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

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:

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

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

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