Viscous flow around spheres

HAMIELEC, A. E. and Johnson, A. I. Can. J. Chem. Eng. 40 (1962) 41. Viscous flow around fluid spheres at intermediate Reynolds numbers. 

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... 

For higher Reynolds numbers, analytical solutions do not exist, so the numerical solutions must be considered. When k->-oo, this problem corresponds to the viscous flow around a rigid particle and was studied by several authors [10—15]. When k = 0, this problem corresponds to the viscous flow around a spherical bubble and was also studied by several authors [15-18]. The significant phenomena are very well explained in the books of Clift et al. [1] and Sadhal et al. [2]. Values of drag coefficients from numerical solutions for bubbles and rigid spheres are presented in Table 5.2, which shows a good agreement between the different studies. 

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... 

Rubinow and Keller [123] calculated the flow around a rotating sphere moving in a viscous fluid for small Reynolds numbers. They determined the drag, torque, and lift force (Magnus) on the sphere to O(Rep). The results were ... 

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 first group of terms on the right-hand-side of Eq. 4 describes particle transport to a collector surface by Brownian diffusion. NPe is the Peclet number, a ratio of particle transport by fluid advection to transport by molecular or viscous processes. The term As is introduced to account for the effects of neighboring collectors or media grains on the fluid flow around a collector of interest. The results here assume Happel s model (Happel, 1958) for flow around a sphere in a packed bed 4S depends only on the porosity of the bed (Table 1). The derivation for diffusive transport is based on the early work of Levich (1962). 

Nakano, Y. and Tien, C., Viscous incompressible non-Newtonian flow around fluid sphere at intermediate Reynolds number, AIChE J., Vol. 16, No. 4, pp. 554-569, 1970. 

Smirnov L. P., Deryaguin B. V., On inertialess electrostatic Deposition of Aerosol Particles on a Sphere at Flow of viscous Fluid around the Sphere, Colloid J., 1967, No. 3 (in Russian). 

In addition to the simple shear, the rheological properties can also be characterized by studying the flow of tiie fluid around fixed obstacles. When the flow plane is parallel to the smectic layers the flow around the sphere involves mainly only fluid directions (see Figure 4.13), and the force acting on a sphere is purely viscous. 

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 ... 

Einstein solved the hydrodynamic equations for flow around hard spheres in a dilute suspension in a viscous fluid. His result was that the viscosity of a suspension of hard... 

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 ... 

Figure 8.4 (a) Rotational motion of a sphere induced by a shear field this motion is resisted by viscous effects over the surface of the sphere, (b) Resultant distortion of the flow field of the liquid dotted line - flow field in the absence of the sphere full line - effect of the presence of the sphere, which slows down the liquid around the upper hemisphere and accelerates it in the lower. 

Even dilute suspensions of repulsive particles will have slightly greater viscosity than hard spheres because of the additional viscous dissipation related to the flow of fluid through the repulsive region around the particle. For particles with EDL repulsion this is known as the primary electro-viscous effect (Hunter, 2001). The total drag on the particle and the double layer is greater than the drag on a hard sphere. The increase in viscosity due to the primary electro-viscous effect is typically minimal. 

On the other hand, shape does matter, since the increase in viscosity comes from the diversion of streamlines in the flow as they are redirected around particles, thus leading to an increase in the viscous energy dissipation, i.e. viscosity. When the particles are non-spherical, this extra dissipation increases, and the incremental amoxmt of viscosity also increases. The measure of the increase is the intrinsic viscosity [rj and this increases when a sphere is pulled out towards a rod, or squashed down towards a disc, with the former giving the greater increase in viscosity. A typical example of a rodlike particle system is paper-fibre suspension, while red blood cells are approximately disc-shaped. Simple formulas for both kinds of particles have been derived by the present author [3] as... 

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