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Boosted by jsonstein@masto.deoan.org ("Jeff Sonstein"):
johncarlosbaez@mathstodon.xyz ("John Carlos Baez") wrote:
Give up on that challenge? Here's how it goes. We're trying to prove
dω/dt = ∇× (v × ω)
so we start with the definition of vorticity
ω = ∇× v
and do
dω/dt = ∇× (dv/dt)
Now we have to use Euler's equation
dv/dt + (v ⋅∇)v = F/ρ
and get
dω/dt = ∇× dv/dt = ∇× (F/ρ - (v ⋅∇)v)
Since we're assuming ∇× F = 0 and the density ρ is constant (since the fluid is incompressible), the first term at right is zero and
dω/dt = -∇× ((v ⋅∇)v)
Now it gets hard for me. I want an identity to help me out! I cheated and looked at this:
https://en.wikipedia.org/wiki/Vector%5Fcalculus%5Fidentities#Vector-dot-Del%5FOperator
where I learned
(v ⋅∇)v = ½∇(v ⋅ v) - v × (∇× v)
Wow! This gives
dω/dt = -∇× ((v ⋅∇)v)
= -∇× (½∇(v ⋅ v) - v × (∇× v))but the curl of a divergence is zero so
dω/dt = ∇× (v × (∇× v))
or in short
dω/dt = ∇× (v × ω)
as desired! Hurrah!
The magic identity I looked up on Wikipedia is actually a special case of a more general one, which I needed when I was learning about magnetohydrodynamics. So I feel the whole subject of fluid dynamics is where fancy vector calculus identities start becoming really important.
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