Suppose you have a container of water as a solvent and you a certain amount of salt as a solute sitting at the bottom of the container that has yet to start dissolving. Supposing temperature and pressure are kept at a constant.
So at this stage: Water has density $\rho_w(x, y, z, t)$. Salt has density $\rho_s(x, y, z, t)$. And as the solution at the beginning is just comprised of the water itself it it has density $\rho_t(x, y, z, t) = \rho_s(x, y, z, t)$.
Now as the salt begins to dissolve the solution becomes a mixture of salt and water and we have $\rho_t = \alpha \rho_w + \beta \rho_s$.
I have a few questions about the state of the system now:
- Is the density of water still the same as it was at the beginning now that salt has begun dissolving? Ie. is the density of water constant? Or will the dissolved salt molecules "squeeze" the water molecules into a smaller volume thus increasing the density of water? And actually we should be talking about concentration of water now instead of density of water as we are dealing with a mixture comprised of two substances?
- Is the concentration/density of salt in the solution constant? This questions seems to have the obvious answer of "no it isn't because the salt is dissolving and thus the concentration of the salt in the solution is changing over space and time as it spreads out"?
- The density of the solution itself, $\rho_t$...as the concentration of salt seems to be non-constant it therefore implies that the density of the solution is non-constant...I.e. it will vary at different points in the fluid over space and time as the salt dissolves until it reaches an equilibrium when all the salt is dissolved?
- So if the solution has non-constant density it means it is compressible? And it will be governed by the compressible Navier-Stokes equations?
I'm trying to program a numerical method for a similar situation to what I described above so I appreciate any help with these questions!