For an inviscid barotropic fluid ( ) under a conservative body force

PHYSICS · Subjective · Fermi University Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow · Kinematics, conservation laws and ideal flow

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

Problem

For an inviscid barotropic fluid (p=p(ρ)p=p(\rho)) under a conservative body force, starting from the three-term vorticity equation DωDt=(ω⋅∇)u−ω(∇⋅u)+∇ρ×∇pρ2\frac{D\boldsymbol{\omega}}{Dt} = (\boldsymbol{\omega}\cdot\nabla)\mathbf{u} - \boldsymbol{\omega}(\nabla\cdot\mathbf{u}) + \frac{\nabla\rho\times\nabla p}{\rho^2} and the mass-continuity equation Dρ/Dt=−ρ∇⋅uD\rho/Dt=-\rho\nabla\cdot\mathbf{u}, show that DDt(ωρ)=(ωρ⋅∇)u,\frac{D}{Dt}\left(\frac{\boldsymbol{\omega}}{\rho}\right) = \left(\frac{\boldsymbol{\omega}}{\rho}\cdot\nabla\right)\mathbf{u}, and explain why this is exactly the equation obeyed by a material line element δl\delta\mathbf{l}, so that vortex lines are frozen into the fluid.

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