Finite element analysis of combined heat and mass transfer in hydromagnetic micropolar flow along a stretching sheet

This paper presents a finite element solution of the problem of heat and mass transfer in a hydromagnetic flow of a micropolar fluid past a stretching sheet. The transformed equations for the flow regime are solved numerically by using finite element method. The effect of important parameters namely magnetic field parameter, material parameter, Eckert number and Schmidt number over velocity, microrotation, temperature and concentration functions has been studied. It has been observed that the magnetic field parameter has the effect of reducing the velocity and increasing the microrotation, temperature and concentration while the micropolar parameter has the opposite effect on these functions except temperature function. Temperature increases with the increase in Eckert number and concentration decreases with the increase in Schmidt number.     

Finite element solution of mixed convection micropolar flow driven by a porous stretching sheet

This paper presents a finite element solution for the mixed convection micropolar flow driven by a porous stretching sheet with uniform suction. The governing partial differential equations are solved numerically by the using finite element method and the results have been compared with those obtained by using the quasi-linearization method. The effect of surface conditions on the velocity, microrotation as well as for temperature functions has been studied. It is noticed that the micropolar fluids help in the reduction of drag forces and also act as a cooling agent.     

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.     

Wall slip induced by a micropolar fluid

Lubrication theory is devoted to the study of thin-film flows, More often, the fluid can be considered as a Newtonian one and no-slip boundary conditions can be retained for the velocity at the fluid solid interface. With these assumptions it is possible to deduce from the (Navier) Stokes system a simplified equation describing the flow: the Reynolds equation. It allows to compute the pressure distribution inside the film and to obtain overall performances of a lubricated device such as load and friction coefficient. For very thin films, however, surface effects at the fluid solid interface become very important and no-slip conditions cannot be retained. Solid surfaces exert some influence on the liquid molecules and the effective shear viscosity along the boundary differs from the classical bulk shear viscosity. Moreover, the microstructure of the fluid cannot be ignored, especially the effects of solid-particle additives in the lubricant. Micropolar theory for fluids is often adopted to account of such microstructure and microrotation. In the present study, a thin micropolar fluid model with new boundary conditions at the fluid–solid interface is considered. This condition links velocity and microrotation at the interface by introducing a so-called “boundary viscosity”. By way of asymptotic analysis, a generalized micropolar Reynolds equation is obtained. Numerical results show the influence of the new boundary conditions for the load and friction coefficients. Comparisons are made with other works that retain the no-slip boundary conditions.     

Numerical study for the effects of thermophoresis and variable thermal conductivity on heat and mass transfer over an accelerating surface with heat source

A modified numerical solution scheme, for local similarity boundary layer analysis, is used to study the effects of thermophoresis and variable thermal conductivity on heat and mass transfer over an accelerating surface with heat source in the presence of suction and blowing. This numerical scheme is efficient and accurate and it can be programmed and applied easily and its application is illustrated, step by step, by studying the above mentioned problem. The resulting boundary layer equations are solved numerically by Chebyshev finite difference method. Numerical results for the velocity, temperature and concentration as well as for the skin friction, Nusselt and Sherwood numbers are obtained and reported graphically for various parametric conditions to show interesting aspects of the solution.     

Non-similar solution for the axisymmetric flow of a third-grade fluid over a radially stretching sheet

Summary The problem of axisymmetric flow of a third grade fluid over a radially stretching sheet is studied. By means of similarity transformation, the governing non-linear partial differential equations are reduced to a non-linear ordinary differential equation. The ordinary differential equation is analytically solved using homotopy analysis method (HAM). The solution for the velocity is obtained. The series solution is developed and the convergence of the results is discussed. Finally, the results are discussed with various graphs.     

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.     

Numerical simulation of unsteady viscous flow around a circular cylinder

A finite difference solution is obtained for the time-dependent viscous incompressible 2-dimensional flow past a circular cylinder by direct integration of the Navier-Stokes equations expressed in a general curvilinear coordinate system. The solution describes the development of the vortex street developed behind the cylinder. Evolution of flow configuration is studied by means of streamlines, pressure contours, and vorticity contours for different Reynolds numbers. The time-dependent lift and drag coefficients are also obtained.     

