Near-wall gas-droplet flows with liquid-film formation

A self-consistent asymptotic theory of a near-wall aerodisperse flow with the formation of a liquid film due to inertial deposition of liquid droplets on the body surface is constructed. With the reference to the examples of low-speed mist flows near plane and curvilinear solid surfaces, asymptotically different situations are analyzed. The two-fluid equations of the mist flow are solved simultaneously with the equations of the liquid film, on the outer edge of which the distributed mass, momentum, and energy fluxes (found from the solution of the outer problem) are specified. It is shown that depending on the values of governing parameters the flow in the film should be described by either boundary-layer or creeping-flow equations in the layer with a priori unknown thickness. For the film on the frontal surface of a blunt body, the film thickness and the friction and heat transfer coefficients are found numerically. The conditions in which the presence of the film significantly reduces the heat fluxes are determined and the possibility of the realization of steady-state flow regimes in the film is demonstrated. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Magnetohydrodynamic Boundary Layers in Czochralski Crystal Growth

The boundary-layer flow close to the crystal-melt interface,and at the free surface of the melt, in Czochralski crystalgrowth in the presence of an axial magnetic field, is examined.Particular attention is devoted to the effective segregationcoefficient and to the inhibiting effect of the magnetic fieldupon the forced convective radial flow.     

复合材料对称层合板单向拉伸与面内剪切下的三维应力分析

本文给出了单向拉伸与面向剪切载荷下复合材料对称层合板中心区域的应力和应变沿板厚的数值计算分布规律。计算结果表明，在斜交对称辅层的层合板中心区域层间界面附近存在着层间边界层效应。层间界面处纤维走向的突变导致局部的三维应力状态和很强的应力集中。     

Influence of the bottom undulation on surface waves in thin film flow

We study gravity-driven viscous thin films flowing down an undulated plane. Applying the integral boundary-layer method we derive a set of two coupled PDEs for the film thickness and the flow rate. The steady state solution shows linear and nonlinear resonance. Based on this analytical solution we carry out a stability analysis with respect to surface waves and study wave generation and annihilation for time dependent flow. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Natural convective boundary-layer flow in a heat generating porous medium with a prescribed wall heat flux

The free convection boundary-layer flow on a vertical surface in a porous medium with local heat generation proportional to (T – T_∞)^p, where T is the local temperature and T_∞ is the ambient temperature, is considered when the surface is thermally insulated. The way in which the flow develops from the leading edge is seen to depend critically on the exponent p. For p ≤ 2 there is a boundary-layer flow for all x > 0, where x measures distance from the leading edge, with the internal heating having a significant effect at large x. For p ≥ 5 there is also a boundary-layer flow to large x but now the internal heating has an increasingly weaker effect as x increases. For 2 < p <  5 the boundary-layer solution breaks down at a finite x, with a singularity developing leading to thermal runaway at a finite distance along the surface.     

Natural convective boundary-layer flow in a heat generating porous medium with a prescribed wall heat flux

The free convection boundary-layer flow on a vertical surface in a porous medium with local heat generation proportional to (T – T_∞)^p, where T is the local temperature and T_∞ is the ambient temperature, is considered when the surface is thermally insulated. The way in which the flow develops from the leading edge is seen to depend critically on the exponent p. For p ≤ 2 there is a boundary-layer flow for all x > 0, where x measures distance from the leading edge, with the internal heating having a significant effect at large x. For p ≥ 5 there is also a boundary-layer flow to large x but now the internal heating has an increasingly weaker effect as x increases. For 2 < p <  5 the boundary-layer solution breaks down at a finite x, with a singularity developing leading to thermal runaway at a finite distance along the surface.     

Thin film limit and film rupture of the visco--capillary gravity--driven channel flow

Starting from an exact solution of a visco-capillary gravity--driven film flow of a Newtonian fluid in an inclined channel, we discuss the special case of thin films. The shape of the free surface, the velocity field and the flow rate are obtained from a pure analytical treatment. Making use of an adequate rescaling with a generalized capillary length we pay special attention to vanishingly thin films where capillary effects become dominant. Our investigations deliver a necessary condition for the flow rate in order to avoid a film rupture.     

Unsteady flow and heat transfer in a thin film of Ostwald–de Waele liquid over a stretching surface

In this paper, the effects of viscous dissipation and the temperature-dependent thermal conductivity on an unsteady flow and heat transfer in a thin liquid film of a non-Newtonian Ostwald–de Waele fluid over a horizontal porous stretching surface is studied. Using a similarity transformation, the time-dependent boundary-layer equations are reduced to a set of non-linear ordinary differential equations. The resulting five parameter problem is solved by the Keller–Box method. The effects of the unsteady parameter on the film thickness are explored numerically for different values of the power-law index parameter and the injection parameter. Numerical results for the velocity, the temperature, the skin friction and the wall-temperature gradient are presented through graphs and tables for different values of the pertinent parameter. One of the important findings of the study is that the film thickness increases with an increase in the power-law index parameter (as well as the injection parameter). Quite the opposite is true with the unsteady parameter. Furthermore, the wall-temperature gradient decreases with an increase in the Eckert number or the variable thermal conductivity parameter. Furthermore, the surface temperature of a shear thinning fluid is larger compared to the Newtonian and shear thickening fluids. The results obtained reveal many interesting behaviors that warrant further study of the equations related to non-Newtonian fluid phenomena, especially the shear-thinning phenomena.     

