A collocation method and its application for solving the linearized Holway equation

O. V. Germider ; V. N. Popov

Matematičeskoe modelirovanie, Tome 32 (2020) no. 9, pp. 3-19.

Résumé

The article proposes a method for solving the linearized ellipsoidal statistical equation of Holway in the framework of the classical problem of the flow of a rarefied gas between two parallel planes. To approximate the desired solution, we use the expansion of the function in a series according to Chebyshev polynomials of the first kind for each variable. The Holway model kinetic equation by the collocation method is reduced to a linear matrix equation. The values of mass and heat flows of gas are calculated.

Détail

Keywords: rarefied gas, kinetic equation, ES-model, collocation method.

Bibliographie

[1] A. A. Frolova, “Chislennoe sravnenie reshenij kineticheskikh modelnykh uravnenii”, Matematika i matematicheskoe modelirovanie, 6 (2015), 61–77

[2] P. L. Bhatnagar, E. P. Gross, M. A. Krook, “Model for collision process in gases”, Physical Review, 94 (1954), 511–525 | Zbl

[3] E. M. Shakhov, “Generalization of the Krook kinetic relaxation equation”, Fluid Dynamics, 3:5 (1968), 95–96 | MR

[4] L. H. Holway, “New statistical models for kinetic theory: Methods of construction”, Physics of Fluids, 9 (1966), 1658–1673

[5] M. N. Kogan, Rarefied Gas Dynamics, Plenum Press, New York, 1969, 515 pp.

[6] A. V. Latyshev, A. A. Yushkanov, “Thirteen-Moment Model Kinetic Equation and Its Parameters”, Mathematical Models and Computer Simulations, 9:2 (2017), 169–175 | MR | Zbl

[7] F. M. Sharipov, V. D. Seleznev, Dvizhenie razrezhennykh gazov v kanalakh i mikrokanalakh, UrO RAN, Ekaterinburg, 2008, 230 pp.

[8] V. A. Titarev, E. M. Shakhov, “Nonisothermal gas flow in a long channel analyzed on the basis of the kinetic S-Model”, Comp. Math. and Math. Physics, 50:12 (2010), 2131–2144 | MR | Zbl

[9] I. Graur, M. T. Ho, “Rarefied gas flow through a long rectangular channel of variable cross section”, Vacuum, 101 (2014), 328–332

[10] P. Andries, J-F. Bourgat, P. Le Tallec, B. Perthame, “Numerical comparison between the Boltzmann and ES-BGK models for rarefied gases”, Comp Methods Appl Mech Eng., 191:31 (2002), 3369–3390 | MR | Zbl

[11] I. A. Graur, A. P. Polikarpov, “Comparison of different kinetic models for the heat transfer problem”, Heat Mass Transfer, 46:2 (2009), 237–244

[12] T. Ohwada, “Heat flow and temperature and density distributions in a rarefied gas between parallel plates with different temperatures. Finite-difference analysis of the nonlinear Boltzmann equation for hard-sphere molecules”, Phys Fluids, 8:8 (1996), 2153–2160 | Zbl

[13] D. C. Wadsworth, “Slip effects in a confined rarefied gas. Part I: temperature slip”, Phys. Fluids A, 5:7 (1993), 1831–1839

[14] D. J. Alofs, R. C. Flagan, G. Springer, “Density distribution measurements in rarefied gases contained between parallel plates at high temperature difference”, Phys. Fluids, 14:3 (1970), 529–533

[15] V. V. Belyi, “Derivation of model kinetic equation”, Europhysics Letters, 111 (2015), 40011

[16] R. D. M. Garcia, C. E. Siewert, “The linearized Boltzmann equation with Cercignani–Lampis boundary conditions: Heat transfer in a gas confined by two plane-parallel surfaces”, Annals of Nuclear Energy, 86, 45–54

[17] A. Baseri, S. Abbasbandy, E. Babolian, “A collocation method for fractional diffusion equation in a long time with Chebyshev functions”, Applied Math. Comp., 322 (2018), 55–65 | MR | Zbl

