Equations of Navier-Stokes

The LGA is a variant of a cellular automaton, introduced as an alternative numerical approximation to the partial differential equation of Navier-Stokes and the continuity equations, whose analytical solution leads to the macroscopic approach of fluid dynamics. The microscopic behavior of the LGA has been shown to be very close to the Navier-Stokes (N-S) equations for incompressible fluids at the macroscopic level. 

Classically, electrokinetic phenomena are described by the equation of Navier-Stokes along with the continuity equation [ 1-3J. The Navier-Stokes equation accounts for the balance of forces in the electrokinetic problem. For steady laminar fluid flow the Navier-Stokes equation takes the following form in electrokinetics ... 

Numerical analysis of the gas inside the dynamic scrubber reduces to solving the Navier-Stokes equations [3]. For the solution of equations of Navier-Stokes equations with a standard (A -8)-turbulence model. To find the scalar parameters k and 8 are two additional model equations containing empirical constants [4-6]. The computational grid was built in the grid generator ANSYS ICEM CFD. The grid consists of 1247 542 elements. 

The hydrodynamic equations of Navier-Stokes were reduced by using variables stream function - vorticity (V , w). The directed finite difference of second order was applied to approximate the vorticity at walls of reactor chamber, for example, in mesh point n,w) ... 

The differ tial equations of Navier-Stokes are basic equations in fluid dynamics. By solving them together with the equation of continuity for each of the phases of a multi-phase flow stem with the corresponding boundary conditions, theoretically, it is possible to describe the hydrodynamic processes in all technical and natiue systems. The equations, written on the basis of a balance of die forces of viscosity, gravity and inertia, are as follows. 

Equation (79) is the differential equation of diffusion (or mass transfor) in a moving flow. In it, besides the concentration, the flow velNavier-Stokes (20) to (22) and the equation of continuity (18). 

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

An Eulerian-Eulerian (EE) approach was adopted to simulate the dispersed gas-liquid flow. The EE approach treats both the primary liquid phase and the dispersed gas phase as interpenetrating continua, and solves a set of Navier-Stokes equations for each phase. Velocity inlet and outlet boundary conditions were employed in the liquid phase, whilst the gas phase conditions consisted of a velocity inlet and pressure outlet. Turbulence within the system was account for with the Standard k-e model, implemented on a per-phase basis, similar to the recent work of Bertola et. al.[4]. A more detailed description of the computational setup of the EE method can be found in Pareek et. al.[5]. 

The momentum equation (the Navier-Stokes equation) for fluid flow (De Groot and Mazur, 1962) is complicated and difficult to solve. It is the subject of fluid mechanics and dynamics and is not covered in this book. When fluid flow is discussed in this book, the focus is on the effect of the flow (such as a flow of constant velocity, or boundary flow) on mass transfer, not the dynamics of the flow itself. 

One important use of the stream function is for the visualization of flow fields that have been determined from the solution of Navier-Stokes equations, usually by numerical methods. Plotting stream function contours (i.e., streamlines) provides an easily interpreted visual picture of the flow field. Once the velocity and density fields are known, the stream function field can be determined by solving a stream-function-vorticity equation, which is an elliptic partial differential equation. The formulation of this equation is discussed subsequently in Section 3.13.1. Solution of this equation requires boundary values for l around the entire domain. These can be evaluated by integration of the stream-function definitions, Eqs. 3.14, around the boundaries using known velocities on the boundaries. For example, for a boundary of constant z with a specified inlet velocity u(r),... 

Boundary-layer behavior is one of several potential simplifications that facilitate channel-flow modeling. Others include plug flow or one-dimensional axial flow. The boundary-layer equations, however, are the ones that require the most insight and effort to derive and to establish the ranges of validity. The boundary-layer equations retain a full two-dimensional representation of all the field variables as well as all the nonlinear behavior of Navier-Stokes equations. Nevertheless, when applicable, they provide a very significant simplification that can be used to great benefit in modeling. 

Electrokinetic phenomena can be understood with the help of two equations The known Poisson equation and the Navier3-Stokes4 equation. The Navier-Stokes equation describes the movement of a Newtonian liquid, i.e., a liquid whose viscosity does not change when it flows and when it is sheared. In order to make the equation plausible we consider an infinitesimal quantity of the liquid having a volume dV = dx dy dz and a mass dm. If we want to write Newtons equation of motion for this volume element we have to consider three forces ... 

Step 1 is purely hydrodynamic and relates the perturbation Q to the velocity near the wall which is the only relevant quantity for the mass transfer response. at are either wall velocity gradients or coefficients involved in the velocity expansion near the wall. This step requires the use of Navier-Stokes equations and will be treated in Chapter 2. 