Effect of reverse dome stretching on dome height and forming limits of sheet materials

Reverse bending and stretching of sheet materials is often employed in press forming of complex automotive components. In this work, hemispherical dome stretching tests were followed by reverse dome tests on automotive aluminum sheet specimens to assess the influence of the strain and shape on dome height at neck formation and limit strains. The above test scheme offers a means of subjecting the sheet material to reverse bending and stretching and thereby changing its strain path during the process. The results indicate significant improvements in dome height as a function of pre-strain (or initial dome height) compared to the simple dome stretching process. The forming limit strains, on the other hand, are lowered. The reason is attributed to the redistribution of strain (more uniform deformation) through the punch/specimen contact during the reverse bending and stretching process.     

A Numerical Comparison of Finite Difference and Finite Element Methods for a Stochastic Differential Equation with Polynomial Chaos

A numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.     

Numerical calculations of acoustic emission

Two computer programs which solve the partial differential equations for sound propagation numerically are applied to the study of problems in acoustic emission. The programs use finite difference and finite element techniques to calculate sound fields due to distributions of sources in complex geometries in two dimensions. The potential to handle more complex geometries and to model more realistic sources is the main advantage of these numerical calculations over the analytic calculations. The main disadvantage of the numerical techniques is the cost of obtaining results since a large main frame computer or supercomputer is required. Both numerical methods are found to agree well with analytic calculations using Green's functions. The finite difference method agrees very well with the analytic calculations but, in its current implementation, is much slower and contains more numerical noise compared to the finite element method. The finite element method has the additional advantage of being capable of handling more complex geometries than the finite difference method.     

$H^1$-Stability and Convergence of the FE,FV and FD Methods for an Elliptic Equation

We obtain the coefficient matrices of the finite element (FE), finite volume (FV) and finite difference (FD) methods based on $P_1$-conforming elements on a quasi-uniform mesh, in order to approximately solve a boundary value problem involving the elliptic Poisson equation. The three methods are shown to possess the same $H^1$-stability and convergence. Some numerical tests are made, to compare the numerical results from the three methods and to review our theoretical results.     

Shape effects of spherical and nonspherical nanoparticles in mixed convection flow over a vertical stretching permeable sheet

In this article, two-dimensional heat transfer mixed convection flow of a nanofluid over a vertical stretching permeable sheet is investigated. Simultaneous effects of spherical and nonspherical shapes of nanoparticles with different sizes in nanolayer are taken into account. The human engineered fluids with Nimonic 80a metal nanoparticles are used as base fluids. Analytic solutions of velocity and temperature under the influence of the Buoyancy force (assists or opposes) are first obtained and then the role of pertinent parameters, such as volume friction, mixed convection, porosity, stretching, power law index, and temperature index, is illustrated through graphs and tables. In addition, correlation of Nusselt number and skin friction corresponding to active parameters are also analyzed.     

A constitutive model for metals over a large range of strain rates Identification for mild-steel and aluminium sheets

A phenomenological model is presented to describe the mechanical behaviour for metals and alloys over a large range of strain rates. It is an elasto-plastic model with a strain rate and temperature-sensitive yield stress. This model partially relies on physical considerations and is specially developed for an easy application in explicit finite element method (FEM) codes. The process for identifying the constants from experimental data is presented, taking into account the exact testing conditions such as temperature increase and strain rate variations during the loading. Applying the model to data obtained for mild steel and commercial aluminium sheets yields satisfactory results.     

