Experimental investigation into convective heat transfer of nanofluids at the entrance region under laminar flow conditions

Heat transfer of aqueous suspensions of carbon nanotubes (CNT nanofluids)

The behavior of suspensions and macromolecular solutions in crossflow microfiltration

Slip occurs in the flow of two-phase systems because of the displacement of the disperse phase away from solid boundaries. This arises from steric, hydrodynamic, viscoelastic and chemical forces and constraints acting on the disperse phase immediately adjacent to the walls. The enrichment of the boundary near the wall with the continuous (and usually low-viscosity) phase means that any flow of the fluid over the boundary is easier because of the lubrication effect. Because this effect is usually confined to a very narrow layer — with typical thickness of 0.1–10 μm—it so resembles the slip of solids over surfaces that it has historically been given the same terminology. The restoring force for all the forces that cause an increase in concentration is usually osmotic, and this will always limit the effective slip. In dilute systems, concentration gradients can be present over relatively large distances out from walls, giving what might be interpreted on an overall basis as a thick solvent-only layer. However, as the concentration of the system increases, the layer gets thinner and thinner because it is more difficult to create with the large reverse osmotic force present. However, the enormous increase in the bulk viscosity with increase in concentration means that although thinner, the layer becomes, paradoxically, even more important. Slip manifests itself in such a way that viscosity measured in different size geometries gives different answers if calculated the normal way — in particular the apparent viscosity decreases with decrease in geometry size (e.g. tube radius). Also, in single flow curves unexpected lower Newtonian plateaus are sometimes seen, with an apparent yield stress at even lower stresses. Sudden breaks in the flow curve can also be seen. Large particles as the disperse phase (remember flocs are large particles), with a large dependence of viscosity on the concentration of the dispersed phase are the circumstances which can give slip, especially if coupled with smooth walls and small flow dimensions. The effect is usually greatest at low speeds/flow rates. When the viscometer walls and particles carry like electrostatic charges and the continuous phase is electrically conducted, slip can be assumed. In many cases we need to characterise the slip effects seen in viscometers because they will also be seen in flow in smooth pipes and condults in manufacturing plants. This is usually done by relating the wall shear stress to a slip velocity using a power-law relationship. When the bulk flow has also been characterized, the flow in real situations can be calculated. To characterise slip, it is necessary to change the size of the geometry, and the results extrapolated to very large size to extract unambigouos bulk-flow and slip data respectively. A number of mathematical manipulations are necessary to retrieve these data. We can make attempts to eliminate slip by altering the physical or chemical character of the walls. This is usually done physically by roughening or profiling, but in the extreme, a vane can be used. This latter geometry has the advantage of being easy to make and clean. In either case—by extrapolation or elimination—we end up with the bulk flow properties. This is important in situations where we are trying to understand the microstructure/flow interactions.

A review of the slip (wall depletion) of polymer solutions, emulsions and particle suspensions in viscometers: its cause, character, and cure

▪ Abstract We review the fluid mechanics and rheology of dense suspensions, emphasizing investigations of microstructure and total stress. “Dense” or “highly concentrated” suspensions are those in which the average particle separation distance is less than the particle radius. For these suspensions, multiple-body interactions as well as two-body lubrication play a significant role and the rheology is non-Newtonian. We include investigations of multimodal suspensions, but not those of suspensions with dominant nonhydrodynamic interactions. We consider results from both physical experiments and computer simulations and explore scaling theories and the development of constitutive equations.

Fluid mechanics and rheology of dense suspensions

A constitutive equation for computing particle concentration and velocity fields in concentrated monomodal suspensions is proposed that consists of two parts: a Newtonian constitutive equation in which the viscosity depends on the local particle volume fraction and a diffusion equation that accounts for shear‐induced particle migration. Particle flux expressions used to obtain the diffusion equation are derived by simple scaling arguments. Predictions are made for the particle volume fraction and velocity fields for steady Couette and Poiseuille flow, and for transient start‐up of steady shear flow in a Couette apparatus. Particle concentrations for a monomodal suspension of polymethyl methacrylate spheres in a Newtonian solvent are measured by nuclear magnetic resonance (NMR) imaging in the Couette geometry for two particle sizes and volume fractions. The predictions agree remarkably well with the measurements for both transient and steady‐state experiments as well as for different particle sizes.

