Vorticity equation, creeping flow

The form of Eq. (5-25) was suggested by noting that the first order curvature corrections to Eqs. (3-47) and (5-35) are near unity and by matching the expression to the creeping flow result, Eq. (3-49), at Re = E Equation (5-25) also represents the results of the application of the thin concentration boundary layer approach (Sc oo) through Eq. (3-46), using numerically calculated surface vorticities. Thus the Schmidt number dependence is reliable for any Sc > 0.25. 

A plot showing lines of constant co (= co ) is also shown in Fig. 7-12. Again, the fore-aft symmetry of the flow is clearly evident. The fact that the constant-vorticity lines all terminate on the body surface is indicative of the fact that the source of vorticity in the flow is the sphere surface. Indeed, the governing equation for vorticity in the creeping-flow limit can be shown from (7-6) to be18... 

Vorticity Equation for Creeping Flow of an Incompressible Newtonian Fluid. Since all scalar components of the velocity vector are exact differentials, it is permissible to reverse the order of mixed second partial differentiation without affecting the final result. If this procedure is performed twice, then inspection of summation representations of the following two vector-tensor operations reveals that they are equivalent ... 

At first glance, three coupled linear third-order PDEs must be solved, as illustrated above. However, each term in the x and y components of the vorticity equation is identically zero because =0 and Vj and Vy are not functions of z. Hence, detailed summation representation of the vorticity equation for creeping viscous flow of an incompressible Newtonian fluid reveals that there is a class of two-dimensional flow problems for which it is only necessary to solve one nontrivial component of this vector equation. If flow occurs in two coordinate directions and there is no dependence of these velocity components on the spatial coordinate in the third direction, then one must solve the nontrivial component of the vorticity equation in the third coordinate direction. 

For creeping viscous flow in spherical coordinates, the -component of the vorticity equation requires that... 

Answer For axisymmetric creeping flow in spherical coordinates, the general solution to the low-Reynolds-number equation of change for fluid vorticity (i.e., E ir = 0) is... 

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