Creeping flow around a spherical bubble, when a liquid flows around a gas bubble, circulation takes place within the bubble. This circulation lowers the interfacial shear stress, and, to a first approximation, we may assume that it is entirely eliminated. Repeat the development of Ex. 4.2-1 for such a gas bubble, assuming it is spherical. 

(a) Show that B.C. 2 of Ex. 4.2-1 is replaced by B.C.2; and that the problem set-up is otherwise the same. 

(b) Obtain the following velocity components:

(c) Next obtain the pressure distribution by using the equation of motion:

(d) Evaluate the total force of the fluid on the sphere to obtain

This result may be obtained by the method of S2.6 or by integrating the z-component of – [n ∙ π] over the sphere surface (n being the outwardly directed unit normal on the surface of sphere).

Part (a) d (1 df\ + 2 at r = R, dr ? dr, Part (b) (e) cos 0 v, = vz| 1 sin 0 Vy = -v% 1 Part (c) (4). p = Po pgh Cos e Part (d) F = {TRpg 4Rv.