This is using 3D flat space,
so raising and lowering is applying the identity transform,
the gradient of the metric is 0,
the metric determinant is 1,
the density of Levi-Civita is 1,
and the gradient of Levi-Civita is 0.

Incompressible Navier-Stokes:
$\partial_t (v^i) + \partial_j (v^i v^j) = -\frac{1}{\rho} \partial^i P + \nu \partial_j \partial^j v^i$
Apply curl:
$\epsilon_{mki} \partial^k (\partial_t (v^i) + \partial_j (v^i v^j)) = \epsilon_{mki} \partial^k (-\frac{1}{\rho} \partial^i P + \nu \partial_j \partial^j v^i)$
$ \epsilon_{mki} \partial^k \partial_t (v^i) + \epsilon_{mki} \partial^k \partial_j (v^i v^j) = -\epsilon_{mki} \partial^k (\frac{1}{\rho} \partial^i P) + \epsilon_{mki} \partial^k (\nu \partial_j \partial^j v^i)$
$ \partial_t (\epsilon_{mki} \partial^k v^i) + \epsilon_{mki} \partial^k (v^j \partial_j v^i + v^i \partial_j v^j) = \epsilon_{mki} (\frac{1}{\rho^2} \partial^k \rho) \partial^i P -\frac{1}{\rho} \epsilon_{mki} \partial^k \partial^i P + \nu \partial_j \partial^j \epsilon_{mki} \partial^k v^i$
Let $w^i = \epsilon^{ijk} \partial_j v_k$
$ \partial_t w_m + \epsilon_{mki} \partial^k \partial_j (v^i v^j) = \epsilon_{mki} \frac{1}{\rho^2} (\partial^k \rho) (\partial^i P) + \nu \partial_j \partial^j w_m$
In vector notation:
$\partial_t \textbf{w} + \boldsymbol{\nabla} \times (\boldsymbol{\nabla} \cdot (\textbf{v} \otimes \textbf{v})) = \frac{1}{\rho^2} \boldsymbol{\nabla} \rho \times \boldsymbol{\nabla} P + \nu \Delta \textbf{w}$
Some further simplifications on the outer product gradient curl:
$ \partial_t w_m + \epsilon_{mki} \partial^k (v^j \partial_j v^i + v^i \partial_j v^j) = \epsilon_{mki} \frac{1}{\rho^2} (\partial^k \rho) (\partial^i P) + \nu \partial_j \partial^j w_m$
$ \partial_t w_m + ( \epsilon_{mki} (\partial^k v^j) (\partial_j v^i) + v^j \partial_j (\epsilon_{mki} \partial^k v^i) + \epsilon_{mki} (\partial^k v^i) (\partial_j v^j) + \epsilon_{mki} v^i (\partial^k \partial_j v^j) ) = \epsilon_{mki} \frac{1}{\rho^2} (\partial^k \rho) (\partial^i P) + \nu \partial_j \partial^j w_m$
$ \partial_t w_m + \epsilon_{mki} \partial^k v^j \partial_j v^i + v^j \partial_j w_m + w_m \partial_j v^j + \epsilon_{mki} v^i \partial^k \partial_j v^j = \epsilon_{mki} \frac{1}{\rho^2} (\partial^k \rho) (\partial^i P) + \nu \partial_j \partial^j w_m$
In vector notation:
$\partial_t \textbf{w} + \star ((\boldsymbol{\nabla} \textbf{v})^T)^2 + \boldsymbol{\nabla}_{\textbf{v}} \textbf{w} + (\boldsymbol{\nabla} \cdot \textbf{v}) \textbf{w} + \boldsymbol{\nabla} (\boldsymbol{\nabla} \cdot \textbf{v}) \times \boldsymbol{v} = \frac{1}{\rho^2} \boldsymbol{\nabla} \rho \times \boldsymbol{\nabla} P + \nu \Delta \textbf{w}$
In an incompressible fluid:
$ \partial_t w_m + \epsilon_{mki} \partial^k v^j \partial_j v^i + v^j \partial_j w_m = \epsilon_{mki} \frac{1}{\rho^2} (\partial^k \rho) (\partial^i P) + \nu \partial_j \partial^j w_m$
In vector notation:
$\partial_t \textbf{w} + \star ((\boldsymbol{\nabla} \textbf{v})^T)^2 + \boldsymbol{\nabla}_{\textbf{v}} \textbf{w} = \frac{1}{\rho^2} \boldsymbol{\nabla} \rho \times \boldsymbol{\nabla} P + \nu \Delta \textbf{w}$