Thermal Creep of a Rarefied Gas on the Basis of Non-linear Korteweg-Theory

Abstract

The study of thermal transpiration, more commonly called thermal creep, is accomplished by use of Korteweg’s theory of capillarity. Incorporation of this theory into the balance laws of continuum mechanics allows resolution of boundary value problems via solutions to systems of ordinary differential equations. The problem was originally considered by Maxwell in his classic 1879 paper Maxwell (Phil Trans Roy Soc (London) 170:231–256, 1879). In that paper Maxwell derived what is now called the Burnett higher order contribution to the Cauchy stress, but was not able to solve his newly derived system of partial differential equations. In this paper the authors note that a more appropriate higher order contribution to the Cauchy stress follows from Korteweg’s 1901 theory Korteweg (Arch Neerl Sci Exactes Nat Ser II 6:1–24, 1901). The appropriateness of Korteweg’s theory is based on the exact summation of the Chapman–Enskog expansion given by Gorban and Karlin. The resulting balance laws are solved exactly, qualitatively, and numerically and the results are qualitatively similar to the numerical and exact results given by Aoki et al., Loyalka et al., and Struchtrup et al.