Material Derivative and Acceleration

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

The above equations tell that the operator for total differential with respect to time, D/Dt in a convective field is related to the partial differential as:

Explanation of equation 7.2 :

The total differential D/Dt is known as the material or substantial derivative with respect to time.
The first term ¶/¶t in the right hand side of is known as temporal or local derivative which expresses the rate of change with time, at a fixed position.
The last three terms in the right hand side of are together known as convective derivative which represents the time rate of change due to change in position in the field.

Explanation of equation 7.1 (a, b, c):

The terms in the left hand sides of Eqs (7.1a) to (7.1c) are defined as x, y and z components of substantial or material acceleration.
The first terms in the right hand sides of Eqs (7.1a) to (7.1c) represent the respective local or temporal accelerations, while the other terms are convectiveaccelerations.

Thus we can write,

(Material or substantial acceleration) = (temporal or local acceleration) + (convective acceleration)

Important points:

In a steady flow, the temporal acceleration is zero, since the velocity at any point is invariant with time.
In a uniform flow, on the other hand, the convective acceleration is zero, since the velocity components are not the functions of space coordinates.
In a steady and uniform flow, both the temporal and convective acceleration vanish and hence there exists no material acceleration.

Existence of the components of acceleration for different types of flow is shown in the table below.