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---
title: "Material derivative"
firstLetter: "M"
publishDate: 2021-03-30
categories:
- Physics
- Fluid mechanics
- Fluid dynamics
- Continuum physics
date: 2021-03-30T19:39:28+02:00
draft: false
markup: pandoc
---
# 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 $\pdv*{f}{t}$,
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.
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