Flow Past a Spherical Particle

The fluid velocity distribution is given by the Stokes equations (2.1.1) with the no-slip boundary conditions on the surface of the solid sphere, 

By passing from the fluid velocity components Vr, V to the stream function according to formulas (2.1.3), we arrive at Eq. (2.1.4). It follows from the remote boundary conditions (2.2.2) that in the general solution (2.1.5) it suffices to retain only the first term (corresponding to the case n = 2). The no-slip conditions (2.2.1) allow us to find the unknown constants A2, B2, C2, and D2. The resulting expression for the stream function, 

The dynamic particle-fluid interaction is characterized by the drag force, which is defined as the projection of the resultant of all hydrodynamic forces on the flow direction  

The stresses on the spherical particle surface are given by the relations 

Using the expressions (2.2.4) and carrying out the integration, we find that the viscous drag acting on the spherical particle is given by 

---

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

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. 

By this technique Acrivos and Taylor obtained the following result for streaming Stokes flow past a spherical particle of radius a maintained at uniform temperature in a fluid whose temperature is uniform at infinity ... 

---