Studies on the Motion of Viscous Flows— V 



V— General Symmetry Properties of 

 Flows and Some Consequences 



Paul Lieber 



University of California 

 ' •' Berkeley, California 



and 



Kirit Yajnik 



Indian Institute of Technology 



Kanpur, India 



ABSTRACT 



Mathematical conditions are formulated for flows displaying symmetry 

 and antisymmetry with respect to a plane. When homogeneous incom- 

 pressible Newtonian fluids manifest such properties, the equations of 

 motion restrict the distribution of the stresses. This report establishes 

 that pressure is symimetrical in symmetrical flows. "When the flow is 

 antisymmetrical, the asymmetrical part of pressure is harmonic. Fur- 

 thermore, each component of vorticity obeys the heat conduction equa- 

 tion in antisymmetrical flows. Thus the vorticity transport equations 

 reduce to linear equations in antisymmetric flows. More than ten exact 

 solutions of Navier-Stokes equations satisfy these linear equations. 

 Finally, such antisymmetrical flows can be represented as a superposi- 

 tion of potential flow on a flow in which each comiponent of velocity sat- 

 isfies the heat conduction equation. 



INTRODUCTION 



Experiments on fluid flow often are designed so that the solid boundaries 

 have properties of symmetry. The departure from symmetry occurring in these 

 boundaries are reduced to a minimum. Yet the flow of fluids in a majority of 

 cases does not enjoy the symmetry property. The wake of a circular cylinder 

 placed symmetrically in a water channel or an air tunnel exhibits a Karman vor- 

 tex street in a limited range of Reynolds numbers, and the flow in the wake is 

 not symmetrical about the central plane of the channel or tunnel. When a water 

 channel or air tunnel bifurcates into two, forming a Y-intersection, there is a 

 regime where the flow takes place in one arm of the bifurcation. Here too the 

 flow does not possess the symmetry of the boundaries. This lack of respect of 

 fluid for symmetry is a striking and puzzling property, 



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