Yih 



REPLY TO DISCUSSION 



ChLa-Shun Yih 



University of Michigan 

 Ann ArboTj Michigan 



It is well known that Rayleigh's sufficient condition for stability 

 of inviscid fluids flowing between rigid boundaries is satisfied by a 

 parabolic velocity profile, whereas plane Poiseuille flow, which has 

 this profile, has been found by Heisenberg and Lin to be unstable at 

 sufficiently large Reynolds numbers , when viscous effects are taken 

 into account. Since the present paper is a study of the stability of 

 inviscid fluids, and, in particular , Rayleigh's criterion for stability 

 is generalized in it. Professor van Wijngaarden's position that the 

 consideration of viscosity may force us to modify some of the con- 

 clusions in the paper is easily acceptable. 



In considering viscous effects, however, it is not entirely 

 self- consistent to assumie a horizontal mean flow with a free surface, 

 as Velthuizen and Professor van Wijngaarden have done [ 1969a, b] , 

 since such a flow obviously cannot be maintained, and must In time 

 attenuate to a state of rest. This Is not to say that any conclusion of 

 Instability reached by them Is without significance, for Instability of 

 a transclent nature may well occur, with the disturbances growing 

 for a short duration of time. In this regard the results of Benjamin 

 [ 1957] and Yih [ 1963] for surface waves in a fluid layer flowing down 

 an inclined plane are relevant. They found that the speed Cr of long 

 surface waves, be they unstable, neutral, or stable, exceeds the 

 mcLxlmum speed of flow. The absence of long waves propagating up- 

 stream supports Professor van Wijngaarden's claim In connection 

 with Burn's result, which is supported by a study [ Yih 1971] of waves 

 In a flowing Inviscid liquid. But the nonexistence of a critical layer 

 renders rather less cogent the argument given in Professor van 

 Wijngaarden's discussion. On the other hand, this nonexistence sub- 

 stantiates the conclusion made in Yih [ 1971] (and similarly in this 

 paper) regarding the nonexistence of singular neutral modes, since 

 the velocity U In laminar flow of a viscous fluid down an Inclined 

 plane is parabolic, with a constant U", 



We also recall that Tollmlen's sufficient condition for Insta- 

 bility [ 1935] of an inviscid fluid Is not much affected by the considera- 

 tion of viscosity, at least when the Reynolds number Is large, and 

 hope that the same is true with the sufficient conditions for instability 

 presented In this paper. 



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