Studies on the Motion of Viscous Flows — III 



By introducing the principle of minimum dissipation as a hypothesis, 

 Lieber and Wan [5] conclude that a two-dimensional flow field for a viscous 

 incompressible fluid is governed by the Biharmonic Equation. Since, in addi- 

 tion, the flow must satisfy the conservation-of- momentum principle in the form 

 of the Vorticity Transport Equation, they regard the vanishing of the convective 

 part of this equation as a compatibility condition. A physical interpretation of 

 this condition based on the consideration of odd and even parts of the stream 

 function brings them essentially to the conclusion stated in the Symmetry 

 Theorem. Our work shows that the flow field cannot be strictly governed by 

 the biharmonic equation, and that the separate vanishing of the biharmonic and 

 convective parts of the vorticity transport equation is not necessary to arrive 

 at the Symmetry Theorem, which is here proved by induction on the basis of a 

 consideration of the solutions of the linear sets of the differential equations and 

 the boundary conditions etc. governing the successive iterations. However, what 

 is significant in our work and the work of others is the clear recognition of the 

 relation between symmetry and time-dependence in viscous flows. We believe 

 that some such relation should hold for all viscous flows in general. 



Using the Fourier representations of Eqs. (1.60), (1.61), (1.62), etc. for the 



stream fxinctions 0^, n = 1, 2, 3, 

 Eqs. (1.29), we get 



together with the expression for i// in 



0(r,0,t) 



ApCr.t) + Co(r,t) + EgCr.t) + 



+ [Aj(r,t) + C/r,t) + Ej(r,t) 

 1 



] cos 



+ Bj(r,t) + D,(r,t) + F,(r,t) + •• 



sin ^ 



+ [A,(r,t) + C,(r,t) + E,(r,t) + 



cos 2t 



+ [B (r,t) + D2(r,t) + F (r,t) + • • •] sin 2( 



+ Y. [A„(r.t) + C„(r,t) + E^(r,t) + •••] cos n< 



+ [B^(r,t) + D^(r,t) + F^ ( r , t ) + •••] sin n( 



(2.37) 



When the stream function ^j is such that its streamline pattern is symmetric 

 about the polar axis, the terms in cos 6, cos 29 and cos nd drop out, giving us 



>A(r,0,t) 



r - - ) + Bj(r, t) + Dj(r, t) + F^(r,t) + •• 

 + [B^Cr.t) + DjCr.t) + F^ ( r , t ) + •••] sin2e 



+ E t^n('-'t) + Dn('-'t) + Fn(r-t) + ■••] sinn( 



sin 9 



(2.38) 



565 



