Hostname: page-component-5db58dd55d-l8wb7 Total loading time: 0 Render date: 2026-07-08T03:59:33.237Z Has data issue: false hasContentIssue false
  • English
  • Français

Onset of thermal convection in non-colloidal suspensions

Published online by Cambridge University Press:  06 April 2021

Changwoo Kang Open the ORCID record for Changwoo Kang [Opens in a new window] ,
Harunori N. Yoshikawa Open the ORCID record for Harunori N. Yoshikawa [Opens in a new window]  and
Parisa Mirbod Open the ORCID record for Parisa Mirbod [Opens in a new window]
Changwoo Kang
Affiliation:
Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, 842 W. Taylor Street, Chicago, IL 60607, USA Department of Mechanical Engineering, Jeonbuk National University, 567 Baekje-daero, Deokjin-gu, Jeonju-si, Jeollabuk-do 54896, Republic of Korea
Harunori N. Yoshikawa
Affiliation:
Institut de Physique de Nice, Université Côte d'Azur, CNRS UMR 7010, 06100 Nice, France
Parisa Mirbod*
Affiliation:
Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, 842 W. Taylor Street, Chicago, IL 60607, USA
*
†Email address for correspondence: pmirbod@uic.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

This study explores thermal convection in suspensions of neutrally buoyant, non-colloidal suspensions confined between horizontal plates. A constitutive diffusion equation is used to model the dynamics of the particles suspended in a viscous fluid and it is coupled with the flow equations. We employ a simple model that was proposed by Metzger, Rahli & Yin (J. Fluid Mech., vol. 724, 2013, pp. 527–552) for the effective thermal diffusivity of suspensions. This model considers the effect of shear-induced diffusion and gives the thermal diffusivity increasing linearly with the thermal Péclet number (Pe) and the particle volume fraction (ϕ). Both linear stability analysis and numerical simulation based on the mathematical models are performed for various bulk particle volume fractions $({\phi _b})$ ranging from 0 to 0.3. The critical Rayleigh number $(R{a_c})$ grows gradually by increasing ${\phi _b}$ from the critical value $(R{a_c} = 1708)$ for a pure Newtonian fluid, while the critical wavenumber $({k_c})$ remains constant at 3.12. The transition from the conduction state of suspensions is subcritical, whereas it is supercritical for the convection in a pure Newtonian fluid $({\phi _b} = 0)$. The heat transfer in moderately dense suspensions $({\phi _b} = 0.2\text{--}0.3)$ is significantly enhanced by convection rolls for small Rayleigh number (Ra) close to $R{a_c}$. We also found a power-law increase of the Nusselt number (Nu) with Ra, namely, $Nu\sim R{a^b}$ for relatively large values of Ra where the scaling exponent b decreases with ${\phi _b}$. Finally, it turns out that the shear-induced migration of particles can modify the heat transfer.

