312] DISTURBED LAMINAR MOTION. 577 



Eliminating % , we find 



(v +kU)(**-W\ - & =0 (viii), 



W / dy* 



which is the fundamental equation. 



If, for any value of y, dUjdy is discontinuous, the equation (viii) must be 

 replaced by 



where A denotes the difference of the values of the respective quantities on 

 the two sides of the plane of discontinuity. This is obtained from (viii) by 

 integration with respect to y, the discontinuity being regarded as the limit of 

 an infinitely rapid variation. The formula (ix) may also be obtained as the 

 condition of continuity of pressure, or as the condition that there should be no 

 tangential slipping at the (displaced) boundary. 



At a fixed boundary, we must have v = Q. 



1. Suppose that a layer of fluid of uniform vorticity bounded by the 

 planes y= h, is interposed between two masses of fluid moving irrotationally, 

 the velocity being everywhere continuous. This is a variation of a problem 

 discussed in Art. 225. 



Assuming, then, U=u for y&amp;gt;h t U=uy/h for h&amp;gt;y&amp;gt; -h, and U= u for 

 y&amp;lt; -h, we notice that d 2 //&amp;lt;% 2 - 0, everywhere, so that (viii) reduces to 



$-&quot;- ....................... ............ 



The appropriate solutions of this are : 



v = Ae- k v, for y&amp;gt;h-, \ 



v = Be-*y + C&amp;lt;*v, for h&amp;gt;y&amp;gt;-k; I ..................... (xi). 



v =De k v, for y&amp;lt;-h 



The continuity of v requires 



With the help of these relations, the condition (ix) gives 

 2 (a- + /hi) Cfe** - 1 (Be-** + C&amp;lt;* h ) = 0, 

 2 (o--/hl) Be kh + 



Eliminating the ratio B : (7, we obtain 



(xiii). 



(xiv). 

 87 



