Note :
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!
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.
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) }
but ln [ x+/- i y ] = .5 ln [ x^2+y^2 ] +/- i tan^-1 { y/x }
To simplify the expression "-m*ln{-r*sin(theta)}", we can use the properties of logarithms. In this case, we have a negative value inside the logarithm.
The natural logarithm of a negative number is not defined in the real number system. However, in complex analysis, we can define the logarithm of a negative number using the principal value of the complex logarithm.
The principal value of the complex logarithm is defined as ln(z) = ln(|z|) + i*arg(z), where arg(z) is the principal value of the argument of z.
In our case, we have -r*sin(theta) as the argument of the logarithm. The principal value of this argument can be obtained by adding or subtracting multiples of 2*pi until the result is within the range (-pi, pi].
Let's assume we add 2*pi to -r*sin(theta) to bring it within the range (-pi, pi]:
-m*ln{-r*sin(theta)} = -m*[ln{-r*sin(theta)} + 2*pi*i]
So, the simplified expression is:
-m*ln{-r*sin(theta)} = -m*ln{|-r*sin(theta)|} - 2*pi*m*i
This matches with the answer you provided: A = uy - m*ln{-r*sin(theta)}. The difference in the sign of the logarithm and the additional term -2*pi*m*i is due to the particular choice of the principal value of the complex logarithm.