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.     

Scaling Transformation for the Effect of Temperature-Dependent Fluid Viscosity with Thermophoresis Particle Deposition on MHD-Free Convective Heat and Mass Transfer Over a Porous Stretching Surface

This article 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 equations 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. Impact of thermophoresis particle deposition in the presence of temperature-dependent fluid viscosity plays an important role on the concentration boundary layer. The results, thus, obtained are presented graphically and discussed.     

Computational study of the axial instability of rimming flow using Arnoldi method

Axial instability of rimming flow has been investigated by solving a linear generalized eigenvalue problem that governs the evolution of perturbations of two‐dimensional base flow. Using the Galerkin finite element method, full Navier–Stokes equations were solved to calculate base flow and this base flow was perturbed with three‐dimensional disturbances. The generalized eigenproblem formulated from these equations was solved by the implicitly restarted Arnoldi method using shift‐invert technique. This study presents instability curves to identify the critical wavelength of the neutral mode and the critical β, which measures the importance of gravity relative to viscosity. The axial instability of rimming flow is examined and three‐dimensional flow was reconstructed by using eigenvector and growth rate at a critical wave number. The critical β value in the axial instability analysis was observed to be comparable to the onset β value of the transition between the bump and the homogeneous film state in 2‐D base flow calculations. Copyright © 2006 John Wiley & Sons, Ltd.     

Scaling group transformation for the effect of temperature-dependent nanofluid viscosity on an mhd boundary layer past a porous stretching surface

This paper deals with a steady two-dimensional flow of an electrically conducting incompressible fluid over a porous 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. The partial differential equations governing the problem under consideration are transformed by a special form of Lie group transformations, namely, scaling group of transformations, into a system of ordinary differential equations, which are solved numerically using the Runge-Kutta-Gill algorithm and the shooting method. The conclusion is drawn that the flow field and temperature profiles are significantly influenced by the Lewis number, Brownian motion number, and thermophoresis number.     

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.     

Free convection from a horizontal moving sheet

In this paper the free convection flow through a thin rigid hot sheet moving horizontally out of a slot is considered. It is found that there is a similarity formulation of the boundary-layer equations so that the problem reduces to solving a system of coupled ordinary differential equations with suitable boundary conditions. This system of equations is solved numerically for various values of the Prandtl number,Pr, namely 0.45≤Pr≤10000. It is found that for the flow under the sheet there is a reverse flow region near the sheet for small values ofPr, whilst in the case of the flow above the sheet there is no reverse flow region for any value ofPr we have investigated. For the flow under the sheet an asymptotic behaviour, which is valid near the minimum value of the Prandtl number for which it is possible to obtain a numerical solution, is proposed.     

Unsymmetrical nonlinear bending problem of circular thin plate with variable thickness

Introduction Manystructuralelements(pole,plate,shell)withunevenandvariablethicknessarewidely usedinallkindsofengineeringfields.Engineerscansavematerialswhentheydesignbecause theseelementshavebetteroptimizedshapeofstructuralfeature,butthisaddsdifficultytotheanalysisoftheirmechanicalcapability.Manypreviouspapers[1-4]havesolvedtheproblemof symmetricalaxis,butnobodyhassolvedtheunsymmetricalnonlineardeformationproblemof circularthinplatewithvariablethicknessandunsymmetricalaxisuptonow,afewworkonly …     

Numerical solution of three‐dimensional velocity–vorticity Navier–Stokes equations by finite difference method

This paper describes the finite difference numerical procedure for solving velocity–vorticity form of the Navier–Stokes equations in three dimensions. The velocity Poisson equations are made parabolic using the false‐transient technique and are solved along with the vorticity transport equations. The parabolic velocity Poisson equations are advanced in time using the alternating direction implicit (ADI) procedure and are solved along with the continuity equation for velocities, thus ensuring a divergence‐free velocity field. The vorticity transport equations in conservative form are solved using the second‐order accurate Adams–Bashforth central difference scheme in order to assure divergence‐free vorticity field in three dimensions. The velocity and vorticity Cartesian components are discretized using a central difference scheme on a staggered grid for accuracy reasons. The application of the ADI procedure for the parabolic velocity Poisson equations along with the continuity equation results in diagonally dominant tri‐diagonal matrix equations. Thus the explicit method for the vorticity equations and the tri‐diagonal matrix algorithm for the Poisson equations combine to give a simplified numerical scheme for solving three‐dimensional problems, which otherwise requires enormous computational effort. For three‐dimensional‐driven cavity flow predictions, the present method is found to be efficient and accurate for the Reynolds number range 100?Re?2000. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical simulation of MHD stagnation point flow and heat transfer of a micropolar fluid towards a heated shrinking sheet

