A Integrating the Stokeslet: Solving Expression 7 from ResearchGate Publication

steve1763
Messages
13
Reaction score
0
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
 
Physics news on Phys.org
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*}
 
From the BCS theory of superconductivity is well known that the superfluid density smoothly decreases with increasing temperature. Annihilated superfluid carriers become normal and lose their momenta on lattice atoms. So if we induce a persistent supercurrent in a ring below Tc and after that slowly increase the temperature, we must observe a decrease in the actual supercurrent, because the density of electron pairs and total supercurrent momentum decrease. However, this supercurrent...
Back
Top