r Figure 216 - Polar coordinates for a small 

 sphere near a wall. 



cos d^ = cos 6 cos a - sin 6 cos co sin a , 

 cos ^2 = <^os sin a + sin cos w cos a. 



The kinetic energy T of the fluid may then be found by substituting in Equation [17c] 

 (f) - 4>i -^ 4>2 ^i*'^ T = a, also q^ = U cos 6 and dS = a^ sin ddddco, and integrating over the 

 sphere. It is 



f = - p/<^ ?„ fifS = - r.p a^ U' 



1 + 



16 



(1 + cos a ) 



[130a] 



The forces on the sphere may now be found by means of Lagrange's equation, 



_d_fdT\ dT 

 dt\dqj dq 



where q stands for any coordinate of the sphere and q = dq./dt. 



In terms of the Cartesian coordinates x and y of the center of the sphere, with 

 velocities x = U cos a , y = U sin a , 



T - — no a'- 

 3 



1 + 



8 „3 



'^T?-;''^ 



[130b] 



For X and y as coordinates, the generalized forces Q are simply the ordinary components of 

 the force on the fluid, or -A" for x and - 7 for y where X and 7 are the components of the 

 force exerted by the fluid on the sphere, respectively away from and parallel to the wall. 



325 



