Abstract
We study pressure-driven suspensions of non-colloidal, non-Brownian, and rigid spheres in a Newtonian solvent where the pipe surface is replaced by porous media using numerical simulations. We examine various values of the permeability of the porous medium K, while we keep the porosity and the thickness of the porous layer constant to clarify the effect of the permeable wall on the suspension flows at bulk particle volume fractions 0.1 ≤ ϕb ≤ 0.5. In the limit of vanishing inertia, the rate of suspension flow decreases as the bulk volume fraction ϕb increases and it builds up as the permeability of the porous media increases. There are also two different regimes characterizing the dimensionless slip velocity normalized by both shear rate and penetration depth, namely, the strong permeability regime and the weak permeability regime. In the former, the solvent penetrates deeper and the streamwise velocity at the interface increases with the porous media permeability, while in the latter, the fluid cannot go through the porous media deeply and the variation of the slip velocity with the permeability is small. Our results might suggest a new passive technique to reduce drag by enhancing the rate of suspension flow in devices where the suspension transport is crucial. It might also offer basic insights for the extension to the flow of suspensions over and through complex porous media.
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References
Agelinchaab M, Tachie MF, Ruth DW (2006) Velocity measurement of flow through a model three-dimensional porous medium. Phys Fluids 18:017105. https://doi.org/10.1063/1.2164847
Ahmed GMY, Singh A (2011) Numerical simulation of particle migration in asymmetric bifurcation channel. J Non-Newton Fluid 166:42–51. https://doi.org/10.1016/j.jnnfm.2010.10.004
Altobelli SA, Givler RC, Fukushima E (1991) Velocity and concentration measurements of suspensions by nuclear magnetic resonance imaging. J Rheol 35:721–734. https://doi.org/10.1122/1.550156
Arthur JK, Ruth DW, Tachie MF (2009) PIV measurements of flow through a model porous medium with varying boundary conditions. J Fluid Mech 629:343–374. https://doi.org/10.1017/S0022112009006405
Battiato I, Bandaru PR, Tartakovsky DM (2010) Elastic response of carbon nanotube forests to aerodynamic stresses. Phys Rev Lett 105:144504. https://doi.org/10.1103/PhysRevLett.105.144504
Beavers GS, Joseph DD (1967) Boundary conditions at a naturally permeable wall. J Fluid Mech 30:197–207. https://doi.org/10.1017/S0022112067001375
Brinkman HC (1949) A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. App Sci Res 1:27. https://doi.org/10.1007/BF02120313
Chun B, Park JS, Jung HW, Won YY (2019) Shear-induce particle migration and segregation in non-Brownian bidisperse suspensions under planar Poiseuille flow. J Rheol 63:437–453. https://doi.org/10.1122/1.5065406
Darcy HPG (1856) Les Fontaines Publique de la Ville de Dijon. Victor Dalmont, Paris
Dbouk T, Lemaire E, Lobry L, Moukalled F (2013) Shear-induced particle migration: Predictions from experimental evaluation of the particle stress tensor. J Non-Newton Fluid 198:78–95. https://doi.org/10.1016/j.jnnfm.2013.03.006
Deng M, Li X, Liang H, Caswell B (2012) Simulation and modelling of slip flow over surfaces grafted with polymer brushes and glycocalyx fibres. J Fluid Mech 711:192–211. https://doi.org/10.1017/jfm.2012.387
Fang Z, Mammoli AA, Brady JF, Ingber MS, Mondy LA, Graham AL (2002) Flow-aligned tensor models for suspension flows. Int J Multiphase Flow 28:137–166. https://doi.org/10.1016/S0301-9322(01)00055-6
