Convective derivative

Given a velocity vector field $v(t,\mathbb{x})$, the **convective derivative** of a function $f(t,\mathbb{x})$ is

$\frac{\partial f}{\partial t} + v \cdot \nabla f$

This is the usual time derivative plus a term expressing how $f$ changes as we move along the flow generated by $v$. This formula works if $f$ is a vector or tensor fields as well. An important special case is the **convective acceleration**, which is the convective derivative of $v$ itself:

$\frac{\partial v}{\partial t} + v \cdot \nabla v$

The convective derivative plays an important role in the Navier-Stokes equations and various related equations such as Burgers' equation.

A good explanation of the convective derivative can be found here:

- Material derivative, Wikipedia.

As the article notes, the convective derivative has many other names! The convective acceleration is explained here:

- Navier-Stokes: convective acceleration, Wikipedia.

category: mathematical methods