Discussion Forum

Hi, I'm trying to model a simplification of fluid flow in a porous pipe. Unknowns are pressure, p, and velocity v. Equations are:

dv/dx = eta (P_R - p); dp/dx = -f v^2; v(0) = v0 ; p(L) = p_L; v0 >=0; p_L<= p <= p_R; 0 < f << 1

If eta, P_R and f are constant, this has an analytic solution.... a fun homework assignment... More generally, I can solve this as an "almost" elliptic formulation with 1D Comsol

d/dx 1/eta dv/dx - f v^2 = 0

No problem. Except it is really p that I want, not v. Or maybe even better: both. Which takes me down the road of a weak formulation:

-\int v dq/dx + \int eta q (p - P_R) = q(0) v_0

- \int p dw/dx + \int f w v^2 = -w(L) p_L

with w(L)=q(0)=0. I take unknowns v,p; draw line [0,1] and create subdomain terms:

-p*test(vx)+f*test(v)*v*v
-v*test(px)+test(p)*(p-p_R)*eta

and boundary terms, at 0,

0 (with strong constraint -v+v_0)
test(p)*v_0

and, at L,

-p_L*test(v)
0 (with strong constraint -p+p_L)

Result is junk. What am I doing wrong? Is this a Babuska-Brezzi thing?

Regards, John