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On the quest for a hyperbolic effective-field model of disperse flows

Published online by Cambridge University Press:  14 August 2013

Daniel Lhuillier ,
Chih-Hao Chang  and
Theo G. Theofanous
Daniel Lhuillier
Affiliation:
Institut Jean le Rond d’Alembert, UPMC Université Paris 6 and CNRS (UMR 7190), F-75005 Paris, France
Chih-Hao Chang
Affiliation:
Chemical Engineering Department, University of California, Santa Barbara, CA 93106, USA
Theo G. Theofanous*
Affiliation:
Chemical Engineering Department, University of California, Santa Barbara, CA 93106, USA
*
†Email address for correspondence: [email protected]
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Abstract

The cornerstone of multiphase flow applications in engineering practice is a scientific construct that translates the basic laws of fluid mechanics into a set of governing equations for effective interpenetrating continua, the effective-field (or two-fluid) model. Over more than half a century of development this model has taken many forms but all of them fail in a way that was known from the very beginning: mathematical ill-posedness. The aim of this paper is to refocus awareness of this problem from a unified fundamental perspective that clarifies the manner in which such failures took place and to suggest the means for a final closure.

JFM classification

Multiphase and Particle-laden Flows: Multiphase and Particle-laden Flows Multiphase and Particle-laden Flows: Multiphase flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 731 , 25 September 2013 , pp. 184 - 194
Copyright
©2013 Cambridge University Press 

