Thermal Creep Flow Research Articles

Sensitivity analysis of the Cercignani - Lampis accommodation coefficients in prototype rarefied gas flow and heat transfer problems via the Monte Carlo method

In rarefied gas dynamics, the Cercignani-Lampis (CL) scattering kernel, containing two accommodation coefficients (ACs), namely the tangential momentum and normal energy ones, is widely employed to characterize gas-surface interaction, particularly in non-isothermal setups, where both momentum and energy may simultaneously be exchanged. Here, a formal and detailed sensitivity analysis of the effect of the CL ACs on the main output quantities of several prototype problems, namely the cylindrical Poiseuille, thermal creep and thermomolecular pressure difference (TPD) flows, as well as the plane Couette flow and heat transfer (Fourier flow), is performed. In each problem, some uncertainties are randomly introduced in the ACs (input parameters) and via a Monte Carlo propagation analysis, the deduced uncertainty of the corresponding main output quantity is computed. The output uncertainties are compared to each other to determine the flow configuration and the gas rarefaction range, where a high sensitivity of the output quantities with respect to the CL ACs is observed. The flow setups and rarefaction regimes with high sensitivities are the most suitable ones for the estimations of the ACs, since larger modeling and experimental errors may be acceptable. In the Poiseuille and Couette flows, the uncertainties of the flow rate and shear stress respectively are several times larger than the input uncertainty in the tangential momentum AC and much smaller than the uncertainty in the normal energy AC in a wide range of gas rarefaction. In the thermal creep flow, the uncertainty of the flow rate depends on the input ones of both ACs, but, in general, it remains smaller than the input uncertainties. A similar behavior with the thermal creep flow is obtained in the TPD flow. On the contrary, in the Fourier flow, the uncertainty of the heat flux may be about the same or even larger than the input ones of both ACs in a wide range of gas rarefaction. It is deduced that in order to characterize the gas-surface interaction via the CL ACs by matching computations with measurements, it is more suitable to combine the Poiseuille (or Couette) and Fourier configurations, rather than, as it is commonly done, the Poiseuille and thermal creep ones. For example, in order to estimate the normal energy AC within an accuracy of 10 %, experimental uncertainties should be less than 4 % in the thermal creep or TPD flows, while may be about 10 % in the Fourier flow.

A theoretical framework of information preservation method and its application to low-speed nonequilibrium gas flows

Simulations of nonequilibrium gas flows have garnered significant interest in modern engineering problems involving rarefied gas flow characteristics. Despite the popularity of the direct simulation Monte Carlo (DSMC) method in simulating such flows, its use in low-speed flows is limited by statistical noises. The information preservation (IP) method is a promising alternative known for its low noise properties. In this study, a new theoretical framework for the IP method based on kinetic theory is introduced to offer complete understanding for the transport properties of the preserved information. Specifically, we introduce a velocity-information joint distribution function (VIJDF) and derive its governing equation as well as the corresponding macroscopic transport equations. To ensure the accuracy of the IP method, the total stress/heat flux in IP, including information stress/heat flux generated during movement and collision steps and compensation stress/heat flux imposed in the compensation step, is matched to the molecular stress/heat flux in DSMC. To this end, a nonequilibrium model for the VIJDF is proposed to evaluate the compensation stress/heat flux. The parameters in the collision model of IP are theoretically determined by equating the transport coefficients associated with the preserved information to the coefficients of viscosity and thermal conductivity in DSMC. Numerical simulations for a variety of nonequilibrium gas flows, including low-speed Couette flow, Fourier flow, high-speed Couette flow, external force-driven Poiseuille flow, lid-driven cavity flow, and thermal creep flow, demonstrate that the IP method can achieve similar accuracy as the DSMC method with a much smaller sampling size.

Some properties of a gas flow submitted to a temperature gradient

The flow of gases through channels generated by a temperature gradient along the channel surfaces is analyzed. This flow is characterized by the Thermomolecular Pressure Difference (TPD) at steady state and by the relaxation time, a characteristic of the transient stage, which indicates the rate at which the flow reaches the steady state. From a simple analytical description an expression to estimate the TPD is proposed. Experimental data obtained in channels with rectangular and circular cross-sections are analyzed, showing good agreement with the estimated TPD. Two normalizations of TPD are suggested that eliminate the gas and cross-section dependences, allowing application of this normalization to a channel with an unknown cross-section. Then, a quasi-analytical expression of the relaxation time is proposed. This expression makes it possible to explicitly calculate the relaxation time for the thermal creep flow in a channel of the rectangular and circular cross-sections using the numerical results obtained from the kinetic theory of gases. The proposed expressions are compared to the experimental data and the good agreement is found. In the case of channels of unknown cross-section, the relaxation time can be related to the channel conductance and the relaxation time can also be calculated using the proposed expression.

