Imposed Shear Flow Research Articles

Shear-Induced Diffusivity in a Dilute Bidisperse Suspension of Hard Spheres

In this paper we calculate the shear-induced self and gradient diffusivities in a dilute bidisperse suspension of hard spheres. Unlike the interaction of identical spheres, the center of mass of a pair of unequally sized spheres does not translate with the imposed shear flow and hence the radial and tangential drifts of the center of mass were tracked using appropriately defined mobility functions. Results of the trajectory calculations show that the minimum surface-to-surface distance ξmin of the limiting closed orbit trajectories was a minimum for equally sized spheres (size ratio λ=1) and monotonically increased for λ→0 and λ→∞. In addition to purely hydrodynamic interactions, trajectories and diffusivities are examined for the case where the hard-sphere radius exceeds the true radius by a factor of (1+ϵs) such as what would be found for rough spheres (da Cunha, F. R., and Hinch, E. J., J. Fluid Mech. 309, 211 (1996)). The force of one sphere on the other at contact was shown to be a strong function of the size ratio λ, with the force being maximum for the equally sized sphere case. The tracer diffusivities were analytically solved for the case where hydrodynamics were absent. For this case, the diffusivities were seen to increase with λ for a dimensionless separation distance, ϵs⪡1, as a result of the increased rate of interaction between a large test sphere and the small background spheres. On the other hand, for ϵs⪢1 and λ the diffusivities were seen to be decreasing functions of λ. A comparison with numerical results shows that the two match in the far-field limit, while they diverge when hydrodynamic interactions become important in the near-field limit. Finally, gradient diffusivities DC were calculated analytically and numerically and were found to be roughly an order of magnitude greater than the corresponding tracer diffusivities.

Morphological instability in a directionally solidifying binary solution with an imposed shear flow

The effect of an imposed shear flow on the stability of directionally solidifying binary alloys is investigated. It is shown that without the imposed shear flow the system is dominated by stationary boundary-layer-mode convection, a convection of salt-finger type confined to the solute boundary layer above the melt/mush interface. When the shear flow (no matter how small) is imposed, the boundary-layer mode becomes a longitudinal mode (roll-axis parallel to the imposed flow) propagating in the direction perpendicular to the shear flow, while the modes containing a transverse component are inhibited. As the shear flow becomes large enough, a transverse mode (roll-axis perpendicular to the imposed flow) of very unstable characteristics is induced. This mode, called the morphological mode, can exist even without buoyancy. It is triggered by the flow induced in the mushy layer through the Bernoulli force, a pressure variation resulting from the imposed flow passing along the corrugated melt/mush interface. It, nonetheless, has no relation to the boundary layer instability of the shear flow. The effect of imposed shear flow is so significant that the stability characteristics can be entirely different when the intensity of the imposed flow is larger than a critical value, which is calculated in the present paper.

Mesoscale Modelling

Increasingly, industrial materials are being designed to have structure on length scales of tens to thousands of nanometers. These structures are crucial to achieving a particular desired material property. Such structures, however, may depend on the underlying chemistry of the material for their existence. For example, a thousandfold increase in the ionic conductivity of a polymer blend may only occur in a narrow region of a hugely complex phase diagram, the location' of which region can be expected to depend on the molecular chemistry and physics from the monomer scale to the coil size. Traditional Computational Chemistry has proved incapable of dealing with the length and time scales involved in the formation of these ‘Mesoscale’ structures. On the other hand, traditional Computational Physics has proved incapable of consistently incorporating the necessary chemical detail for modelling real industrial materials. In this paper we present two novel methods which successfully address both the chemistry and the physics of mesophase formation. The methods, described in detail, are MesoDyn and Dissipative Particle Dynamics (DPD). Unlike phenomenological theories of materials, such as the Landau models which one finds in much of the computational physics literature, the two models mentioned incorporate molecular geometry and connectivity explicitly. We discuss each of the methods briefly. We then give an overview of how these methods are being used in industry to optimise materials and processes. We discuss previous simulation results for triblock Pluronic surfactants in solution studied with MesoDyn, and for diblock copolymers studied with DPD, where the known experimental changes in morphology from micellar to hexagonal to bicontinuous to lamellar have been successfully reproduced. We also present new results for several systems, including binary and ternary blends, where the third component in the latter system is a diblock copolymer, which acts as a compatibiliser. We discuss the effects of changing solvent character on the material properties of these systems, as well as the effects of an externally imposed shear flow.

