Use Of Lagrangian Coordinates Research Articles

One class of MHD equations: Conservation laws and exact solutions

AbstractThe paper analyzes one of the models of equations of magnetohydrodynamics (MHD) derived earlier. The model was obtained as a result of group classification of the MHD equations in mass Lagrangian coordinates, where all dependent variables in Eulerian coordinates depend on time and two spatial coordinates. The use of Lagrangian coordinates made it possible to solve four equations, which led to the form of reduced equations containing four arbitrary functions: entropy and a three‐dimensional vector associated with the magnetic field. The objective of this work is to develop conservation laws and exact solutions for the model. Conservation laws are obtained using Noether's theorem, while exact solutions are obtained either explicitly or by solving a system of ordinary or partial differential equations with two independent variables. Numerical methods are employed for the latter solutions.

Group classification of the two‐dimensional magnetogasdynamics equations in Lagrangian coordinates

The present paper is devoted to the group classification of magnetogasdynamics equations in which dependent variables in Euler coordinates depend on time and two spatial coordinates. It is assumed that the continuum is inviscid and nonthermal polytropic gas with infinite electrical conductivity. The equations are considered in mass Lagrangian coordinates. Use of Lagrangian coordinates allows reducing number of dependent variables. The analysis presented in this article gives complete group classification of the studied equations. This analysis is necessary for constructing invariant solutions and conservation laws on the base of Noether's theorem.

Magnetic reconnection and thermal equilibration

When a magnetic field is forced to evolve on a time scale τev, as by footpoint motions driving the solar corona or non-axisymmetric instabilities in tokamaks, the magnetic field lines undergo large-scale changes in topology on a time scale approximately an order of magnitude longer than τev. But the physics that allows such changes operates on a time scale eight or more orders of magnitude slower. An analogous phenomenon occurs in air. Temperature equilibration occurs on a time scale approximately an order of magnitude longer than it takes air to cross a room, τev, although the physical mechanism that allows temperature equilibration is approximately four orders of magnitude slower than τev. The use of Lagrangian coordinates allows the fundamental equations to be solved and both phenomena explained. The theories of thermal equilibration and magnetic reconnection are developed in parallel to help readers obtain an understanding of the importance and implications of analyses using Lagrangian coordinates.

Group Analysis of the One-Dimensional Gas Dynamics Equations in Lagrangian Coordinates and Conservation Laws

A group analysis of the second-order equation including the one-dimensional gas dynamics equations in Lagrangian coordinates as a particular case is performed. The use of Lagrangian coordinates makes it possible to consider the one-dimensional gas dynamics equations as a variational Euler-Lagrange equation with an appropriate Lagrangian. Conservation laws are derived with the use of the variational presentation and Noether’s theorem. A complete group classification of the Euler-Lagrange equation is obtained; as a result, 18 different classes can be identified.

Fast magnetic reconnection and the ideal evolution of a magnetic field

Regardless of how small non-ideal effects may be, phenomena associated with changes in magnetic field line connections are frequently observed to occur on an Alfvénic time scale. Since it is mathematically impossible for magnetic field line connections to change when non-ideal effects are identically zero, an ideal evolution must naturally lead to states of unbound sensitivity to non-ideal effects. That such an evolution is natural is demonstrated by the use of Lagrangian coordinates based on the flow velocity of the magnetic field lines. The Lagrangian representation of an evolving magnetic field is highly constrained when neither the magnetic field strength nor the forces exerted by the magnetic field increase exponentially with time. The development of a state of fast reconnection consistent with these constraints (1) requires a three-dimensional evolution, (2) has an exponentially increasing sensitivity to non-ideal effects, and (3) has a parallel current density, which lies in exponentially thinning but exponentially widening ribbons, with a magnitude that is limited to a slow growth. The implication is that exponential growth in sensitivity is the cause of fast magnetic reconnection when non-ideal effects are sufficiently small. The growth of the non-ideal effect of the resistivity multiplied by the parallel current density is far too slow to be competitive.