Gas-surface interaction in rarefied gas flows through long capillaries via the linearized Boltzmann equation with various boundary conditions

Reliable gas–surface interaction modeling is of major importance in consistent simulation of rarefied gas flows with applications in vacuum technology and micro-electromechanical systems. The effectiveness of the most prominent kinetic boundary conditions, namely the ones by Maxwell, Cercignani-Lampis and Epstein, is elaborated by numerically solving the classical plane Poiseuille and thermal creep flows via the linearized Boltzmann equation, subject to these boundary conditions. The flow rates are provided in a wide range of the gas rarefaction for many values of the accommodation coefficients of each scattering kernel, identifying in parallel, the sensitivity of the flow rates in small variations of the free parameters. A comparison with available experimental data is performed to deduce that the Cercignani-Lampis and Epstein scattering kernels, compared to the Maxwell one, are more suitable to characterize coupled momentum and energy accommodation modes. With these boundary conditions, the accommodation coefficients may be extracted by fitting computations with measurements, in the Poiseuille flow in a wide range of gas rarefaction, but in the thermal creep flow, experimental data under rarefied conditions, as high as possible, are recommended. The present work may be useful in determining the measurement conditions and uncertainties, allowing accurate extraction of the accommodation coefficients.

Development of explicit formulations of G45-based gas kinetic scheme for simulation of continuum and rarefied flows.

In this work, the explicit formulations of the Grad's distribution function for the 45 moments (G45)-based gas kinetic scheme (GKS) are presented. Similar to the G13 function-based gas kinetic scheme (G13-GKS), G45-GKS simulates flows from the continuum regime to the rarefied regime by solving the macroscopic governing equations based on the conservation laws, which are widely used in conventional Navier-Stokes solver. These macroscopic governing equations are discretized by the finite volume method, where the numerical fluxes are evaluated by the local solution to the Boltzmann equation. The initial distribution function is reconstructed by the G45 distribution function, which is a higher order truncation of the Hermite expansion of distribution function compared with the G13 distribution function. Such high order truncation of Hermite expansion helps the present solver to achieve a better accuracy than G13-GKS. Moreover, the reconstruction of distribution function makes the development of explicit formulations of numerical fluxes feasible, and the evolution of the distribution function, which is the main reason why the discrete velocity method is expensive, is avoided. Several numerical experiments are performed to examine the accuracy of G45-GKS. Results show that the accuracy of the present solver for almost all flow problems is much better than G13-GKS. Moreover, some typical rarefied effects, such as the direction of heat flux without temperature gradients and thermal creep flow, can be well captured by the present solver.

Physics-informed neural networks for rarefied-gas dynamics: Thermal creep flow in the Bhatnagar–Gross–Krook approximation

This work aims at accurately solve a thermal creep flow in a plane channel problem, as a class of rarefied-gas dynamics problems, using Physics-Informed Neural Networks (PINNs). We develop a particular PINN framework where the solution of the problem is represented by the Constrained Expressions (CE) prescribed by the recently introduced Theory of Functional Connections (TFC). CEs are represented by a sum of a free-function and a functional (e.g., function of functions) that analytically satisfies the problem constraints regardless to the choice of the free-function. The latter is represented by a shallow Neural Network (NN). Here, the resulting PINN-TFC approach is employed to solve the Boltzmann equation in the Bhatnagar–Gross–Krook approximation modeling the Thermal Creep Flow in a plane channel. We test three different types of shallow NNs, i.e., standard shallow NN, Chebyshev NN (ChNN), and Legendre NN (LeNN). For all the three cases the unknown solutions are computed via the extreme learning machine algorithm. We show that with all these networks we can achieve accurate solutions with a fast training time. In particular, with ChNN and LeNN we are able to match all the available benchmarks.

A model of micro lubrication between two walls with unequal temperature distribution based on kinetic theory

Micro lubrication of a gas between two walls with arbitrary and independent temperature distributions is studied on the basis of the Bhatnagar–Gross–Krook–Welander (BGKW) model of the Boltzmann equation. The BGKW equation is studied analytically using the slowly varying approximation. Following the author's previous study [T. Doi, “A model of micro lubrication between two walls with an arbitrary temperature difference based on kinetic theory,” Phys. Fluids 32, 052005 (2020)], the leading-order approximation, which ought to be the solution of the nonlinear heat transfer problem, is replaced by its free molecular solution. A lubrication model of the Reynolds-type equation is derived in closed form. A direct numerical analysis of the lubrication flow subject to localized heating or cooling of the walls is conducted for an assessment of the lubrication model. The lubrication lift calculated using the model agrees with that of the direct numerical solution within an error of 5% when the Knudsen number based on the gap size lies between 0.1 and 10. The result of the lubrication model agrees also with that of the Boltzmann equation for a variable hard sphere gas. A sharp peak arises in the pressure distribution for large Knudsen numbers owing to the effect of thermal creep flows induced by localized heating.

