pressure along the a;-axis and over the sphere for positive a;, also along the y-axis, where 

 the pressure is plotted horizontally with negative values toward the left. The pressure is 

 symmetrical on front and rear. For this reason it is obvious that there is no resultant force 

 on the sphere. 



The sphere can be regarded as a Rankine solid for which the source and sink, while 

 increasing indefinitely in strength, have come together to form a dipole. 



The formulas will also represent the flow inside a spherical shell caused by a dipole 

 of moment fi = a^U/2 at the center. In this case both U and p^ represent mathematical 

 constants. 



Changing the sign of U reverses all velocities. (See Reference 2, Section 15.30.) 



129. SPHERE WITHIN A CONCENTRIC SPHERE 



If the moving sphere of Section 126 is surrounded by a fixed concentric spherical 

 shell of radius b, there are two boundary conditions to be satisfied by the field of velocity 

 in the intervening fluid: at r = a, q^ = U cos 6; at r = b, q^ = 0. In order to have two adjustable 

 constants, let a potential function be assumed of such a form as to represent the superposition 

 of uniform and dipole flow, namely, from Section 128, 



/ A\ d(]y I 2A\ 



(^ = I (/'r + — j cos 0, (7r = - — = I - fy ' + — cos 



where the angle 9 is measured from the direction of motion, and the constants 6" and A are 

 to be determined. The boundary conditions require that 



2.4 2A 



- v + — = V,- V ^ — = 



a^ b^ 



Solving for (7 'and A and adding the stream function \jj from [128b] 



a^U I b^ X a^U / , b^ 



lr+ ) cos 0, (A = I '■^ I si 



\ 2r2 / 2(b^-a^) \ ^ I 



\n^d [129a, b] 



&3_^3 \ o;.2 / 2{b^-a^) 



a^U I b^ \ a^U i b^ \ 

 q^ = ( 1 ) cos 0, 00= ( 1 + ) sin (9 [129c,d] 



The possibility of satisfying the boundary conditions in this way for all values of 

 arises from the choice of a suitable function for 4>. The solution is exact, however, only at 

 the instant at which the centers of sphere and shell coincide. Streamlines for equally spaced 

 values of i/f are shown in Figure 214. 



322 



