Studies on the Motion of Viscous Flows — I 



Suppose the swirl at a whirlpoint is negative. Let the radial line 6^=0 be 

 chosen to coincide with the direction of a line element at the origin that has ex- 

 tremum angular velocity. Then one can see from Eq. (24) that a normal element 

 at the origin will also have an extremum value of angular velocity. Let these 

 angular velocities be denoted by Wq and '^tT2- They will have opposite signs, 

 as the swirl is assumed to be negative. The differentiability of velocity permits 

 us to choose a neighborhood where the derivatives of velocity occurring in Eq. 

 (24) at any point differ from the corresponding derivatives at the origin by less 

 than one-sixth of mln ( |wq1 , | w„ 2I ) • Then the angular velocity of a line element 

 in the direction e = situated at any point in the neighborhood differs from the 

 angular velocity of a parallel element at the origin by less than one -half of 

 min (|wo| , |w„ 2' ) • Hence the angular velocity of any line element in the direc- 

 tion 5 = has the same sign as w^ , and similarly the angular velocity of a 

 normal line element has the same sign as w„ 2 • Now consider the segment of 

 streamline in the neighborhood of the whirlpoint. If 9 increases from Oi to 62 

 as h increases from hj and h2, e changes from values in the first quadrant to 

 those in the second at some intermediate point and from the second to those in 

 the third at some other intermediate point. Clearly, do dh cannot be negative 

 at these points. One can see with the help of Eq. (27b) that neither wq nor w^ ^ 

 can be zero, and hence that swirl at the origin, being the product of w^ and w^ 2 > 

 cannot be negative. A similar argument can be made when e decreases as h 

 increases from h^ to h2 . This contradiction leads to the required result. 



Some light is thrown on the question of whether or not the swirl can be zero 

 at a whirlpoint by the conclusion that (Bu Bx) (3v/By) - (Bu'By) Ov 3x) must be 

 different from zero there, provided that all of the derivatives in the expression 

 do not vanish simultaneously. To see this, consider the segment of streamline 

 satisfying the requirements of the definition in any neighborhood of the whirl- 

 point. Now for any given unit vector t , there are two points X3 and x^ such 

 that X3 = kgt , X4 = - k4 t for some positive k3 and k^ , and as a result 



X3 • n = x^ • n = 



for a unit vector n normal to t . Then, by the mean value theorem, 



(dx/dh) • n = 



at an intermediate point. Hence, in any neighborhood of the whirlpoint, there is 

 a particle whose velocity is normal to any specified unit vector n . 



Suppose Ou/Bx)Ov/3y)- Ou/3y)(Bv/3x) is zero at a whirlpoint. Then 

 there is a nontrivial solution (a,b) to the following simultaneous equations 



Ov/Bx)a - Ou/3x)b = 

 (3v/3y)a - (3u/By)b = 0, 

 and for any values of x and y , 



a [Ov/3x) X + (Bv/By)y] - b [(Bu/Bx) x + Ou/By)y] = 0. 



447 



