problem 7.1

Vector potential A

Verify the relation curl v = ? by using the vector potential A defined by (7.5a). State in what cases the div A = 0 is satisfied.

problem 7.2

Invariants of motion

Verify the conservation of the five integrals (7.9) (7.11) of Impulse P, Angular impulse L and Helicity H for the vorticity ? evolving according to the vorticity equation (7.1).

problem 7.3

Invariance of R ²

Verify that the integral (7.31) for R ² is invariant during the vortex motion, according to (7.27) for 2D motions of an incompressible inviscid fluid.

problem 7.4

Vortex sheet

Suppose that a flow is represented by (7.43) having a discontinuous surface at y = 0. Show that its vorticity is given by the expression (7.44).

problem 7.5

Vortex filament

Derive the asymptotic expression (7.68) for the velocity u( x)from the Biot Savart integral (7.67)

problem 7.6

Helical vortex

Show that the following rotating helical vortex x _h = ( x, y, z) ( s, t) satisfies Eq. (7.72):

where a, k, h, ?, ? are constants and t _h is its tangent.

problem 7.7

Lamb s transformation [Lamb32 , Sec. 162]

Consider a thin-cored vortex ring in a fixed coordinate frame in which velocity u( x , t) vanishes at infinity. Suppose that the fluid density is constant and the vortex...