In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material derivative can serve as a link between Eulerian and Lagrangian descriptions of continuum deformation.

For example, in fluid dynamics, the velocity field is the flow velocity, and the quantity of interest might be the temperature of the fluid. In this case, the material derivative then describes the temperature change of a certain fluid parcel with time, as it flows along its pathline (trajectory).

Other names

There are many other names for the material derivative, including:

• advective derivative

• convective derivative

• derivative following the motion

• hydrodynamic derivative

• Lagrangian derivative

• particle derivative

• substantial derivative

• substantive derivative

• Stokes derivative

• total derivative, although the material derivative is actually a special case of the total derivative

History

The material derivative emerged during the mid-18th century, when Jean le Rond d'Alembert and Leonhard Euler first formulated hydrodynamics as a field theory governed by partial differential equations. Euler is believed to have been the first to write the fluid acceleration term as it is understood today, expressing the total acceleration of a fluid particle as the sum of local and convective contributions. The distinction between the two fundamental descriptions of fluid motion, i.e. the Eulerian (field) description and the Lagrangian (material) description, was established during this period. Euler introduced material coordinates in 1762, though they are now commonly called Lagrangian coordinates, while d'Alembert introduced spatial coordinates in 1752, now often called Eulerian coordinates.

The mathematical expression for the material derivative is attributed independently to Euler (around 1770) and Joseph-Louis Lagrange (around 1783). Lagrange, in his 1788 treatise *Mécanique Analytique*, further developed the Lagrangian framework for describing fluid motion, which naturally led to the concept of differentiation following a material particle.

The operator was brought to prominence in the English-speaking world by Sir George Gabriel Stokes notation for the material derivative in his 1845 paper on the motion of incompressible fluids. For this reason, the material derivative is still sometimes called the Stokes derivative.

has been a subject of debate among fluid dynamicists. Stokes first used it in 1845, but it was criticized by Harold Jeffreys and Bertha Jeffreys (1946) as a relic of an obsolete 19th-century notational convention for partial derivatives. for Eulerian coordinates. notation, including Frank M. White and James Lighthill, have argued for its continued use on the grounds of clarity and tradition.

The modern understanding of the material derivative as the link between Eulerian and Lagrangian descriptions was consolidated in the 20th century through the works of continuum mechanicians such as Clifford Truesdell and others. Today, the material derivative is a foundational concept in fluid dynamics, continuum mechanics, and related fields, appearing in the Navier–Stokes equations, the energy equation, and the conservation laws of physics.

Definition

The material derivative is defined for any tensor field that is macroscopic is the covariant derivative is the flow velocity, the one that contains the covariant derivative of the field, can be interpreted both as involving the streamline tensor derivative, or as involving the streamline directional derivative, leading to the same result.. The effect of the time-independent terms in the definitions are for the scalar and tensor case respectively known as advection and convection.

Scalar and vector fields

For example, for a macroscopic scalar field and a macroscopic vector field

is simply the gradient is the covariant derivative of the macroscopic vector (which can also be thought of as the Jacobian matrix). In particular for a scalar field in a three-dimensional Cartesian coordinate system

Development via the total derivative

.

is expanded using the multivariate chain rule

which describes a chosen is chosen, the time derivative becomes equal to the partial time derivative, which agrees with the definition of a partial derivative, then the derivative is taken at some constant position. This static position derivative is called the Eulerian derivative.

is sufficient to describe the rate of change of temperature.

is sufficient to describe the rate of change of temperature. A temperature sensor attached to the swimmer would show temperature varying with time, simply due to the temperature variation from one end of the pool to the other.

Geometrical interpretation

An intuitive, geometrical understanding of the material derivative was provided by Moots and Mavis (1938) in their study of flood waves. in three-dimensional space (Figure 3).

[[File:Material_Derivative_3 thumb.]]

In this geometric view:

• . The slope of the tangent to this curve at any point gives the partial derivative, which is the local rate of change at a fixed spatial location.

• .

• ).

.

Application to unsteady open-channel flow

The practical power of the material derivative is illustrated by its application to flood waves in wide rectangular channels, as developed by Moots and Mavis (1938)..

The equation of motion for a fluid element (Figure 2) was derived by equating the net pressure force, gravity component, and friction force to the mass times acceleration: is the convective acceleration as the particle moves to a region of different velocity.

Coupled with this is the equation of continuity (Figure 1), which states that the difference between inflow and outflow over a small reach equals the change in storage:

Using these equations, the authors derived several important results for flood waves:

• , showing that the wave speed can exceed, equal, or even be less than the fluid velocity, and may become infinite or change sign at certain stages of the flood.

• The **profile of a monoclinal (rising) flood wave** was obtained in closed form by integrating the equations, demonstrating how the wave front gradually steepens or flattens.

• , which is greater than the mean flow velocity.

• .

These classical results highlight the necessity of using the material derivative (rather than just the local derivative) to correctly capture the dynamics of rapidly varying flows, such as those encountered during floods.

Orthogonal coordinates

It may be shown that, in orthogonal coordinates-th component of the convection term of the material derivative of a vector field is given by

are related to the metric tensors

In the special case of a three-dimensional Cartesian coordinate system

is a Jacobian matrix.

There is also a vector-dot-del identity can be expressed as:

See also

• Navier–Stokes equations

• Euler equations (fluid dynamics)

• Derivative (generalizations)

• Lagrangian and Eulerian specification of the flow field

• Lie derivative

• Levi-Civita connection

• Spatial acceleration

• Spatial gradient

References

Further reading