A note on the flow over a stretching sheet

Summary This study deals with the viscous incompressible flow over a stretching sheet. The velocity of the sheet is a quadratic polynomial of the distance from the slit and the sheet is subjected to a linear mass flux. A closed form solution is obtained under some restrictions on the linear mass flux. Stream line patterns are plotted and the effect of mass flux on the flow is also studied.     

Convective heat transfer at a stretching sheet

Summary Analysis of convection flow and heat transfer of a viscous heat-generating fluid near an infinite vertical stretching surface is carried out. The effects of free convection and the presence of heat generation/absorption on the flow and heat transfer characteristics are considered. The equations of conservation of momentum, mass, and energy, which govern the flow and heat transfer problem, are solved numerically by using a variable order, variable step size finite-difference method. The numerical results obtained for the flow and heat transfer characteristics reveal many interesting behaviors. These behaviors warrant further study of the effects of free convection on the flow and heat transfer characteristics.     

Magnetohydrodynamic stagnation-point flow towards a stretching sheet

Steady two-dimensional stagnation-point flow of an incompressible viscous electrically conducting fluid over a flat deformable sheet is investigated when the sheet is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. It is shown that the velocity at a point decreases/increases with increase in the magnetic field when the free stream velocity is less/greater than the stretching velocity. Temperature distribution in the flow is obtained when the surface is held at a constant temperature.     

Boundary layer equations and stretching sheet solutions for the modified second grade fluid

A modified second grade non-Newtonian fluid model is considered. The model is a combination of power-law and second grade fluids in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. The equations of motion are derived for two dimensional incompressible flows. The boundary layer equations are derived from the equations. Symmetries of the boundary layer equations are calculated using Lie Group theory. For a special power law index of m = −1, the principal Lie algebra extends. Using one of the symmetries, the partial differential system is transferred to an ordinary differential system. The ordinary differential equations are numerically integrated for the stretching sheet boundary conditions. Effects of power-law index and second grade coefficient on the boundary layers are shown and solutions are contrasted with the usual second grade fluid solutions. The shear stress on the boundary is also calculated.     

Magnetohydrodynamic heat transfer over a non-isothermal stretching sheet

Summary This paper presents solutions of the energy equation for the boundary layer flow of an electrically conducting fluid under the influence of a constant transverse magnetic field over a linearly stretching non-isothermal flat sheet. Effects due to dissipation, stress work and heat generation are considered. Analytical solutions of the resulting linear nonhomogeneous boundary value problems, expressed in terms of Kummer's functions, are presented for the case of prescribed surface temperature as well as the case of prescribed wall heat flux, both of which are assumed to be quadratic functions of distance. The boundary value problems are also solved by direct numerical integration yielding results in excellent agreement with the analytical solutions.     

Numerical modelling,validation and analysis of multi-pass sheet metal spinning processes

Conventional sheet metal spinning is an incremental forming process which typically involves the cost-effective and high-quality manufacturing of axissymmetric parts. The process is usually executed by highly skilled and experienced personnel which is able of optimizing the process parameters during production. Numerical simulation of the process can substantially help discovering systematic methodologies for optimal parameter determination and thus enable the full automation of the process using CNC machines. The present work aims to assess the quality of numerical modelling techniques by a direct comparison with metal spinning experiments. Based on the geometry and thickness distribution of intermediate and final stages of a spinned component, which are measured using the Optical 3D Digitization technique, the quality and validity of different numerical modeling approaches are assessed. Subsequently, deformation mechanisms occurring during process are identified, analysed and discussed.     

g-Jitter mixed convection adjacent to a vertical stretching sheet

This paper studies the effect of periodical gravity modulation, or g-jitter induced mixed convection, on the flow and heat transfer characteristics associated with a stretching vertical surface in a viscous and incompressible fluid. The velocity and temperature of the sheet are assumed to vary linearly withx, wherex is the distance along the sheet. It is assumed that the gravity vector modulation is given byg*(t)=g_o [1+ɛ cos(πωt)]k, and the resulting non-similar boundary layer equations are solved numerically using an implicit finite-difference scheme. The effects of the amplitude of modulation, frequency of the single-harmonic component of oscillation, mixed convection parameter and Prandtl number on the skin friction and Nusselt number are discussed in detail.     

Numerical study for micropolar flow over a stretching sheet

The paper presents a numerical study of the steady flow of a micropolar fluid flow from a stretching sheet. Approximate analytical solution of high nonlinear momentum, angular momentum and confluent hypergeometric similarity solution of the heat transfer equation are obtained for a particular case when the vortex viscosity is neglected. Accuracy of the analytical solution is verified by numerical solutions obtained by employing finite element and Chebyshev finite difference methods. The good agreement between the numerical results of both methods, together with an excellent agreement with the analytical solutions for the special case, ensures the reliability of the obtained results. The velocity, microrotation and temperature functions are shown graphically and the effect of the permeability parameter is studied.     

