Effects of thermal radiation and magnetic field on unsteady mixed convection flow and heat transfer over an exponentially stretching surface with suction in the presence of internal heat generation/absorption

In this paper, the problem of unsteady laminar two-dimensional boundary layer flow and heat transfer of an incompressible viscous fluid in the presence of thermal radiation, internal heat generation or absorption, and magnetic field over an exponentially stretching surface subjected to suction with an exponential temperature distribution is discussed numerically. The governing boundary layer equations are reduced to a system of ordinary differential equations. New numerical method using Mathematica has been used to solve such system after obtaining the missed initial conditions. Comparison of obtained numerical results is made with previously published results in some special cases, and found to be in a good agreement.     

A numerical study of boundary layer flow of viscous incompressible fluid past an inclined stretching sheet and heat transfer using nonpolynomial spline method

In the present paper, we study the boundary layer flow of viscous incompressible fluid over an inclined stretching sheet with body force and heat transfer. Considering the stream function, we convert the boundary layer equation into nonlinear third-order ordinary differential equation together with appropriate boundary conditions in an infinite domain. The nonlinear boundary value problem has been linearized by using the quasilinearization technique. Then, we develop a nonpolynomial spline method, which is used to solve the flow problem. The convergence analysis of the method is also discussed. We study the velocity function for different angles of inclination and Froude number with the help of various graphs and tables. Then using these in heat convection flow, we obtain the expression for temperature field. Skin friction is also calculated. The various results have been given in tables. At last, we calculated the Nusselt number.     

Response of heat transfer from a moving flat plate in a parabolic flow

This paper deals with the solutions of steady as well as unsteady three-dimensional incompressible thermal boundary layer equations and the study of the response of heat transfer when there is a parabolic flow over a moving flat plate. The components of velocity in boundary layer are discussed by Sarma and Gupta and those results are used to analyse thermal boundary layer equations. A general analysis is made from which we deduce (i) Solutions of two-dimensional thermal boundary layer on a moving flat plate, (ii) Solutions of thermal boundary layer on a yawed flat plate, (iii) Solutions of thermal boundary layer when there is a parabolic flow over a moving flat plate by giving different values to β and Cx. Solutions are developed for large and small times and curves are drawn representing the variations of heat transfer from the plate with time for all the cases. The limiting time is also calculated.     

含开边界二维Stokes问题的Galerkin边界元解法

本文推导了含有开边界的二维有限域上Stokes问题的边界积分方程, 得出基于单层位势的第一类间接边界积分方程.对与之等价的边界变分方程用Galerkin边界元求解以得出单层位势的向量密度. 对于含有开边界端点的边界单元,采用特别的插值函数, 以模拟其固有的奇异性.论文用若干数值算例模拟了含有开边界的有限区域上不可压缩粘性流体的绕流.       

Combined effects of non-uniform heat source/sink and thermal radiation on heat transfer over an unsteady stretching permeable surface

The present paper is concerned with the study of flow and heat transfer characteristics in the unsteady laminar boundary layer flow of an incompressible viscous fluid over continuously stretching permeable surface in the presence of a non-uniform heat source/sink and thermal radiation. The unsteadiness in the flow and temperature fields is because of the time-dependent stretching velocity and surface temperature. Similarity transformations are used to convert the governing time-dependent nonlinear boundary layer equations for momentum and thermal energy are reduced to a system of nonlinear ordinary differential equations containing Prandtl number, non-uniform heat source/sink parameter, thermal radiation and unsteadiness parameter with appropriate boundary conditions. These equations are solved numerically by applying shooting method using Runge–Kutta–Fehlberg method. Comparison of numerical results is made with the earlier published results under limiting cases. The effects of the unsteadiness parameter, thermal radiation, suction/injection parameter, non-uniform heat source/sink parameter on flow and heat transfer characteristics as well as on the local Nusselt number are shown graphically.     

Lie group analysis of natural convection heat and mass transfer in an inclined surface with chemical reaction

In this paper we examine the convective flow, heat and mass transfer of an incompressible viscous fluid past a semi-infinite inclined surface with first-order homogeneous chemical reaction by Lie group analysis. The governing partial differential equations are reduced to a system of ordinary differential equations using scaling symmetries. Numerical solutions of the resulting ordinary differential equations are obtained using the fourth-order Runge–Kutta method. From the numerical results, it is observed that the thickness of the momentum boundary layer increases with increasing the chemical reaction parameter and the Schmidt number. The thicknesses of the thermal and concentration boundary layers are decreased with increasing the chemical reaction parameter and the Schmidt number.     

