Helmholtz’s law
References
Aris [Ari90]: Highly recommended to those who want to learn vector/tensor calculus in fluid mechanics.
Let us derive Helmholtz’s law of vortex motion. The rate of change in \(\boldsymbol{\omega}/\rho\) is expressed as
(185)\[ \frac{D}{Dt} \frac{\boldsymbol{\omega}}{\rho}
= \frac{1}{\rho} \frac{D \boldsymbol{\omega}}{Dt} - \frac{\boldsymbol{\omega}}{\rho^{2}} \frac{D \rho}{Dt}
= \frac{1}{\rho} \left( - \boldsymbol{\omega} ( \nabla \cdot \mathbf{v} ) + \boldsymbol{\omega} \cdot \nabla \mathbf{v} \right) + \frac{\boldsymbol{\omega}}{\rho} \nabla \cdot \mathbf{v}
= \frac{\boldsymbol{\omega} }{\rho} \cdot \nabla \mathbf{v}\]
Let \(\mathbf{X}\) be the initial coordinates of each fluid particle in the system. Therefore,
(186)\[\begin{split}\begin{split}
&\mathbf{x} = \mathbf{x} ( \mathbf{X}, t ) \\
&\mathbf{X} = \mathbf{X} ( \mathbf{x}, t )
\end{split}\end{split}\]
The coordinates \(\mathbf{x}\) and \(\mathbf{X}\) are referred to as the space and material coordinates, respectively. The space coordinate \(\mathbf{x}(\mathbf{X}, t)\) is the trajectory of the fluid particle \(\mathbf{X}\), and indeed
(187)\[ \frac{D \mathbf{x}}{Dt} = \mathbf{v}\]
The difference between \(\mathbf{x}\) of a fluid particle and its neighbor at distance \(d \mathbf{X}\) is written by
(188)\[ d\mathbf{x} = \mathbf{x}( \mathbf{X} + d\mathbf{X}, t ) - \mathbf{x}( \mathbf{X}, t ) \rightarrow \frac{\partial x_{i}}{\partial X_{j}} dx_{j}\]
Let us assume a solution having the following functional form:
(189)\[ \omega_{i} = \rho c_{j} \frac{\partial x_{i}}{\partial X_{j}}\]
Using this, we observe
(190)\[ \frac{D}{Dt} \frac{\omega_{i}}{\rho}
= \frac{D}{Dt} \left( c_{j} \frac{\partial x_{i}}{\partial X_{j}} \right)
= \frac{D c_{j}}{Dt} \frac{\partial x_{i}}{\partial X_{j}} + c_{j}\frac{\partial v_{i}}{\partial X_{j}}\]
(191)\[ \frac{ \omega_{j} }{\rho} \frac{\partial v_{i}}{\partial x_{j}}
= c_{k} \frac{\partial x_{j}}{\partial X_{k}} \frac{\partial v_{i}}{\partial x_{j}}
= c_{k} \frac{\partial v_{i}}{\partial X_{k}}\]
According to Eq. (185),
(192)\[ \frac{D c_{j}}{Dt} \frac{\partial x_{i}}{\partial X_{j}} = 0\]
However, \(| \partial x_{i} / \partial X_{j} |\) is non-zero, we have
(193)\[ \frac{D c_{j}}{Dt} = 0~~~~\text{and}~~~~\mathbf{c} = \mathbf{c} ( \mathbf{X} )\]
Thus, \(\mathbf{c}\) depends only on the initial condition. By writing the initial vorticity \(\boldsymbol{\omega}_{0}\) as
(194)\[ \omega_{i}^{0} = \rho_{0} c_{j} \frac{\partial X_{i}}{\partial X_{j}} = \rho_{0} c_{j} \delta_{ij} = \rho_{0} c_{i}
\rightarrow \mathbf{c} = \frac{\boldsymbol{\omega}_{0}}{\rho_{0}}\]
Thus,
(195)\[ \frac{\omega_{i}}{\rho} = \frac{\omega_{j}^{0}}{\rho_{0}} \frac{\partial x_{i}}{\partial X_{j}}\]
According to this result, if the vorticity of a fluid particle is initially non-zero, that particle will have non-zero vorticity at the later time. Or, if the initial vorticity is zero, the particle will never have non-zero vorticity. This is a part of Helmholtz’s law of vortex motion. For incompressible fluids,
(196)\[ \omega_{i} = \omega_{j}^{0} \frac{\partial x_{i}}{\partial X_{j}}\]
Then, we consider a material line along a vortex line at the initial moment. Let \(\omega_{0}\) be the magnitude of \(\boldsymbol{\omega}_{0}\). We can choose a parameter \(\epsilon\) so as to
(197)\[ dX_{i} = \epsilon \frac{\omega_{i}^{0}}{\rho_{0}} \]
where \(d\mathbf{X} \parallel \boldsymbol{\omega}_{0}\). At a later time,
(198)\[ dx_{i}= \frac{\partial x_{i}}{\partial X_{j}} dX_{j}\]
However, the initial displacement \(d\mathbf{X}\) can be expressed using the initial vorticity:
(199)\[ dx_{i}
= \epsilon \frac{\partial x_{i}}{\partial X_{j}} \frac{\omega_{j}^{0}}{\rho_{0}}
= \epsilon \frac{\omega_{i}}{\rho} \]
This results shows that \(d\mathbf{x} \parallel \boldsymbol{\omega}\), in other words, the vortex line moves with the material line. The lengths, \(ds\) and \(ds_{0}\), for \(d\mathbf{x}\) and \(d\mathbf{X}\), respectively, are
(200)\[\begin{split}\begin{split}
&ds^{2} = dx_{i} dx_{i} = \epsilon^{2} \frac{\omega^{2}}{\rho^{2}} \\
&ds_{0}^{2} = dX_{i} dX_{i} = \epsilon^{2} \frac{\omega_{0}^{2}}{\rho_{0}^{2}}
\end{split}\end{split}\]
where \(\omega = \omega_{i} \omega_{i}\) and \(\omega_{0} = \omega_{i}^{0} \omega_{i}^{0}\). From these expressions we have
(201)\[\begin{split} \frac{ds^{2}}{ds_{0}^{2}} = \frac{\rho_{0}^{2} \omega^{2}}{\rho^{2} \omega_{0}^{2}} \\\end{split}\]
According to the continuity equation,
(202)\[ \rho \sigma ds = \rho_{0} \sigma_{0} ds_{0}\]
where \(\sigma\) is the cross-sectional area occupied by the vortex line. Combining Eqs. (201) and (202) gives
(203)\[ \omega \sigma = \omega_{0} \sigma_{0} \]
This provides the last one of Helmholtz’s law, that is, the strength of the vortex line is constant.