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

In the present paper, we provide an analytical expression for the first- and second-order thermal slip coefficients, σ1,T and σ2,T, by means of a variational technique that applies to the integrodifferential form of the Boltzmann equation based on the true linearized collision operator for hard-sphere molecules. The Cercignani-Lampis scattering kernel of the gas-surface interaction has been considered in order to take into account the influence of the accommodation coefficients (αt, αn) on the slip parameters. Comparing our theoretical results with recent experimental data on the mass flow rate and the slip coefficient for five noble gases (helium, neon, argon, krypton, and xenon), we found out that there is a continuous set of values for the pair (αt, αn) which leads to the same thermal slip parameters. To uniquely determine the accommodation coefficients, we took into account a further series of measurements carried out with the same experimental apparatus, where the thermal molecular pressure exponent γ has been also evaluated. Therefore, the new method proposed in the present work for extracting the accommodation coefficients relies on two steps. First of all, since γ mainly depends on αt, we fix the tangential momentum accommodation coefficient in such a way as to obtain a fair agreement between theoretical and experimental results. Then, among the multiple pairs of variational solutions for (αt, αn), giving the same thermal slip coefficients (chosen to closely approximate the measurements), we select the unique pair with the previously determined value of αt. The analysis carried out in the present work confirms that both accommodation coefficients increase by increasing the molecular weight of the considered gases, as already highlighted in the literature.

Global well-posedness of the Navier-Stokes equations with Navier-slip boundary conditions in a strip domain

<p style='text-indent:20px;'>This paper is concerned with the existence and uniqueness of the strong solution to the incompressible Navier-Stokes equations with Navier-slip boundary conditions in a two-dimensional strip domain where the slip coefficients may not have defined sign. In the meantime, we also establish a number of Gagliardo-Nirenberg inequalities in the corresponding Sobolev spaces which will be applicable to other similar situations.</p>

Twice Order Slip on the Flows of Fractionalized MHD Viscoelastic Fluid

The objective of this article is to investigate the effect of twice order slip on the MHD flow of fractionalized Maxwell fluid through a permeable medium produced by oscillatory movement of an infinite bottom plate. The governing equations are developed by fractional calculus approach. The exact analytical results for velocity field and related shear stress are calculated using Laplace transforms and presented in terms of generalized M-function satisfying all imposed initial and boundary conditions. The flow results for fractionalized Maxwell, traditional Maxwell and Newtonian fluid with and without slips, in the presence and absence of magnetic and porous effects are derived as the limiting cases. The impact of fractional parameter, slip coefficients, magnetic force and porosity parameter over the velocity field and shear stress are discussed and analyzed through graphical illustrations. The outcomes demonstrate that the speed comparing to streams with slip condition is lower than that for stream with non-slip conditions, and the speed with second-slip condition is lower than that with first-order slip condition.

Rarefied transitional flow through diverging nano and microchannels: A TRT lattice Boltzmann study

Rarefied isothermal gaseous flow through long diverging micro and nanochannels is investigated in this paper using the two-relaxation-time (TRT) lattice Boltzmann method (LBM). The simulations are performed over a wide range of Knudsen number, pressure ratio, and divergence angle. The Bounce-Back Specular Reflection (BSR) slip boundary condition is applied and is connected to the second-order slip boundary condition coefficients by means of the antisymmetric relaxation time and the bounce-back portion parameter. The effects of the slip coefficients on the wall and centerline Mach numbers, as well as the mass flow rates, are investigated. The numerical results are validated with those of the direct simulation Monte Carlo (DSMC) reported in the literature. The results show that the local pressure distributions are almost independent of the slip coefficients with excellent agreements with DSMC over a wide range of the divergence angle. Our results demonstrate that there is a specific divergence angle at each pressure ratio where the local unbounded Knudsen and, as a result, Mach numbers remain constant along the channel. This observation is almost independent of the slip coefficients, and the underlying reason is that the pressure drop is compensated by an increase in the channel area.

Assessment and development of the gas kinetic boundary condition for the Boltzmann equation

Gas–surface interactions play important roles in internal rarefied gas flows, especially in micro-electro-mechanical systems with large surface area to volume ratios. Although great progress has been made to solve the Boltzmann equation, the gas kinetic boundary condition (BC) has not been well studied. Here we assess the accuracy of the Maxwell, Epstein and Cercignani–Lampis BCs, by comparing numerical results of the Boltzmann equation for the Lennard–Jones potential to experimental data on Poiseuille and thermal transpiration flows. The four experiments considered are: Ewart et al. (J. Fluid Mech., vol. 584, 2007, pp. 337–356), Rojas-Cárdenas et al. (Phys. Fluids, vol. 25, 2013, 072002) and Yamaguchi et al. (J. Fluid Mech., vol. 744, 2014, pp. 169–182; vol. 795, 2016, pp. 690–707), where the mass flow rates in Poiseuille and thermal transpiration flows are measured. This requires that the BC has the ability to tune the effective viscous and thermal slip coefficients to match the experimental data. Among the three BCs, the Epstein BC has more flexibility to adjust the two slip coefficients, and hence for most of the time it gives good agreement with the experimental measurements. However, like the Maxwell BC, the viscous slip coefficient in the Epstein BC cannot be smaller than unity but the Cercignani–Lampis BC can. Therefore, we propose to combine the Epstein and Cercignani–Lampis BCs to describe gas–surface interaction. Although the new BC contains six free parameters, our approximate analytical expressions for the viscous and thermal slip coefficients provide useful guidance to choose these parameters.