JFM classification

Instability: Nonlinear instability Complex Fluids: Suspensions

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 915 , 25 May 2021 , A128
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abu-Nada, E. 2011 Rayleigh–Bénard convection in nanofluids: Effect of temperature dependent properties. Intl J. Therm. Sci. 50, 1720–1730.CrossRefGoogle Scholar
Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503–537.CrossRefGoogle Scholar
Ahuja, A.S. 1975 a Augmentation of heat transport in laminar flow of polystyrene suspensions. I. Experiments and results. J. Appl. Phys. 46, 3408–3416.CrossRefGoogle Scholar
Ahuja, A.S. 1975 b Augmentation of heat transport in laminar flow of polystyrene suspensions. II. Analysis of the data. J. Appl. Phys. 46, 3417–3425.CrossRefGoogle Scholar
Allen, J.W., Kenward, M. & Dorfman, K.D. 2009 Coupled flow and reaction during natural convection PCR. Microfluid Nanofluid 6, 121–130.CrossRefGoogle Scholar
Ardekani, M.N., Abouali, O., Picano, F. & Brandt, L. 2018 Heat transfer in laminar Couette flow laden with rigid spherical particles. J. Fluid Mech. 834, 308–334.CrossRefGoogle Scholar
Benouared, O., Mamou, M. & Messaoudene, N.A. 2014 Numerical nonlinear analysis of subcritical Rayleigh–Bénard convection in a horizontal confined enclosure filled with non-Newtonian fluids. Phys. Fluids 26, 073101.CrossRefGoogle Scholar
Bodenschatz, E., Pesch, W. & Ahlers, G. 2000 Recent developments in Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 32, 709–778.CrossRefGoogle Scholar
Bosshard, C., Dehbi, A., Deville, M., Leriche, E. & Soldati, A. 2014 Large eddy simulation of particulate flow inside a differentially heated cavity. Nucl. Engng Des. 267, 154–163.CrossRefGoogle Scholar
Buyevich, Y.A. & Kapbasov, S.K. 1996 Fluid dynamics of coarse dispersions. In Advances in Engineering Fluid Mechanics: Multiphase Reactor and Polymerization System Hydrodynamics (ed. Cheremisinoff, N.P.), pp. 119–166. Elsevier.CrossRefGoogle Scholar
Chang, B.H., Mills, A.F. & Hernandez, E. 2008 Natural convection of microparticle suspensions in thin enclosures. Intl J. Heat Mass Transfer 51, 1332–1341.CrossRefGoogle Scholar
Chun, B., Pakr, J.S., Jung, H.W. & Won, Y.-Y. 2019 Shear-induce particle migration and segregation in non-Brownian bidisperse suspensions under planar Poiseuille flow. J. Rheol. 63, 437–453.CrossRefGoogle Scholar
Chung, Y.C. & Leal, L.G. 1982 An experimental study of the effective thermal conductivity of a sheared suspension of rigid spheres. Intl J. Multiphase Flow 8 (6), 605–625.CrossRefGoogle Scholar
Dbouk, T. 2018 Heat transfer and shear-induced migration in dense non-Brownian suspension flows: modelling and simulation. J. Fluid Mech. 840, 432–454.CrossRefGoogle Scholar
Dbouk, T. 2019 A new technology for CPU chip cooling by concentrated suspension flow of non-colloidal particles. Appl. Therm. Engng 146, 664–673.CrossRefGoogle Scholar
Dbouk, T. & Bahrani, S.A. 2021 Modeling of buoyancy-driven thermal convection in immersed granular beds. Intl J. Multiphase Flow 134, 103471.CrossRefGoogle Scholar
Dbouk, T., Lemaire, E., Lobry, L. & Moukalled, F. 2013 Shear-induced particle migration: Predictions from experimental evaluation of the particle stress tensor. J. Non-Newtonian Fluid 198, 78–95.CrossRefGoogle Scholar
Dehbi, A., Kalilainen, J., Lind, T. & Auvinen, A. 2017 A large eddy simulation of turbulent particle-laden flow inside a cubical differentially heated cavity. J. Aerosol Sci. 103, 67–82.CrossRefGoogle Scholar
Denn, M.M. & Morris, J.F. 2014 Rheology of non-Brownian suspensions. Annu. Rev. Chem. Biomol. Engng 5, 203–228.CrossRefGoogle ScholarPubMed
