How to express and solve Darcy's law?

variational formulation
1

Hi everyone,

I’m trying to solve a rather simple expression:

20191018_141943
20191018_141943.jpg1944×782 552 KB

where alpha, kappa and viscosity are constants. The domain is over a cylinder. I know the pressure within the inner lumen and at the outer lumen. I’ve expressed the weak form of the equation (1) as:
F = u*v*dx + ((poro*permeability)/viscosity)*v*grad(p)*dx
My questions are:

  1. How do I express my boundary conditions? Since I only know the pressure and not the velocity

  2. Currently, I’ve worked out an expression for Pressure assuming a linear distribution as follows:

    pressure = Expression("((P_lumen - P_adv)/vessel_thickness)*pow((pow(x[0],2) + pow(x[1],2)),0.5)", degree=1, P_lumen=p_inner_lumen,
    P_adv=p_adventitia, vessel_thickness=vessel_thickness)
    p = interpolate(pressure,V)

    However, I’m not sure if I’m supposed to assume this or treat it as another variable to be solved?

  3. How do I implement the 2nd equation? I’ve been trying to follow this but since my form does not have a temporal domain I find it rather confusing.

2

Hi,

You can follow the tutorial : https://fenicsproject.org/docs/dolfin/1.4.0/python/demo/documented/mixed-poisson/python/documentation.html

I can post a full code of mine if needed (for the boundaries conditions on the pressure and/or on the Darcy velocity).

AMM

3

Hi,

Thanks! Managed to solve it with your help :slight_smile:

4

You’ve linked to an old old version. Latest demo is https://fenicsproject.org/docs/dolfin/2019.1.0/python/demos/mixed-poisson/demo_mixed-poisson.py.html,
or https://fenicsproject.org/docs/dolfin/latest/python/demos/mixed-poisson/demo_mixed-poisson.py.html

5

Could you please share your full code?