Analysis of viscous flow due to a stretching sheet with surface slip and suction

The viscous flow due to a stretching sheet with slip and suction is studied. The Navier–Stokes equations admit exact similarity solutions. For two-dimensional stretching a closed-form solution is found and uniqueness is proved. For axisymmetric stretching both existence and uniqueness are shown. The boundary value problem is then integrated numerically.     

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 incom pressible electrically con- ducting 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 residua 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.     

Flow of a second grade fluid over a sheet stretching with arbitrary velocities subject to a transverse magnetic field

The boundary layer flow of a second grade fluid over a permeable stretching surface with arbitrary velocity and appropriate wall transpiration is investigated. The fluid is electrically conducting in the presence of a constant applied magnetic field. An exact solution to the nonlinear flow problem is presented.     

Analytical solution of stagnation-point flow of a viscoelastic fluid towards a stretching surface

In this paper the problem of stagnation-point flow of a viscoelastic fluid towards a stretching surface [T.R. Mahapatra, A.S. Gupta, Stagnation-point flow of a viscoelastic fluid towards a stretching surface, Int. J. Non-Linear Mech. 39 (2004) 811] is solved analytically by using the homotopy analysis method (HAM). The results for velocity and temperature profiles are obtained. It is noted that the behavior of the HAM solution for velocity and temperature profiles is in good agreement with the numerical solution given in reference [T.R. Mahapatra, A.S. Gupta, Stagnation-point flow of a viscoelastic fluid towards a stretching surface, Int. J. Non-Linear Mech. 39 (2004) 811].     

Flows of a polymer fluid in domain with impermeable boundaries

Nonlinear boundary value problems modeling steady polymer flows in domains with impermeable solid walls are studied. The solvability of a nonhomogeneous boundary value problem for the equations governing a polymer flow in the case of an impermeable boundary is proved. The norms of solutions are estimated. The set of weak solutions is shown to be sequentially weakly closed. Additionally, explicit formulas are found for computing the solution of the boundary value problem describing the polymer flow induced by a stretching (shrinking) sheet.     

Numerical solution of the momentum and heat transfer equations for a hydromagnetic flow due to a stretching sheet of a non-uniform property micropolar liquid

A study of the hydromagnetic flow due to a stretching sheet and heat transfer in an incompressible micropolar liquid is made. Temperature-dependent thermal conductivity and a non-uniform heat source/sink render the problem analytically intractable and hence a numerical study is made using the shooting method based on Runge-Kutta and Newton-Raphson methods. The two problems of horizontal and vertical stretching are considered to implement the numerical method. The former problem involves one-way coupling between linear momentum and heat transport equations and the latter involves two-way coupling. Further, both the problems involve two-way coupling between the non-linear equations of conservation of linear and angular momentums. A similarity transformation arrived at for the problem using the Lie group method facilitates the reduction of coupled, non-linear partial differential equations into coupled, non-linear ordinary differential equations. The algorithm for solving the resulting coupled, two-point, non-linear boundary value problem is presented in great detail in the paper. Extensive computation on velocity and temperature profiles is presented for a wide range of values of the parameters, for prescribed surface temperature (PST) and prescribed heat flux (PHF) boundary conditions.     

Convective heat transfer in a conducting fluid over a permeable stretching surface with suction and internal heat generation/absorption

We consider a convective flow in a porous medium of an incompressible viscous conducting fluid impinging on a permeable stretching surface with suction, and internal heat generation/absorption. Using a similarity transformation the governing equations of the problem are reduced to a coupled third-order nonlinear ordinary differential equations. We first examine a number of special cases for which we may obtain exact solutions. We then obtain analytical solutions (by the Homotopy Analysis Method) and numerical solutions (by a boundary value problem solver), in order to further study the behavior of the nonlinear differential equations, for various values of the physical parameters. Our numerical solutions are shown to agree with the available results in the literature. We then employ the numerical results to bring out the effects of the suction parameter, heat source/sink parameter, stretching parameter, porosity parameter, the Prandtl number and the free convection parameter on the flow and heat transfer characteristics. In the absence of suction and free convection, our findings are in agreement with the corresponding numerical results of Attia [H.A. Attia, On the effectiveness of porosity on stagnation point flow towards a stretching surface with heat generation, Comput. Mater. Sci. 38 (2007) 741-745].     

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.     

MHD power-law fluid flow and heat transfer over a non-isothermal stretching sheet

This article presents a numerical solution for the magnetohydrodynamic (MHD) non-Newtonian power-law fluid flow over a semi-infinite non-isothermal stretching sheet with internal heat generation/absorption. The flow is caused by linear stretching of a sheet from an impermeable wall. Thermal conductivity is assumed to vary linearly with temperature. The governing partial differential equations of momentum and energy are converted into ordinary differential equations by using a classical similarity transformation along with appropriate boundary conditions. The intricate coupled non-linear boundary value problem has been solved by Keller box method. It is important to note that the momentum and thermal boundary layer thickness decrease with increase in the power-law index in presence/absence of variable thermal conductivity.     

