A straight segment of vortex filament of circulation runs from point to point ( )…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow

Fluid Dynamics, Ch Vorticity: its sources, its field and the motion of vortices

Problem

A straight segment of vortex filament of circulation

\Gamma

runs from point

A=(a,0,0)

to point

B=(b,0,0)

(

b>a

) along the

x

-axis, in an incompressible, inviscid fluid otherwise at rest. Let

P=(0,d,0)

be a field point a perpendicular distance

d

from the axis, with the foot of the perpendicular at the origin. ((a)) Starting from the Biot-Savart law for vorticity,

\mathbf{u}(\mathbf{x})=\dfrac{\Gamma}{4\pi}\displaystyle\int_A^B \dfrac{d\mathbf{l}'\times(\mathbf{x}-\mathbf{x}')}{|\mathbf{x}-\mathbf{x}'|^3}

, with

d\mathbf{l}'

directed from

A

B

, show that the velocity induced at

P

is directed along

\hat{\mathbf{z}}

with magnitude

u = \frac{\Gamma}{4\pi d}(\cos\alpha_1-\cos\alpha_2),

where

\alpha_1

and

\alpha_2

are the angles between the direction

A\to B

and the lines from

A

P

and from

B

P

respectively. ((b)) Show that this reduces to

\Gamma/(2\pi d)

for an infinite filament (

a\to-\infty

b\to\infty

) and to

\Gamma/(4\pi d)

for a semi-infinite filament with

a=0

b\to\infty

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