Thanks a lot for being kind and posting this answer. In the meantime I'm trying to learn what differential forms are. They are not easy to swallow.

@Gilgamesh Could you please elaborate in a post? I am interested in understanding how one can apply Reynolds' Transport and Divergence Theorems to transform Equation 3 into its partial differential equation form, what you referred to as Cauchy's Momentum Equation.

@Gilgamesh Secondly, I remain unconvinced that Cauchy's stress tensor model is the most comprehensive representation of a continuum. Simplifying the general control volume to an infinitesimal cuboid results in a loss of information. To verify this, I would like to test whether Cauchy's model adheres to the principle of material frame indifference (MFI). If it does, then Cauchy's tensor should be an objectivity field. Consequently, starting from two different frames of reference, such as two rotated Cartesian coordinate systems, should yield identical results.

@Gilgamesh, I'm hesitant to employ Einstein notation because it doesn't help with my confusion between tensors and matrices. My concerns essentially come down to a few fundamental questions. For instance, we applied the divergence theorem to transform Equation 1 from its integral form to Equation 2, the PDE form. I wonder if there are any generalizations of the divergence theorem that could similarly transform Equation 3?

@ChetMiller please see the picture at the end of this post. This is how I envision the flow to be in the steady state phase.

@ChetMiller I think Anders's suggestion to use elliptic coordinate system is a good idea.