Gadala-Maria F, Acrivos A (1980) Shear-induced structure in a concentrated suspension of solid spheres. J Rheol 24:799–814. https://doi.org/10.1122/1.549584
Ghisalberti M, Nepf H (2009) Shallow flows over permeable medium: the hydrodynamics of submerged aquatic canopies. Trans Porous Med 78:309–326. https://doi.org/10.1007/s11242-008-9305-x
Goharzadeh A, Khalili A, Jørgensen BB (2005) Transition layer thickness at a fluid-porous interface. Phys Fluids 17:057102. https://doi.org/10.1063/1.1894796
Goto H, Kuno H (1982) Flow of suspensions containing particles of two different sizes through a capillary tube. J Rheol 26:387–398. https://doi.org/10.1122/1.549682
Goyeau B, Lhuillier D, Gobin D, Velarde MG (2003) Momentum transport at a fluid-porous interface. Int J Heat Mass Transfer 46:4071–4081. https://doi.org/10.1122/1.549682
Guo P, Weinstein AM, Weinbaum S (2000) A hydrodynamic mechanosensory hypothesis for brush border microvilli. Am J Physiol Renal Physiol 279:698–712. https://doi.org/10.1152/ajprenal.2000.279.4.F698
Haffner EA, Mirbod P (2020) Velocity measurements of a dilute particulate suspension over and through a porous medium model. Phys Fluids 32:083608. https://doi.org/10.1063/5.0015207
Hampton RE, Mammoli AA, Graham AL, Tetlow N (1997) Migration of particles undergoing pressure-driven flow in a circular conduit. J Rheol 41:621–640. https://doi.org/10.1122/1.550863
Han M, Kim C, Kim M, Lee S (1997) Particle migration in tube flow of suspensions. J Rheol 43:1157–1174. https://doi.org/10.1122/1.551019
Jenkins JT, McTigue DF (1990) Transport processes in concentrated suspensions: the role of particle fluctuations. In: Schaeffer DG (ed) Joseph DD. Springer, Two Phase Flows and Waves, pp 70–79. https://doi.org/10.1007/978-1-4613-9022-0_5
Kang C, Mirbod P (2019) Porosity effects in laminar fluid flow near permeable surfaces. Phys Rev E 100:013109. https://doi.org/10.1103/PhysRevE.100.013109
Kang C, Mirbod P (2020) Shear-induced particle migration of semi-dilute and concentrated Brownian suspensions in both Poiseuille and circular Couette flow. Int J Multiphase Flow 126:103239. https://doi.org/10.1016/j.ijmultiphaseflow.2020.103239
Kang C, Yang KS (2011) Heat transfer characteristics of baffled channel flow. ASME J Heat Transfer 133:091901. https://doi.org/10.1115/1.4003829
Kang C, Yang KS (2012) Flow instability in baffled channel flow. Int J Heat Fluid Flow 38:40–49. https://doi.org/10.1016/j.ijheatfluidflow.2012.08.002
Karnis A, Goldsmith HL, Mason SG (1966) The kinetics of flowing dispersions: I. Concentrated suspensions of rigid particles. J Colloid Interface Sci 22:531–553. https://doi.org/10.1016/0021-9797(66)90048-8
Kim J, Moin P (1985) Application of a fractional-step method to incompressible Navier-Stokes equations. J Comp Phys 59:308–323. https://doi.org/10.1016/0021-9991(85)90148-2
Koh CJ, Hookham P, Leal LG (1994) An experimental investigation of concentrated suspension flows in a rectangular channel. J Fluid Mech 266:1–32. https://doi.org/10.1017/S0022112094000911
Krieger IM (1972) Rheology of monodisperse lattices. Adv Colloid Interf Sci 3:111–136. https://doi.org/10.1016/0001-8686(72)80001-0
Kruijt B, Malhi Y, Lloyd J, Norbre AD, Miranda AC, Pereira MGP, Culf A, Grace J (2000) Turbulence statistics above and within two Amazon rain forest canopies. Bound-lay Meteorol 94:297–331. https://doi.org/10.1023/A:1002401829007
