Translational flow past spherical particle

Let us consider mass transfer for a translational flow past a solid spherical particle, where the flow field remote from the particle is the superposition of a translational flow with velocity U and an axisymmetric straining shear flow, the translational flow being directed along the axis of the straining flow. The dimensional fluid velocity components in the Cartesian coordinates relative to the center of the particle have the form... 

Here, we consider Stokes problem of uniform, streaming motion in the positive z direction, past a stationary solid sphere. The problem corresponds to the schematic representation shown in Fig. 7-11 when the body is spherical. This problem may also be viewed as that of a solid spherical particle that is translating in the negative z direction through an unbounded stationary fluid under the action of some external force. From a frame of reference whose origin is fixed at the center of the sphere, the latter problem is clearly identical with the problem pictured in Fig. 7-11. Because we have already derived the form for the stream-function under the assumption of a uniform flow at infinity, we adopt the latter frame of reference. The problem then reduces to applying boundary conditions at the surface of the sphere to determine the constants C and Dn in the general equation (7-149). The boundary conditions on the surface of a solid sphere are the kinematic condition and the no-slip condition,... 

Figure 2.1. Translational Stokes flow past a spherical particle...

For Re > 0.5, asymptotic solutions no longer give an adequate description of translational flow of a viscous fluid past a spherical particle. 

At low Peclet numbers, for the translational Stokes flow past an arbitrarily shaped body of revolution, formula (4.10.8) coincides with the exact asymptotic expression in the first three terms of the expansion [358], Since (4.10.8) holds identically for a spherical particle at all Peclet numbers, one can expect that for particles whose shape is nearly spherical, the approximate formula (4.10.8) will give good results for low as well as moderate or high Peclet numbers. 

The order of magnitude of the characteristic dimensions of diffusion wake regions past a spherical drop and a solid particle in a translational flow is shown in Table 4.9. These estimates remain valid at moderate Reynolds numbers, when the stagnation zones past a drop or a solid particle are absent. 

The dependence of the auxiliary Sherwood number Sho on the Peclet number Pe for a translational Stokes flow past a spherical particle or a drop is determined by the right-hand sides of (4.6.8) and (4.6.17). In the case of a linear shear Stokes flow, the values of Sho are shown in the fourth column in Table 4.4. 

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See also in sourсe #XX -- [ ]

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