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In continuum mechanics, the material derivative [1] [2] describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material derivative can serve as a link between Eulerian and Lagrangian descriptions of continuum ...
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 ...
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 ...
On the other hand, the pressure in thermodynamics is the opposite of the partial derivative of the specific internal energy with respect to the specific volume: (,) = (,) since the internal energy in thermodynamics is a function of the two variables aforementioned, the pressure gradient contained into the momentum equation should be explicited ...
The differential operator is a substantial (material) derivative moving with the fluid particles. [3] Stated more simply, this theorem says that if one observes a closed contour at one instant, and follows the contour over time (by following the motion of all of its fluid elements), the circulation over the two locations of this contour remains ...
The composition of the material contains two types of terms: those involving the time derivative and those involving spatial derivatives. The time derivative portion is denoted as the local derivative, and represents the effects of unsteady flow. The local derivative occurs during unsteady flow, and becomes zero for steady flow.
Reynolds transport theorem can be expressed as follows: [1] [2] [3] = + () in which n(x,t) is the outward-pointing unit normal vector, x is a point in the region and is the variable of integration, dV and dA are volume and surface elements at x, and v b (x,t) is the velocity of the area element (not the flow velocity).
An example of convective acceleration. The flow is steady (time-independent), but the fluid decelerates as it moves down the diverging duct (assuming incompressible or subsonic compressible flow). A significant feature of the Navier–Stokes equations is the presence of convective acceleration: the effect of time-independent acceleration of a ...