A comprehensive study of magneto hydrodynamics two‐dimensional stagnation flow with heat transfer characteristics towards a heated shrinking sheet immersed in an electrically conducting incompressible micropolar fluid in the presence of a transverse magnetic field is analyzed numerically. The governing continuity, momentum, angular momentum and heat equations together with the associated boundary conditions are first reduced to a set of self similar nonlinear ordinary differential equations using a similarity transformation and are then solved by a method based on finite difference discretization. Some important features of the flow and heat transfer in terms of normal and streamwise velocities, microrotation and temperature distributions for different values of the governing parameters are analyzed, discussed and presented through tables and graphs. The results indicate that the reverse flow caused due to shrinking of the sheet can be stopped by applying a strong magnetic field. The magnetic field enhances the shear stresses and decreases the thermal boundary layer thickness. The heat loss per unit area from the sheet decreases with an increase in the shrinking parameter. Micropolar fluids exhibit reduction in shear stresses and heat transfer rate as compared with Newtonian fluids, which may be beneficial in the flow and thermal control of polymeric processing. Copyright © 2011 John Wiley & Sons, Ltd.     

Optimal Microstructures Drag Reducing Mechanism of Riblets

We consider an optimal shape design problem of periodically distributed three-dimensional microstructures on surfaces of swimming bodies in order to reduce their drag. Our model is restricted to the flow in the viscous sublayer of the boundary layer of a turbulent flow. The costs for the optimization problem are very high because the three-dimensional flow equations have to be solved several times. We avoid this problem by approximations: the microscopic optimization problem is reduced applying homogenization. Considering a special geometry (riblets) the resulting so-called macroscopic optimization problem can be additionally reduced to a two-dimensional problem. We analyze the drag reducing mechanism of riblets which are believed to be optimal structures. Therefore we perform direct simulations on the total rough channel for different shapes of microstructures: riblets and fully three-dimensional structures.     

Hydromagnetic thin film flow of Casson fluid in non-Darcy porous medium with Joule dissipation and Navier's partial slip

In this paper, the effects of viscous and Ohmic heating and heat generation/absorption on magnetohydrodynamic flow of an electrically conducting Casson thin film fluid over an unsteady horizontal stretching sheet in a non-Darcy porous medium are investigated. The fluid is assumed to slip along the boundary of the sheet. Similarity transformation is used to translate the governing partial differential equations into ordinary differential equations. A shooting technique in conjunction with the 4 th order Runge-Kutta method is used to solve the transformed equations. Computations are carried out for velocity and temperature of the fluid thin film along with local skin friction coefficient and local Nusselt number for a range of values of pertinent flow parameters. It is observed that the Casson parameter has the ability to enhance free surface velocity and film thickness, whereas the Forchheimer parameter, which is responsible for the inertial drag has an adverse effect on the fluid velocity inside the film. The velocity slip along the boundary tends to decrease the fluid velocity. This investigation has various applications in engineering and in practical problems such as very large scale integration(VLSI) of electronic chips and film coating.     

A low‐dimensional description of transient shear‐thinning free‐surface flow in thin cavities,as applied to injection molding

A spectral methodology is proposed to examine the influence of shear thinning on the transient free‐surface flow inside a three‐dimensional thin cavity. The problem is closely related to the filling stage during the injection molding process. The moving domain is mapped onto a rectangular domain at each time step of the computation. A modified pressure is introduced that is governed by the Laplace's equation. The flow field is expanded in Fourier series along the lateral direction in the mapped domain, and the Galerkin projection is used to derive the equations that govern the expansion coefficients, which are solved using a variable‐step finite‐difference scheme. This approach is valid for simple and complex cavities as illustrated for the cases of a flat plate with variable and constant thickness. It is shown that, even for highly non‐linear shear‐thinning flow, only a few modes are needed for convergence. Shear thinning generally influences the flow behaviour. However, shear thinning may enhance or prohibit the flow, depending whether the flow rate at the entrance of the cavity is fast or slow, respectively. Copyright © 2004 John Wiley & Sons, Ltd.     

Construction of the underground contour of a hydraulic structure with constant flow velocity sections

The underground contour of an embedded rectangular dam, whose corners are rounded in accordance with curves of constant flow velocity and whose water-permeable base is underlain by a confining layer with a curvilinear roof characterized by a constant flow velocity, is constructed. The corresponding boundary value problem is solved by means of the semi-inverse use of the velocity hodograph method. The results of the numerical calculations are given and the effect of the main determining parameters of the model on the shape and dimensions of the underground contour of the dam and the curvilinear confining layer is analyzed. The limiting cases in which the water-permeable base of the dam has an unbounded thickness, namely, a streamlined apron with a horizontal insert and streamlined sheet piling (tooth), are investigated in detail.     

