Modeling granular materials as compressible nonlinear fluids: Heat transfer boundary value problems

Massoudi, Mehrdad; Phuoc, Tran X.

doi:https://doi.org/10.1155/MPE/2006/56046

Mathematical Problems in Engineering

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Volume 2006 | Article ID 056046 | https://doi.org/10.1155/MPE/2006/56046

Modeling granular materials as compressible nonlinear fluids: Heat transfer boundary value problems

Mehrdad Massoudi¹and Tran X. Phuoc¹

Received28 Aug 2006

Accepted13 Sept 2006

Published19 Nov 2006

Abstract

We discuss three boundary value problems in the flow and heat transfer analysis in flowing granular materials: (i) the flow down an inclined plane with radiation effects at the free surface; (ii) the natural convection flow between two heated vertical walls; (iii) the shearing motion between two horizontal flat plates with heat conduction. It is assumed that the material behaves like a continuum, similar to a compressible nonlinear fluid where the effects of density gradients are incorporated in the stress tensor. For a fully developed flow the equations are simplified to a system of three nonlinear ordinary differential equations. The equations are made dimensionless and a parametric study is performed where the effects of various dimensionless numbers representing the effects of heat conduction, viscous dissipation, radiation, and so forth are presented.

References

S. J. Antony, W. Hoyle, and Y. Ding, Granular Materials: Fundamentals and Applications, The Royal Society of Chemistry, Cambridge, 2004.
View at: Google Scholar
T. Astarita and R. Ocone, “Unsteady compressible flow of granular materials,” Industrial & Engineering Chemistry Research, vol. 38, no. 4, pp. 1177–1182, 1999.
View at: Publisher Site | Google Scholar
R. A. Bagnold, “Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear,” Proceedings of the Royal Society of London. Series A, vol. 255, no. 1160, pp. 49–63, 1954.
View at: Google Scholar
R. A. Bagnold, “The flow of cohesionless grains in fluids,” Philosophical Transactions of the Royal Society of London. Series A, vol. 249, pp. 235–297, 1956.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
Y. H. Bashir and J. D. Goddard, “Experiments on the conductivity of suspensions of ionically-conductive spheres,” AIChE Journal, vol. 36, no. 3, pp. 387–396, 1990.
View at: Publisher Site | Google Scholar
R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, John Wiley & Sons, New York, 1960.
View at: Google Scholar
A. J. Chamkha, “Effects of heat absorption and thermal radiation on heat transfer in a fluid-particle flow past a surface in the presence of a gravity field,” International Journal of Thermal Sciences, vol. 39, no. 5, pp. 605–615, 2000.
View at: Publisher Site | Google Scholar
I. F. Collins and G. T. Houlsby, “Application of thermomechanical principles to the modelling of geotechnical materials,” Proceedings of the Royal Society of London. Series A, vol. 453, no. 1964, pp. 1975–2001, 1997.
View at: Publisher Site | Google Scholar
S. C. Cowin, “ Constitutive relations that imply a generalized Mohr-Coulomb Criterion,” Acta Mathematica, vol. 20, no. 1-2, pp. 41–46, 1974.
View at: Publisher Site | Google Scholar
S. C. Cowin, “A theory for the flow of granular materials,” Powder Technology, vol. 9, no. 2-3, pp. 61–69, 1974.
View at: Publisher Site | Google Scholar
P. G. de Gennes, “Reflections on the mechanics of granular matter,” Physica A, vol. 261, no. 3, pp. 267–293, 1998.
View at: Publisher Site | Google Scholar
J. Duran, Sands, Powders, and Grains, Springer, New York, 2000.
View at: Google Scholar | Zentralblatt MATH
S. A. Elaskar and L. A. Godoy, “Constitutive relations for compressible granular materials using non-Newtonian fluid mechanics,” International Journal of Mechanical Sciences, vol. 40, no. 10, pp. 1001–1018, 1998.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
H. U. Fuchs, The Dynamics of Heat, Springer, New York, 1996.
View at: Google Scholar | Zentralblatt MATH
J. D. Goddard, “Dissipative materials as models of thixotropy and plasticity,” Journal of Non-Newtonian Fluid Mechanics, vol. 14, pp. 141–160, 1984.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
I. Goldhirsch, “Rapid granular flows,” Annual Review of Fluid Mechanics, vol. 35, pp. 267–293, 2003.
View at: Publisher Site | Google Scholar