A constitutive equation for concentrated suspensions that accounts for shear‐induced particle migration

Shear‐induced migration of particles occurs in suspensions of neutrally buoyant spheres in Newtonian fluids undergoing shear in the annular space between two rotating, coaxial cylinders (a wide‐gap Couette), even when the suspension is in creeping flow. Previous studies have shown that the rate of migration of spherical particles from the high‐shear‐rate region near the inner (rotating) cylinder to the low‐shear‐rate region near the outer (stationary) cylinder increases rapidly with increasing sphere size. To determine the effect of particle shape, the migration of rods suspended in Newtonian fluids was recently measured. The behavior of several suspensions was studied. Each suspension contained well‐characterized, uniform rods with aspect ratios ranging from 2 to 18 at either 0.30 or 0.40 volume fraction. At the same volume fraction of solids, the steady‐state, radial concentration profiles for rods were independent of aspect ratio and were indistinguishable from those obtained from suspended spheres. Only minor differences near the walls (attributable to the finite size of the rods relative to the curvature of the walls) appeared to differentiate the profiles. Data taken during the transition from a well‐mixed suspension to the final steady state show that the rate of migration increased as the volume of the individual rods increased.

Shear‐induced particle migration in suspensions of rods

We report on experimental measurements and numerical predictions of shear-induced migration of particles in concentrated suspensions subjected to flow in the wide gap between a rotating inner cylinder placed eccentrically within a fixed outer cylinder (a cylindrical bearing). The suspensions consists of large, noncolloidal spherical particles suspended in a viscous Newtonian liquid. Nuclear magnetic resonance (NMR) imaging is used to measure the time evolution of concentration and velocity profiles as the flow induced particle migration from the initial, well-mixed state. A model originally proposed by Phillips et al. (1992) is generalized to two dimensions. The coupled equations of motion and particle migration are solved numerically using an explicit pseudo-transient finite volume formulation. While not all of the qualitative features observed in the experiments are reproduced by this general numerical implementation, the velocity predictions show moderately good agreement with the experimental data.

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Hydrodynamic particle migration in a concentrated suspension undergoing flow between rotating eccentric cylinders

The three‐dimensional motions of particles settling in Stokes flow are tracked numerically and experimentally. The numerical simulations are performed using a boundary element representation of the Stokes flow equations coupled with an Euler parameter representation of the particle kinematics. The experiments provide quantitative, three‐dimensional results for particle trajectories in Stokes flow. The experimental and numerical results are compared and found to agree to within the experimental uncertainty. The numerical results are also benchmarked against previously reported results for two test problems.

Particle tracking in three-dimensional Stokes flow

Individual falling balls were allowed to settle through otherwise quiescent well-mixed suspensions of non-colloidal neutrally buoyant spheres dispersed in a Newtonian liquid. Balls were tracked in three dimensions to determine the variances in their positions about a mean uniform vertical settling path. The primary experimental parameters investigated were the size of the falling ball and the volume fraction and size of the suspended particles. Unlike the horizontal variances, the vertical variances were found to be affected by short-time deterministic behaviour relating to the instantaneous local configurational arrangement of the suspended particles. For sufficiently long intervals between successive observations, the trajectories of the balls were observed to disperse about their mean settling paths in a random manner. This points to the existence of a Gaussian hydrodynamic dispersivity that characterizes the linear temporal growth of the variance in the position of a falling ball. The functional dependence of these horizontal and vertical dispersivities upon the parameters investigated was established.The dispersivity dyadic was observed to be transversely isotropic with respect to the direction of gravity, with the vertical component at least 25 times larger than the horizontal component. The vertical dispersivity Dˆv (made dimensionless with the diameter of the suspended spheres and the mean settling velocity) was observed to decrease with increasing falling ball diameter, but to decrease less rapidly with concentration than theoretically predicted for very dilute suspensions; moreover, for falling balls equal in size to the suspended spheres, Dˆv increased linearly with increasing volume fraction ϕ of suspended solids.In addition to the above experiments performed on suspensions of spheres, previously published settling-velocity data on the fall of balls through neutrally buoyant suspensions of rods possessing an aspect ratio of 20 were re-analysed, and vertical dispersivities calculated therefrom. (These data, taken by several of the present investigators in conjunction with other researchers, had only been grossly analysed in prior publications to extract the mean settling velocity of the ball, no attempt having been made at the time to extract dispersivity data too.) The resulting vertical dispersivities, when rendered dimensionless with the rod length and mean settling velocity, showed no statistically significant dependence upon the falling-ball diameter; moreover, all other things being equal, these dispersivities were observed to increase with increasing rod concentration.

James R. Abbott

Papers

A constitutive equation for concentrated suspensions that accounts for shear‐induced particle migration

Shear‐induced particle migration in suspensions of rods

Hydrodynamic particle migration in a concentrated suspension undergoing flow between rotating eccentric cylinders

Particle tracking in three-dimensional Stokes flow

Dispersion of a ball settling through a quiescent neutrally buoyant suspension