Corrections to Fluid Dynamics

by R. F. Streater, Open Systems and Information Dynamics, 10, 3-30, 2003.

ABSTRACT

We propose a kinetic model of a fluid in which five macroscopic fields, the mass, energy, and three components of momentum, are conserved. The dynamics is constructed using the methods of statistical dynamics, and results in a non-linear discrete-time Markov chain for random fields on a lattice. In the continuum limit we obtain a non-linear coupled parabolic system of field equations, showing a correction to the Navier-Stokes equations: the equation for the conservation of energy (Fourier's law) acquires a cross term, showing the Dufour effect. The system is invariant under the Galilean group. All parameters are predicted in terms of the mass and mean free path of the molecules. It may be that the new equations are more stable as well as more consistent than the Navier-Stokes system. The key to the analysis is the postulate that the state of the system consists of a part in local thermodynamic equilibrium, together with a small correction. We express the dynamics in the form of a master equation, whose expansion in the small parameter, the relaxation time, gives the modified Navier-Stokes. This is rocket science.

The text can be viewed at the archive, math-ph/0105013. Its local preprint number is KCL-MTH-01-28.

REMARKS

Our subject is sometimes known as "compressible Navier-Stokes with temperature" or C-N-S-T. Together with Matheus Grasselli, we have extended a related model to one in an external potential; see Hydrodynamics in an external field",paper [137]. If the diffusion in [137] is put to zero, we get the Euler equations in an external field. If the velocity is put to zero, we get a version of the Smoluchowski equations with temperature, paper [126] with three-dimensional kinematics. If the temperature is then set to a constant, we get the Smoluchowski equation in an external potential.

There have been many studies of the isothermal, incompressible system, usually known as the Navier-Stokes equation: see e.g. Roger Temam, Navier-Stokes equations and non-linear functional analysis, SIAM, 1983, Philadelphia. The isothermal condition can be included in our model by starting with a uniform temperature; after one time-step in the full equations, we ignore the equation governing the conservation of energy, and in the remaining four equations, we re-adjust the changed temperature back to the original one. We then obtain four equations, for the density and the three components of momentum. Energy is not conserved, since any heat produced is removed by the environment...every point of space is assumed to be connected to a heat bath at the uniform temperature. This would be a good model for a good conductor of heat, such as mercury. In an isothermal model, entropy is not increasing as a function of time; instead, the free energy is a decreasing function of time. I call this "the free-energy theorem"; see The F-theorem for stochastic models", Annals of Phys. 218, 255-278, 1992; paper [99], or the chapters on isothermal dynamics in my book, Statistical Dynamics. To compare with the usual Navier-Stokes equations, we must specialise to the case of an incompressible liquid. The condition of incompressibility is harder to model within statistical dynamics. To be strictly incompressible, the state must be such that it is fully occupied, with every site taken up by a particle. This is at the limit of our model, and the pressure is infinite at such configurations. At smaller densities there will always be some compressibility. If we insist that rho is constant in time and space to infinite accuracy, then there is no diffusion and our system reduces to (a special case of) the usual system. It has been remarked (Temam, Navier-Stokes Equations, North Holland, 1977) that the pressure which appears in the velocity equation, is almost determined by the condition of incompressibility div u=0. Thus the pressure acts as a "balancing item", such as petty cash, popular with accountants; the value is adjusted as we go along to compensate for errors in other items. In our model, on the contrary, the pressure is a known function of temperature and density, and the compressibility or not of the fluid is one of the predictables of the model. The condition div u=0 is traditionally imposed in an attempt to set up a tractable model. For consistency, the initial and boundary conditions have to be compatible with div u=0, which precludes some configurations which ought to be arbitrarily assignable on physical grounds. For example, the density of a liquid changes with temperature, which has led e.g. S. Hoyas, H. Herraro and A. M. Mancho (arXiv.math.AP/0110195) to add a buoyancy term to the velocity equation, and the mass being accelerated by the pressure gradient depends on the temperature. This is done so that the temperature on the boundary can be in equilibrium with the nearby fluid. They keep the incompressible equation for the density, however, which is physically inconsistent. This system of equations has been further studied by Kadanoff; he assured us at the Dec 2001 meeting of the Rutgers Conference on Stat Mech. that "these equations have been correct for the last hundred years, and will be correct for the next hundred". Perhaps he hopes that the above physical inconsistency will not be noticed.

Our new, Dufour term does not satisfy Onsager symmetry, which does not hold in the model. In C-N-S-T the principal symbol of the elliptic operator giving time-evolution is a five-by-five matrix of second-order differential operators, with a row of zeros along the top row, and a column of zeros in the first column (the first row/column referring to the mass). Its symmetric part is therefore not invertible. This is what makes the N-S so hard (and interesting, to some). Our Dufour term occurs in the first column, and means that the symmetric part of the symbol could still be uniformly elliptic in some domain. It is therefore possible that our model is easier rather than harder to tackle than the usual system.

There is a CLAY PRIZE of $1M for a satisfactory solution to the incompressible Navier-Stokes equations. See the article by Fefferman.