A boundary element method for viscous flows at low Reynolds numbers

Solutions are presented for the nonlinear biharmonic equation describing steady, two-dimensional viscous flow of an incompressible fluid at low Reynolds number. The governing equation for the flow field is reformulated into a set of coupled nonlinear Poisson-type boundary integral equations. The degree of accuracy of the solution, as compared to existing procedures for this problem, is improved by using linear isoparametric elements combined with analytic expressions for the piecewise integration of the fundamental function and its derivatives over each element. Analytic integration is used to eliminate errors introduced by using Gaussian quadrature, especially those errors associated with internal value calculations very near the boundary. At zero Reynolds number the coupled integral equations are linear, giving rise to a formulation in which it is necessary only to evaluate boundary integrals. At nonzero Reynolds numbers the nonlinear character of the integral equations requires an iterative solution technique and introduces domain integrations that are calculated using an improved volume quadrature method, thereby avoiding the disadvantges associated with explicit domain cell methods. Numerical solutions of the integral formulation are presented in terms of plots of the streamlines at various Reynolds numbers and for several domain geometries. © 1989