Fully-developed free-convective flow of micropolar and viscous fluids in a vertical channel

The problem of fully-developed laminar free-convection flow in a vertical channel is studied analytically with one region filled with micropolar fluid and the other region with a viscous fluid. Using the boundary and interface conditions proposed by previous investigators, analytical expressions for linear velocity, micro-rotation velocity and temperature have been obtained. Numerical results are presented graphically for the distribution of velocity, micro-rotation velocity and temperature fields for varying physical parameters such as the ratio of Grashof number to Reynolds number, viscosity ratio, width ratio, conductivity ratio and micropolar fluid material parameter. It is found that the effect of the micropolar fluid material parameter suppress the velocity whereas it enhances the micro-rotation velocity. The effect of the ratio of Grashof number to Reynolds number is found to enhance both the linear velocity and the micro-rotation velocity. The effects of the width ratio and the conductivity ratio are found to enhance the temperature distribution.     

Solution of the Falkner–Skan equation for wedge by Adomian Decomposition Method

The Adomian Decomposition Method is employed in the solution of the two dimensional laminar boundary layer of Falkner–Skan equation for wedge. This work aims at the solution of momentum equation in the case of accelerated flow and decelerated flow with separation. The Adomian Decomposition Method is provided an analytical solution in the form of an infinite power series. The effect of Adomian polynomials terms is considered on accuracy of the results. The velocity profiles in boundary layer are obtained. Results show a good accuracy compared to the exact solution.     

An Exact Navier-Stokes Solution for Three-Dimensional,Spanwise-Homogeneous Boundary Layers

The plane stagnation flow onto (Hiemenz boundary layer, HBL) and the asymptotic suction boundary layer flow over a flat wall (ASBL) are two boundary layer flows for which the incompressible Navier-Stokes equations are amenable to exact similarity solutions. The Hiemenz solution has been extended to swept Hiemenz flows by superposition of a third, spanwise-homogeneous sweep velocity. This solution becomes singular as the chordwise, tangential base flow component vanishes. In this limit, the homogeneous ASBL solution is valid, which however cannot describe the swept Hiemenz flow, because it does not contain any chordwise velocity. This work presents a generalized three-dimensional similarity solution which describes three-dimensional spanwise homogeneously impinging boundary layers at arbitrary wall-normal suction velocities, using a rescaled similarity coordinate. The HBL and the ASBL are shown to be two limits of this solution. Further extensions consist of oblique impingement or different boundary suction directions, such as slip or stretching walls. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A moving-wall boundary layer flow of a slightly rarefied gas free stream over a moving flat plate

In the current work, the boundary layer flow of a slightly rarefied gas free stream over a moving flat plate is presented and solved numerically. The first-order slip boundary condition is adopted in the derivation. The dimensionless velocity and shear stress profiles are plotted and discussed. A theoretical derivation of the estimated solution domain is developed, which will give a very close estimation to the exact solution domain obtained numerically. The influences of velocity slip at the wall on the velocity and shear stress are also addressed.     

Flow of a micropolar fluid through a circular cylinder subject to longitudinal and torsional oscillations

The internal flow of a micropolar fluid inside a circular cylinder which is subject to longitudinal and torsional oscillations is investigated. Analytical expressions of the fluid velocity and micro-rotation are obtained. Explicit expressions of the shear stresses and drag force acting at the wall of the cylinder are derived as well. A numerical analysis followed to examine the effect of the micropolar fluid on the two components of the velocity field through graphical curves. In addition, the magnitude of the tangential drag is computed and compared with the case of a classical fluid.     

Steady laminar boundary layer of a viscous incompressible fluid in a convergent channel with distributed suction at the wall

In this paper an attempt has been made to find the solution of the boundary layer equations for two-dimensional laminar steady motion of a viscous incompressible fluid in a convergent channel (sink flow) with suction at the wall. Suction velocity v₀ (x) ～ 1/x has been imposed at the wall and an approximate solution has been obtained with the help of similarity transformation. A solution valid at a large distance from the wall and a series solution valid near the wall have been obtained and the two solutions have been joined at a suitable point. It is seen that the boundary layer thickness diminishes as the value of the suction parameter\(\lambda ( = v_0 x/\sqrt {u_1 v} )\) increases. The velocity profile and the boundary layer parameters for solid wall (λ = 0) obtained from this solution are found to be in close agreement with the profile and the parameters calculated from the known exact solution for the solid wall problem.     

Effects of slip on steady Bödewadt flow of a non-Newtonian fluid

The steady flow arising due to the rotation of a non-Newtonian fluid at a larger distance from a stationary disk is extended to the case where the disk surface admits partial slip. The constitutive equation of the non-Newtonian fluid is modeled by that for a Reiner–Rivlin fluid. The momentum equation gives rise to a highly nonlinear boundary value problem. Numerical solution of the governing nonlinear equations are obtained over the entire range of the physical parameters. The effects of slip and non-Newtonian fluid characteristics on the momentum boundary layer are discussed in details. It is observed that slip has prominent effect on the velocity field, whereas a predominant influence of the non-Newtonian parameter is observed on the moment coefficient.     