双曲拉伸面上的流动及其热交换

研究不可压缩粘性流体,在双曲拉伸面上的边界层流动及其热传导．分别使用级数展开法和局部非相似（LNS）法,得到解析结果和数值结果,给出了表面摩擦和Nusselt数的解析结果和数值结果,并进行了互相比较．同时发现动量和热边界层厚度,随着离前缘距离的增加而减小．众所周知,线性拉伸项方程的解,可以作为双曲拉伸首次项方程的解．     

On integration of the unsteady equation of heat exchange between bodies moving in fluid : PMM vol. 43, no. 6, 1979, pp. 1119–1122

The unsteady problem of convective heat exchange of bodies of a arbitrary shape moving in a perfect fluid or of drops moving in a viscous incompressible liquid, is considered in the approximation of thermal boundary layer.     

On the similarity solutions for a steady MHD equation

In this paper, we investigate the similarity solutions for the steady laminar incompressible boundary layer equations governing the magnetohydrodynamic (MHD) flow near the forward stagnation point of two-dimensional and axisymmetric bodies. This leads to the study of a boundary value problem involving a third order autonomous ordinary differential equation. Our main results are the existence, uniqueness and non-existence for concave or convex solutions.     

Effective Boundary Conditions for the Heat Equation with Interior Inclusion

Of concern is the scenario of a heat equation on a domain that contains a thin layer, on which the thermal conductivity is drastically different from that in the bulk. The multi-scales in the spatial variable and the thermal conductivity lead to computational difficulties, so we may think of the thin layer as a thickless surface, on which we impose "effective boundary conditions"(EBCs). These boundary conditions not only ease the computational burden, but also reveal the effect of the inclusion. In this paper, by considering the asymptotic behavior of the heat equation with interior inclusion subject to Dirichlet boundary condition, as the thickness of the thin layer shrinks, we derive, on a closed curve inside a two-dimensional domain, EBCs which include a Poisson equation on the curve, and a non-local one. It turns out that the EBCs depend on the magnitude of the thermal conductivity in the thin layer,compared to the reciprocal of its thickness.     

Retracted: Unsteady flow and heat transfer on a rotating disk in a porous medium

This article has been retracted. See retraction notice DOI: 10.1002/mma.850 . An unsteady flow and heat transfer in a porous medium of a viscous incompressible fluid over a rotating disk in an otherwise ambient fluid are studied. The unsteadiness in the flow field is caused by the angular velocity of the disk which varies with time. The new self‐similar solution of the Navier–Stokes and energy equations is obtained numerically. The solution obtained here is not only the solution of the Navier–Stokes equations, but also of the boundary layer equations. Also, for a simple scaling factor, it represents the solution of the flow and heat transfer in the forward stagnation‐point region of a rotating sphere or over a rotating cone. The asymptotic behaviour of the solution for a large porosity or for a large independent variable is also examined. The surface shear stresses in the radial and tangential directions and the surface heat transfer increase as the acceleration parameter increases. Also, the surface shear stress in the radial direction and the surface heat transfer decrease with increasing porosity, but the surface shear stress in the tangential direction increases. Copyright © 2006 John Wiley & Sons, Ltd.     

Convex solutions of a general similarity boundary layer equation for power-law fluids

This paper is devoted to a general similarity boundary layer equation for power-law fluids, which includes many important similarity boundary layer problems such as the Falker-Skan equation and the magnetohydrodynamic boundary layer equation which arises in the study of self-similar solutions of the two-dimensional steady laminar boundary layer flow for an incompressible electrically conducting power-law fluids along an isolated surface in the presence of an exterior magnetic field orthogonal to the flow. By a rigorous mathematical analysis, the uniqueness, existence and nonexistence results for convex solutions, normal convex solutions and generalized convex solutions to the general similarity boundary layer equation are established. Also the asymptotic behavior of the normal convex solutions at the infinity are displayed.     

On the global existence and uniqueness of solutions to the nonstationary boundary layer system

In this paper, we study the problem of boundary layer for nonstationary flows of viscous incompressible fluids. There are some open problems in the field of boundary layer. The method used here is mainly based on a transformation which reduces the boundary layer system to an initial-boundary value problem for a single quasilinear parabolic equation. We prove the existence of weak solutions to the modified nonstationary boundary layer system. Moreover, the stability and uniqueness of weak solutions are discussed.     