The effect of oscillatory shear flow on step bunching

During crystal growth, an imposed shear flow at the crystal–fluid interface can alter the conditions for the onset of morphological instability. In previous work, we studied the effect of time-independent shear flows and anisotropic interface kinetics on the morphological stability of a crystal growing from supersaturated solution. The model assumes that growth is by the motion of elementary steps, which is treated by a macroscopic anisotropic kinetic law; morphological instability corresponds to the bunching of elementary steps. Predictions from linear stability theory indicate that a solution flowing above a vicinal face of a crystal can either enhance or prevent the development of step bunches, depending on the direction of the steady shear flow in relation to the direction of step motion; this is also observed in experiments. Here we extend the linear stability analysis to include the effect of an oscillatory shear flow on the morphological stability of a crystal growing from solution and present results for a model system for a range of oscillatory shear rate amplitudes and frequencies both with and without a steady shear component. In the presence of a steady shear flow, modulation can either stabilize or destabilize the system, depending on the modulation amplitude and frequency. Numerical solutions of the linearized Navier–Stokes and diffusion equations and an approximate analytical treatment show that the flow oscillations weakens both the stabilization and destabilization induced by steady-state flow. This weakening comes from mixing of solution above the perturbed interface and a modification to the phase shift between the interface perturbation wave and the corresponding concentration and flow waves. Optimal values of modulation frequency and amplitude are found when the steady flow is destabilizing.

Electrohydrodynamic effects on the deformation and orientation of a liquid capsule in a linear flow

The role of a uniform electric field on the deformation and orientation of a liquid capsule with a viscoelastic membrane is considered analytically in the small deformation limit. The capsule is freely suspended either in a quiescent fluid or in a shear flow. The viscoelasticity of the membrane is taken into account by the Kelvin–Voigt model and the electrohydrodynamic flow is analyzed on the basis of the leaky dielectric model. In this article, we consider three different prototype models of capsules; viz., a neo-Hookean (incompressible isotropic) membrane, a red blood cell-type (area-preserving) membrane, and an interfacial-tension droplet. The deformed capsule shape from its initial sphericity and its orientaion are determined from the linearized governing equations and boundary conditions in the limit of small deformations. The asymptotic theory shows that the degree of capsule deformation induced by a uniform electric field alone is independent of the surface viscosity of the capsule as well as the viscosity ratio between the two fluids inside and outside the capsule. Meanwhile, in the presence of an imposed shear flow, the degree of deformation depends on the surface viscosity with preserving still the independence of the viscosity ratio. For an illustrative purpose, experimental results for the role of a uniform electric field on the orientation of an interfacial-tension droplet in a shear flow are discussed briefly.

Friction and mobility for colloidal spheres in Stokes flow near a boundary: The multipole method and applications

We obtain the many-body hydrodynamic friction and mobility matrices describing the motion in a fluid of N hard-spheres with stick boundary conditions in the presence of a planar hard wall or free surface using (1) a multipole expansion of the hydrodynamic force densities induced on the spheres and (2) an image representation to account for the fluid boundary. The coupled multipole equations may be truncated at any order to give positive definite approximations to the exact friction and mobility matrices. An extension of the Bossis–Brady lubrication correction to the friction matrix is also discussed and included. The resulting method for computing the mobility matrix may be used for the Stokesian or Brownian dynamics simulation of N spheres subject to interparticle and external forces and imposed shear flow. We illustrate the method by performing Stokesian dynamics simulation of particles near a hard wall. The simulations exhibit the rapid convergence of the multipole truncation scheme including lubrication corrections.

Hele Shaw Convection with Imposed Shear Flows: Boundary-Layer Formulation

The nonlinear convection forced by the boundaries of a Hele Shaw cell to align perpendicular to an imposed shear flow was analytically investigated by the boundary-layer method. The imposed shear flow may be a Couette flow that extends throughout the convecting layer or flow confined to a boundary, depending on the geometry of the Hele Shaw cell. This study examined the case in which the imposed shear flow has a boundary-layer structure and its interaction with the convecting interior. Analytical solutions for both the boundary layer and interior were obtained. The study revealed the following.For large aspect ratio A, the interaction of the imposed shear flow and convection is confined to the boundary layer. The boundary layer is a viscous rather than a thermal layer. The results showed that the range of validity of the Hele Shaw equations used in the literature is of order 1/A2. For an asymptotically large aspect ratio A up to order 1/A2, the velocity in the y-direction must be zero. The velocity in the x-direction and the z-direction has a parabolic dependence on y, but the temperature perturbation does not depend on y. These results may have implication for convection in porous media.

