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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 ...
Purely Lagrangian methods employ a framework in which a space is discretised into initial subvolumes, whose flowpaths are then charted over time. Purely Eulerian methods, on the other hand, employ a framework in which the motion of material is described relative to a mesh that remains fixed in space throughout the calculation. As the name ...
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 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 ...
In field theory, the independent variable is replaced by an event in spacetime (x, y, z, t), or more generally still by a point s on a Riemannian manifold.The dependent variables are replaced by the value of a field at that point in spacetime (,,,) so that the equations of motion are obtained by means of an action principle, written as: =, where the action, , is a functional of the dependent ...
Lagrangian ocean analysis makes use of the relation between the Lagrangian and Eulerian specifications of the flow field, namely (,) = ((,),) = (,), where (,) defines the trajectory of a particle (fluid parcel), labelled , as a function of the time , and the partial derivative is taken for a given fluid parcel . [6]
Methods have been developed for simulations of viscoelastic fluids, curved fluid interfaces, microscopic biophysical systems (proteins in lipid bilayer membranes, swimmers), and engineered devices, such as the Stochastic Immersed Boundary Methods of Atzberger, Kramer, and Peskin, [6] [7] Stochastic Eulerian Lagrangian Methods of Atzberger, [8 ...