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Boundary integral solution of potential problems arising in the modelling of electrified oil films. (English) Zbl 1333.35182

The author studies a 2D model of electrified oil film \(\Omega_1\) lying over a periodic half-space. The problem reduces to a transmission problem for the Laplace equation in two regions: the film domain \[ \Omega_1=\{(x,y):0<y<h(x)\}, \] and the exterior region \[ \Omega_2=\{(x,y):y>h(x)\}, \] where the periodic function \(h\) is the (given) thickness of the film.
Periodic Dirichlet boundary conditions are assumed on the substrat \(\{y=0\}\) and transmissions conditions are assumed through the film boundary \(\Gamma=\{(x,y):y=h(x)\}\). Finally a decay condition is assumed when \(y\to +\infty\).
The existence of a unique solution is proved by using a boundary integral method, introducing single and double layer potentials on \(\Gamma\). Moreover a discretization method is introduced and numerical experiments are provided.

MSC:

35Q35 PDEs in connection with fluid mechanics
76W05 Magnetohydrodynamics and electrohydrodynamics
76A20 Thin fluid films
78A45 Diffraction, scattering
45B05 Fredholm integral equations
65N38 Boundary element methods for boundary value problems involving PDEs
65R20 Numerical methods for integral equations

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