The bulk electroviscous effect

Electroviscous stresses arise as hydrodynamic flows disturb the ionic (Debye) clouds that screen charged surfaces in electrolyte solutions. The contribution thereof to the effective bulk viscosity (also known as the second or volume viscosity) of two-phase suspensions is quantified here. Specifically, the bulk viscosity of two model suspensions is calculated: (1) a dilute dispersion of rigid charged spherical particles immersed in a compressible electrolyte that undergoes uniform dilatation and (2) a dilute suspension of charged gas bubbles expanding uniformly in an incompressible electrolyte. In both cases, it is assumed that the fluid flow only slightly drives the Debye cloud out of equilibrium, which formally requires that the ratio of the ion diffusion to flow time scales—a Péclet number—is small. For a suspension of rigid particles, the electroviscous contribution to the effective bulk viscosity is proportional to the particle volume fraction and decreases monotonically as the ratio of the particle radius to the Debye length increases. Similar behavior is well known for the electroviscous contribution to the effective shear viscosity of a dilute hard-sphere suspension; a quantitative comparison between the bulk and shear viscosities is made. In contrast, the electroviscous contribution to the bulk viscosity of a dilute suspension of bubbles is independent of the bubble volume fraction and attains a finite value in the limit of vanishing Debye length.     

Slow Motion of an Assemblage of Porous Spherical Shells Relative to a Fluid

The body-force-driven motion of a homogeneous distribution of spherically symmetric porous shells in an incompressible Newtonian fluid and the fluid flow through a bed of these shell particles are investigated analytically. The effect of the hydrodynamic interaction among the porous shell particles is taken into account by employing a cell-model representation. In the limit of small Reynolds number, the Stokes and Brinkman equations are solved for the flow field around a single particle in a unit cell, and the drag force acting on the particle by the fluid is obtained in closed forms. For a suspension of porous spherical shells, the mobility of the particles decreases or the hydrodynamic interaction among the particles increases monotonically with a decrease in the permeability of the porous shells. The effect of particle interactions on the creeping motion of porous spherical shells relative to a fluid can be quite significant in some situations. In the limiting cases, the analytical solution describing the drag force or mobility for a suspension of porous spherical shells reduces to those for suspensions of impermeable solid spheres and of porous spheres. The particle-interaction behavior for a suspension of porous spherical shells with a relatively low permeability may be approximated by that of permeable spheres when the porous shells are sufficiently thick.     

Flowing colloidal suspensions in non-Newtonian suspending fluids: Decoupling the composite birefringence

The rheology of dilute, colloidal suspensions in polymeric suspending fluids can be studied with simultaneous dichroism and birefringence measurements. The dichroism provides a direct measure of the particle dynamics, but the birefringence is a composite property with independent contributions from the suspended particles and the polymer molecules. For suspensions where the contribution from the particles is significant, the composite birefringence must be decoupled in order to analyze the dynamics of the polymeric suspending fluid. A method to perform the decoupling is derived and then demonstrated through transient shear flow experiments with dilute suspensions ofFeOOH particles in semi-dilute, xanthan gum suspending fluids. The birefringence of the xanthan gum suspending fluid is calculated from experimental measurements of the composite birefringence and the dichroism of the suspension. To gather information on particle/polymer interactions, the calculated birefringence is compared to the birefringence of xanthan gum solutions containing no suspended particles and the dirchoism is compared to that of a suspension in a Newtonian fluid.     

A model for the shear viscosity of non-colloidal suspensions with Newtonian matrix fluids

We present a model for the shear viscosity of non-colloidal suspensions with Newtonian matrix fluids. The model is based on the original idea first presented by Brinkman (Applied Sci Research A1:27-34. 1947) for the viscous force exerted by a flowing fluid on a dense swarm of spherical particles. In particular, we consider an inertialess suspension in which the mean flow is driven by a pressure difference, and simultaneously, the suspension is subject to simple shear. Assuming steady state, incompressibility and taking into account a resistance force which is generated due to the presence of the particles in the flow, the three-dimensional governing equations for the mean flow around a single spherical particle are solved analytically. Self-consistency of the model provides a relationship between the resistance parameter and the volume fraction of the solid phase. A volume, or an ensemble, averaging of the total stress gives the bulk properties and an expression for the relative (bulk) viscosity of the suspension. The viscosity expression reduces to the Einstein limit for dilute suspensions and agrees well with empirical formulas from the literature in the semi-dilute and concentrated regimes. Since the model is based on a single particle and its average interaction with the other particles is isotropic, no normal stress differences can be predicted. A possible method of addressing this problem is provided in the paper.     

