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Flow instability and transitions in Taylor–Couette flow of a semidilute non-colloidal suspension

Published online by Cambridge University Press:  06 April 2021

Changwoo Kang Open the ORCID record for Changwoo Kang [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, IL60607, USA Department of Mechanical Engineering, Jeonbuk National University, 567 Baekje-daero, Deokjin-gu, Jeonju-si, Jeollabuk-do54896, Republic of Korea
Parisa Mirbod*
Affiliation:
Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, 842 W. Taylor Street, Chicago, IL60607, USA
*
† Email address for correspondence: pmirbod@uic.edu
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Abstract

Flow of a semidilute neutrally buoyant and non-colloidal suspension is numerically studied in the Taylor–Couette geometry where the inner cylinder is rotating and the outer one is stationary. We consider a suspension with bulk particle volume fraction ${\phi _b} = 0.1$, the radius ratio $(\eta = {r_i}/{r_o} = 0.877)$ and two particle size ratios $\mathrm{\epsilon }\,( = \; d\textrm{/}a) = 60,\;200$, where d is the gap width ($= {r_o} - {r_i}$) between cylinders, a is the suspended particles’ radius and $r_i$ and $r_o$ are the inner and outer radii of the cylinder, respectively. Numerical simulations are conducted using the suspension balance model (SBM) and rheological constitutive laws. We predict the critical Reynolds number in which counter-rotating vortices arise in the annulus. It turns out that the primary instability appears through a supercritical bifurcation. For the suspension of $\mathrm{\epsilon } = 200$, the circular Couette flow (CCF) transitions via Taylor vortex flow (TVF) to wavy vortex flow (WVF). Additional flow states of non-axisymmetric vortices, namely spiral vortex flow (SVF) and wavy spiral vortex flow (WSVF) are observed between CCF and WVF for the suspension of $\mathrm{\epsilon } = 60$; thus, the transitions occur following the sequence of CCF → SVF → WSVF → WVF. Furthermore, we estimate the friction and torque coefficients of the suspension. Suspended particles substantially enhance the torque on the inner cylinder, and the axial travelling wave of spiral vortices reduces the friction and torque coefficients. However, the coefficients are practically the same in the WVF regime where particles are almost uniformly distributed in the annulus by the axial oscillating flow.

