Stokes’ flow around

The uniform flow at time t = 0 rapidly changed into Stokes flow around the obstacle. By / 100 psec a pair of counter-rotating eddies had developed... 

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. 

In the derivation, Stokes flow is assumed for the particle, which assumes the liquid medium around the particle flows as a continuum. Hence, the particle size must be significantly larger than the molecules in the liquid matrix (such as H2O molecules in water). The formulation is not necessarily valid for particles smaller than or about the same size as the matrix molecules themselves. 

Thus, analytic solutions for flow around a spherical particle have little value for Re > 1. For Re somewhat greater than unity, the most accurate representation of the flow field is given by numerical solution of the full Navier-Stokes equation, while empirical forms should be used for C. These results are discussed in Chapter 5. 

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... 

If flow around the drop were laminar, then Stokes Law would govern and ... 

Oll/water settling. Flow around settling oil drops in water or water drops in oil is laminar, so Stokes Law governs. Terminal drop velocity is ... 

Impaction When an air stream containing particles flows around a cylindrical collector, the particle will follow the streamlines until they diverge around the collector. The particles because of their mass will have sufficient momentum to continue to move toward the cylinder and break through the streamlines, as shown in Figure 8.3. The collection efficiency by this inertial impaction mechanism is the function of the Stokes and the Reynolds number as ... 

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 ... 

For the mechanical behavior of two particle-fluid systems to be simitar, it is necessary to have geometric, hydrodynamic, and particle trajectory similarity. Hydrodynamic similarity is achieved by fixing the Reynolds number for the flow around the collector. By (4.26), similarity of the particle trajectories depends on the Stokes number. Trajectory similarity also requires that the particle come within one radius of the surface at the same relative location. This means that the interception parameter, R = dp/L, must also be preserved. 

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. 

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... 

The Reynolds number, which characterizes the importance of inertial forces compared to viscous forces is around unity. This implies that the non-linear terms in the Navier-Stokes equations are weak, easing the task of solving these equations. For some systems, the assumption of Stokes flow may be reasonable, i.e., the inertial terms are set equal to zero this affords a significant simplification of the fluid flow problem [160]. The Reynolds number is independent of pressure, when everything else is held constant. 

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 ... 

The recent development of tensorial schemes for characterizing the intrinsic hydrodynamic resistance of particles of arbitrary shape, and the application of singular perturbation techniques to obtain asymptotic solutions of the Navier-Stokes equations at small Reynolds numbers constitute significant contributions to oim understanding of slow viscous flow around bodies. It is with these topics that this review is primarily concerned. In presenting this material we have elected to use Gibbs polyadics in preference to conventional tensor notation. For in our view, the former symbolism— dealing as it does with direction as a primitive concept—is more closely related to the physical world in which we live than is the latter notation. 

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