Page 100 - Introduction to Continuum Mechanics

Material Derivative 85

so that from Eq. (i)

This is the material description of the velocity field. To obtain the spatial description, we make
use of Eq. (i) again, where we have x^ = X 2, so that

From Eq. (iii), the rate of change of temperature for particular material particles is given by

We note that even though the given temperature field is independent of time, each particle
experiences changes of temperature, since it flows from one spatial position to another.

3.3 Material Derivative

The time rate of change of a quantity (such as temperature or velocity or stress tensor) of
a material particle, is known as a material derivative. We shall denote the material derivative
byD/Dt.

(i)When a material description of the quantity is used, we have

Thus,

(ii) When a spatial description of the quantity is used, we have

where x» the positions of material particles at time t, are related to the material coordinates
by the motion or/ = x^Xi^X^^}- Then,