Boundary layers local coordinates

Such expressions can be extended to permit the evaluation of the distribution of concentration throughout laminar flows. Variations in concentration at constant temperature often result in significant variation in viscosity as a function of position in the stream. Thus it is necessary to solve the basic expressions for viscous flow (LI) and to determine the velocity as a function of the spatial coordinates of the system. In the case of small variation in concentration throughout the system it is often convenient and satisfactory to neglect the effect of material transport upon the molecular properties of the phase. Under these circumstances the analysis of boundary layer as reviewed by Schlichting (S4) can be used to evaluate the velocity as a function of position in nonuniform boundary flows. Such analyses permit the determination of material transport from spheres, cylinders, and other objects where the local flow is nonuniform. In such situations it is not practical at the present state of knowledge to take into account the influence of variation in the level of turbulence in the main stream. 

Damping by wall friction may be addressed on the basis of the theory of oscillatory boundary layers [35], [36]. Under conditions in real motors, the oscillatory boundary layer at the wall is much thinner than the mean-flow boundary layer [37], and the tangential acoustic velocities just outside the oscillatory boundary layer are (TJT ) times those outside the mean-flow boundary layer, where is the wall temperature and the gas temperature in the chamber. Dissipation in the oscillatory boundary layer may be analyzed by considering a flat element of the surface exposed to the complex amplitude V(T JT,) of velocity oscillation parallel to the surface, where equation (24) has been employed for the acoustic field in the chamber, with the coordinate system locally aligned in the direction of velocity oscillation. Neglecting locally the effects of density changes and the spatial variations of acoustic amplitudes, we write the combination of equations (1-2) and (1-5) for the velocity v parallel to the wall as... 

One interesting feature is that the operator V20, which is expressed on the left-hand side of (9-222) in terms of rescaled spherical coordinate variables, takes a form in the limiting approximation (9-225) that appears to be just the normal derivative term in V20 for a Cartesian coordinate system. In fact, we shall see that boundary-layer equations always can be expressed in terms of a local Cartesian coordinate system, with one variable normal to the body surface at each point (Y in this case) and the others tangent to it. This reduction of the equations in the boundary layer to a local, Cartesian form is due to the fact that the dimension of the boundary layer is so small relative to that of the body that surface curvature effects play no role. 

Because this inner boundary-layer region is infinitesimal in thickness relative to a, all curvature terms that appear when the equations of motion are expressed in curvilinear coordinates will drop out to first order in Re thus leaving boundary-layer equations that are effectively expressed in terms of a local Cartesian coordinate system. 

First, however, it is important to recognize that the form of equations (10 28), (10-30), and (10 32) is independent of the geometry of the body (i.e., independent of the cross-sectional shape for any 2D body). Although we started our analysis with the specific problem of flow past a circular cylinder, and thus with the equations of motion in cylindrical coordinates, the equations for the leading-order approximation in the inner (boundary-layer) region reduce to a local, Cartesian form with Y being normal to the body surface and x... 

Adopting the local Cartesian coordinates appropriate to the boundary-layer equations, we can express the result as... 

Adsorption energies on metals calculated in a cluster approach often show considerable oscillations with size and shape of the cluster models because such (finite) clusters describe the surface electronic structure insufficiently [257-260]. These models may yield rather different results for the Pauli repulsion between adsorbate and substrate, depending on whether pertinent cluster orbitals localized at the adsorption site are occupied or empty. The discrete density of states is an inherent feature of clusters that may prevent a correct description of the polarizability of a metal surface and thus hinders cluster size convergence of adsorption energies [257]. Even embedding of metal clusters does not offer an easy way out of this dilemma [260,261]. Anyway, the form of conventional moderately large cluster models may be particularly crucial. Such models are inherently two-dimensional with substrate atoms from two or three crystal layers usually taken into accormt thus, a large fraction of atoms at the cluster boundaries lacks proper coordination. 

Thin-Layer Approximation. Laminar analyses often make the further approximation that the boundary layer is so thin that when the simplified equations of motion are rewritten in terms of local surface coordinates, i.e., in terms of the x and y of Fig. 4.3a, several terms normally associated with curvature effects can be dropped. The Nusselt number equation, based on solutions to such laminar thin-layer equation sets, always takes the form... 

Locally Flat Description of Boundary Layer Mass Transfer in Spherical Coordinates. Since the radial coordinate r does not change much as one moves from the fluid-solid interface to the outer edge of the mass transfer boundary layer, it is acceptable to replace r by in the two-dimensional mass transfer equation and the equation of continuity ... 

The tangential component of the dimensionless equation of motion is written explicitly for steady-state two-dimensional flow in rectangular coordinates. This locally flat description is valid for laminar flow around a solid sphere because it is only necessary to consider momentum transport within a thin mass transfer boundary layer at sufficiently large Schmidt numbers. The polar velocity component Vo is written as Vx parallel to the solid-liquid interface, and the x direction accounts for arc length (i.e., x = R9). The radial velocity component Vr is written... 

Of the main interest is the velocity distribution near the sphere in the diffusion boundary layer. Introduce local system of coordinates y,9), where the y-axis is perpendicular and 6 is tangential to the corresponding area element of the surface. Then r/a = 1 + y/a. Considering the case y/a 1, expand (10.69) as a power series in y/a. In a result, we obtain... 

V along the local coordinates i and n, respectively (the coordinate s is along the surface). Then the boundary layer equation becomes... 

To construct the boundary layer functions we need to introduce new (local) coordinates in the vicinity of the boundary 3ft. Let the parametric... 

Here u k>Hk are the regular terms of the asymptotics, 0 and 0 are transition layer functions, and 11. are boundary functions. The variables p, are defined in an analogous way to that in Section VI.B. The variables t, 6 are introduced as follows. First we introduce new (local) coordinates (r, 0) in the vicinity of the curve F similarly to the introduction of local coordinates (r, ) in the vicinity of dft (see Section VI.B) 0 plays the same role as (we assume that 0 0 Itt), r is the distance from given point M to the curve Fq along the normal to Fq with a plus sign if M E ft, and with a minus sign if Af E ft. ... 

Our aim in this section is to show how it is possible to calculate the defect concentrations in botmdary layers as a function of the control parameters temperature, component activity and doping content, the local coordinate and the materials parameters of both bulk and interface core. Thus, we are going to extend our previous defect chemical analysis to boundary layers, whereby we will, in general, consider a mixed conductor with low defect concentrations [244]. 

We will use in the calculations l = 2.76 A (which corresponds to the distance between molecules in the structure ofice I, as compared to about 2.9 Afor molecules in water), and e = 80. For the local dielectric constant, we will assume e" = 1, which constitutes a lower bound. In a perfect tetrahedral coordination, the average distance between two successive water layers is A = (4/3)1, and the decay length of the hydration interaction calculated using eq 37 is X = 2.96 A. It should be noted that the latter value is in the range determined experimentally for the hydration force between phospholipid bilayers.4 Lower values ofX can be obtained for higher e". For the distance between the center of the ion pair and the interface (located at the boundary of the first organized water layer), the value A = 1.0 A was selected. 

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