Generalized approximation method and a thin film flow of a third grade fluid on a moving belt

We develop a generalized approximation method (GAM) to obtain a solution of a thin film flow of a third grade fluid on a moving belt. The GAM generates a monotone sequence of solutions of linear problems. The sequence of solutions of linear problems converges monotonically and rapidly to a solution of the original nonlinear problem. We present some numerical simulations to illustrate and confirm our results.     

On exact solutions for thin film flows of a micropolar fluid

The present paper contains the exact solutions for thin film flows of a micropolar fluid. Thin films are considered for three different geometries namely: (i) flow down an inclined plane, (ii) flow on a moving belt and (iii) flow down a vertical cylinder. The exact solution is obtained for the velocity and micro-rotation in first two cases for both weak and strong concentrations of the microelements. However, the exact expression for the micro-rotation is obtained for the third case and a numerical solution is presented for the velocity. The micro-rotation and velocity fields are sketched and discussed for different values of the material parameters. It is observed that the rotation of the microelements at the boundary increases the velocity when compared with the case when there is no rotation at the boundary.     

Rheological effect on thermocapillary flow of a liquid film jet painted on a moving boundary

In the present paper, a liquid (or melt) film of relatively high temperature ejected from a vessel and painted on the moving solid film is analyzed by using the second-order fluid model of the non-Newtonian fluid. The thermocapillary flow driven by the temperature gradient on the free surface of a Newtonian liquid film was discussed before. The effect of rheological fluid on thermocapillary flow is considered in the present paper. The analysis is based on the approximations of lubrication theory and perturbation theory. The equation of liquid height and the process of thermal hydrodynamics of the non-Newtonian liquid film are obtained, and the case of weak effect of the rheological fluid is solved in detail.     

Nonlinear stability characterization of thin Newtonian film flows traveling down on a vertical moving plate

The paper presents both the linear and nonlinear stability theories for the characterization of thin Newtonian film flows traveling down along a vertical moving plate. The linear model is first developed to characterize the flow behavior. After showing the inadequacy of the linear model in representing certain flow characteristics, the nonlinear kinematics model is then developed to represent the system. The long-wave perturbation method is employed to derive the generalized kinematic equations with free film surface condition. The linear model is solved by using the normal mode method for three different, namely, the quiescent, up-moving and down-moving, moving conditions. Subsequently, the elaborated nonlinear film flow model is solved by the method of multiple scales. The modeling results clearly indicate that both subcritical instability and supercritical stability conditions are possible to occur in the film flow system. The effect of the down-moving motion of the vertical plate tends to enhance the stability of the film flow.     

Mathematical modeling of coupled turbulent flow and solidification in a single belt caster with electromagnetic brake

A mathematical model has been developed to simulate turbulent fluid flow and solidification in the presence of a DC magnetic field in an extended nozzle for metal delivery to a single belt caster. This paper reports on predicted effects of DC magnetic field conditions in modifying flows and solidification behavior in the metal delivery system. It is shown that the application of a DC magnetic brake to the proposed system can result in a reasonably uniform feeding of melt onto the cooled moving belt. This, in turn, optimises the rate of even shell growth along the chilled substrate. In order to account for the effects of turbulence, a revised low-Reynolds k–ε turbulent model was employed. A Darcy-porosity approach was used to simulate fluid flow within the mushy solidification region. Simulations were carried out for plain carbon steel strip casting. The fully coupled transport equations were numerically solved using the finite volume method. The computed flow patterns were compared with those reported in the literature. The performance of the magnetic flow control device proposed in this work is evaluated and compared with flow modifications obtained by inserting a ceramic filter within the reservoir.     

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.     

A new method for homoclinic solutions of ordinary differential equations

Consideration is given to the homoclinic solutions of ordinary differential equations. We first review the Melnikov analysis to obtain Melnikov function, when the perturbation parameter is zero and when the differential equation has a hyperbolic equilibrium. Since Melnikov analysis fails, using Homotopy Analysis Method (HAM, see [Liao SJ. Beyond perturbation: introduction to the homotopy analysis method. Boca Raton: Chapman & Hall/CRC Press; 2003; Liao SJ. An explicit, totally analytic approximation of Blasius’ viscous flow problems. Int J Non-Linear Mech 1999;34(4):759–78; Liao SJ. On the homotopy analysis method for nonlinear problems. Appl Math Comput 2004;147(2):499–513] and others [Abbasbandy S. The application of the homotopy analysis method to nonlinear equations arising in heat transfer. Phys Lett A 2006;360:109–13; Hayat T, Sajid M. On analytic solution for thin film flow of a forth grade fluid down a vertical cylinder. Phys Lett A, in press; Sajid M, Hayat T, Asghar S. Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt. Nonlinear Dyn, in press]), we obtain homoclinic solution for a differential equation with zero perturbation parameter and with hyperbolic equilibrium. Then we show that the Melnikov type function can be obtained as a special case of this homotopy analysis method. Finally, homoclinic solutions are obtained (for nontrivial examples) analytically by HAM, and are presented through graphs.     