Vehicle Specific Behaviour in Macroscopic Traffic Modelling through Stochastic Advection Invariant

In this contribution a new model to include stochastic vehicle specific behaviour and interaction in traffic flow modelling is presented. The First Order Model with Stochastic Advection (FOMSA) is presented as a first order macroscopic kinematic wave model in a platoon-based Lagrangian coordinate system. The use of Lagrangian coordinates allows characteristics of specific vehicles or vehicle-groups to propagate along with the traffic flow using a vehicle specific invariant. The invariant reflects how vehicle or platoon specific characteristics propagate with the vehicles and influence the local behaviour of a vehicle or platoon on a macroscopic level and in interaction with other surrounding vehicles. It represents a local vehicle specific adjustment to the critical density and makes use of two parameters: a stochastic boundary parameter and a transition parameter. These parameters indicate the extent of differences between vehicles or platoons. A case study is also presented in which a demonstration of the model is given and the face validity and sensitivity of the parameters are shown. Previously, similar approaches have made use of second order model descriptions. The formulation of this model as a first order model makes use of the advantages of first order models and also applies the improved accuracy of Lagrangian coordinates over the Eulerian coordinate system in time-stepping.

A Lagrangian model for spray behaviour within vine canopies

This work concerns the numerical prediction of pesticide deposits on vine by air assisted sprayers. This numerical model consists in two different parts: the air flow characteristics were obtained through a Navier–Stokes solver in which additional terms have been introduced in order to account for the modification of the flow by the vine foliage. The theoretical form of these terms was derived through an averaging procedure. As a result, the canopy effect was modelled by introducing momentum and turbulence source terms in the Navier–Stokes equations. The constants were identified thanks to experimental data obtained by direct measurements of the air flow speeds through an artificial canopy. The second part of the model consists in simulating the droplet cloud by means of a lagrangian stochastic model. The average motion of the droplets was computed through the use of lagrangian coordinates and the turbulence effect on the droplets was interpreted in term of statistical properties of the droplet cloud. The tractor displacements were accounted for through unsteady boundary conditions. Once the size and the droplets cloud locations have been determined, the deposit was predicted for a vine row of one meter length using an efficiency coefficient obtained from simulations. Thanks to this approach we were able to quantify the effect of air turbulence on droplet deposition.

Flow in deformable porous media: Modelling and simulations of compression moulding processes

This paper deals with the mathematical modelling, based on the theory of flows in deformable porous media, and with some numerical simulations of a unidirectional compression molding process. The technological process which we have considered consists of a compression of a pile of preimpregnated layers operated by a piston acting on the top of the pile. The main idea towards the modelling is the use of Lagrangian coordinates fixed on the solid skeleton. This technique allows us to simulate the system evolution either when the dynamics is controlled by the velocity of the piston and when it is controlled by the pressure applied on the piston.

Applications of exact solutions to the Navier–Stokes equations: free shear layers

A family of exact solutions to the Navier—Stokes equations is used to analyse unsteady three-dimensional viscometric flows that occur in the vicinity of a plane boundary that translates and rotates with time-varying velocities. Such flows are important in the study of flows that are produced by rotating machinery. They are also useful in describing local behaviour in more complex global flows, such as that produced in a shear layer by the passage of a disturbance in the mainstream. An example is the flow produced in a turbulent shear layer by the passage of the core of a Rankine vortex. When the effect of viscosity is unimportant, the use of Lagrangian coordinates reduces the mathematical problem to that of solving a set of linear ordinary differential equations.

Subsidence due to intensive pumping from a layered soil : Lee, C K; Fallou, S N; Mei, C C Proc 4th International Conference on Land Subsidence, Houston, 12–17 May 1991P677–690. Publ Wallingford: IAHS Press, 1991

We describe the results of a theoretical study of pumping of groundwater from a layered soil. The model soil system consists of three horizontal layers where a very soft and highly impervious aquitard is sandwiched by two hard and highly porous aquifers. Water is pumped from the bottom aquifer through a vertical well. This type of problem is usu­ ally solved in hydrology by assuming constant total stress in the soil, and that the drawdown can be calculated in advance of the soil stress. By a perturbation theory and the use of Lagrangian coordinates, these common assumptions are found to be inadequate if the aquitard is very soft or thick. Specifi­ cally the drawdown should be nonlinearly coupled to the soil deformation and that the total stress is not quite uniform in depth. By a model constitutive law representative of clays, two patterns of cyclic pumping or recharging are compared.

Solution of the equations for coagulation and recombination in a turbulent medium

Coagulation and recombination in a turbulent medium is studied statistically with the use of Lagrangian coordinates.

The use of Lagrangian coordinates in the water entry and related problems

AbstractLagrangian coordinates are used in conjunction with the self-similarity hypothesis in order to examine the problem of the vertical entry at constant speed of a wedge and also of a cone into an incompressible fluid initially at rest. Certain known properties of the free surface are recovered in a very direct and simple manner and new exact results concerning the inclination of the free surface and the angle of contact with the rigid surface are obtained.