Hk Hendrik Kuiken

The effect of normal blowing on the flow near a rotating disk of infinite extent

The effect of blowing through a porous rotating disk on the flow induced by this disk is studied. For strong blowing the flow is almost wholly inviscid. First-order viscous effects are encountered only in a thin layer at some distance from the disk. The results of an asymptotic analysis are compared with numerical integrations of the full equations and complete agreement is found.

Viscous sintering : the surface-tension-driven flow of a liquid form under the influence of curvature gradients at its surface

A boundary-element method is applied to solve the equations describing the deformation of a two-dimensional liquid region under the influence of gradients of the curvature of its outer boundary. This research is motivated by a desire to obtain a better understanding of viscous sintering processes in which a granular compact is heated to a temperature at which the viscosity of the constituent material becomes low enough for surface tension to cause adjacent particles to deform and coalesce. The boundary-element method is capable of showing how a moderately curved initial shape transforms itself into a circle. Initial shapes showing more extreme curvature gradients, which are relevant in the initial stages of a sintering process, cannot be dealt with by the boundary-element method in its present form. The numerical solution of the continuous model shows a tendency to create oscillations in the outer boundary of the liquid region. On the other hand, an analytical small-amplitude analysis shows that rapid oscillations vanish exponentially fast.

Evaluation of diffusion length and surface‐recombination velocity from a planar‐collector‐geometry electron‐beam‐induced current scan

For performing electron‐beam‐induced current (EBIC) measurements on sufficiently large samples, the use of a ‘‘planar‐collector geometry’’ (i.e., with the collector covering part of the irradiated surface itself) is very attractive. However, the pertinent theoretical EBIC curves for finite surface‐recombination velocities s have so far been lacking. This paper presents the complete theoretical expressions for arbitrary values of s and diffusion length L. Simple asymptotic solutions are given for point‐ and finite‐size generation sources. Easy methods are developed to facilitate the application of these solutions in the practical evaluation of L and s from experimental EBIC curves. These methods are applied to experimental data available through the literature.

Etching: a two-dimensional mathematical approach

A mathematical model for the etching of a semi-infinite active surface is presented. The model assumes that the transport of the active species occurs solely by diffusion. It is shown that, when the diffusion field propagates much faster than the etched surface, the problem can be solved by a singular perturbation technique, which distinguishes a near field in the area where the moving surface and the non-etchable mask meet, and a far field where edge effects may be disregarded to first order. The leading terms of a composite expansion are given, from which the shape of the moving boundary can be determined at all times.

A diffusion problem in semiconductor technology

SummaryThis paper deals with a mathematical model of a SEM-EBIC experiment devised to measure the diffusion length of semiconductor materials. In the model the semiconductor material occupies a half-space, of which the plane bounding surface is partly covered by a semi-infinite current-collecting junction, the Schottky diode. A scanning electron microscope (SEM) is used to inject minority carriers into the material. It is assumed that injection occurs at a single point only. The injected minority carriers diffuse through the material and recombine in the bulk at a rate proportional to their local concentration. Recombination also occurs at the free surface of the material, not covered by the junction, where its rate is determined by the surface recombination velocity v. The mathematical model gives rise to a mixed-boundary-value problem for the diffusion equation, which is solved by means of the Wiener-Hopf technique. An analytical expression is derived for the measurable electron-beam-induced current (EBIC) caused by the minority carriers reaching the junction. The solution obtained is valid for all values of v, and special attention is given to the limiting cases v=∞ and v=0. Asymptotic expansions of the induced current, which are usable when the injection point is more than a few diffusion lengths away from the edge of the junction, are derived as well.

Axisymmetric free convection boundary-layer flow past slender bodies

Radial curvature effects on axisymmetric free convection boundary-layer flow are investigated for vertical cylinders and cones for some special non-uniform temperature differences between the surface and the ambient fluid. The solution is given as a power series expansion, the first term being equal to the solution to be found when no transverse curvature is involved. For a variety of Prandtl numbers numerical integrations have been carried out. An interesting application is the determination of the surface temperature of a free-convection-cooled cylinder with constant heat flux through the surface. For the Prandtl numbers considered, it turns out that the same heat flux induces a lower surface temperature on a cylinder than on a flat plate.

Analysis of the temperature distribution in FZ silicon crystals

An analytical solution for the temperature distribution in freely radiating crystals has been developed. It was possible to solve this problem, within the growing crystal, because the heat transfer due to radiation was small as compared with the heat transfer due to conduction. The solution was applied to the growth of silicon crystals grown by the floating-zone method and the results were found to be in good agreement with experimental data. It is shown that the influence of the crystal growth rate on the temperature profile must be included, when it is higher than 1–2 mm/min. The longitudinal temperature gradient at a concave solid- liquid interface was found to vary in a characteristic manner with the distance from the axis. A similar variation was observed for the height of the defect area formed at the solid-liquid interface in quenched silicon crystals.

The cooling of a low-heat-resistance sheet moving through a fluid

In this paper we consider the cooling of a flat sheet moving through a semiinfinite expanse of viscous fluid. The heat resistance of the sheet is assumed to be so small that the temperature can be considered uniform across the sheet. Precise conditions for this assumption to be valid are derived. The problem is solved first by means of a coordinate expansion, which can be proved to converge for all values of the expansion variable. Since this series cannot be used numerically downstream, an alternative series expansion, which applies downstream, is also derived.Special sections are devoted to deriving solutions valid for small or large values of the Prandtl number. Finally, expressions are obtained for the Nusselt number and the cooling length. It is found that cooling is determined by the smaller of two diffusivities, namely, the kinematic viscosity and the thermal diffusivity.

Solidification of a liquid on a moving sheet

This paper considers the growing of a solid layer on a sheet that moves through a liquid and which is kept at a temperature below freezing. The convection in the liquid is fully taken into account. It is found that the thickness of the layer is proportional to the square root of the distance from the point where the sheet enters the body of liquid. The main difficulty lies in determining the factor of proportionality in this relationship. Asymptotic expressions are derived for this factor in the case where latent heat is much greater than sensible heat. Also presented are approximate solutions valid for very small (liquid metals) and very large (polymers) values of the Prandtl number.

Etching through a slit

The problem of etching through a slit-shaped orifice is formulated in mathematical terms. Although a full solution to this problem can be obtained only by the application of numerical methods, it is shown that certain aspects of it can be dealt with by analytical means. Asymptotic solutions that are valid in certain parts of the time domain are derived. Early, intermediate and late asymptotes are considered. These analytical results are compared with numerical ones, and a satisfactory agreement is found, emphasizing the validity of the two approaches.