Hydrodynamics governing equations

A typical method for thermal analysis is to solve the energy equation in hydrodynamic films and the heat conduction equation in solids, simultaneously, along with the other governing equations. To apply this method to mixed lubrication, however, one has to deal with several problems. In addition to the great computational work required, the discontinuity of the hydrodynamic films due to asperity contacts presents a major difficulty to the application. As an alternative, the method of moving point heat source integration has been introduced to conduct thermal analysis in mixed lubrication. 

The computational code used in solving the hydrodynamic equation is developed based on the CFDLIB, a finite-volume hydro-code using a common data structure and a common numerical method (Kashiwa et al., 1994). An explicit time-marching, cell-centered Implicit Continuous-fluid Eulerian (ICE) numerical technique is employed to solve the governing equations (Amsden and Harlow, 1968). The computation cycle is split to two distinct phases a Lagrangian phase and a remapping phase, in which the Arbitrary Lagrangian Eulerian (ALE) technique is applied to support the arbitrary mesh motion with fluid flow. 

The flow field of the impacting droplet and its surrounding gas is simulated using a finite-volume solution of the governing equations in a 3-D Cartesian coordinate system. The level-set method is employed to simulate the movement and deformation of the free surface of the droplet during impact. The details of the hydrodynamic model and the numerical scheme are described in Sections... 

A mathematical description of an electrochemical system should take into account species fluxes, material conservation, current flow, electroneutrality, hydrodynamic conditions, and electrode kinetics. While rigorous equations governing the system can frequently be identified, the simultaneous solution of all the equations is not generally feasible. To obtain a solution to the governing equations, we must make a number of approximations. In the previous section we considered the mathematical description of electrode kinetics. In this section we shall assume that the system is mass-transport limited and that electrode kinetics can be ignored. 

The second approach is a fractional-step method we call asymptotic timestep-splitting. It is developed by consideration of the specific physics of the problem being solved. Stiffness in the governing equations can be handled "asymptotically" as well as implicitly. The individual terms, including those which lead to the stiff behavior, are solved as independently and accurately as possible. Examples of such methods include the Selected Asymptotic Integration Method (4,5) for kinetics problems and the asymptotic slow flow algorithm for hydrodynamic problems where the sound speed is so fast that the pressure is essentially constant (6, 2). ... 

For each continuous phase k present in a multiphase system consisting of N phases, in principle the set of conservation equations formulated in the previous section can be applied. If one or more of the N phases consists of solid particles, the Newtonian conservation laws for linear and angular momentum should be used instead. The resulting formulation of a multiphase system will be termed the local instant formulation. Through the specification of the proper initial and boundary conditions and appropriate constitutive laws for the viscous stress tensor, the hydrodynamics of a multiphase system can in principle be obtained from the solution of the governing equations. 

Consider hydrodynamics in a rotating disk electrode. [15] [24] The governing equations for velocity distributions are ... 

During our discussion of linear momentum balance in chap. 2, we noted that the fundamental governing equations of continuum mechanics as embodied in eqn (2.32) are indifferent to the particular material system in question. This claim is perhaps most evident in that eqn (2.32) applies just as well to fluids as it does to solids. From the standpoint of the hydrodynamics of ordinary fluids (i.e. Newtonian fluids) we note that it is at the constitutive level that the distinction... 

The designation linear to describe the theoretical study of the fate of small initial disturbances is due to the fact that the dynamics of such small disturbances can be described by means of a linear approximation of the Navier-Stokes, continuity, and other transport equations. Because the governing equations are linear, analytic theory is often possible but this requires that the unperturbed state or flow, whose stability we wish to study, be known analytically. Furthermore, this base solution must be quite simple for even the linear approximation of the equations to be analytically tractable. In practice this reduces significantly the number of problems in which complete analytic results are possible and also explains why hydrodynamic stability theory has been particularly successful in analyzing problems... 

Governing Equations. If the problem is to be solved rigorously, the BTE must be solved for electrons in each valley, optical phonons, and acoustic phonons. The distribution function of each of these depends on six variables—three space and three momentum (or energy). The solution to BTE for this complexity becomes very computer intensive, especially due to the fact that the timescales of electron-phonon and phonon-phonon interactions vary by two orders of magnitude. Monte Carlo simulations are sometimes used although this, too, is very time-consuming. Therefore, researchers have resorted mainly to hydrodynamic equations for modeling electron and phonon transport for practical device simulation. 

In the basic governing equation of advection-diffusion, dispersion refers to the movement of species under the influence of gradient of chemical potential, while advection is the stirring or hydrodynamic transport caused by density gradient or forced convection. A general one-dimensional mass transfer to an electrode is governed by the Nemst-Planck equation ... 

The approximate experimental determination of xl), is based on measurement of the velocity of a charged particle in a solvent subjected to an applied voltage. Such a particle experiences an electrical force that initiates motion. Since a hydrodynamic frictional force acts on the particle as it moves, a steady state is reached, with the particle moving with a constant velocity U. To calculate this electrophoretic velocity U theoretically, it is, in general, necessary to solve Poisson s equation (Equation 3.19) and the governing equations for ion transport subject to the condition that the electric field is constant far away from the particle. The appropriate viscous drag on the particle can be calculated from the velocity field and the electrical force on the particle from the electrical potential distribution. The fact that the sum of the two is zero provides the electrophoretic velocity U. Actual solutions are complex, and the electrical properties of the particle (e.g., polarizability, conductivity, surface conductivity, etc.) come into play. Details are given by Levich (1962) (see also Problem 7.8). 

As for studies of freeboard region hydrodynamics and reactions, Yates and Rowe (1977) developed a simple model of a catalytic reaction based on the assumption that the freeboard contained perfectly mixed, equally dispersed particles derived from bubble wakes. The fraction of wake particles ejected,/, was a model parameter. The governing equation of the model was proposed as... 

Batchelor [167] considered the effect of hydrodynamic interaction of parallel elongated particles in pure extension on the bulk stress. The governing equations were solved under the assumption that the effect on one particle by all the others could be replaced by a cylindrical boundary condition (cell model) and the geometry and concentration of the particles satisfied the following inequality. 

The quasi-one-dimensional model used in the previous sections for analysis of various characteristics of fiow in a heated capillary assumes a uniform distribution of the hydrodynamical and thermal parameters in the cross-section of micro-channel. In the frame of this model, the general characteristics of the fiow with a distinct interface, such as position of the meniscus, rate evaporation and mean velocities of the liquid and its vapor, etc., can be determined for given drag and intensity of heat transfer between working fluid and wall, as well as vapor and wall. In accordance with that, the governing system of equations has to include not only the mass, momentum and energy equations but also some additional correlations that determine... 

The key analysis of hydrodynamic dispersion of a solute flowing through a tube was performed by Taylor [149] and Aris [150]. They assumed a Poiseuille flow profile in a tube of circular cross-section and were able to show that for long enough times the dispersion of a solute is governed by a one-dimensional convection-diffusion equation ... 

The rotating disc electrode is constructed from a solid material, usually glassy carbon, platinum or gold. It is rotated at constant speed to maintain the hydrodynamic characteristics of the electrode-solution interface. The counter electrode and reference electrode are both stationary. A slow linear potential sweep is applied and the current response registered. Both oxidation and reduction processes can be examined. The curve of current response versus electrode potential is equivalent to a polarographic wave. The plateau current is proportional to substrate concentration and also depends on the rotation speed, which governs the substrate mass transport coefficient. The current-voltage response for a reversible process follows Equation 1.17. For an irreversible process this follows Equation 1.18 where the mass transfer coefficient is proportional to the square root of the disc rotation speed. 

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