Sphere In Shear Flow Research Articles

Continuum dynamics of suspensions at low Reynolds number

The dynamics of suspensions of particles has been an active area of research since Einstein first calculated the leading-order correction to the viscosity of a suspension of spherical particles (Einstein, Proc. R. Soc., vol. A102, 1906, pp. 161–179). Since then, researchers have strived to develop an accurate description of the behaviours of suspensions that goes beyond just leading order in the particle volume fraction. Here, we consider the low-Reynolds-number behaviour of a suspension of spherical particles. Working from the Green's functions for the flow due to a single particle, we derive a continuum-level description of the dynamics of suspensions. Our analysis corrects an error in the derivation of these equations in the work of Jackson (Chem. Engng Sci., vol. 52, 1997, pp. 2457–2469) and leads to stable equations of motion for the particles and fluid. In addition, our resulting equations naturally give the sedimentation speed for suspended particles and correct a separate error in the calculation by Batchelor (J. Fluid Mech., vol. 52, 1972, pp. 245–268). Using the pair-correlation function for hard spheres, we are able to compute the sedimentation speed out to seventh order in the volume fraction, which agrees with experimental data up to 30 %–35 %, and also get higher-order corrections to the suspension viscosity, which agree with experiments up to $\sim$ 15 %. Then, using the pair distribution for spheres in shear flow, we find alterations to both the first and second normal stresses.

Drag and Lift Forces Acting on a Sphere in Shear Flow of Power-Law Fluid

Laminar flow of a power-law fluid over a sphere is considered for unbounded shear flow. The Navier–Stokes equations with power-law viscosity are solved numerically using an in-house developed CFD package. Vorticities structures downstream of particle are suppressed for powerlaw fluid. The shear rate influence on drag force is negligible for power index close to unit, and the drag force appreciably decreases with falling power index. For small Reynolds numbers, the lift force coefficient monotonically decreases against the power index and exhibits an opposite behavior for moderate values of Reynolds numbers. The results of the parametric studies are used to derive correlations for the drag force and to detect the hydrodynamic differences from uniform flow. The investigation parameters varied within the following ranges: power-law index 0.3 ≤ n ≤ 1, Reynolds number 0 < Re ≤ 150, and dimensionless shear rate 0.05 ≤ s ≤ 0.4.

Revisiting the use of the immersed-boundary lattice-Boltzmann method for simulations of suspended particles

The immersed-boundary lattice-Boltzmann method (IB-LBM) is increasingly being used in simulations of dense suspensions. These systems are computationally very expensive and can strongly benefit from lower resolutions that still maintain the desired accuracy for the quantities of interest. IB-LBM has a number of free parameters that have to be defined, often without exact knowledge of the tradeoffs, since their behavior in low resolutions is not well understood. Such parameters are the lattice constant Δx, the number of vertices N_{v}, the interpolation kernel ϕ, and the LBM relaxation time τ. We investigate the effect of these IB-LBM parameters on a number of straightforward but challenging benchmarks. The systems considered are (a) the flow of a single sphere in shear flow, (b) the collision of two spheres in shear flow, and (c) the lubrication interaction of two spheres. All benchmarks are performed in three dimensions. The first two systems are used for determining two effective radii: the hydrodynamic radius r_{hyd} and the particle interaction radius r_{inter}. The last system is used to establish the numerical robustness of the lubrication forces, used to probe the hydrodynamic interactions in the limit of small gaps. Our results show that lower spatial resolutions result in larger hydrodynamic and interaction radii, while surface densities should be chosen above two vertices per LU^{2} result to prevent fluid penetration in underresolved meshes. Underresolved meshes also failed to produce the migration of particles toward the center of the domain due to lift forces in Couette flow, mostly noticeable for IBM-kernel ϕ_{2}. Kernel ϕ_{4}, despite being more robust toward mesh resolution, produces a notable membrane thickness, leading to the breakdown of the lubrication forces in larger gaps, and its use in dense suspensions where the mean particle distances are small can result in undesired behavior. r_{hyd} is measured to be different from r_{inter}, suggesting that there is no consistent measure to recalibrate the radius of the suspended particle.

