Charles Peskin developed a method for handling moving boundaries in a fluid flow problem. Here's a link to one of his earlier and much-cited papers. I found this and many other articles by googling "Peskin cadiac valve model". I attended a lecture he gave a few years (decades) ago.
When dealing with a complicated PDE situation, I find it useful to write the forward time, centered space (FTCS) difference equations, and implement them in a script with a simple Euler method. This method is computationally inefficient, but relatively easy to implement. And the relative simplicity of implementation means it may also be easier to add the necessary code for a moving boundary, which will be governed by its own differential equation.
The stability criterion for the numerical solution of the PDE for a Newtonian (constant-viscosity) fluid, when using a simple FTCS method, is
, where k=1/2 or 1/4 or 1.6 for a 1D or 2D or 3D problem, respectively. For a Bingham fluid, I'd use the plastic viscosity, at least as a starting point, when estimating how small 
must be, for a given 
, to assure numerical stability.Do you assume laminar, axisymmetric flow? If so, then you have two spatial dimensions (z=axial and r=radial) in the problem. When you solve it numerically, the boundary condition at the center takes a bit of thought. If v=v(r,z,t) is the velocity, then I would use dv/dr=0 at r=0, at all times, for the boundary condition in the center.
Are the two domains an outer "solid" domain and an inner (circular cross section) core of flowing plastic? And the flowing domain gets bigger when the pressure gradient gets larger?
