﻿Steady Motion oj a Viscous Fluid. 461 



§ 3. Solutions which are independent of the degree 

 of viscosity. 



If we consider the motion of a fluid between two con- 

 centric circular cylinders rotating with given angular 

 velocities, an increase in the viscosity o£ the fluid would 

 necessitate an increase in the couples which maintain the 

 rotation of the cylinders, but otherwise the motion would be 

 unchanged. This will be true for any solution for which i/r 

 is independent of v, i. e. the two sides of the equation (2) 

 vanish separately. Taking the equation in its Cartesian 

 form we have 



B(t ; vy =0 



and V 4 ^ = (6) 



It will be noted that equation (6) is the usual equation for 

 the two-dimensional motion of a viscous fluid on the assump- 

 tion that it is so slow that squares and products of the 

 velocities may be neglected. For an exact solution equation 

 (5) must also be satisfied, and hence 



V 2 ^=/'(t), 



that is the vorticity is constant along each stream-line. 

 Substituting in (6), 



Hence either (1) the resultant velocity is a function of yjr. 

 and is therefore constant along each stream-line, or (2) f f/ (^r) 

 and /'(yfr) are zero. In the latter case 



\/ 2 yjr = const. = 4«, say 



of which the general solution is 



where % is any solution of V 2 % = 0. Hence any solid body 

 rotation superposed upon any irrotational motion is a possible 

 motion of a viscous fluid. The only other solutions of this 

 class are those for which the velocity and the vorticity are 

 constant alone: each stream-line. 



■£5 



§ 4. Discussion of solutions obtained. 



Flow between two planes inclined at any angle. — Wo will 

 irst consider solution V 1 . We may without loss of generality 



