An Examination of the Validity of the Onsager Reciprocal Relations Applied to Flow in the Presence of Thermal Stresses and Thermal Slip

It is experimentally known that a pressure di¤erence is developed in a gas under continuum conditions contained in a closed capillary tube, the ends of which are maintained at di¤erent temperatures. However, in contradiction with this experimentally measurable thermomolecular pressure di¤erence, the use of the methods of irreversible thermodynamics in conjunction with the Navier-Stokes equations subject to the no-slip boundary condition on velocity leads to the conclusion that there can be no thermomolecular pressure di¤erence in a capillary of macroscopic radius. It has been proposed that in order to correctly describe slow, nonisothermal ‡ows, certain Burnett stress terms, namely the thermal stresses, along with the thermal slip of velocity at the wall, reduce to the same order as the Navier-Stokes equations and must be accounted for at the Navier-Stokes level. In this work, the validity of the Onsager reciprocal relations applied to a gas undergoing nonisothermal ‡ow is examined, and it is demonstrated that the Onsager relations are not obeyed when thermal stress and thermal slip e¤ects are accounted for. Introduction It has long been known that there exists a class of problems wherein the Navier-Stokes equations used in conjunction with the no-slip condition on the velocity of the gas at the solid surface yield predictions inconsistent with experimental observations. An example of such a problem is the development of a pressure gradient in a capillary tube containing a gas, with one end of the tube maintained at a higher temperature than the other. This phenomenon was …rst observed in 1879 by Reynolds [1], who termed it thermal transpiration. The …rst theoretical explanation for the development of a thermomolecular pressure gradient in the thermal transpiration tube was provided in 1879 by Maxwell [2]. Maxwell made use of the kinetic theory of gases to derive expressions for the thermal stresses arising in a gas of Maxwellian molecules (i.e., molecules that are point centers of repulsion, with the repulsive force between two molecules being inversely proportional to the …fth power of the distance between them), in the presence of temperature gradients, and an expression for the thermally-induced slip of the gas at its interface with the solid. However, Maxwell, arguing on the basis that the temperature must satisfy the Laplace equation, and hence that the thermal stresses derived by him can cause no motion of the gas, applied the thermal slip condition in conjunction with the incompressible Navier-Stokes equations to derive an expression for the thermomolecular pressure gradient. A rigorous derivation of the boundary conditions applicable at the surface bounding a gas through the asymptotic solution of the Boltzmann equation in the Knudsen layer adjoining the surface is provided in [3]. Further, it is argued by Kogan et al. [4, 5] that the prevalent asymptotic expansions of the Boltzmann equation in terms of the Knudsen number Kn by the methods of Chapman and Enskog [6] and Grad [7], which yield the Navier-Stokes equations at O(Kn), are inapplicable to slow, nonisothermal ‡ows. The Knudsen number, which is the ratio of the mean free path of the gas molecules to the characteristic size of the macroscopic body in contact with the gas, may also be expressed as the ratio of the Mach number to the Reynolds number, i.e., Kn = M=Re : It is now evident that ‡ows of small Knudsen numbers can occur in two situations, when the Mach number is of O(1) and the Reynolds number is large, or when the Mach number is small and the Reynolds number is of O(1). In the latter case, a reexamination of the scaling that produces the Navier-Stokes equations at O(Kn) is required. Speci…cally, the Chapman-Enskog and Grad schemes are based on the non-dimensionalization of the gas velocity using the speed of sound as the characteristic velocity of the gas [3]. As such, the resulting equations apply to the ‡ow of gases at large Reynolds number, whereas for slow ‡ows, where the Reynolds number Re 1; the characteristic gas velocity is in fact the viscous velocity, given by U = =a, where is the kinematic viscosity of the gas and a is the characteristic dimension of the solid in contact with the gas. Upon rescaling the terms resulting from the Chapman-Enskog expansion at various orders, it is found that the thermal stresses, which were previously believed to be at the next level of approximation, reduce to the same order as the terms in the Navier-Stokes equations and hence appear alongside the Navier-Stokes viscous stress tensor. These terms correspond to the third and …fth of the terms derived by Burnett in the third approximation to the velocity distribution function of the Boltzmann equation [6]. The simultaneous ‡ow of gas and heat, particularly with regard to ‡ow in porous media, is typically modelled by assuming that the global ‡uxes of gas volume and heat, averaged over the cross-section of ‡ow, are each linearly related to the gradients in pressure and temperature [8, 9, 10]. The coe¢ cients appearing in these linear relations between the ‡uxes and the driving forces are assumed to follow the Onsager reciprocal relations, and hence the cross-coe¢ cients are assumed to be equal. The Onsager reciprocal relations at the global scale, averaged over the crosssection of ‡ow, have been derived using the Poiseuille ‡ow pro…les and the Onsager symmetry of the Navier-Stokes equations governing the ‡ow pointwise in each individual pore [11]. Thus, the applicability of the Onsager reciprocal relations to the ‡ow rates of gas and heat computed per unit area of cross-section (henceforth referred to as the global Onsager reciprocal relations in this context) presupposes that the gas velocity obeys the no-slip boundary condition, and that the governing hydrodynamic equations possess Onsager symmetry. However, in the present work, we demonstrate that in the presence of thermal stresses and with the imposition of the thermal slip condition on the gas velocity under nonisothermal ‡ow conditions, neither of these assumptions holds. The equations of nonisothermal gas ‡ow The Burnett thermal stresses found by Kogan et al. [4, 5] to reduce to the order of the Navier-Stokes viscous stress tensor are given by