I 



Pressure around Spheres in a Viscous Fluid. 425 



at infinity is increased by an inertia term equivalent to one- 

 halt' the mass of the fluid displaced times the square of the 

 velocity: 



•2T=-p|J</,grf. = |7rp«V a) 



When the sphere moves in a straight line, its motion being- 

 accelerated, and there are no external forces acting on the 

 fluid, the resultant pressure is equivalent to a force 



~8'^^''^JT' ^^^ 



in the direction of motion. If the velocity of the sphere is 

 constant, there being no external force, the force acting on 

 the sphere is zero, the pressure is symmetrical with respect 

 to any axis, and the sphere will move with uniform velocity 

 through the fluid. 



The problem of the motion of two spheres in a perfect 

 fluid was discussed by Stokes in the paper already referred 

 to, and a method for obtaining the solution was suggested. 

 Later a solution was obtained by W. M. Hicks and presented 

 to the Royal Society in 1879 in his paper " On the Motion 

 of Two Spheres in a Fluid "*. 



Hicks finds that the kinetic energy T of two spheres 

 moving in a perfect fluid may be expressed as a very simple 

 function of their relative velocities 2^1, 11-2 : 



2T=AiV--2BHiZ^2 + Ao2r2'; .... (3) 



and that the rate of change of the distance between the 

 centres of gravity of the two spheres is given by the expres- 

 sion 



^t-±Va;a;^b' w 



the positive or negative sign being taken according as the 

 spheres are separating or approaching one another. The 

 spheres will therefore move as though they repelled or 

 attracted one another accordino- as 



BrlAiAo-Bf 



is positive or negative. This condition does not depend on 

 the relative motion of the two spheres at any time, but only 

 on their distance apart and the ratio of the constant energy 



* Phil. Trans, p. 455 (1880). 



