Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-17T17:30:01.187Z Has data issue: false hasContentIssue false
  • English
  • Français

Models for the two-phase flow of concentrated suspensions

Published online by Cambridge University Press:  04 June 2018

TOBIAS AHNERT ,
ANDREAS MÜNCH  and
BARBARA WAGNER
TOBIAS AHNERT
Affiliation:
Institute of Mathematics, Technische Universität Berlin, Strasse des 17. Juni 136, Berlin 10623, Germany emails: ahnert@math.tu-berlin.de, muench@maths.ox.ac.uk, bwagner@math.tu-berlin.de
ANDREAS MÜNCH
Affiliation:
Mathematical Institute, 24–29 St Giles, Oxford OX1 3LB, UK email: muench@maths.ox.ac.uk
BARBARA WAGNER
Affiliation:
Institute of Mathematics, Technische Universität Berlin, Strasse des 17. Juni 136, Berlin 10623, Germany emails: ahnert@math.tu-berlin.de, muench@maths.ox.ac.uk, bwagner@math.tu-berlin.de
Get access Rights & Permissions [Opens in a new window]

Abstract

A new two-phase model for concentrated suspensions is derived that incorporates a constitutive law combining the rheology for non-Brownian suspension and granular flow. The resulting model exhibits a yield-stress behaviour for the solid phase depending on the collision pressure. This property is investigated for the simple geometry of plane Poiseuille flow, where an unyielded or jammed zone of finite width arises in the centre of the channel. For the steady states of this problem, the governing equations are reduced to a boundary value problem for a system of ordinary differential equations and the conditions for existence of solutions with jammed regions are investigated using phase-space methods. For the general time-dependent case a new drift-flux model is derived using matched asymptotic expansions that takes into account the boundary layers at the walls and the interface between the yielded and unyielded region. The drift-flux model is used to numerically study the dynamic behaviour of the suspension flow, including the appearance and evolution of an unyielded or jammed regions.

Keywords

Suspensionsjammingyield stressaveragingmultiphase modelphase-space methodsmatched asymptoticsdrift-flux
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 3 , June 2019 , pp. 585 - 617
Copyright
Copyright © Cambridge University Press 2018 

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.)

Footnotes

AM is grateful for the support by KAUST (Award Number KUK-C1-013-04). TA and BW gratefully acknowledges the support by the Federal Ministry of Education (BMBF) and the state government of Berlin (SENBWF) in the framework of the program Spitzenforschung und Innovation in den Neuen Ländern (Grant Number 03IS2151).

