Stokes’ flow around spheres

The analogous problem of impaction of particles on spheres has been studied for application to gas cleaning In packed beds and aerosol senibbing by droplets. Numerical computations of the impaction efliciency for point particles in in viscid flows around spheres have also been made. Before comparing the numerical computations with the available data, we consider theoretical limiting values for the Stokes number for impaction on cylinders and spheres. 

Thus, the task of calculating the diffusion coefficient of a particle is reduced to the task of computing its drag coefficient y. For comphcated particle shapes, one can simulate Stokes flow around the particle, using a finite element simulation, to calculate y. The simplest diffusion translational and rotational diffusion coefficients are those of a spherical particle its one-dimensional translational diffusion coeffi-cient is and its one-dimensional rotational diffusion coefficient is, where n is the dynamical viscosity of the fluid and r is the radius of the sphere. 

The hydrodynamic radius is defined based on a result from fluid mechanics developed by Stokes long ago, the so-called Stokes flow around a sphere. In Stokes flow, the proportionality constant between the force apphed by the flowing fluid on the moving sphere and the velocity of the sphere is called the friction coefficient g. For rigid bodies such as colloidal particles, the friction coefficient is given by... 

Illustration of stationary flow types, (a) Plug flow at the entrance region of a pipe and (b) the Poiseuille flow in the fuUy developed flow region (c) Stokes flow around a sphere. 

Harper and Chang (H4) generalized the analysis for any three-dimensional body and defined a lift tensor related to the translational resistances in Stokes flow. Lin et al (L3) extended Saffman s treatment to give the velocity and pressure fields around a neutrally buoyant sphere, and also calculated the first correction term for the angular velocity, obtaining... 

In a series of papers, Felderhof has devised various methods to solve anew one- and two-sphere Stokes flow problems. First, the classical method of reflections (Happel and Brenner, 1965) was modified and employed to examine two-sphere interactions with mixed slip-stick boundary conditions (Felderhof, 1977 Renland et al, 1978). A novel feature of the latter approach is the use of superposition of forces rather than of velocities as such, the mobility matrix (rather than its inverse, the grand resistance matrix) was derived. Calculations based thereon proved easier, and convergence was more rapid explicit results through terms of 0(/T7) were derived, where p is the nondimensional center-to-center distance between spheres. In a related work, Schmitz and Felderhof (1978) solved Stokes equations around a sphere by the so-called Cartesian ansatz method, avoiding the use of spherical coordinates. They also devised a second method (Schmitz and Felderhof, 1982a), in which... 

The second boundary condition is related to the transformation of the flow profile in the surface layer into the flow profile for the Stokes problem on the flow around a sphere outside the thin surface layer near the particle surface [4] and can be written as... 

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... 

The stream function and radial velocity distribution function for a low-Reynold.s-number flow around a sphere are given by the following expressions due to Stokes ... 

Problem 7-26. An Alternative Derivation of the Solution for Stokes Flow. In Subsection B.4, we showed that the force acting on a sphere that translates through a fluid at low Reynolds number can be expressed in terms of a resistance tensor A in the form F = A U. A generalization of this idea is that the pressure and velocity fields around the sphere must also be a linear function of the vector U, and thus expressible in terms of a vector pressure and a tensor velocity in the form... 

In the limit case 0 -4 oo (high viscosity of the drop substance), the thermocapillary effect does not influence the motion, B -> the flow around the drop will be the same as for a hard sphere, and (5.11.3) implies the Stokes law (2.2.5). For m = 0 (no heat production or independence of the surface tension on temperature), the thermocapillary effect is absent, and (5.11.3) yields a usual drag force for a drop in the translational flow (2.2.15). 

A similar solution can be obtained for the flow in a cylindrical pipe where y is replaced by the radial distance from the axis of the cylinder. There are a few other simple analytic solutions of the Stokes equation, e.g. for the flow around a sphere, etc. (Lamb, 1932). 

This relation coincides with the boundary condition for a viscous flow around solid spheres. In this approximation the velocity distribution at Re l is expressed by Stokes formula. From Stokes velocity distribution v(z,0) it is easy to calculate the viscous stresses acting on the surface of the sphere and the equilibrating surface tension gradient... 

