Material derivative

From Chemepedia
The material derivative describes the Lagrangian specification to describe change
The local derivative describes the Eulerian specification to describe change

In continuum mechanics, there are primarily two ways of describing the time rate of change of some physical quantity (like density or temperature).

  • In the Lagrangian specification of the field, the differential control volume moves with the fluid parcel along its pathline (See Figure 1). The time rate of change applies to the moving fluid parcel. This also known as the material (or total) derivative.
  • In the Eulerian specification of the flow field, the differential control volume is fixed. The time rate of change only applies for the fixed control volume. (See Figure 2). This is also known as the local derivative.

The relationship between the material and local derivative for a transport quantity is


            (1)

where

  • is the material derivative of
  • is the local derivative of
  • is the velocity field described by its Eulerian specification

The above relationship tells us that the total rate of change of the quantity as the fluid parcels moves through a flow field described by its Eulerian specification is equal to the sum of the local rate of change and the convective rate of change of .

Examples

  • Lagrangian: A Pressure gauge attached to a weather balloon, which transmits information about weather conditions to monitoring stations on the ground. In this example, the pressure gauge moves with the wind field; therefore, it is a Lagrangian frame of reference.


  • Eulerian: A pressure gauge attached right after the discharge of a pump measures pressure as a function of time. In this example, the pressure gauge is stationary, hence is a Eulerian frame of reference.


NOTE: in most chemical engineering processes, measurements are stationary (Eulerian frame of reference).