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diff --git a/source/know/concept/material-derivative/index.md b/source/know/concept/material-derivative/index.md new file mode 100644 index 0000000..6a02a22 --- /dev/null +++ b/source/know/concept/material-derivative/index.md @@ -0,0 +1,115 @@ +--- +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. |