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IMA Journal of Applied Mathematics Advance Access published online on October 18, 2007
IMA Journal of Applied Mathematics, doi:10.1093/imamat/hxm046
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© The Author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A combined BIE-FE method for the Stokes equations

A. V. Lukyanov{dagger}

Department of Mathematics, University of Reading, Reading RG6 6AX, UK

Y. D. Shikhmurzaev and A. C. King

School of Mathematics, University of Birmingham, Birmingham B15 2TT, UK

{dagger} Email: a.lukyanov{at}reading.ac.uk

Received on December 1, 2006; Accepted on March 14, 2007

A numerical algorithm for the biharmonic equation in domains with piecewise smooth boundaries is presented. It is intended for problems describing the Stokes flow in the situations where one has corners or cusps formed by parts of the domain boundary and, due to the nature of the boundary conditions on these parts of the boundary, these regions have a global effect on the shape of the whole domain and hence have to be resolved with sufficient accuracy. The algorithm combines the boundary integral equation method for the main part of the flow domain and the finite-element method which is used to resolve the corner/cusp regions. Two parts of the solution are matched along a numerical ‘internal interface’ or, as a variant, two interfaces, and they are determined simultaneously by inverting a combined matrix in the course of iterations. The algorithm is illustrated by considering the flow configuration of ‘curtain coating’, a flow where a sheet of liquid impinges onto a moving solid substrate, which is particularly sensitive to what happens in the corner region formed, physically, by the free surface and the solid boundary. The ‘moving contact line problem’ is addressed in the framework of an earlier developed interface formation model which treats the dynamic contact angle as part of the solution, as opposed to it being a prescribed function of the contact line speed, as in the so-called ‘slip models’.

Keywords: Dynamic contact angle; finite elements; free surface flows; hybrid numerical technique; Stokes equations.


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