Transient flow in a rotating porous annulus due to suction at the walls

A highly porous material occupies the annular region between two coaxial infinitely long cylinders. A viscous incompressible fluid fills this porous medium and is initially in a state of rigid rotation together with the medium. The flow has been disturbed by imposing suction/injection at the outer/inner cylindrical boundaries respectively. The Brinkman's law has been used to represent the fluid motion. The exact solution for the resulting unsteady flow is obtained by Laplace transformation technique. The transient evolution of the boundary layers and the response of steady boundary layers to the resistance of the medium are discussed.     

Effects of slip on steady Bödewadt flow and heat transfer of an electrically conducting non-Newtonian fluid

The steady flow and heat transfer arising due to the rotation of a non-Newtonian fluid at a larger distance from a stationary disk is extended to the case where the disk surface admits partial slip. The constitutive equation of the non-Newtonian fluid is modeled by that for a Reiner–Rivlin fluid. The fluid is subjected to an external uniform magnetic field perpendicular to the plane of the disk. The momentum equation gives rise to a highly nonlinear boundary value problem. Numerical solution of the governing nonlinear equations are obtained over the entire range of the physical parameters. The effects of slip, non-Newtonian fluid characteristics and the magnetic interaction parameter on the momentum boundary layer and thermal boundary layer are discussed in detail and shown graphically. It is observed that slip has prominent effects on the velocity and temperature fields.     

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.     

具有不同渗透的涨缩管道内微极性流动的同伦分析解

分析了壁面具有不同渗透的涨缩管道内微极性流体的流动．对于壁面的胀缩,考虑常系数和时间函数的膨胀率两种情况．对于第1种情况,应用同伦分析方法得到该问题的速度和微旋转角度的表达式．并且画图分析了各个不同参数,特别是膨胀系数和不同的渗透率对流体的动力特征的影响．可以得到第1个重要的结论:壁面的膨胀率和不同的渗透对流体的动力特征有重要的影响．根据Xu的模型,考虑了第2种也是更具有一般性的情况,假设壁面的膨胀率随时间的变化而变化．在这样的假设下,控制方程被转化成非线性偏微分方程,并且同样也可以应用HAM方法进行求解．应用代数和指数的模型来描述膨胀率从初始状态到最终状态的演变过程．然而,结果表明包含有时间的解很快地趋向于稳态的解．这样可以得到第2个重要的结论,时间在壁面的膨胀收缩中扮演着次要的角色,可以忽略不计．     

Magnetohydrodynamics flow of a Burgers’ fluid in an orthogonal rheometer

The magnetohydrodynamics flow of an electrically conducting, incompressible Burgers’ fluid in an orthogonal rheometer is investigated. An exact solution is obtained. The effects of various dimensionless parameters existing in the model on the velocity field, vorticity and traction are studied graphically. It is noted that boundary layers form for a variety of reasons. It form as the Reynolds number increases. Also, as the Weissenberg number increases a distinct boundary layer formation is observed. It can develop at low Reynolds number provided the Weissenberg number is sufficiently high, however, it is not possible in the case of a Newtonian fluid. It is shown that no torque is exerted by the fluid on one of the disks. Results are compared with Oldroyd-B fluid.     

半渗透涨缩管道内微极性流动解析求解

分析了半渗透涨缩管道内的微极性流体的流动．应用合适的相似变换,将控制方程转化为常微分方程组．为了得到该问题的解析解,应用同伦分析方法得到该问题的速度表达式．并且用图形分析了各个不同参数,特别是膨胀系数对速度场和微旋转角速度的影响．     

The transient retardation of a rectilinear viscoplastic flow when the loading stresses are abruptly removed

The development of a viscoplastic flow in a solid layer of an elastoviscoplastic material on an inclined plane is considered when loading stresses act on its free surface. It is shown that the elastoplastic boundary starts its motion from the rigid inclined plane and, propagating through the elastic core, it can reach the free surface of the layer. An exact solution is obtained for the dynamic problem of the retardation of developed viscoplastic flow after the loading stresses are abruptly removed. The possibility of writing the equation of motion for the unloading wave in terms of the displacements is pointed out. It reduces to an inhomogeneous wave equation where the velocity of the unloading wave is found to be equal to the velocity of the equivoluminal elastic wave. Reflection of the unloading wave from a rigid boundary in the form of an inclined plane is also considered.     

Third-order partial differential equations arising in the impulsive motion of a flat plate

We obtain numerical solutions to a class of third-order partial differential equations arising in the impulsive motion of a flat plate for various boundary data. In particular, we study the case of constant acceleration of the plate, the case of oscillation of the plate, and a case in which velocity is increasing yet acceleration is decreasing. We compare the numerical solutions with the known exact solutions in order to establish the validity of the method. Several figures illustrating both solution forms and the relative strength of the second and third-order terms are presented. The results obtained in this study reveal many interesting behaviors that warrant further study on the non-Newtonian fluid flow phenomena.     

