Question

Consider a two-dimensional ideal flow in the x-y plane (with radial coordinate r2 = x2 + y2). Given the velocity potentials of 1) a uniform flow, 2) a source φ = (q/2π) ln r, and 3) a dipole φ = −d · r/(2πr2):

a) Using the principle of superposition, construct a linear combination of the ingredients above that gives the flow past an infinite cylinder. [10 points]

b) Sketch the streamlines of the flow everywhere in space. [10 points]

Answer 1

Now creating flow over a cylinder :

Uniform Flow potential : flow along +ve X

Dipole along x a-axis

if d > 0 then sink is along -X but we want source along -X i:e our case d<0  

So potential is

Now finding velocity

along r

  

  

If

or

note : d<0

then there is no flow along radial direction for any theta which is the boundary condition required for the cylinder hence above is the required distribution

b Black line is a streamline which marks our cylinder, pardon my drawing this is for understanding only, please let me know if any doubt