Linear stability of plane creeping Couette flow for Burgers fluid

It is well known that plane creeping Couette flow of UCM and Oldroy-B fluids are linearly stable. However, for Burges fluid, which includes UCM and Oldroyd-B fluids as special cases, unstable modes are detected in the present work. The wave speed, critical parameters and perturbation mode are studied for neutral waves. Energy analysis shows that the sustaining of perturbation energy in Poiseuille flow and Couette flow is completely different. At low Reynolds number limit, analytical solutions are obtained for simplified perturbation equations. The essential difference between Burgers fluid and Oldroyd-B fluid is revealed to be the fact that neutral mode exists only in the former.     

Experimental and numerical study of the rotation and the erosion of fillers suspended in viscoelastic fluids under simple shear flow

When a porous agglomerate immersed in a fluid is submitted to a shear flow, hydrodynamic stresses acting on its surface may cause a size reduction if they exceed the cohesive stress of the agglomerate. The aggregates forming the agglomerate are slowly removed from the agglomerate surface. Such a behaviour is known when the suspending fluid is Newtonian but unknown if the fluid is viscoelastic. By using rheo-optical tools, model fluids, carbon black agglomerates and particles of various shapes, we found that the particles had a rotational motion around the vorticity axis with a period which is independent on shape (flat particles not considered), but which is exponentially increasing with the elasticity of the medium expressed by the Weissenberg number (We). Spherical particles are always rotating for We up to 2.6 (largest investigated We in this study) but elongated particles stop rotating for We>0.9 while orienting along the flow direction. Erosion is strongly reduced by elasticity. Since finite element numerical simulation shows that elasticity increases the local stress around a particle, the origin of the erosion reduction is interpreted as an increase of cohesiveness of the porous agglomerate due to the infiltration of a viscoelastic fluid.     

Simple Shear Flow of Suspensions of Elastic Capsules

The simple shear flow of homogeneous suspensions of two-dimensional capsules enclosed by elastic membranes is studied in the limit of vanishing Reynolds number, in the special case where the viscosity of the fluid enclosed by the capsules is equal to the viscosity of the ambient fluid. The deformation of capsules with circular, elliptical, and biconcave unstressed shapes, and the rheological and statistical properties of their infinitely dilute and moderately dense suspensions are investigated by dynamical simulation using the method of interfacial dynamics for Stokes flow. In a preliminary investigation, the behavior of solitary capsules suspended in an infinite fluid is studied as a function of the dimensionless membrane elasticity number expressing the capsule deformability or the strength of the shear flow. It is found that a critical elasticity number above which a capsule exhibits continued elongation does not exist, and an equilibrium configuration is reached no matter how large the shear rate, in agreement with previous results for three-dimensional flow. A correspondence is established between the elasticity numbers for two- and three-dimensional flow at which the capsules undergo the same degree of deformation. Simulations of pairwise capsule interceptions reveal behavior similar to that exhibited by liquid drops with uniform surface tension. Because of strong hydrodynamic interactions in two-dimensional Stokes flow, the concept of hydrodynamic diffusivity in the limit of infinite dilution is ill-defined in the absence of fluid inertia. Dynamical simulations of doubly periodic monodisperse suspensions with up to 50 capsules distributed in each periodic cell at areal fractions of 0.25 and 0.40 provide information on the effective rheological properties of the suspension and on the nature of the statistical properties of the particle motion. The character of the flow is found to be intermediate between that of liquid drops and rigid particles, and this is attributed to the membrane deformability and to the ability of the interfaces to perform tank-treading motion. The results are compared with rheological measurements of blood flow with good agreement. Received 26 April 1999 and accepted 5 October 1999     

Three-dimensional numerical simulation of drops suspended in simple shear flow at finite Reynolds numbers

A three-dimensional study of suspension of drops in simple shear flow has been performed at finite Reynolds numbers. Results are obtained using a finite difference/front tracking method in a periodic domain. The effects of the Reynolds number and the Capillary number are addressed at two volume fractions: 0.195 and 0.34. It is observed that suspensions of deformable drops exhibit a shear-thinning behavior. Similar to the motion of a single drop, drops migrate away from the walls. The effective viscosity, the first and the second normal stress differences oscillate around a mean value in all cases. The first normal stress difference increases with the Capillary number, the Reynolds number and the volume fraction. Results show that drops deform more and orient more in the flow direction as the Capillary number or the volume fraction is increased. Also, the average size of clusters is smaller than for suspension of rigid particles. The radial dependence of the pair distribution function across the channel has been studied. This dependency shows that the tendency to form clusters is reduced as the Capillary number increases or the volume fraction decreases.     

