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The spin-coating process: analysis of the free boundary value problem. (English) Zbl 1229.35331

The purpose of the paper is to describe a model accounting for the description of a spin-coating process. The authors consider a one-phase free boundary value problem for a Newtonian fluid subject to surface tension and rotational effects. A Newtonian fluid fills in the domain \(\Omega (t)\) whose boundary \(\Gamma (t)\) can be split in an “upper” part which evolves according to a mean curvature flow problem and in a “lower” part where a wetting phenomenon occurs. The Navier-Stokes equation which describes the evolution of the fluid in the domain \(\Omega (t)\) contains Coriolis and centrifugal forces. The initial domain \(\Omega (0)\) is supposed to be obtained as \(\Omega (0)=\{(x,y)\in \mathbb R^2:y\in (0,h_{0}(x)+\delta )\}\) for some function \(h_{0}\in W_{p}^{3-2/p}(\mathbb R^2)\), with \(p>5\), and \(\delta >0\). The main result of the paper proves existence and uniqueness results for this problem assuming different regularity hypotheses on the initial data of the problem. The authors also prove that the upper boundary \(\Gamma ^{+}(t)\) of the domain \(\Omega (t)\) remains the graph of a function. The first tool is a Hanzawa transform in order to work in a fixed domain. Then the authors introduce a linearization of this problem for which they prove an existence and uniqueness result and a maximal regularity property. For the nonlinear problem, the authors use a contraction mapping principle in appropriate functional spaces. The paper ends with appendices on the resolution of Laplace, heat and Stokes problems.

MSC:

35R35 Free boundary problems for PDEs
35Q35 PDEs in connection with fluid mechanics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
76U05 General theory of rotating fluids
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