Multidimensional Hele‐Shaw flows modelling Stokesian fluids

Abstract

SUMMARYWe consider here an n-dimensional periodic flow describing the motion of an incompressible Stokesian fluid in a Hele-Shaw cell. The free surface separating the fluid from air, at pressure normalized to be zero, is moving under the influence of gravity and surface tension.We prove the existence of a unique classical Hölder solution for small perturbations of cylinders. Moreover, we evidence the existence of a single steady state and prove its exponential stability.

A generalized Rayleigh–Taylor condition for the Muskat problem

“…Though, when dealing with periodic solutions, we easily get, cf. [12], that h must be constant also in the spatial variable, and if ρ + ≤ ρ − then also f is constant. Whence, equations (5.1) and (5.2) may have nontrivial solutions (f, h) / ∈ R 2 only when γ w > 0 and ρ + > ρ − .…”

Classical solutions and stability results for Stokesian Hele-Shaw flows

“…As a main result we show that circles are exponential stable under small perturbations. However, in contrast to the periodic strip-like geometry considered in [6], steady states are no longer isolated, but form a three-dimensional submanifold M c loc of the phase space. Nevertheless, a centre manifold analysis allows us to prove that M c loc attracts at an exponential rate any solution which is sufficiently close nearby.…”

“…[5]. Given the geometric setting considered in this paper we can weaken slightly the assumptions on the viscosity function µ, because sufficiently small deformations of the unitary disc remains convex, whereas in the strip-like geometry of [5] and [6] the cylinder may lose convexity property by arbitrarily small deformations. The first main result of this paper is proved at the end of Section 3 and guarantees local existence and uniqueness of classical solutions.…”

Steady-state solutions for the Muskat problem

“…Ehrnstrom, Escher and Matioc in [8] found a finite limit value with the property that if the surface tension coefficient remains below this value the curve remains 2π periodic and when the surface tension coefficient approaches to this finite value from below, the maximal slope of the curve tends to infinity. The equation ( 3) in the stable case ρ − > ρ + has only trivial solution (see [9]). In this notes we consider the unstable case, when ρ + > ρ − , that is, when the heavier fluid occupies the upper part and we are interesting in describing a solution with the following initial data z(0) = (0, 0) and z (0) = (−α, 1), α > 0.…”