Stream functions for steady three-dimensional flow.

(a) Show that the velocity functions pv = [∆ x A] and pv = [(∆ψ₁) x (∆ψ₂)] both satisfy the equation of continuity identically for steady compressible flow. The functions ψ₁, ψ₂, and A are arbitrary, except that their derivatives appearing in (∆ ∙ pv) must exist.

(b) Show that, for the conditions of Table 4.2-1, the vector A has the magnitude -pψh₃ and the direction of the coordinate normal to v. Here h₃ is the scale factor for the third coordinate (see SA.7).

(c) Show that the streamlines corresponding to Eq. 4.3-2 are given by the intersections of the surfaces ψ₁ = constant and ψ₂ = constant. Sketch such a pair of surfaces for the flow in Fig. 4.3-1.

(d) Use Stokes' theorem (Eq. A.5-4) to obtain an expression in terms of A for the mass flow rate through a surface S bounded by a closed curve C. Show that the vanishing of v on C does not imply the vanishing of A on C.