Some applications of fluid dynamics consider the linearised Navier-stokes equation where the advection term $(\vec{u}\cdot\vec{\nabla})\vec{u}$ is dropped.
I am trying to build a convincing argument for this based on scale analysis. I tried to find something about this out there but haven't been very lucky so far, hence this question.
Here is what I have come up with so far.
If I set the scale of the velocity as $U_0$, that of length as $L_0$, of time as $T_0$ and pressure as $P_0$. For simplicity I take $P_0=\rho U_0/T_0$ with $\rho$ being the (constant) density. I can then write the dimensionless equation:
$$ \partial_t\vec{u}+\frac{U_0T_0}{L_0}(\vec{u}\cdot\vec{\nabla})\vec{u}=-\vec{\nabla}p+\frac{\nu T_0}{L_0^2}\nabla^2\vec{u} $$
where $\nu$ stands for the kinematic viscosity. The relative scale of each term in the above is contained in their respective pre-factor.
It would then make sense to neglect the second term in the regime $U_0T_0/L_0\ll1$.
Is this argument valid ? If not, any guidance and/or reference would be greatly appreciated.
Thanks.
