Method of Regularized Stokeslet

This work was done as a part of the final project for a special topics course titled Boundary integral method in the fall of 22 by Professor Cortez

The fluid flow problems in tiny scales are usually modeled by the Stokes equations for incompressible flows
\(\mu \Delta \textbf{u}=\nabla p-\textbf{F}, \nabla . \textbf{u}=0\) where $\mu$ is the fluid viscosity, $p$ is the pressure, $\textbf{u}$ is the velocity, and $\textbf{F}$ is force density. A fundamental solution of these equations is called a $Stokeslet$. The particular case of a single force $\mathbf{f}_0$ exerted at $\mathbf{x}_0$ results in a velocity field given by \(\mathbf{u}=\frac{\mathbf{f}_0}{8\pi\mu r}+\frac{(\mathbf{f_0\cdot(x-x_0)})(\mathbf{x-x_0})}{8 \pi\mu r}\) where $r=||\mathbf{x-x_0}||$.

Note this solution is undefined at $r=0$ or $\mathbf{x=x}_0$.

However, the singularities can be eliminated through the function(usually known as blob function) $\phi_{\delta}(\mathbf{x})$ which is radially symmetric and satisfies that the integral over the space is one. So, considering \(\boldsymbol{F}=\boldsymbol f_0 \phi_{\delta}\) the singularity can be removed.

The idea is due to Professor Cortez¹.

With the following choice of blob function: \(\phi_\delta(r)=\dfrac{15\delta^4}{8\pi(r^2+\delta^2)^{\frac{7}{2}}}\) the regularized stokeslet is \(\vec{u}(x)=\boldsymbol f_0 \dfrac{r^2+2\delta^2}{8\pi\mu(r^2+\delta^2)^{\frac{3}{2}}}+\dfrac{(\boldsymbol f_{0} .\boldsymbol x)\boldsymbol x}{8\pi\mu(r^2+\delta^2)^{\frac{3}{2}}}\)

Swimming filament immersed in a viscous fluid

We suppose the slender body is a sine wave, \(y(s)=A\cos(2\pi( s- t)),z(s)=0\) and $x(s)$ such that $\sqrt{(x’)^2+(y’)^2}=1$, and the curvature \(\kappa=\frac{x''y'-y''x'}{(\sqrt{(x')^2+(y')^2})^{3}}=x''y'-y''x'=\frac{-y''}{\sqrt{1-(y')^2}}\) To compute the forces, we’ve utilized the approach discussed in the article A computational model of aquatic animal locomotion ².

Once we’ve computed forces at each point, we can calculate velocity. Then, the locomotion of filament can be found by solving $\dfrac{d\boldsymbol X}{dt}=\boldsymbol u$. To solve this we have used the forward Euler method(other methods like RK can also be used).

It’s worth mentioning that this model has been utilized for various problems; especially those related to the motility of hyper-activated mammalian sperm³.

Source codes are available in RegularizedStokeslet.