旋流喷嘴中空旋转射流近区域流动的研究

从理论上分析了旋流喷嘴喷出的中空射流近区域的液膜的运动,在只考虑液膜表面张力的作用下,应用质量守恒和动量定理,建立了描述液膜运动的非线性常微分（积分）方程组,该方程组可以用数值方法方便地求解,结果表明,液体离开旋流喷嘴后在自由空间形成的液膜呈葫芦形状,其速度和液膜厚度等都周期性地变化。本结果是在液厝受拓动失称碎成液滴前的最基本运动状态,可以在射流的近区域内实验观察到,也是进一步从理论液膜破碎雾化过     

Surfactant spreading on a thin weakly viscoelastic film

A mathematical model is presented for surfactant-driven thin weakly viscoelastic film flows on a flat, impermeable plane. The Oldroyd-B constitutive relation is used to model the viscoelastic fluid. Lubrication theory and a perturbation expansion in powers of the Weissenberg number (We) are employed, which give rise to non-linear coupled evolution equations governing the transport of insoluble surfactant and thin liquid film thickness. Spreading on a Newtonian film is recovered to leading order and corrections to viscoelasticity are obtained at order We. These equations are solved numerically over a wide range of viscosity ratio (ratio of solvent viscosity to the sum of solvent and polymeric viscosities), pre-existing surfactant level and Peclet number (Pe). The effect of viscoelasticity on surfactant transport and fluid flow is investigated and the mechanisms underlying this effect are explored. Shear stress, streamwise normal stress and the temporal rate of change of extra shear stress generated from gradients in surfactant concentration dominate thin viscoelastic film flows whereas only shear stresses play a role in Newtonian thin film flows. Our results also reveal that, for weak viscoelasticity, the influence of viscosity ratio on the evolution of surfactant concentration and film thickness can be significant and varies considerably, depending on the concentration of pre-existing surfactant and surfactant surface diffusivity.     

Cascaded thin lightly loaded airfoils vibrating in a subsonic separated flow

The problem of calculating the nonstationary aerodynamic characteristics of a cascade of thin lightly loaded airfoils in a subsonic flow with the formation of thin separation zones of finite extent is solved approximately. As in [1–5], an approach based on a linear small-perturbation analysis, within which the flow is assumed to be inviscid, is employed and the boundaries of the unsteady separation zones are simulated by oscillating lines of contact discontinuity. However, instead of the requirement of a given fixed pressure at the boundary of the separation zone, used in [1–5], this study proposes a more general condition according to which on each element of length of the thin separation layer the pressure oscillates with an amplitude proportional to the local value of the amplitude of its thickness oscillations. The problem is reduced to a system of two singular integral equations which can be solved numerically.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 181–191, January–February, 1995.     

A co‐ordinate system independent streamwise upwind method for fluid flow computation

A new co‐ordinate invariant streamwise upwind formulation for convection dominated flows is developed. The eddy diffusivity/viscosity is added directly to the equations in order to remove the oscillations in the solution. The equations then can be solved by any high‐order scheme and the solution retains the accuracy of the high‐order scheme. The accuracy and reduced lateral thickness growth rate are demonstrated with several numerical examples, including pure convective flows and lid‐driven cavity flow. The lateral spreading due to the numerical diffusion is controlled by the anisotropic tensor. Copyright © 2004 John Wiley & Sons, Ltd.     

Effect of slip velocity on Casson thin film flow and heat transfer due to unsteady stretching sheet in presence of variable heat flux and viscous dissipation

The aim of the present paper is to study flow and heat transfer characteristics of a viscous Casson thin film flow over an unsteady stretching sheet subject to variable heat flux in the presence of slip velocity condition and viscous dissipation. The governing equations are partial differential equations. They are reduced to a set of highly nonlinear ordinary differential equations by suitable similarity transformations. The resulting similarity equations are solved numerically with a shooting method. Comparisons with previous works are made, and the results are found to be in excellent agreement. In the present work, the effects of the unsteadiness parameter, the Casson parameter, the Eckert number, the slip velocity parameter, and the Prandtl number on flow and heat transfer characteristics are discussed. Also, the local skin-friction coefficient and the local Nusselt number at the stretching sheet are computed and discussed.     

Linearized and non‐linear acoustic/viscous splitting techniques for low Mach number flows

Computation of the acoustic disturbances generated by unsteady low‐speed flow fields including vortices and shear layers is considered. The equations governing the generation and propagation of acoustic fluctuations are derived from a two‐step acoustic/viscous splitting technique. An optimized high order dispersion–relation–preserving scheme is used for the solution of the acoustic field. The acoustic field generated by a corotating vortex pair is obtained using the above technique. The computed sound field is compared with the existing analytic solution. Results are in good agreement with the analytic solution except near the centre of the vortices where the acoustic pressure becomes singular. The governing equations for acoustic fluctuations are then linearized and solved for the same model problem. The difference between non‐linear and linearized solutions falls below the numerical error of the simulation. However, a considerable saving in CPU time usage is achieved in solving the linearized equations. The results indicate that the linearized acoustic/viscous splitting technique for the simulation of acoustic fluctuations generation and propagation by low Mach number flow fields seems to be very promising for three‐dimensional problems involving complex geometries. Copyright © 2003 John Wiley & Sons, Ltd.