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References

Baer, M. R. & Nunziato, J. W. 1986 A two-phase mixture theory for the deflagration-to-detonation transition (ddt) in reactive granular materials. Intl J. Multiphase Flow 12, 861–889.Google Scholar
Bdzil, J. B., Menikoff, R., Son, S. F., Kapila, A. K. & Stewart, D. S. 1999 Two-phase modeling of deflagration to detonation transition in granular materials: a critical examination of modeling issues. Phys. Fluids 11, 378–402.CrossRefGoogle Scholar
Biesheuvel, A. & Spoelstra, S. 1989 The added mass coefficient of a dispersion of spherical gas bubbles in liquid. Intl J. Multiphase Flow 15, 911–924.Google Scholar
Buyevich, Yu. A. & Shchelchkova, I. N. 1978 Flow of dense suspensions. Prog. Aerosp. Sci. 18, 121–150.Google Scholar
Chang, C.-H., Sushchikh, S., Mitkin, V. & Theofanous, T. G. 2011 Shock-induced fluidization. In 20th AIAA Computational Fluid Dynamics Conference. Honolulu, Hawaii.Google Scholar
Chang, C.-H., Sushchikh, S., Nguyen, L., Liou, M.-S. & Theofanous, T. G. 2007 Hyperbolicity, discontinuities, and numerics of the two-fluid model. In 5th ASME/JSME Fluids Engineering Summer Conference, 10th International Symposium on Gas-Liquid Two-phase Flows. San Diego.Google Scholar
Drew, D. A. & Passman, S. L. 1998 Theory of Multicomponents Fluids. Springer.Google Scholar
Fitt, A. D. 1993 The character of two-phase gas/particulate flow equations. Appl. Math. Model. 17, 338–354.Google Scholar
Gallouët, T., Hérard, J. M. & Seguin, N. 2004 Numerical modelling of two-phase flows using the two-fluid two-pressure approach. Math. Models Meth. Appl. Sci. 14, 663–700.CrossRefGoogle Scholar
Geurst, J. A. 1986 Variational principles and two-fluid hydrodynamics of bubbly liquid/gas mixtures. Physica A 135, 455–486.Google Scholar
Hank, S., Saurel, R. & Metayer, O. Le. 2011 A hyperbolic Eulerian model for dilute two-phase suspensions. J. Mod. Phys. 2, 997–1011.Google Scholar
Hérard, J. M. 2003 Numerical modelling of turbulent two-phase flow using the two-fluid approach. AIAA Paper 2003-4113.Google Scholar
Jackson, R. 1997 Locally averaged equations of motion for a mixture of identical spherical particles and a Newtonian fluid. Chem. Engng Sci. 52, 2457–2469.Google Scholar
Jones, A. V. & Prosperetti, A. 1985 On the suitability of first-order differential models for two-phase flow prediction. Intl J. Multiphase Flow 11, 133–148.Google Scholar
Khang, S.-Y., Sangani, A. S., Tsao, H.-K. & Khoch, D. L. 1997 Rheology of dense bubble suspensions. Phys. Fluids 9, 1540–1561.Google Scholar
Lahey, R. T., Cheng, L. Y., Drew, D. A. & Flaherty, J. E. 1980 The effect of virtual mass on the numerical stability of accelerating two-phase flows. Intl J. Multiphase Flow 6, 281–294.Google Scholar
Lhuillier, D. 1985 Phenomenology of inertia effects in a dispersed solid-fluid mixture. Intl J. Multiphase Flow 11, 427–444.Google Scholar
Lhuillier, D. & Theofanous, T. 2010 Multiphase flows: compressible multi-hydrodynamics part i: effective field formulation of multiphase flows. In Handbook of Nuclear Engineering (ed. Cacuci, D. G.), pp. 1813–1857. Springer.Google Scholar
Ndjinga, M. 2007 Influence of interfacial pressure on the hyperbolicity of the two-fluid model. C. R. Acad. Sci. Paris 1 344, 407–412.Google Scholar
Nigmatulin, R. I. 1979 Spatial averaging in the mechanics of heterogeneous and dispersed systems. Intl J. Multiphase Flow 5, 353–385.Google Scholar
Pauchon, C. & Smereka, P. 1992 Momentum interactions in dispersed flow: an averaging and a variational approach. Intl J. Multiphase flow 18, 65–87.Google Scholar
Rakhmatulin, Kh. A. 1956 Fundamentals of gas-dynamics of interpenetrating flows of compressible media (in russian). Prikl. Mat. Mekh. 20, 184–195.Google Scholar
Saurel, R. & Abgrall, R. 1999 A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150, 425–467.CrossRefGoogle Scholar
Saurel, R., Gavrilyuk, S. & Renaud, F. 2003 A multiphase model with internal degrees of freedom: application to shock-bubble interaction. J. Fluid Mech. 495, 283–321.Google Scholar
Smereka, P. & Milton, G. W. 1991 Bubbly flow and its relation to conduction in composites. J. Fluid Mech. 233, 65–81.CrossRefGoogle Scholar
Spelt, P. D. M. & Sangani, A. S. 1998 Properties and averaged equations for flows of bubbly liquids. Appl. Sci. Res. 58, 337–386.Google Scholar
Stewart, H. B. & Wendroff, B. 1984 Two-phase flow: models and methods. J. Comput. Phys. 56, 363–409.CrossRefGoogle Scholar
Stuhmiller, J. H. 1977 The influence of interfacial pressure forces on the character of two-phase flow model equations. Intl J. Multiphase Flow 3, 551–560.CrossRefGoogle Scholar
Voinov, O. V. & Petrov, A. G. 1977 On the stress tensor in a fluid containing disperse particles. Prikl. Mat. Mekh. 41, 368–369.Google Scholar
Wagner, J. L., Kearney, S. P., Beresh, S. J., Pruett, B. O., Wright, E. K. & Stoker, G. C. 2012 Flash x-ray measurement of shock wave interactions with dense particles fields. In 50th AIAA Aerospace Sciences Meeting. Nashville, Tennessee.Google Scholar
Wallis, G. B. 1991 The averaged bernoulli equation and the macroscopic equations of motion for the potential flow of a two-phase dispersion. Intl J. Multiphase Flow 17, 683–695.CrossRefGoogle Scholar
van Wijngaarden, L. 1976 Hydrodynamic interactions between gas bubbles in liquid. J. Fluid Mech. 77, 27–44.Google Scholar
Yurkovetsky, Y. & Brady, J. F. 1996 Statistical mechanics of bubbly liquids. Phys. Fluids 8, 881–895.Google Scholar
Zhang, D. Z. & Prosperetti, A. 1994 Averaged equations for inviscid dispersed two-phase flow. J. Fluid Mech. 267, 185–219.Google Scholar
Zuber, N. 1964 On the dispersed two-phase flow in the laminar flow regime. Chem. Engng Sci. 19, 897–917.Google Scholar