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Convective derivative

Idea

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

ft+vf \frac{\partial f}{\partial t} + v \cdot \nabla f

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

vt+vv \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.

References

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

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