Wall boundary conditions, for

A number of authors have considered channel cross-sections other than rectangular [102-104]. Figure 2.17 shows some examples of cross-sections for which friction factors and Nusselt numbers were computed. In general, an analytical solution of the Navier-Stokes and the enthalpy equations in such channel geometries would be involved owing to the implementation of the wall boundary condition. For this reason, usually numerical methods are employed to study laminar flow and heat transfer in channels with arbitrary cross-sectional geometry. 

For fully developed turbulent flow of liquid metals, the Nusselt number depends on the wall boundary condition. For a constant wall temperature [Notter and Sleicher, Chem. Eng. Science, 27,2073 (1972)],... 

If the convected vortices do not move at the free-stream speed, then the modes do not coalesce as in the previous case and one can simply calculate the wall boundary condition for the wall-mode and call it as 4>bp to distinguish it from the previous case. Thus,... 

This has been used as a wall boundary condition for the solution of Eqs. (36) or (41). 

FIGURE 2.5 Wall boundary conditions for enthalpy/temperature equation. 

FIGURE 4.5 Wall boundary conditions for dispersed phase particles, (a) Reflection, (b) saltation, (c) particle escapes or vanishes. 

The wall boundary condition for kp is specified in accordance with (1.428) ... 

Accordingly, the wall boundary condition for the turbulent energy dissipation rate, e, is given by (1.429) ... 

The wall might also exert a friction force on the bubbles [97]. A wall boundary condition for the gas momentum equations that was formulated by Lopez de Bertodano [94] is thus sometimes used ... 

The governing equations are elliptic so boundary conditions are required at all boundaries. The normal velocity components for both phases are set to zero at the vertical boundaries. The wall boundary conditions for the vertical velocity component, k and e are specified in accordance with the standard wall function approach. The particle phase is allowed to slip along the wall following the boundary condition given by (4.99). 

This appendix considers the appropriate wall boundary condition for temperature when the external resistances to heat transfer are significant. We suppose the tube is jacketed with a fluid at temperature Text that transfers heat to the outer wall of the... 

The top, side and bottom walls of the tank were specified as a zero wall velocity. The wall boundary conditions for the taper and orifice were also specified as zero wall velocity ... 

The above equation indicates that c is a constant following a fluid particle. Suppose we have c = Cq in the free stream and the wall boundary condition for a reacting surface is = 0. The solution of equation (4.75) cannot satisfy the boundary conditions at the reaction surface. Evidently, near the surface, there must be a thin diffusion boundary layer of thickness djy within which the concentration changes rapidly (see Figure4.15). This reasoning parallels the Prandtl boundary layer argument for viscous flow past a solid boundary at high Reynolds number. If is the Prandtl viscous boundary layer thickness for steady unbounded laminar flow, we know that... 

An additional wall lift force that pushes the dispersed phase away from the wall was suggested by Antal et al. [2]. Recently, the defect in the original model that a bubble located far from the wall is attracted to the wall, has been removed by Politano et al. [127]. The modified wall lift force model used by Jakobsen et al. [71] is given by (5.75). The wall might also exert a friction force on the bubbles [107]. A wall boundary condition for the gas momentum equations that was formulated by Lopez de Bertodano [103] is thus sometimes used ... 

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