Why are Multiphase/Multifield/Multifulid Fluid Flow Models Ill-Posed?
December 8th, 2007 by Dan HughesWhen 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.
Posted in PDEs |
December 13th, 2007 at 6:54 am
I am not sure to have quite understood what you wanted to do .
Specifically why focusing on “interfaces” ?
However the mention of red and blue fluids lead me to another thing - the Rayleigh unstabilities (a layer of a heavy fluid above a light fluid) .
This is a well known process exploited in those fascinating devices where you have differently colored very viscous fluids within a moving glass container that you can observe for hours developping ever new patterns .
The thing is obviously chaotic .
But that is not the point .
Applying N-S to that system , nothing happens .
Of course as the initial condition is a perfectly plane and horizontal interface with the velocity field = 0 everywhere , the system stays in the metastable form forever .
To see things happen you must perturbate the initial conditions , if possible very slightly .
Then things begin to happen and look broadly like what happens in reality . Well let’s say very broadly .
Now I remeber having read a paper where people made experiences with this system by using different magnetic properties of the 2 fluids so could maintain the upper layer in position by applying a magnetic field .
The beauty of this is that you can not only control the initial conditions extremely precisely (f.ex a perfect horizontal plane) but you can also apply a perfectly controlled perturbation of the initial conditions by modifying locally the magnetic field .
Well what they saw was that the flow patterns and velocity fields deduced from N-S were very far from what was really happening even if there was a qualitative resemblance (of the kind that any cloud looks a bit like all other clouds :))
So what they did was to develop a simulation bottom up - coming from the atomic level and going to the macroscopic level .
I know it is supposed to be impossible because there are too many atoms/molecules .
Yet this big number is not big enough for a small experiment like the one above .
To make a long story short , the results were much closer to the reality and they didn’t need to tinker anything with the initial conditions .
At the molecular level there are enough random fluctuations to initiate instabilities that grow with time and initiate the process .
What does all this show ?
The obvious - as N-S are continuous equations , they need fluids that are … continuous .
Yet we know that the smaller the scale , the closer to discontinuities .
Once the mean distance between molecules is no more negligeable wrt some characteristic flow scale , the medium is no more continuous and N-S doesn’t apply anymore .
And that is not only a theoretical abstract construct that doesn’t exist in the nature - the cases where that happens are numerous f.ex nuclear fusion , stellar dynamics , space shuttle reentering atmosphere etc .
Coming back to N-S .
They can be applied only then when there is a guarantee that ALL relevant processes and features of the flow happen above a scale where the intermolecular distance can be largely negligible .
That’s a kind of conceptual interface where theories change .
February 15th, 2008 at 11:01 pm
Actually, for gas-solid or flows with droplets, careful attention to what happens to the flow near interfaces between the fluid particles phases does result in the appearance of viscous like terms in the particulate phase. When the equations for the phases are added together, for very small particles, the mixture equation will exhibit the Einstein viscosity correction for a mixture.
February 17th, 2008 at 10:21 am
When I posted this I assumed that I could readily find my notes on the problem. That proved to be an incorrect assumption. I have since found the notes, but in the meantime I got started on several other things.
Maybe the hint about the Euler equations was not sufficiently clear. The single-phase Euler equations have a well-studied case that can be used to illustrate the roles of characteristics, boundary conditions, and numbers of equations and unknowns; shock waves. Chapter V of Numerical Approximation of Hyperbolic Systems of Conservation Laws by Godlewski and Raviart, especially Section 3, has a detailed treatment of characteristics relative to boundary condition specifications. I’m sure other texts also cover the subject in detail.
A key point relative to the general problem that I stated is: If characteristics at a boundary point into the solution domain from outside, data must be specified. The solution cannot be determined from the equations. In contrast if characteristics point to the boundary from inside the solution domain, the equations can be used to calculate the solution there. There are of course nitty-gritty issues of exactly what information in terms of the dependent variables must be supplied to compensate for the missing information, but that is not a primary part of the general issue. The reference above lists the dependent variables, and combinations of those, that can be specified so that a well-posed IBVP is set.
An interesting bit of history about theory of shock waves.
March 15th, 2008 at 7:21 am
The usual approach that is used to resolve one of the the problems that arise in the situations that are the subject of this post is employed in Miskolczi 2004 and Miskolczi 2007.