Shear-induced clustering in a simple driven diffusive model

We study a simple lattice model of shear-induced clustering in two dimensions in which clusters of particles aggregate under an imposed shear flow and fragment stochastically. Two non-equilibrium steady states are identified: an unjammed state and a jammed state characterised by a system-spanning cluster. A discontinuous jamming transition with strong hysteresis occurs as the shear rate is increased or fragmentation rate decreased. We study the kinetics of jamming and measure power law cluster size distributions. We also consider some general simulation issues including the role of Galilean invariance.

Rotating convection in a shear flow

Thermal convection in a horizontal layer of Boussinesq fluid is examined. The fluid layer rotates about a given axis with constant angular velocity, and a constant mean shear is maintained in the fluid by uniform differential motion of the horizontal boundaries. The problem is motivated by convection in geophysical and astrophysical contexts, for which the rotation of the system is often significant, and where there is often a significant background shear flow. The imposed shear flow and the rotation of the layer individually tend to align convective rolls parallel to some preferred direction. It is their combined influence upon the preferred orientation of the convective motion that is investigated here, by means of linear stability analysis. This analysis reveals certain parameter values near which the orientation of the preferred rolls depends very sensitively upon the parameters, and many modes may simultaneously be close to marginal stability. Numerical simulations of the fully nonlinear governing equations near such points of sensitivity reveal nonlinear interactions between these modes. One interaction between three resonant modes results in a stable limit cycle that at different times has a planform approximating rolls, hexagons and rhombs. We analyse this oscillatory form of convection in a three–dimensional phase space and explain the various changes in its symmetry that take place as the Rayleigh number is varied.

Linear stability of rotating convection in an imposed shear flow

In many geophysical and astrophysical contexts, thermal convection is influenced by both rotation and an underlying shear flow. The linear theory for thermal convection is presented, with attention restricted to a layer of fluid rotating about a horizontal axis, and plane Couette flow driven by differential motion of the horizontal boundaries.The eigenvalue problem to determine the critical Rayleigh number is solved numerically assuming rigid, fixed-temperature boundaries. The preferred orientation of the convection rolls is found, for different orientations of the rotation vector with respect to the shear flow. For moderate rates of shear and rotation, the preferred roll orientation depends only on their ratio, the Rossby number.It is well known that rotation alone acts to favour rolls aligned with the rotation vector, and to suppress rolls of other orientations. Similarly, in a shear flow, rolls parallel to the shear flow are preferred. However, it is found that when the rotation vector and shear flow are parallel, the two effects lead counter-intuitively (as in other, analogous convection problems) to a preference for oblique rolls, and a critical Rayleigh number below that for Rayleigh–Bénard convection.When the boundaries are poorly conducting, the eigenvalue problem is solved analytically by means of an asymptotic expansion in the aspect ratio of the rolls. The behaviour of the stability problem is found to be qualitatively similar to that for fixed-temperature boundaries.Fully nonlinear numerical simulations of the convection are also carried out. These are generally consistent with the linear stability theory, showing convection in the form of rolls near the onset of motion, with the appropriate orientation. More complicated states are found further from critical.

Long-wavelength thermal convection in a weak shear flow

The problem of thermal convection in an imposed shear flow is examined for a horizontal layer of fluid between poorly conducting boundaries. The horizontal scale H of the convective motion near its onset is much greater than the depth h of the fluid layer, with h/H being proportional to the one-fourth power of a Biot number appearing in the condition applied to the temperature at the horizontal boundaries. It is known that an asymptotic expansion in powers of h/H yields a nonlinear long-wavelength evolution equation for the depth-averaged temperature field that is spatially isotropic in the absence of an imposed shear flow, but is strongly anisotropic for 'strong' shear. We derive in this paper a nonlocal long-wavelength equation that bridges these two cases, and that contains each case in the zero-shear and large-shear limits. Using this evolution equation, we show how the shear flow stabilizes the longitudinal rolls to the zigzag instability, and how a preference for a square planform on a periodic square lattice gives way to a preference for longitudinal rolls near onset. The longitudinal rolls may then become unstable as the Rayleigh number is increased. The analytical work is illustrated by some numerical simulations of the full three-dimensional Boussinesq Navier-Stokes equations. The problem of pattern selection on a hexagonal lattice is also discussed, and some new results are presented.

