Material Derivative - Development

Development

Consider a scalar quantity φ = φ( x, t ), where t is understood as time and x as position. This may be some physical variable such as temperature or chemical concentration. The physical quantity exists in a fluid, whose velocity is represented by the vector field v( x, t ).

The (total) derivative with respect to time of φ is expanded through the multivariate chain rule:

It is apparent that this derivative is dependent on the vector

which describes a chosen path x(t) in space. For example, if is chosen, the time derivative becomes equal to the partial derivative, which agrees with the definition of a partial derivative: a derivative taken with respect to some variable (time in this case) holding other variables constant (space in this case). This makes sense because if, then the derivative is taken at some constant position. This static position derivative is called the Eulerian derivative.

An example of this case is a swimmer standing still and sensing temperature change in a lake early in the morning: the water gradually becomes warmer due to heating from the sun.

If, instead, the path x(t) is not a standstill, the (total) time derivative of φ may change due to the path. For example, imagine the swimmer is in a motionless pool of water, indoors and unaffected by the sun. One end happens to be a constant hot temperature and the other end a constant cold temperature. By swimming from one end to the other the swimmer senses a change of temperature with respect to time, even though the temperature at any given (static) point is a constant. This is because the derivative is taken at the swimmer's changing location. A temperature sensor attached to the swimmer would show temperature varying in time, even though the pool is held at a steady temperature distribution.

The material derivative finally is obtained when the path x(t) is chosen to have a velocity equal to the fluid velocity:

That is, the path follows the fluid current described by the fluid's velocity field v. So, the material derivative of the scalar φ is:

An example of this case is a lightweight, neutrally buoyant particle swept around in a flowing river undergoing temperature changes, maybe due to one portion of the river being sunny and the other in a shadow. The water as a whole may be heating as the day progresses. The changes due to the particle's motion (itself caused by fluid motion) is called advection (or convection if a vector is being transported).

The definition above relied on the physical nature of fluid current; however no laws of physics were invoked (for example, it hasn't been shown that a lightweight particle in a river will follow the velocity of the water). It turns out, however, that many physical concepts can be written concisely with the material derivative. The general case of advection, however, relies on conservation of mass in the fluid stream; the situation becomes slightly different if advection happens in a non-conservative medium.

Only a path was considered for the scalar above. For a vector, the gradient becomes a tensor derivative; for tensor fields we may want to take into account not only translation of the coordinate system due to the fluid movement but also its rotation and stretching. This is achieved by the upper convected time derivative.