From 07a63237de774b3a57a0975e03cf2c6b68f165b5 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Wed, 31 Mar 2021 19:57:54 +0200
Subject: Expand knowledge base

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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 $\va{U}(\va{r}, t)$:
+
+$$\begin{aligned}
+    \boxed{
+        \frac{\mathrm{D} \va{U}}{\mathrm{D}t}
+        \equiv \pdv{f}{t} + \va{v} \cdot \nabla \va{U}
+    }
+\end{aligned}$$
+
+Where the advective term is to be evaluated in the following way:
+
+$$\begin{aligned}
+    \va{v} \cdot \nabla \va{U}
+    =
+    \begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix}
+    \cdot
+    \begin{bmatrix}
+        \displaystyle\pdv{U_x}{x} & \displaystyle\pdv{U_x}{y} & \displaystyle\pdv{U_x}{z} \\
+        \displaystyle\pdv{U_y}{x} & \displaystyle\pdv{U_y}{y} & \displaystyle\pdv{U_y}{z} \\
+        \displaystyle\pdv{U_z}{x} & \displaystyle\pdv{U_z}{y} & \displaystyle\pdv{U_z}{z}
+    \end{bmatrix}
+    =
+    \begin{bmatrix}
+        v_x \displaystyle\pdv{U_x}{x} & v_y \displaystyle\pdv{U_x}{y} & v_z \displaystyle\pdv{U_x}{z} \\
+        v_x \displaystyle\pdv{U_y}{x} & v_y \displaystyle\pdv{U_y}{y} & v_z \displaystyle\pdv{U_y}{z} \\
+        v_x \displaystyle\pdv{U_z}{x} & v_y \displaystyle\pdv{U_z}{y} & v_z \displaystyle\pdv{U_z}{z}
+    \end{bmatrix}
+\end{aligned}$$
+
+
+
+## References
+1.  B. Lautrup,
+    *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition,
+    CRC Press.
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