Substantial Derivative
The Lagrangian and Eulerian specifications of the kinematics and dynamics of the flow field are related by the substantial derivative (also called the Lagrangian derivative, convective derivative, material derivative, or particle derivative).
Suppose we have a flow field with Eulerian specification u, and we are also given some function F(x,t) defined for every position x and every time t. (For instance, F could be an external force field, or temperature.) Now one might ask about the total rate of change of F experienced by a specific flow parcel. This can be computed as
(where ∇ denotes the gradient with respect to x, and the operator u⋅∇ is to be applied to each component of F.) This tells us that the total rate of change of the function F as the fluid parcels moves through a flow field described by its Eulerian specification u is equal to the sum of the local rate of change and the convective rate of change of F. This is a consequence of the chain rule since we are differentiating the function F(X(a,t),t) with respect to t.
Read more about this topic: Lagrangian And Eulerian Specification Of The Flow Field
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