Studies on the Motion of Viscous Flows —III 



which the fluid element experiences on the side of the fluid must 

 be equal and opposite to the pressure which it experiences on the 

 side of the fluid. Now if the fluid could flow past the solid with a 

 finite velocity, it would follow that the tangential pressure called 

 into play by the continuous sliding of the fluid over itself was no 

 more than counteracted by the abrupt sliding of the fluid over the 

 solid. As this appears exceedingly improbable a priori, it seems 

 reasonable in the first instance to examine the consequences of ' • 



supposing that no such abrupt sliding takes place, more espe- 

 cially as the mathematical difficulties of the problem will thus be 

 materially diminished. I shall assume, therefore, as the condi- 

 tions to be satisfied at the boundaries of the fluid, that the veloc- 

 ity of a fluid particle shall be the same, both in magnitude and 

 direction, as that of the solid particle with which it is in contact. 

 The agreement of the results thus obtained with observation will 

 presently appear to be highly satisfactory. When the fluid, in- 

 stead of being confined within a rigid envelope, extends indefi- 

 nitely around the oscillating body, we must introduce into the so- 

 lution the condition that the motion shall vanish at an infinite 

 distance, which takes the place of the condition to be satisfied at 

 the surface of the envelope. 



These quotations show that in 1845 Stokes was inclined towards the first 

 hypothesis but quite undecided about it and in fact tried Poisson's conditions 

 which in essence represent the second hypothesis. The second hypothesis was 

 deduced from the molecular hypothesis by Navier. However, in 1851, Stokes 

 makes use of the first hypothesis. According to his remarks, the choice of this 

 hypothesis seems to be governed by the following criteria. 



1. Hueristic reasoning when applied to the conditions of equilibrium lead to 

 2l conclusion ivhich seems exceedingly improbable a priori. 



2. Mathematical simplification of the condition. 



3. Experimental justification of the final results. 



From these criteria it is seen that Stokes did not show that the so-called "no- 

 slip condition" was physically the most tenable condition for the equations of the 

 motion of a viscous fluid— not at least conceptually. 



The second hypothesis includes, as a particular case, the first one. Light- 

 hill's discussion [14], which is also based on molecular considerations, shows 

 this to be the case. From this point of view the first hypothesis becomes a valid 

 approximation under ordinary flow conditions. What happens under extreme 

 conditions when thermodynamic equilibrium exists no longer is not so clear. 

 The behavior of the superfluids also raises questions about the nature of this 

 condition at a solid wall. 



Conceptually, it seems that the question about the nature of this condition at 

 a solid wall is an open one. A good discussion on this condition is given by 

 Langlois [17]. We intend to use the "no-slip" condition as a hypothesis on the 



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