-..-■.,-' Lieber and Yajnik 



Here we attempt to obtain some insight into the above problem by posing a 

 slightly different question, viz., what are the general properties of symmetric 

 and antisymmetric flows. The starting point of the present study is the formu- 

 lation of conditions of symmetry and antisymmetry. Their application to the 

 equations of motion immediately provides the information which was sought. 



This study was in part motivated by a result relating to the existence of 

 necessary connections between the geometrical symmetry characteristics of a 



boundary and the production of time-dependent 

 motion under time -independent boundary con- 

 Q^^^.'j ditions [1]. This result was obtained as part 



of a comprehensive study concerned with the 



/ 



y/ formulation and application of variational 



y M/RROR PLA^E principles in the study of viscous flows [6,7, 



8,9,10]. 



Fig. 1 - Velocity vectors Symmetry and Antisymmetry 



about a mirror plane 



Let Q and R in Fig. 1 be mirror images 

 about a plane. Let the velocity vectors at Q 

 and R be also mirror images of each other. Then the components of velocity 

 parallel to the mirror plane must be equal and the components normal to the 

 plane must be equal in magnitude, but opposite in direction. K the velocity vec- 

 tors at any such image points about a mirror plane are images of each other, 

 then the flow is here called symmetrical about the plane. K, on the other hand, 

 the velocity vector at R is equal and opposite to the image of the velocity vector 

 at Q, then such a situation would be an antithesis of the symmetrical flow. We 

 may call a flow antisymmetrical if the velocity vector at the image point of any 

 point is equal and opposite to the velocity vector at the point. 



Let us leave the optical analogy aside and start with analytical definitions. 

 If a flow satisfies the conditions 



Uj(Xj,X2,X3,t) = Uj(Xj,-X2,X3,t) 



U2(Xj,X2,X3,t) = -U^CXj, -X2,X3,t) (1) 



U3(x^,X2,X3,t)3 U3(Uj, - X^ , X3 , t ) 



for all values of x^, X2, Xj, and t, the flow is said to be symmetrical about 

 the plane Xj = 0. Here x^, X2, and X3 are Cartesian coordinates, t is time, 

 and Uj, U2, and U3 are components of velocity parallel to the Xj, Xj, and X3 

 axes. If, on the other hand, the flow satisfies 



U^(Xj,X2,X3,t) = -UjCXj, -X^.Xg.t) 



U2(Xj,X2,X3,t) - U^CXj, -X2,X3,t) (2) 



U3(Xj,X2,X3,t) = -UjCx^, -X2,X3,t) , 



682 