Leighton D, Acrivos A (1987) The shear-induced migration of particles in concentrated suspensions. J Fluid Mech 181:415–439. https://doi.org/10.1017/S0022112087002155
Lyon MK, Leal LG (1998a) An experimental study of the motion of concentrated suspensions in two-dimensional channel flow. Part 1. Monodisperse system. J Fluid Mech 363:25–56. https://doi.org/10.1017/S0022112098008817
Lyon MK, Leal LG (1998b) An experimental study of the motion of concentrated suspensions in two-dimensional channel flow. Part 2. Bidisperse system. J Fluid Mech 363:57–77. https://doi.org/10.1017/S0022112098008829
Miller RM, Morris JF (2006) Normal stress-driven migration and axial development in pressure-driven flow of concentrated suspensions. J Non-Newton Fluid 135:149–165. https://doi.org/10.1016/j.jnnfm.2005.11.009
Mirbod P (2016) Two-dimensional computational fluid dynamical investigation of particle migration in rotating eccentric cylinders using suspension balance model. Int J Multiphase Flow 80:79–88. https://doi.org/10.1016/j.ijmultiphaseflow.2015.11.002
Mirbod P, Wu Z, Ahmadi G (2017) Laminar flow drag reduction on soft porous media. Sci Rep 7:17263. https://doi.org/10.1038/s41598-017-17141-3
Morris JF, Boulay F (1999) Curvilinear flows of noncolloidal suspensions: the role of normal stresses. J Rheol 43:1213–1237. https://doi.org/10.1122/1.551021
Morris JF, Brady JF (1998) Pressure-driven flow of a suspension: buoyancy effects. Int J Multiphase Flow 24:105–130. https://doi.org/10.1016/S0301-9322(97)00035-9
Nott PR, Brady JF (1994) Pressure-driven flow of suspensions: simulation and theory. J Fluid Mech 275:157–199. https://doi.org/10.1017/S0022112094002326
Phillips RJ, Armstrong RC, Brown RA (1992) A constitutive equation for concentrated suspensions that accounts for shear-induced particle migration. Phys Fluids A 4:30–40. https://doi.org/10.1063/1.858498
Rosti ME, Mirbod P, Brandt L (2021) The impact of porous walls on the rheology of suspensions. Chem Eng Sci 230:116178. https://doi.org/10.1016/j.ces.2020.116178
Sinton SW, Chow AW (1991) NMR flow imaging of fluids and solid suspensions in Poiseuille flow. J Rheol 35:735–772. https://doi.org/10.1122/1.550253
Subia SR, Ingber MS, Mondy LA, Altobelli SA, Graham AL (1998) Modelling of concentrated suspensions using a continuum constitutive equation. J Fluid Mech 373:193–219. https://doi.org/10.1017/S0022112098002651
Tachie MF, James DF, Currie IG (2004) Slow flow through a brush. Phys Fluids 16:445–451. https://doi.org/10.1063/1.1637351
Vafai K, Kim SJ (1990) Fluid mechanics of the interface region between a porous medium and a fluid layer-an exact solution. Int J Heat Fluid Flow 11:254–256. https://doi.org/10.1016/0142-727X(90)90045-D
Wu Z, Mirbod P (2018) Experimental analysis of the flow near the boundary of random porous media. Phys. Fluids 30:047103. https://doi.org/10.1063/1.5021903
Wu Z, Mirbod P (2019) Instability analysis of the flow between two parallel where the bottom one coated with porous media. Adv Water Resour 130:221–228. https://doi.org/10.1016/j.advwatres.2019.06.002
Funding
This work has been partially supported by the National Science Foundation award No. 1854376 and partially by Army Research Office award No. W911NF-18-1-0356, and the University of Illinois at Chicago.
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Kang, C., Mirbod, P. Pressure-driven pipe flow of semi-dilute and dense suspensions over permeable surfaces. Rheol Acta 60, 711–718 (2021). https://doi.org/10.1007/s00397-021-01298-w
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DOI: https://doi.org/10.1007/s00397-021-01298-w