Im currently solving a system of mass, heat and momentum equations and I have some problems in trying to couple the momentum and mass equations (for mass transport I am using the diffusion-convection equation since mass velocity is small but not neglible, i.e. mass transport is mostly by diffusion). I am working with a sputtering system which typically operates in the mTorr pressure regime. Is there a valid approximation to the stress tensor which could somehow simplifies the solution of the momentum equations? I was just wondering if I can neglect the shear or normal stress, or if I can use a constant viscosity coefficient. By the way, the working gas is Argon (Im only interested in neutral Argon at the moment) and I am using FVM. Thanks in advance for any answer.

To neglect the stress, you may need to do some dimensional analysis. For example, if the Reynolds number is very small, you may be get rid of the inertia terms(the stress tensor), then you have to keep the viscosity term. Suggest to use the viscosity of Argon, length scale and velocity for the dimensional analysis. I am not sure this will help. But usually that is the way people use to simplify the equations.

Are you even in the continuum limit? At P = 1 mTorr, your mean-free-path should be on the order of 10 cm at room temperature.

Im interested in pressures above 5 mTorr(magnetron sputtering normally takes place from 5 to 40 mTorr). A rough stimation gives a free path of 1.5 cm. The typical distaces considered are 5 to 10 cm, which in turn validates the use of a continuum approach. I am using a MC model for the energetic particles which are impossible to model in a continuum model. Right now I just want to simplify the momentum equation so it is not that complicated to couple with a diffusion-convection equation.

You should calculate a Knudsen number to check the limit.