Cercignani Lampis Boundary Condition Research Articles

Viscous and thermal velocity slip coefficients via the linearized Boltzmann equation with ab initio potential

The viscous and thermal velocity slip coefficients of various monatomic gases are computed via the linearized classical Boltzmann equation, with ab initio potential, subject to Maxwell and Cercignani–Lampis boundary conditions. Both classical and quantum interatomic interactions are considered. Comparisons with hard sphere and Lennard–Jones potentials, as well as the linearized Shakhov model are performed. The produced database is dense, covers the whole range of the accommodation coefficients and is of high accuracy. Using symbolic regression, very accurate closed form expressions of the slip coefficients, easily implemented in the future computational and experimental works, are deduced. The thermal slip coefficient depends, much more than the viscous one, on the intermolecular potential. For example, in the case of diffuse scattering, the relative differences in the viscous slip coefficient data between HS and AI potentials are less than 4%, whilst the corresponding ones in the thermal slip coefficient data are about 6% for He, reaching 15% for Xe. Quantum effects are considered for He, at temperatures 1–104 K to deduce that deviations from the classical behaviour are not important in the viscous slip coefficient, but they become important in the thermal slip coefficient, where the differences between the classical and quantum approaches reach 15% at 1 K. The computational effort of solving the linearized Boltzmann equation with ab initio and Lennard–Jones potentials is the same. Since ab initio potentials do not contain any adjustable parameters, it is recommended to use them at any temperature.

Convergence toward the steady state of a collisionless gas with Cercignani–Lampis boundary condition

We study the asymptotic behavior of the kinetic free-transport equation enclosed in a regular domain, on which no symmetry assumption is made, with Cercignani–Lampis boundary condition. We give the first proof of existence of a steady state in the case where the temperature at the wall varies, and derive the optimal rate of convergence toward it, in the L 1 norm. The strategy is an application of a deterministic version of Harris’ subgeometric theorem, in the spirit of the recent results of Cañizo-Mischler and of the previous study of Bernou. We also investigate rigorously the velocity flow of a model mixing pure diffuse and Cercignani–Lampis boundary conditions with variable temperature, for which we derive an explicit form for the steady state, providing new insights on the role of the Cercignani–Lampis boundary condition in this problem.

Numerical Comparison of the Generalized Maxwell and Cercignani–Lampis Models

An anisotropic gas–surface interaction model is examined for its possible use as a boundary condition in solving a kinetic equation. The surface aerodynamic coefficients and the temperatures obtained by applying a two-parameter anisotropic reflection model and the Cercignani–Lampis boundary condition for external flow problems with various Knudsen numbers and accommodation coefficients are numerically compared. It is shown that the geometry of the body influences the degree of the difference between the solutions for different models of boundary conditions.

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

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.

Local Well-Posedness of Vlasov–Poisson–Boltzmann Equation with Generalized Diffuse Boundary Condition

The Vlasov-Poisson-Boltzmann equation is a classical equation governing the dynamics of charged particles with the electric force being self-imposed. We consider the system in a convex domain with the Cercignani-Lampis boundary condition. We construct a uniqueness local-in-time solution based on an $L^\infty$-estimate and $W^{1,p}$-estimate. In particular, we develop a new iteration scheme along the characteristic with the Cercignani-Lampis boundary for the $L^\infty$-estimate, and an intrinsic decomposition of boundary integral for $W^{1,p}$-estimate.

Cercignani-Lampis boundary in the Boltzmann theory

The Boltzmann equation is a fundamental kinetic equation that describes the dynamics of dilute gas. In this paper we study the local well-posedness of the Boltzmann equation in bounded domain with the Cercignani-Lampis boundary condition, which describes the intermediate reflection law between diffuse reflection and specular reflection via two accommodation coefficients. We prove the local-in-time well-posedness of the equation by establishing an $L^\infty$ estimate. In particular, for the $L^\infty$ bound we develop a new decomposition on the boundary term combining with repeated interaction through the characteristic. Via this method, we construct a unique steady solution of the Boltzmann equation with constraints on the wall temperature and the accommodation coefficient.

