Individual derivative

individual derivative

(Also called material derivative, particle derivative, substantial derivative.) The rate of change of a quantity with respect to time, following a fluid parcel.

For example, if φ(x, y, z, t) is a property of the fluid and x = x(t), y = y(t), z = z(t) are the equations of motion of a certain particle of this fluid, then the total derivative,

(where u is the velocity of the fluid and ∇ is the del operator), is an individual derivative. It gives the rate of change of the property of a given parcel of the fluid as opposed to the rate of change at a fixed geometrical point, which is usually called the local derivative. The term u · ∇φ is called the advective term, expressing the variation of φ in a parcel moving into regions of different φ.

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