How to model the NR eqns in a domain with empty space
 

I am trying to solve the NR eqns in a domain which surrounds an empty space. To make my work a little easier I have created a mesh for the entire domain. So, I have velocity and pressure unknowns in the empty space. Now, I am not sure about the boundary conditions that I should use on the boundaries of the empty space. Certainly, the velocity will be zero, but what about the pressure.

I am solving the NR eqns with the FEM, and the fluid is incompressible.

For the moment, I am only setting the velocity to zero at these boundaries, but I believe that the solution is not very accurate. I believe that the correct way to solve this problem is to modify the code so that the continuity equation on the void boundaries does not have any contribution from the void. In this way, I am having two domains, and there is no interaction between the void and the domain of interest.
Unfortunately, if I do this, the Newton-Raphson scheme does not converge.

Does anybody have an idea about how to model this problem. I would like to say here that I am not interested in a solution which does not consider the empty space, i.e., I do not want to construct a mesh which goes around the empty space.

I thank you in advance for your time.

Newton Raphson would diverge because I had forgotten to update the residuals.
Now, it converges within a couple of iterations.