k , y = 0 :V k = 0, y = h 0 : ψ 2 0V k = T Jk 2V k . (B 27) In § B.1, we considered the eigenvalue k = 0 and the associated double pole. All other eigenvalues are expected to define the simple poles of V in the k-plane: an infinite number of conjugate imaginary and an, at most finite, number of real ones, these associated with isolated neutral capillary modes. To demonstrate these fundamental properties, we first consider two twice differentiable functions U(y), W(y) and the typical inner product h0 0 U W dy; in the following, overbars unambiguously indicate complex-conjugates. One readily confirms that the operator (d 2 /dy 2 − ψ 0 /ψ 0 ) U subject to U(0) = U (h 0 ) = 0 is self-adjoint, but the appearance of the eigenvalue k in the BC at y = h 0 impedes proving its typically expected properties in standard fashion if T > 0: k 2 is real; members of V k for different eigenvalues are orthogonal with respect to the above inner product. Rather, if U and W now denote eigenfunctions for different eigenvalues k 1 and k 2 , say, we obtain from (B 27) via integration by parts This prompts us to seek a transformation of (B 27) such that k no longer enters the BC for y = h 0 . To this end, we introduce the transformed eigenfunctions F k := ψ 0 V k − ψ 0 V k . We then obtain from (B 27) [ψ 2 0 (V k /ψ 0 ) ] ≡ F k = k 2 ψ 0V k (B 29) and, since ψ 0 (h 0 ) = 0, ψ 2 0 F k = T JF k for y = h 0 . Differentiation of (B 29) after division by ψ 2 0 casts (B 27) into the form (−F k /ψ 2 0 ) = −k 2 F k /ψ 2 0 , y = 0 : F k = 0, y = h 0 : ψ 2 0 F k = T JF k . (B 30) Adopting the signs in the usual notation, (B 30) represents a traditional selfadjoint Sturm-Liouville eigenvalue problem with the (for y → 0 singular) weight function ψ −2 0 for the eigenvalues of −k 2 . According to classical results, these indeed form a discrete set k 2 = k 2 i (T ) (i = 0, 1, . . .) bounded from below and satisfying the Weyl asymptotics −k 2 i ∼ (πi/h 0 ) 2 + O(i) (i → ∞), controlled by the right-side of the BC for y = h 0 (cf. Teschl 2012). Here k 2 0 = k 2 u > 0, referring to the single neutral mode considered in § B.1, so that k 2 i is set to −µ 2 i (T ) < 0 for i > 0. It is also noteworthy that this sequence µ i does not collapse in the limiting case T = 1 as µ 1 (1) 0.015569. Supplement C. Outer Rayleigh problem: diffusive overlayer Let us take ψ, p, h + as functions ofX, ξ, . We rectify the Maclaurin expansion of ψ and p for ξ = O(1) justified by (2.3c), (3.10), (3.11) by adding an O( 5/3 )-term (and resultant higher-order corrections) that involves the O(1)-functions Ψ * , P * so as to account for (2.3d): [ψ, p] ∼ [1, P 0 (X)] + ∞ i=1 [Ψ i , P i ](X; ) i/2 ξ i i! + 5/3 [Ψ * , P * ](X, ξ) + O( 12/7 ). (C 1) The structure of (C 1) is explained in the following.
Developed liquid film past a trailing edge: 'teapot effect'
(C 6) The solution to this problem can be expressed in terms of Kummer's confluent hypergeometric function, M : where the functions g i represent the solutions to the hierarchy of inhomogeneous Stokes problems provoked by the inertia terms in (4.4), which cause the remainder term in (E 1). As we now demonstrate, the forcing of these eigensolutions of the Stokes operator by the higher-order, convective terms controls the selection of the leading non-zero coefficient a i of the homogeneous contribution f i to g i . Substituting (E 6) into (4.4) and collecting powers ofr results in the inhomogeneous extension of (E 2) for i > 0: I i,j (ϑ) := (σ j − 2)g k − σ k g k d/dϑ (σ 2 j g j + g j ), k := i − j − 1, (E 7b) g i (0) = g i (0) = g i (π) = g i (π) = 0, (E 7c) where (E 7b) is consistent with the identity σ j + σ k ≡ σ j+k+1 , see (E 3b). The self-adjointness of the homogeneous Stokes operator defined by (E 2) gives 0 = π 0 S i {g i }f i (ϑ) dϑ = S i := π 0 I i (ϑ)f i (ϑ) dϑ.
(E 10) The last statement requires a 2 = 0. This renders the forward-flow case (B) also not possible. Hence, the scenario (C) motivates the following discussion of the special case n = 5. Case: a i = 0 (0 i < 5), a 5 = 0. The result (E 10) includes that the here dominant eigensolution of the Stokes operatorψ 5 given by (E 4), describing a nondegenerate flow profile at separation, trivially generates a vanishing inhomogeneity I 11 . That said, (E 6) then degenerates and reads more accurately, with the help of (E 3b), ψ ∼ψ 5 + ∞ i=qr (1+i)/2 f i +r (7+q)/2 g 6+q + o r (7+q)/2 (q > 5, a q = 0). (E 11) At first, any index q > 5 is conceivable. If (E 11) initiates the solution to the full NS problem, such an index indicating the non-trivial follow-up term tor 3 f 5 must exist. As a central observation, the lowest-order inhomogeneity in (E 7a) specified by (E 7b) is I 6+q = I 6+q,5 + I 6+q,q and produces g 6+q , where the eigenfunction f 6+q corresponds to to the eigenvalue σ q+6 = (7 + q)/2. Inserting these findings into (E 8) yields indeed S 6+q ≡ 0 as for (E 10) but for any q, where we skip the technical details. This guarantees the existences of g q+6 and, in turn, of (E 11).
As an important step, the above analysis determines the least singular (mostdegenerate) local representation ofψ to be given by case (C) extended by (E 11) as the sole reliable option satisfying (4.5). It should be emphasised that the impact of the interactive flow on flow detachment on the NS scales is condensed into the aforementioned (pending) dependence of the coefficient a 5 on Λ 0 .
E.2. Eigensolutions of the Stokes operator having weakly non-algebraic radial variation? Given the absence of a reference length and velocity of the Stokes limit considered, typical dimensional reasoning predicts, in general, algebraic-logarithmic variations of the gauge functions in (E 1) withr. Nonetheless, the following analysis