MHD 3D free convective flow of nanofluid over an exponentially stretching sheet with chemical reaction

This paper deals with a problem where the effect of variable magnetic field and chemical reaction on free convective flow of an electrically conducting incompressible water based nanofluid over an exponentially stretching sheet has been investigated. In the present study, Buongiorno model associated with Brownian motion and thermophoretic diffusion is employed to describe the heat transfer enhancement of nanofluids. Some suitable similarity transformations reduced the governing boundary layer non-linear partial differential equations into a set of ordinary non-linear differential equations. The transformed equations are then solved numerically using fourth order Runga-Kutta method along with Shooting technique. The major outcomes of the present study is that the magnetic field impedes the fluid motion while thermal as well as mass buoyancy forces accelerate it, the thermophoretic diffusion enhances dimensionless fluid temperature as well as concentration leading to thicker thermal and concentration boundary layers. On the other hand, concentration exponent, Brownian motion parameter and chemical reaction parameter exhibit reverse trend on temperature and concentration. In addition, the presence of magnetic field under the influence of thermal as well as mass buoyancies supports to reduce the rate of heat transfer as well as wall shear stress while the first order chemical reaction develops a thinner concentration boundary layer.     

Numerical Methods for Accurate Computation of Design Sensitivities for Two-Dimensional Navier-Stokes Problems

Gradient computations can be a limiting factor in algorithm efficiency and accuracy for optimization based design. In this paper, we present three parameterized flow problems and consider the evaluation of state sensitivities both theoretically and numerically. Existence and uniqueness results are given for the sensitivities of a specific group of two-dimensional Navier-Stokes problems. We then turn our attention to obtaining numerical approximations to state sensitivities. We show convergence of our numerical sensitivities using a problem having an exact solution. Next, two problems, flow around a cylinder and flow over a bump, are used to evaluate several computational schemes. In particular, a local projection scheme for improved state derivative approximations and the use of an adaptive finite element scheme are shown to be important techniques for obtaining accurate sensitivity approximations. Lastly, we evaluate the impact of these computational techniques on cost function and gradient calculation.     

Numerical and experimental analysis of residual stresses for fatigue crack growth

In this paper computational and experimental results are presented concerning residual stress effects on fatigue crack growth in a Compact Tension Shear (CTS) specimen under cyclic mode I loading. For a crack of constant length it is found that hardly any compressive residual stresses or crack closure effects are generated along the crack surfaces behind the crack tip through the considered cyclic mode I loading with a load ratio of R=0.1. Only if fatigue crack growth is modelled during the simulation of the cyclic loading process these well-known effects are found. On the other hand it is shown that they have hardly any influence on the residual stresses ahead of the crack tip and thus on further fatigue crack growth. For all cases considered the computational finite element results agree well with the experimental findings obtained through X-ray diffraction techniques.     

Numerical Methods for Constrained Elliptic Optimal Control Problems with Rapidly Oscillating Coefficients

In this paper we use two numerical methods to solve constrained optimal control problems governed by elliptic equations with rapidly oscillating coefficients: one is finite element method and the other is multiscale finite element method. We derive the convergence analysis for those two methods. Analytical results show that finite element method can not work when the parameter $\varepsilon$ is small enough, while multiscale finite element method is useful for any parameter $\varepsilon$.     

Optimal control of thermal fluid flow using automatic differentiation

The aim of this study is to present a method for numerical optimal control of thermal fluid flow using automatic differentiation (AD). For the optimal control, governing equations are required. The optimal controls that have been previously presented by the present authors’ research group are based on the Boussinesq equations. However, because the numerical results of these equations are not satisfactory, the compressible Navier–Stokes equations are employed in this study. The objective is to determine whether or not the temperature at the objective points can be kept constant by imposing boundary conditions and by controlling the temperature at the control points. To measure the difference between the computed and target temperatures, the square sum of these values is used. The objective points are located at the center of the computational domain while the control points are at the bottom of the computational domain. The weighted gradient method that employs AD for efficiently calculating the gradient is used for the minimization. By using numerical computations, we show the validity of the present method.     

质谱计离子源静电透镜的模拟计算

本文针对某质谱计离子源的设计,建立了离子源静电场数学模型,采用有限差分法对其场域进行离散,半迭代切比雪夫松弛法求解各节点的电位,拉格朗日插值法计算空间任意点的电位和电场,四阶龙格-库塔法计算离子运动轨迹.使用Visual C++ 6.0为平台编程计算程序,结合离子源的具体结构求解了离子源静电透镜整个场域的电位和电场;绘制了等位线和离子运动轨迹.在此基础上,讨论了各电极电压对离子运动轨迹(即对聚焦性能)的影响.