A Boundary Element Method for Oscillating Stokes Flow at Low Frequencies Around a Rigid Body

Abstract

The unsteady creeping flow around a rigid three dimensional body at rest in an incompressible and viscous fluid of Newtonian type is considered. The flow problem is modeled using an indirect boundary integral equation (IBIE), and is numerically solved by using collocation and Galerkin weighting procedures. An IBIE was presented in a previous work for the steady creeping flow case (D’El ´ ıa et al., Mec ´ anica Computacional, vol. XXVIII:1453-1462, 2009), whereas in the present work the attention is focused to the oscillatory creeping flow with an harmonic time dependence. The formulation is specialized to low frequencies and boundary meshes with flat simplex triangles. The double surface integrals in the Galerkin approach that account the pairwise interaction among all boundary elements are computed on using a variation of the scheme proposed by Taylor (D. J. Taylor, IEEE Trans. on Antennas and Propagation, 51(7): 1630-1637, 2003). Numerical examples include the unsteady creeping flow with an harmonic time dependence around the sphere of unit radius and around the cube of unit edge length, both at rest, covering issues on the convergence under mesh refinement and, in the first test case, a comparison against the analytical values as a function of the imposed vibrating frequency.