Note :

Question

Let us combine a source of strength m at the origin with a uniform flow of velocity 𝑢 parallel to the positive x-axis. Then the complex velocity potential(w = -[velocity potential(pi)] - stream function(A)*i) is;

𝑤 = −(𝑢𝑧) − 𝑚*ln(𝑧) , here z = x + iy and i=sqrt(-1)

Question :

Find the stagnation point and equation of the streamline.

The answer is given as follows;

𝑤 = −𝑢𝑧 − 𝑚*ln(𝑧) ( w = -[velocity potential(pi)] - stream function(A)*i ) , here i = sqrt(-1).
Here, 𝑤 = −𝑢𝑧 − 𝑚*ln(𝑧)
𝑑𝑤/𝑑𝑧= −𝑢 − 𝑚𝑧

For the stagnation points
|𝑑𝑤/𝑑𝑧| = 0

i.e. z = -(m/u) , which lies on the x axis.

pi + Ai = −(𝑢𝑧) - [ m*ln(𝑧) ] , i =sqrt(-1)

pi + Ai = -ur*(e^(−𝑖𝜃)) - { m*ln[r*(e^(−𝑖𝜃))] } , r=|z|

A = -[ursin(θ) ] - (m*θ)

A = -(u*y) - m*arctan(y/x) , since z = x + iy

What I'm having confusion is regarding the answer of A;
So, we know (e^(-i*theta))= cos(theta) - isin(theta)

Hence, according to my working,

pi + Ai = -[ur*(e^(−𝑖𝜃))] - { m*ln[r*(e^(−𝑖𝜃))] }
pi + Ai = -{ ur*[cos(theta)] } - isin(theta) ] - m*ln{r*[cos(theta) - isin(theta) ]}

So if we compare real and complex coefficients, we should get,
A = -[ ur(-sin(theta)) ] - m*ln{r*(-sin(theta))

That is, A = ur*sin(theta) - m*ln{-r*sin(theta) }

===> A = uy - m*ln{-r*sin(theta) }

I'm not sure how to simplify " - m*ln{-r*sin(theta) } "

Can anyone explain the reason for this difference in the answers?

Thank you!

Answer 1

𝑤 = −𝑢𝑧 − 𝑚*ln(𝑧)
𝑑𝑤/𝑑𝑧= −𝑢 − 𝑚𝑧
huh?
well, -u makes sense but m z does not
this is a source, in two dimensions the speed has to drop off as 1/radius from source to satisfy continuity
AND
indeed d/dz (ln z) = 1/z not z
to check, the stagnation point is where the velocity on the x axis due to the source is equal to u.

Answer 2

There is a mistake d/dz(ln(z))=1/z

Could you also let me know how they get the answer as follows;

pi + Ai = −(𝑢𝑧) - [ m*ln(𝑧) ] , i =sqrt(-1)

pi + Ai = -ur*(e^(−𝑖𝜃)) - { m*ln[r*(e^(−𝑖𝜃))] } , r=|z|

A = -[ursin(θ) ] - (m*θ)

A = -(u*y) - m*arctan(y/x) , since z = x + iy

and not pi + Ai = -[ur*(e^(−𝑖𝜃))] - { m*ln[r*(e^(−𝑖𝜃))] }
pi + Ai = -{ ur*[cos(theta)] } - isin(theta) ] - m*ln{r*[cos(theta) - isin(theta) ]}

So if we compare real and complex coefficients, we should get,
A = -[ ur(-sin(theta)) ] - m*ln{r*(-sin(theta))

That is, A = ur*sin(theta) - m*ln{-r*sin(theta) }

===> A = uy - m*ln{-r*sin(theta) }

Answer 3

but ln [ x+/- i y ] = .5 ln [ x^2+y^2 ] +/- i tan^-1 { y/x }