Drazin, P.G. & Reid, W. 2004 Hydrodynamic Stability. Cambridge University Press.CrossRefGoogle Scholar
Eckstein, E.C., Bailey, D.G. & Shapiro, A.H. 1977 Self-diffusion of particles in shear flow of a suspension. J. Fluid Mech. 79, 191–208.CrossRefGoogle Scholar
Fang, Z., Mammoli, A.A., Brady, J.F., Ingber, M.S., Mondy, L.A. & Graham, A.L. 2002 Flow-aligned tensor models for suspension flows. Intl J. Multiphase Flow 28, 137–166.CrossRefGoogle Scholar
Fisher, K.M. 1981 The effects of fluid flow on the solidification of industrial castings and ingots. Physico-Chem. Hydrodyn. 2, 311–326.Google Scholar
Flamant, G., Gauthier, D., Benoit, H., Sans, J.-L., Garcia, R., Boissière, B., Ansart, R. & Hemati, M. 2013 Dense suspension of solid particles as a new heat transfer fluid for concentrated solar thermal plants: On-sun proof of concept. Chem. Engng Sci. 102, 567–576.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27–56.CrossRefGoogle Scholar
Guckenheimer, J. & Holmes, P.J. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag.CrossRefGoogle Scholar
Hinch, E.J. 2003 Mixing, turbulence and chaos – an introduction. In Mixing: Chaos and Turbulence (ed. Chaté, H., Villermaux, E. & Chomas, J.-M.), NATO ASI Series, vol. 373. Kluwer Academic/Plenum Publishers.Google Scholar
Jenny, M., Plaut, E. & Briard, A. 2015 Numerical study of subcritical Rayleigh–Bénard convection rolls in strongly shear-thinning Carreau fluids. J. Non-Newtonian Fluid Mech. 219, 19–34.CrossRefGoogle Scholar
Kang, C., Meyer, A., Mutabazi, I. & Yoshikawa, H.N. 2017 Radial buoyancy effects on momentum and heat transfer in a circular Couette flow. Phys. Rev. Fluids 2, 053901.CrossRefGoogle Scholar
Kang, C., Meyer, A., Yoshikawa, H.N. & Mutabazi, I. 2019 a Numerical study of thermal convection induced by centrifugal buoyancy in a rotating cylindrical annulus. Phys. Rev. Fluids 4, 043501.CrossRefGoogle Scholar
Kang, C., Meyer, A., Yoshikawa, H.N. & Mutabazi, I. 2019 b Thermoelectric convection in a dielectric liquid inside a cylindrical annulus with a solid-body rotation. Phys. Rev. Fluids 4, 093502.CrossRefGoogle Scholar
Kang, C. & Mirbod, P. 2020 Shear-induced particle migration of semi-dilute and concentrated Brownian suspensions in both Poiseuille and circular Couette flow. Intl J. Multiphase Flow 126, 103239.CrossRefGoogle Scholar
Kang, C. & Yang, K.-S. 2011 Heat transfer characteristics of baffled channel flow. ASME J. Heat Transfer 133, 091901.CrossRefGoogle Scholar
Kang, C. & Yang, K.-S. 2012 Flow instability in baffled channel flow. Intl J. Heat Fluid Flow 38, 40–49.CrossRefGoogle Scholar
Kim, J., Kang, Y.T. & Choi, C.K. 2004 Analysis of convective instability and heat transfer characteristics of nanofluids. Phys. Fluids 16 (7), 2395–2401.CrossRefGoogle Scholar
Kim, J. & Moin, P. 1985 Application of a fractional-step method to incompressible Navier–Stokes equations. J. Comput. Phys. 59, 308–323.CrossRefGoogle Scholar
Krieger, I.M. 1972 Rheology of monodiperse lattices. Adv. Colloid Interface Sci. 3, 111–136.CrossRefGoogle Scholar
Krishnamurti, R. 1973 Some further studies on the transition to turbulent convection. J. Fluid Mech. 60, 285–303.CrossRefGoogle Scholar
Krishnan, M., Ugaz, V.M. & & Burns, M.A. 2002 PCR in a Rayleigh–Bénard convection cell. Science 298, 793.CrossRefGoogle Scholar
Landau, L.D. & Lifshitz, E.M. 1976 Mechanics, 3rd edn, vol. 1 (Course of Theoretical Physics). Butterworth-Heinemann.Google Scholar
Leal, L.G. 1973 On the effective conductivity of a dilute suspension of spherical drops in the limit of low particle Peclet number. Chem. Engng Commun. 1, 21–31.CrossRefGoogle Scholar
