Categories: Continuum physics, Fluid dynamics, Fluid mechanics, Physics.

# Material derivative

Inside a fluid (or any other continuum), we might be interested in the time evolution of a certain intensive quantity $f$, e.g. the temperature or pressure, represented by a scalar field $f(\va{r}, t)$.

If the fluid is static, the evolution of $f$ is simply $\ipdv{f}{t}$, since each point is motionless. However, if the fluid is moving, we have a problem: the fluid molecules at position $\va{r} = \va{r}_0$ are not the same ones at time $t = t_0$ and $t = t_1$. Those molecules take $f$ with them as they move, so we need to account for this transport somehow.

To do so, we choose an infinitesimal “blob” or **parcel** of the fluid,
which always contains the same specific molecules,
and track its position $\va{r}(t)$ through time as it moves and deforms.
The value of $f$ for this parcel is then given by:

In effect, we have simply made the coordinate $\va{r}$ dependent on time, and have specifically chosen the time-dependence to track the parcel. The net evolution of $f$ is then its “true” (i.e. non-partial) derivative with respect to $t$, allowing us to apply the chain rule:

$\begin{aligned} \dv{}{t}f\big(x(t), y(t), z(t), t\big) &= \pdv{f}{t} + \pdv{f}{x} \dv{x}{t} + \pdv{f}{y} \dv{y}{t} + \pdv{f}{z} \dv{z}{t} \\ &= \pdv{f}{t} + v_x \pdv{f}{x} + v_y \pdv{f}{y} + v_z \pdv{f}{z} \end{aligned}$Where $v_x$, $v_y$ and $v_z$ are the parcel’s velocity components. Let $\va{v} = (v_x, v_y, v_z)$ be the velocity vector field, then we can rewrite this expression like so:

$\begin{aligned} \dv{}{t}f\big(x(t), y(t), z(t), t\big) &= \pdv{f}{t} + (\va{v} \cdot \nabla) f \end{aligned}$Note that $\va{v} = \va{v}(\va{r}, t)$, that is, the velocity can change with time ($t$-dependence), and depends on which parcel we track ($\va{r}$-dependence).

Of course, the parcel is in our imagination:
$\va{r}$ does not really depend on $t$;
after all, we are dealing with a continuum.
Nevertheless, the right-hand side of the equation is very useful,
and is known as the **material derivative** or **comoving derivative**:

The first term is called the **local rate of change**,
and the second is the **advective rate of change**.
In effect, the latter moves the frame of reference along with the material,
so that we can find the evolution of $f$
without needing to worry about the continuum’s motion.

That was for a scalar field $f(\va{r}, t)$, but in fact the definition also works for vector fields $\va{U}(\va{r}, t)$:

$\begin{aligned} \boxed{ \frac{\mathrm{D} \va{U}}{\mathrm{D}t} \equiv \pdv{\va{U}}{t} + (\va{v} \cdot \nabla) \va{U} } \end{aligned}$To evaluate this in various coordinate systems, see orthogonal curvilinear coordinates.

## References

- B. Lautrup,
*Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition, CRC Press.