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

Material derivative

Inside a fluid (or any other continuum), we might be interested in the time evolution of a certain intensive quantity ff, e.g. the temperature or pressure, represented by a scalar field f(r⃗,t)f(\va{r}, t).

If the fluid is static, the evolution of ff is simply ∂f/∂t\ipdv{f}{t}, since each point is motionless. However, if the fluid is moving, we have a problem: the fluid molecules at position r⃗=r⃗0\va{r} = \va{r}_0 are not the same ones at time t=t0t = t_0 and t=t1t = t_1. Those molecules take ff 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 r⃗(t)\va{r}(t) through time as it moves and deforms. The value of ff for this parcel is then given by:

f(r⃗,t)=f(r⃗(t),t)=f(x(t),y(t),z(t),t)\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 r⃗\va{r} dependent on time, and have specifically chosen the time-dependence to track the parcel. The net evolution of ff is then its “true” (i.e. non-partial) derivative with respect to tt, allowing us to apply the chain rule:

ddtf(x(t),y(t),z(t),t)=∂f∂t+∂f∂xdxdt+∂f∂ydydt+∂f∂zdzdt=∂f∂t+vx∂f∂x+vy∂f∂y+vz∂f∂z\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 vxv_x, vyv_y and vzv_z are the parcel’s velocity components. Let v⃗=(vx,vy,vz)\va{v} = (v_x, v_y, v_z) be the velocity vector field, then we can rewrite this expression like so:

ddtf(x(t),y(t),z(t),t)=∂f∂t+(v⃗⋅∇)f\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 v⃗=v⃗(r⃗,t)\va{v} = \va{v}(\va{r}, t), that is, the velocity can change with time (tt-dependence), and depends on which parcel we track (r⃗\va{r}-dependence).

Of course, the parcel is in our imagination: r⃗\va{r} does not really depend on tt; 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:

DfDt≡∂f∂t+(v⃗⋅∇)f\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 ff without needing to worry about the continuum’s motion.

That was for a scalar field f(r⃗,t)f(\va{r}, t), but in fact the definition also works for vector fields U⃗(r⃗,t)\va{U}(\va{r}, t):

DU⃗Dt≡∂U⃗∂t+(v⃗⋅∇)U⃗\begin{aligned} \boxed{ \frac{\mathrm{D} \va{U}}{\mathrm{D}t} \equiv \pdv{\va{U}}{t} + (\va{v} \cdot \nabla) \va{U} } \end{aligned}

To evaluate this in various coordinate systems, see orthogonal curvilinear coordinates.

References

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