Material Derivative and Acceleration

• Let the position of a particle at any instant t in a flow field be given by the space coordinates (x, y, z) with respect to a rectangular cartesian frame of reference.

• The velocity components u, v, w of the particle along x, y and z directions respectively can then be written in Eulerian form as

u = u (x, y, z, t)
v = v (x, y, z, t)
w = w (x, y, z, t)

• After an infinitesimal time interval t , let the particle move to a new position given by the coordinates (x + Δx, y +Δy , z + Δz).

• Its velocity components at this new position be u + Δu, v + Δv and w +Δw.

• Expression of velocity components in the Taylor's series form:

The increment in space coordinates can be written as -

Substituting the values of in above equations, we have

etc

•   In the limit   , the equation becomes