84 Stability of Floiv of an Incompressible Viscous Fluid. 



of flow. Evidently any factor which tends to diminish the 



relative magnitude of the left-hand side of the equation 



tends to reduce the tendency to eddy formation and to 



increase the stability o£ flow and vice versa. Thus, as is 



well known, an increase in /jl or a diminution in p tends to 



increased stability. 



Effect of Free Boundaries. — With "free" as opposed to 



" solid" boundaries, the boundary velocities are increased and, 



assuming flow to take place in the direction o£ increasing x, 



du du 



the magnitudes o£ -7- and of -7- are diminished. Since these- 

 ay dz 



are negative and since the quantity in brackets on the left- 

 hand side of equation (4) is essentially negative, this tends 

 to diminish this term and hence to increase the stability of 

 flow. 



Effect of Converging Boundaries — The same effect may be 

 seen to follow the introduction of converging as opposed to 

 parallel boundaries, or indeed any convergence of stream 

 lines accompanying a change from pressure to kinetic 



T , du dv , dw . , _ . . 



energy. In such a case -7-, 7-, and -7- instead or being 



zero become positive and thus diminish the absolute 

 (negative) value of the term in brackets. Conversely, a 

 retardation in the direction of flow, such as is produced by 

 diverging boundaries, diminishes the stability of flow. 



Effect of Curvature of the Patli. — If, in curvilinear motion, 

 the direction of x be taken as tangential to the path at a 



fi 7* fl II 



given point and if -7- and -~ are positive, the value of 



fi 1 1 



-j- is greater if the velocity is greatest at the outside of the 

 curve, than it is with rectilinear motion. Since, under these 



fin 



conditions, -7- is essentially negative, the absolute value of 



the first term of (4) is greater than with rectilinear motion, 

 and the stability of motion is consequently diminished. 

 Conversely, with the velocity greatest at the inside of the 

 curve the stability is increased. 



University College, Dundee. 



