Studies on the Motion of Viscous Flows — V 



it is said to be antisymmetrical about the plane X2 = 0, 



It is easy to see (see the Appendix for illustration) with the help of the 

 definition of partial derivatives that u^ ^1 ^i,v ^1 a'l "s.tJ "^3,1! "3,3! "2,25 

 V^u^; and v2u3 are even functions of X2 in the symmetrical flow and odd fianc- 

 tions of X2 in the antisymmetrical flow. Here, a comma followed by index 

 1,2, 3, t means partial differentiation with respect to x^, X2, X3, or t. v^f 

 denotes f.^ + f,22 + ^'33- That is, 



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



if the flow is symmetrical, and 



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



if the flow is antisymmetrical. 



Similarly, U2 t! "2,i5 "^2,3; '^1,2; "3,2? ^^'^ "^^^2 are odd in the symmetri- 

 cal flow and even in the antisymmetrical flow. 



Now let us consider a homogenous incompressible Newtonian fluid. The 

 flow obeys the equations 



"i,t + "j^i.j 



-P . + py^u. , (3) 



and 



Ui,i = . (4) 



Here P is the pressure divided by the density, and v the kinematic viscosity. 

 The body forces are assumed to be absent. It is now convenient to define 



P^(Xj,X2,X3, t) = - [P(Xi,X2,X3,t) + P(Xj, -X2,X3,t)] , 



and ... .... 



1 



P^(Xj,X2,X3,t) = - [P(Xj,X2,X3,t) - P(X^, -X2,X3,t)] . 



Hence, 



P^(Xj,X2,X3,t) = PS(Xj, -X2,X3,t) (5) 



P^(Xj,X2,X3,t) = -P^(X^, -X2,X3,t) , (6) 



and 



ps ^ pa ^ p (7) 



683 



