Hydrodynamics and Elasticity: Class 6
Complex potential for steady, two-dimensional, irrotational, inviscid, incompressible flows 1. Find the complex potential for:
(a) A uniform flow inclined at an angle α to the x-axis.
(b) A linear vortex with u =k
reθ in terms of the circulation Γ = I
u · d`.
2. Find the complex potential for a vortex with intensity Γ at a distance d from a wall at x = 0 (the boundary condition at the wall is the vanishing normal velocity component). What are the streamlines for this flow?
Hint: think of an analogue of the method of images.
3. Milne-Thomson’s circle theorem Prove that if w0 is a complex potential in the whole space which has all singularities in the region |z| > a, then
˜
w = w0(z) + w0
a2 z
, (1)
is a complex potential for which |z| = a is a stream line and that ˜w has the same singularities as w0 for
|z| > a.
4. Use the Milne-Thomson theorem with a uniform flow w0= U z to find the flow around a cylinder |z| = a in a stream of an ideal fluid. Show that this is not the only flow satisfying the boundary condition at
|z| = a and a linear vortex can be added at the centre to establish a family of solutions with different circulation, which have the form
˜ w = U
z +a2
z
− iΓ
2πln z. (2)
5. Use the Bernoulli theorem to find that for a 2D steady flow of an ideal fluid around a cylinder computed above, such that u(r) ∼ U exas r → ∞, the force components are given by
Fx= 0, Fy = −ρU Γ. (3)
6. * Blasius theorem Prove that if w(z) is the complex potential describing a steady 2D flow of an ideal, incompressible fluid around a body with a contour C, then the force components acting on the body are given by
Fx− iFy= I
C
dw dz
2
dz. (4)
7. * Use the Blasius theorem to prove the Joukowski theorem which states that the result of the previous problem holds for any body shape. [See: D.J. Acheson, Elementary Fluid Dynamics, Oxford University Press, 1999]
