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. Euclid, detail from The School of Athens by Raphael. ... Continuum mechanics is a branch of physics (specifically mechanics) that deals with continuous matter, including both solids and fluids (i. ... Fluid dynamics is the subdiscipline of fluid mechanics that studies fluids (liquids and gases) in motion. ... In mathematics, the derivative is defined as the instantaneous rate of change of a function. ...

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 as follows

where is the fluid velocity and is the differential operator del. In vector calculus, del is a vector differential operator represented by the nabla symbol, ∇. In the three-dimensional Cartesian coordinate system R3 with coordinates (x, y, z), del can be defined as or alternatively, where is the standard basis in R3. ...

Compare the substantive derivative with the Eulerian derivative (written ) in which fluid with different properties may be advected into the notional infinitesimal control volume. Advection is the transport of a conserved scalar quantity that is transported in a vector field. ... It has been suggested that this article or section be merged with 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. In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In continuum mechanics, including fluid dynamics upper convected time derivative or Oldroyd derivative is the rate of change of some tensor property of a small parcel of fluid that is written in the coordinate system rotating and stretching with the fluid. ...

See also: Navier-Stokes equations, Convective derivative, and Euler equations. The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, are a set of equations that describe the motion of fluid substances like liquids and gases. ... The convective derivative, also known as the Lagrangian derivative, is a derivative taken with a respect to a coordinate system moving with velocity u, and is often used in fluid mechanics. ... In fluid dynamics, the Euler equations govern the motion of a compressible, inviscid fluid. ...