Normal stress differences in Couette flow of granular materials

• Autorzy: Kumar J. , Lakshmana Rao C. , Massoudi M.
• Języki publikacji: EN

Abstrakty

EN

Normal stress differences, a phenomenon which is exhibited by many non-Newtonian fluids or non-linear elastic materials, also arises in the Couette flow of granular materials. This problem is studied using a continuum model proposed by Rajagopal and Massoudi (1990). For a steady, fully developed condition, the governing equations were reduced to a system of coupled non-linear ordinary differential equations, and the resulting boundary value problem was solved numerically (Kumar et al., 2003). The expression for one of the normal stress differences is derived in this paper and from the values of volume fraction obtained in (Kumar et al., 2003), the normal stress difference is calculated. The effect of material parameters, i.e., dimensionless numbers on the normal stress difference is studied. It is observed that the distribution parameter, "B2" and the density parameter "P" affect the normal stress differences most.

Słowa kluczowe

EN

granular materials; normal stress effects; continuum mechanics

PL

materiałoznawstwo; materiał ziarnisty; mechanika

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mechanics and Engineering
• Rocznik: 2003
• Tom: Vol. 8, no 1
• Strony: 5--16
• Opis fizyczny: Bibliogr. 63 poz., wykr.

Twórcy

autor

Kumar J.

• Department of Applied Mechanics Indian Institute of Technology, Madras Chennai - 600036, INDIA

autor

Lakshmana Rao C.

• Department of Applied Mechanics Indian Institute of Technology, Madras Chennai - 600036, INDIA

autor

Massoudi M.

• U.S. Department of Energy National Energy Technology Laboratory P.O. Box 10940, Pittsburgh, PA 15236, USA