Time Evolution of Shear-Induced Structures in Semidilute Polystyrene Solutions

We investigated time evolution of structures induced by imposing shear flow on semidilute solutions of high molecular weight polystyrene (PS) by means of both flow light scattering and rheology. PS with weight-average molecular weights of 5.48 × 106 and 2.89 × 106 were dissolved in dioctyl phthalate (DOP). After imposing shear flow with a shear rate γ̇ > γ̇c (the critical shear rate for shear-induced concentration fluctuations), the unique butterfly-type scattering pattern appeared. The pattern had a scattering maximum along the flow direction. The wave number qm and the intensity Im at the scattering maximum decreased and increased, respectively, with time and eventually reached steady state values. The rheological experiments that were carried out on the same solution under the same shear flow revealed two stress overshoots. The second overshoot was found to be related to the development of butterfly pattern and its time change, i.e., to the formation and growth of shear-induced structures.

The Onset of Thermal Convection Between Poorly Conducting Horizontal Boundaries in the Presence of a Shear Flow

We derive an evolution equation for three-dimensional thermal convection in an imposed shear flow between horizontal boundaries that conduct heat poorly. Through a long-wavelength asymptotic expansion the governing equations are reduced to a single partial differential equation for the depth-averaged temperature field. A linear stability analysis of this equation indicates that the first instability of the quiescent state is to convective rolls whose axes are aligned with the direction of the shear flow. This result complements previous numerical solutions of the linear stability problem for a fluid between perfectly conducting horizontal boundaries, where a similar preference for such rolls has been found.

Kinetics of crystallization in a shearing colloidal suspension.

We report the kinetics of crystallization as observed in simulations of a charged colloidal suspension under shear. We find that the imposed shear flow inhibits crystal nucleation. Crystallization was only observed for shear rates at or below \ensuremath{\gamma}\ifmmode \dot{}\else \.{}\fi{}${\mathrm{\ensuremath{\tau}}}_{0}$=0.07 (where ${\mathrm{\ensuremath{\tau}}}_{0}$ is the time required for a particle to diffuse the averge interparticle distance in a dilute suspension). Under the influence of these low shear rates the suspension crystallizes into a single shear-aligned crystal, as opposed to the polycrystalline structure found in the absence of shear flow. Crystal growth is found to be proportional to the effective supercooling, as measured from the melting temperature of the shearing crystal. We conclude that, by analogy with the case of crystallization at zero shear, the effective supercooling provides an operational measure of the ``thermodynamic'' stability of the shearing crystal. (c) 1995 The American Physical Society

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 the lift force due to the negative shear-translation coefficient, ultimately resulting in the sphere acquiring some steady separation distance. The equilibrium separation distance and velocity are plotted as a function of the parameter Re2/Res, where Res is the sedimentation Reynolds number.

Shear flow at liquid–liquid interfaces

Two nonmiscible liquids separated by planar interfaces and undergoing shear flow have been simulated with nonequilibrium molecular dynamics (NEMD) methods. A homogeneous shear scheme was used for imposing shear flow in the system. The homogeneous shear algorithm needs to be combined with a profile-unbiased thermostat (PUT) in order to assure meaningful results in our nonhomogenous system. Local values of several quantities such as viscosity, local stream velocity, temperature, shear stress, and rate of entropy production were calculated. Planar Couette flow appears in the ‘‘bulk’’ regions of the system with a slip between the two streams of bulk fluid at the interfaces. The shear stress is constant across the system (PUT results) at low strain rates but at high shear rates the shear stress at the interface is lower than in the bulk region. The shear viscosity at the interfaces is lower than in the bulk region showing that the transport of momentum in the former region is less efficient than in the bulk. At high strain rates, the differences in the local rates of viscous heat production and heat removal result in strong temperature gradients. When comparing the viscosity values in the bulk region of the inhomogeneous system with values computed in independent simulations of the bulk, no important differences are found.

Shear flows of nematic polymers. I. Orienting modes, bifurcations, and steady state rheological predictions

A continuum tensorial theory was formulated to describe the isothermal, incompressible flow of uniaxial rodlike nematic liquid crystalline polymers. The tensor theory was reduced to a vector theory that describes the microstructure of the uniaxial phase by specifying the director orientation and the scalar order parameter. The reduction establishes useful relationships between this theory and the Leslie and Ericksen theory. The model was solved for a given steady simple shear flow, assuming spatial homogeneity. Two types of orienting modes are predicted: (a) the simple aligning mode, in which the microstructure reaches a stable shear dependent steady state for all shear rates, and (b) the complex mode, which at sufficiently high nematic potentials exhibits three flow regimes (tumbling, oscillating, and stationary) according to the strength of the imposed shear flow. For sufficiently low nematic potentials, the complex mode predicts the existence of a single stationary mode. Bifurcation theory was used to fully characterize the existence and transitions between the three regimes of the complex mode. Guidelines for future simplifications of the constitutive equations are given. The frequently reported changes in the sign of normal stress differences with shear rate are captured by the model.

