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I have a problem that is isomorphic to the Stokes problem, but with the external force zero and the pressure known.

Specifically, I am trying to find solutions/methods in 3D to solve

$$\nabla^{2}\mathbf{u} - \nabla P = 0$$

$$\nabla\cdot\mathbf{u} = 0$$

where $\mathbf{u}$ is a vector and scalar P is known.

The Papkovich-Neuber solution seems the way to go (using Harmonic functions)

$$\mathbf{u} = \frac{1}{2}\left[\nabla(\mathbf{x}\cdot\Psi + \chi) - 2\Psi\right]$$

$$P = \nabla\cdot\Psi$$

So for the vector harmonic function $\Psi$ I have the following vector Laplace equation

$$\nabla^{2}\Psi=0$$

with

$$\nabla\cdot\Psi=P$$

Are there other well-known methods for solving Stokes flow when external force F is zero and P is known?

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  • $\begingroup$ How $P$ could be known in Stokes flow without solving Stokes equations? $\endgroup$ – Alex Trounev Apr 17 at 17:01
  • $\begingroup$ I mentioned my problem was not a stokes problem but had the same form but with P known and external force zero. I was hoping to gain some insight from the stokes flow to help with my problem $\endgroup$ – phryas Apr 18 at 23:49
  • $\begingroup$ Ok! But what it means "Solution to Stokes flow..." in your headline? It looks like you try to solve equation for the vector potential with given current? $\endgroup$ – Alex Trounev Apr 19 at 0:00
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We can use integral representation of solution for this problem

$$\vec {u}=-\frac{1}{4\pi}\int_V {\frac {\nabla \vec {P}}{r}dV}$$ where $r=\sqrt {(x-x_1)^2+(y-y_1)^2+(z-z_1)^2}$, $dV=dxdydz$. Some application of this approach in comparison with FEM could be found on Current Density In a 3D Loop - Discretising a Model

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