The impact of porous walls on the rheology of suspensions
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 ) of rigid particles in a Newtonian fluid linearly increases with the particle volume fraction , 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 . 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, 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 and z (, and ), and similarly and w (, and ) are the corresponding velocity components. and denote the two
Results
We start our analysis by showing in Fig. 2 (top) the profile of the streamwise component of the mean velocity 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 , the velocity is linear but with a smaller slope than the nominal one of the flow over impermeable walls . The change in the slope grows with the wall permeability , 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.
References (67)
- et al.
Shear-induced resuspension in a couette device
Int. J. Multiph. Flow
(1993) The slow motion of a sphere through a viscous fluid towards a plane surface
Chem. Eng. Sci.
(1961)A second-order accurate immersed boundary method for fully resolved simulations of particle-laden flows
J. Comput. Phys.
(2012)- et al.
Elastoviscoplastic flow in porous media
J. Nonnewton. Fluid Mech.
(2018) - et al.
Application of a fractional-step method to incompressible navier-stokes equations
J. Comput. Phys.
(1985) - et al.
Channel flow of rigid sphere suspensions: Particle dynamics in the inertial regime
Int. J. Multiph. Flow
(2016) - et al.
Normal stress-driven migration and axial development in pressure-driven flow of concentrated suspensions
J. Nonnewton. Fluid Mech.
(2006) - et al.
Suspension flow modeling for general geometries
Chem. Eng. Sci.
(2009) - et al.
Momentum transfer at the boundary between a porous medium and a homogeneous fluid - I. Theoretical development
Int. J. Heat Mass Transf.
(1995) - et al.
Suspensions of deformable particles in a Couette flow
J. Nonnewton. Fluid Mech.
(2018)
Interface-resolved simulations of particle suspensions in Newtonian, shear thinning and shear thickening carrier fluids
J. Fluid Mech.
The effect of Brownian motion on the bulk stress in a suspension of spherical particles
J. Fluid Mech.
The determination of the bulk stress in a suspension of spherical particles to order c 2
J. Fluid Mech.
Elastic response of carbon nanotube forests to aerodynamic stresses
Phys. Rev. Lett.
Unifying suspension and granular rheology
Phys. Rev. Lett.
Predicting the apparent wall slip when using roughened geometries: a porous medium approach
J. Rheol.
Collision model for fully resolved simulations of flows laden with finite-size particles
Phys. Rev. E
Particle pressure in a sheared suspension - A bridge from osmosis to granular dilatancy
Phys. Rev. Lett.
Die viskosität von emulsionen hochviskoser stoffe als funktion der konzentration
Kolloid-Zeitschrift
Investigations on the theory of the Brownian movement
Shear thickening of cornstarch suspensions as a reentrant jamming transition
Phys. Rev. Lett.
Shear viscosity of settling suspensions
Rheol. Acta
Rheology of confined non-Brownian suspensions
Phys. Rev. Lett.
Shear-induced structure in a concentrated suspension of solid spheres
J. Rheol.
Shallow flows over a permeable medium: the hydrodynamics of submerged aquatic canopies
Transp. Porous Media
Transition layer thickness at a fluid-porous interface
Phys. Fluids
A hydrodynamic mechanosensory hypothesis for brush border microvilli
Am. J. Physiol.- Renal Physiol.
Velocity measurements of a dilute particulate suspension over and through a porous medium model
Phys. Fluids
Computational modeling of multiphase viscoelastic and elastoviscoplastic flows
Int. J. Numer. Meth. Fluids
Porosity effects in laminar fluid flow near permeable surfaces
Phys. Rev. E
Turbulence statistics above and within two amazon rain forest canopies
Bound.-Layer Meteorol.
Suspension properties at finite reynolds number from simulated shear flow
Phys. Fluids
Cited by (11)
Effect of porous media models on rheological properties of suspensions
2022, Journal of Non-Newtonian Fluid MechanicsHierarchical data visualization of experimental erythrocyte aggregation employing cross correlation and optical flow applications
2022, Microvascular ResearchCitation Excerpt :The obtained results can then be compared with the blood flow viscosity and rheological properties as have been reported in prior studies (Beris et al., 2021). The model can be then utilized as a tool to analyze erythrocyte aggregation and particle velocity distribution across the vessel as well as shear stress (Mirbod, 2016; Shannon and Mirbod, 2017; Wu and Mirbod, 2018; Haffner and Mirbod, 2020; Kang and Mirbod, 2020; Rosti et al., 2021; Mirbod et al., 2009). Additional extensions of this work will be to investigate the flow instability and clustering of erythrocytes by adding dextran and examining in detail blood flow pressure and velocity profiles experimentally.
Interface-resolved simulations of the confinement effect on the sedimentation of a sphere in yield-stress fluids
2022, Journal of Non-Newtonian Fluid MechanicsCitation Excerpt :The dimensionless parameters employed for the simulations presented here are reported in Table 1. The present three-dimensional numerical solver has been utilized and extensively validated in the past for particulate flows [43–45], non-Newtonian flows [16,42,46–48] and multiphase problems in non-Newtonian fluids [49]. The code has also been recently validated for suspensions of rigid and soft particles and droplets in EVP and viscoelastic fluids [39].
Effect of rheological additives on rheological properties of fly Ash-based sealing coatings
2022, Construction and Building MaterialsCitation Excerpt :However, the addition of dispersant will increase the viscosity and reduce the leveling property of the coating. Therefore, some rheological additives need to be added to improve the rheological properties of the coating[11]. Grigale-Sorocina et al.[12] showed that adding rheological additives in water or solvent system could change the rheological properties of the coating system to a certain extent.
An analysis of non-colloid suspended particles in a Newtonian fluid over porous media
2022, European Journal of Mechanics, B/FluidsStudy of kneading pressure and power consumption in a twin-blade planetary mixer for mixing highly viscous fluids
2021, Chemical Engineering ScienceCitation Excerpt :Fig. 8(a), (d) and (g) display the relative position of blades, the contours of static pressure and strain rate at z = 11.5 mm at the point “a”, respectively. The strain rate SR is a variable that allows estimating the force with which the fluid is deformed and is defined as SR = [2(∂Ui/∂xj)Sij]1/2, where the strain rate tensor Sij = (∂Ui/∂xj+∂Uj/∂xi)/2 (Ramírez-Cruz et al., 2020; Rosti et al., 2021; Yáñez-Varela et al., 2020). It can be seen from Fig. 8(a) that the hollow blade tip is far away from the kneading surface of the solid blade, thus the static pressure and the strain rate near the kneading surface of the solid blade are close to zero as shown in Fig. 8(d) and (g), suggesting that the blade-blade kneading region is not formed.