Pair-particle dynamics and microstructure in sheared colloidal suspensions:Simulation and Smoluchowski theory

The Smoluchowski equation (SE) approach reduced to pair level provides an accepted method for analysis of the pair microstructure, i.e., the pair distribution functiong(r), in sheared colloidal suspensions. Under dilute conditions, the resulting problem is well-defined, but for concentrated suspensions the coefficients of the pair SE are unclear. This work outlines a recently developed theoretical approach for analytical and numerical study of the pair SE for concentrated colloidal suspensions of spheres in shear flow, and then focuses upon evaluation of coefficients and related properties of the problem from Stokesian Dynamics simulation, over a wide range of particle volume fraction, ϕ, and Péclet number (ratio of shear to Brownian motion). The pair distribution functiondetermined from the SE theory is in generally good agreement with Stokesian Dynamics, as are the computed viscosity and normal stresses of the material. The primary focus of the work is to consider the pair relative velocity predicted by the theoryin comparison to Stokesian Dynamics simulations, as well as to evaluate quantities related to the hydrodynamic dispersion needed in the theoretical approach. The pair dynamics for moderate particle volume fraction, 0.20 ⩽ ϕ ⩽ 0.35, are found to be remarkably different from the form for an isolated pair of spheres, and at ϕ ⩾ 0.40 a qualitative change is again seen. Agreement of the theory and simulation on the primary features of the particle motion and structure is good, and discrepancies are clearly delineated.

Hydrodynamic Interactions between Two Equally Sized Spheres in Viscoelastic Fluids in Shear Flow

The effect of using a viscoelastic suspending medium on the in-plane hydrodynamic interaction between two equally sized spheres in shear flow is studied experimentally to understand flow-induced assembly behavior (i.e., string formation). A counterrotating device equipped with a Couette geometry is used together with quantitative videomicroscopy. To evaluate the effects of differences in rheological properties of the suspending media, fluids have been selected that highlight specific constitutive features. These include a reference Newtonian fluid (N), a constant-viscosity, high-elasticity Boger fluid (BF), a wormlike micellar surfactant solution with a single dominant relaxation time (WMS), and a broad spectrum shear-thinning elastic polymer solution (ST). As expected, the trajectories are symmetric in the Newtonian fluid. In the BF, the midpoints of the spheres are observed to remain in the same plane before and after the interaction, as in the Newtonian fluid, although the path lines are in this case no longer symmetric. Interactions in the ST and WMS are highly asymmetric. Two fundamentally different kinds of path lines are observed in the WMS and ST: reversing and open trajectories. The type of trajectory depends on the initial configuration of the spheres with respect to each other and on the shear rate. On the basis of the obtained results, shear-thinning of the viscosity seems to be the key rheological parameter that determines the overall nature of the interactions, rather than the relative magnitude of the normal stress differences.

Effect of viscoelasticity on the rotation of a sphere in shear flow

Abstract When particles are dispersed in viscoelastic rather than Newtonian media, the hydrodynamics will be changed entailing differences in suspension rheology. The disturbance velocity profiles and stress distributions around the particle will depend on the viscoelastic material functions. Even in inertialess flows, changes in particle rotation and migration will occur. The problem of the rotation of a single spherical particle in simple shear flow in viscoelastic fluids was recently studied to understand the effects of changes in the rheological properties with both numerical simulations [D’Avino et al., J. Rheol. 52 (2008) 1331–1346] and experiments [Snijkers et al., J. Rheol. 53 (2009) 459–480]. In the simulations, different constitutive models were used to demonstrate the effects of different rheological behavior. In the experiments, fluids with different constitutive properties were chosen. In both studies a slowing down of the rotation speed of the particles was found, when compared to the Newtonian case, as elasticity increases. Surprisingly, the extent of the slowing down of the rotation rate did not depend strongly on the details of the fluid rheology, but primarily on the Weissenberg number defined as the ratio between the first normal stress difference and the shear stress. In the present work, a quantitative comparison between the experimental measurements and novel simulation results is made by considering more realistic constitutive equations as compared to the model fluids used in previous numerical simulations [D’Avino et al., J. Rheol. 52 (2008) 1331–1346]. A multimode Giesekus model with Newtonian solvent as constitutive equation is fitted to the experimentally obtained linear and nonlinear fluid properties and used to simulate the rotation of a torque-free sphere in a range of Weissenberg numbers similar to those in the experiments. A good agreement between the experimental and numerical results is obtained. The local torque and pressure distributions on the particle surface calculated by simulations are shown.

