The impact of porous walls on the rheology of suspensions

https://doi.org/10.1016/j.ces.2020.116178Get rights and content

Highlights

  • We study rigid particle suspensions in a Couette flow with porous walls.

  • The porous walls induce a decrease in the suspension effective viscosity.

  • The presence of porous walls weakens the wall blocking effect.

  • We provide a closed set of equations for the suspension viscosity.

Abstract

We study the effect of isotropic porous walls on a plane Couette flow laden with spherical and rigid particles. We perform a parametric study varying the volume fraction between 0 and 30%, the porosity between 0.3 and 0.9 and the non-dimensional permeability between 0 and 7.9×10-3 We find that the porous walls induce a progressive decrease in the suspension effective viscosity as the wall permeability increases. This behavior is explained by the weakening of the wall-blocking effect and by the appearance of a slip velocity at the interface of the porous medium, which reduces the shear rate in the channel. Therefore, particle rotation and the consequent velocity fluctuations in the two phases are dampened, leading to reduced particle interactions and particle stresses. Based on our numerical evidence, we provide a closed set of equations for the suspension viscosity, which can be used to estimate the suspension rheology in the presence of porous walls.

Introduction

One of the challenges faced in every aspect of a new technology is how to reduce energy loss and inefficiencies by manufacturing advanced and novel devices at low or no cost. When a suspension transport properties are critical, as of interest here, these devices include, but are not limited to, technologies such as extrusion (shallow screw channels) and thin lubricating films. In such systems, proper boundary conditions play a major role in controlling and driving the flow. In this study, we explore the flow of particle suspensions over porous surfaces in a plane Couette flow in order to evaluate the effect of these walls on the particle laden flow behavior. This understanding may contribute to improving the efficiency and operating lifetime of the abovementioned devices.

Particle-laden flows are encountered in various industrial applications, including blood flow, slurry transport, and pharmaceutical industry applications. Slow flow of non-Brownian suspensions has been analytically and experimentally examined in various geometries, the simplest probably being the Couette flow between impermeable walls (Leighton and Acrivos, 1987, Phillips et al., 1992, Acrivos et al., 1993, Nott and Brady, 1994, Morris and Boulay, 1999, Zarraga et al., 2000, Singh and Nott, 2003, Sierou and Brady, 2002, Miller and Morris, 2006, Yurkovetsky and Morris, 2008, Deboeuf et al., 2009, Miller et al., 2009, Yeo and Maxey, 2010, Guazzelli and Morris, 2011, Lashgari et al., 2014). On the other hand, Newtonian fluid flow past porous surfaces also has many important applications such as flow over sediment beds (Goharzadeh et al., 2005), over crop canopies and in forests (Kruijt et al., 2000, Ghisalberti and Nepf, 2009), in the human body (Guo et al., 2000) and over carbon nanotubes (Battiato et al., 2010). In particular, flow over porous walls is gaining increasing interest due to the possibility of passively controlling the flow and reducing drag in both laminar (Mirbod et al., 2017) and turbulent flows (Rosti et al., 2018a). However, the dynamics and rheological behavior of particles flowing over porous surfaces are qualitatively different from those observed over smooth surfaces due to modifications of the flow and of particle-induced fluid motions by the porous surface, as we will also document here.

Einstein (1956) was the first to show that, the effective viscosity of a dilute suspension (i.e., volume fraction Φ0) of rigid particles in a Newtonian fluid linearly increases with the particle volume fraction Φ, when inertia is negligible. Later on, Batchelor (1977) and Batchelor and Green (1972) extended Einstein’s study to higher volume fractions and added a second-order term in Φ. In general, there is no analytical relation able to predict the suspension viscosity at higher volume fractions, and empirical fits are instead used. Here, we will adopt the so-called Eilers fit (Ferrini et al., 1979, Zarraga et al., 2000, Singh and Nott, 2003, Kulkarni and Morris, 2008). Deviations from the behaviour predicted by this and similar expressions have been found due to inertia (Alghalibi et al., 2018) and at very large volume fractions once friction forces become dominant (Fall et al., 2008, Seto et al., 2013). Herein, we quantitatively characterize the rheological behavior of particles over porous walls across a sheared suspension at semi-dilute concentrations, Φ30% and negligible inertia.

Recently, Rosti et al. (2019a) studied the rheology of a particle suspension in channels with elastic walls and found a shear-thinning behavior of the suspension. This was caused by the particle migration away from the wall towards the channel center due to a lift force (Rallabandi et al., 2018) generated by the particle induced wall deformation. In the present work, we focus on a different kind of wall-modification, rigid porous walls where the fluid is allowed to penetrate through the porous walls.

In particular, we employ direct numerical simulations (DNSs) to explore the particle motion and interactions over rigid porous surfaces for a plane Couette flow where both surfaces are covered with porous media with known permeability and porosity. The chosen set-up is the one typical of fundamental rheology studies, but the results can be extended to more complex and realistic geometries, such as channel and duct flows or Taylor-Couette flows. Here, we quantify the variations in the suspension stresses and slip velocity in a plane Couette flow due to the existence of porous surfaces. We also study the combined effects of particle volume fraction and wall permeability on the effective viscosity of the suspension. The present manuscript is organized as follows: in Section 2 we first present the mathematical and numerical formulations used to model the flow; then, in Section 3 we discuss the results of the simulations in terms of fluid and particle statistics and their variation with the particle volume fraction and with the parameters characterizing the porous media; finally, we collect the main findings in Section 4 and draw some final conclusions.

Section snippets

Methodology

We study the Couette flow of a Newtonian fluid laden with a suspension of rigid particles bounded by two homogeneous and isotropic porous walls. The fluid is incompressible and two flat, isotropic and homogeneous porous layers are attached to the impermeable moving walls, as shown in Fig. 1. The streamwise, wall-normal and spanwise coordinates are denoted by x,y and z (x1,x2, and x3), and similarly u,v and w (u1,u2, and u3) are the corresponding velocity components. y=0 and y=2h denote the two

Results

We start our analysis by showing in Fig. 2 (top) the profile of the streamwise component of the mean velocity u in the absence of particles, where the overbar indicates the average over the homogeneous directions, i.e.  x and z, and over time. We observe that in the purely fluid region y>0, the velocity is linear but with a smaller slope than the nominal one of the flow over impermeable walls γ̇=Uw/h. The change in the slope grows with the wall permeability σ, which is due to the weakening of

Conclusions

We have studied the rheology of suspensions of rigid, spherical particles in a Newtonian fluid in wall-bounded shear flow, i.e., Couette flow, at a sufficiently low Reynolds number so that inertial effects are negligible. The part of the channel filled with particles is bounded by two rigid, homogeneous and isotropic porous layers, fixed on the moving walls and moving with the wall velocity. The problem is solved numerically using an IBM to account for the rigid suspension, while we model the

CRediT authorship contribution statement

Marco E. Rosti: Conceptualization, Data curation, Formal analysis, Writing - original draft, Writing - review & editing. Parisa Mirbod: Conceptualization, Writing - original draft, Writing - review & editing. Luca Brandt: Conceptualization, Writing - original draft, Writing - review & editing, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

PM has been supported in part by the National Science Foundation Award No. 1854376 and in part by the Army Research Office Award No. W911NF-18-1-0356. LB acknowledges financial support from the Swedish Research Council (VR) and the INTERFACE research environment (Grant No. VR 2016-06119) and from Grant No. VR 2014-5001. Computer time was provided by the Swedish National Infrastructure for Computing (SNIC) and by the Scientific Computing section of Research Support Division at OIST.

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