[18] K. Maleknejad, R. Dehbozorgi, “Adaptive numerical approach based upon Chebyshev operational vector for nonlinear Volterra integral equations and its convergence analysis”, Journal of Computational and Applied Mathematics, 344 (2018), 356–366 | MR | Zbl

[19] E. Tohidi, “Application of Chebyshev collocation method for solving two classes of non-classical parabolic PDEs”, Ain Shams Engineering J., 6:230 (2015), 373–379

[20] O.B Arushanyan, N. I. Volchenskova, S. F. Zaletkin, “Metod resheniya zadachi Koshi dlya obyknovennyh differencialnyh uravnenij s ispolzovaniem ryadov Chebysheva”, Vych. met. Programmirovanie, 14:2 (2013), 203–214

[21] J. Mason, D. Handscomb, Chebyshev polynomials, CRC Press, Florida, 2003, 360 pp. | MR | Zbl

[22] O. V. Germider, V. N. Popov, “Primenenie polinomov CHebysheva dlya vychisleniya potokov razrezhennogo gaza v kanalah s cilindricheskoj geometriej”, Sib. ehlektron. matem. izv., 16 (2019), 1947–1959 | Zbl

[23] O. V. Germider, V. N. Popov, “Solution of the Linearized Problem of Heat and Gas Mass Transfer in the Gap between Two Cylindrical Surfaces under a Longitudinal Temperature Gradient”, Comp. Math. and Math. Phys., 58 (2018), 1610–1619 | MR | Zbl

[24] O. V. Germider, V. N. Popov, “Rarefied Gas Flow between Two Coaxial Cylinders Driven by Temperature Gradient in the Case of Specular-Diffuse Reflection”, Comp. Math. Math. Phys., 59 (2019), 1333–1341 | MR | Zbl

[25] V. A. Titarev, “Implicit high-order method for calculating rarefied gas flow in a planarmicrochannel”, Journal of Computational Physics, 231 (2012), 109–134 | MR | Zbl

[26] V. A. Titarev, “Rarefied gas flow in a planar channel caused by arbitrary pressure and temperature drops”, International J. of Heat and Mass Transfer, 55 (2012), 5916–5930

[27] S. Pantazis, D. Valougeorgis, F. Sharipov, “End corrections for rarefied gas flows through capillaries of finite length”, Vacuum, 97 (2013), 26–29

[28] V. A. Titarev, E. M. Sharipov, “Kinetic analysis of the isothermal flow in a long rectangular microchannel”, Comp. Math. and Math. Phys, 50 (2010), 1221–1237 | MR | Zbl

[29] F. M. Sharipov, “Non-isothermal gas flow through rectangular microchannels”, J. Micromech. Microeng., 9:4 (1999), 394–401

[30] C. E. Siewert, “Poiseuille, thermal creep and Couette flow: results based on the CES model of the linearized Boltzmann equation”, European J. of Mechan. B/Fluids, 21 (2002), 579–597 | MR | Zbl

[31] K. A. Hickey, S. K. Loyalka, “Kinetic theory of thermal transpiration and the mechanocaloric effect: Planar flow of a rigid sphere gas with arbitrary accommodation at the surface”, J. Vac. Sci. Technol. A, 9:1 (1991), 158–163

[32] C. E. Siewert, “The linearized Boltzmann Equation: Concise and Accurate Solutions to Basic Flow Problems”, Zeitschrift fur Angewandte Mathem. und Physik, 54 (2003), 273–303 | MR | Zbl

[33] B. T. Porodnov, Teoreticheskoe i ehksperimentalnoe issledovanie ploskogo techeniya razrezhennykh gazov, Dissert. kand. fiz. mat. nauk, UPI, Sverdlovsk, 1969

[34] C. Cercignani, Mathematical Methods in Kinetic Theory, Plenum Press, New York, 1969, 227 pp. | MR | Zbl

[35] M. A. Ramadan, K. R. Raslan, El T. S. Danaf, Abd El M. A. Salam, “On the exponential Chebyshev approximation in unbounded domains: a comparison study for solving high-order ordinary differential equations”, Intern. J. of Pure and Applied Math., 105:3 (2015), 399–413 | MR

[36] C. Cercignani, Theory and application of the Boltzmann equation, Elsevier, New York, 1975, 415 pp. | MR