Potential flow past spheres

Inviscid Flow and Potential Flow Past a Sphere... 

This result may be contrasted with potential flow past a sphere, where the streamlines again have fore-and-aft symmetry but p is an even function of 9 so that there is no net pressure force (see Chapter 1). Additional drag components arise from the deviatoric normal stress ... 

Since the flow is only slightly perturbed from irrotational, a first approximation for the drag on a spherical bubble may be obtained by calculating the viscous energy dissipation for potential flow past a sphere. This gives (Lll) ... 

For CO 0, Eq. (11-7) reduces to the stream function for steady creeping flow past a rigid sphere, i.e., Eq. (3-7) with k = co. The parameter 3 may be regarded as a characteristic length scale for diffusion of vorticity generated at the particle surface into the surrounding fluid. When co is very large, 3 is small, and the flow can be considered irrotational except in the immediate vicinity of the particle. In the limit co go, Eq. (11-7) reduces to Eq. (1-29), the result for potential flow past a stationary sphere. 

These results are useful reference conditions for real flows past spherical particles. For example, comparisons are made in Chapter 5 between potential flow and results for flow past a sphere at finite Re. Other potential flow solutions exist for closed bodies, but none has the same importance as that outlined here for the motion of solid and fluid particles. 

In this regard, it is of interest to contrast the two problems of the streaming motion of a fluid at large Reynolds number past a solid sphere and a spherical bubble. In the case of a solid sphere, the potential-flow solution (10 155)—(10—156) does not satisfy the no-slip condition at the sphere surface, and the necessity for a boundary layer in which viscous forces are important is transparent. For the spherical bubble, on the other hand, the noslip condition is replaced with the condition of zero tangential stress, Tr = 0, and it may not be immediately obvious that a boundary layer is needed. However, in this case, the potential-flow solution does not satisfy the zero-tangential-stress condition (as we shall see shortly), and a boundary-layer in which viscous forces are important still must exist. We shall see that the detailed features of the boundary layer are different from those of a no-shp, sohd body. However, in both cases, the surface of the body acts as a source of vorticity, and this vorticity is confined at high Reynolds number to a thin 0(Re x/2) region near the surface. 

Let the sphere, of radius be at rest at the origin of coordinates and let the liquid flow past in the negative z direction. At any point far from the sphere let the magnitude of the liquid velocity be Wq The equation that must be satisfied by the velocity potential at all points where the motion is irrotational is ... 

Therefore, the complete solution, giving the velocity potential for the flow of an incompressible, inviscid liquid past a fixed sphere under irrotational conditions is ... 

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