Heat transfer in a liquid film over an unsteady stretching surface with viscous dissipation in presence of external magnetic field

This paper presents a mathematical analysis of MHD flow and heat transfer to a laminar liquid film from a horizontal stretching surface. The flow of a thin fluid film and subsequent heat transfer from the stretching surface is investigated with the aid of similarity transformation. The transformation enables to reduce the unsteady boundary layer equations to a system of non-linear ordinary differential equations. Numerical solution of resulting non-linear differential equations is found by using efficient shooting technique. Boundary layer thickness is explored numerically for some typical values of the unsteadiness parameter S and Prandtl number Pr, Eckert number Ec and Magnetic parameter Mn. Present analysis shows that the combined effect of magnetic field and viscous dissipation is to enhance the thermal boundary layer thickness.     

Nonlinear evolution of the travelling waves at the surface of a thin viscoelastic falling film

The problem of hydrodynamic instability of a thin condensate viscoelastic liquid film flowing down on the outer surface of an axially moving vertical cylinder is investigated. In order to improve the accuracy of numerical results, the viscoelastic and heat transfer parameters have been included into the governing equations. Also, the analytical solutions are obtained by utilizing the long-wave perturbation method. The influence of some physical parameters is discussed in both linear and nonlinear steps of the problem. It has been revealed that the stability of the film flow is weakened when the radius of cylinder and the temperature difference are reduced. Moreover, it is found that the increment of down-moving motion of the cylinder can enhance the flow stability. Further, the thin film flow can be destabilized by the viscoelastic property. The results show that both supercritical stability and subcritical instability can take place within the film flow system given appropriate conditions. Moreover, the absence of Reynolds number leads to an obvious difference in the behavior of some physical parameters.     

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.     

具有蒸发液滴两相介质近壁流动的数学分析

在双连续介质理论框架下，采用匹配渐进展开方法导出并求解了具有蒸发液滴的汽雾流中层流边界层方程，给出了控制汽雾流的相似判据。对于沿曲面的流动，边界层方程的形式取决于是否存在液滴的惯性沉积。给出了热钝体驻点附近蒸汽－液滴边界层的数值计算结果。它们表明：由于蒸发，在边界层内近壁处形成了一个无液滴区域；在该区上边界处，液滴半径趋于零而液滴数密度急剧增高。液滴蒸发及聚集的联合效应造成了表面热流的显著增加，甚至在自由来流中液滴质量浓度很低时此效应依然存在。     

Stabilization of horizontal symmetric flow of a liquid film by a heat flux controller

Linearized equations of motion are used to analyze the stability of flow of a liquid film on the walls of a plane-parallel horizontal channel which melt in a stream of hot viscous gas. For materials with a high specific heat of fusion, this flow is shown to be unstable relative to small-amplitude long-wave perturbations.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 74–78, 1986.     

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.     

Steady and Unsteady Jet Wiping Processes

The jet wiping process is widely used in continuous coating applications to remove the excess amount of liquid entrained by a sheet moving out of a liquid bath. Typical fields of applications are hot dip galvanization of metal strips and coating of photographic films. The process is based on the impact of a gas jet onto the liquid film carried by the solid substrate. In the present study the process is investigated for the case of strictly two‐dimensional flow. It is assumed that inertia effects on the film flow can be neglected, whereas the effects of the pressure gradient and the shear stress distribution of the impinging jet and the surface tension of the liquid film are taken into account. As a result it is possible to derive a single kinematic wave equation which governs the distribution of the film thickness. Numerical results for representative steady and unsteady processes including the formation of shock discontinuities are presented.     

The flow pattern in a liquid layer and the spectrum of the boundary-value problem when the surface tension depends non-linearly on the temperature

The flow of a liquid in a plane channel on the bottom of which a specified temperature distribution is maintained while the free surface is thermally isolated is considered. The surface tension of the liquid depends quadratically on the temperature. The system of Navier-Stokes and heat conduction equations possess a self-similar solution which leads to the non-linear eigenvalue problem of finding the flow temperature fields in the channel. The spectrum of this problem is investigated analytically for low Marangoni numbers (the second approximation) and numerically in the limiting case of an ideally heat conducting liquid for any Marangoni number. The pattern of the thermocapillary flow in the layer is analysed as a function of the parameter values. The non-uniqueness of the solution, which is typical for problems of this kind, is established. The results are compared with those obtained previously in the first approximation with respect to the Marangoni number.     