Accurate and efficient computation of the Boltzmann equation for Couette flow: Influence of intermolecular potentials on Knudsen layer function and viscous slip coefficient

The Couette flow is one of the fundamental problems of rarefied gas dynamics, which has been investigated extensively based on the linearized Boltzmann equation (LBE) of hard-sphere molecules and simplified kinetic model equations. However, how the different intermolecular potentials affect the viscous slip coefficient and the structure of Knudsen layer remains unclear. Here, a novel synthetic iteration scheme (SIS) is developed for the LBE to find solutions of Couette flow accurately and efficiently: the velocity distribution function is first solved by the conventional iterative scheme, then it is modified such that in each iteration i) the flow velocity is guided by an ordinary differential equation that is asymptotic-preserving at the Navier–Stokes limit and ii) the shear stress is equal to the average shear stress. Based on the Bhatnagar–Gross–Krook model, the SIS is assessed to be efficient and accurate. Then we investigate the Knudsen layer function for gases interacting through the inverse power-law, shielded Coulomb, and Lennard-Jones potentials, subject to diffuse-specular and Cercignani–Lampis gas-surface boundary conditions. When the tangential momentum accommodation coefficient (TMAC) is not larger than one, the Knudsen layer function is strongly affected by the potential, where its value and width increase with the effective viscosity index of gas molecules. Moreover, the Knudsen layer function exhibits similarities among different values of TMAC when the intermolecular potential is fixed. For Cercignani–Lampis boundary condition with TMAC larger than one, both the viscous slip coefficient and Knudsen layer function are affected by the intermolecular potential, especially when the “backward” scattering limit is approached. With the asymptotic theory by Jiang and Luo (2016) [14] for the singular behavior of the velocity gradient in the vicinity of solid surfaces, we find that the whole Knudsen layer function can be well fitted by the power series ∑n=02∑m=02cn,mxn(xln⁡x)m, where x is the distance to the solid surface. Finally, the experimental data of the Knudsen layer profile are explained by the LBE solution with proper values of the viscosity index and TMAC.

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: Ewartet al.(J. Fluid Mech., vol. 584, 2007, pp. 337–356), Rojas-Cárdenaset al.(Phys. Fluids, vol. 25, 2013, 072002) and Yamaguchiet 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.

The linearized Boltzmann equation with Cercignani–Lampis boundary conditions: Heat transfer in a gas confined by two plane-parallel surfaces

The analytical discrete-ordinates method is used to solve the problem of heat transfer for a single-species gas confined by two plane-parallel surfaces. The formulation of the problem is based on the linearized Boltzmann equation for rigid-sphere interactions between gas particles and the Cercignani–Lampis kernel for gas–surface interactions. Accurate numerical results are reported for the density, temperature, and heat-flow perturbations from a reference (equilibrium) state and are compared with similar results from five kinetic models. An interesting finding of this work is that there are combinations of the four numerical values of the accommodation coefficients used to define the Cercignani–Lampis boundary conditions that give rise to heat flows that are larger for the transition regime than for the free-molecular regime, an effect not observed when the standard (Maxwell) boundary conditions are used.

Gas-surface scattering effect on vacuum gas flows through rectangular channels

Flow of a rarefied gas through a channel of rectangular cross section is investigated numerically and experimentally. Emphasis is given on the study of the molecular scattering law influence at the wall boundary surfaces. The Cercignani-Lampis boundary conditions are considered in detail for the current problem, along with the linearized BGK kinetic model and the discrete velocity method. Numerical conductance values are tabulated for certain aspect ratios and accommodation coefficients in the whole range of the Knudsen number and compared with corresponding experimental results for certain gas-surface combinations. Compared to the classical Maxwell diffuse boundary conditions, the Cercignani-Lampis gas-surface interaction allows a better agreement with the measurements.

