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On “Inertia effects in viscous flow” by J. L. Livesey and “An experimental investigation of inertia effects in viscous flow” by P. G. Morgan and A. Saunders
Int. J. Mech.Sci. PergamonPressLtd. 1961. Vol.1, pp. 369-370. Printedin GreatBritain
LETTERS
TO THE
EDITOR
On "Inertia effects in viscous flow" by J. L. Livesey and "An experimental investigation of inertia effects in viscous flow" by P. G. Morgan and A. Saunders (Received 21 February 1963) INERTIA effects in a radial viscous flow between two parallel plates h a v e been studied b y different authors, one o f which is Professor Comolet I who used the same integration m e t h o d as J. L. Livesey ~ and who, of course, f o u n d the same result. H o w e v e r , we m u s t consider t h a t this m e t h o d of calculation is p r o b a b l y only valid for l a m i n a r flow; when the radial gradient of pressure 8p/~r changes sign, which means t h a t inertia effects become i m p o r t a n t , we m a y t h e n e x p e c t the appearance of a t u r b u l e n t flow. I n fact, using Livesey's notation, equation (2) (p. 85) when applied to the wall gives 8r ~-P"[Sy2J~=±h
=
0
dy2 Jy=o is surely negative as we have a m a x i m u m of speed. I f ~p/Sr is positive (inertia effects prevailing), 8y2J y=±h m u s t be positive, and, as a consequence, the velocity profile has an inflexion p o i n t : the corresponding flow is unstable and should become t u r b u l e n t . This state o f affairs is realized in the experiments o f P. G. Morgan and A. Saunders 3, whose results a p p e a r in Fig. 1 (p. 9): in the p a r t where the fall of pressure P l - P ~ is negative (and p r o b a b l y even before) the theoretical calculations with inertia terms have certainly no physical meaning.
P. L. PEUBE Universit~ de Paris Mgcanique Expdrimentale des Fluides Facult~ des Sciences, Centre d'Orsay Orsay (S & O) REFERENCES 1. R. COMOLET, Ecoulement d'un Fluide entre deux Plans parall~les. Publications Scientifiques et Techniques du Minist~re de FAir, No. 334 (1957). 2. J. L. LIVESEy, Int. J. Mech. Sci. 1, 84 (1960). 3. P. G. MORGA~and A. SAUlqDERS, Int. J. Mech. Sci. 2, 8 (1960). 369
LETTERS
TO THE
EDITOR
On "Inertia effects in viscous flow" by J. L. Livesey and "An experimental investigation of inertia effects in viscous flow" by P. G. Morgan and A. Saunders (Received 21 February 1963) INERTIA effects in a radial viscous flow between two parallel plates h a v e been studied b y different authors, one o f which is Professor Comolet I who used the same integration m e t h o d as J. L. Livesey ~ and who, of course, f o u n d the same result. H o w e v e r , we m u s t consider t h a t this m e t h o d of calculation is p r o b a b l y only valid for l a m i n a r flow; when the radial gradient of pressure 8p/~r changes sign, which means t h a t inertia effects become i m p o r t a n t , we m a y t h e n e x p e c t the appearance of a t u r b u l e n t flow. I n fact, using Livesey's notation, equation (2) (p. 85) when applied to the wall gives 8r ~-P"[Sy2J~=±h
=
0
dy2 Jy=o is surely negative as we have a m a x i m u m of speed. I f ~p/Sr is positive (inertia effects prevailing), 8y2J y=±h m u s t be positive, and, as a consequence, the velocity profile has an inflexion p o i n t : the corresponding flow is unstable and should become t u r b u l e n t . This state o f affairs is realized in the experiments o f P. G. Morgan and A. Saunders 3, whose results a p p e a r in Fig. 1 (p. 9): in the p a r t where the fall of pressure P l - P ~ is negative (and p r o b a b l y even before) the theoretical calculations with inertia terms have certainly no physical meaning.
P. L. PEUBE Universit~ de Paris Mgcanique Expdrimentale des Fluides Facult~ des Sciences, Centre d'Orsay Orsay (S & O) REFERENCES 1. R. COMOLET, Ecoulement d'un Fluide entre deux Plans parall~les. Publications Scientifiques et Techniques du Minist~re de FAir, No. 334 (1957). 2. J. L. LIVESEy, Int. J. Mech. Sci. 1, 84 (1960). 3. P. G. MORGA~and A. SAUlqDERS, Int. J. Mech. Sci. 2, 8 (1960). 369