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.

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

The study of the mass and heat transfer phenomena in micro-nano devices and systems (MEMS/NEMS) very often involves polyatomic gas flows in long capillaries. In the case of polyatomic gases, the internal degrees of freedom of gas molecules may have a significant effect on the macroscopic quantities of practical interest. The scope of the present study is two-fold. First, an accurate methodology for calculating the mass and heat flow rate coefficients is proposed based on a simple kinetic model proposed by Holway. The proposed methodology is based on the available experimental data in the literature and allows overcoming the well-known limitation of the Holway model to describe all the transport coefficients simultaneously. Second, the proposed methodology is applied to investigate the rarefied polyatomic gas flows through long micro- and nanotubes under temperature and pressure differences taking into account the translational, rotational and vibrational degrees of freedom of the gas molecules on the basis of the Holway kinetic model. Results are presented for N2, CO2, CH4 and SF6 representing linear and non-linear, but always non-polar molecules in a wide range of the gas temperature. The mass flow rates of the polyatomic gases in the temperature-driven flow differ significantly from the corresponding monatomic ones, with the difference reaching a maximum of about 25% for the examined gases and temperatures varied between 300 and 1000 K. The maximum relative deviation between monatomic and polyatomic heat flow rates can reach 39%, 64%, 73% and 87% for N2, CO2, CH4 and SF6, respectively. The results are compared against the corresponding published ones based on the Rykov model highlighting the very good agreement between the two models and the accuracy of the proposed methodology for the calculation of the mass and heat flow coefficients is confirmed.

Effect of the internal degrees of freedom of the gas molecules on the heat and mass transfer in long circular capillaries

In this paper we present the results of an experimental analysis concerning non-isothermal gas flow through a porous medium made of a packed bed of spheres. This problem is prototypal for more complex geometries which can be found in Micro Electro-Mechanics Systems (MEMS) where the heat transfer is a crucial issue. The characteristics length can be so tiny that gas flow regime could fall in almost rarefied regime where the continuum procedures are not valid. The experimental investigation covers a regime which spans from the continuum to the rarefied one and the Knudsen number varies from 0.0054 to 2.9. In particular here are reported results concerning gas flows through two concentric cylinders with surfaces at given temperatures. The mass flow rate and the radial and longitudinal distributions of the temperature are presented for different values of the Knudsen number. Finally some comparisons with numerical predictions are provided and discussed thoroughly.

Rarefied gas flow through porous media: Experiments and numerical simulation

Abstract The present study aims at estimating the heat and the mass transfer coefficients in the case of the polyatomic gas flows through long rectangular microchannels driven by small and large pressure (Poiseuille flow) and temperature (Thermal creep flow) drops. The heat and mass transfer coefficients are presented for all gas flow regimes, from free molecular up to hydrodynamic ones, and for channels with different aspect ratios as well as for various values of translational and rotational Eucken factors. The applied values of the Eucken factors were extracted based on the Rayleigh-Brillouin experiments and the kinetic theory of gases. The numerical study has been performed on the basis of a kinetic model for linear and non-linear gas molecules considering the translational and rotational degrees of freedom. The solution of the obtained system of the kinetic equations is implemented on the Graphics Processing Units (GPUs), allowing the reduction of the computational time by two orders of magnitude. The results show that the Poiseuille mass transfer coefficient is not affected by the internal degrees of freedom and the non-dependence of the previous observed deviations with the experimental data on the molecular nature of the gas molecules is confirmed. However, the study shows that the deviation between monatomic and polyatomic values of the mass transfer coefficient in the thermal creep flow is increased as the gas rarefaction is decreased, and for several polyatomic gases met in practical applications in the temperature range from 300 to 900 K might reach 15%. In addition, the effect of the internal degrees of freedom on the heat transfer coefficient is found to be rather significant. The polyatomic heat transfer coefficients are obtained essentially higher than the monatomic ones, with the maximum difference reaching about 44% and 67% for linear and non-linear gas molecules. In view of the large differences between monatomic and polyatomic gases, the present results may be useful in the design of technological devices in which the thermal creep phenomenon plays a dominant role. 

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Kinetic modeling of polyatomic heat and mass transfer in rectangular microchannels

Light-induced drift of a one-component gas in a flat channel

The authors base the description of heat and mass transfer processes in a polyatomic gas in a flat channel on a model equation, which takes into account the possibility of the excitation of two types of internal energy of the molecules: rotational and vibrational. Numerical estimations are made of the contribution of the internal degrees of freedom of the molecules to the fluxes. The results of these studies can be used to justify and design experimental methods for determining the characteristic relaxation times in polyatomic gases.

Heat and mass transfer processes of a polyatomic gas in a flat channel for arbitrary knudsen numbers

Processes of heat and mass transfer of a multiatomic gas in a cylindrical capillary due to the longitudinal pressure and temperature gradients are considered, with arbitrary Knudsen numbers. The problem is solved on the basis of a linearized kinetic equation with a model collisional term taking account of the excitation of rotational and vibrational degrees of freedom of the molecules. The contributions of internal degrees of freedom of the molecules to the thermal fluxes are analyzed. The dependence of these fluxes on the character of the interaction between the molecules and the channel surface is considered.

I. V. Chermyaninov

Papers

Heat and mass transfer processes of a polyatomic gas in a flat channel for arbitrary knudsen numbers

Accomodational dependence of the heat- and mass-transfer processes in a multiatomic gas in a capillary with arbitrary knudsen numbers