3.4: Kelvin

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3.4 Kelvins circulation theorem
For any flow governed by Eulers equation ,
circulation round a closed chain of fluid
particles is conserved. This is

• Remarks
• A closed chain of fluid particles
• a loop which consists continuously of the
  same fluid particles i.e. each element dl is
  moving with the fluid.
• b) has been extended slightly from its
  application at a single point to indicate that
  each point of the loop is moving with the fluid.
  

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Prove the theorem
Using Eulers eq. for the 1st term on RHS
Where,
( using the fact that the line integral about a
closed loop of a perfect differential is zero.)
2nd term on RHS
Since is a position vector,
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Application of the Kelvins Circulation Theorem
--------the generation of lift on an
aerofoil
The measured pressure shows that pressures above
the wing are less than those below. From the
Bernoulli Theorem for irrotational flow,
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If pupltplow, u upgtulow must be satisfied.
The Kutta-Joukowski Lift Theorem The lift force
perpendicular to the streamline is
Where u is flow speed at infinity. So,
and
Why u upgtulow
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KCT explains the formation of negative
circulation ?
At
Consider a loop abcda which is large enough to
be away thin boundary layer on the aerofoil and
away a thin wake
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At tt1, portition of curve aecda has positive
vorticity from votex and
from Stokess theorem. And, from KCT,
Therefore,
from K-J Lift Theorem
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3.5 The persistence of irrotational flow from
Stokes theorem
Hence, the vorticity flux through a close chain
of fluid particles is conserved in inviscid flow.

----------- Cauchy---Lagrange theorem
For a 2-D flow, vorticity eq.
C-L theorem is obvious.
For a 3-D flow , C-L theorem is not as obvious as
in 2-D flow.
(see Lec. 6)
If at t0,
at is one solution for (2).
If a portion of the fluid is initially in
irrotational motion, that portion will always be
in irrotational motion. C-L theorem
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Physical explanation of C-L theorem In
Eulers eq. (i.e. inviscid) , only stresses are
, which act normally to the particle surface
and cannot apply a couple (cant generate torque
as pressure act through the center of mass of an
element) to the particle to bring it into
rotation since i. HENCE, the study of inviscid
motion may be reduced to the study of
irrotational motion.
For irrotational flow ,
velocity field can be expressed by velocity
potential .
And for incompressible
fluid . ( )
(only when vorticity0, can velocity write in
velocity potential form)
Stream function for 2-D flow
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is known as the stream function because it is
constant along a streamline.
On a streamline (i.e. definition).
dy
v
Or
dx
u
Streamline give a snapshot of the velocity
field at any instant (steady flow) . For 2_D
irrotationality flow.
We have
Cauchy-Riemann conditions
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Equipotential lines (on which is constant)
and streamlines are orthogonal, as This
demonstration fails at stagnation points where
the velocity is zero.