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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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