﻿2 Mr. A. R. McLeod on Unsteady Motion produced 



for the " stopping " than for the " starting " experiments. 

 The work was done at the Royal Aircraft Establishment, 

 Farnborough, during the months Feb.-Sept. 1919. 



In some later work, not yet published, the discrepancy 

 in the case of the " starting " experiments is traced to 

 the effect of the base ; and the theory will probably apply 

 to this motion in very long cylinders, provided eddies 

 do not form owing to initial instability of the water. In 

 the case of the " stopping " experiments, the discrepancy 

 is due to instability and turbulence. 



§ 1. Theoretical. 



In dealing with a rectilinear two-dimensional eddy in an 

 incompressible fluid which contains no sources or sinks, 

 the usual assumption is that particles of the fluid move 

 in circles about the axis of the eddy. This makes the 

 problem one of complete symmetry, and the radius vector r 

 and the time t are the two independent co-ordinates. The 

 equations of motion, when written in cylindrical co-ordinates 

 with the axis of z along the axis of symmetry, reduce to tlie 

 following forms, in which p denotes pressure, p is density, 

 v is the kinetic viscosity, and <fi is the angular velocity 

 about the axis : — 



^ = p^' 2 (i) 



for the pressure, and 



dr* r "dr v ~dt 



(2) 



giving the angular velocity. Let us suppose that the 

 angular velocity <j> satisfies the conditions 



= F(r) for t= 0, . . (3) 



</> = <f)(t) „ r — c = radius of cylinder ; . (4) 



that is, at the initial instant the angular velocity in the 

 cylinder is known to be F(V) at radius r, while thereafter 

 the rotation of the boundary is prescribed to be <j>(t). The 

 solution of (2) satisfying the conditions (3) and (4) is, in 



