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.
Read more about this topic: Material Derivative
Famous quotes containing the word definition:
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)