If I have a velocity integral defined by the following:
$$\mathbf{v}(\mathbf{r}) = \int \mathbf{H}(\mathbf{r} - \mathbf{r}') \cdot \mathbf{f}(\mathbf{r}')~\mathrm d^3 \mathbf r' $$
where $\mathbf{H}$ is a Greens function (called the Oseen tensor) with components given by
$$H_{\alpha \beta}(\mathbf{r}) = {1\over{8 \pi \mu r}} \left(\delta_{\alpha\beta} + {r_{\alpha}r_{\beta}\over{r^2}}\right)$$
where $r = |\mathbf{r}|$. To derive the Stokeslet, we find the velocity $\mathbf{v}(\mathbf{r})$ for a point force $\mathbf{f} = \delta(\mathbf{r})\hat{e}$.
My question is whether I'd be correct in the following derivation:
We can re-write the Oseen tensor in vector-form
$$\mathbf{H}(\mathbf{r}) = {1\over{8 \pi \mu |\mathbf{r}|}} \left(\delta(\mathbf{r}) + {\mathbf{r}\otimes\mathbf{r}\over{|\mathbf{r}|^2}}\right)$$ thereby we can get $$\mathbf{H}(\mathbf{r - r'}) = {1\over{8 \pi \mu |\mathbf{r - r'}|}} \left(\delta(\mathbf{r - r'}) + {(\mathbf r - \mathbf r')\otimes(\mathbf r - \mathbf r')\over{|\mathbf{r - r'}|^2}}\right)$$ thus the velocity integral becomes
$$ \textbf{v}(\mathbf{r}) = \int{1\over{8 \pi \mu |\mathbf{r} - \mathbf {r}'|}} \left( \delta(\mathbf r-\mathbf r') + {(\mathbf r - \mathbf r')\otimes(\mathbf r - \mathbf r')\over{|\mathbf{r} - \mathbf{r}'|^2}}\right) \cdot \delta(\textbf{r}')\hat{e} \mathrm d^3\mathbf r'$$ and idk what to really do from here. I'm not entirely sure how to treat the delta functions.
The original text is here (equations 9-11): https://iopscience.iop.org/article/10.1088/0034-4885/78/5/056601/pdf
Thank you
They are overloading the $\delta$ symbol and using it for two different purposes here. You say that
$$H_{\alpha \beta}(\mathbf{r}) = {1\over{8 \pi \mu r}} \left(\delta_{\alpha\beta} + {r_{\alpha}r_{\beta}\over{r^2}}\right)$$
implies
$$\mathbf{H}(\mathbf{r}) = {1\over{8 \pi \mu |\mathbf{r}|}} \left(\delta(\mathbf{r}) + {\mathbf{rr}\over{|\mathbf{r}|^2}}\right)$$
However, this is incorrect. If we follow the chain of citations, the next paper down is more explicit in defining
$$\mathbf{H}(\mathbf{r}) = {1\over{8 \pi \mu |\mathbf{r}|}} \left(\mathbf{1} + {\mathbf{rr}\over{|\mathbf{r}|^2}}\right)$$
Then
$$\textbf{v}(\mathbf{r}) = \int{1\over{8 \pi \mu |\mathbf{r} - \mathbf {r}'|}} \left( \mathbf{1} + {(\mathbf r - \mathbf r')(\mathbf r - \mathbf r')\over{|\mathbf{r} - \mathbf{r}'|^2}}\right) \cdot \delta(\textbf{r}')\hat{e} \mathrm d^3\mathbf r'$$
The $\delta(\mathbf{r^{'}})$ here refers to the Dirac-Delta function, a tricky object which itself is not actually a function but which has the effect of evaluating an integral at the point where it's input is $0$. That is
$$=\left. \frac{1}{8 \pi \mu |\mathbf{r} - \mathbf {r}'|} \left( \mathbf{1} + {(\mathbf r - \mathbf r')(\mathbf r - \mathbf r')\over{|\mathbf{r} - \mathbf{r}'|^2}}\right) \cdot \hat{e} \right|_{\mathbf{r^{'}}=\mathbf{0}}=\frac{1}{8\pi\mu|\mathbf{r}|}\left(\hat{e}+\frac{\mathbf{r}\mathbf{r}\cdot \hat{e}}{|\mathbf{r}|^2}\right)=\frac{1}{8\pi\mu|\mathbf{r}|}\left(\hat{e}+\frac{\mathbf{r}(\mathbf{r}\cdot \hat{e})}{|\mathbf{r}|^2}\right)$$
which is what the original paper states.