The Attempt at a Solution
Part a is simple. The first step is to find the complex potential. I wasn't 100% sure, but it seems like I need to add an image of the source, so I added an additional source located at z = -ia and then added the three complex potentials:
W(z) = source + image + uniformFlow
W(z) = m/(2*pi) * ln(z-ia) + m/(2*pi) * ln(z+ia) + Uz
W(z) = m/(2*pi) * ln(z^2 + a^2) + Uz
Then, to satisfy the no-penetration boundary condition along the plate, we need to ensure that the x-axis is a streamline. We can substitute z = x+iy and then plug in y=0 to simplify our equation. From there, we know that the imaginary part of W(z) is the stream function, and this is clearly zero along the x-axis, so we conclude that the x-axis is indeed a streamline.
Part b is where I'm having trouble. I need to find the stagnation points, which occur when dW/dz = 0. I take the derivative and set it equal to zero as follows:
mz/(pi*(z^2 + a^2)) + U = 0
Then, with some manipulation:
z^2 + mz/(pi*U) + a^2 = 0
I then apply the quadratic formula to find:
z = -m/(2*pi*U) +/- (a/2)*sqrt( (m/(pi*U*a))^2 - 4
Since the question asks for the solution in terms of m/(U*a), I can manipulate the result to find:
z = a* (-XY +/- sqrt((XY)^2 - 1))
with X = 1/(2*pi)
and Y = m/(U*a)
I'm not sure if this is right, because the multiplication of the entire expression by 'a' seems odd. Is this correct so far?
Part c also gives me trouble. We need to figure out how strong the source needs to be in order to have no flow from the uniform flow pass underneath the source. I'm not sure how to proceed here. My first idea was to figure out under what conditions the y-axis is a streamline, but that lead to a dead end. I then tried to find a stagnation point along the x-axis, but this introduces a new variable (x-coordinate of the stagnation point) that is giving me trouble. Can someone push me in the right direction here? Thanks!