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.

A study on Knudsen pump driven by thermal creep flow around Peltier devices

A study on Knudsen pump driven by thermal creep flow around Peltier devices

Numerical investigation of thermally generated gas flow between saw-tooth like surfaces

The two-dimensional thermally generated gas flow between two saw-tooth like surfaces is numerically investigated using the kinetic approach based on the S-model of the Boltzmann kinetic equation. An implicit scheme for the solution of the S-model kinetic equation is applied and the algorithm is optimized for the use of massive parallelization in both physical and velocity spaces. Both surfaces are assumed to be isothermal and kept at different temperatures. Top peaks are shifted with respected to bottom peaks in the x-direction. The perpendicular to surfaces temperature gradient and the asymmetric distribution of saw tooth along surfaces may lead to an occurrence of temperature gradients tangential to surfaces causing a thermal creep flow in the x-direction. The rarefaction effect on the thermal transportation flow is estimated by varying Knudsen number from 0.02 to 40, i.e. covering slip, transition and free-molecular regimes. The influence of the surface geometry, such as the distribution of top and bottom peaks, the inclination angle of saw-tooth structure, on flow is analyzed. It is found that rarefaction, as well as geometry configuration, have a crucial effect on the occurrence and strength of gas flow.

Uncertainty analysis of computed flow rates and pressure differences in rarefied pressure and temperature driven gas flows through long capillaries

Abstract The computation of the mass flow rate in the Poiseuille and thermal creep flows and of the pressure difference in the thermomolecular pressure difference (TPD) flow through long capillaries is well-known and available in the literature. Here, the uncertainty in the solution due to induced uncertainties in the input data, namely, the capillary radius and length, the pressure and/or temperature imposed at the capillary ends and the accommodation coefficient of the Maxwell diffuse-specular boundary conditions, is investigated. The uncertainty analysis is performed by the Monte Carlo Method. Conducting the required number of trials the distribution function of the output quantity and its associated uncertainty are obtained. In most cases, the uncertainty in the input quantity driving the flow has the largest effect on the output uncertainties. This is always true in the TPD flow and for small pressure and temperature differences in the Poiseuille and thermal creep flows respectively. In the case of large driving thermodynamic forces, in the latter two flows, the radius becomes the most important source of uncertainty. The accommodation coefficient uncertainty is always the less important one. Documenting the expected effect of the uncertainty in each input parameter is certainly beneficial in computational as well as experimental work.

Polyatomic thermal creep flows through long microchannels at large temperature ratios

Rarefied polyatomic gas flows through long microchannels of circular cross section due to small temperature and pressure gradients have been studied on the basis of the Rykov model in a wide range of the gas rarefaction and for various values of the reference flow temperature. Results are presented for N2, CO2, CH4, and SF6 representing linear and nonlinear polyatomic gas molecules. The present numerical results for N2 and CO2 are in good agreement with the corresponding results of previous studies. In addition, a simple method of calculation of the thermal creep under large temperature differences, which has been proposed in previous studies for monatomic gases, is extended in the present work in the case of polyatomic gases. The results based on the polyatomic modeling differ significantly from the corresponding monatomic ones and the differences depend on the gas rarefaction, the working gas, and the flow temperature. Special attention is also given to the computation of the thermomolecular pressure effect in the case of polyatomic gases under large temperature ratios. Furthermore, the dependence of both the thermal creep and the thermomolecular pressure effect on the viscosity variation with temperature along the microchannel is pointed out. Finally, the numerical data are provided as supplementary material for modeling any polyatomic gas flow in a wide range of the gas rarefaction and flow temperatures.

Thermal-driven flow inside graphene channels for water desalination

A novel concept of membrane process in a thermal-driven system is proposed for water desalination. By means of molecular dynamics simulations, we show fast water transport through graphene galleries at a temperature gradient. Water molecules are driven to migrate through nanometer-wide graphene channels from cold reservoir to hot reservoir by the effect of thermal creep flow. Reducing the interlayer spacing to 6.5 Å, an abrupt escalation occurs in water permeation between angstrom-distance graphene slabs. A change from disordered bulklike water to quasi-square structure has been found under this extremely confined condition. This leads to a transition to subcontinuum transport. Water molecules perform collective diffusion behaviors inside graphene channels. The special transport processes with structure change convert thermal energy into motion without dissipation, resulting in unexpectedly high water permeability. The thermal-driven system reaches maximum flowrate at temperature variance of 80 K, corresponding to the quantity at pressure difference up to 105 bar in commercial reverse osmosis processes and 230 bar in pressure-driven slip flow. Our results also reveal the movement of saline ions influenced by thermophoretic effect, which complements the geometry limitation at greater layer spacing, enhancing the blockage of ions. This finding aims to provide an innovational idea of developing a high-efficiency desalination technology able to utilize various forms of energy.