JFM classification

Flow Control: Instability control Complex Fluids: Suspensions
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 916 , 10 June 2021 , A12
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Ali, M.E., Mitra, D., Schwille, J.A. & Lueptow, R.M. 2002 Hydrodynamic stability of a suspension in cylindrical Couette flow. Phys. Fluids 14, 1236–1243.CrossRefGoogle Scholar
Andereck, C.D., Liu, S.S. & Swinney, H.L. 1986 Flow regimes in a circular Couette system with independently rotating cylinders. J. Fluid Mech. 164, 155–183.CrossRefGoogle Scholar
Asmolov, E.S. 1999 The inertial lift on a spherical particle in a plane Poiseuille flow at large channel Reynolds number. J. Fluid Mech. 381, 63–87.CrossRefGoogle Scholar
Baroudi, L., Majji, M.V. & Morris, J.F. 2020 Effect of inertial migration of particles on flow transitions of a suspension Taylor–Couette flow. Phys. Rev. Fluids 5, 114303.CrossRefGoogle Scholar
Boubnov, B.M., Gledzer, E.B. & Hopfinger, E.J. 1995 Stratified circular Couette flow: instability and flow regimes. J. Fluid Mech. 292, 333–358.CrossRefGoogle Scholar
Caton, F., Janiaud, B. & Hopfinger, E.J. 1999 Primary and secondary Hopf bifurcations in stratified Taylor–Couette flow. Phys. Rev. Lett. 82, 4647–4650.CrossRefGoogle Scholar
Chan, P.C.-H. & Leal, L.G. 1981 An experimental study of drop migration in shear flow between concentric cylinders. Intl J. Multiphase Flow 7, 83–99.CrossRefGoogle Scholar
Climent, E., Simonnet, M. & Magnaudet, J. 2007 Preferential accumulation of bubbles in Couette–Taylor flow patterns. Phys. Fluids 19, 083301.CrossRefGoogle Scholar
Coles, D. 1965 Transition in circular Couette flow. J. Fluid Mech. 21, 385–425.CrossRefGoogle Scholar
Cox, R.G. & Brenner, H. 1968 The lateral migration of solid particles in Poiseuille flow: I. Theory. Chem. Engng Sci. 23, 147–173.CrossRefGoogle Scholar
Dherbécourt, D., Charton, S., Lamadie, F., Cazin, S. & Climent, E. 2016 Experimental study of enhanced mixing induced by particles in Taylor–Couette flows. Chem. Engng Res. Des. 108, 109–117.CrossRefGoogle Scholar
Diprima, R.C., Eagles, P.M. & Ng, B.S. 1984 The effect of radius ratio on the stability of Couette flow and Taylor vortex flow. Phys. Fluids 27, 2403–2411.CrossRefGoogle Scholar
Drew, D.A. & Lahey, R.T. 1993 Analytical modeling of multiphase flow. In Particular Two-Phase Flow (ed. Roco, M.), pp. 509. Butterworths.Google Scholar
Eckhardt, B., Grossmann, S. & Lohse, D. 2007 Torque scaling in turbulent Taylor–Couette flow between independently rotating cylinders. J. Fluid Mech. 581, 221–250.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. Int. J. Multiphase Flow 28, 137–166.CrossRefGoogle Scholar
Gillissen, J.J.J. & Wilson, H.J. 2019 Taylor–Couette instability in sphere suspensions. Phys. Rev. Fluids 4, 043301.CrossRefGoogle Scholar
Grossmann, S., Lohse, D. & Sun, C. 2016 High-Reynolds number Taylor–Couette turbulence. Annu. Rev. Fluid Mech. 48, 53–80.CrossRefGoogle Scholar
Guillerm, R., Kang, C., Savaro, C., Lepiller, V., Prigent, A., Yang, K.-S. & Mutabazi, I. 2015 Flow regimes in a vertical Taylor–Couette system with a radial thermal gradient. Phys. Fluids 27, 094101.CrossRefGoogle Scholar
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer-Verlag.CrossRefGoogle Scholar
Ho, B.P. & Leal, L.G. 1974 Inertial migration of rigid spheres in two-dimensional unidirectional flows. J. Fluid Mech. 65, 365–400.CrossRefGoogle Scholar
Hogg, A.J. 1994 The inertial migration of non-neutrally buoyant spherical particles in two-dimensional shear flows. J. Fluid Mech. 272, 285–318.CrossRefGoogle Scholar
Hristova, H., Roch, S., Schmid, P. & Tuckerman, L.S. 2002 Transient growth in Taylor–Couette flow. Phys. Fluids 14, 3475–3484.CrossRefGoogle Scholar
Huisman, S.G., Van Gils, D.P.M., Grossmann, S., Sun, C. & Lohse, D. 2012 Ultimate turbulent Taylor–Couette flow. Phys. Rev. Lett. 108, 024501.CrossRefGoogle ScholarPubMed
Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 69–94.CrossRefGoogle Scholar
Kang, C., Meyer, A., Mutabazi, I. & Yoshikawa, H.N. 2017 a 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. 2017 b Numerical simulation of circular Couette flow under a radial thermo-electric body force. Phys. Fluids 29, 114105.CrossRefGoogle Scholar