On explicit,purely analytic solutions of off-centered stagnation flow towards a rotating disc by means of HAM

In this article, the off-centered stagnation flow towards a rotating disc is studied analytically. The governing non-linear equations and their associated boundary conditions are transformed into coupled non-linear ordinary differential equations. The series solution of the problem is obtained by utilizing the HAM. The convergence of the obtained series solutions is carefully checked. Graphical results are presented to investigate the influence of the rotation ratio on the flow field. An important point to note is that the non-alignment complicates the flow field and surface shear, but does not affect the torque. It is noted that the behavior of the HAM solution for velocity components is in good agreement with the numerical solution given in reference [C.Y. Wang, Off-centered stagnation flow towards a rotating disc, Int. J. Eng. Sci. 46 (2008) 391–396].     

Analytical approximate solution of the cooling problem by Adomian decomposition method

The Adomian decomposition method (ADM) can provide analytical approximation or approximated solution to a rather wide class of nonlinear (and stochastic) equations without linearization, perturbation, closure approximation, or discretization methods. In the present work, ADM is employed to solve the momentum and energy equations for laminar boundary layer flow over flat plate at zero incidences with neglecting the frictional heating. A trial and error strategy has been used to obtain the constant coefficient in the approximated solution. ADM provides an analytical solution in the form of an infinite power series. The effect of Adomian polynomial terms is considered and shows that the accuracy of results is increased with the increasing of Adomian polynomial terms. The velocity and thermal profiles on the boundary layer are calculated. Also the effect of the Prandtl number on the thermal boundary layer is obtained. Results show ADM can solve the nonlinear differential equations with negligible error compared to the exact solution.     

Free vibration and stability analysis of a multi-span pipe conveying fluid using exact and variational iteration methods combined with transfer matrix method

In this paper, a new formulation based on the variational iteration method (VIM) is applied to investigate the dynamic behavior and stability of a multi-span pipe conveying fluid. Transfer matrix method (TMM) is used to assemble the system of equations resulting from applying the boundary conditions. The natural frequencies of the pipe system are obtained for different flow velocities. Results from VIM are compared with those predicted by the exact solution method and also with published literature. The influence of the number of spans on the VIM convergence is investigated. Also, the effects induced by varying the value and location of an intermediate elastic support on the critical velocity and stability are studied. It is shown that using VIM yields highly accurate results that are in very well agreement with the exact solution. The main advantage of the VIM is that it successfully overcomes well-known computational difficulties that are usually encountered during complex root finding step maintaining high precision as well.     

Viscous flow over a shrinking sheet with an arbitrary surface velocity

In this paper, an analytical solution in a closed form for the boundary layer flow over a shrinking sheet is presented when arbitrary velocity distributions are applied on the shrinking sheet. The solutions with seven typical velocity profiles are derived based on a general closed form expression. Such flow is usually not self-similar and the solution can only be implemented when the mass transfer at the wall is prescribed and determined by the moving velocity of the wall. The characteristics of the flows with the typical velocity distributions are discussed and compared with previous similarity solutions. The flow is observed to have quite different behavior from that of the self-similar flow reported in the literature and the results demonstrate distinctive momentum and energy transport characteristics. Some plots of the stream functions are also illustrated to show the difference in flow field between the shrinking sheet and the stretching sheet. An integral approach to solve boundary layer flow over a shrinking or stretching sheet with uncoupled arbitrary surface velocity and wall mass transfer velocity is outlined and the effectiveness of this approach is discussed.     

Micro/nano sliding plate problem with Navier boundary condition

For Newtonian flow through micro or nano sized channels, the no-slip boundary condition does not apply and must be replaced by a condition which more properly reflects surface roughness. Here we adopt the so-called Navier boundary condition for the sliding plate problem, which is one of the fundamental problems of fluid mechanics. When the no-slip boundary condition is used in the study of the motion of a viscous Newtonian fluid near the intersection of fixed and moving rigid plane boundaries, singular pressure and stress profiles are obtained, leading to a non-integrable force on each boundary. Here we examine the effects of replacing the no-slip boundary condition by a boundary condition which attempts to account for boundary slip due to the tangential shear at the boundary. The Navier boundary condition, possesses a single parameter to account for the slip, the slip length ℓ, and two solutions are obtained; one integral transform solution and a similarity solution which is valid away from the corner. For the former the tangential stress on each boundary is obtained as a solution of a set of coupled integral equations. The particular case solved is right-angled corner flow and equal slip lengths on each boundary. It is found that when the slip length is non-zero the force on each boundary is finite. It is also found that for a suffciently large distance from the corner the tangential stress on each boundary is equal to that of the classical solution. The similarity solution involves two restrictions, either a right-angled corner flow or a dependence on the two slip lengths for each boundary. When the tangential stress on each boundary is calculated from the similarity solution, it is found that the similarity solution makes no additional contribution to the tangential stress of that of the classical solution, thus in agreement with the findings of the integral transform solution. Values of the radial component of velocity along the line θ = π /4 for increasing distance from the corner for the similarity and integral transform solutions are compared, confirming their agreement for sufficiently large distances from the corner. (Received: November 9, 2005)