David T. Leighton

The characterization of the total stress of concentrated suspensions of noncolloidal spheres in Newtonian fluids

The total stress of a concentrated suspension of noncolloidal spheres in a Newtonian fluid was characterized by independent measurements in viscometric flows. Using a suspension balance formulation, the normal stress in the vorticity direction (Σ33) for a suspension undergoing simple shear was extracted from Acrivos et al.’s [Int. J. Multiphase Flow 19, 797 (1993)] resuspension data in a Couette device. Employing a new correlation for the relative viscosity μr which obeys the Einstein relation in the dilute limit while diverging at random close packing, it was found that Σ33/τ (where τ is the magnitude of the shear stress) was a strong function of the solid volume fraction φ, scaling as φ3e2.34φ. The relative viscosity, measured in a parallel plate viscometer, was in good agreement with the proposed correlation, while the normal stress differences N1 and N2 for concentrated suspensions (φ=0.30–0.55) were characterized using parallel plate and cone-and-plate geometries, as well as laser profilometry measur...

Inertial lift on a moving sphere in contact with a plane wall in a shear flow

In this paper we calculate the lift force on a smooth sphere rotating and translating in a simple shear flow in contact with a rigid wall. The calculation involves only known creeping flow solutions and is presented in terms of six different coefficients, each arising as a result of a pairwise combination of the translational velocity, rotational velocity, and the imposed shear flow. The results obtained agree well with those of Cherukat and McLaughlin [J. Fluid Mech. 263, 1 (1994a); and (personal communication, 1994b)], extrapolated for the case of zero separation distance. The calculated lift is further integrated into a force and torque balance on a non‐neutrally buoyant rough sphere moving in contact with a plane. It is found that if the shear Reynolds number Re is sufficiently large, the lift force exceeds the gravitational force and the sphere separates from the plane. The increased separation is accompanied by an increase in the translational velocity U of the sphere and a corresponding decrease in...

Measurement of the hydrodynamic surface roughness of noncolloidal spheres

A new technique for measuring the surface roughness of noncolloidal spheres is presented. The time for a sphere initially in contact with a smooth surface to fall away under the influence of gravity through a viscous fluid is shown to be related to the largest scale of surface roughness of sufficient surface coverage to support the particle. The ratio of the time taken for a sphere to fall one particle diameter from a smooth mica plane to that for the sphere to fall between one radius and one diameter from the plane thus provides a means of measuring the effective hydrodynamic surface roughness of spheres. This technique was employed to measure the roughness of eight types of particles ranging from 43–6350 μm in diameter. The roughnesses were found to be on the order of 10−2 to 10−3 particle radii, and were in agreement with independent observations using a scanning electron microscope and an optical profilometer.

Measurement of the drift of a droplet due to the presence of a plane

The drift of a deformable droplet of low viscosity (viscosity ratio λ=0.08) in a Couette device is examined. The drift is measured both in the plane of shear (due to the rigid outer bounding walls of the Couette device) and also normal to the plane of shear (due to the upper bounding stress‐free surface). A general relationship between normal stresses induced by the deformation of a droplet in an arbitrary shear flow and the leading‐order drift normal to rigid and stress‐free plane surfaces is described theoretically. This relationship is consistent with previous theoretical predictions for droplet migration in shear flows, and is used to compare results from the drift measurement experiments with first‐order deformation theories. The measured drift velocities are in reasonable agreement with the theory of Schowalter et al. [J. Colloid Interface Sci. 26, 152 (1968)].

Normal stresses and free surface deformation in concentrated suspensions of noncolloidal spheres in a viscoelastic fluid

Concentrated suspensions of noncolloidal spheres in a constant viscosity elastic fluid were characterized rheologically using rotating plate viscometers and profilometry of the suspension surface deflection near a rotating rod. It was found that the relative viscosity was quantitatively consistent with a previously determined correlation for suspensions based on Newtonian fluids. Moreover, the first normal stress difference N1 was found to be positive and the second normal stress difference N2 negative. Although the magnitude of N1 and N2 increased with the solids volume fraction φ, in general the ratio |N1/N2| decreased as loading increased. Analysis of the normal stress data suggests that the rheological contribution of the solids microstructure was in large part independent of that of the dissolved polymers at high solids loading (φ⩾0.3). The magnitude of N2 at high concentrations approached that measured for similar suspensions in Newtonian fluids, while the magnitude of N1 could be attributed to the ...

