^ = A '^'Pi ^ -Ax 



^ (a;2+y2^ 22)1/2' dx (^2^^2^22)3/2 



and d(fi^/dx is also a solution of the Laplace equation. Here x can also be replaced by t cos 6 

 in terms of spherical coordinates with origin at P and the a;-axis as axis. Thus, another solu- 

 tion of Laplace's equation is {-n/ A){d4>-^^/dx) or 



ulx u cos 6 



4> = = [12d] 



{X^ + y2 ^ ^2)3/2 ^2 



where ^ is a new constant. 



The type of flow thus defined is said to be due to a point dipole or double source at P, 

 also called a point doublet, because it can be produced by placing a source and sink of equal 

 strength close together and letting their distance apart decrease to zero while the product of 

 distance and the strength of the positive source is kept equal to p.. The line from which 6 is 

 measured is called the axis of the doublet. 



13. TWO-DIMENSIONAL FLOW 



The flow is two-dimensional when there is no variation of anything in a certain direction, 

 and when the component of the velocity in that direction is everywhere zero. Thus, along any 

 line having this direction, the pressure and the particle velocity are uniform. Each fluid 

 particle moves in a plane perpendicular to the direction of uniformity, and the motion is the 

 same in all of these planes. It suffices to study the motion in a single plane, which may be 

 taken as the ary-plane. Then the 3-component of the velocity w is 0, and the components u and 

 V, like the pressure, are functions of x, y and perhaps the time t. 



Alternatively, it is sometimes convenient to consider the fluid between the rry-plane and 

 a parallel plane at unit distance from it. This part of the fluid remains permanently between 

 the two planes, and its motion is typical of the motion of the whole. 



For the two-dimensional flow of incompressible fluids it is convenient to define another 

 function known as the stream function. Choose a fixed line perpendicular to the a;?/-plane, 

 intersecting it in the point A, and a parallel line intersecting the ajy-plane in P, as in Figure 

 10a. Let the lines be joined by an open cylindrical surface parallel to z and having the lines 

 as two of its generators; this cylinder will intersect the xy plane in a curve, as illustrated by 

 one of the curves in Figure 10a. Let ib denote the volume of fluid that passes per second 

 across the part of the cylindrical surface that lies between the a;y-plane and a parallel plane 

 unit distance away; let lA be called positive when the fluid crosses in the positive direction 

 of rotation about A, or in the direction from Ox toward Oy. 



The quantity ih thus defined may be described briefly as the volume of fluid that passes 

 per second across any curve, per unit of thickness in the 2-direction. Its value must be the 

 same for all curves joining A and P, since no fluid is created or destroyed between the 



20 



