Density driven flow
I'm doing some work on density induced flow in porous media. My problem contains a single phase fluid with 2 components (water and a solute). I'm solving the continuity equation along with the advectiondiffusion/dispersion eq., Darcy, the equation of state (links between the concentration and density) and a viscosity function.
Currently I'm trying to figure out whether the Boussinesq approximation is valid in my case. I've scaled the equations and while doing so, revealed a gap in my Understanding of the physics involved.
The continuitymass conservation for the fluid phase:
$$φ\frac{∂ ρ}{∂t}+∇\cdot(ρq)=0$$
I've decided to scale my system of equations using a diffusive time scale $x_0^2/D$. The other relevant scaling factors are the flux $q_0$ e and the density is scaled as $${ρ^*}=({ρ}{ρ_0})/(ρ_{max}ρ_0)$$
The scaling produced:
$$\frac{ε}{Pe}\frac{∂ρ}{∂t}+ε\cdot ρ∇\cdot q+∇\cdot q+ε\cdot q∇ρ=0$$
where $ε\ll1$ and $Pe\ll ε$
For that case I can see that the accumulation term has the largest magnitude and therefore $∂ρ/∂t=0$.It does not make sense to me since density does change in respect to time due to diffusive processes. Clearly i'm missing something basic here.
Any ideas?
Thanks!