Relative Viscosity of Suspensions One of the most interesting derivations of the T vs. (() dependence (covering the full range of concentration) was published by Simha [1952]. He considered the effects of concentration on the hydrodynamic interactions between suspended particles of finite size. (Note that previously the particles were simply considered point centers of force that decayed with cube of the distance.) Simha adopted a cage model, placing each solid, spherical particle of radius a inside a spherical enclosure of radius b. At distances x < b, the presence of other particles does not influence flow around the central sphere and the Stokes relation is satisfied. This assumption leads to a modified Einstein [1906, 1911] relation ... 

Boundary conditions far from the sphere suggest that only L2(cos0) is required for Stokes s flow around solid spheres and gas bubbles. Hence, A = B = C = = 0 for n 2 and... 

In flow around a sphere, for example, the fluid changes velocity and direction in a complex manner. If the inertia effects in this case were important, it would be necessary to keep all the terms in the three Navier-Stokes equations. Experiments show that at a Reynolds number below about 1, the inertia effects are small and can be omitted. Hence, the equations of motion, Eqs. (3.7-36)-(3.7-39) for creeping flow of an incompressible fluid, become... 

Only in the Stokes law regime, (Re)p < 0.2, have theoretieal methods of evaluating Cd met with much success. The theoretical analysis starts with the viscous flow around a rigid sphere, which can be expressed as... 

A second important flow at small Reynolds numbers is Stokes flow — the flow of a viscous fluid around a sphere of radius a, which is moving with a speed V. The derivation of the actual velocity field and the resulting drag force f D are complicated, and we will just quote the result here, which reads ... 

The time required for a ball to fall a given distance in a fluid is probably the simplest and certainly one of the oldest viscosity tests (Stokes, 1851). Unfortunately, creeping flow around a sphere is very complex. Thus the falling ball is really an index test and requires a constitutive equation for complete analysis. Analyses of the flow have been made for inelastic (Gottlieb, 1979 Beris et al., 1985) and viscoelastic fluids (Hassager and Bisgaard, 1983 Graham et al., 1989). Usually an apparent viscosity based on the Newtonian analysis is reported. 

If the relative velocity is sufficiently low, the fluid streamlines can follow the contour of the body almost completely all the way around (this is called creeping flow). For this case, the microscopic momentum balance equations in spherical coordinates for the two-dimensional flow [vr(r, 0), v0(r, 0)] of a Newtonian fluid were solved by Stokes for the distribution of pressure and the local stress components. These equations can then be integrated over the surface of the sphere to determine the total drag acting on the sphere, two-thirds of which results from viscous drag and one-third from the non-uniform pressure distribution (refered to as form drag). The result can be expressed in dimensionless form as a theoretical expression for the drag coefficient ... 

A number of authors from Ladenburg (LI) to Happel and Byrne (H4) have derived such correction factors for the movement of a fluid past a rigid sphere held on the axis of symmetry of the cylindrical container. In a recent article, Brenner (B8) has generalized the usual method of reflections. The Navier-Stokes equations of motion around a rigid sphere, with use of an added reflection flow, gives an approximate solution for the ratio of sphere velocity in an infinite space to that in a tower of diameter Dr ... 

If the spherical particle were not present in Figure 2.3, the volume elements of the flowing fluid would move upward in straight lines. In the presence of the particle, however, the flow profile is distorted around the sphere in the manner suggested by Figure 2.3. It is apparent that the velocity of any volume element passing the sphere is a function of both time and location and must be described as such in any quantitative treatment. The trajectory of such a volume element is called the flow streamline function. For spherical particles, this was analyzed by G. G. Stokes in 1850. 

For B > the flow pattern around the drop is similar to the Hadamard-Rybczynski flow (Figure 2.2). As B decreases, the fluid circulation intensity decreases within the drop, and vanishes for B = -f. Under further decrease (B < - ), a circulation zone is produced around the drop. The direction of the inner circulation becomes opposite to the direction in the Hadamard-Rybczynski case. It follows from (5.11.3) that the drag force acting on the drop exceeds the Stokes force for a hard sphere. 

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