Abstract
A biharmonic boundary integral equation (BBIE) method is used to solve a two dimensional contained viscous flow problem. In order to achieve a greater accuracy than is usually possible in this type of method analytic expressions are used for the piecewise integration of all the kernel functions rather than the more time-consuming method of Gaussian quadrature.
Because the boundary conditions for the problem under consideration — commonly referred to as the ‘stick-slip’ problem — give rise to a singularity in the solution domain for the biharmonic stream function, we find that the rate of convergence of the solution is poor in the neighbourhood of the singularity. Hence a modified BBIE (MBBIE) method is presented which takes into account the analytic nature of the aforementioned singularity. This modification is seen to produce rapid convergence of the results throughout the solution domain.
The BBIE and MBBIE also provide information concerning the pressure and velocity fields of the flow and these properties are seen to be in excellent agreement with the analytical results of Watson.
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Ingham, D.B., Kelmanson, M.A. (1984). An Integral Equation Method for the Solution of Singular Slow Flow Problems. In: Boundary Integral Equation Analyses of Singular, Potential, and Biharmonic Problems. Lecture Notes in Engineering, vol 7. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-82330-5_2
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DOI: https://doi.org/10.1007/978-3-642-82330-5_2
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