# Rationale behind the linearised Navier-Stokes equation

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.

## 1 Answer

This is exactly what we do to for the Stokes flow.

Instead of $$P_0$$, we use the dimensionless quantity, Reynolds number (Re), for linearising the Navier-Stokes in the limit $$Re \rightarrow 0$$. Refer to the wiki page on Stokes flow and Reynolds number for more details.