8.2. Singularities in Poisson equation

8.2.1. 3-dimensional Poisson equation

Green’s function reads \(\mathbf{G}(\mathbf{r}; \mathbf{r}_0) = \frac{1}{4 \pi |\mathbf{r} - \mathbf{r}_0| }\).

Point Source.

\[\begin{split}\begin{aligned} \varphi (\mathbf{r}; \mathbf{r}_0) & = G(\mathbf{r}; \mathbf{r}_0) = \frac{1}{4 \pi |\mathbf{r} - \mathbf{r}_0| } \\ \mathbf{u}(\mathbf{r}; \mathbf{r}_0) & = \nabla G(\mathbf{r}; \mathbf{r}_0) = \frac{1}{4 \pi} \frac{\mathbf{r} - \mathbf{r}_0}{ |\mathbf{r} - \mathbf{r}_0|^3 } \end{aligned}\end{split}\]

having exploited

\[\nabla |\mathbf{r}-\mathbf{r}_0| = \hat{\mathbf{x}}_i \partial_i |\mathbf{r} - \mathbf{r}_0| = \hat{\mathbf{x}}_i \frac{x_i - x_i^0}{|\mathbf{r}-\mathbf{r}_0|} = \frac{\mathbf{r}-\mathbf{r}_0}{|\mathbf{r}-\mathbf{r}_0|} \ .\]

Poisson’s equation \(\ \nabla^2 \varphi = 0\), for \(\ \mathbf{r} \ne \mathbf{r}_0\)

…

Velocity flux, \(\ \oint_{\partial V} \mathbf{u} \cdot \hat{\mathbf{n}} = E_V(\mathbf{r}_0)\)

…

Point doublet.

\[\begin{split}\begin{aligned} \varphi (\mathbf{r}; \mathbf{r}_0) & = \hat{\mathbf{n}}_0 \cdot \nabla G(\mathbf{r}; \mathbf{r}_0) = \\ \mathbf{u}(\mathbf{r}; \mathbf{r}_0) & = \dots \end{aligned}\end{split}\]

Poisson’s equation \(\ \nabla^2 \varphi = 0\), for \(\ \mathbf{r} \ne \mathbf{r}_0\)

\[\begin{aligned} \partial_{ii} \left( n^0_k \partial_k G \right) & = n^0_k \partial_k \partial_{ii} G = 0 \ , \end{aligned}\]

for \(\mathbf{r} \ne \mathbf{r}_0\), as \(\hat{\mathbf{n}}_0\) is independent from the spatial coordinate \(\mathbf{r}\).

Point vortex.

\[\begin{aligned} \mathbf{u}(\mathbf{r}; \mathbf{r}_0) & = \nabla G(\mathbf{r}; \mathbf{r}_0) \times \hat{\mathbf{t}}_0 \end{aligned}\]

There’s no irrotational point vortex solution. This is somehow linked to vorticity dynamics, and Hemlholtz’s theorem about vortices: whatever a vortex is, vortices must be either lines or surfaces (either closed or with extreme points on solid boundaries), or a volume distribution of vorticity.

Non-zero vorticity field for point vortices

\[\boldsymbol\omega = \nabla \times \mathbf{u} = \nabla \times \left( \nabla G \times \hat{\mathbf{t}}_0 \right)\]

\[\begin{split}\begin{aligned} \varepsilon_{ijk} \partial_j \left( \varepsilon_{klm} \partial_l G t_m \right) & = \left( \delta_{il} \delta_{jm} - \delta_{im} \delta_{jl} \right) \partial_j \left( \partial_l G t_m \right) = \\ & = \partial_m \left( \partial_i G t_m \right) - \partial_m \left( \partial_m G t_i \right) \ , \end{aligned}\end{split}\]

with \(\hat{\mathbf{t}}_0\) independent from the spatial coordinate \(\mathbf{r}\). Here \(\nabla^2 G = 0\), for \(\mathbf{r} \ne \mathbf{r}_0\), and

\[\begin{aligned} \left( \hat{\mathbf{t}}_0 \cdot \nabla \right) \nabla G & = \dots \ne \mathbf{0} \ . \end{aligned}\]

Line vortex.

\[\mathbf{u}(\mathbf{r}) = \int_{\mathbf{r}_0 \in \ell_0} \nabla G(\mathbf{r}; \mathbf{r}_0) \times \hat{\mathbf{t}}_0 \Gamma_0 \ ,\]

with constant \(\Gamma_0\) over the vortex line \(\ell_0\).

Vorticity field of a line vortex

If \(\ell_0\) is not closed, …

If \(\ell_0\) is closed, (and \(\mathbf{r} \notin \ell_0\) so that \(G\) is not singular anywhere for \(\mathbf{r}_0 \in \ell_0\)), the velocity field can be recast as

\[\begin{split}\begin{aligned} \oint_{\mathbf{r}_0 \in \ell_0} \nabla G \times \hat{\mathbf{t}}_0 & = \oint_{\mathbf{r}_0 \in \ell_0} \varepsilon_{ijk} \partial_j G t^0_k = \\ & = \int_{S_{\ell_0}} n_a^0 \varepsilon_{abk} \partial_b \left( \varepsilon_{ijk} \partial_j G \right) = \\ & = \int_{S_{\ell_0}} n_a^0 \left( \delta_{ia} \delta_{jb} - \delta_{ib} \delta_{ja} \right) \partial_{bj} G = \\ & = \int_{S_{\ell_0}} n_i^0 \underbrace{\partial_{jj} G}_{=0} - \int_{S_{\ell_0}} n_j^0 \partial_{ji} G = \\ \end{aligned}\end{split}\]

while the vorticity field is identically equal to zero in every point \(\mathbf{r} \notin \ell_0\),

\[\begin{split}\begin{aligned} \nabla \times \oint_{\mathbf{r}_0 \in \ell_0} \nabla G \times \hat{\mathbf{t}}_0 & = - \varepsilon_{ijk} \partial_j \int_{S_{\ell_0}} n_l^0 \partial_{lk} G = \\ & = - \int_{S_{\ell_0}} n_l^0 \varepsilon_{ijk} \partial_{ljk} G = \\ & = - \int_{S_{\ell_0}} \hat{\mathbf{n}}_0 \cdot \nabla \underbrace{\left( \nabla \times \nabla G \right)}_{= \mathbf{0}} \end{aligned}\end{split}\]