Shear-induced Particle Diffusion Research Articles

Shear-induced diffusion and dynamic heterogeneities in dense granular flows

We study two-dimensional dense granular flows by molecular dynamics simulations. We quantify shear-induced diffusion of granular particles by the transverse component of particle displacements. In long time scales, the transverse displacements are described as normal diffusion and obey Gaussian distributions, where time correlations of particle velocities entirely vanish. In short time scales, the transverse displacements are strongly non-Gaussian if the system is dense and sheared quasistatically though memory effects on the particle velocities are further suppressed. We also analyze spatio-temporal structures of the transverse displacements by self-intermediate scattering functions and dynamic susceptibilities. We find that the relation between the maximum intensity and characteristic time scale for dynamic heterogeneities is dependent on the models of contact damping (which exhibit different rheological properties such as the Newtonian fluids’ behavior and shear thickening). In addition, the diffusion coefficient over the shear rate is linear (sub-linear) in the maximum of dynamic susceptibility if the damping force is not restricted (restricted) to the normal direction between the particles in contact.

Microstructural changes of concentrated Newtonian suspensions in the first oscillation cycles probed with linear and non-linear rheology.

Carotenuto et al. (Rheol Acta, 2021, 60, 309) recently showed that the complex viscosity of a Newtonian non-Brownian suspension is smaller than the steady shear one, whatever the imposed strain amplitude. Oscillatory shear can alter the microstructure through a shear induced particle diffusion mechanism. This mechanism needs time to show its effect and cannot be invoked to explain the observed mismatch between the steady shear and the complex viscosity. Moreover, in the limit of vanishing strain amplitudes and of very large ones, where the oscillatory shear is equivalent to consecutive steady flow reversals, the oscillatory shear should not alter the microstructure and the Cox-Merz rule should hold. With a combination of approaches exploiting the Lissajous-Bowditch plots, the Fourier transform rheology and the Sequence of Physical Processes, we investigate the microstructure changes induced in the first oscillatory cycles. The results from the different analyses agree with the microstructure rearranging mechanisms proposed by Carotenuto et al.: at small amplitudes, the oscillatory shear rotates couples of touching particles towards the flow direction, at medium amplitudes it breaks particle clusters and at very large amplitudes it reshuffles and reorients all the particles. We show that the vast majority of the microstructure rearrangement occurs soon after the flow inversion of the first cycle, while before it the microstructure is not altered. This allows us to suggest a procedure to "recover" the Cox-Merz rule: a single cycle of oscillation must be imposed and the stress response of the sole first quarter of oscillation must be analysed.

Particle-induced miscible fingering: Continuum limit

Oil is injected into a mixture of the same oil and noncolloidal particles inside a Hele-Shaw cell to investigate the connection between miscible fingering and the interfacial structure that develops inside the thin gap. By tuning the channel confinement relative to the particle size, it is demonstrated that shear-induced diffusion of particles can be enhanced and the interfacial shape caused to become more rounded, which results in changes in fingering morphologies. The results of the study suggest a potential use of the wall confinement to control hydrodynamic instabilities uniquely in suspensions.

Interfacial instabilities in sediment suspension flows

AbstractWe report the existence of interfacial instability in the two-dimensional channel flow of a sediment suspension whose particles diffuse in the carrier fluid due to shear-induced collisions. We derive partial differential equations that govern the deformations of the interface between the sediment suspension and the clear fluid, and devise a perturbation method that preserves the positivity of the particle volume fraction. We solve perturbed momentum, particle transport and deforming interface equations to show that a Kelvin–Helmholtz-type unstable wave develops at the interface for wavelengths longer than a critical value. Short-wavelength oscillations of the interface are damped due to shear-induced diffusion of particles. We also show that the lowest critical Reynolds number, above which the interface is unstable, occurs for intermediate values of the total volume fraction of particles.

A two-phase flow model of sediment transport: transition from bedload to suspended load

AbstractThe transport of dense particles by a turbulent flow depends on two dimensionless numbers. Depending on the ratio of the shear velocity of the flow to the settling velocity of the particles (or the Rouse number), sediment transport takes place in a thin layer localized at the surface of the sediment bed (bedload) or over the whole water depth (suspended load). Moreover, depending on the sedimentation Reynolds number, the bedload layer is embedded in the viscous sublayer or is larger. We propose here a two-phase flow model able to describe both viscous and turbulent shear flows. Particle migration is described as resulting from normal stresses, but is limited by turbulent mixing and shear-induced diffusion of particles. Using this framework, we theoretically investigate the transition between bedload and suspended load.

