Thin film flow over a non-linear stretching sheet in presence of uniform transverse magnetic field

A thin viscous liquid film flow is developed over a stretching sheet under different non-linear stretching velocities in presence of uniform transverse magnetic field. Evolution equation for the film thickness is derived using long-wave approximation of thin liquid film and is solved numerically by using the Newton–Kantorovich method. It is observed that all types of stretching produces film thinning, but non-monotonic stretching produces faster thinning at small distance from the origin. Effect of the transverse magnetic field is to slow down the film thinning process. Observed flow behavior is explained physically.     

Thin-film flow over a nonlinear stretching sheet

A thin viscous liquid film flow is developed over a stretching sheet under different nonlinear stretching velocities. An evolution equation for the film thickness, is derived using long-wave approximation of thin liquid film and is solved numerically by using the Newton–Kantorovich method. A comparison is made with the analytic solution obtained in [B. S. Dandapat, A. Kitamura, B. Santra, “Transient film profile of thin liquid film flow on a stretching surface”, ZAMP, 57, 623-635 (2006)]. It is observed that all types of stretching produce film thinning but non-monotonic stretching produces faster thinning at small distance from the origin. The velocity u along the stretching direction strongly depends on the distance along the stretching direction and the Froude number.     

Thin-film flow over a nonlinear stretching sheet

A thin viscous liquid film flow is developed over a stretching sheet under different nonlinear stretching velocities. An evolution equation for the film thickness, is derived using long-wave approximation of thin liquid film and is solved numerically by using the Newton–Kantorovich method. A comparison is made with the analytic solution obtained in [B. S. Dandapat, A. Kitamura, B. Santra, “Transient film profile of thin liquid film flow on a stretching surface”, ZAMP, 57, 623-635 (2006)]. It is observed that all types of stretching produce film thinning but non-monotonic stretching produces faster thinning at small distance from the origin. The velocity u along the stretching direction strongly depends on the distance along the stretching direction and the Froude number.     

Transient film profile of thin liquid film flow on a stretching surface

In this paper we have studied a non-planar thin liquid film flow on a planar stretching surface. The stretching surface is assumed to stretch impulsively from rest and the effect of inertia of the liquid is considered. Equations describing the laminar flow on the stretching surface are solved analytically. It is observed that faster stretching causes quicker thinning of the film on the stretching surface. Velocity distribution in the liquid film and the transient film profile as functions of time are obtained.     

Transient film profile of thin liquid film flow on a stretching surface

In this paper we have studied a non-planar thin liquid film flow on a planar stretching surface. The stretching surface is assumed to stretch impulsively from rest and the effect of inertia of the liquid is considered. Equations describing the laminar flow on the stretching surface are solved analytically. It is observed that faster stretching causes quicker thinning of the film on the stretching surface. Velocity distribution in the liquid film and the transient film profile as functions of time are obtained. (Received: May 4, 2004; revised: February 2/August 24, 2005)     

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.     

Hydromagnetic instability of a power-law liquid film flowing down a vertical cylinder using numerical approximation approach techniques

The long-wave perturbation method is employed to investigate the hydromagnetic stability of a thin electrically-conductive power-law liquid film flowing down the external surface of a vertical cylinder in a magnetic field. The validity of the numerical results is improved through the introduction of the flow index and the magnetic force into the governing equation. In contrast to most previous studies presented in the literature, the solution scheme employed in this study is based on a numerical approximation approach rather than an analytical method. The normal mode approach is used to analyze the stability of the film flow. The modeling results reveal that the stability of the film flow system is weakened as the radius of the cylinder is reduced. However, the flow stability can be enhanced by increasing the intensity of the magnetic field and the flow index, respectively. In general, the optimum conditions can be found through the use of a system to alter stability of the film flow by controlling the applied magnetic field.     

Heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet with non-uniform heat source/sink

This paper presents the study of momentum and heat transfer characteristics in a hydromagnetic flow of viscoelastic liquid over a stretching sheet with non-uniform heat source, where the flow is generated due to a linear stretching of the sheet and influenced by uniform magnetic field applied vertically. Here an analysis has been carried out to study the effect of magnetic field on the visco-elastic liquid flow and heat transfer over a stretching sheet with non-uniform heat source. The non-linear boundary layer equation for momentum is converted into ordinary differential equation by means of similarity transformation and is solved exactly. Heat transfer differential equation is also solved analytically. The effect of magnetic field on velocity, skin friction and temperature profiles are presented graphically and discussed.     

Radiation effects on the MHD flow near the stagnation point of a stretching sheet

The radiation effects using the Rosseland approximation on the flow of an incompressible viscous electrically conducting fluid over a stretching sheet near the stagnation point in the presence of uniform transverse magnetic field is studied. The governing equations transform to ordinary differential equation by using suitable similarity transformation and then by a perturbation technique the numerical results for temperature distribution were obtained and discussed graphically.     

弹性动脉血管中血液流动特性的模拟和分析

研究了两个不同的非牛顿血液流动模型:低粘性剪切简单幂律模型和低粘性剪切及粘弹性振荡流的广义Maxwell模型．同时利用这两个非牛顿模型和牛顿模型,研究了磁场中刚性和弹性直血管中血液的正弦型脉动．在生理学条件下,大动脉中血液的弹性对其流动性态似乎并不产生影响,单纯低粘性剪切模型可以逼真地模拟这种血液流动．利用高剪切幂律模型模拟弹性血管中的正弦型脉动流,发现在同一压力梯度下,与牛顿流体相比较,幂律流体的平均流率和流率变化幅度都更小．控制方程用Crank-Niclson方法求解．弹性动脉中血液受磁场作用是产生此结果的直观原因．在主动脉生物流的模拟中,与牛顿流体模型比较,发现在匹配流率曲线上,幂律模型的平均壁面剪切应力增大,峰值壁面剪切应力减小．讨论了弹性血管横切磁场时的血液流动,评估了血管形状和表面不规则等因素的影响．     

Heat and mass transfer in MHD non-Darcian flow of a micropolar fluid over a stretching sheet embedded in a porous media with non-uniform heat source and thermal radiation

A mathematical analysis has been carried out to study magnetohydrodynamic boundary layer flow, heat and mass transfer characteristic on steady two-dimensional flow of a micropolar fluid over a stretching sheet embedded in a non-Darcian porous medium with uniform magnetic field. Momentum boundary layer equation takes into account of transverse magnetic field whereas energy equation takes into account of Ohmic dissipation due to transverse magnetic field, thermal radiation and non-uniform source effects. An analysis has been performed for heating process namely the prescribed wall heat flux (PHF case). The governing system of partial differential equations is first transformed into a system of non-linear ordinary differential equations using similarity transformation. The transformed equations are non-linear coupled differential equations which are then linearized by quasi-linearization method and solved very efficiently by finite-difference method. Favorable comparisons with previously published work on various special cases of the problem are obtained. The effects of various physical parameters on velocity, temperature, concentration distributions are presented graphically and in tabular form.     

Perturbation analysis of a magnetohydrodynamic boundary layer flow

This paper presents a perturbation analysis study of the flow of an electrically conducting power-law fluid in the presence of a uniform transverse magnetic field over a stretching sheet. The perturbation solutions for small and large values of the mixed convection parameter are explored. The asymptotic behavior of the solutions was examined for different values of the power-law index and the magnetic parameter.     

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.     

Transition of MHD boundary layer flow past a stretching sheet

In this paper a study is carried out to understand the transition effect of boundary layer flow: (1) due to a suddenly imposed magnetic field over a viscous flow past a stretching sheet and (2) due to sudden withdrawal of magnetic field over a viscous flow past a stretching sheet under a magnetic field. In both the cases the sheet stretches linearly along the direction of the fluid flow. Governing equations have been non-dimensionalised and the non-dimensionalised equations have been solved using the implicit finite difference method of Crank–Nicholson type. Comparison between the steady state exact solutions and the steady state computed solutions has been carried out. Graphical representation of the dimensionless horizontal velocity, vertical velocity and local skin friction profiles of the steady state and unsteady state has been presented. Computation has been carried out for various values of the magnetic parameter M. The obtained results has been interpreted and discussed.     