Analytical solutions of the boundary layer flow of power-law fluid over a power-law stretching surface

This article discusses analytical solutions for a nonlinear problem arising in the boundary layer flow of power-law fluid over a power-law stretching surface. Using perturbation method analytical solution is presented for linear stretching surface. This solution covers large range of shear thinning and shear thickening fluids and matches excellently with the numerical solution. Furthermore, some new exact solutions are found for particular combination of m (power-law stretching index) and n (power-law fluid index). This leads to generalize the case of linear stretching to nonlinear stretching surface. The effects of fluid index n on the boundary layer thickness and the skin friction for nonlinear stretching surface is analyzed and discussed. It is observed that the boundary layer thickness and the skin friction coefficient increase as non-linear parameter n decreases. This study gives a new dimension to obtain analytical solutions asymptotically for highly nonlinear problems which to the best of our knowledge has not been examined so far.     

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.     

On the exact solutions of laminar MHD flow over a stretching flat plate

The magnetohydrodynamic steady-state laminar flow of a viscous incompressible and electrically conducting fluid over a continuous permeable stretching surface is considered. It is shown that in the presence of a vertical inverse-linear magnetic field, we establish a sufficient condition for the existence of exact solutions of this problem with respect to the three parameters: the magnetic parameter M, the suction/injection parameter γ, and the stretching parameter ξ. Numerical results are also obtained and give the effect of the suction parameter and the magnetic parameter on the velocity.     

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)     

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.     

纳米流体流经热分层线性多孔伸展平面时的MHD自然对流及其Lie对称群变换

就不可压缩粘性纳米流体,流经半无限垂直伸展平面并计及热分层时,研究该流体的MHD自然对流和热交换.通过特定形式的Lie对称群变换,即单参数群变换,将所考虑问题的偏微分控制方程变换为常微分方程组.然后,使用基于打靶法的Runge Kutta Gill法进行数值求解.最后得到结论:流场、温度和纳米颗粒体积率受热分层和磁场的影响很显著.     

On the similarity solutions of magnetohydrodynamic flows of power-law fluids over a stretching sheet

A rigorous mathematical analysis is given for a magnetohydrodynamics boundary layer problem, which arises in the two-dimensional steady laminar boundary layer flow for an incompressible electrically conducting power-law fluid along a stretching flat sheet in the presence of an exterior magnetic field orthogonal to the flow. In the self-similar case, the problem is transformed into a third-order nonlinear ordinary differential equation with certain boundary conditions, which is proved to be equivalent to a singular initial value problem for an integro-differential equation of first order. With the aid of the singular initial value problem, the uniqueness and existence results for (generalized) normal solutions are established and some properties of these solutions are explored.     

Existence and a priori bounds for steady stagnation flow toward a stretching cylinder

We investigate the nonlinear boundary value problem (BVP) that is derived from a similarity transformation of the Navier-Stokes equations governing fluid flow toward a stretching permeable cylinder. Existence of a solution is proven for all values of the Reynolds number and for both suction and injection, and uniqueness results are obtained in the case of a monotonic solution. A priori bounds on the skin friction coefficient are also obtained. These bounds achieve any desired order of accuracy as the injection parameter tends to negative infinity.     

The motion of a submerged sphere in a liquid under an ice sheet

The solution of the linear steady problem of the flow of an inviscid, incompressible and infinitely deep liquid around a sphere under an ice sheet, which is modelled by a thin elastic stressed plate of constant thickness is constructed. Special cases of this problem are the motion of a submerged sphere under broken ice, a membrane, and also under the free surface both in the presence and absence of capillary effects. The method of multipole expansions is used in the framework of the linear potential wave theory. The hydrodynamic loads (the wave drag and the buoyancy) acting on the body and also the distribution of the deflections of the ice sheet are calculated as a function of the body velocity, the ice thickness and the value of the compressing or stretching forces. It is shown that all the flow characteristics depend considerably on the ratio of the body velocity and the critical velocity of flexural-gravitational waves.     

Flow,heat, and species transfer over a stretching plate considering coupled Stefan blowing effects from species transfer

In this paper, we investigate the flow, heat and mass transfer of a viscous fluid flow over a stretching sheet by including the blowing effects of mass transfer under high flux conditions. Mass transfer in this work means species transfer and is different from mass transpiration for permeable walls. The new contribution from this work is, for the first time, to consider the coupled blowing effects from massive species transfer on flow, heat, and species transfer for a stretching plate. Based on the exact solutions of the momentum equations, which are valid for the whole Navier–Stokes equations, the energy and mass transfer equations are solved exactly and the effects of the blowing parameter, the Schmidt number, and the Prandtl number on the flow, heat and mass transfer are presented and discussed. The solution is given in terms of an incomplete Gamma function. It is found the coupled blowing effects due to mass transfer can have significant influences on velocity profiles, drag, heat flux, as well as temperature and concentration profiles. These solutions provide rare results with closed form analytical expressions and can be used as benchmark problem for numerical code validation.