Page 102 - Introduction to Continuum Mechanics

Material Derivative 8?

Example 3.3.1

J~WC|
Use Eq. (3.3.4), obtain —p— for the motion and temperature field given in the previous
example.
Solution. From Example 3.2.1, we have

and

The gradient of © is simply

Therefore,

which agrees with the previous example.

3.4 Acceleration of a Particle In a Continuum

The acceleration of a particle is the rate of change of velocity of the particle. It is therefore
the material derivative of velocity. If the motion of a continuum is given by Eq. (3.1.1), i.e..

then the velocity v, at time t, of a particle X is given by

and the acceleration a, at time t, of a particle X is given by

Thus, if the material description of velocity, v(X,f) is known (or is obtained from Eq. (3.4.1),
then the acceleration is very easily computed, simply taking the partial derivative with respect
to tim£ of the function v££f). On the other hand, if only the spatial description of velocity [i.e.,
v = v(*,f)] is known, the computation of acceleration is not as simple.
(A)RectanguIar Cartesian Coordinates (jcj^^s)- With