Lamb's form of Euler's equation for an inviscid fluid of possibly varying density can…

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

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

Problem

Lamb's form of Euler's equation for an inviscid fluid of possibly varying density can be written ∂u∂t+ω×u=−∇(u22+χ)−1ρ∇p,\frac{\partial \mathbf{u}}{\partial t} + \boldsymbol{\omega}\times\mathbf{u} = -\nabla\left(\frac{u^2}{2}+\chi\right) - \frac{1}{\rho}\nabla p, where χ\chi is the potential of a conservative body force (g=−∇χ\mathbf{g}=-\nabla\chi) and u2=u⋅uu^2=\mathbf{u}\cdot\mathbf{u}. (a) Take the curl of both sides. Use the identity ∇×(ω×u)=ω(∇⋅u)+(u⋅∇)ω−(ω⋅∇)u\nabla\times(\boldsymbol{\omega}\times\mathbf{u}) = \boldsymbol{\omega}(\nabla\cdot\mathbf{u}) + (\mathbf{u}\cdot\nabla)\boldsymbol{\omega} - (\boldsymbol{\omega}\cdot\nabla)\mathbf{u}, valid because ∇⋅ω=0\nabla\cdot\boldsymbol{\omega}=0 always, and the identity ∇×(f∇g)=∇f×∇g\nabla\times(f\nabla g)=\nabla f\times\nabla g for scalar fields f,gf,g. (b) Assemble the result into the three-term inviscid 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 name each of the three terms on the right.

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