The example is the fuel cell module of commercial CFD software FLUENT . As was already mentioned, it is a set of additional subroutines (on top of Navier-Stokes equations and other transport models in CFD) to account for electrochemistry inside... 

The hydrodynamics is then described by the system of Navier-Stokes equations in the film-flow approximation (Shilkin et al., 2006) ... 

The exact solution of the convection-diffusion equations is very complicated, since the theoretical treatments involve solving a hydrodynamic problem, i.e., the determination of the solution flow velocity profile by using the continuity equation or -> Navier-Stokes equation. For the calculation of a velocity profile the solution viscosity, densities, rotation rate or stirring rate, as well as the shape of the electrode should be considered. 

Navier-Stokes equation — The Navier-Stokes equation is one of the equations used to derive the velocity profile of an incompressible fluid ... 

CAST, a two-dimensional FEM computer code solves the mass-balance equation, the Navier-Stokes-equations which result from the momentum balance for flows with friction and the energy balance in each control volume of the calculation grid. All equations can be written in a general form [9] ... 

A direct simulation of the flow field was also attempted in Sengupta et al. (2002), where the following stream function- vorticity formulation of Navier- Stokes equation was used. 

Another approach that has promise for study of turbulence structure is the fluctuating velocity field (FVF) closure, adopted by Deardorff (D3). Using the analog of a MVF closure for turbulent motions of smaller scale than his computational mesh, Deardorff carried out a three-dimensional unsteady solution of Navier-Stokes equations, thereby calculating the structure of the larger-scale eddy motions. While it is likely that calculations of such complexity will remain beyond the reach of most for some time to come, results like Deardorff s should serve as guides for framing closure models. 

The starting point for the description of the fluid dynamics of the film-coating process is the Navier-Stokes equation and the continuity equation. The Navier-Stokes equation reads ... 

The application of Navier-Stokes equations to turbulent flows are discribed in sect 1.2.7. The Reynolds averaged equations for incompressible flows are normally adopted deriving the transfer coefficients for heat and mass. 

The only nonzero component of Navier-Stokes equation, (2-91), is the z component that governs u(x, y, t). Taking account of (3-5), we find that this equation is... 

Proudman and Pearson [382] tried to obtain analytic solutions in a wider range of Reynolds numbers. They solved the system of Navier-Stokes equations... 

In the buffer zone, i.e., between the physical domain and the open boundary, the damping terms are added to the right-hand side (RHS) of Navier Stokes equations ... 

Craya and Curtet [4] have used averaged integrated values of Navier-Stokes equation as well as the continuity equation to predict the confined jet behavior. The dimensionless Craya-Curtet parameter m is given by ... 

The problem of swirling flows leads to the solution of Navier-Stokes equations, which often involve a dimension reduction and are obtained numerically (Grcar, 1996). By assuming that the flow has rigid rotational symmetry, axial velocity and temperature independent of the radius, density independent of the pressure variation, and pressure quadratic with respect to the radius and independent of the axial profile, the general model is... 

More frequently, instead of the equilibrium pattern sketched so far, one observes electroconvection (EC) patterns in nematics, which present dissipative structmes characterized by director distortions, space charges and material flow. A necessary requirement for their existence is the presence of charge carriers in the nematic. In a distorted nematic, where n is neither parallel nor perpendicular to E, the generation of a non-zero space charge, pei, by charge separation is then inevitable. The resulting Coulomb force in the flow equations (generalized Navier-Stokes equations) drives a... 

In the small nanochaimels (from a few to about 100 nm), the electric double layer (EDL) thickness becomes larger or at least comparable with the nanochaimels lateral dimensions. It affects the balance of bulk ionic concentrations of co-ions and counterions in the nanochannels. Thus, many conventional approaches such as the Poisson—Boltzmann equation and the Helmholtz-Smoluchowski slip velocity, which are based on the thin EDL assumption and equal number of co-ions and counterions, lose their credibility and cannot be utilized to model the electrokinetic effects through these nanoscale channels. The Poisson equation, the Navier-Stokes equations, and the Nemst-Planck equation should be solved directly to model the electrokinetic effects and find the electric... 

When there is an electric field parallel to the double layer, the external electric field will drive the ions to move. Because of the viscosity of the liquid, the bulk liquid will move with the diffuse layer and the resulted motion of the bulk liquid is termed as electroosmotic flow (EOF), which is also referred to as the classic electroosmotic flow comparing with the induced-charge electroosmotic flow. In a microchannel (Fig. 2), the value of the electroosmotic velocity can be solved by continuity equation and Navier-Stokes equation ... 

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