Effects of temperature dependent fluid properties and variable Prandtl number on the transient convective flow due to a porous rotating disk

In this paper we have studied the effects of temperature dependent fluid properties such as density, viscosity and thermal conductivity and variable Prandtl number on unsteady convective heat transfer flow over a porous rotating disk. Using similarity transformations we reduce the governing nonlinear partial differential equations for flow and heat transfer into a system of ordinary differential equations which are then solved numerically by applying Nachtsheim–Swigert shooting iteration technique along with sixth-order Runge–Kutta integration scheme. Comparison with previously published work for steady case of the problem were performed and found to be in very good agreement. The obtained numerical results show that the rate of heat transfer in a fluid of constant properties is higher than in a fluid of variable properties. The results further show that consideration of Prandtl number as constant within the boundary layer for variable fluid properties lead unrealistic results. Therefore, modeling thermal boundary layers with temperature dependent fluid properties Prandtl number must treated as variable inside the boundary layer.     

MHD flow and heat transfer over a stretching surface with variable thermal conductivity and partial slip

In this paper we analyze the flow and heat transfer of an MHD fluid over an impermeable stretching surface with variable thermal conductivity and non-uniform heat source/sink in the presence of partial slip. The governing partial differential equations of the problem are reduced to nonlinear ordinary differential equations by using a similarity transformation. The temperature boundary conditions are assumed to be linear functions of the distance from the origin. Analytical solutions of the energy equations for Prescribed Surface Temperature (PST) and Prescribed Heat Flux (PHF) cases are obtained in terms of a hypergeometric function, without applying the boundary-layer approximation. The effects of the governing parameters on the flow and heat transfer fields are presented through tables and graphs, and they are discussed. Furthermore, the obtained numerical results for the skin friction, wall-temperature gradient and wall temperature are analyzed and compared with the available results in the literature for special cases.     

Basic equations of turbulent flow for variable density and variable viscosity Newtonian fluid in open channel

I.Intr0ducti0nManyscholarshaveresearchedtheturbulenceproblemsofnon-compressiblefluidll'2l,e.g.inRef[1],DouGu0renhaspresentedtheturbulentmotionaldifferentialequationsofconstantdensityandconstantvisc0sityfluidin0penchannel;inRef[3],XiaoTianduohasstudiedthet…     

Modeling of in-situ biorestoration of organic compounds in groundwater

A convergent numerical method for modeling in situ biorestoration of contaminated groundwater is outlined. This method treats systems of transport-biodegradation equations by operator splitting in time. Transport is approximated by a finite element modified method of characteristics. The biodegradation terms are split from the transport terms and treated as a system of ordinary differential equations. Numerical results for vertical cross-sectional flow are presented. The effects of variable hydraulic conductivity and variable linear adsorption are studied.     

一类偏微分方程的Hamilton正则表示

主要给出一系列关于力学中的偏微分方程的无穷维Ｈａｍｉｌｔｏｎ正则表示．其中包括变系数线性偏微分方程,ＫｄＶ方程,ＭＫｄＶ方程,ＫＰ方程,Ｂｏｕｓｉｎｅｓｑ方程等的无穷维Ｈａｍｉｌｔｏｎ正则表示．     

Nonlinear stability of truncated shallow spherical shell with variable thickness under uniformly distributed load

I.Intr0ducti0nTheclampedtruncatedshailowsphericalshellwithanondeformablerigidbodyatthecenterisoftenusedinstructureandelasticcomponentofprecisioninstrument.Inthebuildingengineering,wemustpreventtheshelllosingstability.Butintheelasticcomponent0fprecisionins…     

集中载荷作用下变厚度开顶扁球壳的非线性稳定问题

首先应用逐步加载法将具有硬中心的开顶扁球壳在集中载荷作用下的非线性微分方程组线性化，然后利用样条配点法解线性微分方程组，得到了临界载荷的大小。     

Effects of thermal buoyancy and variable thermal conductivity on the MHD flow and heat transfer in a power-law fluid past a vertical stretching sheet in the presence of a non-uniform heat source