Parameter-dependent systems on the half-line and free convection boundary-layers in porous media

In this paper we use a recently elaborated abstract method, for parameter-dependent ODE systems over the interval (0, ∞), to obtain existence results for the problem of self-similar solutions in boundary-layer free convection in porous media. Using a generalization of the method to exponentially decaying solutions, we are able to recover some known results, and to obtain a new branch of solutions in the case of the so-called backward boundary-layer. The arguments involve the derivation of suitable a priori estimates for the solutions of the problem.     

The effect of shock-wave / boundary-layer interaction on turbulent spot propagation

We performed highly resolved large-eddy simulations (LES) of transitional shock-wave/boundary-layer interactions (SW/ BLI) in which a turbulent spot passes through a laminar shock-induced separation bubble. The initial condition consists of a laminar boundary-layer solution over a flat plate with a superimposed oblique shock which induces a separation bubble on the plate. An upstream-positioned initial disturbance triggers the turbulent spot that develops and encounters the SW/BLI region. Unlike the laminar boundary layer it does not separate but tunnels the SW/BLI region. Compared to a simulation without the SW/BLI region the spot growth is increased significantly during the passage. This finding supports the results of previous direct numerical simulations (DNS) in the literature. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Breakup of finite fluid films

We study the dewetting process of thin fluid films that partially wet a solid surface. Using long wave (lubrication) approximation, we formulate a nonlinear partial differential equation governing the evolution of the film thickness, h. This equation includes the effects of capillarity, gravity, and additional conjoining/disjoining pressure term to account for intermolecular forces. We perform standard linear stability analysis of an infinite flat film, and identify the corresponding stable, unstable and metastable regions. Within this framework, we analyze the evolution of a semi-infinite film of length L in one direction. The numerical simulations show that for long and thin films, the dewetting fronts of the film generate a pearling process involving successive formation of ridges at the film ends and consecutive pinch-off behind these ridges. On the other hand, for shorter and thicker films, the evolution ends up by forming a single drop. The time evolution as well as the final drops pattern shows a competition between the dewetting mechanisms caused by nucleation and by free surface instability. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the thread problem for minimal surfaces

For a given one-dimensional fixed boundary $\Gamma$ in and a given constant we consider any one-dimensional free boundary $F$ in subject to the conditions that the length of is equal to , that and form a closed boundary, and that the minimal surface of dimension two being bounded by and minimizes the area among all comparison surfaces being bounded by and some with length equal to . This variational problem is known as the thread problem for minimal surfaces and stems from soap film experiments, in which the fixed boundary parts are pieces of wires and the free boundary parts are threads. The new result of this article will be that has no singular points in , provided the admissible surfaces and boundary parts are supposed to be rectifiable flat chains modulo two. Received February 16, 1995 / Accepted October 20, 1995     

Maximization of the lift/drag ratio of airfoils with a turbulent boundary layer: Sharp estimates,approximation, and numerical solutions

The lift/drag ratio of an airfoil placed in an incompressible attached flow is maximized taking into account the viscosity in the boundary-layer approximation. An exact solution is constructed. The situation when the resulting solutions are not in the admissible class of univalent flows is discussed. A procedure is proposed for determining physically feasible airfoils (with a univalent flow region) with a high lift/drag ratio. For this purpose, a class of airfoils is constructed that are determined by a twoparameter function approximating the found exact solution to the variational problem. For this class, the ranges of free parameters leading to physically feasible flows are found. The results are verified by computing a turbulent boundary layer using Eppler’s method, and airfoils with a high lift/drag ratio in an attached flow are detected.     

On one model problem for the reaction–diffusion–advection equation

The asymptotic behavior of the solution with boundary layers in the time-independent mathematical model of reaction–diffusion–advection arising when describing the distribution of greenhouse gases in the surface atmospheric layer is studied. On the basis of the asymptotic method of differential inequalities, the existence of a boundary-layer solution and its asymptotic Lyapunov stability as a steady-state solution of the corresponding parabolic problem is proven. One of the results of this work is the determination of the local domain of the attraction of a boundary-layer solution.     