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.     

Analytical solution of magnetohydrodynamic stagnation-point flow of a power-law fluid towards a stretching surface

A new kind of analytic technique, namely the homotopy analysis method (HAM), is employed to give an explicit analytical solution of the steady two-dimensional stagnation-point flow of an electrically conducting power-law fluid over a stretching surface when the surface is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. A uniform transverse magnetic field is applied normal to the surface. An explicit analytical solution is given by recursive formulae for the first-order power-law (Newtonian) fluid when the ratio of free stream velocity and stretching velocity is not equal to unity. For second and real order power-law fluids, an analytical approach is proposed for magnetic field parameter in a quite large range. All of our analytical results agree well with numerical results. The results obtained by HAM suggest that the solution of the problem under consideration converges.     

Lie group analysis for the effect of temperature-dependent fluid viscosity with thermophoresis and chemical reaction on MHD free convective heat and mass transfer over a porous stretching surface in the presence of heat source/sink

This paper concerns with a steady two-dimensional flow of an electrically conducting incompressible fluid over a vertical stretching sheet. The flow is permeated by a uniform transverse magnetic field. The fluid viscosity is assumed to vary as a linear function of temperature. A scaling group of transformations is applied to the governing equations. The system remains invariant due to some relations among the parameters of the transformations. After finding three absolute invariants a third-order ordinary differential equation corresponding to the momentum equation and two second-order ordinary differential equation corresponding to energy and diffusion equations are derived. The equations along with the boundary conditions are solved numerically. It is found that the decrease in the temperature-dependent fluid viscosity makes the velocity to decrease with the increasing distance of the stretching sheet. At a particular point of the sheet the fluid velocity decreases with the decreasing viscosity but the temperature increases in this case. It is found that with the increase of magnetic field intensity the fluid velocity decreases but the temperature increases at a particular point of the heated stretching surface. Impact of thermophoresis particle deposition with chemical reaction in the presence of heat source/sink plays an important role on the concentration boundary layer. The results thus obtained are presented graphically and discussed.     

Combined free and forced convection effects on the magnetohydrodynamic flow through a porous channel

The combined effect of free and forced convection on the flow of an electrically conducting liquid between two horizontal parallel porous walls has been studied. There is a transverse magnetic field at the walls. The equations of motion and energy have been solved by a small perturbation method. The flow phenomenon has been characterized by the non-dimensional numbers like R (cross-flow Reynolds number), K (Brinkman number), G (Grashof number), M (magnetic number) and the effects of these numbers on the velocity and temperature fields, induced magnetic field, electric field and shearing stress at the walls have been studied.     

MHD flow and heat transfer of a micropolar fluid over a stretching surface with heat generation (absorption) and slip velocity

In this work, the effects of slip velocity on the flow and heat transfer for an electrically conducting micropolar fluid over a permeable stretching surface with variable heat flux in the presence of heat generation (absorption) and a transverse magnetic field are investigated. The governing partial differential equations describing the problem are converted to a system of non-linear ordinary differential equations by using the similarity transformation, which is solved numerically using the Chebyshev spectral method. The effects of the slip parameter on the flow, micro-rotation and temperature profiles as well as on the local skin-friction coefficient, the wall couple stress and the local Nusselt number are presented graphically. The numerical results of the local skin-friction coefficient, the wall couple stress and the local Nusselt number are given in a tabular form and discussed.     

MHD flow of a Newtonian fluid over a stretching sheet: an approximate solution

An approximate solution to the problem of steady laminar flow of a viscous incompressible electrically conducting fluid over a stretching sheet is presented. The approach is based on the idea of stretching the variables of the flow problem and then using least squares method to minimize the residual of a differential equation. The effects of the magnetic field on the flow characteristics are demonstrated through numerical computations with different values of the Hartman number.