Integrating the Stokeslet: Solving Expression 7 from ResearchGate Publication

[steve1763]

TL;DR Summary

I'm having trouble figuring out this integral. The fact that we are integrating vectors, absolute values, tensor products etc doesnt help.

https://www.researchgate.net/publication/301874096_Emergent_behavior_in_active_colloids/link/5730bb3608ae08415e6a7c0a/download (expression 9 on this document derivation). I understand the need for substitution etc into the integral. What puzzles me is how the integral equals what it does. If somebody could show me how to solve the integral that would be brilliant.

Perhaps a quicker question would be, how does one integrate the Stokeslet? If its value is that of expression 7.

Thank you

 

[ergospherical]

The tensor field $\mathbf{O}(\mathbf{r})$ is defined in equation $(7)$,\begin{align*}
\mathbf{O}(\mathbf{r}) \equiv \frac{1}{8\pi \eta}\left( \frac{1}{r} \mathbf{1} + \frac{1}{r^3} \mathbf{r} \otimes \mathbf{r} \right)
\end{align*}The Green's function solution to the Navier-Stokes equation in terms of a forcing term $\mathbf{f}$ is \begin{align*}
\mathbf{v}(\mathbf{r}, t) = \int \mathbf{O}(\mathbf{r} - \mathbf{r}') \mathbf{f}(\mathbf{r}',t) d^3 x'
\end{align*}They consider a monopole forcing term $\mathbf{f} = f\mathbf{e} \delta(\mathbf{r} - \mathbf{r}_0)$ localised at $\mathbf{r}_0$. In this case you have\begin{align*}
\mathbf{v}(\mathbf{r}, t) &= \int \mathbf{O}(\mathbf{r} - \mathbf{r}') f\mathbf{e} \delta(\mathbf{r}' - \mathbf{r}_0)d^3 x' \\
&= \mathbf{O}(\mathbf{r} - \mathbf{r}_0) f \mathbf{e}
\end{align*}For brevity one can define $r \equiv |\mathbf{r} - \mathbf{r}_0|$ and $\hat{\mathbf{r}} \equiv (\mathbf{r} - \mathbf{r}_0)/r$. Then, using the definition of $\mathbf{O}(\mathbf{r})$, you have\begin{align*}
\mathbf{v}(\mathbf{r}, t) &= \frac{f}{8\pi \eta r}\left( \mathbf{1} + \hat{\mathbf{r}} \otimes \hat{\mathbf{r}} \right) \mathbf{e}
\end{align*}Recall from tensor algebra that $\mathbf{1}\mathbf{e} = \mathbf{e}$, and also $(\hat{\mathbf{r}} \otimes \hat{\mathbf{r}})(\mathbf{e}) \equiv (\hat{\mathbf{r}} \cdot \mathbf{e}) \hat{\mathbf{r}}$. Then you get the quoted result,\begin{align*}
\mathbf{v}(\mathbf{r}, t) &= \frac{f}{8\pi \eta r}\left( \mathbf{e} +(\hat{\mathbf{r}} \cdot \mathbf{e}) \hat{\mathbf{r}} \right)
\end{align*}