Kang, C., Meyer, A., Yoshikawa, H.N. & Mutabazi, I. 2019 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. & Mutabazi, I. 2015 Thermal effect on large-aspect-ratio Couette–Taylor system: Numerical simulation. J. Fluid Mech. 771, 57–78.CrossRefGoogle Scholar
Kim, J. & Moin, P. 1985 Application of a fractional-step method to incompressible Navier-Stokes equations. J. Comp. Phys 59, 308–323.CrossRefGoogle Scholar
Krieger, I.M. 1972 Rheology of monodisperse lattices. Adv. Colloid Interface Sci. 3, 111–136.CrossRefGoogle Scholar
Landau, L.D. & Lifshitz, E.M. 1976 Mechanics, 3rd edn. Elsevier Butterworth-Heinemann.Google Scholar
Lathrop, D.P., Fineberg, J. & Swinney, H.S. 1992 a Transition to shear-driven turbulence in Couette–Taylor flow. Phys. Rev. A 46, 6390–6405.CrossRefGoogle ScholarPubMed
Lathrop, D.P., Fineberg, J. & Swinney, H.S. 1992 b Turbulent flow between concentric rotating cylinders at large Reynolds numbers. Phys. Rev. Lett. 68, 1515–1518.CrossRefGoogle Scholar
Leighton, D. & Acrivos, A. 1987 The shear-induced migration of particles in concentrated suspensions. J. Fluid Mech. 181, 415–439.CrossRefGoogle Scholar
Lim, T.T., Chew, Y.T. & Xiao, Q. 1998 A new flow regime in a Taylor–Couette flow. Phys. Fluids 10, 3233–3235.CrossRefGoogle Scholar
Majji, M.V., Banerjee, S. & Morris, J.F. 2018 Inertial flow transitions of a suspension in Taylor–Couette geometry. J. Fluid Mech. 835, 936–969.CrossRefGoogle Scholar
Majji, M.V. & Morris, J.F. 2018 Inertial migration of particles in Taylor–Couette flows. Phys. Fluids 30, 033303.CrossRefGoogle Scholar
Marques, F. & Lopez, J. 1997 Taylor–Couette flow with axial oscillations of the inner cylinder: Floquet analysis of the basic flow. J. Fluid Mech. 348, 153–175.CrossRefGoogle Scholar
Mclaughlin, J.B. 1993 The lift on a small sphere in wall-bounded linear shear flows. J. Fluid Mech. 226, 249–265.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
Morris, J.F. & Brady, J.F. 1998 Pressure-driven flow of a suspension: buoyancy effects. Int. J. Multiphase Flow 24, 105–130.CrossRefGoogle Scholar
Morris, J.F. & Boulay, F. 1999 Curvilinear flows of noncolloidal suspensions: the role of normal stresses. J. Rheol. 43, 1213–1237.CrossRefGoogle Scholar
Mullin, T., Cliffe, K.A. & Pfister, G. 1987 Unusual time-dependent phenomena in Taylor–Couette flow at moderately low Reynolds numbers. Phys. Rev. Lett. 58, 2212–2215.CrossRefGoogle ScholarPubMed
Nott, P.R. & Brady, J.F. 1994 Pressure-driven flow of suspensions: simulation and theory. J. Fluid Mech. 275, 157–199.CrossRefGoogle Scholar
Pfister, G. & Rehberg, I. 1981 Space dependent order parameter in circular Couette flow transitions. Phys. Lett. A 83, 19–22.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
Ramesh, P. & Alam, M. 2020 Interpenetrating spiral vortices and other coexisting states in suspension Taylor–Couette flow. Phys. Rev. Fluids 5, 042301(R).CrossRefGoogle Scholar
Ramesh, P., Bharadwaj, S. & Alam, M. 2019 Suspension Taylor–Couette flow: co-existence of stationary and travelling waves, and the characteristics of Taylor vortices and spirals. J. Fluid Mech. 870, 901–940.CrossRefGoogle Scholar
Resende, M.M., Tardioli, P.W., Fernandez, V.M., Ferreira, A.L.O., Giordano, R.L.C. & Giordano, R.C. 2001 Distribution of suspended particles in a Taylor–Poiseuille vortex flow reactor. Chem. Engng Sci. 56, 755–761.CrossRefGoogle Scholar
Richardson, J.F. & Zaki, W.N. 1954 Sedimentation and fluidization: part 1. Trans. Inst. Chem. Engrs 32, 35–53.Google Scholar
Rida, Z., Cazin, S., Lamadie, F., Dherbécourt, D., Charton, S. & Climent, E. 2019 Experimental investigation of mixing efficiency in particle-laden Taylor–Couette flows. Exp. Fluids 60, 61.CrossRefGoogle Scholar
Rudman, M. 2004 Mixing and particle dispersion in the wavy vortex regime of Taylor–Couette flow. AIChE J. 44, 1015–1026.CrossRefGoogle Scholar
Sierou, A. & Brady, J.F. 2004 Shear-induced self-diffusion in non-colloidal suspensions. J. Fluid Mech. 506, 285–314.CrossRefGoogle Scholar
Tetlow, N., Graham, A.L., Ingber, M.S., Subia, S.R., Mondy, L.A. & Altobelli, S.A. 1998 Particle migration in a Couette apparatus: Experiment and modeling. J. Rheol. 42, 307–327.CrossRefGoogle Scholar
Vasseur, P. & Cox, R.G. 1976 The lateral migration of a spherical particle in two-dimensional shear flows. J. Fluid Mech. 78, 385–413.CrossRefGoogle Scholar
Wereley, S.T. & Lueptow, R.M. 1999 Inertial particle motion in a Taylor–Couette rotating filter. Phys. Fluids 11, 325–333.CrossRefGoogle Scholar