積雲対流による運動量輸送のパラメータ化による全球モデルの系統的誤差の減少

Laboratory experiments on Rayleigh-Benard convection with no imposed mean flow have shown that at sufficiently high Rayleigh numbers a mean horizontal flow u(z) is spontaneously generated. This flow is maintained against viscous dissipation by the countergradient momentum transport of the Reynolds stress uw associated with the tilted plumes. The shear of the mean flow ∂u/∂z at mid-depth may be of either sign, depending upon initial conditions. However, for either sign, the tilt of the plumes is always such that horizontal momentum is transported up the mean gradient in the interior. In similar experiments, but with an externally imposed shear flow, it is found that for a sufficiently large shear and sufficiently large Rayleigh number, the tilt of the plumes is determined by the imposed shear. For either sign of shear the plumes tilt in such a way as to transport momentum up the mean gradient. Sufficient data now exists to express the momentum flux as a function of the Rayleigh number and the imposed shear.In the atmosphere measurements of vertical transport of horizontal momentum, counter to the mean gradient, include those in the sub-cloud layer under fair weather cumulus, and in deep cumulonimbus convection in several squall lines observed during GATE.Although the parameterization of heat and moisture flux by cumulus convection has been successfully incorporated into numerical weather prediction models, momentum flux has generally not been so included. We attempt to include such a parameterization of the vertical momentum flux by convection, using laboratory measurements as a guide. In this application the conventional Rayleigh number is replaced by an atmospheric parameter which is proportional to the vertical gradient of the equivalent potential temperature. The momentum flux by the convective clouds is then parameterized as a function of the modified Rayleigh and Reynolds numbers.The results from a control experiment without the inclusion of this parameterization are compared with an experiment where it is included. Of specific interest to the study are the errors in the trade winds of the Atlantic and Pacific Oceans and the low level monsoon circulation over the Indian Ocean. This formulation is shown to reduce the systematic errors of the low level winds over the central Pacific Ocean quite dramatically by over 10 ms-1. However the reduction of errors over tropical land areas, especially around regions of elevation exceeding 500mb, are smaller.

Equivalent Modons in Simple Shear

Abstract We use a perturbation technique to construct modon solutions that are valid in the presence of weak, linear north-south shear to help assess the application of modons to atmospheric blocking. The model equation is equivalent barotropic and describes the quasi-geostrophic motion in the thin upper layer of a two-layer system in which the lower layer is deep and has no motion other than the imposed shear flow. This model is shown to be a geophysically relevant limit of the more complicated model adopted by McWilliams, who uses a truncated modal projection to establish the connection between the quasi-geostrophic equations and the atmosphere. The new solutions are found to be stable and robust in numerical calculations, but we do not explore the full parametric range of interest.

Nonequilibrium molecular dynamics study of shear flow in soft disks

Nonequilibrium molecular dynamics calculations of homogeneous shear flow in two dimensions have been performed on soft disks close to the freezing transition. Simulations at discrete shear rates, γ̇, reveal complex phase changes and dynamical behavior during extensive shear thinning and a difference in the values of the diagonal components of the pressure tensor. A shear thickening regime was also found near to γ̇=10 but it had disappeared at γ̇=50, in line with recent work by Woodcock. The calculations reveal that there is a change from an amorphous liquid structure at γ̇=1.0 to a fluid microscopically layered along the stream lines at γ̇=10 and higher shear rates. An intermediate two-phase region was observed at γ̇=5 in which both amorphous and ordered phases coexist within the MD cell, containing 896 disks. In the ordered phase the particle dynamics are dominated by the imposed shear flow and can be reproduced in large part by simple models which exclude thermal motion. The time correlation functions are highly oscillatory which reflect well defined motion on a local distance scale. The soft disks spend most of their time trapped between similar disks in adjacent layers, in between rapid periods of ‘‘free-flight’’. Instantaneous pictures of the high shear rate (γ̇≳10) structure consequently show the presence of oscillating transitory strings of highly overlapping soft disks formed roughly perpendicular to those stably formed along the streamlines.