References

[1] Ahmadpour, A. & Sadeghy, K. (2013) An exact solution for laminar, unidirectional flow of Houska thixotropic fluids in a circular pipe. J. Non-Newton. Fluid Mech. 194, 23–31.Google Scholar
[2] Ahnert, T. (2015) Mathematical Modeling of Concentrated Suspensions: Multiscale Analysis and Numerical Solutions. PhD Thesis, Technical University Berlin, November.Google Scholar
[3] Ahnert, T., Münch, A., Niethammer, B. & Wagner, B. (2018) Stability of concentrated suspensions under Couette and Poiseuille flow. J. Eng. Math. 1–27. https://doi.org/10.1007/s10665-018-9954-xGoogle Scholar
[4] Batchelor, G. K. & Green, J. T. (1972) The determination of the bulk stress in a suspension of spherical particles to order c 2. J. Fluid Mech. 56 (03), 401–427.Google Scholar
[5] Boyer, F., Guazzelli, É. & Pouliquen, O. (2011) Unifying suspension and granular rheology. Phys. Rev. Lett. 107 (18), 188301.Google Scholar
[6] Brennen, C. E. (2005) Fundamentals of Multiphase Flow, Cambridge University Press, New York.Google Scholar
[7] Brinkman, H. C. (1949) A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. 1 (1), 27–34.Google Scholar
[8] Cartellier, A., Andreotti, M. & Sechet, P. (2009) Induced agitation in homogeneous bubbly flows at moderate particle Reynolds number. Phys. Rev. E 80(6).Google Scholar
[9] Cassar, C., Nicolas, M. & Pouliquen, O. (2005) Submarine granular flows down inclined planes. Phys. Fluids 17 (10), 103301.Google Scholar
[10] W. Chow, A., Sinton, S. W., Iwamiya, J. H. & Stephens, T. S. (1994) Shear-induced particle migration in Couette and parallel-plate viscometers: NMR imaging and stress measurements. Phys. Fluids 6 (8), 2561–2576.Google Scholar
[11] Cook, B. P., Bertozzi, A. L. & Hosoi, A. E. (2008) Shock solutions for particle-laden thin films. SIAM J. Appl. Math. 68 (3), 760–783.Google Scholar
[12] de Bruyn, J. (2011) Unifying liquid and granular flow. Physics 4, 86.Google Scholar
[13] DeGiuli, E., Düring, G., Lerner, E. & Wyart, M. (2015) Unified theory of inertial granular flows and non-Brownian suspensions. Phys. Rev. E 91 (6), 062206.Google Scholar
[14] Drew, D. A. (1983) Mathematical modeling of two-phase flow. Annu. Rev. Fluid Mech. 15 (1), 261–291.Google Scholar
[15] Drew, D. A. (2001) A turbulent dispersion model for particles or bubbles. J. Eng. Math. 41 (2–3), 259–274.Google Scholar
[16] Drew, D. A. & Passman, S. L. (1999) Theory of Multicomponent Fluids, Applied Mathematical Sciences, Vol. 135, Springer, New York.Google Scholar
[17] Drew, D. A. & Segel, L. A. (1971) Averaged equations for two-phase media. Stud. Appl. Math. 50 (2), 205–231.Google Scholar
[18] Einstein, A. (1906) Eine neue Bestimmung der Moleküldimensionen. Ann. Phys. 324 (2), 289–306.Google Scholar
[19] Fox, R. O. (2014) On multiphase turbulence models for collisional fluid-particle flows. J. Fluid Mech. 742, 368–424.Google Scholar
[20] Gadalamaria, F. & Acrivos, A. (1980) Shear-induced structure in a concentrated suspension of solid spheres. J. Rheol. 24 (6), 799–814.Google Scholar
[21] Garrido, P., Concha, F. & Bürger, R. (2003) Settling velocities of particulate systems: 14. Unified model of sedimentation, centrifugation and filtration of flocculated suspensions. Int. J. Mineral Process. 72, 57–74.Google Scholar
[22] Hampton, R. E. (1997) Migration of particles undergoing pressure-driven flow in a circular conduit. J. Rheol. 41 (3), 621–640.Google Scholar
[23] Hermes, M., Guy, B. M., Poon, W. C. K., Poy, G., Cates, M. E. & Wyart, M. (2016) Unsteady flow and particle migration in dense, non-Brownian suspensions. J. Rheol. 60 (5), 905–916.Google Scholar
[24] Hormozi, S. & Frigaard, I. A. (2017) Dispersion of solids in fracturing flows of yield stress fluids. J. Fluid Mech. 830, 93–137.Google Scholar
[25] Isa, L., Besseling, R. & Poon, W. C. K. (2007) Shear zones and wall slip in the capillary flow of concentrated colloidal suspensions. Phys. Rev. Lett. 98, 198305.Google Scholar
[26] Ishii, M. & Hibiki, T. (2011) Thermo-Fluid Dynamics of Two-Phase Flow, Springer, New York.Google Scholar
[27] James, N., Han, E., Jureller, J. & Jaeger, H. (2017) Interparticle hydrogen bonding can elicit shear jamming in dense suspensions. arXiv:1707.09401v1 [cond-mat].Google Scholar
[28] Jenkins, J. T. & McTigue, D. F. (1990) Transport processes in concentrated suspensions: The role of particle fluctuations. In: Joseph, D. D. and Schaeffer, D. G. (editors), Two Phase Flows and Waves, The IMA Volumes in Mathematics and Its Applications, Vol. 26, Springer, New York, pp. 70–79.Google Scholar
