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Hmm..your equation is wrong (there shouldn't be a minus sign in front of the first term on the right-hand side).
Restricting myself to the case of constant density&dynamic viscosity and conservative volume forces, taking the curl of Navier-Stokes yields:
\frac{D\vec{c}}{dt}=(\vec{c}\cdot\nabla)\vec{v}+\nu\nabla^{2}\vec{c},\vec{c}\equiv\nabla\times\vec{v},\frac{D}{dt}\equiv\frac{\partial}{\partial{t}}+\vec{v}\cdot\nabla
In the inviscid case (with volume forces being a gradient field), therefore, we have:
\frac{D\vec{c}}{dt}=(\vec{c}\cdot\nabla)\vec{v}
(Note that this is also exact in the inviscid, barotropic case, \rho=\rho(p))
As for the interpretation of the right-hand side, we note that if two points A,B in the fluid is separated by the vector \vec{AB}=dq\vec{s} (where dq is an infinitesemal quantity), we have:
\vec{v}_{B}-\vec{v}_{A}=((\vec{AB}\cdot\nabla)\vec{v})\mid_{A}=dq(\vec{s}\cdot\nabla\vec{v})\mid_{A}
This is proven as follows:
\vec{v}_{B}=\vec{v}(x_{A}+dx,y_{A}+dy,z_{A}+dz,t)=\vec{v}_{A}+dq(\vec{s}\cdot\nabla)\vec{v}\mid_{A},(dx,dy,dz)=dq\vec{s}
If therefore A, B is fluid-particles at the same instantaneous vortex line (i.e, defined with \vec{c} as the tangent vector at time t), we have, by the vorticity equation:
\vec{v}_{B}-\vec{v}_{A}=dq\frac{D\vec{c}}{dt}\mid_{A}
(at time t)
That is, the vorticity change, as experienced by particle A is only affected by the perceived velocity difference along the vortex line to which A instantaneously belongs.
To proceed further, let \vec{AB}_{t+\bigtriangleup{t}} be the vector joining A, B an instant later.
We have therefore the equality:
\vec{AB}_{t+\bigtriangleup{t}}=\vec{AB}+(\vec{v}_{B}-\vec{v}_{A})\bigtriangleup{t}=dq(\vec{c}+\frac{D\vec{c}}{dt}\bigtriangleup{t})\mid_{A}=dq\vec{c}(t+\bigtriangleup{t})\mid_{A}
That is, particles A and B remain joined to the SAME vortex line at time t+\bigtriangleup{t} as they were on time t.
(At time t+\bigtriangleup{t} the vorticity experienced by particle A is \vec{c}(t+\bigtriangleup{t})
Hence, we may conclude that in the inviscid case, vortex lines are MATERIAL curves, in that they consist of the same particles throughout time.
Restricting myself to the case of constant density&dynamic viscosity and conservative volume forces, taking the curl of Navier-Stokes yields:
\frac{D\vec{c}}{dt}=(\vec{c}\cdot\nabla)\vec{v}+\nu\nabla^{2}\vec{c},\vec{c}\equiv\nabla\times\vec{v},\frac{D}{dt}\equiv\frac{\partial}{\partial{t}}+\vec{v}\cdot\nabla
In the inviscid case (with volume forces being a gradient field), therefore, we have:
\frac{D\vec{c}}{dt}=(\vec{c}\cdot\nabla)\vec{v}
(Note that this is also exact in the inviscid, barotropic case, \rho=\rho(p))
As for the interpretation of the right-hand side, we note that if two points A,B in the fluid is separated by the vector \vec{AB}=dq\vec{s} (where dq is an infinitesemal quantity), we have:
\vec{v}_{B}-\vec{v}_{A}=((\vec{AB}\cdot\nabla)\vec{v})\mid_{A}=dq(\vec{s}\cdot\nabla\vec{v})\mid_{A}
This is proven as follows:
\vec{v}_{B}=\vec{v}(x_{A}+dx,y_{A}+dy,z_{A}+dz,t)=\vec{v}_{A}+dq(\vec{s}\cdot\nabla)\vec{v}\mid_{A},(dx,dy,dz)=dq\vec{s}
If therefore A, B is fluid-particles at the same instantaneous vortex line (i.e, defined with \vec{c} as the tangent vector at time t), we have, by the vorticity equation:
\vec{v}_{B}-\vec{v}_{A}=dq\frac{D\vec{c}}{dt}\mid_{A}
(at time t)
That is, the vorticity change, as experienced by particle A is only affected by the perceived velocity difference along the vortex line to which A instantaneously belongs.
To proceed further, let \vec{AB}_{t+\bigtriangleup{t}} be the vector joining A, B an instant later.
We have therefore the equality:
\vec{AB}_{t+\bigtriangleup{t}}=\vec{AB}+(\vec{v}_{B}-\vec{v}_{A})\bigtriangleup{t}=dq(\vec{c}+\frac{D\vec{c}}{dt}\bigtriangleup{t})\mid_{A}=dq\vec{c}(t+\bigtriangleup{t})\mid_{A}
That is, particles A and B remain joined to the SAME vortex line at time t+\bigtriangleup{t} as they were on time t.
(At time t+\bigtriangleup{t} the vorticity experienced by particle A is \vec{c}(t+\bigtriangleup{t})
Hence, we may conclude that in the inviscid case, vortex lines are MATERIAL curves, in that they consist of the same particles throughout time.
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