Measurement of shear-induced dispersion in a dilute emulsion

The time-dependent drop distribution of a dilute, polydisperse emulsion is measured in a simple shear flow. The suspending fluid is much more viscous than the dispersed phase (1:1000). Drops are found to drift away from either bounding wall and accumulate near the center of the gap, due to the anisotropy of droplet–plane interactions. An expression for this drift velocity has been derived for single drops by Chan and Leal [J. Fluid Mech. 92, 131 (1979)] and was in agreement with isolated drop migration observed in our work. Eventually the inward drift is balanced by a shear-induced gradient diffusivity, and a steady-state concentration distribution is reached. When the drops are sufficiently far from either wall a self-similar, parabolic concentration profile is predicted at all times. Droplet diffusivities were determined for capillary numbers Ca=γāμ/σ between 0.17 and 0.92, where γ is the shear rate, ā is the mean drop radius, μ is the viscosity of the suspending fluid, and σ the interfacial tension. ...

Normal stress and diffusion in a dilute suspension of hard spheres undergoing simple shear

The complete set of normal stresses in a dilute suspension of hard spheres undergoing simple shear at low Reynolds number is calculated using a path integration approach for the cases where the concentration is uniform and where a small gradient in concentration is present. As expected, the normal stresses are seen to be a strong function of es=2(b−a)/a, where b is the hard sphere radius and a is the particle radius. The normal stress differences N1 and N2, are negative while the osmotic pressure is large and positive, with Π>|N2| and N1→0 as es→∞. For es≪1, the asymmetry in the pair distribution function due to a depletion of particles in the extensional side of a pair interaction leads to |N1|>|N2|. On the other hand, for es→∞, the additional stresslet induced when hard sphere radii touch dominates the stress generated in the suspension, and N2 becomes the prevailing normal stress difference. The self and gradient diffusivities are calculated using da Cunha and Hinch’s [J. Fluid Mech. 309, 211 (1996)] t...

Measurement of an unexpectedly large shear-induced self-diffusivity in a dilute suspension of spheres

We report the measurement of unexpectedly large shear-induced diffusivities for various sized tracers in a dilute suspension of noncolloidal spheres in simple shear. The suspension was sheared in a narrow gap Couette device at low Reynolds number, and the tracer diffusivities parallel to the velocity gradient D were obtained using an orbit-time technique. It is shown that the presence of even a dilute concentration φ of particles renders an otherwise smooth tracer trajectory strongly stochastic and the resulting diffusivity is linear in φ in this limit, indicating irreversible displacements at the pair interaction level. The measured values of the diffusivity, however, are at least an order of magnitude larger than that predicted by current theories. In a previous study, Beimfohr et al. [Proc. DOE/NSF Workshop, Ithaca, NY (1993)] likewise obtained values for D larger than that predicted by theory, but the discrepancy was attributed to the large eccentricity (average aspect ratio=1.19) of the particles use...

Stability of oscillatory two-phase Couette flow: Theory and experiment

The interfacial instability due to viscosity stratification is studied experimentally in a closed Couette geometry. A vertical interface is formed between two concentric cylinders with density-matched fluids of unequal viscosity. The outer cylinder is rotated with a time-harmonic motion, causing spatially periodic disturbances of the interface. The wavelengths and growth rates predicted by linear theory agree well with experimental results. Application of Fjo/rtoft’s inflection point theorem shows the neutral stability curves to be consistent with an internal instability occurring in the less viscous phase. Because the standard Floquet theory yields only time-averaged growth rates, the instantaneous behavior of the system is examined numerically. This reveals the flow to be unstable to a disturbance which has a maximum that oscillates between the interface and a location within the less viscous fluid. Surprisingly, it is found that interfacial wave amplification originates with the internal disturbance, a...

The effect of gravity on the meniscus accumulation phenomenon in a tube

It is well known that when a suspension of noncolloidal, neutrally buoyant rigid particles is drawn into an empty tube, the inward shear-induced migration of the particles results in a continuously growing packed layer of particles at the advancing meniscus. For non-neutrally buoyant suspensions, however, gravity perpendicular to the flow direction modifies the fully developed concentration profile and can lead to depletion of particles at the meniscus. This accumulation/depletion phenomenon was studied experimentally as a function of the buoyancy number Nb, the ratio of gravitational to viscous forces. The rate of accumulation or depletion was used to determine the area average concentration as a function of Nb and the feed concentration. These results are in good agreement with the profiles calculated theoretically using the suspension balance model of Nott and Brady [J. Fluid Mech. 275, 157 (1994)] coupled with the anisotropic constitutive equations of Zarraga et al. [J. Rheol. 44, 185 (2000)].