The paper considers the flow of a power-law fluid past a vertical stretching sheet. Effects of variable thermal conductivity and non-uniform heat source/sink on the heat transfer are addressed. The thermal conductivity is assumed to vary linearly with temperature. Similarity transformation is used to convert the governing partial differential equations into a set of coupled, non-linear ordinary differential equations. Two different types of boundary heating are considered, namely Prescribed power-law Surface Temperature (PST) and Prescribed power-law Heat Flux (PHF). Shooting method is used to obtain the numerical solution for the resulting boundary value problems. The effects of Chandrasekhar number, Grashof number, Prandtl number, non-uniform heat source/sink parameters, wall temperature parameter and variable thermal conductivity parameter on the dynamics are shown graphically in several plots. The skin friction and heat transfer coefficients are tabulated for a range of values of the parameters. Present study reveals that in a gravity affected flow buoyancy effect has a significant say in the control of flow and heat transfer.     

Flow of variable thermal conductivity fluid due to inclined stretching cylinder with viscous dissipation and thermal radiation

The aim of the present study is to investigate the flow of the Casson fluid by an inclined stretching cylinder. A heat transfer analysis is carried out in the presence of thermal radiation and viscous dissipation effects. The temperature dependent thermal conductivity of the Casson fluid is considered. The relevant equations are first simplified under usual boundary layer assumptions, and then transformed into ordinary differential equations by suitable transformations. The transformed ordinary differential equations are computed for the series solutions of velocity and temperature. A convergence analysis is shown explicitly. Velocity and temperature fields are discussed for different physical parameters by graphs and numerical values. It is found that the velocity decreases with the increase in the angle of inclination while increases with the increase in the mixed convection parameter. The enhancement in the thermal conductivity and radiation effects corresponds to a higher fluid temperature. It is also found that heat transfer is more pronounced in a cylinder when it is compared with a flat plate. The thermal boundary layer thickness increases with the increase in the Eckert number. The radiation and variable thermal conductivity decreases the heat transfer rate at the surface.     

Study of differential control method for solving chaotic solutions of nonlinear dynamic system

In this paper, we focus on the need to solve chaotic solutions of high-dimensional nonlinear dynamic systems of which the analytic solution is difficult to obtain. For this purpose, a Differential Control Method (DCM) is proposed based on the Mechanized Mathematics-Wu Elimination Method (WEM). By sampling, the computer time of the differential operator of the nonlinear differential equation can be substituted by the differential quotient of solving the variable time of the sample. The main emphasis of DCM is placed on substituting the differential quotient of a small neighborhood of the sample time of the computer system for the differential operator of the equations studied. The approximate analytical chaotic solutions of the nonlinear differential equations governing the high-dimensional dynamic system can be obtained by the method proposed. In order to increase the computational efficiency of the method proposed, a thermodynamics modeling method is used to decouple the variable and reduce the dimension of the system studied. The validity of the method proposed for obtaining approximate analytical chaotic solutions of the nonlinear differential equations is illustrated by the example of a turbo-generator system. This work is applied to solving a type of nonlinear system of which the dynamic behaviors can be described by nonlinear differential equations.     

Differential transformation method for studying flow and heat transfer due to stretching sheet embedded in porous medium with variable thickness,variable thermal conductivity,and thermal radiation

This article presents a numerical solution for the flow of a Newtonian fluid over an impermeable stretching sheet embedded in a porous medium with the power law surface velocity and variable thickness in the presence of thermal radiation. The flow is caused by non-linear stretching of a sheet. Thermal conductivity of the fluid is assumed to vary linearly with temperature. The governing partial differential equations （PDEs） are transformed into a system of coupled non-linear ordinary differential equations （ODEs） with appropriate boundary conditions for various physical parameters. The remaining system of ODEs is solved numerically using a differential transformation method （DTM）. The effects of the porous parameter, the wall thickness parameter, the radiation parameter, the thermal conductivity parameter, and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction and the Nusselt numbers are presented. Comparison of the obtained numerical results is made with previously published results in some special cases, with good agreement. The results obtained in this paper confirm the idea that DTM is a powerful mathematical tool and can be applied to a large class of linear and non-linear problems in different fields of science and engineering.     

