Completing the shock theory of inertial dewetting: viscous shock structure, two-dimensional rim shapes, and capillary-ripple coupling

When a millimetric film of a low-viscosity liquid dewets a non-wetting solid, the liquid collects into a rim whose leading edge travels as a hydraulic shock. Brochard-Wyart and de Gennes (1997) modeled this inertial dewetting with shallow-water theory: a plug-flow rim of equilibrium thickness $h^*$ set by the spreading coefficient through $\frac{1}{2}\rho g h^{*2}=-S$, a shock obeying Rankine-Hugoniot jump conditions, front velocities of order $\sqrt{g h^*}$, and a rim Reynolds number of order $10^2$. The model explicitly left three issues open: (1) the thin viscous boundary layers near the leading shock, their role in regularizing the hydraulic discontinuity, and their matching to the plug-flow rim; (2) the validity of the one-dimensional approximation, which requires rim thickness, rim width, and hole radius to be well separated ($h_1\ll\ell\ll R$) and hence very large experiments, and the transition to two-dimensional rim shapes; (3) capillary effects, in particular ripples observed near the front at intermediate Reynolds numbers, and their interplay with the hydraulic shock. Later work settled parts of these issues: the axisymmetric annular rim was treated by the authors themselves (1999), and the wavelength selection of the ripples on liquid substrates follows capillary-gravity dispersion (2006). For the original solid-substrate configuration, the shock's internal structure and its matching to the rim, non-axisymmetric rim shapes and the regime where rim width is comparable to hole radius, and a quantitative theory of the ripple-shock coupling at intermediate Reynolds numbers remain open. The completion of the 1997 shock theory is therefore open.

Background

When a liquid film rests on a surface it does not wet, the film is metastable: if a dry patch is nucleated it grows, and the displaced liquid piles up into a raised ridge, the rim, at the edge of the growing hole. The thermodynamic driving force is the spreading coefficient $S=\gamma_{SO}-(\gamma_{SL}+\gamma)$, the difference between the surface energy of the dry solid and that of the wetted solid, built from the solid-vapor, solid-liquid, and liquid-vapor interfacial tensions $\gamma_{SO}$, $\gamma_{SL}$, $\gamma$; $S<0$ favors dewetting. For microscopic polymer films (micron thickness) the rim motion is resisted by viscous dissipation. For millimetric films of low-viscosity liquids such as water, inertia dominates instead, and Brochard-Wyart and de Gennes (1997) proposed the description used here.

They modeled the front in the shallow-water (Saint-Venant) approximation: the depth-averaged mass and momentum balances for a thin liquid layer, the same framework used for rivers, bores, and tsunamis. The rim, of uniform thickness $h_1$ and moving at plug velocity $V$ (velocity uniform over the depth), is preceded by a hydraulic shock - a moving bore, directly analogous to a tidal bore, traveling at speed $V_S$ into the undisturbed film of thickness $h_0$. Across such a jump the Rankine-Hugoniot conditions of shallow-water theory enforce conservation of mass and momentum, giving $V=V_S(1-h_0/h_1)$ and $V_S^2=\frac{1}{2}g\,h_1(h_0+h_1)/h_0$, where $g$ is gravity. Balancing the growth of rim momentum against the capillary and hydrostatic energy release, $\frac{1}{2}\rho g(h^{*2}-h_0^2)=\rho h_0 V V_S$, yields the compact prediction that the rim carries the equilibrium pancake thickness $h^*$ of a gravity-flattened puddle, set by $\frac{1}{2}\rho g h^{*2}=-S$, so that $h_1=h^*$, with front velocities of order $\sqrt{g h^*}$ and a rim Reynolds number $Re=\rho V h_1/\eta$ (the ratio of inertial to viscous forces; $\rho$ is density, $\eta$ dynamic viscosity) of order $10^2$ for water-like liquids at $h^*\sim1$ mm. A geometric identity links rim width $\ell$ and hole radius $R$, $\ell/R=h_0/(h_1-h_0)$, so a flat, one-dimensional rim requires $h_1\ll\ell\ll R$ - rim thickness much smaller than rim width much smaller than hole radius - which for thin films demands very large experiments, $R>(h_1-h_0)h_1/h_0$.