Flows of a Rarefied Gas between Coaxial Circular Cylinders with Nonuniform Surface Properties

Flows of a rarefied gas between coaxial circular cylinders with nonuniform surface properties are studied on the basis of kinetic theory. It is assumed that the outer cylinder is a diffuse reflection boundary and the inner cylinder is a Maxwell-type boundary whose accommodation coefficient varies in the circumferential direction. Three fundamental flows are studied: 1) a flow caused by the rotation of the outer cylinder (Couette flow), 2) a flow induced between the cylinders at rest kept at different temperatures (heat transfer problem), and 3) a flow induced by the circumferential temperature distribution along the cylindrical surfaces (thermal creep flow). The linearized ES-BGK model of the Boltzmann equation is numerically analyzed using a finite difference method. The time-independent behavior of the gas is studied over a wide range of the gas rarefaction degree, the radii ratio, and a parameter characterizing the distribution of the accommodation coefficient. Due to an effect of nonuniform surface properties, a local heat transfer occurs between the gas and the cylindrical surfaces in Couette flow; a local tangential stress arises in the heat transfer problem. However, the total heat transfer between the two cylinders in Couette flow and the total torque acting on the inner cylinder in the heat transfer problem vanish irrespective of the flow parameters. Two nondegenerate reciprocity relations arise due to the effect of nonuniform surface properties. The reciprocity relations among the above-mentioned three flows are numerically confirmed over a wide range of the flow parameters. The force on the inner cylinder, which also arises due to the effect of nonuniform surface properties in Couette flow and the heat transfer problems, is studied.

Making Monte Carlo Simulation of Thermal Creep Flow around Discs with Slits

Making Monte Carlo Simulation of Thermal Creep Flow around Discs with Slits

Application of discrete unified gas kinetic scheme to thermally induced nonequilibrium flows

In this article, we present the applications of the discrete unified gas kinetic scheme (DUGKS) for simulating thermal induced non-equilibrium flows. Four different types of thermally induced flows, including the thermal creep flow, thermal edge flow, radiometric flow and temperature discontinuity induced flow have been simulated in a wide range of Knudsen numbers. The numerical results have been compared with direct simulation Monte Carlo (DSMC) solutions and show that the Shakhov model based DUGKS can be faithfully used for such thermally induced nonequilibrium flows. In particular, due to the asymptotic preserving property of the DUGKS, the flow features in the near continuum flows can be captured efficiently. The extremely low-speed character of such flows is also in favor of the current deterministic model equation solver.

Mathematical modelling of the mass transfer process between two coaxial cylinders in the problem of thermal creep

An analytic solution to the Williams equation in the thermal creep flow problem in the channel, formed by two coaxial cylinders, has been constructed using the kinetic approach. For the boundary condition on the channel walls, the diffusion model is used. Taking into account the constructed distribution function for various values of the ratio of radii of cylinders and Knudsen number, the value of the mass flow rate in channel is calculated. The analysis of the obtained results during the transition to free molecular and hydrodynamic regimes is done.

Mass flow rate measurement of thermal creep flow from transitional to slip flow regime

Measurements of the thermal creep flow through a single rectangular microchannel connected to two tanks maintained initially at the same pressure, but at different temperatures, are carried out for five noble gas species, over a large range of pressure and for two temperature differences between the tanks. The time-dependent pressure variations in both cold and hot tanks are investigated, and the temperature-driven (thermal creep) mass flow rate between two tanks is calculated from these data for the rarefaction parameter ranging from the transitional to slip flow regime. The measured mass flow rate is compared with the numerical solution of the S-model kinetic equation, and they show good agreement. A novel approximate expression to calculate the temperature-driven mass flow rate in the transitional and slip flow regimes is proposed. This expression provides results in good agreement with the measured values of the mass flow rate. In the slip flow regime, the thermal slip coefficient is calculated by employing the previously reported methodology, and the influence of the nature of the gas on this coefficient is investigated. The measured values of the thermal slip coefficient agree well with the values available in the literature, indicating that this coefficient is independent of the shape of a channel.

ラチェット構造によって誘起される熱ほふく流に関する研究

ラチェット構造によって誘起される熱ほふく流に関する研究

Thermal creep flow for the Boltzmann equation

It is known that the Boltzmann equation has close relation to the classical systems in fluid dynamics. However, it provides more information on the microscopic level so that some phenomena, like the thermal creep flow, can not be modeled by the classical systems of fluid dynamics, such as the Euler equations. The author gives an example to show this phenomenon rigorously in a special setting. This paper is completely based on the author’s recent work, jointly with Wang and Yang.