Boundary layer momentum

Here, the dimensionless thickness of the momentum boundary layer A, and the dimensionless quantity, a, relating to the direction of local flow are functions of 0 they are shown graphically in Fig. 3. The average Sherwood number may be obtained by integrating Eq. (51) over the surface the result for a hemisphere may be given as [22] ... 

For kinetic disequilibrium partitioning of trace elements, equation (9.6.6) after Burton et al. (1953) is commonly presented as an alternative to equation (9.6.5) due to Tiller et al. (1953) (e.g., Magaritz and Hofmann, 1978 Lasaga, 1981 Walker and Agee, 1989 Shimizu, 1981). However, the relative values of viscosity and chemical diffusivity in common liquids and silicate melts make the momentum boundary-layer (i.e., the liquid film which sticks to the solid) orders of magnitude thicker than the chemical boundary layer. It is therefore quite unlikely that, except for rare cases of transient state, liquid from outside the momentum boundary-layer may encroach on the chemical boundary-layer, i.e., <5 may actually be taken as infinite. As a simple description of steady-state disequilibrium fractionation, the model of Tiller et al. (1953) has a much better physical rationale. A more elaborate discussion of these processes may be found in Tiller (1991a, b). 

EXAMPLE 4.4 Development of a momentum boundary layer over a solid surface (Blasius solution)... 

Momentum boundary layer calculations are useful to estimate the skin friction on a number of objects, such as on a ship hull, airplane fuselage and wings, a water surface, and a terrestrial surface. Once we know the boundary layer thickness, occurring where the velocity is 99% of the free-stream velocity, skin friction coefficient and the skin friction drag on the solid surface can be calculated. Estimate the laminar boundary layer thickness of a 1-m-long, thin flat plate moving through a calm atmosphere at 20 m/s. 

The gas film coefficient is dependent on turbulence in the boundary layer over the water body. Table 4.1 provides Schmidt and Prandtl numbers for air and water. In water, Schmidt and Prandtl numbers on the order of 1,000 and 10, respectively, results in the entire concentration boundary layer being inside of the laminar sublayer of the momentum boundary layer. In air, both the Schmidt and Prandtl numbers are on the order of 1. This means that the analogy between momentum, heat, and mass transport is more precise for air than for water, and the techniques apphed to determine momentum transport away from an interface may be more applicable to heat and mass transport in air than they are to the liquid side of the interface. 

The entry-length region is characterized by a diffusive process wherein the flow must adjust to the zero-velocity no-slip condition on the wall. A momentum boundary layer grows out from the wall, with velocities near the wall being retarded relative to the uniform inlet velocity and velocities near the centerline being accelerated to maintain mass continuity. In steady state, this behavior is described by the coupled effects of the mass continuity and axial momentum equations. For a constant-viscosity fluid,... 

Figure 4.14 illustrates the transient solution to a problem in which an inner shaft suddenly begins to rotate with angular speed 2. The fluid is initially at rest, and the outer wall is fixed. Clearly, a momentum boundary layer diffuses outward from the rotating shaft toward the outer wall. In this problem there is a steady-state solution as indicated by the profile at t = oo. The curvature in the steady-state velocity profile is a function of gap thickness, or the parameter rj/Ar. As the gap becomes thinner relative to the shaft diameter, the profile becomes more linear. This is because the geometry tends toward a planar situation. 

The momentum boundary layer thickness is represented by 8, and the thermal boundary layer thickness is represented by 8,. 

To have a better control on the stability of the explicit method by monitoring a single criterion, the second term in the x-momentum boundary layer equation can be depicted as... 

Por the same initial conditions, one can expect the laminar thermal and momentum boundary layers on a fiat plate to have the same thickness when the Prandtl number of the flowing fluid is... 

The influence of a wall on the turbulent transport of scalar (species or enthalpy) at the wall can also be modeled using the wall function approach, similar to that described earlier for modeling momentum transport at the wall. It must be noted that the thermal or mass transfer boundary layer will, in general, be of different thickness than the momentum boundary layer and may change from fluid to fluid. For example, the thermal boundary layer of a high Prandtl number fluid (e.g. oil) is much less than its momentum boundary layer. The wall functions for the enthalpy equations in the form of temperature T can be written as ... 

In this context the momentum boundary layer thickness y = d is conveniently defined as the point beyond which the velocity takes on its free stream value The second condition stating that the velocity gradient vanishes a,t y = S, ensures that we obtain a continuous gradient as the velocity attains its free stream value. 

It has been shown that there exists a continuous change in the physical behavior of the turbulent momentum boundary layer with the distance from the wall. The turbulent boundary layer is normally divided into several regions and sub-layers. It is noted that the most important region for heat and mass transfer is the inner region of the boundary layer, since it constitutes the major part of the resistance to the transfer rates. This inner region determines approximately 10 — 20% of the total boundary layer thickness, and the velocity distribution in this region follows simple relationships expressed in the inner variables as defined in sect 1.3.4. 

The conclusion from the previous paragraph is that similarity solutions of the momentum boundary-layer equations should not generally be expected. An interesting question is whether similarity solutions can be obtained in any case other than the flat plate problem in the previous section. To answer this question, we start with the boundary-layer equations in their most general form ... 

By now the reader may have come to associate the momentum boundary layer as a thin region of ()(Re l/2) adjacent to a body surface across which the dimensionless tangential velocity changes by 0(1) in order to satisfy the no-slip condition at the body surface. To be sure, many boundary layers do exhibit this structure, but it is overly restrictive as a general description. A more accurate and generally applicable description is that the boundary layer is a thin region of ()(Re l/2) adjacent to a body surface inside of which the vorticity generated at the body surface is confined. 

Although the dependence of the thermal boundary-layer thickness on the independent parameters Re and Pr (or Pe) remains to be determined, we may anticipate that the magnitudes of Re and Pe will determine the relative dimensions of the two boundary layers. If Pe yp Re yp 1, both the momentum and thermal layers will be thin, but it seems likely that the thermal layer will be much the thinner of the two. Likewise, if Pe Pe 1, we can guess that the momentum boundary layer will be thinner than the thermal layer. In the analysis that follows in later sections of this chapter, we consider both of the asymptotic limits Pr —> oc (Pe yy Re y> D and Pr 0 (Re yy> Pe p> 1). We shall see that the relative dimensions of the thermal and momentum layers, previously anticipated on purely heuristic grounds, will play an important and natural role in the theory. 

The conditions (1 l-7a) and (1 l-7c) correspond to the assumption that the surface of the body is heated (or cooled) to a constant temperature T0 beyond a certain position denoted as x. Upstream of x, the body is not heated. Hence, at steady state in the present boundary layer limit, both the body surface and the fluid remain at the ambient temperature T,x, for x < x. If the body is heated over its whole surface, then x = 0. In this case, the leading edges of the thermal and momentum boundary layers are coincident. 

In particular, at leading order in Re 1, the matching condition for the momentum boundary-layer solution,... 

For all Re, Pr ()( ), the value of the number on the right-hand side of (11-10) is also 0(1). Here, however, we consider the limit Re >> 1. Hence, applying the momentum boundary-layer scaling (11-5) to the definition (11-9), we see that... 

In the high-T3 case, we obtain the leading-order term in an asymptotic expansion, for the part of the domain where the momentum boundary-layer scaling is applicable, by taking the limit Pr —oo in (11-6). The result is... 

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