Analytic solutions for a liquid film on an unsteady stretching surface

The momentum and heat transfer in a laminar liquid film on a horizontal stretching sheet is analyzed by the Homotopy analysis method (HAM). Analytic series solutions are given and compared with numerical results given by other authors. The good agreement between them shows the effectiveness of HAM to the problem of liquid film on an unsteady stretching surface.     

Heat transfer in a power-law fluid film over a unsteady stretching sheet

In this paper the problem of momentum and heat transfer in a thin liquid film of power-law fluid on an unsteady stretching surface has been studied. Numerical solutions are obtained for some representative values of the unsteadiness parameter S and the power-law index n for a wide range of the generalized Prandtl number, 0.001 ≤ Pr ≤ 1000. Typical temperature and velocity profiles, the dimensionless film thickness, free-surface temperature, and the surface heat fluxes are presented at selected controlling parameters. The results show that increasing the value of n tends to increase the boundary-layer thickness and broadens the temperature distributions. The free-surface temperature of a shear thinning fluid is larger than that of a Newtonian fluid, but the opposite trend is true for a shear thickening fluid. For small generalized Prandtl numbers, the surface heat flux increases with a decrease in n, but the impacts of n on the heat transfer diminish for Pr greater than a moderate value (approximately 1 ≤ Pr ≤ 10, depending on the magnitude of S).     

不稳定伸展表面上的薄液膜流动分析

探讨了不稳定伸展表面上的薄液膜流动问题.利用相似变换将边界层流动控制方程转化为常微分方程边值问题.利用同伦分析方法获得解析解,讨论不稳定参数对液膜流动的影响,得到一般性规律.将部分级数解与前人的数值解进行比较,结果具有较高的一致性.该方法还可以用于其他科学工程问题.     

Liquid film flow due to an unsteady stretching sheet

We have studied two-dimensional flow of a thin liquid film over an impulsively stretching sheet under assumption of uniform initial film thickness. Using singular perturbation technique both momentum and film evolution equations are solved analytically for small Reynolds number and these solutions are verified numerically. Numerical computation for large Reynolds number shows an anomalous behaviour of film thinning rate in different time zone. These results are explained physically and the crucial role-played by viscosity in this case is highlighted. It is found that faster rate of thinning can be obtained if the sheet is stretched impulsively with continuously increasing stretching speed.     

Thermal radiation effects on the flow and heat transfer in a liquid film on an unsteady stretching sheet

The influence of thermal radiation on the flow and heat transfer within Newtonian liquid film over an unsteady stretching sheet with and without thermocapillarity is examined. The governing non‐linear partial differential equations describing the problem are reduced to a system of nonlinear ordinary differential equations using similarity transformation, which is solved numerically for different values of the thermal radiation parameter and the thermocapillarity parameter. The results show that the dimensionless velocity, the film thickness and the local Nusselt number increase as the thermocapillarity parameter increases, while the free surface temperature decreases with increasing the thermocapillarity parameter. Also, both the dimensionless temperature and the free surface temperature increase and the local Nusselt number decreases as the thermal radiation parameter increases. Copyright © 2010 John Wiley & Sons, Ltd.     

Computational modeling of heat transfer over an unsteady stretching surface embedded in a porous medium

The present paper deals with the study of heat transfer characteristics in the laminar boundary layer flow of an incompressible viscous fluid over an unsteady stretching sheet which is placed in a porous medium in the presence of viscous dissipation and internal absorption or generation. Similarity transformations are used to convert the governing time dependent nonlinear boundary layer equations into a system of non-linear ordinary differential equations containing Prandtl number, Eckert number, heat source/sink parameter, porous parameter and unsteadiness parameter with appropriate boundary conditions. These equations are solved numerically by applying shooting method using Runge-Kutta-Fehlberg method. Comparison of numerical results is made with the earlier published results under limiting cases. The effects of the parameters which determine the velocity and temperature fields are discussed in detail.     

