Partial Density Homework: Integrate Momentum in Navier-Stokes

[womfalcs3]

Homework Statement

I am trying to integrate the momentum equation in the Navier-Stokes equations, and I need to make sure I have the density formulation correct. I'm unsure of my work because a lot of the terms in the end are canceling out.

It's a two-phase mixture, so the sum of the partial pressures is:

P = P_a+P_v

Where a and v denote air and vapor, respectively.

Homework Equations

Ideal gas law

density=P/(R_gas*T)

The Attempt at a Solution

Using the sum of partial pressures, I'm getting this for total density:

\rho=\rho_{a}*R_{a}/R+\rho_{v}*R_{v}/R

Where R is the total gas constant.I am trying to find an equation for the total density in terms of the partial densities.

 

[womfalcs3]

Does the total density relation to the partial densities look correct?

Intuitively, I was thinking total rho = rho_a + rho_v, but the equation, obviously doesn't support that.

 

[womfalcs3]

EDIT:I realize the images are big. I'm the process of scaling them down and re-uploading them.

EDIT 2: Re-hosted and included updated images.

I apologize for the bump, but can somebody please just go through this formulation and verify my work?

It seems I am in fact eventually getting total density being the sum of partial densities.

That kind of makes sense given Dalton's Law, but the specific gas constants throw my intuitive thinking off.