Boundary-layer flow pressure gradient

For boundary layer flows on bodies of other shape, the transition Reynolds number based on the distance around the surface from the leading edge of the body is usually increased if the pressure is decreasing, i.e., if there is a favorable pressure gradient, and is usually decreased if the pressure is increasing, i.e., if there is an unfavorable pressure gradient. 

The oldest description of CVD uses the boundary-layer model, which assumes that between the bulk gas phase (uniform in composition) and the substrate there is a stagnant boundary layer in which gradients develop in temperature (cold wall reactors) and in the partial pressures of the reactants and the gaseous reaction products. The boundary-layer model presents some difficulties. Stagnant gas layers have a variable thickness in the reactor as long as unmodified bulk gas concentrations exist, if they exist at all. Moreover the flow is always laminar. Conceptually, however, the model has its advantages.The boundary-layer thickness is an effective parameter by which to characterize the deposition regime. This model will be used here for a simple overview of reaction conditions. 

Processing variables that affect the properties of the thermal CVD material include the precursor vapors being used, substrate temperature, precursor vapor temperature gradient above substrate, gas flow pattern and velocity, gas composition and pressure, vapor saturation above substrate, diffusion rate through the boundary layer, substrate material, and impurities in the gases. Eor PECVD, plasma uniformity, plasma properties such as ion and electron temperature and densities, and concurrent energetic particle bombardment during deposition are also important. 

Diffusion-blading loss. This loss develops because of negative velocity gradients in the boundary layer. Deceleration of the flow increases the boundary layer and gives rise to separation of the flow. The adverse pressure gradient that a compressor normally works against increases the chances of separation and causes significant loss. 

The outlet diffuser is used to eonvert the high absolute veloeity leaving the exdueer into statie pressure. If this eonversion is not done, the effieieney of the unit will be low. This eonversion of the flow to a statie head must be done earefully, sinee the low-energy boundary layers eannot tolerate great adverse pressure gradients. 

Derive the momentum equation for the flow of a fluid over a plane surface for conditions where the pressure gradient along the surface is negligible. By assuming a sine function for the variation of velocity with distance from the surface (within the boundary layer) for streamline flow, obtain an expression for the boundary layer thickness as a function of distance from the leading edge of the surface. 

Unfortunately, these equations cannot be modeled using the simple parallel-flow assumptions. In the entry region the radial velocity v and the pressure gradient will have an important influence on the axial-velocity profile development. Therefore we defer the detailed discussion and solution of this problem to Chapter 7 on boundary-layer approximations. 

Fig. 6.4 Streamlines for two axisymmetric Hiemenz stagnation flow situations having different outer velocity gradients, one at a = 1 s 1 and the other at a = 5 s—. Both cases are for air flow at atmospheric pressure and T = 300 K. The streamlines are plotted to an axial height of 3 cm and a radius of 10 cm. However, the solution itself has infinite radial extent in both the axial and radial directions. In both cases the streamlines are separated by 2jt A l = 2.0 x 10-5 kg/s. The shape of the scaled radial velocities V = v/r is plotted on the right of the figures. The maximum value of the scaled radial velocity is Vmax = a/2. Even though streamlines show curvature everywhere, the viscous region is confined to the boundary layer defined by the region of V variation. Outside of this region the flow behaves as though it is inviscid.

The pressure-gradient term Ar requires either a further boundary condition or a determination of the domain size. In the semi-infinite cases, Ar is a constant that is specified in terms of the outer potential-flow characteristics. However, the extent of the domain end must be determined in such a way that the viscous boundary layer is entirely contained within the domain. In the finite-gap cases, the inlet velocity n(zend) is specified at a specified inlet position. Since the continuity equation is first order, another degree of freedom must be introduced to accommodate the two boundary conditions on u, namely u — 0 at the stagnation surface and u specified at the inlet manifold. The value of the constant Ar is taken as a variable (an eigenvalue) that must be determined in such a way that the two velocity boundary conditions for u are satisfied. 

The magnitude of the Schmidt number indicates the relative thicknesses of the velocity and concentration boundary layers under some conditions, such as laminar low Re flow. And finally, the Womersley number (Wo) which tells us to what extent an unsteady or oscillating pressure gradient will be reflected in unsteady flow between hairs on an antenna ... 

In combined convective flow over a horizontal flat surface, the buoyancy forces are at right angles to the flow direction and lead to pressure changes across the boundary layer, i.e., there is an induced pressure gradient in the boundary layer despite the fact that flow over a flat plate is involved. Under some circumstances, this can lead to complex three-dimensional flow in the boundary layer. This type of flow will not be considered here, more information being available in [17] to [23]. 

While the engineer may frequently be interested in the heat-transfer characteristics of flow systems inside tubes or over flat plates, equal importance must be placed on the heat transfer which may be achieved by a cylinder in cross flow, as shown in Fig. 6-7. As would be expected, the boundary-layer development on the cylinder determines the heat-transfer characteristics. As long as the boundary layer remains laminar and well behaved, it is possible to compute the heat transfer by a method similar to the boundary-layer analysis of Chap. 5. It is necessary, however, to include the pressure gradient in the analysis because this influences the boundary-layer velocity profile to an appreciable extent. In fact, it is this pressure gradient which causes a separated-flow region to develop on the back side of the cylinder when the free-stream velocity is sufficiently large. 

For incompressible flow without viscous heating and for zero pressure gradient, the boundary-layer equations to be solved are the familiar ones presented in Chap. 5 when the injected fluid is the same as the free-stream fluid ... 

It is well known that for a zero pressure gradient flat plate boundary layer, the skin friction for laminar flow is given by, Cf = that at... 

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