---
title: "Material derivative"
date: 2021-03-30
categories:
- Physics
- Fluid mechanics
- Fluid dynamics
- Continuum physics
layout: "concept"
---

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 of the fluid is motionless.
However, if the fluid is moving, we have a problem:
the fluid molecules at position $\va{r} = \va{r}_0$ are not necessarily
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:

$$\begin{aligned}
    f(\va{r}, t)
    = f(\va{r}(t), t)
    = f\big(x(t), y(t), z(t), t\big)
\end{aligned}$$

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**:

$$\begin{aligned}
    \boxed{
        \frac{\mathrm{D}f}{\mathrm{D}t}
        \equiv \pdv{f}{t} + (\va{v} \cdot \nabla) f
    }
\end{aligned}$$

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 $\va{U}(\va{r}, t)$:

$$\begin{aligned}
    \boxed{
        \frac{\mathrm{D} \va{U}}{\mathrm{D}t}
        \equiv \pdv{\va{U}}{t} + (\va{v} \cdot \nabla) \va{U}
    }
\end{aligned}$$

Where the advective term is to be evaluated in the following way in Cartesian coordinates:

$$\begin{aligned}
    (\va{v} \cdot \nabla) \va{U}
    =
    \begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix}
    \cdot
    \begin{bmatrix}
        \displaystyle\pdv{U_x}{x} & \displaystyle\pdv{U_x}{y} & \displaystyle\pdv{U_x}{z} \\
        \displaystyle\pdv{U_y}{x} & \displaystyle\pdv{U_y}{y} & \displaystyle\pdv{U_y}{z} \\
        \displaystyle\pdv{U_z}{x} & \displaystyle\pdv{U_z}{y} & \displaystyle\pdv{U_z}{z}
    \end{bmatrix}
    =
    \begin{bmatrix}
        v_x \displaystyle\pdv{U_x}{x} & v_y \displaystyle\pdv{U_x}{y} & v_z \displaystyle\pdv{U_x}{z} \\
        v_x \displaystyle\pdv{U_y}{x} & v_y \displaystyle\pdv{U_y}{y} & v_z \displaystyle\pdv{U_y}{z} \\
        v_x \displaystyle\pdv{U_z}{x} & v_y \displaystyle\pdv{U_z}{y} & v_z \displaystyle\pdv{U_z}{z}
    \end{bmatrix}
\end{aligned}$$



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