where p is the density of the medium and p ' is the density of the sphere. 

 For the fluid sphere: 



(J =2l111{ r - p ') 3 /*' + 3 f* Hadamard-Rybczynski's Law 



f 1 3»'+2, 



V =l!-l{p- p ') e + 2r(n + y.') Boussinesq's Law 



For the case of bubbles, where p.' « p. and p' « p, the last two expressions reduce to t 



U= I r2 9P 

 3 p 



and 



v =— r2 yp e + 3r ^ 



9 p e + 2rp 



In the last equation, the factor e + f 1 approaches 1 for r approaching zero or e very 



e + Irp 



large; it approaches 3/2 for e « rp, i.e., for large r or for c approaching zero. Hence, 

 for very small bubbles, Boussinesq's solution approaches Stokes' law as a limit, while the 

 other limit is Hadamard-Rybczynski's solution. 



From the boundary conditions as stated above, it is obvious that, for the Hadamard- 

 Rybczynski and Boussinesq solutions, circulation exists inside the bubble. For the Stokes 

 solution, of course, there is no circulation inside the sphere. 



DIMENSIONAL ANALYSIS 



Since an analytical solution for the drag of fluid bodies over a large range of sizes is 

 hardly attainable, dimensional analysis of the phenomenon may serve to correlate the experi- 

 mental results. 



In the case of fluid bodies the following physical variables are usually considered as 

 pertinent: 



V Velocity of the body 



g Acceleration due to gravity 



p Density of the fluid medium 



p' Density of the fluid body 



I Length parameter of the fluid body 



p Coefficient of dynamic viscosity of the fluid medium 



