Thermal slip of a nonuniformly heated gas along a solid planar surface

Two methods are used to study the solution of a linearized model of the Boltzmann equation in the problem of thermal slip of a nonuniformly heated gas along a solid flat wall.The first method involves analytic solution of the integral equation for the average gas velocity. In the case of purely diffuse or purely specular reflection of the molecules from the wall surface the first method makes it possible to obtain analytically two important results; namely, the average gas velocity at the surface and at a large distance from the wall. The average gas velocity profile cannot be constructed analytically with this method. The second approximate method involves expanding the distribution function into a series in Sonine polynomials in velocity space and formulation of half-space moment equations from which the correction to the distribution function is determined. This method is used to obtain a simple analytic expression for the distribution function, from which we can find the average velocity profile for the gas for any arbitrary tangential momentum accommodation coefficient. In particular cases in which analytic solution of the problem by the first method is possible, good agreement is obtained between the two computational methods.It is known that a gas in a temperature gradient field tangent to the wall must begin to move in the direction of the temperature gradient (thermal slip). The first attempt to solve the thermal slip problem was made by Maxwell [1]. In his analysis Maxwell assumed that the distribution function of the molecules incident on the wall near the surface does not differ from the bulk distribution at a large distance from the wall. As a result Maxwell obtained the following expression for the thermal slip velocity for any tangential momentum accommodation coefficientu^*=3/4 grad lnT.Here is the kinematic viscosity.However, in the case of molecular reflection from the wall which is not purely specular, the distribution of the incident molecules in the Knudsen layer differs from the bulk distribution because of collisions with the molecules reflected from the wall. Thus, Maxwell's assumption is not valid in the general case.For the exact solution of the problem it is necessary to find the distribution function in the Knudsen layer by solving the Boltzmann equation. Several investigators have used the Grad method [2] to find the distribution function in the Knudsen layer. However, the use of Grad's method in the thermal-slip problem leads to Maxwell's result [3].The solution of the thermal-slip problem obtained by Sone [4] is more exact than the analyses noted above. A comparison of the results obtained by Sone with those of this investigation is given at the end of our paper.