Free convection above a heated horizontal circular disk

The free convection boundary layer flow above a heated horizontal disk is considered. The equations of motion are solved numerically starting at the circumference of the disk where the flow is basically the same as that above a flat plate. The importance of the curvature effects increases as the centre is approached. It is shown that near the centre, the boundary-layer thickness is very large, and that the flow splits up into two distinct regions. There is a thin viscous region next to the disk of thickness ofO(r ^2/3), wherer measures distance from the centre and a thick outer inviscid region of thickness ofO(r ^?2/3).     

Computational treatment of free convection effects on perfectly conducting viscoelastic fluid

We introduced a magnetohydrodynamic model of boundary-layer equations for a perfectly conducting viscoelastic fluid. This model is applied to study the effects of free convection currents with one relaxation time on the flow of a perfectly conducting viscoelastic fluid through a porous medium, which is bounded by a vertical plane surface. The state space approach is adopted for the solution of one-dimensional problems. The resulting formulation together with the Laplace transform technique is applied to a thermal shock problem and a problem for the flow between two parallel fixed plates, both without heat sources. Also a problem for the semi-infinite space in the presence of heat sources is considered. A discussion of the effects of cooling and heating on a perfectly conducting viscoelastic fluid is given. Numerical results are illustrated graphically for each problem considered.     

The thick free-convective boundary-layer along a semi-infinite isothermal vertical cylinder

Summary The free convection boundary-layer flow along a vertical isothermal semi-infinite cylinder is solved for the case that the boundary-layer is thick in comparison with the radius of the cylinder. Contrary to assertions in the literature that is has been solved, this problem is still awaiting solution. It is shown that a rather unorthodox boundary-layer transformation should be applied to obtain the solution. some numerical calculations are carried out and these are displayed in graphical form.

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Zusammenfassung Das Problem der freien konvektiven Grenzschichtströmung entlang einem vertikalen isothermen halbunendlichen Zylinder wird für den Fall gelöst, dass die Grenzschicht dick ist im Vergleich zum Radius des Zylinders. Obwohl in der Literatur Behauptungen zu finden sind, dass dieses Problem gelöst ist, so zeigt sich doch, dass dies nicht zutrifft. Es wird gezeigt, dass man mit Hilfe einer noch wenig bekannten Grenzschichttransformation die Lösung des Problems finden kann. Einige numerische Berechnungen sind durchgeführt worden, und die Ergebnisse werden in grafischer Form dargestellt.

     

Asymptotic solutions of non-classical boundary-value problems of the natural vibrations of orthotropic shells

The natural vibrations of orthotropic shells are considered in a three-dimensional formulation for different versions of the boundary conditions on the faces: rigid clamping rigid clamping, rigid clamping free surface, and mixed conditions. Asymptotic solutions of the corresponding dynamic equations of the three-dimensional problem of the theory of elasticity are obtained. The principal values of the frequencies of natural vibrations are determined. It is shown that three types of natural vibrations occur in the shell: two shear vibrations and a longitudinal vibration, which are due solely to the boundary conditions on the faces. It is proved that each boundary layer has its own natural frequency. The boundary-layer functions are determined and the rates at which they decrease with distance from the faces inside the shell are established.     

Instabilität einer von unten erwärmten Flüssigkeitsschicht bei gleichzeitiger Berücksichtigung von Oberflächenspannung und Auftriebskraft

Combined effects of surface tension and buoyancy force on the thermal instability in a horizontal liquid film heated from below, which is bounded by a rigid wall and a free surface, are considered under a generalized boundary condition for the temperature disturbance at the free surface, which is introduced by consideration of the continuity of heat flow through the free surface to a gas above the liquid film. The results show that the critical Rayleigh number varies with the Marangoni number and with the Nusselt number.

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Die Arbeit ist während eines Aufenthaltes als Dozentenstipendiat der Alexander-von-Humboldt-Stiftung bei Herrn Prof. Dr.H. Görtler im Institut für Angewandte Mathematik der Albert-Ludwigs-Universität, Freiburg i. Br. entstanden.     

Vertical free convective boundary-layer flow in a porous medium using a thermal nonequilibrium model: Elliptical effects

In this paper we study the effect of adopting a two-temperature model of microscopic heat transfer on the classical Cheng &; Minkowycz [1] vertical free convection boundary-layer flow in a porous medium. Such a model, which allows the solid and fluid phases not to be in local thermal equilibrium, is found to modify substantially the behaviour of the flow relatively close to the leading edge. A companion paper deals with the (parabolic) boundary-layer theory, but the present work investigates in detail how elliptical effects are manifested. This is undertaken by solving the full equations of motion, rather than the boundary-layer approximation. In general, it is found that at any point in the flow, the temperature of the solid phase is higher than that of the fluid phase, and therefore that the thermal field of the solid phase is of greater extent than that of the fluid phase. The microscopic inter-phase heat transfer is characterised by the coefficient, H,and it is shown that these thermal non-equilibrium effects are strongest when H is small.