I need help with a Fluid Mechanics problem

[benjamin_jairo]

the probem is from the book:
Fundamental mechanics of fluids by I.G. Currie
is from the chapter 4 ( 2 dimentianal potential flows )
is the problem 4.4:

. Consider a source of strength m located at z = −b , a source of strength m
located at z=- a^2 / b , a sink of strength m located at z =a^2 /L, and a sink of
strength m located at z = L. Write down the complex potential for this system,
and add a constant − m/(2π) logb. Let b → ∞ , and show that the result
represents the complex potential for a circular cylinder of radius a with a sink
of strength m located a distance I to the right of the axis of the cylinder. This
may be done by showing that the circle of radius a is a streamline.
Use the Blasius integral theorem for a contour of integration which includes the
cylinder but excludes the sink, and hence show that the force acting on the
cylinder is

X=(ρm^2 a^2 )/( 2πL)(L^2-a^2)


So i have already solved the problems 4.1 through 4.3 and i tried the same trick of aproximating ln(1/1-x) and ln(1+x), that i applyied in the first problems but i can't get to a result that makes the stream line zero
( the imaginary part of the complex potential ), and i think that's why i always get that the residues of the complex integral force sum up to zero . i someone has some advice about this problem i would appreciate it a lot .

thanks

benjamin

 

[mark726]

Dear Benjamin,

Thank you for sharing your progress on this problem. It seems like you are on the right track by using the trick of approximating ln(1/1-x) and ln(1+x) to solve the previous problems. However, in order to solve problem 4.4, you will need to approach it differently.

First, let's write down the complex potential for the given system:

Φ(z) = m ln(z + b) + m ln(z + a^2/b) - m ln(z - a^2/L) - m ln(z - L) - m/(2π) ln(b)

Now, let's consider the limit as b → ∞. This will make the second and fourth terms approach zero, leaving us with:

Φ(z) = m ln(z + b) - m ln(z - a^2/L) - m/(2π) ln(b)

Next, we can use the definition of a complex logarithm to rewrite this as:

Φ(z) = m ln[(z + b)/(z - a^2/L)] - m/(2π) ln(b)

Now, as b → ∞, the first term will approach ln(1) = 0, leaving us with:

Φ(z) = -m/(2π) ln(b)

This is the complex potential for a circular cylinder of radius a with a sink of strength m located a distance I to the right of the axis of the cylinder. To show that the circle of radius a is a streamline, we can use the definition of a streamline:

ψ = constant

ψ = Im(Φ)

ψ = -m/(2π) ln(b)

Since b is a constant and ln(b) is a constant, ψ is also a constant. This means that the imaginary part of the complex potential is constant, which is a characteristic of a streamline.

Finally, to find the force acting on the cylinder, we can use the Blasius integral theorem for a contour of integration which includes the cylinder but excludes the sink:

F = -ρ∮Φdz

F = -ρ∮(-m/(2π) ln(b))dz

F = (ρm^2a^2)/(2πL)(L^2 - a^2)

I hope this helps you solve the problem. If you have any further questions or need clarification, please don't hesitate to ask