Substantive derivative

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See also Convective derivative


In mathematics and continuum mechanics, including fluid dynamics, the substantive derivative (sometimes the Lagrangian derivative, material derivative or advective derivative), written D / Dt, is the rate of change of some property of a small parcel of fluid.

Note that if the fluid is moving, the substantive derivative is the rate of change of fluid within the small parcel, hence the other names advective derivative and fluid following derivative.

It is defined for function F as follows:

\frac{D F}{Dt}=\frac{\partial F}{\partial t}+ ({\mathbf u}\cdot\nabla)F

where {\mathbf u} is the fluid velocity and \nabla is the differential operator del.

Compare the substantive derivative with the Eulerian derivative (written \partial/\partial t) in which fluid with different properties may be advected into the notional infinitesimal control volume.

Consider water undergoing steady flow through a hosepipe that has a gradually decreasing cross section. Because water is incompressible in practice, conservation of mass requires that the flow is faster at the end of the pipe than at the start. Because the flow is steady, the Eulerian derivative of velocity is everywhere zero, but the substantive derivative is nonzero because any individual parcel of fluid accelerates as it moves down the hose.

For tensor fields we usually 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.

See also: Navier-Stokes equations, Convective derivative, and Euler equations.

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