Hello, I'm trying out the escript python FEM software package which is so far rather impressive, if for no other reason than the developers have included a Stokes Flow solver. The problem I'm having, however, is that they have formulated the problem in a manner I have not encountered before, nor can seem to make it "work" in the manner I would expect it to. In particular, we have from from section 6.1 of the users manual:

My basic problem is that I have not encountered what would normally be the Laplacian on the LHS of the above statement. That is, I would typically expect Stokes problem to be stated as
[tex]\Delta v - \nabla p = f[/tex]
which, components aside, does not seem to be an equivalent statement. Due to my application, the inclusion of the initial condition [itex]\sigma_{ij,j}[/itex] is unimportant, and conservation of mass ([itex]\nabla\cdot v=0[/itex]) is assumed in both cases.

So, can anyone tell me what I'm doing wrong, or where I might find a derivation of the quoted formulation so that I can actually apply it?

The problem I'm having is not one of indices vs. operator, what I'm failing to see is how
[tex]\nabla\cdot\left(\eta\left(\nabla v + \nabla^T v \right)\right)[/tex]
is equivalent (in some sense?) to
[tex]\eta\Delta v[/tex]
That is, I'm assuming that they mean that [itex]\nabla^T[/itex] denotes the adjoint to [itex]\nabla[/itex], but even then that doesn't seem to add up...

[itex]\nabla^Tv[/itex] denotes the TRANSPOSE of [itex]\nabla v[/itex]

If you sum them both and divide by 2, you get a symmetrical tensor called the "rate of stain tensor", let's call it ε

For an incompressilble flow ([itex]\nabla · v = 0[/itex]) the law that relates the "viscous stress tensor σ" (I think this one is also called deviatoric stress tensor) to the "rate of strain tensor ε" is:

σ= 2η·ε

Now, in the equation of conservation of momentum, σ doesn't appear as such, but through its divergence. If you calculate its divergence (or just look it up, Navier-Poisson's Law), you get to the conclusion: