A new family of unsteady boundary layers over a stretching surface

In this paper, a new family of unsteady boundary layers over a stretching flat surface was proposed and studied. This new class of unsteady boundary layers involves the flows over a constant speed stretching surface from a slot, and the slot is moving at a certain speed. Depending on the slot moving parameter, the flow can be treated as a stretching sheet problem or a shrinking sheet problem. Both the momentum and thermal boundary layers were studied. Under special conditions, the solutions reduce to the unsteady Rayleigh problem and the steady Sakiadis stretching sheet problem. Solutions only exist for a certain range of the slot moving parameter, α. Two solutions are found for −53.55° < α < −45°. There are also two solution branches for the thermal boundary layers at any given Prandtl number in this range. Compared with the upper solution branch, the lower solution branch leads to simultaneous reduction in wall drag and heat transfer rate. The results also show that the motion of the slot greatly affects the wall drag and heat transfer characteristics near the wall and the temperature and velocity distributions in the fluids.     

Three-dimensional flow over a stretching surface in a viscoelastic fluid

This article looks at the hydrodynamic elastico-viscous fluid over a stretching surface. The equations governing the flow are reduced to ordinary differential equations, which are analytically solved by applying an efficient technique namely the homotopy analysis method (HAM). The solutions for the velocity components are computed. The numerical values of wall skin friction coefficients are also tabulated. The present HAM solution is compared with the known exact solution for the two-dimensional flow and an excellent agreement is found.     

Effect of wall compliance on compressible fluid transport induced by a surface acoustic wave in a microchannel

Effects of complaint wall properties on the flow of a Newtonian viscous compressible fluid has been studied when the wave propagating (surface acoustic wave, SAW) along the walls in a confined parallel‐plane microchannel is conducted by considering the slip velocity. A perturbation technique has been employed to analyze the problem where the amplitude ratio (wave amplitude/half width of channel) is chosen as a parameter. In the second order approximation, the net axial velocity is calculated for various values of the fluid parameters and wall parameters. The phenomenon of the “mean flow reversal” is found to exist both at the center and at the boundaries of the channel. The effect of damping force, wall tension, and compressibility parameter on the mean axial velocity and reversal flow has been investigated, also the critical values of the tension are calculated for the pertinent flow parameters. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 621–636, 2011 Keywords:     

A new model for study the effect of wall properties on peristaltic transport of a viscous fluid

The present study extends the two-dimensional analysis of peristaltic motion to include a compliant wall. The fluid-solid interaction problem is investigated by considering equations of motion of both the fluid and the deformable boundaries. The driving mechanism of the muscle is represented by assuming the channel walls to be compliant. A perturbation solution of the stream function for zeroth, first and second order in a small amplitude ratio is obtained. The phenomenon of the “mean flow reversal” is found to exist both at the center and at the boundaries of the channel. The effect of wall damping, wall elastance and wall tension on the mean axial velocity and reversal flow has been investigated. The numerical results show that the possibility of flow reversal increases by increasing the wall damping and decreases by increasing the wall elastance and wall tension.     

Dense underflow into a lake or reservoir – supercritical flow solutions

Steady two-dimensional flow from an angled structure into a lake or a reservoir where the interface between the intrusion and the ambient fluid separates from a solid wall is considered. The fluid is assumed to be of finite depth and the incoming channel makes a downward angle α with the horizontal axis. This simple configuration provides a model for the plunging inflow and subsequent underflow of dense water in a reservoir or lake. Exact solutions are presented at infinite Froude number and compared with the solutions to the full nonlinear problem for supercritical flow. Limiting flows are found to separate from the upper boundary at a stagnation point, and regions of non-uniqueness in the solution domain are found.     

考虑到渗透效应的一种血液流动的计算方法

得到了定常情况下,狗二分叉动脉横截面的三维Navier＿Stokes方程的有限元处理方法,并考虑到管壁的渗透影响,数值方法还包括直角坐标和曲线坐标的变换·　详细讨论了渗透性影响下的定常流、分叉流以及切应力情况·　以分支和主干血管的速度比为参量,计算雷诺数为1000情况下管壁切应力,数值结果和先前的实验结果符合得很好·　该文的工作是Sharma等(2001)工作的改进,使计算量更小,能够处理的雷诺数范围更大·     

Singularities on Free Surfaces of Fluid Flows

Isolated singularities on free surfaces of two-dimensional and axially symmetric three-dimensional steady potential flows with gravity are considered. A systematic study is presented, where known solutions are recovered and new ones found. In two dimensions, the singularities found include those described by the Stokes solution with a 120° angle, Craya's flow with a cusp on the free surface, Gurevich's flow with a free surface meeting a rigid plane at 120° angle, and Dagan and Tulin's flow with a horizontal free surface meeting a rigid wall at an angle less than 120°. In three dimensions, the singularities found include those in Garabedian's axially symmetric flow about a conical surface with an approximately 130° angle, flows with axially symmetric cusps, and flows with a horizontal free surface and conical stream surfaces. The Stokes, Gurevich, and Garabedian flows are exact solutions. These are used to generate local solutions, including perturbations of the Stokes solution by Grant and Longuet-Higgins and Fox, perturbations of Gurevich's flow by Vanden-Broeck and Tuck, asymmetric perturbations of Stokes flow and nonaxisymmetric perturbations of Garabedian's flow. A generalization of the Stokes solution to three fluids meeting at a point is also found.     