Effect of non-uniform heat source on MHD heat transfer in a liquid film over an unsteady stretching sheet

An analysis has been carried out to study the magnetohydrodynamic boundary layer flow and heat transfer characteristics of a laminar liquid film over a flat impermeable stretching sheet in the presence of a non-uniform heat source/sink. The basic unsteady boundary layer equations governing the flow and heat transfer are in the form of partial differential equations. These equations are converted to non-linear ordinary differential equations using similarity transformation. Numerical solutions of the resulting boundary value problem are obtained by the efficient shooting technique. The effects of magnetic and the non-uniform heat source/sink parameters on the dynamics are discussed. Findings of the paper reveal that non-uniform heat sinks are better suited for effective cooling of the stretching sheet. Skin friction coefficient and the local Nusselt number are also explored for typical values of magnetic and non-uniform heat source/sink parameters. The results are in excellent agreement with the earlier published works, under some limiting cases.     

MHD flow and heat transfer in a thin liquid film on an unsteady stretching sheet by the homotopy analysis method

The magnetohydrodynamics flow and heat transfer in a thin liquid film over an unsteady elastic stretching surface are analyzed by the homotopy analysis method. A more general surface temperature is taken into consideration. The effects of various parameters in this study are discussed and presented graphically. The good agreement between the analytic series solutions and the previous numerical results shows the effectiveness of HAM to this problem. Copyright © 2009 John Wiley & Sons, Ltd.     

Flow and heat transfer over an unsteady stretching surface with Hall effect

The present investigation is concerned with the effect of Hall currents on the flow and heat transfer of an electrically conducting fluid over an unsteady stretching surface in presence of a strong magnetic field. The induced magnetic field is neglected while the electron-atom collision frequency is assumed to be relatively high, so that the Hall effect is assumed to exist. The incorrect similarity transformation of Elbashbeshy and Bazid (Heat Mass Transfer 41:1–4, 2004). is corrected and a physically realistic distribution of the velocity and temperature is obtained. Using a similarity transformation the governing time dependent boundary layer equations for momentum and thermal energy are reduced to a set of coupled ordinary differential equations which are then solved numerically by the shooting method. Effects of the magnetic field, M , Hall parameter, m, and the unsteadiness parameter, S, on the velocity and temperature profiles as well as the local skin friction coefficients and the heat transfer rate are shown graphically.     

Waves on the surface of a falling power-law fluid film

Waves that occur at the surface of a falling film of thin power-law fluid on a vertical plane are investigated. Using the method of integral relations an evolution equation is derived for two types of waves equation which are possible under long wave approximation. This equation reveals the presence of both kinematic and dynamic wave processes which may either act together or singularly dominate the wave field depending on the order of different parameters. It is shown that, at a small flow rate, kinematic waves dominate the flow field and the energy is acquired from the mean flow during the interaction of the waves, while, for high flow rate, inertial waves dominate and the energy comes from the kinematic waves. It is also found that this exchange of energy between kinematic and inertial waves strongly depends on the power-law index n. Linear stability analysis predicts the contribution of different terms in the wave mechanism. Further, it is found that the surface tension plays a double role: for a kinematic wave process, it exerts dissipative effects so that a finite amplitude case may be established, but for a dynamic wave process it yields dispersion. Further, it is shown that the non-Newtonian character n plays a vital role in controlling the role of the term that contains surface tension in the above processes.     

Numerical studies for flow and heat transfer of the Powell-Eyring fluid thin film over an unsteady stretching sheet with internal heat generation using the chebyshev finite difference method

An analysis is carried out to study the unsteady two-dimensional Powell-Eyring flow and heat transfer to a laminar liquid film from a horizontal stretching surface in the presence of internal heat generation. The flow of a thin fluid film and subsequent heat transfer from the stretching surface is investigated with the aid of a similarity transformation. The transformation enables to reduce the unsteady boundary layer equations to a system of nonlinear ordinary differential equations. A numerical solution of the resulting nonlinear differential equations is found by using an efficient Chebyshev finite difference method. A comparison of numerical results is made with the earlier published results for limiting cases. The effects of the governing parameters on the flow and thermal fields are thoroughly examined and discussed.     