Leighton, D. & Acrivos, A. 1987 The shear-induced migration of particles in concentrated suspensions. J. Fluid Mech. 181, 415–439.CrossRefGoogle Scholar
Lyon, M.K. & Leal, L.G. 1998 a An experimental study of the motion of concentrated suspensions in two-dimensional channel flow. Part 1. Monodisperse system. J. Fluid Mech. 363, 25–56.CrossRefGoogle Scholar
Lyon, M.K. & Leal, L.G. 1998 b An experimental study of the motion of concentrated suspensions in two-dimensional channel flow. Part 2. Bidisperse system. J. Fluid Mech. 363, 57–77.CrossRefGoogle Scholar
Mackie, C. 2000 Convective stability of a particle-laden fluid system in the presence of solidification. Intl J. Heat Mass Transfer 43, 1617–1627.CrossRefGoogle Scholar
Metzger, B., Rahli, O. & Yin, X. 2013 Heat transfer across sheared suspensions: role of the shear-induced diffusion. J. Fluid Mech. 724, 527–552.CrossRefGoogle Scholar
Miller, R.M. & Morris, J.F. 2006 Normal stress-driven migration and axial development in pressure-driven flow of concentrated suspensions. J. Non-Newtonian Fluid 135, 149–165.CrossRefGoogle Scholar
Mirbod, P. 2016 Two-dimensional computational fluid dynamical investigation of particle migration in rotating eccentric cylinders using suspension balance model. Intl J. Multiphase Flow 80, 79–88.CrossRefGoogle Scholar
Morize, C., Herbert, E. & Sauret, A. 2017 Resuspension threshold of a granular bed by localized heating. Phys. Rev. E 96, 032903.CrossRefGoogle ScholarPubMed
Morris, J.F. & Boulay, F. 1999 Curvilinear flows of noncolloidal suspensions: The role of normal stresses. J. Rheol. 43, 1213–1237.CrossRefGoogle Scholar
Morris, J.F. & Brady, J.F. 1998 Pressure-driven flow of a suspension: buoyancy effects. Intl J. Multiphase Flow 24, 105–130.CrossRefGoogle Scholar
Mulligan, J.C., Colvin, D.P. & Bryant, Y.G. 1996 Microencapsulated phase-change material suspensions for heat transfer in spacecraft thermal systems. J. Spacecr. Rockets 33, 278–284.CrossRefGoogle Scholar
Nield, D.A. & Kuznetsov, A.V. 2010 The onset of convection in a horizontal nanofluid layer of finite depth. Eur. J. Mech. B/Fluids 29, 217–223.CrossRefGoogle Scholar
Nott, P.R. & Brady, J.F. 1994 Pressure-driven flow of suspensions: simulation and theory. J. Fluid Mech. 275, 157–199.CrossRefGoogle Scholar
Phillips, R.J., Armstrong, R.C. & Brown, R.A. 1992 A constitutive equation for concentrated suspensions that accounts for shear-induced particle migration. Phys. Fluids A 4, 30–40.CrossRefGoogle Scholar
Schlüter, A., Lortz, D. & Busse, F. 1965 On the stability of steady finite amplitude convection. J. Fluid Mech. 23, 129–144.CrossRefGoogle Scholar
Shin, S. & Lee, S.-H. 2000 Thermal conductivity of suspensions in shear flow fields. Int. J. Heat Mass Transfer 43, 4275–4284.CrossRefGoogle Scholar
Sierou, A. & Brady, J.F. 2004 Shear-induced self-diffusion in non-colloidal suspensions. J. Fluid Mech. 506, 285–314.CrossRefGoogle Scholar
Sohn, C.W. & Chen, M.M. 1981 Microconvective thermal conductivity in disperse two-phase mixtures as observed in a low velocity Couette flow experiment. J. Heat Transfer 103, 47–51.CrossRefGoogle Scholar
Subia, S.R., Ingber, M.S., Mondy, L.A., Altobelli, S.A. & Graham, A.L. 1998 Modelling of concentrated suspensions using a continuum constitutive equation. J. Fluid Mech 373, 193–219.CrossRefGoogle Scholar
Tuckerman, L.S. & Barkley, D. 1990 Bifurcation analysis of the Eckhaus instability. Physica D 46, 57–86.CrossRefGoogle Scholar
Yoshikawa, H.N., Fogaing, M.T., Crumeyrolle, O. & Mutabazi, I. 2013 Dielectrophoretic Rayleigh–Bénard convection under microgravity conditions. Phys. Rev. E 87, 043003.CrossRefGoogle ScholarPubMed