Some singular,nonlinear differential equations arising in boundary layer theory

The nonlinear, second-order differential equation which describes boundary layer flow when Crocco variables are employed is examined under several different endpoint conditions. Uniqueness and analyticity are established for boundary conditions corresponding to flow behind weak expansion and shock waves and for the flow above a moving conveyor belt.     

Self-excited oscillations of a two-mass oscillator with dry “stick-slip” friction

A system of two masses, moving along a single straight line, is considered. The first is connected by a spring to a fixed point, while the second is connected by a spring to the first and is in contact with a belt with dry friction moving with constant velocity. A piecewise-constant model of dry friction with different coefficients of friction, sliding and at rest, is used. The limit “stick-slip” type cycles are investigated analytically. It is shown numerically that in the case of equal masses there are forward and reverse limit cycles. The period of the oscillations of the forward and reverse cycles increases as the ratio of the stick and slip coefficients of friction increases, and decreases when the velocity of the belt increases. The reverse cycle exists for all values of the parameters of the problem, while the forward cycle exists up to a certain critical value of the ratio of the stick and slip coefficients of friction, and this critical value increases when the velocity of the belt increases.     

微极流体薄膜层通过以滑移速度移动的可渗透无限平板时流体特性变化和热辐射对流动和热传导的影响

微极流体薄膜层通过按滑移速度移动的可渗透无限竖直平板时,研究热辐射对混合对流薄膜层流动和热传导的影响.假定流体粘度和热传导率变化是温度的一个函数.对一些典型的可变参数值,应用Chebyshev谱方法,数值求解流动的控制方程.将所得结果与已发表文献的结果进行比较,结果是一致的.绘出并讨论了可变参数对速度、微旋转速度、温度分布曲线、表面摩擦因数和Nusselt数的影响.     

Critical velocities for flow induced vibrations of a U-shaped belt

Two excitation mechanisms for flow induced vibrations of a U-shaped belt are described. Firstly, the belt can be excited by vortices forming in the cavity of the U. This mechanism is relevant at low reduced (dimensionless) flow velocities. Secondly, an excitation through the so called motion induced vortices can occur at larger reduced velocities. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

粘弹性运动带动力响应分析

基于Kelvin粘弹性材料本构模型及带运动方程,建立了运动带非线性动力学分析模型.基于该模型和Lie群分析方法推导了匀速运动及简谐运动带线性问题的解析解;基于该非线性模型的数值仿真讨论了运动带材料参数、带稳态运动速度、扰动速度对系统动态响应的影响.结果表明:1)当带匀速运动时,无论系统是线性还是非线性,运动带横向振动"频率"都随着带运动稳态速度增加而减小.2)随着材料粘性增加,系统耗散能力逐渐增强,动态响应逐渐减小.3)当带运动速度简谐波动时,系统动态响应随扰动速度增大而增大.扰动频率对带横向振动影响较大.     

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.     

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.     

Influence of the contact model on the onset of sprag-slip

In systems with sliding-friction often strong self-excited vibrations do occur. One of the possible underlying mechanisms is the so-called sprag-slip instability. In the present work the onset of sprag-slip is investigated by a simple model in which an inclined elastic beam slides over a rigid belt moving with constant velocity. For a Coulomb friction law and a contact model with constant contact stiffness for a certain range of parameters the system loses its static solution corresponding to the steady sliding state. Simultaneously with this loss of existence of the static solution the qualitative properties of the system's flow field in phase space change, resembling a transition from stable to unstable behavior. To investigate the influence of contact models and related parameters on the details of this onset of sprag-slip also Hertz theory of elastic contact is applied. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the solution of boundary value problems on an inhomogeneous plane with a crack and a barrier connected in series

We consider boundary value problems for an equation in divergence form on a plane divided into two inhomogeneous half-planes by a film inclusion in the form of a strongly permeable crack and a weakly permeable barrier connected in series; this models a contact of heterogeneous media under inhomogeneous external conditions. The desired potentials have prescribed singular points (sources, drains, etc.). The coefficients of the equation are nonconstant and may increase or decrease when moving away from the film inclusion along a family of parabolas. We obtain representations of solutions of the considered problems via harmonic functions with the corresponding singular points on the plane.     

Some Self-Similar Two-Phase Flows of Non-Newtonian Fluids Through a Porous Medium

Pascal This paper addresses the question of the rheological effects of non-Newtonian fluids in a flow system, in which a two-phase flow zone is coupled to a single-phase flow zone by a moving fluid interface. This flow system is involved in a technique for oil displacement in a porous medium, where a non-Newtonian displacing fluid (a polymer solution) is used to displace a non-Newtonian heavy oil. The self-similar solutions of the equations governing the dynamics of the moving interface, separating the displacing and displaced fluids, are obtained for the one-dimensional and plane radial flows. The effects associated with the presence of a two-phase flow zone, behind the moving interface, on the interface movement are analyzed. The existence of a pressure front ahead of the moving interface, moving with a finite velocity, is also shown. The relevance of this result to the propagation of pressure disturbances in a non-Newtonian fluid flowing through a porous medium is discussed with regard to interpretation of the transient pressure response in an injection well for polymer-solution floods.     

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)     

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.