Microscale Flow and Heat Transfer pp 189-258 | Cite as

# Grad Equations: Derivation and Analysis

- 431 Downloads

## Abstract

In certain physical problems, the Navier–Stokes equations are insufficient to describe the flow physics accurately, making it necessary to take recourse to higher order continuum transport equations. Grad pioneered the moment method by which successive approximations of the microscopic Boltzmann equation can be obtained. Grad proposed to expand the particle distribution function *f*(**c**, **x**, *t*) in terms of orthogonal Hermite polynomials, with an infinite set of Hermite coefficients being equivalent to the particle distribution function *f* itself. The number of Hermite coefficients to be considered in the expansion depends upon the degree of accuracy desired. The first few Hermite coefficients are the state variables, stress tensor, and heat flux vector. These variables are not expressed in terms of other thermodynamic variables but are on par with them, and satisfy their own differential equation. This distinct approach of Grad along with the resulting equations is expounded in this chapter. The linearized form of the Grad equations is applied to two practical problems and the obtained solutions are compared against that obtained through the Navier–Stokes equations. The Grad equations can be used to obtain the Navier–Stokes equations and Cattaneo’s equation as discussed here. The differences as well as the similarities between the Grad approach and the Chapman–Enskog approach (introduced in Chap. 5) are also brought forward.

## References

- 3.Agrawal A (2016) Higher-order continuum equation based heat conduction law. INAE Lett 1(2):35–39CrossRefGoogle Scholar
- 7.Ai DK (1960) Cylindrical Couette flow in a rarefied gas according to Grad’s equations. PhD thesis, California Institute of TechnologyGoogle Scholar
- 28.Cercignani C (2000) Rarefied gas dynamics: from basic concepts to actual calculations, vol 21. Cambridge University Press, CambridgezbMATHGoogle Scholar
- 61.Grad H (1949) Note on
*N*-dimensional Hermite polynomials. Commun Pure Appl Math 2(4):325–330MathSciNetCrossRefGoogle Scholar - 62.Grad H (1949) On the kinetic theory of rarefied gases. Commun Pure Appl Math 2(4):331–407MathSciNetCrossRefGoogle Scholar
- 63.Grad H (1952) The profile of a steady plane shock wave. Commun Pure Appl Math 5(3):257–300MathSciNetCrossRefGoogle Scholar
- 64.Grad H (1958) Principles of the kinetic theory of gases. Springer, Berlin, pp 205–294Google Scholar
- 67.Gross EP, Jackson EA (1958) Kinetic theory of the impulsive motion of an infinite plane. Phys Fluids 1(4):318–328MathSciNetCrossRefGoogle Scholar
- 83.Kogan MN (1969) Rarefied gas dynamics. Plenum Press, New YorkCrossRefGoogle Scholar
- 84.Kremer GM (2010) An introduction to the Boltzmann equation and transport processes in gases. Springer, BerlinCrossRefGoogle Scholar
- 85.Kundu PK (1990) Fluid mechanics. Academic Press, New YorkzbMATHGoogle Scholar
- 102.Muralidhar K, Biswas G (2005) Advanced engineering fluid mechanics. Narosa Publishing House, New DelhiGoogle Scholar
- 141.Struchtrup H (2005) Macroscopic transport equations for rarefied gas flows. Springer, BerlinCrossRefGoogle Scholar
- 149.Truesdell C, Muncaster RG (1980) Fundamentals of Maxwel’s kinetic theory of a simple monatomic gas: treated as a branch of rational mechanics, vol 83. Academic Press, New YorkGoogle Scholar
- 172.Yang HT (1955) Rayleigh’s problem at low Mach number according to the kinetic theory of gases. Doctoral dissertation, California Institute of TechnologyGoogle Scholar