Thin film flow and heat transfer on an unsteady stretching sheet with internal heating

This research studied the influence of internal heat generation on flow and heat transfer in a thin liquid film on an unsteady stretching sheet. The velocity and temperature fields were solved using the Homotopy Analysis Method (HAM), taking a general surface temperature into consideration. The analytical series solution are presented and the numerical results obtained are tabulated. The effects of unsteadiness parameter, Prandtl number and temperature-dependent parameter in this study are discussed and presented graphically via the velocity and temperature profiles.     

Approximate solutions for the problem of liquid film flow over an unsteady stretching sheet with thermal radiation and magnetic field

The proposed method is based on replacement of the unknown function by a truncated series of the shifted Legendre polynomial expansion. An approximate formula of the integer derivative is introduced. Special attention is given to study the convergence analysis and derive an upper bound of the error for the presented approximate formula. The introduced method converts the proposed equation by means of collocation points to a system of algebraic equations with shifted Legendre coefficients. Thus, after solving this system of equations, the shifted Legendre coefficients are obtained. This efficient numerical method is used to solve the system of ordinary differential equations which describe the thin film flow and heat transfer with the effects of the thermal radiation, magnetic field, and slip velocity.     

The flow of a power-law liquid in a continuous-flow squeeze film

Consideration is given to the flow of an inelastic ‘power-law’ liquid in a continuous flow squeeze film. This simulates the flow in a conventional squeeze film by continuously injecting fluid into the narrow gap between two plates through the lower plate (Oliver et al. [6]). To zero order in the usual lubrication approximation the results are identical with those for the conventional squeeze film. To first order, useful corrections to the normal force due to the effects of inertia are obtained.     

Convection from a stretching surface with suction and power-law variation in species concentration

The problem of laminar natural convection flow from a permeable semi-infinite accelerating vertical surface that is coated with a reacting chemical species is studied. The plate velocity and the species concentration vary as power laws. The fundamental parameters of the problem are the Schmidt number, the surface permeability and the reaction rate. The governing equations were transformed to a non-similar form and then solved analytically and numerically using the Keller box method. A parametric study illustrating the effects of the flow parameters on the velocity and the concentration fields was conducted and the physical aspects of the problem discussed. The study found, inter alia, that the fluid motion is decelerated by increases in the permeability of the accelerating surface and that the rate of mass transfer increases with Schmidt numbers but reduces with increasing reaction rates and the porosity of the accelerating surface.     

Computational modeling of an MHD flow of a non-Newtonian fluid over an unsteady stretching sheet with viscous dissipation effects

An investigation has been conducted on the MHD Casson fluid and heat transfer over an unsteady stretching sheet with viscous dissipation effects. With suitable dimensionless variables, partial differential equations are reduced to ordinary differential equations, which are then solved by the homotopy analysis method. Dependences of flow characteristics on various parameters involved into the equations are obtained.     

On the numerical solution for the flow and heat transfer in a thin liquid film over an unsteady stretching sheet in a saturated porous medium in the presence of thermal radiation

The problem of the flow and heat transfer over an unsteady stretching sheet embedded in a porous medium in the presence of thermal radiation is studied theoretically and numerically. The continuity, momentum, and energy equations, which are coupled nonlinear partial differential equations, are reduced to a set of two nonlinear ordinary differential equations. Special attention is given to study the convergence of the proposed method. The error estimation is also given. The effects of various parameters, such as the Darcy parameter, the radiation parameter, and the Prandtl number, on the flow and temperature profiles, as well as on the local skin-friction coefficient and the local Nusselt number are presented and discussed. The results obtained agree very well with the data obtained by the Runge-Kutta method coupled with the shooting technique.