Studies on the Motion of Viscous Flows— I 



line elements, cannot be zero. Eddies may thus arise in a region of sensible 

 vorticity such as the boundary layer or the wake of a body. 



One may also expect that if the velocities in the vicinity of a particle are 

 parallel, it cannot be in an eddy. With the z axis in the direction of the veloci- 

 ties, v^ , Vxz , V ^ , and V can be seen to be zero and so also whirlicity. 

 Hence rectilinear flows, including plane Couette and Poisuelle flows, do not 

 have any eddies. -■ - ■- - - . -■ - ....■ 



It should be remarked in passing that an eddy is thought to be a connected 

 set of particles of positive whirlicity. For the sake of definiteness, we will take 

 it to be a maximal connected set. This avoids the possibility of intersection of 

 eddies. 



Consider the boundary of an eddy where whirlicity vanishes. The velocity 

 of propagation normal to itself is then given by 



3(4M3 + 27N2)/Bt + c|V(4M3 + 27N2)| =0, '''/;_ - 



or , 



' 2M2 3M/3t + 9NBN/3t ' ''^ ' ' / \_ /g^x 



|(2M2vM + 9NVN)| 



if the denominator is different from zero. Since this velocity will in general be 

 different from fluid velocity, particles may enter or leave an eddy. 



Now let us consider eddies of an incompressible fluid. The invariants M 

 and N reduce to 



M = -Vu : Vu/2 = -(V • CJ)/2 , (36a) 



N = det|Vu| . (36b) 



Thus M is proportional to the divergence of acceleration, whereas N^ is a local 

 measure of three -dimensionality of the flow. If N is zero, there exists a non- 

 zero vector a such that (Vu)a vanishes and the flow in the vicinity of the parti- 

 cle relative to an observer travelling with the particle is normal to a . Note 

 that three -dimensionality always favors the occurrence of eddies. This conclu- 

 sion is compatible with the observations of more prevalent eddies in three- 

 dimensional turbulent flows than in two-dimensional flows. 



Since the equations of motion of a Newtonian fluid of constant properties are 



pu = - Vp + /oF -I mV^u , (37) 



p, fj. , p , and F being respectively, density, coefficient of viscosity, pressure, 

 and body force, the invariant M can be calculated with the help of a continuity 

 equation as 



M = (V-p - pVF)/2 . (38) 



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