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An analytical solution to Boltzmann equation of dilute granular flow with homotopy analysis method. (English) Zbl 1184.76892

Summary: The homotopy analysis method (HAM), as a new mathematical tool, has been employed to solve many nonlinear problems. As a fundamental equation in non-equilibrium statistical mechanics, the Boltzmann integro-differential equation (BE) describing the movement of particles is of strong nonlinearity. In this work, HAM is preliminarily applied to dilute granular flow which is relatively simple. By choosing the Maxwell velocity distribution function as the initial solution, the concrete expression of the first-order approximate solution to BE with collision term being the BGK model is given. Furthermore it is consistent with the solution using Chapman-Enskog method but does not rely on little parameters.

MSC:

76T25 Granular flows
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
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References:

[1] Campbell C S. Granular material flows: An overview. Powder Tech, 2006, 162: 208–229 · doi:10.1016/j.powtec.2005.12.008
[2] Campbell C S. Stress controlled elastic granular shear flows. J Fluid Mech, 2005, 539: 273–297 · Zbl 1075.74027 · doi:10.1017/S0022112005005616
[3] Wang G Q, Xiong G, Fang H W. The general constitutive relation of particle flow (in Chinese). Sci China Ser E (Chinese Ver), 1998, 28: 282–288
[4] Wang G Q, Ni J R, Fei J X. The kinetic model for collisional stresses in rapid granular flows. Chin J Theor Appl Mech, 1993, 25: 348–355
[5] Jop P, Forterre Y, Pouliquen O. A constitutive law for dense granular flows. Nature, 2006, 441: 727–730 · doi:10.1038/nature04801
[6] Sun Q C, Wang G Q. Introduction of Particle Mechanics (in Chinese). Beijing: Science Press, 2009
[7] Ge W, Li J H. Macro-scale pseudo-particle modeling for particle-fluid systems. Chinese Sci Bull, 2001, 46: 1503–1506 · doi:10.1007/BF02900568
[8] Li J H. Nonlinearity and computer simulation of particle-fluid systems (in Chinese). Chinese Sci Bull (Chinese Ver), 1996, 41(S1): 10
[9] Wang G Q. The motion theory and experimental study of solid-liquid two-phase flows and granular flows (in Chinese). PhD Dissertation. Beijing: Tsinghua University, 1989
[10] Chapman S, Cowling T G. The Mathematical Theory of Non-uniform Gases. Cambridge: Cambridge University Press, 1970 · Zbl 0063.00782
[11] Wang G Q, Hu C H. Research Development of Sediment (in Chinese). Beijing: China Waterpower Press, 2006
[12] Fu X D, Wang G Q. Kinetic model of particle phase in dilute solid-liquid two-phase flows (in Chinese). Chin J Theor Appl Mech, 2003, 35: 650–659
[13] Ying T C. The Theory and Application of Gas Transportation (in Chinese). Beijing: Tsinghua University Press, 1990
[14] Cercignani C. The Boltzmann Equation and its Applications. New York: Springer-Verlag, 1988 · Zbl 0646.76001
[15] Chen W F, Wu Q F. Methods for solving Boltzmann equation (in Chinese). J Nat Univ Defense Tech, 1999, 21: 4–7
[16] Huang Z Q, Ding E J. A kind of singular perturbation method for solving the Boltzmann equation with small Knudsen number (in Chinese). Acta Phys Sin, 1984, 33: 722–727 · Zbl 0616.76090
[17] Huang Z Q, Ding E J. The singular perturbation solution of Boltzmann equation I– normal solution (in Chinese). Acta Phys Sin, 1985, 34: 65–76 · Zbl 0605.76087
[18] Huang Z Q, Ding E J. The singular perturbation solution of Boltzmann equation II– initial layer solution (in Chinese). Acta Phys Sin, 1985, 34: 77–87 · Zbl 0605.76088
[19] Huang Z Q, Ding E J. The singular perturbation solution of Boltzmann equation III– boundary layer solution (in Chinese). Acta Phys Sin, 1985, 34: 213–224 · Zbl 0605.76089
[20] Huang Z Q, Ding E J. The singular perturbation solution of Boltzmann equation IV– extension to non-Maxwell molecules (in Chinese). Acta Phys Sin, 1985, 34: 225–234 · Zbl 0605.76090
[21] Huang Z Q, Ding E J. Theory and applications of the Boltzmann equation (in Chinese). Prog Phys, 1986, 6: 300–351
[22] Wei J B, Zhang X W. Eternal solutions of the Boltzmann equation (in Chinese). Acta Math Sci, 2007, 27: 240–247 · Zbl 1174.82331
[23] Kawashima S, Asano K. Global solution of the initial value problem for a discrete velocity model of the Boltzmann equation. Proc Japan Acad, 1981, 57: 19–24 · Zbl 0476.76071 · doi:10.3792/pjaa.57.19
[24] Peirano E, Leckner B. Fundamentals of turbulent gas-solid flows applied to circulating fluidized bed combustion. Prog Energy Combust Sci, 1998, 24: 259–296 · doi:10.1016/S0360-1285(98)00002-1
[25] Shen H C. Exact solution of the Boltzmann equation (in Chinese). Appl Math Mech, 1987, 8: 419–431 · Zbl 0645.73039 · doi:10.1007/BF02019628
[26] Liu T P, Yang T. Energy method for Boltzmann equation. Physica D, 2004, 188: 178–192 · Zbl 1098.82618 · doi:10.1016/j.physd.2003.07.011
[27] Wang G Q, Ni J R. Research review of particle flows (in Chinese). Mech Eng, 1992, 14: 7–19
[28] Xuan Y M, Li Q, Yao Z P. Application of lattice Boltzmann scheme to nanofluids. Sci China Ser E, 2004, 47: 129–140 · Zbl 1183.82072 · doi:10.1360/03ye0163
[29] Huang W F, Li Y, Liu Q S. Application of the lattice Boltzmann method to electrohydrodynamics: Deformation and instability of liquid drops in electrostatic fields. Chinese Sci Bull, 2007, 52: 3319–3324 · Zbl 1134.76052 · doi:10.1007/s11434-007-0530-4
[30] Liao S J. Beyond Perturbation: Introduction to the Homotopy Analysis Method (in Chinese). Beijing: Science Press, 2006
[31] Sajid M, Hayat T. Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt. Nonlin Dyn, 2007, 50: 27–35 · Zbl 1181.76031 · doi:10.1007/s11071-006-9140-y
[32] Abbasbandy S. The application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation. Phys Lett A, 2007, 361: 478–483 · Zbl 1273.65156 · doi:10.1016/j.physleta.2006.09.105
[33] Tan Y, Abbasbandy S. Homotopy analysis method for quadratic Riccati differential equation. Comm Nonlin Sci Numer Simul, 2008, 13: 539–546 · Zbl 1132.34305 · doi:10.1016/j.cnsns.2006.06.006
[34] Gidaspow D, Ding J M, Jayaswal U K. Multiphase Navier-Stokes equation solver-numerical methods for multiphase flows. ASME, 1990, 91: 47–56
[35] Wang G Q, Ni J R. The kinetic theory for dilute solid/liquid two-phase flow. Int J Multiph Flow, 1991, 17: 273–281 · Zbl 1134.76702 · doi:10.1016/0301-9322(91)90020-4
[36] Wang G Q, Ni J R. Kinetic theory for particle concentration distribution in two-phase flow. J Eng Mech, 1990, 116: 2738–2748 · doi:10.1061/(ASCE)0733-9399(1990)116:12(2738)
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