Material derivative

The material derivative is the rate of change of some physical quantity (heat, or momentum, or concentration of a material) for a material element subjected to a velocity field, with respect to time.

In fluid dynamics, the velocity field is the flow velocity itself (i.e., a vector field), and the physical quantity is the temperature of the fluid. Then the material derivative describes the temperature evolution of a certain fluid parcel in time, as it is being moved along its trajectory while following the fluid flow.

So, given a vector filed $v$

• the material derivative of a function $f=f(x,t)$:

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

• the material derivative of a vector field $F=F(x,t)$:

$$\frac{DF}{Dt} = \frac{\partial F}{\partial t} + v\cdot\nabla F,$$ where $\nabla f$ is the gradient of a scalar function, while $\nabla F$ is the covariant derivative (the curl in particular) of a vector field.

The second term is also recognizable as the directional derivative. They are the same, therefore, for a time-independent function.

In terms of differential forms, the analogue should be as follows.

Suppose $v$ is a $1$-form, then the material (exterior) derivative of a $0$-form $f=f(x,t)$ is: $$Df=d_tf+v\cdot d_xf,$$ where $d_t, d_x$ are the exterior derivatives with respect to $t$ and $x$ respectively. So this is a $1$-form.

Where does this come from?

WE need to evaluate the rate of change, with respect to $t$, of the function $G(t)=f(x+tv,t)$. Then $G=fh$, where $h(x,t)=x+tv$. Then by the Chain Rule, we have $$dG=df\cdot dh=(d_tf+d_xf)\cdot (1+v).$$ Since each pair of terms in the two factors are orthogonal (as $t$ and $x$ are), we have $$dG=d_tf+v\cdot d_xf.$$

In case of discrete differential forms, $(Df)(a)=(d_tf)(a)$ or $v\cdot (d_xf)(a)$, depending on the location of the $1$-cell $a$.

Note that in case if a trivial $v$, we have $$Df=df,$$ the exterior derivative of the form -- with respect to $(x,t)$. This is the reason that the material derivative is also called the total derivative.

See also Lie derivative.