variational form of navier-stokes equation with variable density

Question

Hi
Can any body guide me how to write variational form of following navier-stokes system?

ρ(ut + u. ∇u) -ηΔu + ∇p = ρg in Ω
∇.u = 0 in Ω
ρt + u . ∇ρ = 0 in Ω
u(x; 0) = u0 in Ω
ρ(x; 0) = ρ0 in Ω
u(x; t) = ud on ∂Ω
where the third equation indicates that the density of the fluid is advected with the fluid, which remains incompressible (second equation).

Thanks alot

Answer 1

I would start by looking at the documented demo:

http://fenicsproject.org/documentation/dolfin/1.4.0/python/demo/documented/navier-stokes/python/documentation.html

Comments

many thanks for replying. I have read the documentation. actually I could operate navier-stokes code for cavity and flow past a cylinder problem when the density was constant. But I don't know how to write it in weak form when density is a function of x and t. could you show me an example of such situation?
saeedeh

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So basically, you want the weak formulation of a system of equations, is that correct? In that case, you should consider a mixed formulation. Take a look at:

http://fenicsproject.org/documentation/dolfin/1.4.0/python/demo/documented/mixed-poisson/python/documentation.html

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It was useful, thank you.
But in this system of equations , I have 3 equations. Does it mean that, I need 1 trial function and 2 test function? and as a consequence 3 function space?

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If you want 3 equations in your variational formulation, then you need 3 trial functions and 3 test functions, something like:

V = FunctionSpace(mesh, ..., ...)
W = MixedFunctionSpace([V]*3)

# Define trial and test functions
(u1,u2,u3)  = TrialFunctions(W)  # solution
(v1,v2,v3)  = TestFunctions(W)