A fluid field can change because conditions at a fixed point change with time and because a moving fluid parcel travels into a region where the field has a different value.

The material derivative combines both effects.

For a scalar field $f(\mathbf r,t)$ carried by a velocity field $\mathbf v$,

$$\frac{Df}{Dt}=\frac{\partial f}{\partial t}+\mathbf v\cdot\nabla f.$$

Local change

The term

$$\frac{\partial f}{\partial t}$$

measures how the field changes at one fixed position.

Advective change

The term

$$\mathbf v\cdot\nabla f$$

measures how the moving parcel experiences spatial variation as it travels through the field.

Fluid acceleration

Applying the same idea to velocity gives the acceleration of a fluid parcel:

$$\frac{D\mathbf v}{Dt}=\frac{\partial\mathbf v}{\partial t}+(\mathbf v\cdot\nabla)\mathbf v.$$

A steady velocity field can therefore still accelerate fluid parcels through the nonlinear advective term.