Heat transfer in unsteady axisymmetric second grade fluid

This work looks at the heat transfer effects on the flow of a second grade fluid over a radially stretching sheet. The axisymmetric flow of a second grade fluid is induced due to linear stretching of a sheet. Mathematical analysis has been carried out for two heating processes, namely (i) with prescribed surface temperature (PST case) and (ii) prescribed surface heat flux (PHF case). The modelled non-linear partial differential equations in two dependent variables are reduced into a partial differential equation with one dependent variable. The resulting non-linear partial differential equations are solved analytically using homotopy analysis method (HAM). The series solutions are developed and the convergence is properly discussed. The series solutions and graphs of velocity and temperature are constructed. Particular attention is given to the variations of emerging parameters such as second grade parameter, Prandtl and Eckert numbers.     

Hiemenz flow and heat transfer of a third grade fluid

The laminar flow and heat transfer of an incompressible, third grade, electrically conducting fluid impinging normal to a plane in the presence of a uniform magnetic field is investigated. The heat transfer analysis has been carried out for two heating processes, namely, (i) with prescribed surface temperature (PST-case) and (ii) prescribed surface heat flux (PHF-case). By means of the similarity transformation, the governing non-linear partial differential equations are reduced to a system of non-linear ordinary differential equations and are solved by a second-order numerical technique. Effects of various non-Newtonian fluid parameters, magnetic parameter, Prandtl number on the velocity and temperature fields have been investigated in detail and shown graphically. It is found that the velocity gradient at the wall decreases as the third grade fluid parameter increases.     

Gas dynamics applications of a characteristic Cauchy problem

Some initial-boundary-value problems for a system of quasilinear partial differential equations of gas dynamics with the initial data prescribed on the characteristic surface (characteristic Cauchy problem) are considered. The following three-dimensional flow problems are investigated: the flow produced by a motion of an impermeable piston; the flow produced by a permeable piston with a given pressure; and the flow produced by the moving free boundary. In the first two problems, the piston motion is prescribed; in the last problem, the free boundary motion cannot be prescribed in advance and must be determined as a part of the problem. It is shown that those problems can be reduced to a characteristic Cauchy problem of a certain standard type that satisfies the analog of Cauchy-Kowalewski's existence theorem in the class of analytical functions (Differential Equations 12 (1977) 1438-1444). Thus, it is proved that, in the case of the analyticity of the input data, the considered problems have unique piecewise analytic solutions which may be expressed by infinite power series (the procedure of constructing the power series solution is described in Differential Equations 12 (1977) 1438-1444 as a part of the proof of the theorem).     

Perturbation analysis of a modified second grade fluid over a porous plate

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 flow of this fluid is considered over a porous plate. Equations of motion in dimensionless form are derived. When the power-law effects are small compared to second grade effects, a regular perturbation problem arises which is solved. The validity criterion for the solution is derived. When second grade effects are small compared to power-law effects, or when both effects are small, the problem becomes a boundary layer problem for which the solutions are also obtained. Perturbation solutions are contrasted with the numerical solutions. For the regular perturbation problem of small power-law effects, an excellent match is observed between the solutions if the validity criterion is met. For the boundary layer solution of vanishing second grade effects however, the agreement with the numerical data is not good. When both effects are considered small, the boundary layer solution leads to the same solution given in the case of a regular perturbation problem.     

Continuous dependence results for a class of nonlinear heat equations in a cylindrical region

In this paper, we derive bounds for the solutions of a quasilinear heat equation in a finite cylindrical region if the far end and the lateral surface are held at zero temperature, and a nonzero temperature is applied at the near end. Some continuous dependence inequalities are also obtained. We also investigate the case in which a given heat flux is prescribed at the near end, instead of a given temperature. Copyright © 2008 John Wiley & Sons, Ltd.     

Helical flows of second grade fluid due to constantly accelerated shear stresses

The helical flows of second grade fluid between two infinite coaxial circular cylinders is considered. The motion is produced by the inner cylinder that at the initial moment applies torsional and longitudinal constantly accelerated shear stresses to the fluid. The exact analytic solutions, obtained by employing the Laplace and finite Hankel transforms and presented in series form in term of usual Bessel functions of first and second kind, satisfy both the governing equations and all imposed initial and boundary conditions. In the limiting case when α → 0, the solutions for Newtonian fluid are obtained for the same motion. The large-time solutions and transient solutions for second grade fluid are also obtained, and effect of material parameter α and kinematic viscosity ν is discussed. In the last, the effects of various parameters of interest on fluid motion as well as the comparison between second grade and Newtonian fluids are analyzed by graphical illustrations.     

Unsteady flow with heat and mass transfer of a third grade fluid over a stretching surface in the presence of chemical reaction

This paper describes the unsteady flow with heat and mass transfer characteristics in a third grade fluid bounded by a stretching sheet. The resulting problems are solved by means of homotopy analysis method (HAM). Convergence of derived series solutions is explicitly discussed. Graphical results for various interesting parameters are presented and analyzed.     

