Conservation energy-momentum

If a beam of monoenergetic ions of mass A/, is elastically scattered at an angle 6 by surface atoms of mass Mg, conservation of momentum and energy requires that... 

J. C. Simo, N. Tarnow, and K. K. Wang. Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Computer Methods in Applied Mechanics and Engineering, 100 63-116, 1994. 

J. C. Simo and N. Tarnow. The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics. ZAMP, 43 757-793, 1992. 

O. Gonzales and J. C. Simo. On the stability of symplectic and energy-momentum conserving algorithms for nonlinear Hamiltonian systems with symmetry. Comp. Meth. App. Mech. Engin., 134 197, 1994. 

The relationship between the two conditions is estabUshed by conservation of energy and by conservation of momentum across the shock front. 

The conservation of mass gives comparatively Httle useful information until it is combined with the results of the momentum and energy balances. Conservation of Momentum. The general equation for the conservation of momentum is... 

Direct and Indirect Energy Gap. The radiative recombination rate is dramatically affected by the nature of the energy gap, E, of the semiconductor. The energy gap is defined as the difference in energy between the minimum of the conduction band and the maximum of the valence band in momentum, k, space. Eor almost all semiconductors, the maximum of the valence band occurs where holes have zero momentum, k = 0. Direct semiconductors possess a conduction band minimum at the same location, k = O T point, where electrons also have zero momentum as shown in Eigure la. Thus radiative transitions that occur in direct semiconductors satisfy the law of conservation of momentum. 

Macroscopic and Microscopic Balances Three postulates, regarded as laws of physics, are fundamental in fluid mechanics. These are conservation of mass, conservation of momentum, and con-servation of energy. In addition, two other postulates, conservation of moment of momentum (angular momentum) and the entropy inequality (second law of thermodynamics) have occasional use. The conservation principles may be applied either to material systems or to control volumes in space. Most often, control volumes are used. The control volumes may be either of finite or differential size, resulting in either algebraic or differential consei vation equations, respectively. These are often called macroscopic and microscopic balance equations. 

D.E. Burton, Conservation of Energy, Momentum, and Angular Momentum in Lagrangian Staggered-Grid Hydrodynamics, UCRL-JC-105926, Lawrence Livermore National Laboratory, Livermore, CA, 1990. 

Conservation relations are used to derive mechanical stress-volume states from observed wave profiles. Once these states have been characterized through experiment or theory they may, in turn, predict wave profiles for the material in question. For the case of a well-defined shock front propagating at constant speed L/ to a constant pressure P and particle velocity level u, into a medium at rest at atmospheric pressure, with initial density, p, the conservation of momentum, mass, and energy leads to the following relations ... 

In kinetics, Newton s second law, the principles of kinematics, conservation of momentum, and the laws of conservation of energy and mass are used to develop relationships between the forces acting on a body or system of bodies and the resulting motion. 

It is easy to invent rules that conserve particle number, energy, momentum and so on, and to smooth out the apparent lack of structural symmetry (although we have cheated a little in our example of a random walk because the circular symmetry in this case is really a statistical phenomenon and not a reflection of the individual particle motion). The more interesting question is whether relativistically correct (i.e. Lorentz invariant) behavior can also be made to emerge on a Cartesian lattice. Toffoli ([toff89], [toffSOb]) showed that this is possible. 

When the product ion moved with a higher kinetic energy than predicted by the stripping model, the collision apparently was more elastic— i.e., less kinetic energy of the incident ion was used for internal excitation of the products. In an ideal elastic collision with H transfer the products carry no internal energy at all. If the secondary ion moves forward and the H atom moves backwards, conservation of momentum requires that the primary ion has a velocity ... 

It is easy to verify that multiparticle collisions conserve mass, momentum, and energy in every cell. Mass conservation is obvious. Momentum and energy conservation are also easily established. For momentum conservation in cell E, we have... 

Multiparticle collision dynamics can be generalized to treat systems with different species. While there are many different ways to introduce multiparticle collisions that distinguish between the different species [16, 17], all such rules should conserve mass, momentum, and energy. We suppose that the A-particle system contains particles of different species a=A,B,... with masses ma. Different multiparticle collisions can be used to distinguish the interactions among the species. For this purpose we let V 1 denote the center of mass velocity of particles of species a in the cell i ,3... 

Here va and va are the stoichiometric coefficients for the reaction. The formulation is easily extended to treat a set of coupled chemical reactions. Reactive MPC dynamics again consists of free streaming and collisions, which take place at discrete times x. We partition the system into cells in order to carry out the reactive multiparticle collisions. The partition of the multicomponent system into collision cells is shown schematically in Fig. 7. In each cell, independently of the other cells, reactive and nonreactive collisions occur at times x. The nonreactive collisions can be carried out as described earlier for multi-component systems. The reactive collisions occur by birth-death stochastic rules. Such rules can be constructed to conserve mass, momentum, and energy. This is especially useful for coupling reactions to fluid flow. The reactive collision model can also be applied to far-from-equilibrium situations, where certain species are held fixed by constraints. In this case conservation laws... 

Multiparticle collision dynamics describes the interactions in a many-body system in terms of effective collisions that occur at discrete time intervals. Although the dynamics is a simplified representation of real dynamics, it conserves mass, momentum, and energy and preserves phase space volumes. Consequently, it retains many of the basic characteristics of classical Newtonian dynamics. The statistical mechanical basis of multiparticle collision dynamics is well established. Starting with the specification of the dynamics and the collision model, one may verify its dynamical properties, derive macroscopic laws, and, perhaps most importantly, obtain expressions for the transport coefficients. These features distinguish MPC dynamics from a number of other mesoscopic schemes. In order to describe solute motion in solution, MPC dynamics may be combined with molecular dynamics to construct hybrid schemes that can be used to explore a variety of phenomena. The fact that hydrodynamic interactions are properly accounted for in hybrid MPC-MD dynamics makes it a useful tool for the investigation of polymer and colloid dynamics. Since it is a particle-based scheme it incorporates fluctuations so that the reactive and nonreactive dynamics in small systems where such effects are important can be studied. 

Extraction of the speed distribution is achieved in an analogous manner by integrating over all angles for each speed. The speed distributions can be further transformed, using the law of conservation of momentum, into total translational energy distributions for the O3 — O2(X3S ) + 0(3Pj) dissociation. 

Mixture conservation of mass equation Mixture conservation of momentum equation Mixture conservation of energy equation Slip equation (concerning the difference in velocity)... 

This expression applies to the transport of any conserved quantity Q, e.g., mass, energy, momentum, or charge. The rate of transport of Q per unit area normal to the direction of transport is called the flux of Q. This transport equation can be applied on a microscopic or molecular scale to a stationary medium or a fluid in laminar flow, in which the mechanism for the transport of Q is the intermolecular forces of attraction between molecules or groups of molecules. It also applies to fluids in turbulent flow, on a turbulent convective scale, in which the mechanism for transport is the result of the motion of turbulent eddies in the fluid that move in three directions and carry Q with them. 

For steady flow of a gas (at a constant mass flow rate) in a uniform pipe, the pressure, temperature, velocity, density, etc. all vary from point to point along the pipe. The governing equations are the conservation of mass (continuity), conservation of energy, and conservation of momentum, all applied to a differential length of the pipe, as follows. 

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