M. A. Goodman and S. C. Cowin, “Two problems in the gravity flow of granular materials,” Journal of Fluid Mechanics, vol. 45, pp. 321–339, 1971.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
M. A. Goodman and S. C. Cowin, “A continuum theory for granular materials,” Archive for Rational Mechanics and Analysis, vol. 44, no. 4, pp. 249–266, 1972.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. Gudhe, K. R. Rajagopal, and M. Massoudi, “Fully developed flow of granular materials down a heated inclined plane,” Acta Mechanica, vol. 103, no. 1–4, pp. 63–78, 1994.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. Gudhe, R. C. Yalamanchili, and M. Massoudi, “Flow of granular materials down a vertical pipe,” International Journal of Non-Linear Mechanics, vol. 29, no. 1, pp. 1–12, 1994.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
K. Hutter and K. R. Rajagopal, “On flows of granular materials,” Continuum Mechanics and Thermodynamics, vol. 6, no. 2, pp. 81–139, 1994.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
K. Hutter, F. Szidarovszky, and S. Yakowitz, “Plane steady shear flow of a cohesionless granular material down an inclined plane: a model for flow avalanches part I: theory,” Acta Mechanica, vol. 63, no. 1–4, pp. 87–112, 1986.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
K. Hutter, F. Szidarovszky, and S. Yakowitz, “Plane steady shear flow of a cohesionless granular material down an inclined plane: a model for flow avalanches part II: numerical results,” Acta Mechanica, vol. 65, no. 1–4, pp. 239–261, 1987.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
D. J. Jeffrey, “Conduction through a random suspension of spheres,” Proceedings of the Royal Society of London. Series A, vol. 335, no. 1602, pp. 355–367, 1973.
View at: Google Scholar
G. M. Johnson, M. Massoudi, and K. R. Rajagopal, “Flow of a fluid-solid mixture between flat plates,” Chemical Engineering Science, vol. 46, no. 7, pp. 1713–1723, 1991.
View at: Publisher Site | Google Scholar
M. Kaviany, Principles of Heat Transfer in Porous Media, Springer, New York, 2nd edition, 1995.
View at: Google Scholar | Zentralblatt MATH
I. S. Liu, Continuum Mechanics, Advanced Texts in Physics, Springer, Berlin, 2002.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
M. Massoudi, “On the heat flux vector for flowing granular materials - part I: effective thermal conductivity and background,” Mathematical Methods in the Applied Sciences, vol. 29, no. 13, pp. 1585–1598, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
M. Massoudi, “On the heat flux vector for flowing granular materials - part II: derivation and special cases,” Mathematical Methods in the Applied Sciences, vol. 29, no. 13, pp. 1599–1613, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
M. Massoudi and M. M. Mehrabadi, “A continuum model for granular materials: considering dilatancy and the Mohr-Coulomb criterion,” Acta Mechanica, vol. 152, no. 1–4, pp. 121–138, 2001.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
M. Massoudi and T. X. Phuoc, “Flow and heat transfer due to natural convection in granular materials,” International Journal of Non-Linear Mechanics, vol. 34, no. 2, pp. 347–359, 1999.
View at: Publisher Site | Google Scholar
M. Massoudi, K. R. Rajagopal, and T. X. Phuoc, “On the fully developed flow of a dense particulate mixture in a pipe,” Powder Technology, vol. 104, no. 3, pp. 258–268, 1999.
View at: Publisher Site | Google Scholar
D. F. McTigue, “A non-linear constitutive model for granular materials: applications to gravity flow,” Journal of Applied Mechanics, vol. 49, pp. 291–296, 1982.
View at: Google Scholar | Zentralblatt MATH
A. Mehta, Ed., Granular Matter, A. Mehta, Ed., Springer, New York, 1994.
View at: Google Scholar
J. R. Miller and P. M. Weaver, “Temperature profiles in composite plates subject to time-dependent complex boundary conditions,” Composite Structures, vol. 59, no. 2, pp. 267–278, 2003.
View at: Publisher Site | Google Scholar
I. Müller, “On the entropy inequality,” Archive for Rational Mechanics and Analysis, vol. 26, pp. 118–141, 1967.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
T. Y. Na, Computational Methods in Engineering Boundary Value Problems, vol. 145 of Mathematics in Science and Engineering, Academic Press, New York, 1979.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
