Boundary layer approach

Johnson DA and Amidon GL (1988) Determination of Intrinsic Membrane Transport Parameters from Perfused Intestine Experiments a Boundary Layer Approach to Estimating the Aqueous and Unbiased Membrane Permeabilities. J Theor Biol 131 pp 93-106. 

The form of Eq. (5-25) was suggested by noting that the first order curvature corrections to Eqs. (3-47) and (5-35) are near unity and by matching the expression to the creeping flow result, Eq. (3-49), at Re = E Equation (5-25) also represents the results of the application of the thin concentration boundary layer approach (Sc oo) through Eq. (3-46), using numerically calculated surface vorticities. Thus the Schmidt number dependence is reliable for any Sc > 0.25. 

Attempts to obtain theoretical solutions for deformed bubbles and drops are limited, while no numerical solutions have been reported. A simplifying assumption adopted is that the bubble or drop is perfectly spheroidal. SalTman (SI) considered flow at the front of a spheroidal bubble in spiral or zig-zag motion. Results are in fair agreement with experiment. Harper (H4) tabulated energy dissipation values for potential flow past a true spheroid. Moore (Mil) applied a boundary layer approach to a spheroidal bubble analogous to that for spherical bubbles described in Chapter 5. The interface is again assumed to be completely free of contaminants. The drag is given by... 

Consequently, the boundary layer approach suggests that the drag coefficient and the heat transfer coefficient should indeed become independent of the velocity at sufficiently high velocities. 

Kinetic equations for reversible adsorption and reversible coagulation are established when the interaction potential has primary and secondary minima of comparable depths. The process is assumed to occur in two successive steps. First the particles move from the bulk of the fluid to the secondary minimum. A fraction of the particles which have arrived al the secondary minimum move further to the primary minimum. Quasi-steady state is assumed for each of the steps separately. Conditions are identified under which rates of reversible adsorption or coagulation at the primary minimum can be computed by neglecting the rate of accumulation at the secondary minimum. The interaction force boundary layer approach has been improved by introducing the tangential velocity of the particles near the surface of the collector into the kinetic equations. To account for reversibility a short-range repulsion term is included in the interaction potential. 

In the interaction force boundary layer approach the tangential velocity of the particles in the layer is neglected. Because the center of sufficiently large particles is far from the wall (even when the minimum distance h between wall and particle is small) this velocity may be appreciable. A second goal of the paper is to account for this effect. 

The boundary layer approach is used to investigate the mode I plane strain fields near the crack. The symmetry of the problem allows consideration of only half the geometry (see Fig. 7), which consists of an initial blunted crack of radius n with traction-free surfaces along the crack. Along the boundary of the remote region at a distance R with R % 200rt, the mode I elastic field at stress intensity factor Ki is prescribed [8,22],... 

The coordinate transformation makes all the velocities coincide. The boundary layer approaches the core flow asymptotically and in principle stretches into infinity. The deviation of the velocity wx from that of the core flow is, however, negligibly small at a finite distance from the wall. Therefore the boundary layer thickness can be defined as the distance from the wall at which wx/wx is slightly different from one. As an example, if we choose the value of 0.99 for Wj/uico, the numerical calculation yields that this value will be reached at the point r]+ m 4.910. 

The problems considered so far were one-dimensional in essence. Here, a two-dimensional problem will be considered. It should be also noted that no reduction of any term of the complete Navier—Stokes equations was made yet. This means that no groundless assumptions were employed apart from the representation of EPR by a distributed force. In contrast, the Navier—Stokes equations reduced to the boundary-layer approach (1.4), (1.5) were employed by meteorologists. 

The boundary-layer approach neglects some terms of the momentum equations (3.29) and yields two simplified equations instead of three ... 

Integral relations are common for the boundary layer approach and express the fundamental conservation laws for momentum, heat, and mass. Similar laws hold in easily penetrable roughness boundary layers. 

The previous problem made use of the Prandtl s boundary layer approach. Although it is widely applied, it would be worth to examine this approximation on a test problem, where the complete Navier-Stokes equations should be solved. Here is such a test problem that has also its own significance. 

