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Micro-scale flow is a very important and prominent problem in the design and application of micro-electromechanical systems. With the decrease of the scale, effects, such as viscous dissipation, compression work and boundary slip etc., which are ignored in a large-scale flow, play important roles in a microfluidic system. #br#With its certain advantages such as high numerical efficiency, easy implement, parallel algorithms etc., the lattice Boltzmann method is a powerful numerical technique for simulating fluid flows and modeling the physics in fluids. The double-distribution-function lattice Boltzmann method has been widely used in a micro-scale thermal flow system, since it utilizes two different distribution functions to take account of the viscous dissipation and compression work. However, most of the existing double-distribution-function lattice Boltzmann methods are “decoupling” models, and decoupling will cause the models to be limited to Boussinesq flows in which temperature variation is small. In order to overcome the above problem, based on the low-order Hermite expansion of the continuous equilibrium distribution function, we propose a coupling double-distribution-function thermal lattice Boltzmann method. This method introduces temperature changes into the lattice Boltzmann momentum equation in the form of the momentum source, which can affect the distribution of flow velocity and density, so as to realize the coupling between the momentum field and the energy field. In the process of fluid flow, the temperature change of the energy field includes two parts: one is for different times at the same lattice which can cause the change of the fluid characteristic parameters, such as the viscosity coefficient and the thermal diffusivity; the other is for the same time at different lattices which mainly affects the distribution of the velocity. In the collision and the migration processes, temperature change is introduced into the fluid flow to achieve the effect of temperature changes on the flow field and the coupling between the energy field and the momentum field. This method can break up the limitation of the Boussinesq flows and expand the application scope of the lattice Boltzmann method. #br#Two natural convection models (one takes into consideration the viscous dissipation and compression work, and the other does not) are studied in this paper to verify the effectiveness and accuracy of the coupling double-distribution-function thermal lattice Boltzmann method. Flow field and the changing trend in temperature, velocity and the averaged Nusselt number are analyzed emphatically at different Rayleigh number and Prandtl number. Results of this paper are excellently consistent with those in papers published, confirming the validity and accuracy of this method. Results also show that the convective heat transfer gradually enhances with increasing Rayleigh number and Prandtl number in the cavity, and the boundary layer is obviously formed in the regions very close to the walls; the heat transfer is greatly enhanced if viscous dissipation and compression work are considered; and these effects should not be neglected in the micro-scale flow system.
[1] Jiang H Y, Ren Y K, Ao H R, Ramos A 2008 Chin. Phys. B 17 4541
[2] Esfahanian V, Dehdashti E, Dehrouye A M 2014 Chin. Phys. B 23 084702
[3] Alexander F J, Chen S, Sterling J D 1993 Phys. Rev. E 47 R2249
[4] Gan Y, Xu A, Zhang G, Li Y 2011 Phys. Rev. E. 83 056704
[5] Chikatamarla S S, Karlin I V 2008 Comput. Phys. Commun. 179 140
[6] Gan Y, Xu A, Zhang G, Yu X, Li Y 2008 Physica A 387 1721
[7] Pierre L, Luo L S 2003 Phys. Rev. E 68 036706
[8] Pierre L, Luo L S 2003 Int. J Mod. Phys. B 17 41
[9] Liu F F, Wei S S, Wei C Z, Ren X F 2014 Acta Phys. Sin. 63 1947041 (in Chinese) [刘飞飞, 魏守水, 魏长智, 任晓飞 2014 物理学报 63 194704]
[10] Li Q, He Y L, Tang G H, Tao W Q 2009 Phys. Rev. E 80 037702
[11] Chen S, Tölke J, Krafczyk M 2009 Phys. Rev. E 79 016704
[12] He X, Chen S, Doolen G D 1998 J. Comput. Phys. 146 282
[13] Dixit H N, Babu V 2006 Int. J. Heat Mass Trans. 49 727
[14] Wang C H, Yang R 2006 Appl. Math. Comput. 173 1246
[15] Shi Y, Zhao T S, Guo Z L 2006 Comput. Fluids 35 1
[16] Peng Y, Shu C, Chew Y T 2003 Phys. Rev. E 68 143