Short-term and long-term irreversibility in particle suspensions undergoing small and large amplitude oscillatory stress

The short-term and long-term irreversible behaviors of suspensions of rigid particles in oscillatory shear flow are studied by measuring the evolution of complex viscosity in time and applying of nonlinear analysis of the responded strain signal under the controlled-stress mode, and complemented by optical measurements on the particle motion. The short-term transition time for the system to reach a quasisteady state is an approximately bell-shaped function of the amplitude of the strain response, thus showing a critical strain amplitude accounting for the peak transition time. The short-term behavior is caused by the particle self-organization due to collisions between particles. At longer time scales, the complex viscosity of the suspension increases when probed by forces that elicit small strain amplitudes and decreases when stresses that result in large strain amplitudes are applied. It is proposed that the long-term behavior for stresses eliciting small strain amplitude is induced by the shear-induced diffusion of particles which self-organize into a crystal-like microstructure that can be easily annulled in oscillatory flow with large strain amplitude, while for stresses causing large strain amplitude the dominant microstructure is formed immediately via the oscillation.

Heat transfer across sheared suspensions: role of the shear-induced diffusion

AbstractSuspensions of non-Brownian spherical particles undergoing shear provide a unique system where mixing occurs spontaneously at low Reynolds numbers. Through a combination of experiments and simulations, we investigate the effect of shear-induced particle diffusion on the transfer of heat across suspensions. The influence of particle size, particle volume fraction and applied shear are examined. By applying a heat pulse to the inner copper wall of a Couette cell and analysing its transient temperature decay, the effective thermal diffusivity of the suspension, $\alpha $, is obtained. Using index matching and laser-induced fluorescence imaging, we measured individual particle trajectories and calculated their diffusion coefficients. Simulations that combined a lattice Boltzmann technique to solve for the flow and a passive Brownian tracer algorithm to solve for the transfer of heat are in very good agreement with experiments. Fluctuations induced by the presence of particles within the fluid cause a significant enhancement (${\gt }200\hspace{0.167em} \% $) of the suspension transport properties. The effective thermal diffusivity was found to be linear with respect to both the Péclet number ($\mathit{Pe}= \dot {\gamma } {d}^{2} / {\alpha }_{0} \leq 100$) and the solid volume fraction ($\phi \leq 40\hspace{0.167em} \% $), leading to a simple correlation $\alpha / {\alpha }_{0} = 1+ \beta \phi \mathit{Pe}$ where $\beta = 0. 046$ and ${\alpha }_{0} $ is the thermal diffusivity of the suspension at rest. In our Couette cell, the enhancement was found to be optimum for a volume fraction, $\phi \approx 40\hspace{0.167em} \% $, above which, due to steric effects, both the particle diffusion motion and of the effective thermal diffusion dramatically decrease. No such correlation was found between the average particle rotation and the thermal diffusivity of the suspension, suggesting that the driving mechanism for enhanced transport is the translational particle diffusivity. Movies are available with the online version of the paper.

A low cost implementation of the Phillips model in solid–liquid flow modeling

AbstractThe interpolation requirements for the loosely coupled finite element solution of the Navier–Stokes equations and Phillips shear‐induced particle diffusion model are discussed. It is shown that a second‐order approximation of the fluid velocity field is required to adequately capture the spatial derivatives of the rate‐of‐strain tensor. To circumvent this limitation, a shear‐rate smoothing procedure is introduced, thereby allowing the use of lower‐order approximations for the fluid phase. Numerical experiments comparing the convergence and CPU cost of the different tetrahedral interpolation bases are presented. Copyright © 2009 John Wiley & Sons, Ltd.

Shear-Induced Diffusion of Platelike Particles in Microchannels

We exploit the recent developments of microfluidic technologies to investigate the collective shear-induced diffusion in suspensions of micron-sized particles. Whereas spherical particles do not diffuse on the time scale of our experiments, the results with platelike clay particles show a strong cross-stream shear-induced diffusivity at low volume fraction (phi{0}<or=0.01). Moreover, we find a linear dependence of the collective diffusion coefficient with the average shear rate (in the range 10;{2}-10;{4} s;{-1}) and the particle concentration. These data are in good agreement with previous experimental and theoretical results for spheres when rescaled with the particle number density.

Stripes in sheared non-Brownian suspensions with a free surface

The experiments of Tirumkudulu et al. (Phys. Fluids 11 (1999) 507; Phys. Fluids 12 (2000) 1615) showed spontaneous segregation and stripe formation in neutrally buoyant suspensions of particles in a partially filled, horizontal, rotating cylinder. Here, we propose a mechanism for this phenomenon, using two well-known properties of suspensions—shear-induced diffusion of particles and the concentration dependence of the viscosity.

Theory of suspension segregation in partially filled horizontal rotating cylinders

It is shown that a suspension of particles in a partially-filled, horizontal, rotating cylinder is linearly unstable towards axial segregation and an undulation of the free surface at large enough particle concentrations. Relying on the shear-induced diffusion of particles, concentration-dependent viscosity, and the existence of a free surface, our theory provides an explanation of the experiments of Tirumkudulu et al. [Phys. Fluids 11, 507 (1999); 12, 1615 (2000)].

