2 research outputs found
Simple and efficient representations for the fundamental solutions of Stokes flow in a half-space
We derive new formulas for the fundamental solutions of slow, viscous flow,
governed by the Stokes equations, in a half-space. They are simpler than the
classical representations obtained by Blake and collaborators, and can be
efficiently implemented using existing fast solvers libraries. We show, for
example, that the velocity field induced by a Stokeslet can be annihilated on
the boundary (to establish a zero slip condition) using a single reflected
Stokeslet combined with a single Papkovich-Neuber potential that involves only
a scalar harmonic function. The new representation has a physically intuitive
interpretation
Fast Ewald summation for free-space Stokes potentials
We present a spectrally accurate method for the rapid evaluation of
free-space Stokes potentials, i.e. sums involving a large number of free space
Green's functions. We consider sums involving stokeslets, stresslets and
rotlets that appear in boundary integral methods and potential methods for
solving Stokes equations. The method combines the framework of the Spectral
Ewald method for periodic problems, with a very recent approach to solving the
free-space harmonic and biharmonic equations using fast Fourier transforms
(FFTs) on a uniform grid. Convolution with a truncated Gaussian function is
used to place point sources on a grid. With precomputation of a scalar grid
quantity that does not depend on these sources, the amount of oversampling of
the grids with Gaussians can be kept at a factor of two, the minimum for
aperiodic convolutions by FFTs. The resulting algorithm has a computational
complexity of O(N log N) for problems with N sources and targets. Comparison is
made with a fast multipole method (FMM) to show that the performance of the new
method is competitive.Comment: 35 pages, 15 figure