Modeling natural convection boundary layer flow of micropolar nanofluid over vertical permeable cone with variable wall temperature

This paper discusses the natural convection boundary layer flow of a micropolar nanofluid over a vertical permeable cone with variable wall temperatures. Non-similar solutions are obtained. The nonlinearly coupled differential equations under the boundary layer approximations governing the flow are solved numerically using an efficient, iterative, tri-diagonal, implicit finite difference method. Different experimental correlations for both nanofluid effective viscosity and nanofluid thermal conductivity are considered.It is found that as the vortex-viscosity parameter increases, both the velocity profiles and the local Nusselt number decrease. Also, among all the nanoparticles considered in this investigation, Cu gives a good convection.     

Solution of free vibration equations of semi-rigid connected Reddy–Bickford beams resting on elastic soil using the differential transform method

The literature regarding the free vibration analysis of Bernoulli–Euler and Timoshenko beams under various supporting conditions is plenty, but the free vibration analysis of Reddy–Bickford beams with variable cross-section on elastic soil with/without axial force effect using the Differential Transform Method (DTM) has not been investigated by any of the studies in open literature so far. In this study, the free vibration analysis of axially loaded and semi-rigid connected Reddy–Bickford beam with variable cross-section on elastic soil is carried out by using DTM. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments in this study. The governing differential equations of motion of the rectangular beam in free vibration are derived using Hamilton’s principle and considering rotatory inertia. Parameters for the relative stiffness, stiffness ratio and nondimensionalized multiplication factor for the axial compressive force are incorporated into the equations of motion in order to investigate their effects on the natural frequencies. At first, the terms are found directly from the analytical solutions of the differential equations that describe the deformations of the cross-section according to the high-order theory. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the governing differential equations of the motion. The calculated natural frequencies of semi-rigid connected Reddy–Bickford beam with variable cross-section on elastic soil using DTM are tabulated in several tables and figures and are compared with the results of the analytical solution where a very good agreement is observed.     

Viscometric flows of polymeric melts treated according to the rational theory of Eckart's continuum. I. General theory

The purely rational theory of Eckart continua (i.e. elastic bodies with a variable relaxed state) is applied to viscometric flows of polymeric melts. The main assumptions are thermodynamic non-interaction of inelastic behaviour and of non-elastic stress, as well as elastic isotropy. After establishing the time-dependent differential equations of viscometric flow, these equations are simplified to a set of algebraic equations describing steady-state flow. From this we deduce two general equations connecting the three elastic steady-state viscometric functions which do not depend upon the elastic behaviour. The law of rubber elasticity used in this paper is described in the Appendix.     

Free Convection Boundary Layer Flow Past a Vertical Surface in a Porous Medium with Temperature-Dependent Properties

Numerical solutions for the free convection heat transfer in a viscous fluid at a permeable surface embedded in a saturated porous medium, in the presence of viscous dissipation with temperature-dependent variable fluid properties, are obtained. The governing equations for the problem are derived using the Darcy model and the Boussinesq approximation (with nonlinear density temperature variation in the buoyancy force term). The coupled non-linearities arising from the temperature-dependent density, viscosity, thermal conductivity, and viscous dissipation are included. The partial differential equations of the model are reduced to ordinary differential equations by a similarity transformation and the resulting coupled, nonlinear ordinary differential equations are solved numerically by a second order finite difference scheme for several sets of values of the parameters. Also, asymptotic results are obtained for large values of | f _w|. Moreover, the numerical results for the velocity, the temperature, and the wall-temperature gradient are presented through graphs and tables, and are discussed. It is observed that by increasing the fluid variable viscosity parameter, one could reduce the velocity and thermal boundary layer thickness. However, quite the opposite is true with the non-linear density temperature variation parameter.     

