Marker and cell method

The calculation algorithm was developed using the artificial compressibility and markers and cells methods [537], Equidistant steps Ax and Az as well as a fictitious time step At should have been chosen to ensure the accurate numerical results. The last parameters were considered as satisfactory if (i) discharge in the duct was conserved with the accuracy of 0.001%, and (ii) velocity profiles at the outlet really converged to the steady-state shape. We recall that the latter profile for any parameters 6 and A can be represented analytically by formula (3.7) for a linear EPR, k = 0, so that this case suits for the examination of the algorithm. The number of numerical cells was chosen 80 x 40 for moderate values of A and Re 160 x 80 cells are required if A Re < 1000, but one needs 320 x 160 cells if A Re > 1000 - 2000. Bigger values of A Re caused the algorithmic difficulties which are not overcome yet. 

A numerical analysis of cavity filling was developed to evaluate and optimize the use of reactive fluids in RIM. This method, which has been previously described in detail O), employs the marker and cell method for treating transient fluid flows in conjunction with finite difference solutions of the conversion and temperature fields in the pre-polymer and the mold wall. The time-temperature-conversion-vlscoslty correlations shown earlier for epoxy + AEP were then used in the mold filling simulation. 

This approach ( FE + CV ) is repeated in a loop until the mold cavity is filled completely. This approach is based on the marker and cell method. One of the major advantages of this approach is that it does... 

Modeling Concepts for Multiphase Flow 3.3. 1 The Marker and Cell Method... 

However, predictions from simple flows, such as those described in Example 7.3 above, cannot be generalized to more realistic systems. Bigg and Middleman (23) analyzed a somewhat more realistic system of flow in a rectangular channel, shown in Fig. 7.15. The motion of the upper surface induces partial mixing of the fluids, and the interfacial area, which was calculated as a function of time, is used as a quantitative measure of the laminar mixing. The marker and cell calculation method, developed by Harlow and Welch (24), was used to solve the flow field and calculate the position of the interface. The evolution of the interface of two fluids of equal viscosities and densities in a channel with an aspect ratio of 0.52, and a... 

This paper reports the mathematical modelling of electrochemical processes in the Soderberg aluminium electrolysis cell. We consider anode shape changes, variations of the potential distribution and formation of a gaseous layer under the anode surface. Evolution of the reactant concentrations is described by the system of diffusion-convection equations while the elliptic equation is solved for the Galvani potential. We compare its distribution with the C02 density and discuss the advantages of the finite volume method and the marker-and-cell approach for mathematical modelling of electrochemical reactions. 

It follows from the above discussion and numerical results that even a simple convective-diffusive model of concentration behaviour mechanism gives realistic results and yields a satisfactory description of the formation of the gaseous layer under the anode surface. The model may be improved by adding the electrolyte circulation and electromagnetic forces yet we hope that it will not change the main conclusions. The finite volume method proves to be a flexible and sufficiently accurate numerical technique for solving both the equations for the Galvani potential and the reactant concentrations. The marker-and-cell approach makes it possible to outline the electrode surfaces easily. 

Marker and Cell (MAC) Method The first method capable of modeling gas-hquid flows, separated by a moving interface, was the MAC of Harlow and Welch [4]. In the MAC method, as shown in Fig. 2b, the massless markers are used to define the location and track the movement of fi ee surface. This is in fact a combination of a Eulerian solution of the basic flow field, with Lagrangian tracking of marker particles. The computational cycle in the MAC method consists of the advancement of discrete field variables fi om an initial time to to the subsequent time to + At by accomplishing the steps below. 

For non-Newtonian fluids the viscosity p is fitted to flow curves of experimental data. The models for this fit are discussed in the next chapter. The energy equation is also implemented in the code and can be used for temperature-dependent problems, but it is not needed for the simulation of fluid dynamic problems like jet breakup due to the uncoupling of the density in the incompressible formulation. The finite volume scheme uses the Marker and Cell (MAC) method to discretize the computational domain in space. The convective and diffusive terms are discretized with second-order accuracy and the fluxes are calculated with a Godunov-type scheme. 

The geometry and mesh arrangement in the fluid region are exactly the same as those of the steady-state subchannel analysis code. Figure 6.60 shows the entire algorithm. The momentum conservation equations for three directions and a mass conservation equation are solved with the Simplified Marker And Cell (SMAC) method [32]. In the SMAC method, a temporary velocity field is calculated, the Poison equation is solved, and then the velocity and pressure fields are calculated as shown in Fig. 6.61. The Successive Over-Relaxation (SOR) method is used to solve a matrix. 

Another 3-D cornea model, comprising rabbit primary cultures of epithelial and stromal cells as well as mouse immortalized endothelial cells, was described in 1994 by Zieske and coworkers [70], They showed the influence of endothelial cells on the formation of a tightly packed, multilayered epithelium as well as the expression of laminin, type VII collagen, a6 integrin, keratin K3, and a-enolase. Furthermore, their findings suggested that the formation of an in vivo-like epithelium requires the cultivation of the 3-D corneal construct under AIC conditions. By contrast, LCC methods of cultivating corneal equivalents in the absence of endothelial cells failed to promote the expression of differentiation markers and basement membrane components. 

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