The authors stressed that this picture is an idealization with three unresolved issues. First, at $Re\sim10^2$ viscosity acts only in thin boundary layers - the thin regions near solid and free surfaces across which the velocity adjusts from the plug flow to its boundary values - but the boundary-layer structure near the shock, its role in regularizing the hydraulic discontinuity (an exactly sharp jump cannot persist in a viscous liquid), and the matching to the plug-flow rim were not computed. Second, the one-dimensional description rests on $h_1\ll\ell\ll R$; the transition to genuinely two-dimensional rim shapes, when the rim width is not small compared with the hole radius, was left open (only the radial-flow correction for a circular hole was anticipated). Third, the model contains gravity but no capillarity at the front: capillary effects, in particular the ripples that experiments near the front at intermediate Reynolds numbers (between the viscous and fully inertial limits) had reported, lay outside the treatment, and their interplay with the hydraulic shock was deferred. The companion experiments of Buguin, Vovelle and Brochard-Wyart (1999) confirmed the shock picture - the dewetting velocity follows a Culick-like law (the velocity law familiar from the retraction of burst soap films, $V\sim\sqrt{\gamma/(\rho h)}$ up to a numerical factor, here with capillarity and gravity combined in the driving force), the rim profile is flat, and ripples are emitted ahead of the rim as the "signature of the shock" whenever the front outruns the long gravity-wave speed $\sqrt{g h_0}$ - which sharpens rather than removes all three issues.

Problem Statement

The open problem is the completion of the 1997 shock theory of inertial dewetting on a rigid non-wetting solid, at the generality its authors acknowledged. It asks to resolve three linked issues concerning one physical object, the inertially dewetting front; all three issues are required, and each stands as an independently stated deliverable. Throughout, $h_0$ denotes the initial film thickness, $h_1$ the rim thickness behind the leading shock, $h^*$ the equilibrium pancake thickness defined by $\frac{1}{2}\rho g h^{*2}=-S$ (with $S<0$ the spreading coefficient, $\rho$ the liquid density, and $g$ gravity), $\ell$ the rim width, $R$ the hole radius, $V$ the rim velocity, and $V_S$ the shock speed.

(1) Internal structure of the leading shock. Determine the viscous (and, where relevant, capillary) structure that regularizes the hydraulic discontinuity at the front: the boundary layers near the shock, their thickness scalings with Reynolds number and geometry, the way dissipation there is compatible with the jump relations, and the asymptotic matching between the resolved front region, the plug-flow rim behind it, and the undisturbed film ahead - such that in the outer limit the Rankine-Hugoniot jumps $V=V_S(1-h_0/h_1)$, $V_S^2=\frac{1}{2}g h_1(h_0+h_1)/h_0$ and the rim-thickness prediction $h_1=h^*$, $\frac{1}{2}\rho g h^{*2}=-S$, are recovered.

(2) Rim dynamics beyond one dimension. Extend the rim description to two-dimensional shapes: beyond the radial-divergence correction for a circular hole (the axisymmetric annular rim, already treated), determine the rim shape and velocity field when the rim width $\ell$ is not small compared with the hole radius $R$ (i.e. outside the $h_1\ll\ell\ll R$ regime and the size bound $R>(h_1-h_0)h_1/h_0$), and determine the azimuthal dynamics - shape selection and stability - of the rim, which a one-dimensional model cannot address.

(3) Capillary ripples and the shock. Construct the quantitative theory of the capillary ripples emitted ahead of the rim on solid substrates - their wavelength and amplitude selection, their damping, and their back-reaction on the shock and rim speed - in the regime where they are observed (front speed exceeding the relevant wave speed; intermediate Reynolds numbers between the viscous and inertial limits), where the purely gravitational shallow-water treatment does not apply.

For each issue, the theory is to state its domain of validity, and its predictions - front velocities, rim thickness and profile, shock-front structure, and ripple characteristics as functions of film thickness $h_0$, spreading coefficient $S$, liquid properties, and geometry - are to be confronted with the published experimental observations.

References

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