Boundary layer, viscous

The above set of equations is similar to that used in traditional computational fluid dynamics (CFD) (except for the electron energy balance and the EM equations), and advances made in that field can be used to benefit the plasma reactor simulation problem. As an example, the sheath near the wall can be thought of as similar to a boundary layer in fluid flow (chemically reacting or not) [130]. Separating the flow into bulk (inviscid) and boundary layer (viscous) and then patching the two solutions (asymptotic analysis) has long been practiced in fluid mechanics and may also be applied to the plasma problem [102, 103, 151]. 

For external flow such as flow over a flat stationary plate whose surface temperature is different from the bulk fluid temperature, both hydrodynamic and thermal boundary layers develop along the direction of the flow. Inside the hydrodynamic boundary layer viscous forces are dominant resulting in velocity profile. Similarly as thermal boundary layer develops along the flow direction and heat is being transferred to or from the surface results in a temperature profile. In Figure 22.8 the hydrod5mamic and thermal boundary layers are shown for flow over a heated flat plate. Both velocity and temperature inside the boundary layer reach 99% of the free stream velocity (Vf) and temperature (T ), respectively, at the edge of the boundary layer. 

The concentration boundary layer forms because of the convective transport of solutes toward the membrane due to the viscous drag exerted by the flux. A diffusive back-transport is produced by the concentration gradient between the membranes surface and the bulk. At equiUbrium the two transport mechanisms are equal to each other. Solving the equations leads to an expression of the flux ... 

The flow in the diffuser is usually assumed to be of a steady nature to obtain the overall geometric configuration of the diffuser. In a channel-type diffuser the viscous shearing forces create a boundary layer with reduced kinetic energy. If the kinetic energy is reduced below a certain limit, the flow in this layer becomes stagnant and then reverses. This flow reversal causes... 

The primary cause of efficiency losses in an axial-flow turbine is the buildup of boundary layer on the blade and end walls. The losses associated with a boundary layer are viscous losses, mixing losses, and trailing edge losses. To calculate these losses, the growth of the boundary layer on a blade must be known so that the displacement thickness and momentum thickness can be computed. A typical distribution of the displacement and momentum thickness is shown in Figure 9-26. The profile loss from this type of bound-ary-layer build-up is due to a loss of stagnation pressure, which in turn is... 

In die streamline boundary layer the only forces acting within the fluid are pure viscous forces and no transfer of momentum takes place by eddy motion. 

When a viscous fluid flows over a surface it is retarded and the overall flowrate is therefore reduced. A non-viscous fluid, however, would not be retarded and therefore a boundary layer would not form. The displacement thickness 8 is defined as the distance the surface would have to be moved in the 7-direction in order to obtain the same rate of flow with this non-viscous fluid as would be obtained for the viscous fluid with the surface retained at x = 0. 

Schlichting, H. Boundary Layer Theory (trails, by KESTIN, J.) 6th edn (McGraw-Hill, New York, 1968). White, F. M. Viscous Fluid Flow (McGraw-Hill, New York, 1974). 

Derive the momentum equation for the flow of a viscous fluid over a small plane surface. Show that the velocity profile in the neighbourhood of the surface may be expressed as a sine function which satisfies the boundary conditions at the surface and at the outer edge of the boundary layer. 

These steps occur in the sequence shown and the slowest step determines the deposition rate. The rules of the boundary layer apply in most CVD depositions in the viscous flow range where pressure is relatively high. In cases where very low pressure is used (i.e., in the mTorr range), the rules are no longer applicable. 

This chapter presents a physical description of the interaction of flames with fluids in rotating vessels. It covers the interplay of the flame with viscous boundary layers, secondary flows, vorticity, and angular momentum and focuses on the changes in the flame speed and quenching. There is also a short discussion of issues requiring further studies, in particular Coriolis acceleration effects, which remain a totally unknown territory on the map of flame studies. 

The hydrodynamic boundary layer has an inner part where the vertical velocity increases to a maximum determined by a balance of viscous and buoyancy forces. In fluids of high Schmidt number, the concentration diffusion layer thickness is of the same order of magnitude as this inner part of the hydrodynamic boundary layer. In the outer part of the hydrodynamic boundary layer, where the vertical velocity decays, the buoyancy force is unimportant. The profile of the vertical velocity component near the electrode can be shown to be parabolic. 

Fig. 16 Lines of flow of liquid in the Levich rotating disc method of determining dissolution rates. There is a transition from (A) flow essentially normal to the surface to (B) flow parallel with the surface, pointing to the existence of a viscous boundary layer. (Reproduced with permission of the copyright owner, the Royal Society of Chemistry, from Ref. 101.)...

As the fluid s velocity must be zero at the solid surface, the velocity fluctuations must be zero there. In the region very close to the solid boundary, ie the viscous sublayer, the velocity fluctuations are very small and the shear stress is almost entirely the viscous stress. Similarly, transport of heat and mass is due to molecular processes, the turbulent contribution being negligible. In contrast, in the outer part of the turbulent boundary layer turbulent fluctuations are dominant, as they are in the free stream outside the boundary layer. In the buffer or generation zone, turbulent and molecular processes are of comparable importance. 

Equation 2.40 is an empirical equation known as the one-seventh power velocity distribution equation for turbulent flow. It fits the experimentally determined velocity distribution data with a fair degree of accuracy. In fact the value of the power decreases with increasing Re and at very high values of Re it falls as low as 1/10 [Schlichting (1968)]. Equation 2.40 is not valid in the viscous sublayer or in the buffer zone of the turbulent boundary layer and does not give the required zero velocity gradient at the centre-line. The l/7th power law is commonly written in the form... 

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