How to update coefficients for transient flow with variable coefficients
 

Hi,

I am learning SIMPLE algorithm and writing a toy code to implement it based on FVM.
In geernal, coefficients are variable and need to move with the flow field, e.g.:
1. density, viscosity for N-S equations;
2. density, heat capacity, thermal cunductivity for heat equations.

I have been wondering how to update coefficients for every time step. For instance, givening a cavity with flow driven by a dense ball (as shown in the attached figure), the density and viscosity of ball differ from thoese of the background medium.

After each time step, new velocity is obtained via SIMPLE algorithm, and after this, the ball (embodied by the spatially larger value of density and viscosity) moves downwards.

To "move" (or spatially "update") the coefficients, is it necessary to solve another (linear) advection equation of the field of this coefficient?

If so, how to deal with boudary conditions (BC) of this advection eqaution, esp., what is the appropriate type of "BC" of the density (e.g., Dirichelt, Nuemann, etc)?

Thanks a lot!

f