[17] Li Q, He Y, Wang Y, Tang G 2008 Int. J. Mod. Phys. C 19 125
[18] Guo Z L, Zheng C G, Shi B C, Zhao T S 2007 Phys. Rev. E 75 3654
[19] Mo J Q, Cheng Y 2009 Acta Phys. Sin. 58 4379 (in Chinese) [莫嘉琪, 程燕 2009 物理学报 58 4379]
[20] Shan X, Yuan X F, Chen H 2006 J. Fluid Mech. 550 413
[21] Hung L H, Yang J Y 2011 Ima J. Appl. Math . 76 774
[22] Li Q, Luo K H, He Y L, Gao Y J, Tao W Q 2012 Phys. Rev. E 85 016710
[23] Basu R, Layek G C 2013 Chin. Phys. B 22 054702
[24] Sun D K, Zhu M F, Yang C R, Pan S Y, Dai T 2009 Acta Phys. Sin. 58 S285 (in Chinese) [孙东科, 朱鸣芳, 杨朝蓉, 潘诗琰, 戴挺 2009 物理学报 58 S285]
[25] Abdel R G, Khader M M, Megahed A M 2013 Chin. Phys. B 22 030202
[26] Liu F F, Wei S S, Wang S W, Wei C Z, Ren X F 2014 J. Nanoengin. Nanosys. 228 189
[27] Tang G H, Tao W Q, He Y L 2005 Phys. Rev. E 72 6435
[28] Sun L, Sun Y F, Ma D J, Sun D J 2007 Acta Phys. Sin. 56 6503
[29] Costa V A F 2005 Int. J. Heat Mass Tran. 48 2333
[30] Barakos G, Mitsoulis E, Assimacopoulos D 1994 Int. J. Numer. Meth. Fl. 18 695
[31] Cheng T S 2011 Int. J. Therm. Sci. 50 197
[32] Arcidiacono S, Dipiazza I, Ciofalo M 2001 Int. J. Heat Mass Tran. 44 537
[33] Chatterjee D, Biswas G 2011 Numer. Heat Tr. A-Appl. 59 421
[34] MacGregor R, Emery A 1969 J. Heat Tran. 91 391
[35] He Y, Yang W, Tao W 2005 Numer. Heat Tr. A-Appl. 47 917
[36] Pesso T, Piva S 2009 Int. J. Heat Mass Tran. 52 1036
[37] Karimipour A, Nezhad A H, D’Orazio A, Shirani E 2013 J Theor. App. Mech-pol 51 447
[38] Kawamura H, Abe H, Matsuo Y 1999 Int. J. Heat Fluid Fl. 20 196
[39] Dipiazza I, Ciofalo M 2000 Int. J. Heat. Mass Tran. 43 3027
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[1] Jiang H Y, Ren Y K, Ao H R, Ramos A 2008 Chin. Phys. B 17 4541
[2] Esfahanian V, Dehdashti E, Dehrouye A M 2014 Chin. Phys. B 23 084702
[3] Alexander F J, Chen S, Sterling J D 1993 Phys. Rev. E 47 R2249
[4] Gan Y, Xu A, Zhang G, Li Y 2011 Phys. Rev. E. 83 056704
[5] Chikatamarla S S, Karlin I V 2008 Comput. Phys. Commun. 179 140
[6] Gan Y, Xu A, Zhang G, Yu X, Li Y 2008 Physica A 387 1721
[7] Pierre L, Luo L S 2003 Phys. Rev. E 68 036706
[8] Pierre L, Luo L S 2003 Int. J Mod. Phys. B 17 41
[9] Liu F F, Wei S S, Wei C Z, Ren X F 2014 Acta Phys. Sin. 63 1947041 (in Chinese) [刘飞飞, 魏守水, 魏长智, 任晓飞 2014 物理学报 63 194704]
[10] Li Q, He Y L, Tang G H, Tao W Q 2009 Phys. Rev. E 80 037702
[11] Chen S, Tölke J, Krafczyk M 2009 Phys. Rev. E 79 016704
[12] He X, Chen S, Doolen G D 1998 J. Comput. Phys. 146 282
[13] Dixit H N, Babu V 2006 Int. J. Heat Mass Trans. 49 727
[14] Wang C H, Yang R 2006 Appl. Math. Comput. 173 1246
[15] Shi Y, Zhao T S, Guo Z L 2006 Comput. Fluids 35 1
[16] Peng Y, Shu C, Chew Y T 2003 Phys. Rev. E 68 143
[17] Li Q, He Y, Wang Y, Tang G 2008 Int. J. Mod. Phys. C 19 125
[18] Guo Z L, Zheng C G, Shi B C, Zhao T S 2007 Phys. Rev. E 75 3654
[19] Mo J Q, Cheng Y 2009 Acta Phys. Sin. 58 4379 (in Chinese) [莫嘉琪, 程燕 2009 物理学报 58 4379]
[20] Shan X, Yuan X F, Chen H 2006 J. Fluid Mech. 550 413
[21] Hung L H, Yang J Y 2011 Ima J. Appl. Math . 76 774
[22] Li Q, Luo K H, He Y L, Gao Y J, Tao W Q 2012 Phys. Rev. E 85 016710
[23] Basu R, Layek G C 2013 Chin. Phys. B 22 054702
[24] Sun D K, Zhu M F, Yang C R, Pan S Y, Dai T 2009 Acta Phys. Sin. 58 S285 (in Chinese) [孙东科, 朱鸣芳, 杨朝蓉, 潘诗琰, 戴挺 2009 物理学报 58 S285]
[25] Abdel R G, Khader M M, Megahed A M 2013 Chin. Phys. B 22 030202
[26] Liu F F, Wei S S, Wang S W, Wei C Z, Ren X F 2014 J. Nanoengin. Nanosys. 228 189
[27] Tang G H, Tao W Q, He Y L 2005 Phys. Rev. E 72 6435
[28] Sun L, Sun Y F, Ma D J, Sun D J 2007 Acta Phys. Sin. 56 6503
[29] Costa V A F 2005 Int. J. Heat Mass Tran. 48 2333
[30] Barakos G, Mitsoulis E, Assimacopoulos D 1994 Int. J. Numer. Meth. Fl. 18 695
[31] Cheng T S 2011 Int. J. Therm. Sci. 50 197
[32] Arcidiacono S, Dipiazza I, Ciofalo M 2001 Int. J. Heat Mass Tran. 44 537
[33] Chatterjee D, Biswas G 2011 Numer. Heat Tr. A-Appl. 59 421
[34] MacGregor R, Emery A 1969 J. Heat Tran. 91 391
[35] He Y, Yang W, Tao W 2005 Numer. Heat Tr. A-Appl. 47 917
[36] Pesso T, Piva S 2009 Int. J. Heat Mass Tran. 52 1036
[37] Karimipour A, Nezhad A H, D’Orazio A, Shirani E 2013 J Theor. App. Mech-pol 51 447
[38] Kawamura H, Abe H, Matsuo Y 1999 Int. J. Heat Fluid Fl. 20 196
[39] Dipiazza I, Ciofalo M 2000 Int. J. Heat. Mass Tran. 43 3027
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