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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 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 ...
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 its values are different.
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.
This "special" derivative is in fact the ordinary derivative of a function of many variables along a path following the fluid motion; it may be derived through application of the chain rule in which all independent variables are checked for change along the path (which is to say, the total derivative). For example, the measurement of changes in ...
Among the 11 sectors represented in the S&P 500, materials is usually the most overlooked and it is easy to understand why. Simply put, at a weight of 2.61%, materials is the smallest sector ...
where D / Dt is the material derivative operator, u is the flow velocity, ρ is the local fluid density, p is the local pressure, τ is the viscous stress tensor and B represents the sum of the external body forces. The first source term on the right hand side represents vortex stretching.