Appendix C Useful Theoretical Approaches and Formulas
489
x
Fig. C.3: Definition of spherical coordinate system
and
2 (au aw) 2
/LV'
- - - =/LV''f)=0.
az ax
(C.47)
As was shown in Sect. C.2, the quantities ~, 'f) and ( are the components of the
vorticity vector.
Let us now differentiate Eqs. (C.44) with respect to x, y and z, respectively,
and summarize. Thus, after using the equation of mass conservation (C.19),
we obtain (Batchelor, 1967):
(C.48)
Applying the Laplacian operator V'2 to Eqs. (C.44) and using Eq. (C.48) gives:
(C.49)
Because of a sphere geometry, it is convenient to use the cylindrical coordinate
system (r, cp, B) - see Fig. C.3. At infinity there is undisturbed flow when:
w = V, Vr = V cos cp, v = V sin cp.
(C.SO)
On the surface of the sphere (r = D/2) non-slip conditions apply, i.e.:
Vr = v = 0,
(C.S1)