606 Differential Equations of Mathematical Physics. 



Theorem. 



There is no expression, when JJljv is small, for the stream 

 function of the viscous steady-motion equations in two 

 dimensions, expressible in a finite number of terms in powers 

 of Jjl/v other than those explicitly independent of JJl/v. 

 For if so, let 



and insert this as before in the differential equation. 

 Equating the coefficient of C 2 " +1 in the equation to zero, 

 we find for yjr n 





(21) 



with the conditions that 



^ and I* and fd*|-(f*) 



B^' Oj/ Jbody Ort\0#/ 



are zero round the boundaries. 



Equation (21) is clearly the limiting case of our funda- 

 mental equation (8) when v tends to zero. There is, however, 

 this difference from the common conception of the perfect 

 fluid — that here the no-slip condition is maintained by the 

 presence of some distribution of vorticity. In fact, (21) may 

 be written 





ft = Ws^ 



which says that the lines of constant values of V 2 ^« are 

 coincident with those of constant -^r n ; i. e., 



where /is a function to be determined from the boundary 

 conditions. We may regard equation (21) as the limiting- 

 case of the flow of a fluid of exceedingly small viscosity ; 

 but since there are no externally applied forces or pressures 

 causing motion of the fluid, and no motion of any boundaries, 

 there cannot possibly be flow of any nature. Under these 

 circumstances ty n will be constant, and may accordingly be 

 ignored. This clearly applies to all values of n down to 

 n = l when the boundary conditions are now no longer 

 zero. 



It may oe remarked in passing that the solutions of 

 problems in steady motion independent of JJl/v must 