Shear-induced particle diffusion in inelastic hard sphere fluids

Shear-induced particle diffusion in inelastic hard sphere fluids

A convection–shear induced resuspension model for crossflow microfiltration

The steady laminar flow of a suspension of neutrally buoyant particles of non-colloidal dimensions, through a pressurized open-ended channel having microporous membrane walls, is studied in the context of crossflow microfiltration ( Davis and Leighton, 1987, Chem. Engng Sci. 42, 275–281). Fully developed, Poiseuille flow is assumed in the bulk of the suspension whereas the filtrate crossflow velocity through the membrane walls, V w , is taken to be much smaller than the average downchannel velocity, U 0, and is assumed to be constant for all practical purposes. It is shown that in the limit as a/ L→0, where a is the characteristic particle radius and L is the characteristic length scale along the channel walls, the particle concentration remains uniform throughout the channel except for two O( a 2 L) 1/3 regions, near the two channel walls, where convection and shear induced particle diffusion balance each other. The system of equations within these two regions admits a treatment analogous to that presented by Pelekasis and Acrivos, 1995, ( J. Fluid Mech. 294, 301–321) in their study of sedimentation and convection past a flat plate. In the present study particle filtration takes the place of sedimentation while the constant shear rate imposed by the poiseuille bulk flow replaces the Blasius velocity profile. The main goal of the study to be presented is to predict the critical length along the channel walls needed for the particle concentration at the wall to reach its maximum value ϕ m =0.58, and consequently for the formation of a stagnant sediment layer. A similarity solution is obtained near the entrance of the channel which predicts an increase of the particle concentration by an amount proportional to X 1/3, with X measuring the distance from the entrance. For values of X beyond the entrance region a numerical solution is obtained which provides the prediction for the critical length over a wide range of bulk particle concentrations, ϕ s . Asymptotic results are derived for the cases ϕ s →0, and ϕ s → ϕ m . It should be noted that, in the limit ϕ s →0, the model employed by Davis and Leighton (1987, Chem. Engng Sci 42, 275–281), in one of the first theoretical approaches that accounted for shear induced particle diffusion in crossflow microfiltration, is recovered. Finally, as a demonstration of the validity of our approach, the results of the present study are shown to compare favorably with the experimental findings by Davis and Birdsell.

Transverse shear-induced gradient diffusion in a dilute suspension of spheres

We study the shear-induced gradient diffusion of particles in an inhomogeneous dilute suspension of neutrally buoyant spherical particles undergoing a simple shearing motion, with all inertia and Brownian motion effects assumed negligible. An expansion is derived for the flux of particles due to a concentration gradient along the directions perpendicular to the ambient flow. This expression involves the average velocity of the particles, which in turn is expressed as an integral over contributions from all possible configurations. The integral is divergent when expressed in terms of three-particle interactions and must be renormalized. For the monolayer case, such a renormalization is achieved by imposing the condition of zero total macroscopic flux in the transverse direction whereas, for the three-dimensional case, the additional constraint of zero total macroscopic pressure gradient is required. Following the scheme of Wang, Mauri &amp; Acrivos (1996), the renormalized integral is evaluated numerically for the case of a monolayer of particles, giving for the gradient diffusion coefficient 0.077γa2c¯2, where is the applied shear rate, a the radius of the spheres and c¯ their areal fraction.

Steady suspension flows into two-dimensional horizontal and inclined channels

In this study a model which was developed previously is used to theoretically investigate the steady flow of a particulate suspension into two-dimensional horizontal and inclined channels. The continuity equation for the fluid and the simplified two-dimensional Navier-Stokes equations are then solved together with a particle concentration equation. This latter equation is formulated by considering the balance between the particle flux due to gravity with the corresponding particle fluxes due to convection and shear-induced diffusion. The resulting coupled system of equations is solved numerically using a specialised finite-difference method. It is found, for the parameter range for flows of proppants in hydraulic fractures, that when the suspension enters the channel with a uniform velocity profile it almost instantaneously becomes parabolic. In addition, the effects of particle sedimentation are most dominant in the entrance region, but further downstream such effects are balanced as shear-induced particle diffusion becomes more important. It is also shown that the suspension flow depends critically on the choice of the parameters used, e.g. the ratio of the particle radius to the height of the channel.

