2.4: Continuum Coordinates and the Convective Derivative

There are two commonly used representations of continuum variables. One of these is familiar from classical mechanics, while the other is universally used in electrodynamics. Because electromechanics involves both of these subjects, attention is now drawn to the salient features of the two representations.

Consider first the "Lagrangian representation." The position of a material particle is a natural example and is depicted by Figure 2.4.1a. When the time t is zero, a particle is found at the position \(\overrightarrow{r_o}\). The position of the particle at some subsequent time is \(\overrightarrow{\xi}\). To let \(\overrightarrow{\xi}\) represent the displacement of a continuum of particles, the position variable ro is used to distinguish particles. In this sense, the displacement \(\overrightarrow{r_o}\) then also becomes a continuum variable capable of representing the relative displacements of an infinitude of particles.

Figure 2.4.1. Particle motions represented in terms of (a) Lagrangian coordinates, where the initial particle coordinate \(\overrightarrow{r_o}\) designates the particle of interest, and (b) Eulerian coordinates, where \((x,y,z)\) designates the spatial position of interest.

In a Lagrangian representation, the velocity of the particle is simply

\[ \overrightarrow{v} = \frac{\partial{\overrightarrow{\xi}}}{\partial{t}} \label{1} \]

If concern is with only one particle, there is no point in writing the derivative as a partial derivative. However, it is understood that, when the derivgtive is taken, it is a particular particle which is being considered. So, it is understood that \(\overrightarrow{r_o}\) is fixed. Using the same line of reasoning, the acceleration of a particle is given by

\[ \overrightarrow{a} = \frac{\partial{\overrightarrow{v}}}{\partial{t}} \label{2} \]

The idea of representing continuum variables in terms of the coordinates \((x,y,z)\) connected with the space itself is familiar from electromagnetic theory. But what does it mean if the variable is mechanical rather than electrical? We could represent the velocity of the continuum of particles filling the space of interest by a vector function \(\overrightarrow{v}(x,y,z,t) = \overrightarrow{v}(\overrightarrow{r},t)\). The velocity of particles having the position \((x,y,z,)\) at a given time \(t\) is determined by evaluating the function \(\overrightarrow{v}(\overrightarrow{r},t)\). The velocity appearing in Sec. 2.2 is an example. As suggested by Figure 2.4.1b, if the function is the velocity evaluated at a given position in space, it describes whichever particle is at that point at the time of interest. Generally, there is a continuous stream of particles through the point \((x,y,z)\).

Computation of the particle acceleration makes evident the contrast between Eulerian and Lagrangian representations. By definition, the acceleration is the rate of change of the velocity computed for a given particle of matter. A particle having the position \((x,y,z)\) at time \(t\) will be found an instant At later at the position \((x + v_x\Delta t,y + v_y \Delta t,z + v_z \Delta t)\). Hence the acceleration is

\[ \overrightarrow{a} = \lim_{\Delta t \to 0} \frac{\overrightarrow{v} (x + v_x\Delta t,y + v_y \Delta t,z + v_z \Delta t,t + \Delta t) - \overrightarrow{v} (x,y,z,t)}{ \Delta t} \label{3} \]

Expansion of the first term in Equation \ref{3} about the initial coordinates of the particle gives the convective derivative of \(\overrightarrow{v}\):

\[ \overrightarrow{a} = \frac{\partial{\overrightarrow{v}}}{\partial{t}} + v_x \frac{\partial{\overrightarrow{v}}}{\partial{x}} + v_y \frac{\partial{\overrightarrow{v}}}{\partial{y}} + v_z \frac{\partial{\overrightarrow{v}}}{\partial{z}} \equiv \frac{\partial{\overrightarrow{v}}}{\partial{t}} + \overrightarrow{v} \cdot \nabla \overrightarrow{v} \label{4} \]

The difference between Equation \ref{2} and Equation \ref{4} is resolved by recognizing the difference in the significance of the partial derivatives. In Equation \ref{2}, it is understood that the coordinates being held fixed are the initial coordinates of the particle of interest. In Equation \ref{4}, the partial derivative is taken, holding fixed the particular point of interest in space.

The same steps .show that the rate of change of any vector variable \(\overrightarrow{A}\), as viewed from a particle having the velocity \(\overrightarrow{v}\), is

\[ \frac{D\overrightarrow{A}}{Dt} \equiv \frac{\partial{\overrightarrow{A}}}{\partial{t}} + (\overrightarrow{v} \cdot \nabla) \overrightarrow{a}; \quad \overrightarrow{A} = \overrightarrow{A}(x,y,z,t) \label{5} \]

The time rate of change of any scalar variable for an observer moving with the velocity v is obtained from Equation \ref{5} by considering the particular case in which t has only one component, say \(\overrightarrow{A} = f(x,y,z,t) \overrightarrow{1_x}\). Then Equation \ref{5} becomes

\[ \frac{Df}{Dt} \equiv \frac{\partial{f}}{\partial{t}} + \overrightarrow{v} \cdot \nabla f \label{6} \]

Reference 3 of Appendix C is a film useful in understanding this section.