Navier-Stokes:
$\partial_t (v^i) + \partial_j (v^i v^j) = -\frac{1}{\rho} \partial^i P + \nu \partial_j \partial^j v^i$
$\partial_t (v^i) + \partial_j (v^i v^j) - \nu \partial_j \partial^j v^i = -\frac{1}{\rho} \partial^i P$
$\partial_t (v^i v^t) + \partial_j (v^i v^j) - \nu \partial_j \partial^j v^i = -\frac{1}{\rho} \partial^i P$
$\partial_\mu (v^\mu v^i) - \nu \partial_j \partial^j v^i + \frac{1}{\rho} \partial^i P = 0$
$\partial_\mu (v^\mu v^i) - \nu \partial_\mu \partial^\mu v^i + \nu \partial_t \partial^t v^i + \frac{1}{\rho} \partial^i P = 0$
$\partial_\mu ((v^\mu - \nu \partial^\mu) v^i) + \nu \partial_t \partial^t v^i + \frac{1}{\rho} \partial^i P = 0$
Let $D^\mu = v^\mu - \nu \partial^\nu$
$\partial_\mu D^\mu v^i + \nu \partial_t \partial^t v^i + \frac{1}{\rho} \partial^i P = 0$

Or just consider the spatial covariant derivative.
Let $D^i = v^i - \nu \partial^i$
$\partial_t (v^t - \nu \partial^t) v^i + \partial_j D^j v^i + \nu \partial_t \partial^t v^i + \frac{1}{\rho} \partial^i P = 0$
$\partial_t (v^t v^i) - \nu \partial_t \partial^t v^i + \partial_j D^j v^i + \nu \partial_t \partial^t v^i + \frac{1}{\rho} \partial^i P = 0$
$\partial_t (v^t v^i) + \partial_j D^j v^i + \frac{1}{\rho} \partial^i P = 0$