Hamiltonian Fluid Mechanics - Irrotational Barotropic Flow

Irrotational Barotropic Flow

Take the simple example of a barotropic, inviscid vorticity-free fluid.

Then, the conjugate fields are the mass density field ρ and the velocity potential φ. The Poisson bracket is given by

and the Hamiltonian by:

where e is the internal energy density, as a function of ρ. For this barotropic flow, the internal energy is related to the pressure p by:

where an apostrophe ('), denotes differentiation with respect to ρ.

This Hamiltonian structure gives rise to the following two equations of motion:

\begin{align} \frac{\partial \rho}{\partial t}&=+\frac{\delta\mathcal{H}}{\delta\varphi}= -\vec{\nabla}\cdot(\rho\vec{v}), \\ \frac{\partial \varphi}{\partial t}&=-\frac{\delta\mathcal{H}}{\delta\rho}=-\frac{1}{2}\vec{v}\cdot\vec{v}-e', \end{align}

where is the velocity and is vorticity-free. The second equation leads to the Euler equations:

after exploiting the fact that the vorticity is zero:

Read more about this topic:  Hamiltonian Fluid Mechanics

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