Navier numerical diffusion

Reviews of concentration polarization have been reported (14,38,39). Because solute wall concentration may not be experimentally measurable, models relating solute and solvent fluxes to hydrodynamic parameters are needed for system design. The Navier-Stokes diffusion—convection equation has been numerically solved to calculate wall concentration, and thus the water flux and permeate quaUty (40). 

The strategies discussed in the previous chapter are generally applicable to convection-diffusion equations such as Eq. (32). If the function O is a component of the velocity field, the incompressible Navier-Stokes equation, a non-linear partial differential equation, is obtained. This stands in contrast to O representing a temperature or concentration field. In these cases the velocity field is assumed as given, and only a linear partial differential equation has to be solved. The non-linear nature of the Navier-Stokes equation introduces some additional problems, for which special solution strategies exist. Corresponding numerical techniques are the subject of this section. 

Perhaps even more important is die fact that LEM does not require a numerical solution to die Navier-Stokes equation. Indeed, even a three-dimensional diffusion equation is generally less computationally demanding than the Poisson equation needed to find die pressure field. 

P 61] The numerical simulations were based on the solution of the incompressible Navier-Stokes equation and a convection-diffusion equation for a concentration field by means of the finite-volume method [152], The Einstein convention of summation over repeated indices was used. For pressure-velocity coupling, the SIMPLEC algorithm and for discretization of the species concentration equation the QUICK differencing scheme were applied. Hybrid and the central differencing schemes referred to velocities and pressure, respectively (commercial flow solvers CFX4 and CFX5). 

In problems of heat convection, the most complex equations to solve are the fluid flow equations. Often times, the governing equations for the fluid flow are the Navier-Stokes equations. It is useful, therefore, to study a model equation that has similar characteristics to the Navier-Stokes equations. This model equation has to be time-dependent and include both convection and diffusion terms. The viscous Burgers equation is an appropriate model equation. In the first few sections of this chapter, several important numerical schemes for the Burgers equation will be discussed. A simple physical heat convection problem is solved as a demonstration. 

In thermally non-homogeneous supercritical fluids, very intense convective motion can occur [Ij. Moreovei thermal transport measurements report a very fast heat transport although the heat diffusivity is extremely small. In 1985, experiments were performed in a sounding rocket in which the bulk temperature followed the wall temperature with a very short time delay [11]. This implies that instead of a critical slowing down of heat transport, an adiabatic critical speeding up was observed, although this was not interpreted as such at that time. In 1990 the thermo-compressive nature of this phenomenon was explained in a pure thermodynamic approach in which the phenomenon has been called adiabatic effect [12]. Based on a semi-hydrodynamic method [13] and numerically solved Navier-Stokes equations for a Van der Waals fluid [14], the speeding effect is called the piston effecf. The piston effect can be observed in the very close vicinity of the critical point and has some remarkable properties [1, 15] ... 

These equations have to be solved simultaneously with the Reynolds averaged Navier-Stokes equations and the Ergun-equation given in Section 11.7.5 to account for the radial void profile and generate both axial and radial flow velocity components, Ur and Uz. In these equations it is assumed that the heat transfer through the fluid and solid phase occur in parallel [de Wasch and Froment, 1971 Dixon and Cresswell, 1979 Dixon, 1985]. and AV can be calculated from the correlations of de Wasch and Froment [1972] and Zehner and Schliinder [1972], The internal diffusion limitations appear in those equations by means of the effectiveness factor r , which is obtained by numerical integration of the diffusion and reaction equations inside the particles, as discussed in Chapter 3. 

In 2001, Ferrigno and Girault proposed a hydrodynamic approach to ion-transfer reactions [122]. In the same way that the friction factor for the diffusion coefficient in the Stokes-Einstein equation can be given by solving the Navier-Stokes equation for a sphere in a laminar flow, the Navier-Stokes equation was solved numerically to account for the passage of a sharp boundary between two continuum media. These data show that the friction coefficient varies from to in a continuous manner over a distance one order of magnitude larger... 

The numerical modeling methods for polymer blends have been reviewed in this chapter, with different categories such as volume-of-fluid, molecular dynamics and diffusion-controlled methods being introduced. Use of the Cahn-Hilliard method was emphasized for binary and ternary polymer systems with no obvious mechanical flux, while specific factors such as elastic energy and functionalized substrate were considered for purposes of comparison. The diffusion-controlled model described, using the Cahn-Hilliard equation as the constitutive equation, can be used to depict the gradient of the interface as well as the composition profile of partially miscible blends hence, it is feasible to implement this equation in a polymer blend system. It should be noted that although these examples do not consider mechanical flux, additional constitutive equations (e.g., Navier-Stokes) can easily be added to this diffusion-controlled model. 

Any experimental validation of that assumption is difficult, but recently direct numerical simulation of turbulent reactive flows appeared possible, for very moderate Reynolds number, however. Direct numerical simulations use the primitive equations themselves, the continuity and Navier Stokes equations, jointly with one or two diffusion-reaction equations like (1) a very big computer is required for the integration... 

This paper presents a mathematical model and numerical analysis of momentum transport and heat transfer of polymer melt flow in a standard cooling extruder. The finite element method is used to solve the three-dimensional Navier-Stokes equations based on a moving barrel formulation a semi-Lagrangian approach based on an operator-splitting technique is used to solve the heat transfer advection-diffusion equation. A periodic boundary condition is applied to model fully developed flow. The effects of polymer properties on melt flow behavior, and the additional effects of considering heat transfer, are presented. 

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