Method of regularized stokeslets: Flow analysis and improvement of convergence

Since their development in 2001, regularized stokeslets have become a popular numerical tool for low-Reynolds-number flows since the replacement of a point force by a smoothed blob overcomes many computational difficulties associated with flow singularities [Cortez, SIAM J. Sci. Comput. 23, 1204 (2001)]. The physical changes to the flow resulting from this process are, however, unclear. In this paper, we analyze the flow induced by general regularized stokeslets. An explicit formula for the flow from any regularized stokeslet is first derived, which is shown to simplify for spherically symmetric blobs. Far from the center of any regularized stokeslet we show that the flow can be written in terms of an infinite number of singularity solutions provided the blob decays sufficiently rapidly. This infinite number of singularities reduces to a point force and source dipole for spherically symmetric blobs. Slowly decaying blobs induce additional flow resulting from the nonzero body forces acting on the fluid. We also show that near the center of spherically symmetric regularized stokeslets the flow becomes isotropic, which contrasts with the flow anisotropy fundamental to viscous systems. The concepts developed are used to identify blobs that reduce regularization errors. These blobs contain regions of negative force in order to counter the flows produced in the regularization process but still retain a form convenient for computations.

Figure 1

Streamlines (lines) and flow strength (density plot) from an example of a regularized stokeslet with $\epsilon =0$ (left), $\epsilon =0.5$ (middle), and $\epsilon =1$ (right). The net dimensionless force in each case is $\mathbf{F}=(1,1,0)$. The regularized stokeslet has the algebraic blob $f\left(r\right)=15/\left(8\pi \sqrt{1+r^2}\right)$. The units for $X$ and $Y$ are arbitrary.

Figure 2

Streamlines (black lines) and velocity speed (density plot) of a stokeslet (left) and source dipole (right) solutions aligned with the arrows. The stokeslet has dimensionless strength $\mathbf{F}=(1,1,0)$, and the source dipole has the sink placed to the top right of the source. The units for $X$ and $Y$ are arbitrary.

Figure 3

Plots of the three blobs (a) The original blob used by Cortez, plotted with different $\epsilon$ [41]. (b) Plot of the Gaussian and a blob of compact support. All lengths have been scaled by $\epsilon$ as it is the only relevant length scale.

Figure 4

The difference in flow between the three regularized stokeslets and the true singularity. (a) The difference in the power law regularization, $({\mathbf{S}}_{\epsilon }^p-\mathbf{S})·\mathbf{F}$. (b) The difference in the Gaussian regularization $({\mathbf{S}}_{\epsilon }^g-\mathbf{S})·\mathbf{F}$. (c) The difference in the supported regularization $({\mathbf{S}}_{\epsilon }^s-\mathbf{S})·\mathbf{F}$. Arrows display the direction of the deviation flow while contours display the strength. In all cases $z=0, \mathbf{F}=6\pi \widehat{\mathbf{x}}$ and all lengths have been scaled by $\epsilon$.

Figure 5

Plot of the example cutoff function ${\eta }_0\left(r\right)$, Eq. (26).

Figure 6

Streamlines (lines) and flow strength (density plot) of the $n=7$ flow, $-3\mathbf{I}/5r^5+\mathbf{xx}/r^7$, with strength (1,1). The units for $X$ and $Y$ are arbitrary.

Figure 7

The regularizations and flows from the improved-accuracy regularized stokeslets. (a)–(c) Compact supported regularization; (d)–(f) exponential regularization. Panels (a) and (d) show the improved regularizations. The insets show the transition where the blobs become negative; panels (b) and (e) plot the strength of the $\mathbf{I}$ term in the flow; panels (c) and (f) show the strength of the $\mathbf{r}\mathbf{r}$ term in the flow. The red dashed lines are the flow from a stokeslet. All lengths have been scaled by $\epsilon$.