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A thermal lattice Boltzmann model with diffuse scattering boundary condition for micro thermal flows. (English) Zbl 1177.76321
Summary: A lattice Boltzmann model for simulating isothermal micro flows has been proposed by us recently [X. D. Niu, A lattice Boltzmann BGK model for simulation of micro flows. Europhys. Lett. 67, No. 4, 600 (2004)]. In this paper, we extend the model to simulate the micro thermal flows. In particular, the thermal lattice Boltzmann equation (TLBE) [X. He et al., J. Comput. Phys. 146, No. 1, 282–300 (1998; Zbl 0919.76068)] is used with modification of the relaxation times linking to the Knudsen number. The diffuse scattering boundary condition (DSBC) derived in our early model is extended to consider temperature jump at wall boundaries. Simple theoretical analyses of the DSBC are presented and the results are found to be consistent with the conventional velocity slip and temperature jump boundary conditions. Numerical validations are carried out by simulating two-dimensional thermal Couette flows and developing thermal flows in a microchannel, and the obtained results are found to be in good agreement with those given from the direct simulation Monte Carlo (DSMC), the molecular dynamics (MD) approaches and the Maxwell theoretical prediction.

MSC:
76M28 Particle methods and lattice-gas methods
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[1] Tuckerman, D.B.; Pease, R.F.W., High-performance heat sinking for VLSI, IEEE electron dev lett, EDL-2, 126, (1981)
[2] Wu, P.Y.; Little, W.A., Measurement of the heat transfer characteristics of gas flow in fine channel heat exchangers for micro miniature refrigerators, Cryogenics, 24, 415, (1984)
[3] Peng, X.F.; Peterson, G.P.; Wang, B.X., Heat transfer characteristics of water flowing through microchannels, Exp heat transfer, 7, 265, (1994)
[4] Yang, C.; Li, D.; Masliyah, J.H., Modeling forced liquid convection in rectangular microchannels with electrokinetics effects, Int J heat mass transfer, 41, 4229, (1998) · Zbl 0962.76614
[5] Sharipov, F.; Seleznev, V., Data on internal rarefied gas flows, J phys chem ref data, 27, 3, 657, (1998)
[6] Loyalka, S.K.; Hamoodi, S.A., Poiseuille flow of a rarefied gas in a cylindrical tube: solution of linearized Boltzmann equation, Phys fluids A, 2, 11, 2061, (1990) · Zbl 0713.76089
[7] Ohwada, T.; Sone, Y.; Aoki, K., Numerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann equation for hard sphere molecules, Phys fluids A, 1, 12, 2042, (1989) · Zbl 0696.76092
[8] Alder, B.J.; Waiwright, T.E., J chem phys, 27, 1208, (1957)
[9] Bird, G., Molecular gas dynamics and the direct simulation of gas flows, (1994), Oxford Science Publications
[10] Higuera, F.; Succi, S.; Benzi, R., Lattice gas dynamics with enhanced collision, Europhys lett, 9, 345, (1989)
[11] Benzi, R.; Succi, S.; Vergassola, M., The lattice Boltzmann equation: theory and applications, Phys rep, 222, 145, (1992)
[12] Qian, Y.H.; d’Humiéres, D.; Lallemand, P., Lattice BGK models for the navier – stokes equation, Europhys lett, 17, 479, (1992) · Zbl 1116.76419
[13] Chen, S.; Doolen, G.D., Lattice Boltzmann method for fluid flows, Ann rev fluid mech, 30, 329, (1998) · Zbl 1398.76180
[14] Bhatnagar, P.L.; Gross, E.P.; Krook, M., A model for collision processes in gases. I. small amplitude processes in charged and neutral one-component systems, Phys rev, 94, 511, (1954) · Zbl 0055.23609
[15] Nie, X.; Doolen, G.D.; Chen, S., Lattice Boltzmann simulations of fluid flows in MEMS, J stat phys, 107, 1/2, 279, (2002) · Zbl 1007.82007
[16] Lim, C.Y.; Shu, C.; Niu, X.D.; Chew, Y.T., Application of lattice Boltzmann method to simulate microchannel flows, Phys fluids, 14, 7, 2299, (2002) · Zbl 1185.76227
[17] Succi, S., Mesoscopic modeling of slip motion at fluid-solid interfaces with heterogeneous catalysis, Phys rev lett, 89, 64502, (2002)
[18] Niu, X.D.; Chew, Y.T.; Shu, C., A lattice Boltzmann BGK model for simulation of micro flows, Europhys lett, 67, 4, 600, (2004)
[19] Tretheway DC, Zhu L, Petzold L, Meinhart CD. Examination of the slip boundary condition by μ-PIV and lattice Boltzmann simulations. In: Proceedings of IMECE’ 2002, ASME international mechanical engineering congress & exposition, 2002, p. 33704.
[20] Jiaung, W.S.; Ho, J.-R., Lattice Boltzmann study on size effect with geometrical bending on phonon heat conduction in a nanoduct, J appl phys, 95, 958, (2004)
[21] Tang, G.H.; Tao, W.Q.; He, Y.L., Lattice Boltzmann method for gaseous microflows using kinetic theory boundary conditions, Phys fluids, 17, 5, 058101, (2005) · Zbl 1187.76518
[22] Zhang, Y.H.; Qin, R.S.; Emerson, D.R., Lattice Boltzmann simulation of rarefied gas flows in microchannels, Phys rev E, 71, 4, 047702, (2005)
[23] Toschi, F.; Succi, S., Lattice Boltzmann method at finite Knudsen numbers, Europhys lett, 69, 4, 549, (2005)
[24] Ansumali, S.; Karlin, I.V., Entropy function approach to the lattice Boltzmann method, J stat phys, 107, 291, (2002) · Zbl 1007.82019
[25] Cercignani, C., Mathematical methods in kinetic theory, (1969), Plenum New York · Zbl 0191.25103
[26] Maxwell, J.C., ()
[27] Cercignani, C.; Illner, R.; Pulvirenti, M., The mathematical theory of dilute gases, (1994), Springer Berlin · Zbl 0813.76001
[28] He, X.; Chen, S.; Doolen, G.D., A novel thermal model for the lattice Boltzmann method in incompressible limit, J comput phys, 146, 282, (1998) · Zbl 0919.76068
[29] He, X.; Luo, L.-S., A priori derivation of the lattice Boltzmann equation, Phys rev E, 55, 6333, (1997)
[30] Gatignol, R., Kinetic theory boundary conditions for discrete velocity gases, Phys fluids, 20, 2022, (1977) · Zbl 0373.76065
[31] Ansumali, S.; Karlin, I.V., Kinetic boundary conditions in the lattice Boltzmann method, Phys rev E, 66, 026311-1, (2002)
[32] Karniadakis, G.E.; Beskok, A., Micro flows: fundamentals and simulation, (2002), Springer-Verlag New York · Zbl 0998.76002
[33] Shu, C.; Chew, Y.T.; Niu, X.D., Least square-based LBM: A meshless approach for simulation of flows with complex geometry, Phys rev E, 64, 045701-1, (2001)
[34] Willis, D.R., Comparison of kinetic theory analyses of linearized Couette flow, Phys fluids, 5, 127, (1962) · Zbl 0102.41101
[35] Morris, D.L.; Hannon, L.; Garcia, A.L., Slip length in a dilute gas, Phys rev A, 46, 8, 5279, (1992)
[36] Kavehpour, H.P.; Faghri, M.; Asako, Y., Effects of compressibility and rarefaction on gaseous flows in microchannels, Numer heat transfer, part A, 32, 677, (1997)
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