Referential and Spatial Descriptions

When a continuum is in motion, quantities (such as temperature $\theta$ , velocity $\mathbf{v}$ ) that are associated with specific particles change with time. There are two ways to describe these changes.

(i)

Following the particles, that is, we express $\theta$ ,

as functions of the coordinates of a particle in a fixed reference configuration and time

. Said differently, we express

	$\displaystyle \theta = \theta (X_1,\ X_2,\ X_3,\ t),$
	$\displaystyle v_i = v_i (X_1,\ X_2,\ X_3,\ t).$

Such a description is known as Lagrangian or referential or material description.

(ii)

Observing the changes at fixed locations, that is, we express $\theta,\ v_i$ etc. as functions of

and

. Thus

	$\displaystyle \theta = \theta(x_1,\ x_2,\ x_3,\ t),$
	$\displaystyle v_i = v_i (x_1,\ x_2,\ x_3,\ t).$

Such a description is known as spatial or Eulerian. We note that in this description, what is described (or measured) is the change of quantities at a fixed point in space (not a specific material particle) as a function of time. The same spatial position is occupied by different particles at different times. Therefore, the spatial description does not provide direct information regarding the changes in the values of a quantity associated with a material particle as it moves about in space.

The referential and spatial descriptions are, of course, related by the motion. That is, if the motion is known then, one description can be obtained from the other as illustrated by the following example.

Example: Given the motion of a body to be

$\displaystyle x_i = X_i + 0\cdot 2tX_2\delta_{1i}. \nonumber$

(i)

For the temperature field given by

$\displaystyle \theta = 2x_1 + (x_2)^2, \nonumber$

(ii)

(a): find the material description of temperature, and
(b): the rate of change of temperature of a particle which at time was at the place $(0,\ 1,\ 0)$ .

Solution

(a)

By substituting (i) into (ii), we obtain

$\displaystyle \theta =\$	$\displaystyle 2x_1 + (x_2)^2,$
$\displaystyle =\$	$\displaystyle 2(X_1 + 0\cdot 2tX_2) + (X_2)^2,$
$\displaystyle =\$	$\displaystyle 2X_1 + (X_2 + 0\cdot 4t)X_2.\nonumber$	(iii)

(b)

The temperature of the desired particle at different times is given by

$\displaystyle \theta =\$	$\displaystyle (1 + 0\cdot 4t).$
$% latex2html id marker 11946 $\displaystyle \therefore\ \frac{d\theta}{dt}=\ $$	$\displaystyle 0\cdot 4.$

Note that even though the temperature is independent of time in the spatial description, a particle experiences a change in temperature as it moves from one spatial position to another; this becomes clear from eqn. (iii).

Whereas spatial description is often used in fluid mechanics, referential description is employed in solid mechanics and in formulating laws of mechanics.