I’m trying to model a coolant fluid flowing through a porous fuel medium in 1D. The model is currently very simple: the fuel generates heat and distributes it through internal conduction. The coolant is heated by contact with the fuel, and eliminates heat by leaving the volume.
I’m using a Dirichlet BC on one side, and a Neumann on the other.
Fuel:
-kA\Delta u_{f} - hA_{bubbles} (u_{f} - u_{c}) = q
Coolant:
hA_{bubbles} (u_{f} - u_{c}) = C_{p}\dot{m} \nabla u_{c}
Where:
k is the conductivity of the fuel.
A is the conductive area.
u_{f} is the temperature distribution of the fuel.
h is the convective heat transfer coefficient.
A_{bubbles} is the area of coolant exposed to the fuel.
u_{c} is the temperature distribution of the coolant.
q is the rate of heat generation.
C_p is the coolant’s specific heat capacity.
\dot{m} is the coolant flow rate.
Setup:
mesh = IntervalMesh(n_shells, 0, r_total)
Ve = FiniteElement("CG", mesh.ufl_cell(), 2)
V = FunctionSpace(mesh, MixedElement([Ve, Ve]))
vf, vc = TestFunctions(V)
u = Function(V)
uf, uc = split(u)
Could someone help me translate these two equations into a variational formula?