What is the easiest way to treat d(rho)/dt in the Poisson equation for multiphase, incompressible flows, as d(rho)/dt is not equal to zero for mixtures.

can you give more details please!

Su, I would like to model two phase flows, specifically, liquid-liquid or liquid-gas. There will of course be two different densities, and assuming that we are only looking at the liquid-liquid mixing, with two different densities, the Poisson equation is solved for in the calculation of the pressure. In doing that, there is actually a d(rho)/dt in the equation, which is zero if there is only 1 phase and incompressibility. However, for multi-phase flows, I can't see how this is zero since it is actually the mixture density now, which does change with time. Saying that, I am not sure how to model d(rho)/dt in that equation. Once the density field is solved for, then the volume fraction of will be calculated.

The pressure equation is derived from the continuity equation. The details depend on the method. For instance, the formulation used in the Simple and Simple-like techniques is slightly different from the earlier work (MAC, SMAC) out of Los Alamos.

In your case, you'll have two continuity equations, one for each component (or you can write one in terms of a total density and the other in terms of a mass fraction).

Your best bet is to start with the momentum and continuity expressions appropriate to your problem and work through the details following the outline of one of the established methods. The d(rho)/dt term will appear as a part of that development.

Good luck.

If you mean 'mass derivative' by d/dt, d(rho)/dt=0 when fluid is incompressible, even for multiphase flow without phase change. If you want partial(rho)/partial(t), then you can calculate it from volume fraction( VOF method), level set function, etc.

Good luck.

Let us think in this way:

first we need to write your equations then think of modeling liq-liq or gas-liq. etc. this is a simple example or just a keywork or an idea in general but let see; as i know you'll have a non-conservative form of the equations which is a MaJoR problem in two-phase flow area.

once you accept that; you've 2 different densities; so you've 2 different conservation of mass, momentum and energy etc. so the frist equations may be written as (i'm writing \partial not d...)

\partial_{t}(\alpha_{j} \rho_{j} + \partial_{x} (\alpha_{j} \rho_{j} u_{j}) = Sorce term (or 0).

and then start solving it etc. with j = 1, 2 for phase 1 and phase 2. using mixture relations e.g. \rho = \alpha_{2} \rho_{2} + \alpha_{1} \rho_{1}, where

we can write it as

\partial_{t}(\rho) + \partial_{x} (\rho u) = Sorce term (or 0).

so you can have a system of mixture-type two-phase flow;

As the poisson eq for pressure comes from cont.eqn I gues s one could write it as (ddt means partial ddt)

[1] ddt(rho)+div(rho u)=0 [2] ddt(rho)+u div(rho)+rho div(u)=0

but if assuming that rho is proportional to eg. a convected vof or level set scalar Q ( ddt(Q)+u div(Q)=0 ) the first two terms in [2] vanishes. So no ddt(rho) left for the cont.eqn it seems.