Effective viscosity of a concentrated suspension of composite porous spherical particles

Meccanica - We present an analytical study of the effective viscosity of concentrated suspension of porous spherical particles with a rigid core, under the creeping flow conditions. It is assumed...     

Uniqueness and drag for fluids of second grade in steady motion

For incompressible fluids of second grade that are compatible with the Clausius-Duhem inequality, non-uniqueness of steady flows with small Reynolds number (i.e. creeping flows) is possible provided the ‘absorption number’ is also small. We discuss this uniqueness question, generalize a well-known theorem of Tanner concerning how solutions of the Stokes equations may be used to generate solutions of the creeping flow equations for fluids of second grade, and give a new uniqueness theorem appropriate to a class of problems for the steady creeping flow of fluids of second grade. Under the conditions for uniqueness, we obtain a simple formula for the drag force on a fixed body which is immersed in a fluid of second grade which is undergoing uniform creeping flow. For bodies with certain geometric symmetries, the non-Newtonian nature of the fluid has no effect upon the drag.     

Stokes flow in a cylindrical tube containing a line of spheroidal particles

Viscous flow in a circular cylindrical tube containing an infinite line of rigid spheroidal particles equally spaced along the axis of the tube is considered for (a) uniform axial translation of the spheroids (b) flow past a line of stationary spheriods and (c) flow of the suspending fluid and spheroids under an imposed pressure gradient. The fluid is assumed to be incompressible and Newtonian. The Reynolds number is assumed to be small and the equations of creeping flow are used. Two types of solutions are developed: (i) an exact solution in the form of an infinite series which is valid for ratios of the spheroid diameter to the tube diameter up to 0.80, (ii) an approximate solution using lubrication theory which is valid for spheroids which nearly fill the tube. The drag on each spheroid and the pressure drop are computed for all cases. Both prolate and oblate spheroids are considered. The results show that the drag and pressure drop depend on the spheroidal diameter perpendicular to the axis of tube primarily and the effects of the spheroidal thickness and spacing are secondary. The results are of interest in connection with mechanics of capillary blood flow, sedimentation, fluidized beds, and fluid-solid transport.     

Effects of shear thinning on migration of neutrally buoyant particles in pressure driven flow of Newtonian and viscoelastic fluids

The pattern of cross stream migration of neutrally buoyant particles in a pressure driven flow depends strongly on the properties of the suspending fluid. These migration effects have been studied by direct numerical simulation in planar flow. Shear thinning has a large effect when the inertia or elasticity is large, but only a small effect when they are small. At moderate Reynolds numbers, shear thinning causes particles to migrate away from the centerline, creating a particle-free zone in the core of the channel, which increases with the amount of shear thinning. In a viscoelastic fluid with shear thinning, particles migrate either toward the centerline or toward the walls, creating an annular particle-free zone at intermediate radii. The simulations also give rise to precise determination of slip velocity distributions in the various cases studied.     

Thermal analysis of a flow of immiscible couple stress fluids in a channel

An analytical study of the entropy generation rate and heat transfer in a flow of immiscible couple stress fluids between two horizontal parallel plates under a constant pressure gradient is performed. Both plates are kept at different and constant temperatures higher than that of the fluid. The Stokes couple stress flow model is employed. The classical no-slip condition is prescribed at the plates, and continuity of the velocity, rotation, couple stress, shear stress, temperature, and heat flux is imposed at the interfaces. The velocity and temperature distributions are found analytically, and they are used to compute the entropy generation number and Bejan number. The effects of the couple stress parameter and Reynolds number on the velocity, temperature, entropy generation number, and Bejan number are investigated. It is observed that the friction near the plates in couple stress fluids decreases as the couple stress increases.     

Particle-liquid mass transfer in viscoelastic media

The problem of mass transfer from a solid sphere to a viscoelastic fluid has been examined theoretically. It is shown that fluid elasticity increases marginally the mass transfer rate in the creeping flow regime. This will have serious implications on the mass transfer from bubbles if impurities are present. Some conclusions on mass transfer at high Reynolds numbers are also offered.     

