Harold Grad

Bio: Harold Grad is an academic researcher from New York University. The author has contributed to research in topics: Boltzmann equation & Boundary value problem. The author has an hindex of 18, co-authored 27 publications receiving 5661 citations.

Principles of the Kinetic Theory of Gases

Abstract: For the purposes of this article, the subject of the kinetic theory of gases is considered to be coextensive with the theory of the Boltzmann equation We consider only the original equation of Maxwell and of Boltzmann for classical point molecules and short range forces, putting aside the equally interesting but distinct questions which arise from the inclusion of internal degrees of freedom, quantum interactions, inverse square forces, and imperfect gases The special case of a Knudsen gas of freely streaming particles is only touched on, mainly for purposes of comparison

Asymptotic Theory of the Boltzmann Equation

Abstract: The precise mathematical relation that the Hilbert and Chapman‐Enskog expansions bear to the manifold of solutions of the Boltzmann equation is described. These expansions yield inherently imprecise descriptions of a gas in terms of macroscopic fluid variables instead of a molecular distribution function. It is shown that these expansions are asymptotic to a very special class of solutions of the Boltzmann equation for sufficiently small mean free path. Next, a generalization of the Hilbert and Chapman‐Enskog expansions is described in terms of extended sets of macroscopic state variables, each governed by partial differential equations similar to those found in fluid dynamics, but sufficiently general to approximate an arbitrary distribution function. The generalized expansions are shown to be asymptotic to quite arbitrary solutions of the Boltzmann equation. It is then shown that the ordinary Hilbert and Chapman‐Enskog expansions can also be made asymptotic to very general solutions of the Boltzmann equation by reinterpreting the variables that enter these expansions as certain well‐defined replacements for the actual fluid state of the gas. In this way the scope of the Euler, Navier‐Stokes, Burnett equations, etc., is greatly extended by interpreting them as governing the artificial variables. Not only are general solutions of the Boltzmann equation shown to be approximated by fluid dynamics (in the limit of small mean free path), but the rapid decay of an arbitrary initial distribution function to a special Hilbert distribution function is also governed by sets of partial differential equations similar to those found in fluid dynamics.

Note on N-dimensional hermite polynomials

Abstract: I t is well known that a complete set of orthonormal polynomials in N variables can be obtained by using products of such polynomials in a single variahle. Such a procedure lacks symmetry, and there is sometimes an advantage to be gained by expressing the polynomials in tensor invariant notation. The -\--vector z,(i = 1, 2, . . . , X ) is denoted by x. The second order tensor (dyad) X , X , is denoted by x'. Similarly, xn is used for the n-th order tensor x,,xi2 x,, . For the scalar product I t is convenient to introduce a special notation.

Toroidal containment of a plasma.

Abstract: The question of plasma containment in a torus is much more complicated than in an open‐ended mirror system. Serious questions arise of the nonexistence of flux surfaces, of noncontained particle drifts, and of nonexistence of self‐consistent equilibria at small gyroradius.

The non-linear field theories of mechanics

Abstract: Matter is commonly found in the form of materials. Analytical mechanics turned its back upon this fact, creating the centrally useful but abstract concepts of the mass point and the rigid body, in which matter manifests itself only through its inertia, independent of its constitution; “modern” physics likewise turns its back, since it concerns solely the small particles of matter, declining to face the problem of how a specimen made up of such particles will behave in the typical circumstances in which we meet it. Materials, however, continue to furnish the masses of matter we see and use from day to day: air, water, earth, flesh, wood, stone, steel, concrete, glass, rubber, ... All are deformable. A theory aiming to describe their mechanical behavior must take heed of their deformability and represent the definite principles it obeys.

Multiple-relaxation-time lattice Boltzmann models in three dimensions.

Abstract: This article provides a concise exposition of the multiple-relaxation-time lattice Boltzmann equation, with examples of 15-velocity and 19-velocity models in three dimensions. Simulation of a diagonally lid-driven cavity flow in three dimensions at Re = 500 and 2000 is performed. The results clearly demonstrate the superior numerical stability of the multiple-relaxation-time lattice Boltzmann equation over the popular lattice Bhatnagar-Gross-Krook equation.