Selfsimilar solutions of the problem of heat and mass transfer in a saturated porous medium with a volume heat source

Selfsimilar solutions of a system of stationary equations of heat condunction and filtration of molten material in the presence of a volume heat source generated by absorption of the energy of electromagnetic radiation, are considered. The possibility of the existence of a self-similar solution in the case of various (plane, cylindrical and spherical) spatial symmetries is studied. The existence of a selfsimilar solution is shown for the axisymmetric case when the radiation obeys a prescribed law. The influence of the surface volume heating and convective heat transfer due to filtration is studied. A solution for the case when the filtration of the molten phase is quasistationary is also investigated.     

Truss topology optimization including unilateral contact

This work extends the ground structure approach of truss topology optimization to include unilateral contact conditions. The traditional design objective of finding the stiffest truss among those of equal volume is combined with a second objective of achieving a uniform contact force distribution. Design variables are the volume of bars and the gaps between potential contact nodes and rigid obstacles. The problem can be viewed as that of finding a saddle point of the equilibrium potential energy function (a convex problem) or as that of minimizing the external work among all trusses that exhibit a uniform contact force distribution (a nonconvex problem). These two formulations are related, although not completely equivalent: they give the same design, but concerning the associated displacement states, the solutions of the first formulation are included among those of the second but the opposite does not necessarily hold.In the classical noncontact single-load case problem, it is known that an optimal truss can be found by solving a linear programming (LP) limit design problem, where compatibility conditions are not taken into account. This result is extended to include unilateral contact and the second objective of obtaining a uniform contact force distribution. The LP formulation is our vehicle for proving existence of an optimal design: by standard LP theory, we need only to show primal and dual feasibility; the primal one is obvious, and the dual one is shown by the Farkas lemma to be equivalent to a condition on the direction of the external load. This method of proof extends results in the classical noncontact case to structures that have a singular stiffness matrix for all designs, including a case with no prescribed nodal displacements.Numerical solutions are also obtained by using the LP formulation. It is applied to two bridge-type structures, and trusses that are optimal in the above sense are obtained.This work was supported by The Center for Industrial Information Technology (CENIIT) and the Swedish Research Council for Engineering Sciences (TFR).     

MHD three-dimensional flow of viscoelastic fluid over an exponentially stretching surface with variable thermal conductivity

This study models the magnetohydrodynamic (MHD) three-dimensional boundary layer flow of viscoelastic fluid. The flow is due to the exponentially stretching surface. The heat transfer analysis is performed through prescribed surface temperature (PST) and prescribed surface heat flux (PHF). The thermal conductivity is taken temperature dependent. Series solutions of velocities and temperatures are constructed. Graphical results for PST and PHF cases are plotted and analyzed. Numerical values of skin-friction coefficients and Nusselt numbers are presented and discussed.     

Homotopy analysis method for MHD viscoelastic fluid flow and heat transfer in a channel with a stretching wall

In this paper, we analyze the flow and heat transfer characteristics of a magnetohydrodynamic (MHD) viscoelastic fluid in a parallel plate channel with a stretching wall. Homotopy analysis method (HAM) is used to obtain analytical solutions of the governing nonlinear differential equations. The analytical solutions are obtained in the form of infinite series and the convergence of the series solution is discussed explicitly. The obtained results are presented through graphs for several sets of values of the parameters, and the salient features of the solutions are analyzed. A comparison of our HAM results (for a special case of the study) with the available results in the literature (obtained by other methods) shows that our results are accurate for a wide range of parameters. Further, we point that our analysis finds application to the study of the haemodynamic flow of blood in the cardiovascular system subject to external magnetic field.     

Partial slip effects on the flow and heat transfer characteristics in a third grade fluid

This article looks at the slip effects on the flow and heat transfer of a third grade fluid past a porous plate. The resulting equations and boundary conditions are non-linear. The non-linear boundary condition is reduced into a linear one and a series solution of the problem is obtained using the homotopy analysis method (HAM). Variations of interesting parameters are seen on the velocity and temperature profiles.     

Heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet with thermal radiation and non-uniform heat source/sink

An analysis has been carried out to study the flow and heat transfer characteristics for MHD viscoelastic boundary layer flow over an impermeable stretching sheet with space and temperature dependent internal heat generation/absorption (non-uniform heat source/sink), viscous dissipation, thermal radiation and magnetic field due to frictional heating. The flow is generated due to linear stretching of the sheet and influenced by uniform magnetic field, which is applied vertically in the flow region. The governing partial differential equations for the flow and heat transfer are transformed into ordinary differential equations by a suitable similarity transformation. The governing equations with the appropriate conditions are solved exactly. The effects of viscoelastic parameter and magnetic parameter on skin friction and the effects of viscous dissipation, non-uniform heat source/sink and the thermal radiation on heat transfer characteristics for two general cases namely, the prescribed surface temperature (PST) case and the prescribed wall heat flux (PHF) case are presented graphically and discussed. The numerical results for the wall temperature gradient (the Nusselt number) are presented in tables and are discussed.     

