Transient free convection flow between long vertical parallel plates with ramped wall temperature at one boundary in the presence of thermal radiation and constant mass diffusion

The unsteady laminar free convection flow between two long vertical parallel plates with ramped wall temperature at one boundary has been investigated in the presence of thermal radiation and chemical species concentration. The exact solutions of the momentum, energy and concentration equations have been obtained using the Laplace transform technique. The velocity and temperature profiles, skin-friction and Nusselt number variations are shown graphically and the numerical values of the volume flow rate, the total heat rate and species rate added to the fluid are presented in a table. The influence of different system parameters such as the radiation parameter (R), buoyancy ratio parameter (N), Schmidt number (Sc) and time (t) has been analyzed carefully. A critical analysis of the coupled heat and mass transfer phenomena is provided. The free convective flow due to ramped wall temperature has also been compared with the baseline case of flow due to constant wall temperature.     

Laminar Mixed Convection Adjacent to Vertical Continuously Stretching Sheets With Variable Viscosity and Variable Thermal Diffusivity

The paper presents a study of the laminar mixed convection adjacent to vertical continuously stretching sheets, taking into account the effects of variable viscosity and variable thermal diffusivity. The similarity solutions are reported for isothermal sheet moving with a velocity of the form u_w=Bx^0.5 and a continuous linearly stretching sheet with a linear surface temperature distribution. The equations of conservation of mass, momentum and energy, which govern the flow and heat transfer, are solved numerically by using the shooting method. The numerical results obtained for the flow and heat transfer characteristics reveal many interesting behaviors. The numerical results show that, variable viscosity, variable thermal diffusivity, the velocity exponent parameter, the temperature exponent parameter and the buoyancy force parameter have significant influences on the velocity and temperature profiles, shear stress and Nusselt number in two cases air and water.     

Melting heat transfer in the stagnation‐point flow of an upper‐convected Maxwell (UCM) fluid past a stretching sheet

The steady laminar boundary layer flow and heat transfer past a stretching sheet arre considered. Upper‐convected Maxwell (UCM) fluid is treated as a rheological model. The resulting nonlinear differential system is solved by homotopy analysis method (HAM). The influence of melting parameter (M), Prandtl number (Pr), Deborah number (β) and stretching ratio (A = a/c) on the velocity and temperature profiles is thoroughly examined. It is noticed that fields are effected appreciably with the variation of parameters. Furthermore, it is seen that the local Nusselt number is a decreasing function of melting parameter. Copyright © 2011 John Wiley & Sons, Ltd.     

Dual solutions of Casson fluid flows over a power-law stretching sheet

A steady boundary layer flow of a non-Newtonian Casson fluid over a power-law stretching sheet is investigated. A self-similar form of the governing equation is obtained, and numerical solutions are found for various values of the governing parameters. The solutions depend on the fluid material parameter. Dual solutions are obtained for some particular range of these parameters. The fluid velocity is found to decrease as the power-law stretching parameter β in the rheological Casson equation increases. At large values of β, the skin friction coefficient and the velocity profile across the boundary layer for the Casson fluid tend to those for the Newtonian fluid.     

Dual solutions of a mixed convection flow near the stagnation point region over an exponentially stretching/shrinking sheet in nanofluids

The aim of this paper is to study the development of mixed convection flow near the stagnation point region over an exponentially stretching/shrinking sheet in nanofluids. The external flow, stretching velocity and wall temperature are assumed to vary as prescribed exponential functions. Using the local similarity method, it has been shown that dual solutions of velocity and temperature exist for certain values of suction/injection, mixed convection, nanoparticle volume fraction and stretching/shrinking parameters. The transformed non-linear ordinary differential equations along with the boundary conditions form a two point boundary value problem and are solved using Shooting method, by converting into an initial value problem. In this method, the system of equations is converted into a set of first order system which is solved by fourth-order Runge–Kutta method. Three different types of nanoparticles, namely copper (Cu), aluminum oxide (Al₂O₃) and titanium oxide (TiO₂) are considered by using water-based fluid with Prandtl number Pr = 6.2. It is also found that the skin friction coefficient and the heat transfer rate at the surface are highest for Copper–water nanofluids as compared to Al₂O₃. The effect of the solid volume fraction parameter φ of the nanofluids on the heat transfer characteristics is also investigated. The results indicate that dual solutions exist only for shrinking sheet. The effects of various parameters on the velocity and temperature profiles are also presented here.     