Forced convection and sedimentation past a flat plate

The steady laminar flow of a well-mixed suspension of monodisperse solid spheres, convected steadily past a horizontal flat plate and sedimenting under the action of gravity, is examined. It is shown that, in the limit as Re → ∞ and ∈ → 0, where Re is the bulk Reynolds number and ∈ is the ratio of the particle radius a to the characteristic length scale L, the analysis for determining the particle concentration profile has several aspects in common with that of obtaining the temperature profile in forced-convection heat transfer from a wall to a fluid stream moving at high Reynolds and Prandtl numbers. Specifically, it is found that the particle concentration remains uniform throughout the O(Re−1/2) thick Blasius boundary layer except for two O(∈2/3) thin regions on either side of the plate, where the concentration profile becomes non-uniform owing to the presence of shear-induced particle diffusion which balances the particle flux due to convection and sedimentation. The system of equations within this concentration boundary layer admits a similarity solution near the leading edge of the plate, according to which the particle concentration along the top surface of the plate increases from its value in the free stream by an amount proportional to X5/6, with X measuring the distance along the plate, and decreases in a similar fashion along the underside. But, unlike the case of gravity settling on an inclined plate in the absence of a bulk flow at infinity considered earlier (Nir &amp; Acrivos 1990), here the concentration profile remains continuous everywhere. For values of X beyond the region near the leading edge, the particle concentration profile is obtained through the numerical solution of the relevant equations. It is found that, as predicted from the similarity solution, there exists a value of X at which the particle concentration along the top side of the plate attains its maximum value ϕm and that, beyond this point, a stagnant sediment layer will form that grows steadily in time. This critical value of X is computed as a function of ϕs, the particle volume fraction in the free stream. In contrast, but again in conformity with the similarity solution, for values of X sufficiently far removed from the leading edge along the underside of the plate, a particle-free region is predicted to form adjacent to the plate. This model, with minor modifications, can be used to describe particle migration in other shear flows, as, for example, in the case of crossflow microfiltration.

Shear-induced resuspension in a couette device

The viscous resuspension of an initially settled bed of particles resulting from the application of a laminar shear flow was studied experimentally in a narrow-gap Couette device. The measured height of the resuspended layer was found to be in excellent agreement with that predicted theoretically using a model developed by Leighton & Acrivos, in which the downward gravitational particle flux is balanced by a corresponding upward flux due to shear-induced particle diffusion.

Shear-induced particle diffusion and longitudinal velocity fluctuations in a granular-flow mixing layer

In flows of granular material, collisions between individual particles result in the movement of particles in directions transverse to the bulk motion. If the particles were distinguishable, a macroscopic overview of the transverse motions of the particles would resemble a self-diffusion of molecules as occurs in a gas. The present granular- flow study includes measurements of the self-diffusion process, and of the corresponding profiles of the average velocity and of the streamwise component of the fluctuating velocity. The experimental facility consists of a vertical channel fed by an entrance hopper that is divided by a splitter plate. Using differently-coloured but otherwise identical glass spheres to visualize the diffusion process, the flow resembles a classic mixing-layer experiment. Unlike molecular motions, the local particle movements result from shearing of the flow; hence, the diffusion experiments were performed for different shear rates by changing the sidewall conditions of the test section, and by varying the flow rate and the channel width. In addition, experiments were also conducted using different sizes of glass beads to examine the scaling of the diffusion process. A simple analysis based on the diffusion equation shows that the thickness of the mixing layer increases with the square-root of downstream distance and depends on the magnitude of the velocity fluctuations relative to the mean velocity. The results are also consistent with other studies that suggest that the diffusion coefficient is proportional to the particle diameter and the square-root of the granular temperature.

Global model of crossflow microfiltration based on hydrodynamic particle diffusion

A global model of crossflow microfiltration is presented which predicts filtration flux and concentrated particle layer thickness as they vary along the axial coordinate of a crossflow microfilter under steady or quasi-steady operation. The model incorporates the concepts of shear-induced hydrodynamic diffusion of particles and the transport along the membrane surface of a fluid-like concentrated particle layer whose viscosity is a function of particle volume fraction. Conditions for which a stagnant layer of packed particles exists beneath a flowing layer are also described. This particle layer may impart a hydraulic resistance to flow of permeate which is modeled by standard filtration theory. The particle volume fraction and the suspension velocity within the flowing layer are predicted as they vary with both distance from the membrane surface and with distance from the channel entrance. The effects of shear rate, transmembrane pressure drop, particle size and other quantities on the permeate flux are discussed.

Viscous resuspension

The resuspension of a settled layer of particles in a shear flow, typically associated with turbulence, has also been found to occur under viscous conditions (Gadala-Maria, F.A., 1979, Ph.D. Thesis, Stanford University). In this work we demonstrate that viscous resuspension, characterized by the equilibrium height achieved by an initially settled bed of particles expanding in the presence of an applied shear stress, results from a balance between the downward flux of particles due to gravity and an upward flux arising from shear-induced particle diffusion. Predictions of this equilibrium height using independent measurements of the shear-induced diffusion coefficient are shown to be in approximate agreement with experimental observations.