Mixed convection turbulent film condensation on a sphere

The present paper reports a research on condensation heat transfer of an isothermal sphere with an external flow of vapor. The high tangential velocity of the vapor flow is determined from potential flow theory. The transition criterion of the onset turbulence has been given in the local film Reynolds number (Re_Γ). An eddy diffusivity model along with an expression by [H. Kato, N.N. Shiwaki, M. Hirota, On the turbulent heat transfer by free convection from a vertical plate, Int. J. Heat Mass Transfer, 11(1968) 1117–1125] is used to model turbulence. And the local liquid–vapor interfacial shear which occurs for high velocity vapor flow across a sphere surface is defined by the Colburn analogy. The paper then presents analytical analysis for the local dimensionless film thickness and heat transfer characteristics for the film condensation. And a comparison with those generated by previous theoretical of laminar condensation is discussed. The comparison shows the heat transfer coefficient of turbulent film condensation is higher than laminar film condensation under the high vapor velocity.     

Analytical estimation of liquid film thickness in two-phase annular flow using electrical resistance measurement

This paper presents an analytical solution to estimate the liquid film thickness in two-phase annular flow through a circular pipe using electrical resistance tomography. Gas–liquid flow with circular gas core surrounded by a liquid film is considered. Conformal mapping is employed to obtain the analytic solution for annular flow with an eccentric circular gas core. The liquid film thickness for an arbitrary annular flow is estimated by comparing the resistance values for concentric and eccentric annular flows. The film thickness estimation has a good performance when the normalized distance between the gas core center and the flow center is less than 0.2 and the void fraction is greater than 0.4, the estimated error of the normalized thickness is less than 0.04.     

The motions of liquid films in a gas

The motions in a gas of thin films of a viscous incompressible liquid acted upon by capillary forces are considered. The surface tension depends on the impurity concentration of a surface-active material, soluble or insoluble in the liquid, and the liquid is non-volatile. The inertia of the liquid, viscous stresses, the Laplace pressure and the surface-tension gradients, impurity transfer and also the particular properties of super-thin films are taken into account. The motions of the films are described using the model of quasi-steady viscous flow. Systems of equations are obtained in the approximation of an ideal compressible medium and for small Mach numbers. The conditions for the incompressible film surface approximation to hold are obtained. The severe limitations of the gas-dynamic approximation in the case of a soluble impurity due to attenuation of the waves related to diffusion are investigated. A continuum model of the film as a compressible medium with a non-equilibrium pressure is constructed. The asymptotic form of the solutions of unsteady problems of impurity transfer in the limit of weak non-equilibrium is obtained. Integrals of the equations of motion of the films in steady one-dimensional problems are derived. Integral forms of the equations of momentum and its moment for an arbitrary contour of the film are presented, which hold for steady flows in a film and in quasi-statics. The boundary conditions for the solutions of the system of equations of motion of films are given.     

Effects of the magnetic field on direct contact melting transport processes during rotation

Contact melting heat transfer occurs via relative motion between the heating source and a phase change material (PCM) during melting in various applications. In this study, we investigated the physics of the close contact melting process generated by rotation and when subjected to an applied magnetic field. We transformed the physical model comprising the three-dimensional mass, momentum, and energy equations of the liquid melt layer in the cylindrical coordinate system, including the effects of the Lorentz forces and coupled with an interfacial energy jump condition, into a set of nonlinear similarity equations. Various characteristic dimensionless variables were identified, including an external force parameter σ, which defines the relationship between the external load on the PCM and the centrifugal force due to rotation, and a magnetic field parameter M. Numerical results were obtained and we systematically studied and interpreted the effects of various dimensionless variables on the contact melting and heat transfer processes during rotation, including the structures of the flow and thermal fields, melt layer thickness, and the melting and heat transfer rates. In particular, our results demonstrate that the melting and heat transfer rates increase while the liquid melt film becomes thinner as the external force parameter σ increases. By contrast, an increase in the magnetic field parameter M decreases the melting and heat transfer rates, while yielding relatively thicker melt layers.     

Predictions of the gas–liquid flow in wet electrostatic precipitators

Based on Computational Fluid Dynamics (CFD), the present paper aims to simulate several important phenomena in a wet type ESP from the liquid spray generation to gas-droplet flow in electric field. A single passage between the adjacent plates is considered for the simulation domain. Firstly, the electric field intensity and ion charge density are solved locally around a corona emitter of a barbed wire electrode, which are applied to the entire ESP using periodic conditions. Next, the Euler–Lagrange method is used to simulate the gas-droplet flow. Water droplets are tracked statistically along their trajectories, together with evaporation and particle charging. Finally, the deposition density on the plate is taken as the input for the liquid film model. The liquid film is simulated separately using the homogenous Eulerian approach in ANSYS-CFX. In the current case, since the free surface of the thin water film is difficult to resolve, a special method is devised to determine the film thickness.As parametric study, the variables considered include the nozzle pressure, initial spray spreading patterns (solid versus hollow spray) and plate wettability. The droplet emission rate and film thickness distribution are the results of interest. Main findings: electric field has strong effect on the droplet trajectories. Hollow spray is preferred to solid spray for its lower droplet emission. The liquid film uniformity is sensitive to the plate wettability.     