[29] Keyfitz, B. L., Sanders, R. & Sever, M. (2003) Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow. Discrete Continuous Dyn. Syst. – Series B 3 (4), 541–563.Google Scholar
[30] Kolev, N. I. (2005) Multiphase Flow Dynamics 1: Fundamentals, Multiphase Flow Dynamics, Springer-Verlag, Berlin Heidelberg.Google Scholar
[31] Leighton, D. & Acrivos, A. (1987) Shear-induced migration of particles in concentrated suspensions. J. Fluid Mech. 181 (1), 415–439.Google Scholar
[32] Lhuillier, D., Chang, C.-H. & Theofanous, T. G. (2013) On the quest for a hyperbolic effective-field model of disperse flows. J. Fluid Mech. 731, 184–194.Google Scholar
[33] Miller, R. M. & Morris, J. F. (2006) Normal stress-driven migration and axial development in pressure-driven flow of concentrated suspensions. J. Non-Newton. Fluid Mech. 135 (2–3), 149–165.Google Scholar
[34] Miller, R. M., Singh, J. P. & Morris, J. F. (2009) Suspension flow modeling for general geometries. Chem. Eng. Sci. 64 (22), 4597–4610.Google Scholar
[35] Morris, J. F. & Boulay, F. (1999) Curvilinear flows of noncolloidal suspensions: The role of normal stresses. J. Rheol. 43, 1213–1237.Google Scholar
[36] Murisic, N., Pausader, B., D. Peschka & Bertozzi, A. L. (2013) Dynamics of particle settling and resuspension in viscous liquid films. J. Fluid Mech. 717, 203–231.Google Scholar
[37] Nott, P. R. & Brady, J. F. (1994) Pressure-driven flow of suspensions: simulation and theory. J. Fluid Mech. 275 (1), 157–199.Google Scholar
[38] Oh, S., Song, Y.-Q., Garagash, D. I., Lecampion, B. & Desroches, J. (2015) Pressure-driven suspension flow near jamming. Phys. Rev. Lett. 114(8).Google Scholar
[39] Phillips, R. J., Armstrong, R. C., Brown, R. A., Graham, A. L. & Abbott, J. R. (1992) A constitutive equation for concentrated suspensions that accounts for shear-induced particle migration. Phys. Fluids A 4 (1), 30–40.Google Scholar
[40] Prosperetti, A. & Jones, A. (1987) The linear stability of general two-phase flow models – II. Int. J. Multiphase Flow 13 (2), 161–171.Google Scholar
[41] Quemada, D. (1997) Rheological modelling of complex fluids. I. The concept of effective volume fraction revisited. Eur. Phys. J. Appl. Phys. 1, 119–127.Google Scholar
[42] Quemada, D. (1998) Rheological modeling of complex fluids: III. Dilatant behavior of stabilized suspensions. Eur. Phys. J. Appl. Phys. 3, 309–320.Google Scholar
[43] Quemada, D. (1998) Rheological modelling of complex fluids: II. Shear thickening behavior due to shear induced flocculation. Eur. Phys. J. Appl. 2, 175–181.Google Scholar
[44] Quemada, D. (1999) Rheological modelling of complex fluids: IV: Thixotropic and “thixoelastic” behaviour. Start-up and stress relaxation, creep tests and hysteresis cycles. Eur. Phys. J. Appl. Phys. 5 (2), 191–207.Google Scholar
[45] Ramachandran, A. (2013) A macrotransport equation for the particle distribution in the flow of a concentrated, non-colloidal suspension through a circular tube. J. Fluid Mech. 734, 219–252.Google Scholar
[46] Snook, B., Butler, J. E. & Guazzelli, É. (2016) Dynamics of shear-induced migration of spherical particles in oscillatory pipe flow. J. Fluid Mech. 786, 128–153.Google Scholar
[47] Stickel, J. J. & Powell, R. L. (2005) Fluid mechanics and rheology of dense suspensions. Annu. Rev. Fluid Mech. 37 (1), 129–149.Google Scholar
[48] Trulsson, M., Andreotti, B. & Claudin, P. (2012) Transition from the viscous to inertial regime in dense suspensions. Phys. Rev. Lett. 109 (11), 118305.Google Scholar
[49] Whitaker, S. (1986) Flow in porous media I: A theoretical derivation of Darcy's law. Trans. Porous Media 1 (1), 3–25.Google Scholar
[50] Whitaker, S. (1998) The Method of Volume Averaging, Vol. 13, Springer, Netherlands, in: “Theory and Applications of Transport in Porous Media”.Google Scholar
[51] Wyart, M. & Cates, M. E. (2014) Discontinuous shear thickening without inertia in dense non-brownian suspensions. Phys. Rev. Lett. 112, 098302.Google Scholar
[52] Wylie, J. J., Koch, D. L. & Ladd, A. J. C. (2003) Rheology of suspensions with high particle inertia and moderate fluid inertia. J. Fluid Mech. 480, 95–118.Google Scholar
[53] Yapici, K., Powell, R. L. & Phillips, R. J. (2009) Particle migration and suspension structure in steady and oscillatory plane Poiseuille flow. Phys. Fluids 21 (5), 053302.Google Scholar
[54] Zhou, J., Dupuy, B., Bertozzi, A. & Hosoi, A. (2005) Theory for shock dynamics in particle-laden thin films. Phys. Rev. Lett. 94 (11), 117803.Google Scholar