# Fluid Mechanics and order of magnitude calculation

1. Feb 16, 2014

### Niles

Hi

In my lecture notes we making some calculations and all terms $\mathcal O(M^3)$ are to be thrown away. Here M is the Mach number. Now, there is the expression (u denotes the velocity):
$$uu\partial_t \rho \approx \rho_0 uu\nabla u$$
which in my notes are thrown away because they claim it is $\mathcal O(M^3)$. But is it really true, I mean the derivative of u will not necessarily be on the same order as Ma, right?

2. Feb 17, 2014

### olivermsun

That's right. The gradient also contains a length scale in each direction. In many cases one simply asserts on physical grounds that du/dx is same order as u (so the flow is sufficiently "smooth") or that there is some characteristic length scale of order one. Does the problem assume M << 1 and also no viscous effects?

3. Feb 17, 2014