Variational derivation of thermal slip coefficients on the basis of the Boltzmann equation for hard-sphere molecules and Cercignani–Lampis boundary conditions: Comparison with experimental results

Abstract

In the present paper, a variational method is applied to solve the Boltzmann equation based on the true linearized collision operator for hard-sphere molecules and the Cercignani–Lampis boundary conditions. This technique allows us to obtain an explicit relation between the first- and second-order thermal slip coefficients and the tangential momentum and normal energy accommodation coefficients, defined in the frame of the Cercignani–Lampis scattering kernel. Comparing the theoretical results with the experimental data from the work of Yamaguchi et al. [“Mass flow rate measurement of thermal creep flow from transitional to slip flow regime,” J. Fluid Mech. 795, 690 (2016)], a pair of accommodation coefficients has been extracted for each noble gas considered in the experiments. Then, these values have been used to compute, by means of our variational technique, the temperature-driven mass flow rates, and the outputs have been compared with the measurements for helium, neon, and argon. Good agreement has been obtained between the theoretical and the experimental data, within the range of validity of the proposed second-order slip model. For all the gases analyzed, the tangential accommodation coefficient is found to be much larger than the normal energy coefficient. The general trend, according to which, by increasing the molecular weight of the different gases, the values of both accommodation coefficients also increase, is confirmed in this study.

Extraction of Tangential Momentum and Normal Energy Accommodation Coefficients by Comparing Variational Solutions of the Boltzmann Equation with Experiments on Thermal Creep Gas Flow in Microchannels

“…The main objective of this work is to extend the previous findings reported in [31] and to propose a methodology to uniquely determine α t and α n .…”

“…Both walls are fixed at x = ±d/2. If the pressure and temperature gradients are taken to be small, the Boltzmann equation can be linearized [31]:…”

“…The nondimensional coefficients appearing in Equations ( 25)-( 27) have been explicitly reported in [31]. The derivatives of J (1) ( h)/(kθ) 2 , J (2) ( h)/(τθ) 2 , and J (3) ( h)/[(kθ)(τθ)], with respect to A p , A T , B, C p and C T vanish in correspondence of the optimal values of these constants [31].…”

“…The reliability of a "second-order" solution for the pressure-and temperature-driven mass flow rate (Equations ( 33) and ( 34)) has been well assessed in Refs. [31,40,41].…”

“…Recently, in Ref. [31], the thermal-creep problem of a rarefied gas between two parallel plates has been investigated by using a variational approach that applies to the integrodifferential form of the linearized Boltzmann equation for hard-sphere molecules. In the frame of the Cercignani-Lampis boundary conditions, new analytical expressions for the firstand second-order thermal slip coefficients have been derived in terms of the tangential momentum α t and the normal energy α n accommodation coefficients.…”