2.5 Time derivatives

The conservation of mass Eq. (2.8 ) included a partial derivative in time eqn relating to a fixed region of space. This is the local rate of change of eqn, relating to the change in eqn in the fluid measured by an observer at a fixed location. It is not the time rate of change experienced by a mass of fluid particles as they move through space. In the same way, eqn is not the acceleration experienced by the fluid.

Acceleration relates to the material, or substantive, derivative which describes the time rate of change of a fixed mass of moving material. It is denoted by eqn and is related to the local rate of change, using eqn as an example tensor of any rank, by

|--------------------| D| @ | -Dt- = -@t-+ u r | --------------------- \relax \special {t4ht=

PICT\relax \special {t4ht=

The relation is derived from the chain rule of differential. In one dimension, it is illustrated by two particles of fluid that occupy positions 1 and 2 at some initial time, then positions 3 and 4 at a later time eqn. The particles move in the eqn-direction at speed eqn, such that the particle at 1 later occupies the position 3 and the particle at 2 occupies position 4.

The material time derivative of eqn, following the mass from 2eqn4, is the sum of: the local change in eqn at a fixed position eqn (2eqn3); and, the change due to the gradient of eqn between positions 3 and 4, fixing time eqn. This equates to

D @ dx =ux @ Dt-- = -@t- + dt- @x- -: x x t \relax \special {t4ht=
The material derivative relation in Eq. (2.14) is simply the 3D equivalent of Eq. (2.15 ).


The last term in Eq. (2.14) introduces the gradient denoted by eqn. If eqn is a scalar, the gradient produces a vector whose magnitude and direction is that of the steepest gradient.

PICT\relax \special {t4ht=

The figure above illustrates the gradient using a surface that represents a distribution of a scalar field in 2 directions. The gradients at 3 locations are in the direction of steepest ascent.

When eqn is a vector, the gradient produces a tensor, representing the direction and magnitude of steepest ascent for each of the 3 components of the vector.

Notes on CFD: General Principles - 2.5 Time derivatives