Regularized Stokeslet Surfaces

Abstract

An extension of the Method of Regularized Stokeslets (MRS) in three dimensions is developed for triangulated surfaces with a piecewise linear force distribution. The method extends the regularized Stokeslet segment methodology used for piecewise linear curves. By using analytic integration of the regularized Stokeslet kernel over the triangles, the regularization parameter $ε$ is effectively decoupled from the spatial discretization of the surface. This is in contrast to the usual implementation of the method in which the regularization parameter is chosen for accuracy reasons to be about the same size as the spatial discretization. The validity of the method is demonstrated through several examples, including the flow around a rigidly translating/rotating sphere, a rotating spheroid, and the squirmer model for self-propulsion. Notably, second order convergence in the spatial discretization for fixed $ε$ is demonstrated. Considerations of mesh design and choice of regularization parameter are discussed, and the performance of the method is compared with existing quadrature-based implementations.

Figures (19)

• Figure 1: A sketch of the triangle in physical space and in the parameter space.
• Figure 2: Geometry of line segment integrals \ref{['eq:Slm1']} and \ref{['eq:Slp1']}. The length of the line segment is $L$ and the $\bm{\hat{\ell}}$ points from $\mathbf{y}(1)$ to $\mathbf{y}(0)$.
• Figure 3: A picture displaying the important geometric terms used in the derivation of $T_{0,0,3}$.
• Figure 4: A comparison of Stokeslet surfaces and MRS $\ell_2$ errors for the forward problem of computing the velocity on a sphere using the hydrodynamic traction $\mathbf{f} = - \frac{3 \mu}{2a}(1,0,0).$ (a) Error as a function of $\epsilon/h$. The spatial discretization is fixed for both methods as $h=0.1243$. (b) Error as a function of $h$. For the MRS, $\epsilon$ was set to $0.05$ while for the Stokeslet surface method it was set to $10^{-4}$.
• Figure 5: A visualization of the error in the velocity field for the unit sphere (spatial discretization $h=0.1398$, regularization $\epsilon=10^{-4}$) with imposed traction $\mathbf{f}=-\frac{3 \mu}{2a}(1,0,0)$. The traction induces rigid translation of the sphere with unit velocity in the x direction. The errors are shown in the equatorial plane ($z=0$). (a) illustrates that the largest errors (by absolute magnitude) are concentrated near the sphere on the sides of the sphere orthogonal to the translational direction. (b) shows the largest relative errors are about 0.6% in the yellow band orthogonal to the translational direction. White arrows show the fluid velocity in the lab frame.