Semi - infinite plane

Several models have been proposed to predict adhesion force—the maximum force required to pull off the surfaces. Among these, the JKR theory is one receiving the greatest attention [2], which says that for an elastic spherical body in contact with a semi-infinite plane, the adhesion force can be estimated by... 

When the molecule diffuses to or from the semi-infinite plane, the rate of diffusion (dC/dt) in direction x is governed by the first Fick s law. 

The system of equations with initial and boundary conditions formulated above allows us to find the velocity distributions and pressure drop for the filled part of the mold. In order to incorporate effects related to the movement of the stream front and the fountain effect, it is possible to use the velocity distribution obtained285 for isothermal flow of a Newtonian liquid in a semi-infinite plane channel, when the flow is initiated by a piston moving along the channel with velocity uo (it is evident that uo equals the average velocity of the liquid in the channel). An approximate quasi-stationary solution can be found. Introduction of the function v /, transforms the momentum balance equation into a biharmonic equation. Then, after some approximations, the following solution for the function jt was obtained 285... 

The diffusion was assumed to occur from a semi-infinite plane sheet with a uniform initial concentration (C ) throughout and a constant surface concentration (C,) of zero. For this case the solution to the error function equation takes the form ... 

Figure 12. Theoretical hydrogen evolution curves using diffision equation for semi-infinite plane sheet (based on erf(x) five different initial diffusible hydrogen concentrations Dtrt = 7.5 x 105 cm2/sec).

The solubility product is defined for a semi-infinite plane surface where the interfacial energy between the crystal and the solution makes a negligible contribution to the free-energy of formation of the particle. The definition also necessitates that the solid phase is homogeneous in structure and that a chemical potential may be assigned to the components irrespective of their position within the solid. When the crystals are small this may not be true because the imbalance of interionic forces at the surface produce... 

Consider diffusion with a first order reaction in a semi-infinite plane ... 

I. C. Sheng and Y. Chen, Thermoelastic Analysis for a Semi-Infinite Plane Subjected to a Moving Gaussian Heat Source, J. Thermal Stresses, 14, pp. 129-141,1991. 

The geometrical process for creating a disclination in smectic A is as follows. Cut the material by a semi-infinite plane that runs parallel to the layers, the limit of the cut being the disclination line L. Rotate the two faces... 

Planar but chemically inhomogeneous substrates are encountered in various practical application, in particular those related to open nanofluidics. One tries to understand the wetting and adsorption properties of such systems in terms of the wetting properties of separate, chemically homogeneous regions. A simple chemical step consists of two semi-infinite planes, each composed of a different material, say, (1) for x < 0 and (2) for... 

Continuous Point Source on a Semi-Infinite Plane Emitting into Half Space... 

The analysis of the diffusion data may be criticized because of the implicit assumption that the diffusion coefficients are not functions of depth into the oxide. However, because oxidation is occurring there must be a gradient in oxygen potential, with an implied gradient in oxygen vacancy concentration, so that the diffusion coefficient D may be written Z)o(l + ax), where X is the distance into the oxide. This is formally equivalent to considering diffusion into a cylinder rather than diffusion into a semi-infinite plane, and will be discussed in a later work. 

Strictly speaking, this expression is correct for a semi-infinite region bounded by a plane wall and containing a gas at rest. Here it is applied to a bounded region surrounded by a curved wall, and the molecules have a drift velocity parallel to che wall. Knudsen was concerned that this drift velocity might invalidate the treatment, but Pollard and Present [8] showed Chat this is not che case. 

Sphere—radiation to surface Infinite circular cylinder— radiation to curved surface Semi-infinite cylinder— radiation to base Cylinder of equal height and diameter — radiation to entire surface Infinite parallel planes — radiation to planes... 

FIGURE 26.49 Calculated stored elastic energy and the horizontal stress component due to two line forces at an angle a to the plane of the rubber surface and a fixed distance x apart for different depths from the surface of a semi-infinite body. 

Experimental measurements of DH in a-Si H using SIMS were first performed by Carlson and Magee (1978). A sample is grown that contains a thin layer in which a small amount (=1-3 at. %) of the bonded hydrogen is replaced with deuterium. When annealed at elevated temperatures, the deuterium diffuses into the top and bottom layers and the deuterium profile is measured using SIMS. The diffusion coefficient is obtained by subtracting the control profile from the annealed profile and fitting the concentration values to the expression, valid for diffusion from a semiinfinite source into a semi-infinite half-plane (Crank, 1956),... 

Some transient problems tend to a trivial (and useless) steady-state solution without flux and concentration profiles. For instance, concentration profiles due to limiting diffusion towards a plane in an infinite stagnant medium always keep diminishing. Spherical and disc geometries sustain steady-state under semi-infinite diffusion, and this can be practically exploited for small-scale active surfaces. 

The leading boundary conditions correspond with semi-infinite diffusion with an instantaneous sink of the diffusing species at the plane x = 0 ... 

Let us assume parallel flux in a semi-infinite medium bound by the plane x=0. Diffusion of a given element takes place from the plane x=0 kept at concentration Cint. Introducing a Boltzmann variable u with constant diffusion coefficient such as... 

