Thin-film approximations of the two-phase Stokes problem

On the Thin Film Muskat and the Thin Film Stokes Equations

“…Furthermore, they are O(1) of the physical parameters in the problem. In this respect, we improved the results in [13,14] where the size restrictions were assumed in higher order Sobolev norms and not explicit. Eventually, we would like to point out that our technique does not rely on a gradient flow structure or a particular form of an energy functional.…”

“…The above system was derived by Escher, Matioc & Matioc in [14] using lubrication approximation and cross sectional averaging. We remark that for fluids in the Stokes regime gravitational and capillary effects appear at different order in the approximation.…”

“…We remark that for fluids in the Stokes regime gravitational and capillary effects appear at different order in the approximation. As a consequence, the case where both capillary and gravitational effects are taken into account simultaneously appears to be physically not relevant [14]. [14] proved local existence of solutions in the Bessel potential space H s,p , 2 ≤ p and s ∈ ( 3 2 , 2], and exponential convergence towards the flat equilibrium for initial data sufficiently close to their mean in H 2,p , p ≥ 2.…”

Analysis of a thin film approximation for two-fluid Taylor-Couette flows

“…The dynamics of a coupled system for a thin film approximation of the two phase Stokes flow has been considered in [15] as well as [8]. In particular, in [15] has been proved that the interfaces converge exponentially to a planar stationary solution in the particular setting considered in that paper. Section 4 is devoted to analyse two different kind of steady solutions of (1.1) and (1.2).…”

On a thin film model with insoluble surfactant

“…Pioneering works in this direction in absence of surfactant effects are due to Greenspan [32], Constantin, Dupont, Goldstein, Kadanoff, Shelley & Zhou [13], Bernis & Friedman [4], Beretta, Bertsch & Dal Passo [3] and Bertozzi & Pugh [5]. Also, Escher, Matioc & Matioc [23] considered the flow in porous media (see also Escher & Matioc [25], Matioc [39], Escher, Laurençot & Matioc [21], Laurençot & Matioc [35][36][37][38] and Bruell & Granero-Belinchón [10]) while the Stokes flow was considered by Escher, Matioc & Matioc [24] (see also Escher & Matioc [26] and Bruell & Granero-Belinchón [10]). A more recent reference is Pernas-Castaño & Velázquez [40], where the authors study the evolution of the interface between two different fluids in two concentric cylinders when the velocity is given by the Navier-Stokes equation and one of the fluids is thin.…”