﻿Steady Motion of a Viscous Fluid. 45D 



opting M = (g) 2 + (|f) 2 , 



we see that a, J3, f are restricted by the condition that M 

 must satisfy the equation 



+ vf"" («)M = 0. . . . (3) 



This will be satisfied by any system a, /3 whatever, if 

 f /! = 0, which corresponds to the otherwise obvious fact that 

 any solution of V 2 ^ = is a solution of (2). Thus any 

 irrotational motion is a possible motion of a viscous fluid. 



Suppose * + #=(* + *)*, 



then „ . n ~ l 



M = n 2 (a 2 + /3<) n , 



equation (3) gives 

 2v(n-l)(n-2)/ v, (*)+4i;n(n-l)a/'- // («)--2n(n-l)i8/ («)/"(") 



If n=l, it is sufficient that /'""(a) = 0, which leads to the 

 well-known solution for the motion between infinite parallel 

 planes : 



Otherwise, equating coefficients of powers of separately to 

 zero, 



/""(«) =0, /(«)/•"(«) =0, /'»=(>. 



Hence f(a) is a linear function of a, which corresponds to 

 the case of irrotational motion. It appears, therefore, that 

 for no value of n other than unity is there any new solution 

 of this class. 



Next consider polar coordinates r 9 6. Equation (2) 

 becomes 



W ="'•* « 



and 1 J) / h\ 1 y 



V ~r3rV M + ?btf»' 



First seek a solution of the form 



t=/(.-). 