Temperature fields in a hollow cylinder in presence of heat source under the boundary conditions of the second kind

General expressions have been derived for unsteady temperature distribution in a finite hollow cylinder under the influence of a time dependent volume heat source and prescribed heat fluxes at the boundaries. By introducing certain artificial additional heat source functions, corresponding Pseudo-steady solutions are defined and by means of which the temperature fields are expressed in the form of uniformly convergent series solutions. By the application of integral transform techniques the expressions for temperature distribution are obtained in various forms which may be applicable to various cases of technological importance.     

Stretching a plane surface in a viscoelastic fluid with prescribed skin friction

A model of forced convection flow due to stretching surface is derived to represent the physical system with prescribed skin friction. To achieve the similar solutions, the partial differential equations are reduced into ordinary differential equations. The analytic solutions of the resulting problems have been obtained by a homotopy analysis method. The convergence of the developed series solution is seen. Finally, the results of velocity, temperature, the stretching velocity, and Nusselt number are analyzed. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Mixed convection heat transfer in the boundary layers on an exponentially stretching surface with magnetic field

An analysis has been carried out to describe mixed convection heat transfer in the boundary layers on an exponentially stretching continuous surface with an exponential temperature distribution in the presence of magnetic field, viscous dissipation and internal heat generation/absorption. Approximate analytical similarity solutions of the highly non-linear momentum and energy equations are obtained. The present results are found to be in excellent agreement with previously published work on various special cases of the problem. Numerical results for temperature distribution and the local Nusselt number have been obtained for different values of the governing parameters. The numerical solutions are obtained by considering an exponential dependent stretching velocity and prescribed boundary temperature on the flow directional coordinate. The effects of various physical parameters like Prandtl number, Hartman number, Grashof number on dimensionless heat transfer characteristics are discussed in detail. In particular, it has been found that increase in Prandtl number decreases the skin-friction coefficient at the stretching surface, while increase in the strength of the magnetic field leads to increase in the local Nusselt number.     

Heat transfer analysis of the steady flow of an Oldroyd 8-constant fluid due to a suddenly moved plate

This paper concentrates on the heat transfer analysis of the steady flow of an Oldroyd 8-constant fluid due to a suddenly moved plate. The heat transfer analysis has been carried out for the prescribed surface temperature. Employing homotopy analysis method, the developed system of equations are solved analytically. The convergence of the obtained series solution is established. The influence of pertinent parameters on temperature profiles and Nusselt number is shown and discussed through several graphs. Further, a comparison between temperature profiles of Newtonian and Oldroyd fluids is also made.     

Analytical solutions for pipe flow of a fourth grade fluid with Reynold and Vogel’s models of viscosities

In this paper, the steady incompressible flow of a fourth grade fluid down a vertical cylinder with variable viscosity and heat transfer analysis are considered. The governing equations of motion of fourth grade fluid in cylindrical coordinates have been derived. The complete analytic solutions of momentum and energy equations have been obtained by homotopy analysis method (HAM). The entropy generation number for two cases of viscosities namely (i) Reynold’s model and (ii) Vogel’s model have also been computed. The results are compared with the available finite difference and perturbation solutions of third grade fluid. At the end the pressure gradient for the fourth grade fluid has also been calculated.     

Purely analytic approximate solutions for steady three-dimensional problem of condensation film on inclined rotating disk by homotopy analysis method

The similarity transform for the steady three-dimensional problem of a condensation film on an inclined rotating disk gives a system of nonlinear ordinary differential equations which are analytically solved by applying a newly developed method namely the homotopy analysis method (HAM). The analytic solutions of the system of nonlinear ordinary differential equations are constructed in the series form. The convergence of the obtained series solutions is carefully analyzed. The velocity and temperature profiles are shown and the influence of the Prandtl number on the heat transfer and the Nusselt number is discussed in detail. The validity of our results is verified by numerical results.     

Some Convergence Results for a Class of Nonlinear Phase-Field Evolution Equations

Two heat diffusion problems in the framework of the parabolic phase-field model are presented. The first problem is related to a single isotropic fluid and the other describes the heat transmission between two different substances in contact. Some known existence and uniqueness results are briefly recalled. Then, an asymptotic analysis of both situations is carried out as the kinetic equation collapses to a temperature-phase relation of Stefan type, in the first case in the whole material, and in the second in only one of the substances. In both cases, a convergence result for the solutions is proved. The second problem shows some more mathematical difficulties that are due to the presence of nontrivial terms on the common boundary. In order to control the latter, some tools are used from the Γ-convergence theory for convex functionals.