Hele–Shaw flow in thin threads: A rigorous limit result

Abstract

Abstract. We rigorously prove the convergence of appropriately scaled solutions of the 2D Hele-Shaw moving boundary problem with surface tension in the limit of thin threads to the solution of the formally corresponding Thin Film equation. The proof is based on scaled parabolic estimates for the nonlocal, nonlinear evolution equations that arise from these problems.

Existence and stability of weak solutions for a degenerate parabolic system modelling two-phase flows in porous media

“…Indeed, the system (1.1) has been obtained in [4] by passing to the limit of small layer thickness in the Muskat problem studied in [3] (with homogeneous Neumann boundary condition). Similar methods to those presented in [4] have been used in [6] and [8], where it is rigorously shown that, in the absence of gravity, appropriate scaled classical solutions of the Stokes' and one-phase Hele-Shaw problems with surface tension converge to solutions of thin film equations ∂ t h + ∂ x (h a ∂ 3 x h) = 0, with a = 3 for Stoke's problem and a = 1 for the Hele-Shaw problem.…”

Non-negative global weak solutions for a degenerate parabolic system modelling thin films driven by capillarity

“…We refer the reader to the survey papers [1,10], in which many aspects concerning the thin film equation are discussed. It should be noted that similar methods to those in [7] have been used in [9,11] to rigorously show that, in the limit of thin fluid threads, the solutions of the moving boundary-value problems for Stokes and Hele-Shaw flows converge towards the corresponding solutions (determined by the initial data) of the thin film equation (1.4), with n = 3 for Stokes and n = 1 for the Hele-Shaw flow. Compared with the thin film equation, system (1.1) is more involved because it is strongly coupled, both equations of (1.1) containing highestorder derivatives of f and g, and, furthermore, there are two sources of degeneracy, because both f and g may equal zero.…”

Thin-film approximations of the two-phase Stokes problem

“…Stokes) moving boundary problem. However, a rigorous convergence result along the lines of [2,3] is not possible since the approach for the flows driven by surface tension uses in an essential way the ellipticity of the curvature operator. A further impediment is due to the fact that the gravity driven Stokes problem is not a parabolic evolution equation.…”

“…The authors of [2] prove in fact that sufficiently regular solutions of the latter problem converge, when the thickness of the fluid layer tends to zero, towards corresponding solutions of the thin film equation (1.2) with m = 3. The methods of [2] have been adapted later on in [3] to show that the solutions of the capillary driven Hele-Shaw problem approximate solution of the thin film equation (1.2) with m = 1 in the thin thread limit (see also [4] for an alternative approach). For a review of the extensive literature on these and related equations we refer to [5].…”