Figure 10a Figure 10b 



Figure 10 - Illustrating the definition of the stream function ijj. 



corresponding cylindrical surfaces. If A is held fixed, therefore, th is a function of the coor- 

 dinates ar, y of P, and also perhaps of the time, or \h{x, y, t). This function is called the 

 stream function. Its dimensions are those of volume per unit time per unit of length parallel 

 to 2, or L^f-i. 



If the base point is moved from 4 to some other point 6, then all values of xb are changed 

 by a fixed value representing the flow across BA. Thus 6 contains an arbitrary additive con- 

 stant. 



If the values of th at two points Pj, P^ are lA^ and i//^, the rate of flow across any curve 

 P-iP^i ^^ i'^ Figure 10b, in the positive direction around P , per unit of length in the 3-direction, 

 is l'/, - (//,. If P. and P. lie on the same streamline, the rates of flow across AP. and across 

 APj must be the same, since there is no flow across a streamline, llence ih has a constant 

 value along any given streanUine. The family of curves defined by i// = constant is thus the 

 set of streamlines, and the streamlines themselves can be identified by means of the 

 associated values of lA. It follows in particular that i/f must have a constant value over any 

 stationary boundary, which is necessarily composed of streamlines. 



Simple relations exist between the stream function and the particle velocity. For, if P 

 is displaced an infinitesimal distance dx in the x direction, (/> increases by the flow across dx 

 or by vdx\ whereas if P is displaced a distance dy in the y direction, t/i/; = -udy, in view of 

 the convention as to the sign of i/*; see Figure 10a. Thus 



difj 



dip 



[13a, b] 



dy dx 



Or, more generally, let dih/ds denote the space rate of variation of t/f in a chosen positive 

 direction along any curve drawn on the a;y-plane, and let q^ denote the component of the 

 velocity normal to the curve, taken positive in a direction rotated counterclockwise through 

 90 degrees from the positive direction along the tangent to the curve. Thus, if ^^ > 0, the fluid 



21 



