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 variations of that physical quantity. 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 which case, the material derivative then describes the temperature change of a certain fluid parcel with time, as it flows along its pathline (trajectory).

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

The material derivative is defined for any tensor field y that is macroscopic, with the sense that it depends only on position and time coordinates, y = y(x, t):

where ∇y is the covariant derivative of the tensor, and u(x, t) is the flow velocity. Generally the convective derivative of the field u·∇y, the one that contains the covariant derivative of the field, can be interpreted both as involving the streamline tensor derivative of the field u·(∇y), or as involving the streamline directional derivative of the field (u·∇) y, leading to the same result. Only this spatial term containing the flow velocity describes the transport of the field in the flow, while the other describes the intrinsic variation of the field, independent by the presence of any flow. Confusingly, sometimes the name "convective derivative" is used for the whole material derivative D/Dt, instead for only the spatial term u·∇, which is also a redundant nomenclature. In the nonredundant nomenclature the material derivative only equals the convective derivative for absent flows. The effect of the time-independent terms in the definitions are for the scalar and tensor case respectively known as advection and convection.

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