# Solving inhomogeneous Stokes equation

I want to solve the Stokes inhomogeneous equation, i.e.

$$\nabla^2 \vec v -\nabla P = \vec f(r,\theta)$$ $$\nabla\cdot\vec v=0$$

where $\vec f$ is irrotational, i.e. $\partial_y f_x - \partial_x f_y = 0$

The Oseen tensor solution (i.e. using the open plane Green's function) doesn't get me anywhere because the integral is very tricky, so I am forced to use eigenfunctions expansions.

My way of proceeding has been like this:

1) I find a solution $\vec v^{(H)}$ for the homogeneous problem, by defining such that $$\vec v=(\partial_y\psi,-\partial_x \psi)$$ and solving thus $$\nabla^4\psi=0$$

2) Once I have found $\psi^{(H)}$, I use its definition (I differentiate) to get $\vec v^{(H)}$

3) Now I need, in order to have a general solution to the full problem, and exploiting the nonlinearity, a particular integral of the full inhomogeneous equation. But, what confuses me is the presence of the pressure $P$. Since my $\vec f$ has a form

$$f_i(r,\theta)=r^n(\cos m\theta, \sin m \theta)$$

it is easy to guess a form for $\vec v$ and plug it into the equation, but what about the form for $P$? And how should I use the incompressibility condition?

• Maybe you can take the rotational of the first equation and obtain $\nabla^2\omega=0$ then you can find $\psi$ by solving poisson equation and for finding $P$ you can take the divergence of the first equation to obtain $\nabla^2P=-\nabla\cdot\boldsymbol{f}$. – C. Esparza Mar 1 '15 at 2:28