Material derivative

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, who introduced the now-standard D / D t {\displaystyle D/Dt} 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. The notation D / D t {\displaystyle D/Dt} 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. Some authors prefer to write the full expression ∂ / ∂ t + u ⋅ ∇ {\displaystyle \partial /\partial t+\mathbf {u} \cdot \nabla } without using an abbreviation, while others use d / d t {\displaystyle d/dt} for Lagrangian coordinates and reserve D / D t {\displaystyle D/Dt} for Eulerian coordinates. Defenders of the D / D t {\displaystyle D/Dt} 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 y {\displaystyle y} that is macroscopic, with the sense that it depends only on position and time coordinates, y = y ( x , t ) {\displaystyle y=y(x,t)} :

D y D t ≡ ∂ y ∂ t + u ⋅ ∇ y , {\displaystyle {\frac {\mathrm {D} y}{\mathrm {D} t}}\equiv {\frac {\partial y}{\partial t}}+\mathbf {u} \cdot \nabla y,}

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