Sphere in shear flow

A rigid sphere in shear flow experiences a force that moves the particle normal to the flow direction. 

Dandy and Dwyer [30] computed numerically the three-dimensional flow around a sphere in shear flow from the continuity and Navier-Stokes equations. The sphere was not allowed to move or rotate. The drag, lift, and heat flux of the sphere was determined. The drag and lift forces were computed over the surface of the sphere from (5.28) and (5.33), respectively. They examined the two contributions to the lift force, the pressure contribution and the viscous contribution. While the viscous contribution always was positive, the pressure contribution would change sign over the surface of the sphere. The pressure... 

Dandy DS, Dwyer HA (1990) A sphere in shear flow at finite Reynolds number Effect of shear on particle lift, drag, and heat transfer. Journal of Fluid Mechanics 216 381-410... 

To specify the velocity field u, we must solve the Navier-Stokes equations subject to the boundary condition (9-160) at infinity. For present purposes, we follow the example of Section C and assume that the Reynolds number, defined here as Re = a2yp/ii, is very small so that the creeping-flow solution for a sphere in shear flow (obtained in Chap. 8, Section B) can be applied throughout the domain in which 6 differs significantly from unity. Hence, from (8-51) and (8-57), we have... 

External streamlines around a fluid sphere in shear flow. Solid lines for = 0 (a gas bubble) dotted line for = 00 (a rigid sphere) Note Aat the flow is less disturbed for a gas bubble. From Baitok and Mason (1958). 

At large inter-droplet distances, each droplet follows a streamline of the external flow field. In shear flow, each of the streamlines corresponds to a different velocity, thereby enabling approach of droplets on different streamlines. The collision frequency per unit volume of monodisperse spheres in shear flow was first derived by Smoluchowski [45] ... 

At high enough electrolyte concentrations the electric potentials are quickly dissipated and this effect vanishes. Since droplets and bubbles are not rigid spheres, they may deform in shear flow. Also, with the presence of emulsifying agents at the interface, the drops will not be non-interacting, as is assumed in the theory. 

Since tangential diffusion has been neglected, only fluid motion contributes to the tangential particle velocity. Near to the collector the fluid appears to be experiencing uniform, linear, shear how. Goldman et al. (1967b) studied the viscous motion of a sphere in Couette flow and found the particle velocity to be proportional to, hut lew than, the undisturbed fluid velocity at the particle s center ... 

The aerodynamic diameter is one of the most common equivalent diameters. It can be defined as the diameter of a unit den.sity sphere with the same terminal settling velocity as the particle being measured. The aerodynamic diameter is commonly used to describe the mt)lioii of particles in collection devices such as cyclone separators and impactors. However, in shear flows, the motion of irregular particles may not be characterized accurately by the equivalent diameter alone because of the complex rotational and translational motion of inegular particles compared with spheres. That is, the path of the irregular particle may not follow that of a particle of the same aerodynamic diameter. It is of course possible that there may be a. sphere of a certain diameter and unit density that deposits at the same point this could be an average point of deposition because of the effects of turbulence or the. stochastic behavior of irregular particles. 

J. A. Schonberg and E. J. Hinch, Inertial migration of a sphere in Poiseuille flow, J. Fluid Mech. 203, 517-524 (1989) E. S. Asmolov, The inertial lift on a small particle in a weak-shear parabolic flow, Phys. of Fluids 14, 15-28 (2002) P. Cherukat, J. B. McLaughlin, and D. S. Dandy, A computational study of the inertial lift on a sphere in a linear shear flow field, Int. J. of Multiphase Flow, 25, 15-33 (1999). 

Sphere in Linear Flows Axisymmetric Extensional Flow and Simple Shear... 

N. A. Frankel and A. Acrivos, Heat and mass transfer from small spheres and cylinders freely suspended in shear flow, Phys. Fluids 11, 1913-18 (1968). 

The rheological properties of the suspension are strongly influenced by the spatial distributiOTi of the particles. The relationship between microstructure and rheology of suspensions has been smdied extensively (Brader 2010 Morris 2009 Vermant and Solomon 2005). Most of earlier smdies dealt with the simplest form of suspensions, in which dilute hard-sphere suspensions are subjected only to hydro-dynamic and thermal forces near the equilibrium state (i.e., Peclet number << 1) (Bergenholtz et al. 2002 Brady 1993 Brady and Vicic 1995). In shear flows of such suspensions, the structure is governed only by the particle volume fraction and the ratio of hydrodynamic to thermal forces, as given by the Peclet number. 

Now consider a coiled polymer molecule as being impenetrable to solvent in the first approximation. A hydrodynamic sphere of equivalent radius Re will be used to approximate the coil dimensions (see Figure 3.13). In shear flow, it exhibits a frictional coefficient of /o. Then according to Stokes law. 

Recently R. L. Hoffman(i5) studied optical diffraction from uniform sphere dispersions in shear flow. As shear rate increased, he observed progressive formation of hexagonally packed particle layers parallel to the shear planes. At the high-shear limiting viscosity, he observed diffraction patterns of hexagonally packed layers of particles. At still higher shear rates, the... 

In 1922 G. B. Jeffery [38] sought to generalize Einstein s analysis for a dilute suspension of spheres to ellipsoids. Like Einstein [27], he considered the sphere to be in shear flow... 

The total drag on the sphere may be obtained, as in steady flow, by integrating the normal and shear stresses over the surface. In terms of the instantaneous velocity U the result is (L4) ... 

Woods,M.E., Krieger,I.M. Rheological studies on dispersions of uniform colloidal spheres. I. Aqueous dispersions in steady shear flow. J. Colloid Sci. 34,91-99 (1970). 

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