Studies on the Motion of Viscous Flows — III 



(2.46) 



The relations of Eqs. (2.45) and (2.46) show that, if at any generic point P(r,0) 

 in the flow field the velocities U^, v^, uj, vj have the same sign so that u^ + 

 U2 and Vj + Vj represent a strengthened velocity field, then at a point <I>(r,0), 

 which is the mirror- image of the point P (r,i9) about the line at right angles to 

 the polar axis, the velocities Uj and Vj will have signs opposite to the veloci- 

 ties Uj 2-^^ ^2J ^^^ hence their sums Uj + uj and vj + vj will represent a 

 weakened velocity field. The streamline patterns for ^^ = constant and ^^ - 

 constant are sketched roughly in Figs. D and E respectively. The arrows indi- 

 cate the direction of the velocity vector, which is tangential to the streamlines, 

 and hence the direction of the flow at any instant. 



Fig. D - Streamline pat- 

 terns for vj = constant 



Fig. E - Streamline pat- 

 terns for ^2 - constant 



Let us consider the superposition of the two streamline patterns given in 

 Figs. D and E due to ^j - constant and 02 - constant, wherein we have as- 

 sumed that in front of the cylinder ui, Vj, u^, and V2 are all positive. What- 

 ever the relative magnitudes of Uj, vj, U2, and vj, the resultant flow due to 

 the superposition in front of the cylinder will be moving with higher velocities, 

 and its appearance will be similar to that in Fig. D, but with the streamlines 

 displaced outwards from the cylinder. Specifically, there cannot be any closed 

 streamline in front of the cylinder. In the rear of the cylinder the resultant 

 flow will be moving with lower velocities, and its appearance will depend on 

 whether or not |u,| > |u,| and |v,| > |v,| everywhere. K lu,| > lu,| and 

 |vi| > IV2I everywhere in the flow field, then the flow in the rear will resemble 

 the flow pattern in Fig. D, but with very widely separated streamlines as indi- 

 cated by lowered velocities. The streamlines, moreover, cannot be closed and 

 there cannot be any separation. However, if in some velocity domain luj < |u2| 

 and I Vjl < I V2I , then the flow in that part of the domain which is at the rear of 

 the cylinder will resemble the flow pattern in Fig. E, and this region will be 

 characterized by the appearance of closed streamlines and points of separation. 



567 



