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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 ...
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 ...
In the Lagrangian, the position coordinates and velocity components are all independent variables, and derivatives of the Lagrangian are taken with respect to these separately according to the usual differentiation rules (e.g. the partial derivative of L with respect to the z velocity component of particle 2, defined by v z,2 = dz 2 /dt, is ...
Derivation of the Lagrangian and Eulerian finite strain tensors A measure of deformation is the difference between the squares of the differential line element d X {\displaystyle d\mathbf {X} \,\!} , in the undeformed configuration, and d x {\displaystyle d\mathbf {x} \,\!} , in the deformed configuration (Figure 2).
In scientific visualization, Lagrangian–Eulerian advection is a technique mainly used for the visualization of unsteady flows. The computer graphics generated by the technique can help scientists visualize changes in velocity fields. This technique uses a hybrid Lagrangian and Eulerian specification of the flow field.
Original file (WebM audio/video file, VP8, length 11 s, 1,280 × 480 pixels, 1.93 Mbps overall, ... Lagrangian and Eulerian specification of the flow field;
The Euler equations can be formulated in a "convective form" (also called the "Lagrangian form") or a "conservation form" (also called the "Eulerian form"). The convective form emphasizes changes to the state in a frame of reference moving with the fluid.
ξ(α, t) is the Lagrangian position vector of a fluid parcel, u(x, t) is the Eulerian velocity, x is the position vector in the Eulerian coordinate system, α is the position vector in the Lagrangian coordinate system, t is time. Often, the Lagrangian coordinates α are chosen to coincide with the Eulerian coordinates x at the initial time t ...