Heat transfer through rarefied gases between coaxial cylindrical surfaces with arbitrary temperature difference

The problem of nonlinear heat transfer through a rarefied gas confined between concentric cylinders maintained at different temperatures is investigated. The formulation is based on the nonlinear Shakhov kinetic model subject to Cercignani–Lampis boundary conditions, while molecular interaction is modelled by the inverse power law. The detailed behaviour of the radial heat flow, density, temperature and pressure distributions in terms of the normalized temperature difference between the cylindrical walls, the ratio of the two cylindrical radii and the gas rarefaction is investigated and certain interesting characteristics are revealed. The study includes small, moderate and large temperature differences and various radius ratios and is extended in the whole range of the Knudsen number. It is verified that the type of molecular interaction plays an important role when the heat transfer configuration becomes strongly nonlinear, while the influence of the gas–surface scattering law has similar effects both in linear and nonlinear conditions. By comparing linear and nonlinear results corresponding to the same conditions, it is concluded that linearized analysis can capture the correct behaviour of the heat flow configuration not only for infinitesimally small but also for finite temperature differences and that its range of applicability is wider than expected.

Viscous-slip, thermal-slip, and temperature-jump coefficients based on the linearized Boltzmann equation (and five kinetic models) with the Cercignani–Lampis boundary condition

A polynomial expansion procedure and the ADO (analytical discrete-ordinates) method are used to compute the viscous-slip coefficient, the thermal-slip coefficient, and the temperature-jump coefficient from the linearized Boltzmann equation (LBE) for rigid-sphere interactions and the Cercignani–Lampis (CL) boundary condition. These same quantities are also computed from five kinetic models, with the CL condition, and compared to the LBE result. Equivalent results for the LBE and the kinetic models, all based on the usual Maxwell boundary condition, are also reported.

The linearized Boltzmann equation with Cercignani–Lampis boundary conditions: Basic flow problems in a plane channel

A polynomial expansion procedure and the ADO (analytical discrete-ordinates) method are used to solve a collection of basic flow problems based on the linearized Boltzmann equation for rigid-sphere interactions and the Cercignani–Lampis boundary conditions with a free choice of the accommodation coefficients at each boundary. In particular, three classical problems defined by flow in a plane-parallel channel (Poiseuille, thermal-creep, and Couette flow) are solved (essentially) analytically and evaluated to a very high numerical standard. Some comparisons with known kinetic models are also reported.

Surface effects in rarefied gas dynamics: an analysis based on the Cercignani–Lampis boundary condition

The flow of a rarefied gas, in a plane channel, is investigated, with special attention to the gas–surface interaction, modeled by the Cercignani–Lampis kernel that is defined in terms of normal and tangential accommodation coefficients. An analytical version of the discrete-ordinates method is used to solve, in an unified approach, the Poiseuille flow, thermal-creep flow and Couette flow problems with kinetic equations defined in terms of the BGK model. Numerical results for the velocity and heat-flow profiles and flow rates are reported, for a wide range of the accommodation coefficients and the Knudsen number.

Viscous-slip, thermal-slip, and temperature-jump coefficients as defined by the linearized Boltzmann equation and the Cercignani–Lampis boundary condition

A polynomial expansion procedure and an analytical discrete-ordinates method are used to evaluate the viscous-slip coefficient, the thermal-slip coefficient, and the temperature-jump coefficient as defined by a rigorous version of the linearized Boltzmann equation for rigid-sphere interactions and the Cercignani–Lampis boundary condition.

Model equations in rarefied gas dynamics: Viscous-slip and thermal-slip coefficients

Various model equations are used to define the viscous-slip and the thermal-slip coefficients in rarefied gas dynamics. More specifically, the BGK model, the S model, the variable collision model and the CES model are used to establish the slip coefficients basic to Kramers’ problem and the half-space problem of thermal creep. While the most general results are developed from use of the Maxwell boundary condition, results for the BGK model and the S model as defined by the Cercignani–Lampis boundary condition are also reported. An analytical discrete-ordinates method is used to establish the reported numerical results, and when available results from a numerical solution of the linearized Boltzmann equation are used as reference values. In addition to the numerical work based on model equations, the important issue of how to define meaningful ways (appropriate mean-free paths) to compare the results for the various models is discussed.