Steady potential flow around a stationary sphere. 2 In Example 4.2-1 we worked through the creeping flow around a sphere. We now wish to consider the flow of an incompressible, inviscid fluid in irrotational flow around a sphere. For such a problem, we know that the velocity potential must satisfy Laplace's equation (see text after Eq. 4.3-11). 

(a) State the boundary conditions for the problem.

(b) Give reasons why the velocity potential Φ can be postulated to be of the form Φ (r, θ) = f(r) cos θ.

(c) Substitute the trial expression for the velocity potential in (b) into Laplace's equation for velocity potential. 

(d) Integrate the equation obtained in (c) and obtain the function f(r) containing two constants of integration; determine these constants from the boundary conditions and find

(e) Next show that

(f) Find the pressure distribution, and then show that at the sphere surface

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Part (d) 1(R $ = -v,R cos e Part (e) - (4)]• cos 6 v, = v% 1 1 (R Vo = -v 1 + 2 sin 0 Part (f) P - P, = pv.(1 - sin? 0)