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For integrable systems in the sense of multidimensional consistency (MDC) we can consider the Lagrangian as a form, which is closed on solutions of the equations of motion. For 2-dimensional systems, described by partial difference…
A significant drawback of Lagrangian (particle-tracking) reactive transport models has been their inability to properly simulate interactions between solid and liquid chemical phases, such as dissolution and precipitation reactions. This…
It is shown that the Euler hydrodynamics for vortical flows of an ideal fluid coincides with the equations of motion of a charged {\it compressible} fluid moving due to a self-consistent electromagnetic field. Transition to the Lagrangian…
Lagrangian transport structures for three-dimensional and time-dependent fluid flows are of great interest in numerous applications, particularly for geophysical or oceanic flows. In such flows, chaotic transport and mixing can play…
In many engineering systems operating with a working fluid, the best efficiency is reached close to a condition of flow separation, which makes its prediction very crucial in industry. Providing that wall-based quantities can be measured,…
Active solids consume energy to allow for actuation, shape change, and wave propagation not possible in equilibrium. Whereas active interfaces have been realized across many experimental systems, control of three-dimensional (3D) bulk…
A phenomenological theory of the fluctuations of velocity occurring in a fully developed homogeneous and isotropic turbulent flow is presented. The focus is made on the fluctuations of the spatial (Eulerian) and temporal (Lagrangian)…
We consider viscous two-dimensional steady flows of incompressible fluids past doubly periodic arrays of solid obstacles. In a class of such flows, the autocorrelations for the Lagrangian observables decay in accordance with the power law,…
In this paper we use the finite size Lyapunov Exponent (FSLE) to characterize Lagrangian coherent structures in three-dimensional (3d) turbulent flows. Lagrangian coherent structures act as the organizers of transport in fluid flows and are…
Recent advancements have demonstrated that fully Eulerian methods can effectively model frictionless contact between deformable solids. Unlike traditional Lagrangian approaches, which require contact detection and resolution algorithms, the…
Certain systems of inviscid fluid dynamics have the property that for solutions that are only slightly better than differentiable in Eulerian variables, the corresponding Lagrangian trajectories are analytic in time. We elucidate the…
An arbitrary Lagrangian--Eulerian (ALE) finite element method for arbitrarily curved and deforming two-dimensional materials and interfaces is presented here. An ALE theory is developed by endowing the surface with a mesh whose in-plane…
We present an Eulerian vortex method based on the theory of flow maps to simulate the complex vortical motions of incompressible fluids. Central to our method is the novel incorporation of the flow-map transport equations for line elements,…
An optimization method used in image-processing (metamorphosis) is found to imply Euler's equations for incompressible flow of an inviscid fluid, without requiring that the Lagrangian particle labels exactly follow the flow lines of the…
The study of transport and mixing processes in dynamical systems is particularly important for the analysis of mathematical models of physical systems. We propose a novel, direct geometric method to identify subsets of phase space that…
The log-homotopy particle flow filter resolves the Bayesian update by transporting particles along a continuous trajectory in pseudo-time. However, the governing partial differential equation for the flow velocity is fundamentally…
Methods of dynamical system's theory are used for numerical study of transport and mixing of passive particles (water masses, temperature, salinity, pollutants, etc.) in simple kinematic ocean models composed with the main Eulerian coherent…
Direct numerical simulation of microscale fluid--structure interactions in multicomponent and multiphase flows requires methods that can represent moving boundaries together with fields constrained to evolving interfaces. Diffuse-domain…
This article reviews several recently developed Lagrangian tools and shows how their combined use succeeds in obtaining a detailed description of purely advective transport events in general aperiodic flows. In particular, because of the…
A computational approach is introduced for the study of the rheological properties of complex fluids and soft materials. The approach allows for a consistent treatment of microstructure elastic mechanics, hydrodynamic coupling, thermal…