Properties of Solutions for the Problem of a Joint Slow Motion of a Liquid and a Binary Mixture in a Two-Dimensional Channel

Under study is a conjugate boundary value problemdescribing a joint motion of a binary mixture and a viscous heat-conducting liquid in a two-dimensional channel, where the horizontal component of the velocity vector depends linearly on one of the coordinates. The problemis nonlinear and inverse because the systems of equations contain the unknown time functions—the pressure gradients in the layers. In the case of small Marangoni numbers (the so-called creeping flow) the problem becomes linear. For its solutions the two different integral identities are valid which allow us to obtain a priori estimates of the solution in the uniform metric. It is proved that if the temperature on the channel walls stabilizes with time then, as time increases, the solution of the nonstationary problem tends to a stationary solution by an exponential law.     

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.     

New asymptotic heat transfer model in thin liquid films

In this article, we present a model of heat transfer occurring through a liquid film flowing down a vertical wall. This new model is formally derived using the method of asymptotic expansions by introducing appropriately chosen dimensionless variables. In our study the small parameter, known as the film parameter, is chosen as the ratio of the flow depth to the characteristic wavelength. A new Nusselt solution is obtained, taking into account the hydrodynamic free surface variations and the contributions of the higher order terms coming from temperature variation effects. Comparisons are made with numerical solutions of the full Fourier equations in a steady state frame. The flow and heat transfer are coupled through Marangoni and temperature dependent viscosity effects. Even if these effects have been considered separately before, here a fully coupled model is proposed. Another novelty consists in the asymptotic approach in contrast to the weighted residual approach which have been formerly applied to these problems.     

Unsteady Marangoni convection heat transfer of fractional Maxwell fluid with Cattaneo heat flux

Fractional shear stress and Cattaneo heat flux models are introduced in characterizing unsteady Marangoni convection heat transfer of viscoelastic Maxwell fluid over a flat surface. Governing equations and boundary condition are formulated firstly via the balance between the surface tension and shear stress. Numerical solutions are obtained by new developed numerical technique and some novel phenomena are found. Results shown that the fractional derivative parameters, Marangoni number and power law exponent have significant influence on characteristics velocity and temperature fields. As fractional derivative parameters increase, the temperature profiles rise remarkably and the viscoelastic effects of the fluid enhance with delayed response to surface tension, however the temperature profiles decline significantly with a thinner thickness of thermal boundary layer with the increase of Marangoni number. The average skin friction coefficient increases with the augment of Marangoni number, while the average Nusselt number decreases for larger values of power law exponent.     

Peristaltic transport of a Newtonian fluid in a vertical asymmetric channel with heat transfer and porous medium

The problem of peristaltic flow of a Newtonian fluid with heat transfer in a vertical asymmetric channel through porous medium is studied under long-wavelength and low-Reynolds number assumptions. The flow is examined in a wave frame of reference moving with the velocity of the wave. The channel asymmetry is produced by choosing the peristaltic wave train on the walls to have different amplitudes and phase. The analytical solution has been obtained in the form of temperature from which an axial velocity, stream function and pressure gradient have been derived. The effects of permeability parameter, Grashof number, heat source/sink parameter, phase difference, varying channel width and wave amplitudes on the pressure gradient, velocity, pressure drop, the phenomenon of trapping and shear stress are discussed numerically and explained graphically.     

A three-dimensional steady state thermal fluid model of jumbo ingot casting during electron beam re-melting of Ti–6Al–4V

A 3-D coupled thermal-fluid model describing mass, momentum and energy transport within a Ti–6Al–4V rolling ingot cast in an (Electron Beam Cold Hearth Remelting) EBCHR process has been developed to describe steady state casting conditions. The model incorporates a number of the physical phenomena inherent to the industrial process, including a metal inlet in the center of one of the narrow faces, complex boundary conditions based on industrial practice, buoyancy driven flow within the liquid and flow attenuation using a Darcy momentum source term within the mushy zone. The model ignores turbulence in the liquid pool and Marangoni (surface tension) driven surface flows. The model has been validated against liquid pool depth and profile measurements made on an experimental casting seeded with insoluble dense markers and doped with dense alloy additions. Comparisons have also been made to video images taken of the top surface during casting. The results indicate that the model is able to quantitatively predict the steady state sump depth and profile and is able to qualitatively predict aspects of the top surface temperature distribution. The model has also been used to conduct a process heat balance and sensitivity analyses. The process heat balance conducted on the model domain indicates that at steady state the liquid metal inlet contributes 88% of the total power input, while the electron beam provides net 12% after accounting for radiation losses from the top surface; 62% of the heat is lost through the ingots sides and the balance is lost via bulk transport of sensible heat through the bottom of the domain. The results of the sensitivity analysis on pool depth indicate that casting rate has the largest effect followed by metal inlet superheat. The thermal, flow and pressure fields predicted by the steady state model serves as the initial conditions for a transient hot-top model, which is the subject of a forth-coming paper.     

