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Boundary integral solutions of coupled Stokes and Darcy flows. (English) Zbl 1188.76232

This work presents an accurate computational method based on the boundary integral formulation for solving boundary value problems for Stokes and Darcy flows. The method is applied to problems where the equations are coupled across the interface through appropriate boundary conditions. First, the adopted technique reformulates singular integrals of the boundary integral formulation for the fluid flows as single and double layer potentials. Then the layer potentials are regularized and discretized using standard quadratures. At the final step, the leading term in the regularization error is eliminated in order to increase the order of accuracy.

Various presented test cases show that the results are consistent with theoretical predictions. In the first example, the authors compute the Stokes flow in a channel driven by a uniform pressure gradient. Assuming the porous lower boundary and imposing the Beavers-Joseph-Saffman slip boundary condition, the authors obtain a velocity profile that is non-zero on the boundary. Using corrected expressions, the slip velocity can be calculated more accurately than with uncorrected expressions. The error as a function of regularization parameter $\delta$ reduces in magnitude, and the convergence rate increases from $O\left(\delta \right)$ to $O\left({\delta }^{2}\right)$. Then the authors test the Darcy solution by prescribing an exact velocity on a circle. The kernels in the Darcy velocity have singularities of higher order, and the authors indicate numerical errors in the uncorrected case as large as 20% in velocity and 70% in pressure. The corrections reduce the error by high factors of 100–1000, and the accuracy increases by one order.

The solved equations are steady-state, but once the fluid velocity is known, the position of the Lagrangian particles can be updated in time, thus extending the method to time-dependent problems.

##### MSC:
 76M15 Boundary element methods (fluid mechanics) 76D07 Stokes and related (Oseen, etc.) flows 76S05 Flows in porous media; filtration; seepage