MHD flow of a viscoelastic fluid past a stretching sheet with suction

Summary The laminar flow of a viscoelastic fluid past a stretching sheet in the presence of a magnetic field, when the fluid is extracted from the sheet at a uniform rate, is considered. An exact analytical solution exists for the problem. It is shown that when there is a suction of the fluid, the solutions are possible only upto a critical value of the viscoelastic parameter. Also, for values less than this critical value, dual solutions exist.     

Heat transfer from a stretching sheet with uniform suction and blowing

Summary The conjugate problem of heat transfer from an accelerating surface in presence of suction and blowing has been solved analytically. The effects of different parameters on cooling of the surface have been discussed.With 3 Figures     

Heat transfer in the flow of a viscoelastic fluid over a stretching sheet

Summary The problem of heat transfer in the viscoelastic fluid flow over a stretching sheet is examined. The important physical quantities such as the skin-friction coefficient and the heat transfer coefficient, are determined. It is found that the heat transfer coefficient decreases with the non-Newtonian parameter.     

A note on the equivalence of shock manifold equations

Summary The shock manifold equation is a first order nonlinear partial differential equation, which describes the kinematics of a shockfront in an ideal gas with constant specific heats. However, it was found that there was more than one of these shock manifold equations, and the shock surface could be embedded in a one parameter family of surfaces, obtained as a solution of any of these shock manifold equations. Associated with each shock manifold equation is a set of characteristic curves called shock rays. This paper investigates the nature of various associated shock ray equations.On leave from Beloit College, Beloit, WI, 535111 U.S.A.     

Unsteady boundary layer flow of power-law fluid on stretching sheet surface

Two-dimensional unsteady boundary layer equations of non-Newtonian fluids are treated. Flow of a thin fluid film of a power-law caused by stretching of surface is investigated by using a similarity transformation. By using this transformation, we reduce the unsteady boundary layer equations to a non-linear ordinary differential equation system. Numerical solutions of outcoming nonlinear differential equations are found by using a combination a Runge–Kutta algorithm and shooting technique. Boundary layer thickness is explored numerically for different values of power-law index.     

On the stability of MHD flow of a viscoelastic fluid past a stretching sheet

Summary The effects of a magnetic field on the stability characteristics of a viscoelastic fluid flow due to the stretching sheet are investigated. A three-dimensional linear stability analysis is performed by means of the Method of Weighted Residuals for disturbances of the Taylor-Görtler type. It is found that the magnetic field exerts a stabilizing influence on the flow.     

The nonuniqueness of the MHD flow of a viscoelastic fluid past a stretching sheet

Summary Two closed-form solutions were found for the boundary layer equations of the title problem. Discussions are made to trace among them the physically realistic solution.     

Nonlinear streaming due to the oscillatory stretching of a sheet in a viscous fluid

Summary An elastic sheet is stretched back and forth in a viscous fluid. The problem is governed by a nondimensional parameterS which represents the relative magnitude of frequency to stretching rate. The Navier-Stokes equations are solved by matched asymptotic expansions for largeS. Due to nonlinearity there exists boundary layers ofO(S ^–1/2). The unsteady oscillatory flow contains both basic and higher harmonic oscillations. The induced steady streamlines show a saddle like flow which is different from that of acoustic streaming.With 5 Figures     

The flow of an elastico-viscous fluid past a stretching sheet with partial slip

Summary An analysis is carried out to study the flow characteristics in an elastico-viscous fluid (Walters' liquid-B model) over a stretching sheet with partial slip. The flow is generated due to linear stretching of the sheet. Using suitable similarity transformations on the highly non-linear partial differential equations we derive exact analytical solution with appropriate boundary conditions. The important finding in this communication is the effect of partial slip on the velocity and skin friction coefficient.     

Evaluation of the in-plane effective elastic moduli of single-layered graphene sheet

In this paper, an equivalent continuum-structural mechanics approach is used to characterize the mechanical behaviour of nanostructured graphene. The in-plane elastic deformation of armchair graphene sheets is simulated by using finite element modelling. The model is based on the assumption that force interaction among carbon atoms can be modelled by load-carrying beams in a representative two-dimensional honeycomb lattice structure. The elastic properties of beam elements are determined by equating the energies of the molecular structure and the continuum beam model subjected to small strain deformation. Then an equivalent continuum technique is adopted to estimate effective elastic moduli from which elastic constants are extracted. A comparison of elastic constants obtained from current modelling concur with results reported in literature. With the multifunctional properties of graphene sheets as manifested in a broad range of industrial applications, determination of their elastic moduli will facilitate a better design of the corresponding materials at macroscopic level.     

Heat transfer in a fluid with variable thermal conductivity over a linearly stretching sheet

Summary This paper considers the boundary layer heat transfer in a two-dimensional Newtonian fluid flow caused by a porous and linearly stretching sheet in the presence of blowing/suction. The thermal conductivity is assumed to vary linearly with temperature as is found in liquid metals. The resulting nonlinear energy equation forms a boundary value problem which is solved by a shooting method. A perturbation method is also used to derive a set of uncoupled, linear boundary value problems which are solved by superposition of solutions.