Stokes flow solutions in infinite and semi-infinite porous channels

We derive a class of exact solutions for Stokes flow in infinite and semi-infinite channel geometries with permeable walls. These simple, explicit, series expressions for both pressure and Stokes flow are valid for all permeability values. At the channel walls, we impose a no-slip condition for the tangential fluid velocity and a condition based on Darcy's law for the normal fluid velocity. Fluid flow across the channel boundaries is driven by the pressure drop between the channel interior and exterior; we assume the exterior pressure to be constant. We show how the ground state is an exact solution in the infinite channel case. For the semi-infinite channel domain, the ground-state solutions approximate well the full exact solution in the bulk and we derive a method to improve their accuracy at the transverse wall. This study is motivated by the need to quantitatively understand the detailed fluid dynamics applicable in a variety of engineering applications including membrane-based water purification, heat and mass transfer, and fuel cells.     

Heat transfer of a generalized stretching/shrinking wall problem with convective boundary conditions

In this paper, we investigate the heat transfer of a viscous fluid flow over a stretching/shrinking sheet with a convective boundary condition. Based on the exact solutions of the momentum equations, which are valid for the whole Navier–Stokes equations, the energy equation ignoring viscous dissipation is solved exactly and the effects of the mass transfer parameter, the Prandtl number, and the wall stretching/shrinking parameter on the temperature profiles and wall heat flux are presented and discussed. The solution is given as an incomplete Gamma function. It is found the convective boundary conditions results in temperature slip at the wall and this temperature slip is greatly affected by the mass transfer parameter, the Prandtl number, and the wall stretching/shrinking parameters. The temperature profiles in the fluid are also quite different from the prescribed wall temperature cases.     

A new analytical solution branch for the Blasius equation with a shrinking sheet

A similarity equation of the momentum boundary layer is analytically studied for a moving flat plate with mass transfer in a stationary fluid by a newly developed technique namely homotopy analysis method (HAM). The equation shows its significance for the practical problem of a shrinking sheet with a constant velocity, and only admits the existence of the solution with mass suction at the wall surface. The present work provides analytically new solution branch of the Blasius equation with a shrinking sheet in different solution areas, including both multiple solutions and unique solution with the aid of an introduced auxiliary function. The analytical results show that quite complicated behavior with three different solution areas controlled by two critical mass transfer parameters exists, which agrees well with the numerical techniques and greatly differs from the continuously stretching surface problem and the Blasius problem with a free stream. The new analytical solution branch of the Blasius equation with a shrinking sheet enriches the solution family of the Blasius equation, and helps to deeply understand the Blasius equation.     

On sinc discretization and banded preconditioning for linear third‐order ordinary differential equations

Some draining or coating fluid‐flow problems and problems concerning the flow of thin films of viscous fluid with a free surface can be described by third‐order ordinary differential equations (ODEs). In this paper, we solve the boundary value problems of such equations by sinc discretization and prove that the discrete solutions converge to the true solutions of the ODEs exponentially. The discrete solution is determined by a linear system with the coefficient matrix being a combination of Toeplitz and diagonal matrices. The system can be effectively solved by Krylov subspace iteration methods, such as GMRES, preconditioned by banded matrices. We demonstrate that the eigenvalues of the preconditioned matrix are uniformly bounded within a rectangle on the complex plane independent of the size of the linear system. Numerical examples are given to illustrate the effective performance of our method. Copyright © 2010 John Wiley & Sons, Ltd.     

Falkner-Skan equation for flow past a stretching surface with suction or blowing: Analytical solutions

The simultaneous effects of suction and injection on tangential movement of a nonlinear power-law stretching surface governed by laminar boundary layer flow of a viscous and incompressible fluid beneath a non-uniform free with stream pressure gradient is considered. The self-similar flow is governed by Falkner-Skan equation, with transpiration parameter γ, wall slip velocity λ and stretching sheet (or pressure gradient) parameter β. The exact solution for β = −1 and three closed form asymptotic solutions for β large, large suction γ, and λ → 1 have also been presented. Dual solutions are found for β = −1 for each value of the transpiration parameter, including the non-permeable surface, for each prescribed value of the wall slip velocity λ. The large β asymptotic solution also dual with respect to wall slip velocity λ, but do not depend on suction and blowing. The critical values of γ, β and λ are obtained and their significance on the skin friction and velocity profiles is discussed. An approximate solution by integral method for a trial velocity profile is presented and results are compared with the exact solutions.     

精确的磁流体动力学汇流解析解

就一个特殊的磁流体动力学(MHD)流动,即速度幂指数为-1时的汇流,得到著名的Falkner-Skan方程精确的解析解.解析解是封闭的,并有多重解分支.分析了磁场参数和壁面伸长参数的影响.发现了有趣的速度分布现象:即使壁面固定,回流区域依然出现.在一个罕见的FalknerSkan MHD流动中,得到了一组解,以精确封闭的解析公式表示,极大地丰富了著名的Falkner-Skan方程的解析解,也加深了对这重要又有趣方程的理解.     

特殊坐标系中特殊非牛顿流边界层方程的相似解

给出了在一个特殊坐标系中三阶流体的二维定常运动方程组．该坐标系中由无粘流体的势流确定，即以环绕任意物体的非粘性流动的流线为Ф-坐标，速度势线为ψ-坐标，构成正交曲线坐标系．结果表明，边界层方程与浸没在流体中的物体的形状无关．第一次近似假定第二梯度项与粘性项和第三梯度项相比，可以忽略不计．第二梯度项的存在，将防碍第三梯度流相似解的比例变换的导出．利用李群方法计算了边界层方程的无穷小生成元．将边界层方程组变换为常微分方程组．利用Runge-Kutta法结合打靶技术求解了该非线性微分方程组的数值解．     

On some unsteady flows of a non-Newtonian fluid

Some properties of unsteady unidirectional flows of a fluid of second grade are considered for flows impulsively started from rest by the motion of a boundary or two boundaries or by sudden application of a pressure gradient. Flows considered are: unsteady flow over a plane wall, unsteady Couette flow, flow between two parallel plates suddenly set in motion with the same speed, flow due to one rigid boundary moved suddenly and one being free, unsteady Poiseuille flow and unsteady generalized Couette flow. The results obtained are compared with those of the exact solutions of the Navier–Stokes equations. It is found that the stress at time zero on the stationary boundary for the flows generated by impulsive motion of a boundary or two boundaries is finite for a fluid of second grade and infinite for a Newtonian fluid. Furthermore, it is shown that for unsteady Poiseuille flow the stress at time zero on the boundary is zero for a Newtonian fluid, but it is not zero for a fluid of second grade.     

Comment on: “Fundamental flows with nonlinear slip conditions: exact solutions”, by R. Ellahi, T. Hayat, F. M. Mahomed and A. Zeeshan, Z. Angew. Math. Phys. 61 (2010) 877–888

In their article (Fundamental flows with nonlinear slip conditions: exact solutions, R. Ellahi, T. Hayat, F. M. Mahomed and A. Zeeshan, Z. Angew. Math. Phys. 61 (2010) 877–888.), the authors considered three simple cases of the steady flow of a third grade fluid between parallel plates with slip conditions; namely, Couette flow, Poiseuille flow, and generalized Couette flow. They obtained exact solutions, which were utilized in a way that did not lead to useful results. Their conclusion that the Couette flow cannot be obtained from the generalized Couette flow, by dropping the pressure gradient, is incorrect. Meaningful results based on their solution are herein presented.     

Stagnation point flow and heat transfer over a non-linearly moving flat plate in a parallel free stream with slip

An analysis is presented for the steady boundary layer flow and heat transfer of a viscous and incompressible fluid in the stagnation point towards a non-linearly moving flat plate in a parallel free stream with a partial slip velocity. The governing partial differential equations are converted into nonlinear ordinary differential equations by a similarity transformation, which are then solved numerically using the function bvp4c from Matlab for different values of the governing parameters. Dual (upper and lower branch) solutions are found to exist for certain parameters. Particular attention is given to deriving numerical results for the critical/turning points which determine the range of existence of the dual solutions. A stability analysis has been also performed to show that the upper branch solutions are stable and physically realizable, while the lower branch solutions are not stable and, therefore, not physically possible.     

Comment on: “Fundamental flows with nonlinear slip conditions: exact solutions”, by R. Ellahi,T. Hayat,F. M. Mahomed and A. Zeeshan,Z. Angew. Math. Phys. 61 (2010) 877–888

In their article (Fundamental flows with nonlinear slip conditions: exact solutions, R. Ellahi, T. Hayat, F. M. Mahomed and A. Zeeshan, Z. Angew. Math. Phys. 61 (2010) 877–888.), the authors considered three simple cases of the steady flow of a third grade fluid between parallel plates with slip conditions; namely, Couette flow, Poiseuille flow, and generalized Couette flow. They obtained exact solutions, which were utilized in a way that did not lead to useful results. Their conclusion that the Couette flow cannot be obtained from the generalized Couette flow, by dropping the pressure gradient, is incorrect. Meaningful results based on their solution are herein presented.     

Couette and Poiseuille flows of an Oldroyd 6-constant fluid with magnetic field

In this paper, we develop a set of differential equations describing the steady flow of an Oldroyd 6-constant magnetohydrodynamic fluid. The fluid is electrically conducting in the presence of a uniform transverse magnetic field. The developed non-linear differential equation takes into account the effect of the material constants and the applied magnetic field. We presented the solution for three types of steady flows, namely,

(i)

Couette flow

(ii)

Poiseuille flow and

(iii)

generalized Couette flow.

Homotopy analysis method (HAM) is used to solve the non-linear differential equation analytically. It is found from the present analysis that for steady flow the obtained solutions are strongly dependent on the material constants (non-Newtonian parameters) which is different from the model of Oldroyd 3-constant fluid. Numerical solutions are also given and compared with the solutions by HAM.