Motion of Two Spheres

Motion of two spheres along a line passing through their centers. In the Stokes approximation, an exact closed-form solution of the axisymmetric problem about the motion of two spheres with the same velocity was obtained in [463]. This solution is practically important and can be used for estimating the accuracy of approximate methods, which are applied for solving more complicated problems on the hydrodynamic interaction of particles. 

The force acting on each of the spheres is described by the formula [179] 

Since A 1, it follows from (2.9.1) that the velocity of the steady-state motion of each of the spheres in the ensemble is greater than the velocity of a single sphere. 

In the gravitational field, the steady-state velocities of particles and drops of various shapes (or mass) are different [179, 417]. Therefore, the distance between the centers of particles is not constant, and hence, the entire problem about the hydrodynamic interaction is, strictly speaking, nonsteady. It was shown in [417] that for Re l/a this problem can be treated as quasisteady. 

Motion of two spheres arbitrarily located with respect to each other. Let us consider two spherical particles of equal radius remote from each other and moving with the same velocity U. The Stokes force acting on each particle is determined by the formula [179] 

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Happel, J., and Pfeffer, R. The Motion of Two Spheres Following Each Other in a Viscous Fluid. AIChEJ. 6 (1960) 129-33. 

As a first approximation to the motion of two spheres in a solvent (which can be regarded as a continuum), the spheres can be presumed to move about the solvent sufficiently slowly that the very much simplified Navier—Stokes equation of fluid flow is applicable. The application of a pressure gradient VP(r) in the fluid develops velocity gradients within the fluid, Vv(r). If another force F(r) is included in the fluid, this can generate a pressure gradient and further affect the velocity gradients. The Navier— Stokes equations [476] becomes... 

Of special interest is the motion of two spheres with different radii and velocities. The approach used to solve this problem is the same as before. In a special case, when one of the spheres is much bigger then the other, the latter can be considered as moving near a solid plane. If the small sphere moves perpendicular to flat wall, the coefflcient X in the expression (8.36) for resistance force is determined by... 

O Neill (1970) and coworkers solved the asymmetrical motion of two equal spheres and the motion of two spheres approaching each other or a solid wall. With these works and that of Goldman, Cox, and Brenner (1967), a complete solution of forces acting on two equal solid psheres moving in an unbounded medium along their line of centers in known. 

The comprehensive work of Happel and Brenner (1965) treats motion of two spheres as well as many particles. An approximate solution for the case of two spherical particles of the same diameter that are isotropic with respect to translation and rotation in an unbounded medium at rest at infinity gives the force as... 

Fig. 2 Hydrodynamic interactions from LB simulations with particles of radius a = 2.5b. The solid symbols are the LB friction coefiBcients, and for the relative motion of two spheres along the line of centers (left) and perpendicular to the line of centers (right). Results are compared with essentially exact results from a multipole code [21] in the same geometry (stdid lines)...

A. Nir and A. Acrivos, On the Creeping Motion of Two Arbitrary-sized Touching Spheres in a Linear Shear Field, J. Fluid Mech., 59, 209-223 (1973). 

Attraction of Two Spheres—An expression for the attraction between two or more particles, based on collision theory rather than attraction due to the motion of spheres, may be developed by the methods of dimensional analysis. Let the force of attraction between two particles of diameters d and d2 be F and assume that when the particles are close enough so that the gaseous film enveloping each particle coalesces over a region about the point of contact, then the whole attraction is due to a free surface energy a. If the average surface of contact of the particles is denoted by Sc then... 

When the gap width between two particles becomes very small, numerical calculations involved in both the bispherical coordinate method and the boundary collocation technique are computationally intensive because the number of terms in the series required to be retained to achieve a desired accuracy increases tremendously. To solve this near-contact motion more effectively and accurately, Loewenberg and Davis [43] developed a lubrication solution for the electrophoretic motion of two spherical particles in near contact along their line of centers with the assumption of infinitely thin ion cloud. The axisymmetric motion of the two particles in near contact can be approximated as the pairwise motion of the spheres in point contact plus a deviation stemming from their relative motion caused by the contact force. The lubrication results agree very well with those obtained from the collocation method. It is shown that near contact electrophoretic interparticle... 

The applicability of the Enskog theory for high pressures is explained by the vortical character of the thermal motion of molecules. For molecular motions presented in Fig. 1 the relative motion of two neighboring molecules is only essential. In this case all molecules being on some sphere (circle) interact with their neighbors on the next spheres (circles) identically. So, the conditions for the applicability of two-particle approximation arise. 

At present, the only multiparticle system for which exact values of the resistance tensors can be determined is that of two spheres. It tnms out that all types of hydrodynamic flows related to the motion of two spherical particles (of radii and R can be expressed as superpositions of the elementary processes depicted in Fignre 5.35.278,412,421,422,447-456... 

In this section we consider the detailed analysis for two applications of lubrication theory the classic slider-block problem that was depicted in the previous section and the motion of a sphere toward an infinite plane wall when the sphere is very close to the wall. It is the usual practice in lubrication theory to focus directly on the motion in the thin gap using (5-69)-(5-72), or their solutions (5-74) and (5-79), without any mention of the asymptotic nature of the problem or of the fact that these equations (and their solutions) represent only a first approximation to the full solution in the lubrication layer. We adopt the same approach here but with the formal justification of the preceding section. 

G5b. Goldman, A. J., Cox, R. G., and Brenner, H., Slow viscous motion of two identical arbitrarily oriented spheres through a viscous fluid. Chem. Eng. Sci. (in press) see also Goldman, A. J., Investigations in low Reynolds number fluid-particle dynamics. Ph.D. Dissertation, New York University, New York, 1966. 

Another method consists in the exact solution of the problem of motion of two spherical particles at any value of a/l. The solution is derived in a special bipolar system of coordinates [5). As an illustration of the method, consider the motion of two identical solid spheres with constant and equal velocities along the line of centers. Introduce the vorticity vector... 

Kynch G. J., The slow motion of two or more spheres through a viscous fluid, J. Fluid Mech, 1959, Vol. 5,... 

Reed L. D., Morrison F. A., The slow Motion of two touching Fluid Spheres along their Line of Centres,... 

Fig.1 Simulation of two-dimensional Brownian motion of a sphere The initial and final position of the sphere is shown by the green and red circle, respectively. Although the mean deviation is zero (since every direction is equally probable), the mean squared deviation is nonzero (4Dt, in two dimensions)...

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