热传导对气-液射流界面不稳定性的作用机理研究

研究了平面分层气-液射流在非线性温度分布条件下的界面不稳定性性质.考虑了气体的可压缩性、液体的粘性、以及气体热导率和密度随温度变化等事实.并应用正则模态方法将问题转化为四阶变系数常微分方程,用数值积分和多重打靶法对模型的空间模式进行了计算,研究了不稳定模态随各物理参量的变化趋势.计算表明模型所体现的不稳定性特征与其它模型的计算结果是一致的.同时计算还得出气体和液体的温差越小、雷诺数越大、热导率变大均将有利于液体射流有效雾化的结果.该结论与HJE.Co.Inc(Glens Falls,NY,USA)的实验数据是定性吻合的.     

On the Navier-Stokes Equations with Energy-Dependent Nonlocal Viscosities

We discuss the mathematical modeling of incompressible viscous flows for which the viscosity depends on the total dissipation energy. In the two-dimensional periodic case, we begin with the case of temperature-dependent viscosities with very large thermal conductivity in the heat convective equation, in which we obtain the Navier-Stokes system coupled with an ordinary differential equation involving the dissipation energy as the asymptotic limit. Letting further the latent heat to vanish, we derive the Navier-Stokes equations with a nonlocal viscosity depending on the total dissipation of energy. Bibliography: 7 titles.Dedicated to V. A. Solonnikov on the occasion of his 70th birthday__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 306, 2003, pp. 71–91.     

Free convection heat and mass transfer in a power law fluid past an inclined surface with thermophoresis

The problem of heat and mass transfer in a power law, two-dimensional, laminar, boundary layer flow of a viscous incompressible fluid over an inclined plate with heat generation and thermophoresis is investigated by the characteristic function method. The governing non-linear partial differential equations describing the flow and heat transfer problem are transformed into a set of coupled non-linear ordinary differential equation which was solved using Runge–Kutta shooting method. Exact solutions for the dimensionless temperature and concentration profiles, are presented graphically for different physical parameters and for the different power law exponents 0 < n < 0.5 and for n > 0.5.     

Local-in-time well-posedness for compressible MHD boundary layer

In this paper, we are concerned with the motion of electrically conducting fluid governed by the two-dimensional non-isentropic viscous compressible MHD system on the half plane with no-slip condition on the velocity field, perfectly conducting wall condition on the magnetic field and Dirichlet boundary condition on the temperature on the boundary. When the viscosity, heat conductivity and magnetic diffusivity coefficients tend to zero in the same rate, there is a boundary layer which is described by a Prandtl-type system. Under the non-degeneracy condition on the tangential magnetic field instead of monotonicity of velocity, by applying a coordinate transformation in terms of the stream function of magnetic field as motivated by the recent work [27], we obtain the local-in-time well-posedness of the boundary layer system in weighted Sobolev spaces.     

Self-similar decay of localized perturbations in the integral boundary layer equation

The integral boundary layer equation (IBLe) arises as a long wave approximation for the flow of a viscous incompressible fluid down an inclined plane. The trivial solution of the IBLe is linearly at best marginally stable, i.e., it has essential spectrum at least up to the imaginary axis. Here, we show that in the stable case this trivial solution is in fact nonlinearly stable, with a Burgers like self-similar decay of localized perturbations. The proof uses renormalization theory and the fact that in the stable case Burgers equation is the amplitude equation for long small amplitude waves in the IBLe.     

Flow past an elliptic cylinder

A theoretical investigation of the unsteady two-dimensional flow of a viscous, incompressible fluid normal to a thin elliptic cylinder is described. The cylinder, which is started impulsively from rest in an open field, continues to move with uniform velocity for the remainder of the problem. Using a vorticity-streamfunction formulation of the full Navier-Stokes equations, transformation techniques are employed to find the initial flow. Strategies which employ boundary layer theory and series expansions of the flow variables to find flow solutions for small values of time are outlined.     

Buoyancy and chemical reaction effects on MHD mixed convection heat and mass transfer in a porous medium with thermal radiation and Ohmic heating

The combined effect of mixed convection with thermal radiation and chemical reaction on MHD flow of viscous and electrically conducting fluid past a vertical permeable surface embedded in a porous medium is analyzed. The heat equation includes the terms involving the radiative heat flux, Ohmic dissipation, viscous dissipation and the internal absorption whereas the mass transfer equation includes the effects of chemically reactive species of first-order. The non-linear coupled differential equations are solved analytically by perturbation technique. The results obtained show that the velocity, temperature and concentration fields are appreciably influenced by the presence of chemical reaction, thermal stratification and magnetic field. It is observed that the effect of thermal radiation and magnetic field is to decrease the velocity, temperature and concentration profiles in the boundary layer. There is also considerable effect of magnetic field and chemical reaction on skin-friction coefficient and Nusselt number.