Simulating deformable particle suspensions using a coupled lattice-Boltzmann and finite-element method

A novel method is developed to simulate suspensions of deformable particles by coupling the lattice-Boltzmann method (LBM) for the fluid phase to a linear finite-element analysis (FEA) describing particle deformation. The methodology addresses the need for an efficient method to simulate large numbers of three-dimensional and deformable particles at high volume fraction in order to capture suspension rheology, microstructure, and self-diffusion in a variety of applications. The robustness and accuracy of the LBM–FEA method is demonstrated by simulating an inflating thin-walled sphere, a deformable spherical capsule in shear flow, a settling sphere in a confined channel, two approaching spheres, spheres in shear flow, and red blood cell deformation in flow chambers. Additionally, simulations of suspensions of hundreds of biconcave red blood cells at 40% volume fraction produce continuum-scale physics and accurately predict suspension viscosity and the shear-thinning behaviour of blood. Simulations of fluid-filled spherical capsules which have red-blood-cell membrane properties also display deformation-induced shear-thinning behaviour at 40% volume fraction, although the suspension viscosity is significantly lower than blood.

Effect of free rotation on the motion of a solid sphere in linear shear flow at moderate Re

The effect of free rotation on the drag and lift forces on a solid sphere in unbounded linear shear flow is investigated. The sphere Reynolds number, Re=|ur|d/ν, is in the range 0.5–200, where ur is the slip velocity. Direct numerical simulations of three-dimensional flow past an isolated sphere are performed using spectral methods. The sphere is allowed to rotate and translate freely in the shear flow in response to the hydrodynamic forces and torque acting on it. The effect of free rotation is studied in a systematic way by considering three sets of simulations. In the first set of simulations, we study how fast a pure rotational or translational motion of the sphere approaches steady state. The “history” effect of rotational and translational motions are compared. Results at high Re are found to be significantly different from the analytical prediction based on low Re theory. In steady simulations, the sphere is allowed to rotate in a torque-free condition. The torque-free rotation rate and the drag and lift forces under such a condition are reported. Comparisons are drawn with the forces on a nonrotating sphere. The effect of rotation is observed to be high in the range 5≲Re≲100. The total lift force is shown to be the sum of the lift force on a nonrotating sphere in shear flow and the lift force on a sphere that is forced to spin at the torque-free rotation rate in a uniform stream (Magnus effect). Finally, we consider the effect of combined free rotation and translation. It is observed that under such combined motion, the sphere achieves translational equilibrium with the local fluid much earlier than it can achieve the zero torque state. The sphere rotates at a rate much lower than the torque-free rotation rate. Free rotation is shown to have a negligible effect on the unsteady drag and lift forces.

Behavior of multiple spheres in shear and poiseuille flow fields at low Reynolds number

The multipole truncation technique developed previously by the authors for describing the hydrodynamic interaction of three-dimensional finite clusters of spherical particles at low Reynolds number is used to obtain solutions for the motion of freely suspended particles in planar shear and/or Poiseuille flow fields. Instantaneous configurations containing up to 13 particles are studied and quasi-steady trajectories are obtained for time-varying configurations of two or three particles. Interesting applications of the theory presented in this paper include the time-dependent motion of a chain of spheres with fixed interparticle spacings in shear flow which may serve as a model to study the deformation of a polymer chain and the motion of neutrally buoyant configurations in planar Poiseuille flow to study the lateral migration of particles. The motion of a neutrally buoyant sphere in the presence of a rigidly held sphere in shear flow is also examined. This study reveals a very intriguing behavior in which the free sphere rolls along the fixed sphere but an adverse pressure gradient forces a retrograde motion of its center.

A sphere in shear flow at finite Reynolds number: effect of shear on particle lift, drag, and heat transfer

Three-dimensional numerical solutions have been obtained for steady, linear shear flow past a fixed, heated spherical particle over a wide range of Reynolds number (0.1 [les ] R [les ] 100) and dimensionless shear rates (0.005 [les ] α [les ] 0.4). The results indicate that at a fixed shear rate, the dimensionless lift coefficient is approximately constant over a wide range of intermediate Reynolds numbers, and the drag coefficient also remains constant when normalized by the known values of drag for a sphere in uniform flow. At lower values of the Reynolds number, the lift and drag coefficients increase sharply with decreasing R, with the lift coefficient being directly proportional to R−½. For the range of shear rates studied here, the rate of heat transfer to the particle surface was found to depend only on the Reynolds number, that is, it was insensitive to the shear rate. The dimensionless rate of heat transfer, the Nussel number Nu, was seen to increase monotonically with R.

Interactions between colloidal particles in simple shear flow

Recent progress in experimental and theoretical developments dealing with colloidal. interactions between two spheres in shear flow is reviewed. A systematic comparison is made between spheres suspended in simple electrolyte and in cationic polyelectrolye solutions. Microrheological observations, performed with the traveling microtube, make an in-depth investigation possible of the colloidal forces and the mechanisms of polymer bridge formation. Reasons are discussed for the often-observed aging of colloidal aggregates. Finally, coagulation rates are presented for systems with weak and strong Brownian motion. It is shown that the often-used assumption of additivity of the ortho- and perikinetic coagulation rates is incorrect.