A number of numerical calculations was carried out for various Reynolds numbers up to 2000, and the duct s initial region length was found as a graphical function Lx = f(Re h, A) represented in Fig. 3.11. The dependence Lx vs A significantly differs for small and big Reynolds numbers Re. The value A = 0 corresponds to the case where the EPR is absent. It can be seen that, for A -> 0, all the curves arrive at the constant value c 0.03, but the curves associated with small Reynolds numbers 1, 5, and 10. Hence, it can be concluded that the principal Schlichting s formula (3.45) has been justified for large Re despite it was derived from the boundary-layer approach, rather than from the complete Navier - Stokes equations. 

In the simplest case, all the droplets are of the same size and the droplet canopy affects the wind flow like an easily penetrable roughness mathematically expressed by the conjugation problem (3.33)—(3.35). The boundary layer approach is thus accepted. The distributed mass force / should depend, however, not on the local velocity P of the carried medium alone, but on the relative velocity between the two media V - T. To get /, the individual force (1.14) should be multiplied by the concentration of droplets n. 

Consider now the boundary layer flow over the surface with a droplet EPR of the height h over it Fig. 3.8 illustrates the physical situation. Both the carying and carried media are governed by the system of equations (3.74) determining the two-dimensional fields U(x,z), V(x,z) and u(x,z) in the boundary-layer approach (with r(x,z) = P vt1U )-The non-slip velocity on the surface z = 0, and the asymptotics lim U = along... 

There have been fewer studies in electrochemistry where the flow is known but the boundary-layer approach is inapplicable. One example has been recently analyzed and compared with experiment. In this case, mass transfer to a line electrode or an array of line electrodes in the presence of an oscillatory shear flow was treated. A finite-volume approach was used for the numerical analysis and a ferri/ferrocyanide redox couple was used to measure the mass-transfer rate. The studies show that boundary-... 

DAL generates surface forces which can be naturally named non-equilibrium forces since they arise due to a deviation of the adsorption layer from equilibrium. The effect of non-equilibrium surface forces on the dynamics of these layers is substantially different to that of equilibrium ones. In many cases, the radius of their actions is much greater than the radius of action of equilibrium surface forces since they are localised within the diffusion boundary layer. Approaching a surface, particles pass first of all the diffusion layer, so that in many cases the possibility of coagulation is determined by the action of non-equilibrium surface forces. In other situations it is connected with the action of equilibrium surface forces while nonequilibrium forces influence the rate of the process. 

The topic of the boundary layer is introduced in Sec. 10.1. The topic is a very large one. It is an active field of fluid mechanics research, so new results are constantly being published. Here we cannot hope to cover the entire topic rather, we intend to show by a few examples what types of solution are obtainable and to impart some feeling for the results of the boundary-layer approach. More terminology is introduced than is necessary for the subject actually treated. This terminology is in common use in the boundary-layer literature it is introduced here to show the student how these common terms relate to the other subjects treated in this book. 

Johnson DA, Amidon GL. Determination of intrinsic transport parameters from perfused intestine experiments a boundary layer approach to estimating the aqueous and unbiased membrane permeability. J Theor Biol 1988 131 93-106. 

Figure 7-30. Schematic of two approaches to the ac-ti ve-to-passi ve oxidation transition (a) Wagner boundary layer approach and (b) Si02 smoke formation. (Adapted from Narushima et al., 1997.)...

J. Goldsmith, H. Guo, S.N. Hunt, Mingjiang Tao, and S. Koehler. Drag on intruders in granular beds A boundary layer approach. Physical Review E, 88 030201, September 2013. 

By choosing a time step for the transient solid temperature equation long enough for gas-phase equilibration, the discretized, time-independent set of the Navier-Stokes equations under the boundary layer approximation can be solved for the flow field inside the catalytic channels [12] (quasisteady assumption for the gas-phase) using the CRESLAF package [13]. The applicability of the boundary layer approach in catalytic combustion at sufficiently large Reynolds numbers (Re > 20) has already been demonstrated [14]. The simplified equations thus become ... 

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