The Dilute Rheology of Swimming Suspensions: A Simple Kinetic Model

A simple kinetic model is presented for the shear rheology of a dilute suspension of particles swimming at low Reynolds number. If interparticle hydrodynamic interactions are neglected, the configuration of the suspension is characterized by the particle orientation distribution, which satisfies a Fokker-Planck equation including the effects of the external shear flow, rotary diffusion, and particle tumbling. The orientation distribution then determines the leading-order term in the particle extra stress in the suspension, which can be evaluated based on the classic theory of Hinch and Leal (J Fluid Mech 52(4):683–712, 1972), and involves an additional contribution arising from the permanent force dipole exerted by the particles as they propel themselves through the fluid. Numerical solutions of the steady-state Fokker-Planck equation were obtained using a spectral method, and results are reported for the shear viscosity and normal stress difference coefficients in terms of flow strength, rotary diffusivity, and correlation time for tumbling. It is found that the rheology is characterized by much stronger normal stress differences than for passive suspensions, and that tail-actuated swimmers result in a strong decrease in the effective shear viscosity of the fluid.     

On the rheology of a dilute suspension of rigid spheres in a weakly viscoelastic matrix fluid

The complete solution for the pressure and the velocity field up to O(De) of a dilute suspension of neutrally buoyant, non-Brownian rigid spheres suspended in an unbounded, weakly viscoelastic matrix fluid, where is the solid volume fraction and De is the Deborah number of the matrix fluid, is presented. The spheres are subjected to an arbitrary linear velocity profile at infinity. The analytical solution is used for the prediction of the bulk stress, and specifically for the calculation of the first and the second normal stress differences in simple shear and uniaxial elongational flows. A comparison of the results with available values reported in the literature is also offered. The final expressions for the bulk normal stress differences in shear and uniaxial elongational flow fully agree with those reported earlier by Greco et al., J. Non-Newton. Fluid Mech., 147 (2007) 1–10.     

Squeeze flow of a power-law fluid between two rigid spheres with wall slip

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A correlation for yield stress fluid flow through packed beds

Relatively few correlations are available for non-Newtonian fluid flows through packed beds, even though such fluids are frequently used in industry. In this paper, a correlation is presented for yield stress fluid flow through packed beds. The correlation is developed by introducing the yield stress model in place of the Newtonian model used in deriving Erguns equation. The resulting model has three parameters that are functions of the geometry and roughness of the particle surfaces. Two of the parameters can be deduced in the limit as the yield stress becomes negligible and the model reduces to Erguns equation for Newtonian fluids. The third model parameter is determined from experimental data. The correlation relates a defined friction factor to the dimensionless Reynolds and Hedstrom numbers and can be used to predict pressure drop for flow of a yield stress fluid through a packed bed of spherical particles. Conditions for flow or no-flow are also determined in the correlation. Comparison of model calculations, between a Newtonian and a yield stress fluid for flow penetration into a packed bed of spheres, shows the yield stress fluid initially performs similar to the Newtonian fluid, at large Reynolds numbers. At lower Reynolds numbers the yield stress effect becomes important and the flow rate significantly decreases when compared to the Newtonian fluid.     

On the interplay between inertial and viscoelastic effects for the flow in weakly modulated channels

The flow inside a spatially modulated channel is examined for viscoelastic fluids of the Oldroyd‐B type. The lower wall is flat and the upper wall is sinusoidally modulated. The modulation amplitude is assumed to be small. Thus, a regular perturbation expansion of the flow field coupled to a variable‐step finite‐difference scheme is used to solve the problem. Convergence and accuracy assessment against earlier experimental results indicate that there is a significant range of validity of the perturbation approach. The influences of wall geometry, inertia and viscoelasticity on the flow kinematics and stresses are investigated systematically. In particular, the interplay between the flow and fluid parameters effects on the conditions for the onset of backflow, number of vortices, their size and location is revealed. The distance between the flow separation and reattachment locations identifies the vortex size. Non‐monotonic dependence of the vortex size on elasticity is reported. The critical conditions for the onset of negative elasticity effects on vortex size are identified. The critical Reynolds number for the onset of backflow initially decreases then levels off or even increases as elasticity increases. For highly elastic fluid and large enough Reynolds number, more than one vortex appear near the lower wall. Copyright © 2005 John Wiley & Sons, Ltd.     

The flow of closely fitting particles in tapered tubes

The flow of rigid spheres, truncated cones and elastic incompressible spheres in tapered tubes is investigated assuming that the Reynolds equation is valid in the fluid and the linear theory of elasticity is applicable in the solid. It is shown that leading terms in the asymptotic expansion of pressure drop in terms of minimum fluid film thickness for neutrally buoyant rigid spheres and truncated cones are of higher order of magnitude compared to the corresponding terms for the flow of these particles in circular cylindrical tubes. The effect of taper angle on pressure drop is reduced in the case of soft elastic particles because of particle deformations and significant velocities at the particle surface.