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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.
The derivative of a field with respect to a fixed position in space is called the Eulerian derivative, while the derivative following a moving parcel is called the advective or material (or Lagrangian [2]) derivative. The material derivative is defined as the linear operator:
The Lagrangian and Eulerian specifications of the kinematics and dynamics of the flow field are related by the material derivative (also called the Lagrangian derivative, convective derivative, substantial derivative, or particle derivative). [1] Suppose we have a flow field u, and we are also given a generic field with Eulerian specification F ...
On the other hand, the two second-order partial derivatives of the specific internal energy in the momentum equation require the specification of the fundamental equation of state of the material considered, i.e. of the specific internal energy as function of the two variables specific volume and specific entropy: = (,).
Note that the material derivative consists of two terms. The first term ∂ ρ ∂ t {\displaystyle {\tfrac {\partial \rho }{\partial t}}} describes how the density of the material element changes with time.
The local derivative occurs during unsteady flow, and becomes zero for steady flow. The portion of the material derivative represented by the spatial derivatives is called the convective derivative. It accounts for the variation in fluid property, be it velocity or temperature for example, due to the motion of a fluid particle in space where ...
In the case, a shareholder derivative action brought against Match Group , Delaware's top court ruled that transactions could avoid the heightened standard so long as fully informed, disinterested ...
where D / Dt is the material derivative, which is the sum of local and convective derivatives. This additional constraint simplifies the governing equations, especially in the case when the fluid has a uniform density.