594 Messrs. Cowley and Levy: Method of Analysis suitable 



Stanton and Pannell * have borne out the fact that the 

 motion o£ air in pipes obeyed the same law and likewise 

 exhibited a corresponding critical state for the same value of 

 .Reynold's number. These two researches and the vast 

 additional amount of aeronautical evidence accumulated 

 during the past few years indicate that the whole process 

 of the motion in any problem centres round the value of 

 the non-dimensional expression vl/v. This vital fact will 

 be utilized as suggesting a means for analysing the equation. 

 With regard to the state of affairs existing at the surface of 

 a body in a fluid, it has long been recognized that on general 

 grounds no relative motion may be expected to exist, and 

 much confirmatory evidence for this view has been found 

 for slow motions of fluids. For the present it will be assumed 

 that the fluid with which we deal has density (p) , viscosity (v), 

 and is incompressible ; and these are the only properties of 

 the fluid that affect the question apart from the " no slip " 

 condition at the boundary. 



§ 2. Under these circumstances the accepted equations of 

 motion restricting ourselves to two dimensions : 



v 1 op ~du , dw d?/ 

 p ox at ox oy 



v 1 op "dv ov "dv , 



P oy ot ox &y 



along with the equation of continuity, 



|^ + |^=0, (3) 



ox oy 



where the usual notation is employed. 



The last equation admits of the introduction of the stream- 

 function i|r without loss of generality, so that the fundamental 

 equations may be re-written : 



P ox ~dt V oy ) oy ox L oy J ox oy 2 



+ "| / V 2 ^ • • (4) 



P~dy ^y\bx) oy ~dx 2 oxoyydx) 



-"&*+' ' ' (5) 



* Phil. Trans. A, vol. 214. 