J. S. Patton, R. H. Sabersky, and C. E. Brennen, “Convective heat transfer to rapidly flowing granular materials,” International Journal of Heat and Mass Transfer, vol. 29, no. 8, pp. 1263–1269, 1986.
View at: Publisher Site | Google Scholar
K. R. Rajagopal and M. Massoudi, “A method for measuring material moduli of granular materials: flow in an orthogonal rheometer,” Tech. Rep. DOE/PETC/TR-90/3, USDOE Pittsburgh Energy Technology Center, Pennsylvania, 1990.
View at: Google Scholar
K. R. Rajagopal, M. Massoudi, and A. S. Wineman, “Flow of granular materials between rotating disks,” Mechanics Research Communications, vol. 21, no. 6, pp. 629–634, 1994.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
K. R. Rajagopal and T. Y. Na, “Natural convection flow of a non-Newtonian fluid between two vertical flat plates,” Acta Mechanica, vol. 54, no. 3-4, pp. 239–246, 1985.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
K. R. Rajagopal and L. Tao, Mechanics of Mixtures, vol. 35 of Series on Advances in Mathematics for Applied Sciences, World Scientific, New Jersey, 1995.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
K. R. Rajagopal, W. Troy, and M. Massoudi, “Existence of solutions to the equations governing the flow of granular materials,” European Journal of Mechanics, B/Fluids, vol. 11, no. 3, pp. 265–276, 1992.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
M. Reiner, “A mathematical theory of dilatancy,” American Journal of Mathematics, vol. 67, pp. 350–362, 1945.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
O. Reynolds, “On the dilatancy of media composed of rigid particles in contact with experimental illustrations,” Philosophical Magazine, vol. 20, no. 5, pp. 469–481, 1885.
View at: Google Scholar
O. Reynolds, “Experiments showing dilatancy, a property of granular material, possibly connected with gravitation,” Proceedings of the Royal Institution of Great Britain, vol. 11, pp. 354–363, 1886.
View at: Google Scholar
R. M. Saldanha da Gama, “On the conduction/radiation heat transfer problem in a body with wavelength-dependent properties,” Applied Mathematical Modelling, vol. 28, no. 9, pp. 795–816, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
S. B. Savage, “Gravity flow of cohesionless granular materials in chutes and channels,” Journal of Fluid Mechanics, vol. 92, no. 1, pp. 53–96, 1979.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
S. B. Savage, “The mechanics of rapid granular flows,” Advances in Applied Mechanics, vol. 24, pp. 289–366, 1984.
View at: Google Scholar | Zentralblatt MATH
D. G. Schaeffer, “Instability in the evolution equations describing incompressible granular flow,” Journal of Differential Equations, vol. 66, no. 1, pp. 19–50, 1987.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. J. M. Spencer, “Deformation of ideal granular materials,” in Mechanics of Solids, pp. 607–652, Pergamon Press, New York, 1982.
View at: Google Scholar | Zentralblatt MATH
A. Z. Szeri, Fluid Film Lubrication, Cambridge University Press, Massachusetts, 1998.
View at: Google Scholar | Zentralblatt MATH
A. Z. Szeri and K. R. Rajagopal, “Flow of a non-Newtonian fluid between heated parallel plates,” International Journal of Non-Linear Mechanics, vol. 20, no. 2, pp. 91–101, 1985.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
G. I. Tardos, “A fluid mechanistic approach to slow, frictional flow of powders,” Powder Technology, vol. 92, no. 1, pp. 61–74, 1997.
View at: Publisher Site | Google Scholar
C. Truesdell, “The meaning of viscometry in fluid mechanics,” Annual Review of Fluid Mechanics, vol. 6, pp. 111–146, 1974.
View at: Google Scholar | Zentralblatt MATH
C. Truesdell and W. Noll, The Nonlinear Field Theories of Mechanics, Springer, Berlin, 1992.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
O. R. Walton and R. L. Braun, “Stress calculations for assemblies of inelastic spheres in uniform shear,” Acta Mechanica, vol. 63, no. 1–4, pp. 73–86, 1986.
View at: Publisher Site | Google Scholar
J. W. Wu and H. S. Chu, “Combined conduction and radiation heat transfer in plane-parallel packed beds with variable porosity,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 61, no. 4, pp. 443–452, 1999.
View at: Publisher Site | Google Scholar

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Copyright © 2006 Mehrdad Massoudi and Tran X. Phuoc. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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