﻿454 Equations of Motion of a Viscous Fluid. 



which may be written 



(*-£-£ 3r)*w«°. 



2) 

 where — has the same meaning as before (p. 452). 

 Qt B vr ' 



These equations can readily be transformed to curvilinear 

 coordinates should occasion arise since, apart from transfor- 

 mations already discussed, we require only the form of the 

 operator <E> and this can be obtained from the identity 



"S3" 0"UJ 



§ 4. Two-dimensional Motion. 



The equations for two-dimensional motion may be obtained 

 as a particular case of the general equations, but they are 

 more readily obtained from the Cartesian equations of 

 motion. Using the same notation as before and taking 

 the plane of xy as the plane of motion, we have 



B" ~du "du By , ^o 

 dt B# oy ox 



and 



Bv . B* 1 , B^ B% . „ 2 

 B^ o# By B// 



where >|r is Earnsbaw's current function. 



Substituting these values for ??, t* and eliminating ^ we 

 have 



s(v«*)+ * ( £; v y = »vv- 



B* o(.i', ,'/) 



Taking a, /3, conjugate functions of #, y, as orthogonal 

 curvilinear coordinates, we have from (4) 



also 



Bty, W ) = Bty, V ! | ) . dp, ff ) 