Leveling a capillary ridge generated by substrate geometry

The formation of capillary ridges is typical of thin viscous films flowing over a topographical feature. This process is studied by using a two-dimensional model describing the slow motion of a thin viscous nonisothermal liquid film flowing over complex topography. The model is based on the Navier-Stokes equations in the Oberbeck-Boussinesq approximation. The density, surface tension, and viscosity of the liquid are linear functions of temperature. For a nonisothermal flow over a planar substrate with a local heater, the influence of the heater on the free surface is analyzed numerically depending on the buoyancy effect, Marangoni stresses, and variable viscosity. The analysis shows that the film can create its own ridges or valleys depending on the heater and the dominating liquid properties. It is shown that the capillary ridges generated by the substrate features can be optimally leveled by using various types of heaters consistent with the dominating liquid properties. Numerical results for model problems are presented.     

The forced convection flow of a uniform stream over a flat surface with a convective surface boundary condition

The forced convection heat transfer resulting from the flow of a uniform stream over a flat surface on which there is a convective boundary condition is considered. In previous papers [5], [6], [7], [8] it was assumed that the convective heat transfer parameter h_f associated with the hot surface depended on x, where x measures distance along the surface, so that problem could be reduced to similarity form. Here it is assumed that this heat transfer parameter h_f is a constant, with the result that the temperature profiles and overall heat transfer characteristics evolve as the solution develops from the leading edge. The heat transfer near the leading edge (small x), which we find to be dominated by the surface heat flux, the solution at large distances along the surface (large x), which dominated by the surface temperature, are discussed. A numerical solution to the full problem is then obtained for a range of values of the Prandtl number to join these two solution regimes.     

The effect of vertical vibration on the onset of thermocapillary convection in a horizontal liquid layer

The effect of vertical vibration on the onset of Marangoni convection in a horizontal layer of a viscous incompressible uniform liquid with a free surface and a hard (solid) or soft (impermeable and stress-free) wall is investigated. In the case of harmonic vibration, a dispersion relation is constructed in explicit form using continued fractions. From this, equations are obtained for determining the critical values of the parameters for all three main types of loss of stability. Neutral curves of the monotonic and oscillatory instability are constructed, for fixed frequency and amplitude of the vibration, in the form of a graph of the Marangoni number against the wave number. The regions of parametric resonances, corresponding to synchronous and subharmonic modes are determined. The frequency values for which a high-frequency asymptotic form is reached are obtained. The long-wave Marangoni oscillatory instability is investigated, and it is shown that in this case the Marangoni numbers are negative and depend only on the Prandtl and Biot numbers.     

Justification of the Ginzburg–Landau approximation for an instability as it appears for Marangoni convection

The Ginzburg–Landau equation appears as a universal amplitude equation for spatially extended pattern forming systems close to the first instability. It can be derived via multiple scaling analysis for the Marangoni convection problem that is driven by temperature‐dependent surface tension and is the subject of our interest. In this paper, we prove estimates between this formal approximation and true solutions of a scalar pattern forming model problem showing the same spectral picture as the Marangoni convection problem in case of a thin fluid. The new difficulties come from neutral modes touching the imaginary axis for the wave number k = 0 and from identical group velocities at the critical wave number k = k_c and the wave number k = 0. The problem is solved by using the reflection symmetry of the system and by using the fact that the modes concentrate at integer multiples of the critical wave number k = k_c. The paper presents a method that is applicable whenever this kind of instability occurs. Copyright © 2012 John Wiley & Sons, Ltd.     

Mixed convective heat and mass transfer in an asymmetric channel with peristalsis

This paper describes the fluid mechanics effects of mixed convective heat and mass transfer in an asymmetric channel with peristalsis. The flow is examined in a wave frame of reference moving with the velocity of the wave. The channel asymmetry is produced by choosing the peristaltic wave train on the walls to have different amplitudes and phase. The momentum, energy and concentration equations have been linearized under long wavelength approximation. The analytical solutions for temperature, concentration, velocity and stream function are obtained. The effects of various parameters such as local temperature Grashof number, local mass Grashof number and geometrical parameters on flow variables have been discussed numerically and explained graphically.     

Thin-layer version of the moment equations in the boundary layer problem

The two-dimensional problem of a hypersonic kinetic boundary layer developing on a thin body in the case of a monatomic gas is considered. The model of the flow arises from the kinetic theory of gases and, within its accuracy, i.e., in the approximation of a hypersonic boundary layer, takes into account the strong nonequilibrium of the flow with respect to translational degrees of freedom. A method for representing the solution of the problem in terms of the solution of a similar classical (Navier-Stokes) hypersonic boundary layer problem is described. For the kinetic version of the problem, it is shown that the shear stress and the specific heat flux on the body surface are equal to their counterparts in the Navier-Stokes boundary layer.     

