Dynamic and static stability of a drop attached to an inhomogeneous plane wall

International audienceThis article concerns the stability of a drop on a wall for which the contact angle, w , varies from place to place. Such a wall may allow unstable equilibria of the drop, i.e. ones for which small perturbations to equilibrium grow, making the equilibrium unrealisable in practice. This will be referred to as dynamic instability and is one of the two versions of instability considered. The other arises from consideration of potential energy, which is the sum of surface (liquid/gas, liquid/solid and solid/gas) components and the gravitational potential energy. Equilibria are extrema of the potential energy with respect to variations of drop geometry which preserve its volume. An equilibrium is said to be statically stable if it is a local minimum of the potential energy for volume-preserving perturbations of the drop. The relationship between static and dynamic stability is the main subject of this paper. The liquid flow is governed by the incompressible Navier-Stokes equations. To allow for the moving contact line, a Navier slip condition with slip length is used at the wall, as is a prescribed contact angle, , w w x y , at the contact line, where x , y are Cartesian coordinates on the wall. The perturbation is assumed small, allowing linearization of the governing equations and, in the usual manner of stability analysis, complex modes having the time dependency st e are introduced. This leads to an eigenvalue problem with eigenvalue s , the sign of whose real part determines dynamic stability/instability. A quite different eigenvalue problem, which describes static stability/instability is also derived. It is shown that, despite this difference, the conditions for dynamic and static instability are in fact the same. This conclusion is far from evident a priori, but should be good news for interested numerical analysts because determination of static stability is much less numerically costly than a dynamic stability study, whereas it is the latter which gives a true determination of stability