APPENDIX I 



MODEL OF THE POLYNYA CIRCULATORY SYSTEM 



Figure 7 illustrates streamlines generated by the motion of a 

 solid sphere in an infinite mass of frictionless fluid. If we take the 

 origin at the center of the sphere and the x axis in the direction of 

 motion, the normal velocity (Vp ) at the surface of the sphere is U cos 9, 

 where U is the velocity of the center. 



Lamb shows that the stream function due to the sphere is 

 ,/, = -J-U-&- sin^e 



r 2 r 



where a is the radius of the sphere, r is the radius vector from the 

 center to points on or exterior to the sphere (r-a), and 9 is the angle 

 between the radius vector and the x axis. At any given instant the 

 trajectories of the fluid particles are tangent to the streamlines. 



The total flux through a curved surface S is JgVf^dS. 

 Arbitrarily making this value equal to -27n//, we have 



-27r<//=rVndS. 



In the case where S is the surface of the above sphere (r=a) substitu- 

 tion of 2 7ryds for dS yields 



-2 



>=/Vn27ryds, 



where ds, as shown in Figure. 8 , is an infinitesimal length of arc sub- 

 tended by an infinitesimal angle, d 9 , on the surface S. Substitution 

 of U cos 9,3. sin 9 , and a dfi for Vp, y, and ds, respectively^; and 

 integrating between the limits and 9 yields 



-v// = Ua2 Pcos 9Q\r\ 9 6 9. 



Therefore, 



4f = -j\Ja^s\n^ 9. 



Lamb shows that the stream function from an n pole is given by 



^ = K-^ COS 9 



Since the sphere acts as a dipole. 



From the boundary value r=a, 



K 



^sm^9 = --^\Ja''s\n'' 9. 

 a 2 



17 