Bibliografia

• [1] Ahmadi G. (1982): A generalized continuum theory for granular materials. - Int. J. Non-Linear Mech., vol. 17, pp.21-33.
• [2] Bagnold R.A. (1954): Experiments on a gravity free dispersion of large solid spheres in a Newtonian fluid under shear. - Proc. Roy. Soc. Lond., vol.225A, pp.49-63.
• [3] Boateng A. A. and Barr P.V. (1996): Modelling of particie mixing and segregation in the transverse piane of a rotary kiln. - Chem. Engng. Sci., vol.51, pp.4167-4181.
• [4] Carnahan B., Luther H.A. and Wilkes J.O. (1969): Applied Numerical Methods. - New York: Wiley.
• [5] Coleman B.D. and Noll W. (1961): Normal stresses in second-order viscoelasticity. - Trans. Soc. Rheol., vol.5, pp.41-46.
• [6] Cowin S.C. (1974a): A theory for the flow of granular materials. - Powder Technology, vol.9, pp.61-69.
• [7] Cowin S.C. (1974b): Constitutive relations that imply a generalized Mohr-Coulomb criterion. - Acta Mechanica, vol.20, pp.41-46.
• [8] Craig K., Buckholz R.H. and Domoto G. (1984): An experimental study of the rap id flow of dry cohesionless metal powders. - ASME J. Appl. Mech., vol.53, pp.935-942.
• [9] Debbaut B., Avalosse T., Dooley J. and Hughes K. (1997): On the development of secondary motions in straight channels induced by the second normal stress difference: experiments and simulations. - J. Non-Newtonian Fluid Mech., vol.69, pp.251-271.
• [10] de Gennes P.G. (1998): Reflections on the mechanics of granular matter. - Physica A, vol.241, pp.267-293.
• [11] Edwards S.F. (1990): The rheology of powders. - Rheol. Acta, vol.29, pp.493-499.
• [12] Elliott K.E., Ahmadi G. and Kvasnak W. (1998): Couette flows of a granular mono-layer - an experimental study. - J. Non-Newtonian Fluid Mech., vol.74, pp. 89-111.
• [13] Ericksen J.L. (1956): Over de termination of the speed in rectilinear motion of non-Newtonian fluids. - Q. Appl. Math., vol.l4, pp.318-321.
• [14] Goddard J.D. (1991): Granular dilatancy and the plasticity of glassy lubricants. - Ind. Eng. Chem. Res., vol.38, pp.820-822.
• [15] Goldhirsch I. and Sela N. (1996): Origin of normal stress differences in rapid granular flows. - Phys. Rev. E, vol.54, pp.445 8-4461.
• [16] Gray J.M.N.T. (2001): Granular flow in partially filled slowly rotating drums. - J. Fluid Mech., vol.441, pp.1-29.
• [17] Gudhe R., Yalamanchili R.C. and Massoudi M. (1994a): Flow of granular materials down a vertical pipe. - Int. J. Non- Linear Mech., vol.29, pp. 1-12.
• [18] Gudhe R., Rajagopal K.R. and Massoudi M. (1994b): Fully developed flow of granular materials down a heated inclined plane. - Acta Mechanica, vol.l03, pp.63-78.
• [19] Hanes D.M. and Inman D.L. (1985): Observations of rapidly flowing granular fluid materials. - J. Fluid Mech., vol.l50, pp.357-380.
• [20] Hermann H.J. and Luding S. (1998): Modeling of granular media on the Computer. - Continuum Mech. Thermodyn., vol.l0, pp.189-231.
• [21] Hutter K. and Rajagopal K.R. (1994): On flows of granular materials. - Continuum Mech. Thermodyn., vol.6, pp.81-139.
• [22] Jin S. and Slemrod M. (2001): Regularization of the Burnett equations for rapid granular flows via relaxation. - Physica D, vol.l50, pp.207-218.
• [23] Johnson G., Massoudi M. and Rajagopal K.R. (1991): Flow of fluid-solid mixture between fiat plates. - Chem. Engng. Sci., vol.46, pp.1713-1723.
• [24] Karion A. and Hunt M.L. (2000): Wall stresses in granular Couette flows of mono-sized particles and binary mixtures. - Powder Tech., vol.l09, pp. 145-163.
• [25] Knight P.C., Seville J.P.K., Wellm A.B. and Instone T. (2001): Prediction of impeller torque in high shear powder mixers. - Chem. Engng. Sci., vol.56, pp.4457-4471.
• [26] Kumar J., Lakshmana Rao C. and Massoudi M. (2003): Couette flow of granular materials. - Int. J. of Non-linear Mechanics, vol.38, pp. 11-20.
• [27] Larson R.G. (1999): The Structure and Rheology of Complex Fluids. - New York: Oxford University Press.
• [28] Markovitz H. (1965): Normal stress measurements on polymer Solutions, In: Proc. the Fourth Int. Cong. Rheol. (A.L. Copley and E.H. Lee, Eds.). - NY: Interscience, Part 1, pp. 189-211.
• [29] Markovitz H. (1967): Nonlinear steady-flow behavior, In: Rheology: Theory and Applications (F.R. Eirich, Ed.). - New York: Academic Press, vol.4, pp.347-410.
• [30] Massoudi M. (2001): On the flows of granular materials with variable material properties. - Int. J. Non-Linear Mech., vol.36, pp.25-37.
• [31] Massoudi M. and Boyle EJ. (2001): A continuum-kinetic theory approach to the flow of granular materials: The effects of volume fraction gradient. - Int. J. Non-Linear Mech., vol.36, pp.637-648.
• [32] Massoudi M. and Mehrabadi M.M. (2001): A continuum model for granular materials: Considering dilatancy, and the Mohr-Coulomb criterion. - Acta Mech., vol.l52, pp. 121-138.
• [33] Massoudi M. and Phuoc T.X. (2000): The effect of slip boundary condition on the flow of granular materials: A continuum approach. - Int. J. Non-Linear Mech., vol.35, pp.745-761.
• [34] Maxwell B. and Chartoff R.P. (1965): Studies of a polymer melt in an orthogonal rheometer. - Trans. Soc. Rheol., vol.9, pp.41-52.
• [35] McCarthy J.J., Shinbrot T., Metcalfe G., Wolf J.E. and Ottino J.M. (1996): Mixing of granular materials in slowly rotated containers. - AIChE J., vol.42, pp.3351-3363.
• [36] McTigue D.F. (1982): A non-linear constitutive model for granular materials: applications to gravity flow. - J. Appl. Mech., vol. 49, pp.291-296.
• [37] Mehrabadi M.M., Loret B. and Nemat-Nasser S. (1993): Incremental constitutive relations for granular materials based on micromechanics. - Proc. Royal Soc. Lond., vol.A44, pp.433-463.
• [38] Mehta A. (Ed.) (1994): Granular Matter: An Interdisciplinary Approach. - New York: Springer-Verlag.
• [39] Metzner A.B., Houghton W.T., Sailor R.A. and White J.L. (1961): A method for the measurement of normal stresses in simple shearing flow. - Trans. Soc. Rheol., vol.5, pp.133-147.
• [40] Mollica F. and Rajagopal K.R. (1997): Secondary deformations due to axial shear of the annular region between two eccentrically placed cylinders. - J. Elasticity, vol.48, pp. 103-123.
• [41] Mollica F. and Rajagopal K.R. (1999): Secondary flows due to axial shearing of a third grade fluid between two eccentrically placed cylinders. - Int. J. Engng. Sci., vol.37, pp.411-429.
• [42] Ogden R.W. (1997): Non-Linear Elastic Deformation. - Mineola, NY: Dover Publications, Inc.
• [43] Passman S.L., Nunziato J.W., Bailey P.B. and Thomas Jr. J.P. (1980): Shearing flows of granular materials. - J. Engg. Mech. Div. ASCE, vol. 106, pp.773-783.
• [44] Pipkin A.C. and Tanner R.I. (1972): A survey of theory and experiment in viscometric flows of viscoelastic liquids. - Mechanics Today, vol.l, pp.262-321.
• [45] Polashenski W. and Chen J.C. (1997): Normal solid stress in fluidized beds. - Powder Tech., vol.90, pp. 13-23.
• [46] Pritchard W.G. (1971): Measurement of the viscometric functions for a fluid in steady shear flows. - Phil. Trans. Roy. S. London, vol.A270, pp.507-556.
• [47] Rajagopal K.R. and Massoudi M. (1990): A method for measuring material moduli for granular materials: flow in an orthogonal rheometer. - Topical Report U.S. Department of Energy, Pittsburgh, DOE/PETC/TR90/3.
• [48] Rajagopal K.R., Gupta G. and Yalamanchili R.C. (2000): A rheometer for measuring the properties of granular materials. - Particulate Sci. Tech., vol.l8, pp.39-55.
• [49] Rajchenbach J. (2000): Granular flows. - Adv. Phys., vol.49, pp.229-256.
• [50] Reiner M. (1945): A mathematical theory of dilatancy. - Amer. J. Math., vol.67, pp.350-362.
• [51] Reynolds O. (1886): Experiments showing dilatancy a property of granular material, possibly connected with gravitation. - Proc. R. Inst. Gr. Britain, vol.ll, pp.356-363.
• [52] Rovaglio M., Manca D. and Biardi G. (1998): Dynamic modeling of waste incineration plants with rotary kilns: Comparisons between experimental and simulation data. - Chem. Engng. Sci., vol.53, pp.2727-2742.
• [53] Savage S.B. (1979): Gravity flow of cohesion-less granular materials in chutes and channels. - J. Fluid Mech., vol.92, pp.53-96.
• [54] Savage S.B. and Sayed M. (1984): Stresses developed by dry cohesionless granular materials sheared in an annular shear celi. - J. Fluid Mech., vol.l42, pp.391-430.
• [55] Sela N. and Goldhirsch I. (1998): Hydrodynamic equations for rap id flows of smooth inelastic spheres, to Burnett order. - J. Fluid Mech., vol.361, pp.41-74.
• [56] Shahinpoor M. and Lin S.P. (1982): Rapid Couette flow of cohesionless granular materials. - Acta Mech., vol.42, pp.183-196.
• [57] Slemrod M. (2001): Constitutive relations for rapid granular flow of smooth spheres: Generalized rational approximation to the sum of the Chapman-Enskog expansion. - J. Diff. Eqn., vol.l69, pp.549-566.
• [58] Stewart R.L., Bridgwater J. and Parker D.J. (2001): Granular flow over a fiat -bladed stirrer. - Chem. Engng. Sci., vol.56, pp.4257-4271.
• [59] Tardos G.I. (1997): A fluid mechanistic approach to slow, frictional flow of powders. - Powder Technol., vol.92, p.61-74.
• [60] Truesdell C. and Noll W. (1992): The Non-Linear Field Theories of Mechanics. - New York: Springer-Verlag.
• [61] Truesdell C. (1974): The meaning of viscometry in fluid dynamics. Annual Rev. Fluid Mech., vol.6, pp.l 11-146.
• [62] Wieghardt K. (1975): Experiments in granular flow. - Annual Rev. Fluid Mech., vol.7, pp.89-114.
• [63] Zheng X.M. and Hill J.M. (1996): Boundary effects of Couette flow granular materials: Dynamical modeling. - Appl. Math. Modeling, vol.20, pp.82-92.