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the.drizzle

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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?

Thanks!

We want to calculate the velocity ﬁeld v and pressure p of an incompressible ﬂuid. They are given as the solution of the Stokes problem

[tex]-\left( \eta \left( v_{i,j} + v_{j,i} \right) \right)_{,j} + p_{,i} = f_i + \sigma_{ij,j}[/tex]

where [itex]f_i[/itex] deﬁnes an internal force and [itex]\sigma_{ij,j}[/itex] is an initial stress. The viscosity may weakly depend on pressure and velocity. If relevant we will use the notation [itex]\eta\left(v,p\right)[/itex]to express this dependency.

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?

Thanks!

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