If the plane source is on the surface of a semi-infinite medium, the problem is said to be a thin-film problem. The diffusion distance stays the same, but the same mass is distributed in half of the volume. Hence, the concentration must be twice that of Equation 3-45a ... 

A3.1.2. Plane source for one-dimensional semi-infinite medium (0 < x> co)... 

Unsteady transfer with Pe oo has been treated using the thin concentration boundary layer assumptions. With this approximation, the last term in Eq. (3-56) is deleted. Hence, for small t where the convection term is negligible, the transfer rate for rigid or circulating spheres is identical to that for diffusion from a plane into a semi-infinite region ... 

The factors determining the appearance of ordered cell-like motions were first investigated by Sternling and Scriven (S33) who considered the two-dimensional stability of a plane interface separating two immiscible semi-infinite fluid phases with mass transfer occurring between the phases. This system was shown to be unstable for mass transfer in one direction, but stable for transfer in the opposite direction. For an interfacial tension-lowering solute, instability... 

For example, the treatment of diffusion that is to follow is solely restricted to semi-infinite linear diffusion, i.e., diffusion that occurs in the region between x = 0 and x —> +oo, to a plane of infinite area. Thus, diffusion to a point sink—called spherical diffusion—is not treated, though it has been shown to be relevant to the particular problem of the electrolytic growth of dendritic crystals from ionic melts. 

Fig. 6.3 Nondimensional axial and radial velocity profiles for the axisymmetric stagnation flow in the semi-infinite half plane above a solid surface. The flow is approaching the surface axially (i.e., u < 0) and flowing radially outward (i.e., V > 0). The temperature profile, which is the result of solving the thermal-energy equation, is discussed in Section 6.3.6.

Fig. 6.6 Comparison of two alternative stagnation-flow configurations. The upper illustration shows the streamlines that result from a semi-infinite potential flow and the lower illustration shows streamlines that result from a uniform inlet velocity issuing through a manifold that is parallel to the stagnation plane. Both cases are for isothermal air flow at atmospheric pressure and T = 300 K. In both cases the axial inlet velocity is u = —5 cm/s. The separation between the manifold and the substrate is 3 cm. For the outer-potential-flow case, the streamlines are plotted over the same domain, but the flow itself varies in the entire half plane above the stagnation surface. The stagnation plane is illustrated as a 10 cm radius, but the solutions are for an infinite radius.

Fig. 6.13 Comparison of streamlines from rotating-disk solutions at two rotation rates. Both cases are for air flow at atmospheric pressure and T = 300 K. The induced inlet velocity is greater for the higher rotation rate. In both cases the streamlines axe separated by 27tA4< = 1.0 x 10-6 kg/s. The solutions are illustrated for a 2 cm interval above the stagnation plane and a 3 cm radius rotation plane. The similarity solutions themselves apply for the semi-infinite half plane above the surface.

Consider the two axial-velocity profiles in Fig. 17.10 that correspond to the low-strain solution in Fig. 17.9. While at the symmetry plane both solutions must have zero velocity, the inlet-velocity boundary conditions are quite different. In the finite-gap case (here the gap is 3.5 mm), the inlet velocity is specified directly (here as 250 cm/s). In the semiinfinite case, the inlet cannot be specified. Instead, the velocity gradient a = du/dz is specified, with the velocity itself growing linearly away from the surface. In the finite-gap case the strain rate is determined by evaluating the velocity gradient just ahead of the flame, where there is a region in which the velocity gradient is reasonably linear. In the semi-infinite case, the velocity gradient is specified directly, whereas in the finite-gap case it must be evaluated from the solution. 

Fig. 17.11 Extinction behavior of strained, opposed-flow, premixed, methane-air flames. The left-hand panel shows the dependence of the maximum temperature at the symmetry plane as a function of the semi-infinite strain-rate parameter a, for five different mixture stoichiometries. The right-hand panel compares measured extinction strain rates [238] with predictions for both the semi-infinite and finite-gap model formulations. The nozzle separation distance is 7 mm (i.e., 3.5 mm from nozzle to symmetry plane).

A plane-parallel stellar atmosphere is a semi-infinite medium. In the numerical calculation, we divide it into n finite elements and 1 semi-infinite element. Let us define a node as a point between two elements. Node 0 is defined as the boundary between the surface element and the vacuum. In total, we have n+1 nodes. The distribution of any physical quantity is represented by a vector of n+1 dimensions with its values at the n+1 nodes as elements. The mean intensity of radiation J is written in the ordinary expression as... 

In semi-infinite planar geometry, the electrode occupies the x = 0 plane and transport occurs perpendicularly to that plane from a limitless unimpeded medium as shown in Fig. 29. To prevent radial diffusion to the edge of the electrode, it is necessary to have walls of some kind to constrain the transport direction to be normal to the electrode. Because of this requirement, electrodes with precise semi-infinite planar geometry are difficult to fabricate and are, in fact, rather rare in practice. Nevertheless, because theoretical derivations are simplest for this geometry and because many practical geometries closely approximate the semi-infinite planar one as a limiting case, the geometry of this section is of paramount importance. 

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