Download Conservation of Energy Due to the fact that in a nonviscous flow

Conservation of Energy
Due to the fact that in a nonviscous flow there is no mechanism for heat generation or the motion is
too rapid for heat conduction, a nonviscous flow is nondissipative. The absence of a local heat flow
ensures a constant local entropy. Therefore it is stated that a nonviscous flow is isentropic, which
makes the ‘conservation of energy’ to take a simple form. The energy density consists of two
contributions: the kinetic energy density and the internal energy density. In this way the
conservation law can be written in differential form as:
(
With
)
as the kinetic energy density and
[
]
as the internal energy density. Given a element of
fluid, three sources contribute to the increase of the total energy in V: convection of kinetic and
internal energy, work done by the pressure force on the surface, and work done by the volume force
f.
Bernoulli’s Theorem
To apply Bernoulli’s Theorem on a irrotational flow, one can take an exact first integral of the Euler’s
Equation. However, the following restrictions have to be made:
1. The motion is irrotational,
(where v is velocity field)
2. The external force is conservative,
(where U is potential energy/unit mass)
3. The fluid is incompressible with fixed constant density,
For an isentropic irrotational flow, Bernoulli’s Theorem can be written as:
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Thomson’s circulation theorem:
It states that, if the external forces acting on an inviscid fluid are conservative and if the fluid density
is a function of the pressure only, then the circulation along a closed curve which moves with the
fluid does not change with time.
=0
The principal use of Kelvin's theorem is in the study of incompressible, inviscid fluid flows. If a body is
moving through such a fluid, the vorticity far from the body is, by definition, zero. Then according to
Kelvin's theorem, the vorticity in the fluid will everywhere be zero and the flow will be irrotational.
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Irrotational Flow
Flow is said to be an irrotational flow. The vorticity of an irrotational flow is zero. continuity
equation states that, in any steady state process, the rate at which mass enters a system is equal to
the rate at which mass leaves the system. Using continuity equation and Bernoulli’s theorem for
isentropic irrotational flow we get Euler-Lagrange equation of the lagrangian density. From
lagrangian, we get the Hamiltonian density and also the Energy flux.
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Daan Wilmink
Majid Rasool
Ali Hassan