Figure 62 — A line source near a wall. 



first assumed to be the same as it would be in unbounded fluid, the velocity has a component 

 normal to the wall. Upon this flow let there be superposed the flow that would be associated 

 in an infinite mass of fluid with an equal and parallel line source located on the opposite side 

 of the wall, at a distance h from it and on the perpendicular from the given source to the wall 

 produced. Then at the wall the normal components of these two flows will cancel and the 

 boundary condition will thus be satisfied. These two partial flows are assumed to exist only 

 on the side of the wall on which the original source lies. The imaginary second source is 

 called an image of the first in the plane of the wall. 



Let the a;-axis be drawn through the source and its image, being thus perpendicular to 

 the wall, and the y-axis along the wall. Then, from Section 40, the complex potential is as 

 given above and the resultant potential and streamfunction yp are 



<^ = - A\n{r^T^), lb ^- A{6^ + d^) [43b,c] 



the significance of r,, '"oi ^i» ^9 ^^ exhibited in Figure 62. If it is desired to balance the equa- 

 tions dimensionally, r. t„ can be replaced by r, t^/o? in cji, thereby merely adding a constant to 

 all values of 0. The volume of fluid emitted by unit length of the line source per second is 

 2ttA. 



Some of the streamlines are shown in Figure 63, above the a;-axis only, relative to which 

 the flow is symmetrical. The source is at S. The streamlines are arcs of rectangular hyperbolas 

 with centers at the origin 0, given by 



9 "^ 9 



t^ + 2 xy cot y^ 



A 



h^ 



[43d] 



as is easily verified by writing out tan {d. +62)' '^^® equipotential curves are BernouUian 

 lemniscates. 



Since z = x + iy 



j2 ^ =4^2 



= ^A' 



x^ + y 



(a.2_y2_^2) +4a;2y2 



whence 



? = 



2 At 



[43e] 