Spatial Velocity and Temperature Measurements in Non‐Isothermal Liquid Flows

A surface tension driven flow in the liquid vicinity of an air bubble on a heated wall is studied experimentally. The liquid flow caused by the temperature gradient along the surface of the bubble is termed thermocapillary convection. The surface tension force and the buoyancy force oppose one another. The measurement technique is the 3D particle tracking velocity and thermometry, 3D PTV/T, using thermochromic liquid crystals and digital image processing. The paper describes the method in some detail and presents quantitative results for different Marangoni numbers. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Influence of eddies on heat transfer in Couette flow over undulated substrates

In Couette flows over undulated substrates eddies can be generated under creeping flow conditions. In contrast to free surface flows on undulated substrates even smooth bottom undulations allow for eddy generation due to the kinematical constraints. The subject of our paper is how these flow patterns interact with the temperature field in non–isothermal flows. Our analysis of the thermo–mechanical coupling is focused on the two dominant effects, namely convection and thermoviscosity, whereas dissipation heat, buoyancy and temperature–dependence of the remaining material parameters are neglected. We solve the problem in two steps: First, the influence of the eddies on the convective heat transfer is considered by solving the heat conduction equation with convection. For the velocity field we take the solution resulting analytically from Reynolds' lubrication approximation for the isothermal flow. The thermoviscous feedback of the resulting temperature field to the flow is considered in forthcoming papers. For the construction of the solution an analytical approach based on a nonorthogonal series representation of the fundamental fields and a variational formulation of the field equations is used. The results are visualised and the physical effects they reveal are discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonisothermic flow of gas mixture in a channel at intermediate Knudsen numbers

Solution of the problem of gas mixture flow in a plane channel at intermediate Knudsen numbers is considered on the basis of the 20-moment approximation as a function of distribution. The applied method consists of averaging moment equations valid throughout the flow region (including the Knudsen layers) with the determination of boundary values of macroscopic parameters on the wall using the approximate Loyalka method /1,2/. Expressions are obtained for a binary mixture for the mean molar velocity averaged over the channel cross section, difference of component velocities, and the relative heat flux in the presence of longitudinal gradients of partial pressures, and for the temperature gradients. Respective kinetic coefficients of the Onsager matrix are calculated. Dependence of these coefficients on the Knudsen number, and the properties of molecule scatter on the channel wall are analyzed in detail in the case of one-component gas and of a binary mixture with small relative difference of mass and diameters of molecule scatter.     

FOURIER REGULARIZATION FOR DETERMINING SURFACE HEAT FLUX FROM INTERIOR OBSERVATION BASED ON A SIDEWAYS PARABOLIC EQUATION

In this paper we consider a non-standard inverse heat conduction problem for determining surface heat flux from an interior observation which appears in some applied subjects. This problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. A Fourier method is applied to formulate a regularized approximation solution, and some sharp error estimates are also given.     

A Priori Estimates of the Solution of the Problem of the Unidirectional Thermogravitational Motion of a Viscous Liquid in the Plane Channel

We consider an initial boundary-value problem describing the unidirectional motion of a liquid in the Oberbeck–Boussinesq model in a plane channel with rigid immovable walls on which the temperature distribution is given (or the upper wall is heat-insulated). For this problem, we obtain a priori estimates, find an exact stationary solution, and determine conditions under which the solution converges to its stationary regime.     

Influence of thermal load on the characteristics of a flow with evaporation

The two-layer flows of a liquid and a gas in a horizontal channel are investigated under condition of given gas flow rate. Evaporation on the thermocapillary interface is taken into account. An exact solution is constructed of the Navier–Stokes equations in the Boussinesq approximation, taking into account the Dufour effect in the gas-vapor layer. Within the framework of linear theory, the stability of the obtained solutions and the characteristics of the arising perturbations are studied. The influence is considered of the thickness of the liquid layer and the magnitude of the longitudinal temperature gradient on the structure of the basic flow and perturbations.     

THE GLOBAL ARTIFICIAL BOUNDARY CONDITIONS FOR NUMERICAL SIMULATIONS OF THE 3D FLOW AROUND A SUBMERGED BODY

We consider the numerical approximations of the three-dimensional steady potential flow around a body moving in a liquid of finite constant depth at constant speed and distance below a free surface in a channel. One vertical side is introduced as the up-stream artificial boundary and two vertical sides are introduced as the downstream arti-ficial boundaries. On the artificial boundaries, a sequence of high-order global artificial boundary conditions are given. Then the original problem is reduced to a problem defined on a finite computational domain, which is equivalent to a variational problem. After solving the variational problem by the finite element method, we obtain the numerical approximation of the original problem. The numerical examples show that the artificial boundary conditions given in this paper are very effective.     

Thermal stress state of a bimaterial with a closed interfacial crack having rough surfaces

We have formulated the problem of thermoelasticity for a bimaterial whose components differ only in their shear moduli, with a closed interfacial crack having rough surfaces. The bimaterial is subjected to the action of compressive loads and heat flow normal to the interfacial surface. We have taken into account the dependence of thermal conductance of the defect on the contact pressure of its faces and heat conductivity of the medium that fills it. The problem is reduced to a Prandtl-type nonlinear singular integro-differential equation for temperature jump between the crack surfaces. An analytical solution of this problem has been constructed for the case of action of the heat flow only. We have analyzed the dependence of contact pressure of the defect faces, temperature jump between them, and the intensity factor of tangential interfacial stresses on the value of given heat flow, roughness of the surfaces, and ratio between the shear moduli of joined materials.