非均匀Winkler弹性地基上变厚度矩形板自由振动的DTM求解

针对非均匀Winkler弹性地基上变厚度矩形板的自由振动问题,通过一种有效的数值求解方法——微分变换法（DTM）,研究其无量纲固有频率特性。已知变厚度矩形板对边为简支边界条件,其他两边的边界条件为简支、固定或自由任意组合。采用DTM将非均匀Winkler弹性地基上变厚度矩形板无量纲化的自由振动控制微分方程及其边界条件变换为等价的代数方程,得到含有无量纲固有频率的特征方程。数值结果退化为均匀Winker弹性地基上矩形板以及变厚度矩形板的情形,并与已有文献采用的不同求解方法进行比较,结果表明,DTM具有非常高的精度和很强的适用性。最后,在不同边界条件下分析地基变化参数、厚度变化参数和长宽比对矩形板无量纲固有频率的影响,并给出了非均匀Winkler弹性地基上对边简支对边固定变厚度矩形板的前六阶振型。     

Integration of Special Linear Equations of the Second Order

Under certain conditions on the coefficients, the Chazy equation with constant coefficients reduces to a second-order linear differential equation with six singular points. Investigating this equation with the use of the Schwarz derivative, we obtain linear equations in which some of these six singular points coincide. An integration procedure for these equations is considered. Their general solutions are obtained in explicit form.     

Numerical study on a vertical plate with variable viscosity and thermal conductivity

Free convection over an isothermal vertical plate immersed in a fluid with variable viscosity and thermal conductivity is studied in this paper. We consider the two-dimensional, laminar and unsteady boundary layer equations. Using the appropriate variables, the basic governing equations are transformed to non-dimensional governing equations. These equations are then solved numerically using a very efficient implicit finite difference scheme known as Crank–Nicolson scheme. The fluid considered in this study is of viscous incompressible fluid of temperature dependent viscosity and thermal conductivity. The effect of varying viscosity and thermal conductivity on velocity, temperature, shear stress and heat transfer rate are discussed. The velocity and temperature profiles are compared with previously published works and are found to be in good agreement.     

Effects of thermophoresis particle deposition and of the thermal conductivity in a porous plate with dissipative heat and mass transfer

Network simulation method(NSM) is used to solve the laminar heat and mass transfer of an electricallyconducting,heat generating/absorbing fluid past a perforated horizontal surface in the presence of viscous and Joule heating problem. The governing partial differential equations are non-dimensionalized and transformed into a system of nonlinear ordinary differential similarity equations,in a single independent variable,η. The resulting coupled,nonlinear equations are solved under appropriate transformed boundary conditions. Computations are performed for a wide range of the governing flow parameters,viz Prandtl number,thermophoretic coeffcient(a function of Knudsen number),thermal conductivity parameter,wall transpiration parameter and Schmidt number. The numerical details are discussed with relevant applications. The present problem finds applications in optical fiber fabrication,aerosol filter precipitators,particle deposition on hydronautical blades,semiconductor wafer design,thermo-electronics and problems including nuclear reactor safety.     

The bending of elastic circular ring of non-homogeneous and variable cross section under the actions of arbitrary loads

On the basis of the stepped reduction method suggested in [1], we investigate the problem of the bending of elastic circular ring of non-homogeneous and variable cross section under the actions of arbitrary loads. The general solution of this problem is obtained so that it can be used for the calculations of strength and rigidity of practical problems such as arch, tunnel etc. In order to examine results of this paper and explain the application of this new method, an example is brought out at the end of this paper. Circular ring and arch are commonly used structures in engineering. Timoshenko, S.^[2], Barber, J. R.^[3], Tsumura Rimitsu^[4] et al. have studied these problems of bending, but, so far as we know, it has been solely restricted to the general solution of homogeneous uniform cross section ring. The only known solution for the problems with variable cross section ones has been solely restricted to the solution of special case of flexural rigidity in linear function of coordinates. On account of fundamental equations of the non-homogeneous variable cross section problem being variable coefficients, it is very difficult to solve them. In this paper, we use the stepped reduction method suggested in [1] to transform the variable coefficient differential equation into equivalent constant coefficient one. After introducing virtual internal forces, we obtain general solution of an elastic circular ring with non-homogeneity and variable cross section under the actions of arbitrary loads.