1D moving boundary problem

I have the following problem: A two-phase 1D Poiseuille flow in cylindrical coordinates in a tube with radius R, the interface of which (k1) is defined via a scalar function. Since it is a moving boundary problem, I am trying to use an ALE formulation. The unknowns are the fluid velocity (v), the stress of the fluid (trz) and the velocity of the mesh (w). I am solving it in a fixed space divided into two regions, dx_f1 (fluid1) and dx_f2 (fluid2). According to FSI BOOK in page 41, the ALE formulation for the fluid (this book refers to FSI so it has fluid and solid) is given by Problem 4 (only the terms of {Ω}_f ). I have also two extra equations for the unknowns k1 & k2. k1 defines the area occupied by fluid1 (or fluid2 (R-k1) it is the same) and it is given by

k1 = \sqrt{\frac{f(R,k2)}{k2}} (Eq.1)

while k2 is given by an integral constraint as

k2 \int_{0}^R v r dr = d \int_{0}^{k1} v r dr (Eq.2), where d is a constant. I introduced k1 and k2 as Lagrangian multipliers in space R.

I have two main problems:

  1. I suppose that the driven force of the movement of the mesh is Eq.1 (correct me if I am wrong). How can I introduce it in the problem to get a moving interface?
  2. I tried to solve the problem for k2 with a pre-determined k1. So I solved Eq.2 for k1 = 0.5.
    This is the sample of my code as far as the weak forms

# Fluid 1
inertia  =  Re*J(w,U,dt)*( (1./dt)*(v-v0) + Finv(w,U,dt)*(v-w)*v.dx(0) ) * phi[0]
viscous  =  J(w,U,dt)*FinvT(w,U,dt)*( - phi[0] - (1./x[0])*trz*phi[0] + trz*phi[0].dx(0) )

Res1	 = (  inertia + viscous ) * dx_f1
Res2     = (trz - 2.0*v.dx(0)) * phi[1] * dx_f1
Res3     = - ( dt*(1.0/J(w,U,dt))*w.dx(0)*phi[2].dx(0) + (1.0/J(w,U,dt))*U.dx(0)*phi[2].dx(0)  ) * dx_f1


# Fluid 2
inertia  =  Re*J(w,U,dt)*( (1./dt)*(v-v0) + Finv(w,U,dt)*(v-w)*v.dx(0) ) * phi[0]
viscous  =  J(w,U,dt)*FinvT(w,U,dt)*( - phi[0] - (1./x[0])*trz*phi[0] + trz*phi[0].dx(0) )

Res4	 = (  inertia + viscous ) * dx_f2
Res5     = (trz - v.dx(0)) * phi[1] * dx_f2
Res6     = - ( dt*(1.0/J(w,U,dt))*w.dx(0)*phi[2].dx(0) + (1.0/J(w,U,dt))*U.dx(0)*phi[2].dx(0)  ) * dx_f2


# Global Uknowns
Res7     = ( k1 - 0.5  ) * phi[3] * dx_f1
Res8     = k2*v*x[0]*phi[4]*dx - v*x[0]*phi[4]*dx_f1

I get “nan” which is originated from Eq.2 (Res8)

Res1 & Res4 → equation for v in fluid1 and fluid2
Res2 & Res5 → equation for trz in fluid1 and fluid2
Res3 & Res6 → equation for w in fluid1 and fluid2
Res7 → equation for k1
Res8 → equation for k2

Any idea for problems 1) or 2) ?

Thanks

Please let me know if my post and the problems are clear. Thank you