Desai and Lieber 



figure employed for comparison and upon which the experimental 

 case was projected. By this means, it was proved that the two 

 were in absolute agreement. 



Mathematicians, however, predicted with absolute certainty, that 

 with stream-line motion the water should flow round and meet at 

 the back, a state of things that, however slow we make the motion 

 in the present case, does not occur owing to the effect of inertia. 

 They have drawn with equal confidence the lines along which this 

 should take place. We could either effect this result with the ex- 

 periment you have just seen, by using a much more viscous liquid, 

 such as treacle, or, what comes to the same thing, bringing the 

 two sheets of glass nearly close together. 



In these quoted passages we see that there are three controlling factors by 

 which a real flow configuration is created so that its streamline field is exactly 

 the same as that of a corresponding potential flow. These are (a) the pressure, 

 (b) the viscosity, and (c) the distance separating the two plates. 



Following Professor Hele-Shaw's experiments. Professor Stokes (1898) in 

 his paper "Mathematical Proof of the Identity of the Stream-Lines Obtained by 

 Means of a Viscous Film with Those of a Perfect Fluid Moving in Two Dimen- 

 sions" [29], starts out with the equations for a creep motion in which the non- 

 linear inertia terms are neglected, and shows that, when the distance separating 

 two plates is small, the stream function for the flow satisfies the harmonic 

 equation and is uniquely determined by the condition that the boundaries shall be 

 streamlines. And since a stream function for a potential flow meets these re- 

 quirements, the identity is established. He continues: "It may be objected that 

 the streamlines cannot be the same in the two cases, inasmuch as the perfect 

 liquid glides over the surface of the obstacle, whereas in the case of the viscous 

 liquid the motion vanishes at the surface of the obstacle. This is perfectly true, 

 and forms the qualification above referred to; but it does not affect the truth of 

 the proposition, which applies only to the limiting case of a viscid liquid con- 

 fined between walls which are infinitely close. Any finite thickness of the stra- 

 tum of liquid will entail a departure from the identity of the streamlines in the 

 two cases, which, however will be sensible only to a distance from the obstacle 

 comparable with the distance between the walls, and therefore capable of being 

 indefinitely reduced by taking the walls closer and closer together." 



In 1938, F. Riegels [30] carried out further experiments on Hele-Shaw flows 

 and gave a theoretical representation of the same, based on an iterative scheme. 

 As a starting point he takes a solution of the equations of creeping motion, which 

 may be written as 



2 

 h^ 



Uj = Uo(x,y) (l - 77 



524 



