Page 68 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Dynamics of Inviscid Fluids 53

THEOREM 2.1. The rotation (vorticity) flux across a certain part of
a fluid surface (which is watched during its motion) is constant.
As direct consequences of this theorem we have the following results
which can be proved by “reductio ad absurdum”’:
- A fluid surface %, which at a certain instant ¢g is arotation (vorticity)
surface will preserve this quality all the time, i.e., it will be a rotation
(vorticity) surface during the motion. A similar result could also be

formulated for the vorticity (rotation) lines, these lines being defined as
the intersection of two vorticity (rotation) surfaces;
- If, at a certain moment, the fluid flow is irrotational (potential),
then this quality will be kept at any later moment.
This last result, known as the Lagrange theorem and which 1s valid in
the above mentioned hypotheses, could be obtained either by reductio ad
absurdum (supposing that the flux of rotation across a certain surface,

with w 4 0, would be different from zero which leads obviously to a
contradiction) or by remarking that the equation th (+) = (gradv) >

has the solution (in Lagrangian coordinates) @ a= ORF ot , where wo (we)
oo

is the vorticity vector at the moment fg and p = pg is Pte mass density
at the same moment.
If the fluid flow is rotational, then there will be a velocity potential
® such that v = grad®. As 2 ot = grad, from Euler’s equation in
Helmholtz form, in the same hypotheses of a barotropic fluid and of the
conservative character of the external forces, we also get

Ob ov? dp
grad (7 De +5+/2- } = 0.

In other words, in an irrotational flow of an inviscid barotropic fluid
with external forces coming from a potential U, we have 2 Sr P40 ey J4 ap _

U = C(t), where C(t) is a function depending only on time (in the steady
case this function becomes a constant, which does not change its value
in the whole fluid domain). This result, known as the second Bernoulli
theorem (integral) could be also extended in the case of a rotational
fluid flow. Precisely, by considering the inner product of both sides of
Euler’s equation with v, we will have that pZe +pv-gradK = 0,where
2 d.
K=5+f?-U.

If the flow is steady, then we will have at once v- gradK = 2& =0,
Dt ~
e., the quantity K = we +f ap — U is constant at any path line, the
value of this constant being different when we change the trajectory.

This last result is known as the first Bernoulli theorem (integral).