How to solve complex potential for a circular cylinder with sources and sinks?

[benjamin_jairo]

Homework Statement

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

 

[Abhi]

Homework Statement

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

Hello Benjamin,

Could you please share how you arrived at the result .

I tried this by adding the source potentials and subtracting the sink potential and then differented it to get the complex velocity.But I could not arrive at the answer it represents the complex potential for a circle of radius a.

Pls Help.

Thanks!
Abi