Figure 11 — Illustrating the 

 relationship between the 

 space rate of variation of 

 stream function di///ds 

 and particle velocity q^. 



crosses from right to left as the curve is traced positively; 

 see Figure 11. Then 



.„ = - [13c] 



The stream function exists for any type of flow in an 

 incompressible fluid, even when the motion is rotational. 



If the flow is irrotational, then the velocity potential 

 also exists, and the two families of curves, = constant and 

 i// = constant, cut orthogonally, since the velocity at any point 

 is perpendicular to a curve = constant through that point and 

 tangential to a curve dr = constant. Then also, as in [6b, c] 



, V = 



dx By 



[13d, e] 



Comparison of these equations with Equation [13a, b] leads to the following relations between 



i6 and i/c 



d<p dill d(h d\!f 



dx d y dy dx 



The Laplace equation for <^ or Equation [7a] becomes, in two dimensions, 



[13f, g] 



^ + 



(9^0 



0; 



[13h] 



(9a;2 ay2 

 and differentiation and subtraction of Equations [13f, g] yields also ttie result that 



dx'^ 



d^il; 

 dy^ 



= 



[13i] 



Thus in irrotational, two-dimensional flow the velocity potential and the stream function 

 are both solutions of the two-dimensional Laplace equation. Solutions of this equation, 

 related as stated in Equations [13f, g], are called conjugate solutions or functions. If either 

 d> or (Ij is known, the other can be found, except for an arbitrary constant, by integrating 

 Equations [13 e, f] 



The two orthogonal families of curves, = constant and ^ = constant, are called a flow 

 pattern. If closely spaced curves of both types are drawn, they divide the plane into small 

 areas approximately rectangular in shape; such a diagram is called a flow net. If the same 

 equal spacing is used for both sets of curves, the rectangles become squares; for, by the 

 definitions of (f) and </j, between two adjacent curves 86 = - q8s where 8s is the distance 

 between them, and similarly between two lA curves 8x1/ = gSs', hence if \Scf)\ = \Si'j\, 8s = 5s'. 



This property of flow nets is sometimes made the basis of a graphical method for the 

 construction of an approximate flow net to satisfy given boundary conditions. The <p and ilr 

 curves are sketched in smoothly by estimation and are then corrected repeatedly while keeping 

 them in harmony with the boundary conditions, until they divide the area as nearly as possible 



22 



