Momentum terms Links

In Eq. (38) the hydrodynamic velocity is that used to evaluate the momentum of the phase. In viscous flow it is the term used in establishing the shear from a knowledge of the viscosity when effects of cross linking of fluxes (01, 02) are neglected. The hydrodynamic velocity and the diffusional velocity are related by... 

On one level it is a quantum effect, and can be described in terms of photon—phonon scattering. The incident NIR beam is a source of photons, and the energy from the piezotransducer provides a source of lattice phonons that propagate through the crystal. As in all collision processes, the twin principles of conservation of momentum and conservation of energy apply. The momentum of a quantum particle is linked to its wavevector by hk. The energy is linked to its frequency by hjj. 

But any complete description of the evolution of perturbations in the universe will link all of these terms initial velocity and density perturbations to the various components (baryons, dark matter, photons) evolve prior to last scattering as discussed above, and so photon overdensities occur in potential wells, and velocity perturbations occur in response to gravitational and pressure forces. Indeed, to solve this problem in its most general form, we must resort to the Boltzmann equation. The Boltzmann equation gives the evolution of the distribution function, fi(xp,Pp) for a particle of species i with position Xp, and momentum p/(. In its most general form, the Boltzmann equation is formally... 

The energy conservation equation is intimately linked to momentum conservation equations via the fourth and fifth terms. For most reacting systems, the contribution of energy released or absorbed by chemical reactions usually dominates the other terms originating from pressure and viscous effects. For highly viscous flows with low heats of reaction, it may be important to consider the viscous heating terms. An order of magnitude analysis is often used to examine the relative importance of different terms. 

The main objective of performing kinetic theory analyzes is to explain physical phenomena that are measurable at the macroscopic level in a gas at- or near equilibrium in terms of the properties of the individual molecules and the intermolecular forces. For instance, one of the original aims of kinetic theory was to explain the experimental form of the ideal gas law from basic principles [65]. The kinetic theory of transport processes determines the transport coefficients (i.e., conductivity, diffusivity, and viscosity) and the mathematical form of the heat, mass and momentum fluxes. Nowadays the kinetic theory of gases originating in statistical mechanics is thus strongly linked with irreversible- or non-equilibrium thermodynamics which is a modern held in thermodynamics describing transport processes in systems that are not in global equilibrium. 

To initiate the iteration process a pressure field p" is guessed. The discretized momentum component equations on the form like (12.160) are solved using the guessed pressure field to obtain the corresponding approximate values for the velocity components. Substitution of the correct pressure held into the momentum equations yields the correct velocity held. The discretized equations on the form (12.160) link the correct velocity fields with the correct pressure held. Subtraction of the approximate momentum equations from the exact ones (neglecting any changes in the source term S ), we obtain the necessary relations between the velocity and pressure corrections on the form ... 

At this point it is useful to define creation (at) and annihilation (a) operators, which are analogous to angular momentum raising and lowering operators. These at, a operators profoundly simplify the algebra needed to set up the polyad Heff matrices, to apply some of the dynamics diagnostics discussed in Sections 9.1.4 and 9.1.7, and to transform between basis sets (e.g., between normal and local modes). They also provide a link between the quantum mechanical Heff model, which is expressed in terms of at, a operators and adjustable molecular constants (evaluated by least squares fits of spectra), and a reduced-dimension classical mechanical HeS model. 

The three subjects of heat, mass and momentum transfer are inter-linked heat flows down a temperature gradient, mass flows down a concentration gradient and momentum passes down a velocity gradient. These inter-relationships, which are firmly based on fundamental equations, are useful. In the previous chapter some introductory aspects of fluid flow were examined but it is important to realise that the three subjects mentioned above are all interdependent and an umbrella term transport phenomena is often used to group them together. 

In a third major task, the velocity components respond to changes in the guessed or estimated pressure are approximated. Substitution of the correct pressure field into the momentum equation components (12.178)-(12.180) presumably holding the correct coefficients and source terms the outcome will be the correct velocity field components n + and The correct velocity field is thus linked with the correct pressure field through the discretized momentum equation components on the form ... 

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