Consider creeping flow of a liquid at velocity v=Uex relative to a spherical bubble of radius R, with coordinates as in the Figure 1 below (presume the figure to depict an equatorial cross-section of a sphere). The general solution given by Deen Eqs. 8.3-28 through 8.3-30 is applicable here. Figure 1: Applicable coordinate system Figure 2: Streamlines for flow past a sphere. (a) What are the boundary conditions for this problem? (b) In the bonus problem you will show that the velocity and pressure in the surrounding liquid are given by: vr(r,)=Ucos(1rR)v(r,)=Usin(121rR)P(r,)=r2URcos where is the viscosity of the liquid. Use these equations to calculate the drag force for the bubble. - Hint 1: Look at Example 6.6-3 and Eqn 6.6-30 for the drag on a sphere of radius R. - Hint 2: In contrast to the case of a solid sphere, the normal stress rr is no longer zero for a fluid or gas bubble. (why?) (c) The drag coefficient, CD, is defined as (Welty Eqn 12-3): APFDCD(21v2) where V is the incoming fluid velocity (U in this problem), AP is the projected area of an object perpendicular to the flow, and is the fluid density. For a sphere, Ap=R2. What is the drag coefficient for the bubble as a function of the Renolds number (Re)? (e) Perform a force balance on a rising bubble where the buoyancy force balances the drag force and the density difference between the fluid and the gas in the bubble is =0. What will be the terminal velocity of the bubble in this case