CHAPTER V. 



ON THE MOTION OF SOLIDS THROUGH A LIQUID. 



99. THE chief subject treated of in this chapter is the 

 motion of a solid through an infinite mass of liquid under the 

 action of any given forces. The same analysis applies with little 

 or no alteration to the case of a liquid occupying a cavity in a 

 moving solid. We shall consider, though less fully, cases where 

 we have more than one moving solid, or where the fluid does not 

 extend in all directions to infinity, being bounded externally by 

 fixed rigid walls. 



We shall assume in the first instance that the motion of the 

 fluid is entirely due to that of the solid, and is therefore charac 

 terized by the existence of a single-valued velocity-potential (/&amp;gt; 

 which besides satisfying the equation of continuity 



V = (1) 



fulfils the following conditions: (a) the value of - , dn denoting 



as usual an element of the normal at any point of the surface of 

 the solid drawn towards the fluid, must be equal to the velocity of 

 the surface at that point normal to itself, and (b) the differential 



coefficients ??-. j -f- must vanish at an infinite distance, in 

 asc ay az 



every direction, from the solid. The latter condition is rendered 

 necessary by the consideration that a finite velocity at infinity 

 would imply an infinite kinetic energy, which could not be gener 

 ated by finite forces acting for a finite time on the solid. It is 

 also the condition to which we are led by supposing the fluid 

 enclosed within a fixed vessel infinitely large and infinitely dis 

 tant all round from the moving body. For on this supposition 



