Consider a system of singularities consisting of the followi

Consider a system of singularities consisting of the following: 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 A sink of strength m located at z a^2/l A constant term of magnitude -[m/(2 pi)] log b Write down the complex potential for this system and let b rightarrow infinity. Show that the result represents the complex potential for flow around a circular cylinder of radius a due to a sink of strength m located a distance l to the right of the center of the cylinder. This may be done by showing that the circle of radius a is a streamline. Use the Blasius integral theorem around a contour of integration that includes the cylinder but excludes the sink, and hence show that the force acting on the cylinder, due to the presence of the sink, is X = rho m^2 a^2/2 pil (l^2 - a^2)

Solution

The complex potential W for the sink is -m/2pi ln( z-z0)

for the source is +m/2pi ln(z-z0)

Plugging in the values, the total potential for the four singularities is W1+W2+W3+W4

Adding the constant term -m/2pi*ln(b)

Total potential can be expressed as m/2pi *ln[ {(z+b)*(z+a^2/b)}/{((z-a^2/l)* (z-l)*b}]

where W1 = m/2pi * ln ( z+b)

W2 = m/2pi* ln ( z+ a^2/b)

W3 = -m/2pi * ln( z-a^2/b)

W4 = -m/2pi* ln (z-l)

PART 2:   DIVIDING BY b, the expression simplifies when b---infinity to

W= m/2pi* ln[(Z/( Z- a^2\'L)(Z-L)]

It can be seen to be the superposition of a source (Z term in numerator) with a quadratic term in denom corresponding to a cylinder.. This can be shown by using the argument of the

function Z/(Z-L)*(Z-a^2/L)

This can be reduced to   polar form using Z= a exp(itheta), and we find

the imaginary terms disappear, giving argument of Z is a constant, leading to a stream line.

For Blasius, integrate bkeeping in mind that force - rho *U * Gamma, Gamma being circukation