Material Derivative - Definition

Definition

The material derivatives of a scalar field φ( x, t ) and a vector field u( x, t ) are defined respectively as:

where the distinction is that is the gradient of a scalar, while is the covariant derivative of a vector. In case of the material derivative of a vector field, the term v•∇u can both be interpreted as v•(∇u) involving the tensor derivative of u, or as (v•∇)u, leading to the same result.

Confusingly, the term convective derivative is both used for the whole material derivative Dφ/Dt or Du/Dt, and for only the spatial rate of change part, v•∇φ or v•∇u respectively. For that case, the convective derivative only equals D/Dt for time independent flows.

These derivatives are physical in nature and describe the transport of a scalar or vector quantity in a velocity field v( x, t ). The effect of the time independent terms in the definitions are for the scalar and vector case respectively known as advection and convection.