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.     

Mathematical framework for predicting solar thermal build-up of spectrally selective coatings at the Earth’s surface

A transient finite element thermal model is formulated valid for surface coatings on any substrate material and based on the continuum conduction equations with solar loading as a heat source. The model allows cooling to be applied at outer surfaces of the body, by natural convection and accounts for ambient radiative heat loss. Hemispherical spectral reflectivities are obtained for various polymer-based coatings on a steel substrate using spectrophotometers in the 0.1 μm to 25 μm wavelengths. A time-dependent solar irradiation energy source (blackbody equivalent) is applied to an object with spectrally diffuse outer surfaces, and the incoming heat flux is split by a band approximation into reflected and absorbed energy and finally integrated over the complete spectrum to provide thermal source terms for the finite element model.     

Rimming flow of a power-law fluid: Qualitative analysis of the mathematical model and analytical solutions

Rimming flow of a non-Newtonian fluid on the inner surface of a horizontal rotating cylinder is investigated. Simple lubrication theory is applied since the Reynolds number is small and liquid film is thin. For the steady-state flow of a power-law fluid the mathematical model reduces to a simple algebraic equation regarding the thickness of the liquid film. The qualitative analysis of this equation is carried out and the existence of two possible solutions is rigorously proved. Based on this qualitative analysis, different regimes of the rimming flow are defined and analyzed analytically. For the particular case, when the flow index in a power-law constitutive equation is equal to 1/2, the problem reduces to the fourth order algebraic equation which is solved analytically by Ferrari method.     

Dynamic method of determining the mechanical characteristics of the surface layer of polymers

A thin polymer film is treated as a three-layered plate, the inner layer being characterized by the bulk and the outer layers by the surface properties of the polymer. Using the equations describing the vibrations of a cantilever element of rectangular section, the authors propose formulas for the thickness, modulus of elasticity, and mechanical loss factor of the surface layer. The mechanical characteristics of the surface layers of PMM and nitrocellulose have been determined by electromagnetically exciting and photometrically recording vibrations in film specimens of varying thickness.Institute of the Chemistry of High-Molecular Compounds, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 3, pp. 548–550, May–June, 1969.     

Numerical modelling of bubbly flows with and without heat and mass transfer

In this paper, modelling gas–liquid bubbly flows is achieved by the introduction of a population balance equation combined with the three-dimensional two-fluid model. For gas–liquid bubbly flows without heat and mass transfer, an average bubble number density transport equation has been incorporated in the commercial code CFX5.7 to better describe the temporal and spatial evolution of the geometrical structure of the gas bubbles. The coalescence and breakage effects of the gas bubbles are modelled according to the coalescence by the random collisions driven by turbulence and wake entrainment while for bubble breakage by the impact of turbulent eddies. Local radial distributions of the void fraction, interfacial area concentration, bubble Sauter mean diameter, and gas and liquid velocities, are compared against experimental data in a vertical pipe flow. Satisfactory agreements for the local distributions are achieved between the predictions and measurements. For gas–liquid bubbly flows with heat and mass transfer, boiling flows at subcooled conditions are considered. Based on the formulation of the MUSIG (multiple-size-group) boiling model and a model considering the forces acting on departing bubbles at the heated surface implemented in the computer code CFX4.4, comparison of model predictions against local measurements is made for the void fraction, bubble Sauter mean diameter, interfacial area concentration, and gas and liquid velocities covering a range of different mass and heat fluxes and inlet subcooling temperatures. Good agreement is achieved with the local radial void fraction, bubble Sauter mean diameter, interfacial area concentration and liquid velocity profiles against measurements. However, significant weakness of the model is evidenced in the prediction of the vapour velocity. Work is in progress through the consideration of additional momentum equations or developing an algebraic slip model to account for the effects of bubble separation.     

Theoretical model for swirling thin film flows inside nozzles with converging-diverging shapes

A quasi-one-dimensional model was developed to describe a swirling, thin, liquid film inside nozzles with different wall profiles. The model quantifies the effects of swirl strength, initial film thickness, and Reynolds and Weber numbers on the film thickness along the nozzle surface. Moreover, the model allows for a rapid (at least, qualitative) evaluation of different effects, e.g. of the swirl strength and nozzle geometry, and can serve as a benchmark case for the subsequent more involved numerical simulations. Steady-state solutions are presented as a function of various parameters. The effect of the nozzle geometry on film thickness is explored. As swirling flow entered the expanding (diverging) section of the nozzle, film thickness decreased to satisfy continuity (to conserve mass). Conversely, film thickness increased upon entering the contracting (converging) region of the nozzle. Geometric effects controlled film thicknesses much more than other flow parameters. This quasi-one-dimensional model for a swirling thin film can be useful for designing a swirl jet used in various industrial applications.