Page 101 - Introduction to Continuum Mechanics

86 Kinematics of a Continuum

Cartesian coordinates are used, these are the velocity components v/ of the particle JQ. Thus,
the material derivative in rectangular coordinates is

or,

wherejt should be emphasized that these equations are for 0 in a spatial description, i.e.,
© = ©(jt lr *2,#3,0- Note that if the temperature field is independent of time and if the velocity
of a particle is perpendicular to V© (i.e, the particle is moving along the path of constant ©)

Note again that Eq. (33.4a) is valid only for rectangular Cartesian coordinates, whereas
Eq. (3.3.4b) has the advantage that it is valid for all coordinate systems. For a specific
coordinate system, all that is needed is the appropriate expression for the gradient. For
example, in cylindrical coordinate (r, 0, z),

and from Eq. (2D2.3)

Thus,

In spherical coordinates (r,0,0)

and from Eq. (2D3.9)

Thus,