﻿1 wo- Dimensional Steady Motion of a Viscous Fluid. 455 

 so that the equation for -yjr becomes 



L. TAe Two-Dimensional Steady Motion of a Viscous 

 Fluid. By G. B. Jeffery, J/.JL, B.Sc , Assistant in the 

 Department of Applied Mathematics, University College, 

 London*. 



r |^HE object of this paper is to search for some exact 

 JL solutions of the equations of motion of a viscous' fluid. 

 Much has been accomplished by assuming that the motion is 

 slow, and that the squares and products of the velocity compo- 

 nents may therefore be neglected. It has indeed been held 

 that this is the only useful proceeding, since the equations of 

 motion are themselves formed on the assumption of a linear 

 stress-strain relation, and this is probably only justifiable if 

 the motion is sufficiently slow. On the other hand, there is 

 very little evidence of the breakdown of the linear law in 

 the case of fluids, and in any case it is only possible to test 

 its validity by an investigation of solutions which do not 

 require the motion to be slow. It is, therefore, of some 

 importance to obtain some solutions which are free from this 

 limitation. In the present paper we confine our attention 

 to plane motion. Orthogonal curvilinear coordinates are 

 employed, and we discuss the possibility of so choosing them 

 that either the stream-lines or the lines of constant vorticity 

 are identical with one family of the coordinate curves. The 

 most important solutions obtained are those which correspond 

 to (1) the motion round a canal in the form of a circular arc r 

 (2) the motion between rotating circular cylinders with a 

 given normal flow over the surfaces, as in a centrifugal 

 pump, (3) the flow between two infinite planes inclined at 

 any angle. 



If u, v be the components of velocity, p the mean pressure, 

 V the potential of the external forces, v the kinematic 

 viscosity, and p the density of the fluid, the equations of 



* Communicated by Prof. Karl Pearson, F.R.S, 