The microrheology of colloidal dispersions

Abstract Observations of the spread in the period of rotation of doublets of touching spheres in shear flow due to rotary Brownian motion and of the second moment of the orientation distribution function at high Peclet numbers are reported, and results compared with our recently developed theory. Despite the stringent experimental limitation, the measurements are in general agreement with the predictions.

The microrheology of colloidal dispersions: IV. Pairs of interacting spheres in shear flow

The effect of interaction forces on the trajectories of pairs of equal-sized spheres in shear flow is calculated. Depending on the functional form of the interaction potential there are three modes of behavior: (i) the spheres can collide and form a touching doublet, (ii) they can be captured in an energy minimum, orbiting around each other after capture, or (iii) the interactions are transitory and the spheres separate. For interactions determined jointly by van der Waals attraction and electrostatic repulsion, we have calculated the conditions under which (i) touching doublets from (i.e., in the so-called primary energy minimum) and the system coagulates, (ii) nontouching doublets form (in a secondary minimum) and the system flocculates, or (iii) no permanent doublets form and the system is stable. The results indicate that when Brownian motion can be neglected, dilute spherical sols can be unstable at low and high shear rates, but stable in between.

The microrheology of colloidal dispersions: VI. Chains of spheres in shear flow

The behavior in shear flow of chains of equal-sized interacting spheres depends on their shape and whether or not the spheres touch. Straight nontouching linear aggregates will change length periodically with their orientation, curved ones will bend. When the spheres touch, however, the chains will rotate as rigid bodies. A theory of the behavior of straight chains of nontouching spheres is developed and applied with reasonable success to experiments with chains of polystyrene latex spheres. A theory for the bending of nontouching chains is not yet available, but observations show that the variation in bending angle of a flexible triplet is, under the experimental conditions used, about 25°. A large difference is observed in the behavior in shear flow of nontouching doublets and triplets, thus indicating that caution must be exercised in applying a two-sphere model in calculations of orthokinetic flocculation and coagulation of sols.

The microrheology of colloidal dispersions: V. Primary and secondary doublets of spheres in shear flow

In simple shear flow the period of rotation T of primary doublets (two spheres in contact) should be significantly lower than for secondary doublets: two nontouching spheres, which orbit each other around a gap width heq at which the interaction energy exhibits a minimum value; it follows that heq can, in principle, be calculated from the measured T. Whereas the measured periods of rotation of primary doublets in hydrosols of polystyrene latex spheres (2 and 4 μm diam) were in reasonably good agreement with theory, observations on secondary doublets showed discrepancies which are believed due to nonuniformity of the surface charge distribution on the spheres. By measuring the conditions under which secondary doublets either break apart or are converted into primary doublets, one can test aspects of existing theories of colloid stability. The advantage of the method is that observations can be made on two-sphere interactions only, which form the basis of most such theories. Calculation of the Hamaker constant of polystyrene in water based on the observed transition of secondary to primary doublets of polystyrene latex spheres yields a value of 5 × 10 −19 J, an order of magnitude higher than that observed by all other methods. Reasons for this discrepancy are discussed.

The displacement effect of a sphere in a two-dimensional shear flow

This paper contains a theoretical investigation of the displacement effect of a pitot tube in a shear flow. Viscosity is neglected throughout so that the vorticity field alone is considered.It is first shown that a two-dimensional approach does not produce a large enough displacement effect because it does not include the stretching of vortex tubes that takes place around a three-dimensional pitot tube. Then the three-dimensional problem is considered. A solution is obtained in the plane of symmetry for a sphere in a shear flow. This solution is found by making an assumption about the rate of stetching of vortex tubes perpendicular to the plane of symmetry and then considering the shear flow as a small perturbation of a uniform flow. A solution in the plane of symmetry is sufficient to obtain the displacement effect, which is found to be of the same order as the experimental result obtained by Young &amp; Maas (1936) for a conventional pitot tube. The sphere may be considered to represent a conventional pitot tube (of slightly smaller diameter), so it is concluded that a large part of the displacement effect of a pitot tube may be accounted for without the inclusion of viscosity, i.e. by consideration of the vorticity field alone.To a first approximation, the vorticity in the plane of symmetry is found to depend only on the distance from the centre of the sphere.An outline of shear flows past some two-dimensional bodies is given in an appendix. The bodies considered are a circular cylinder and a two-dimensional ‘pitot-tube’ consisting of two parallel semi-infinite plates.