Force on a plate from a fluid line source

[Raptor112]

Homework Statement

A line source of strength 2$\pi$m is located $a$ distance a above a horizontal plate. Find the force per unit width on the plate, ignoring gravity and taking the pressure below the plate to be uniform and equal to the stagnation pressure of the fluid. (You may nd the substitution $x = a tan(\theta)$ useful in evaluating any integral that arises.)

2. The attempt at a solution
$F = - \int p \hat{n} dl = - \int_{-\infty}^{\infty} p \hat{n} dx, \hat{n} = \hat{j}$ $p$- pressure

 

[Chestermiller]

Are you familiar with the method of images?

 

[Raptor112]

Are you familiar with the method of images?

Place a source of equal strength at a distance $a$ below the horizontal plate so the total complex potential becomes

$w(z) = mlog(z^2+a^2)$

but how will that give me the pressure at the boundary?

 

[Chestermiller]

Place a source of equal strength at a distance $a$ below the horizontal plate so the total complex potential becomes

$w(z) = mlog(z^2-a^2)$

but how will that give me the pressure at the boundary?

If you have a single line source, then it is easy to get the velocity at a radial distance r from the source. You can then use Bernoulli's equation to get the pressure at that location, assuming that the pressure and velocity at infinity are zero.