Models, Methods, Software

When the flows mentioned in the title are approached by use of model conservation/balance equations for the individual constituents, or separate fluid regions, in the flow field, many (all?) are ill-posed as hyperbolic mathematical problems. We can get into issues associated with parabolic and elliptic systems, but let’s try to stick to the hyperbolic case to start off. In fact, one method to regularize (?) the hyperbolic case is to convert it to the parabolic case by use of a vanishingly small viscosity-like parameter. And by all means let’s avoid discussions of the effects of what discrete approximations to the continuous equations do to the whole messy situation.

Why is that?

Could it be that the fundamental continuum mechanics equations for fluid flow, usually taken to be the Navier-Stokes equations, are missing something when physical interfaces are present in the flow field? To be specific, the Euler equations, I think, are the hyperbolic case of interest.

Let’s attempt to count the number of characteristics that are pointing to a physical interface between two constituents or regions. We can do this even when the constituents are red and blue water, for example. Then we’ll have to find enough equations for the unknowns we count. That’ll be the fun part.

Who wants to start?

Let me know if you find incorrectos in the above.