Mixed convection stagnation-point flow on vertical stretching sheet with external magnetic field

The problem of steady laminar magnetohydrodynamic （MHD） mixed con- vection stagnation-point flow of an incompressible viscous fluid over a vertical stretch- ing sheet is studied. The effect of an externally magnetic field is taken into account. The transformed boundary layer equations are solved numerically by using an implicit finite-difference scheme. Numerical results are obtained for various values of the mixed convection parameter, Hartmann number, and Prandtl number. The effects of an exter- nally magnetic field on the skin friction coefficient, local Nusselt number, velocity, and temperature profiles for both A 〉 1 and A ~ 1, where A is the velocity ratio parameter, are presented graphically and discussed in detail. Both assisting and opposing flows are considered, and it is found that dual solutions exist for the opposing flow.     

Slip effects on unsteady stagnation-point flow and heat transfer over a shrinking sheet

This paper investigates the unsteady boundary layer stagnation-point flow and heat transfer over a linearly shrinking sheet in the presence of velocity and thermal slips. Similarity solutions for the transformed governing equations are obtained and the reduced equations are then solved numerically using fourth order Runge-Kutta method with shooting technique. The numerical results show that multiple solutions exist for certain range of the ratio of shrinking velocity to the free stream velocity (i.e., α) which again depend on the unsteadiness parameter β and the velocity slip parameter (i.e., δ). An enhancement of the velocity slip parameter δ causes more increment in the existence range of similarity solution. Fluid velocity at a point increases increases with the increase in the value of the velocity slip parameter δ, resulting in a decrease in the temperature field. The effects of the velocity and thermal slip parameters, unsteadiness parameter (β) and the velocity ratio parameter (α) on the velocity and temperature distributions are computed, analyzed and discussed. The reported results are in good agreement with the available published results in the literature.     

Biorthogonal stretching and shearing of an impermeable surface in a uniformly rotating fluid system

The flow induced by an impermeable flat surface executing orthogonal stretching and orthogonal shearing in a rotating fluid system is investigated. Both the stretching and shearing are linear in the coordinates. An exact similarity reduction of the Navier–Stokes equations gives rise to a pair of nonlinearly-coupled ordinary differential equations governed by three parameters. In this study we set one parameter and analyze the problem which leads to flow for an impermeable surface with shearing and stretching due to velocity u along the x-axis of equal strength a while the shearing and stretching due to velocity v along the y-axis of equal strength b. These solutions depend on two parameters—a Coriolis (rotation) parameter \(\sigma = \Omega /a\) and a stretching/shearing ratio \(\lambda =b/a\). A symmetry in solutions is found for \(\lambda = 1\). The exact solution for \(\sigma = 0\) and the asymptotic behavior of solutions for \(|\sigma | \rightarrow \infty\) are determined and compared with numerical results. Oscillatory solutions are found whose strength increases with increasing values of \(|\sigma |\). It is shown that these solutions tend to the well-known Ekman solution as \(|\sigma | \rightarrow \infty\).     

Melting effect and Cattaneo-Christov heat flux in fourth-grade material flow through a Darcy-Forchheimer porous medium

The melting phenomenon in two-dimensional (2D) flow of fourth-grade material over a stretching surface is explored. The flow is created via a stretching surface. A Darcy-Forchheimer (D-F) porous medium is considered in the flow field. The heat transport is examined with the existence of the Cattaneo-Christov (C-C) heat flux. The fourth-grade material is electrically conducting subject to an applied magnetic field. The governing partial differential equations (PDEs) are reduced into ordinary differential equations (ODEs) by appropriate transformations. The solutions are constructed analytically through the optimal homotopy analysis method (OHAM). The fluid velocity, temperature, and skin friction are examined under the effects of various involved parameters. The fluid velocity increases with higher material parameters and velocity ratio parameter while decreases with higher magnetic parameter, porosity parameter, and Forchheimer number. The fluid temperature is reduced with higher melting parameter while boosts against higher Prandtl number, magnetic parameter, and thermal relaxation parameter. Furthermore, the skin friction coefficient decreases against higher melting and velocity ratio parameters while increases against higher material parameters, thermal relaxation parameter, and Forchheimer number.

     

Heat transfer characteristics of a continuous stretching surface

The similarity solutions for the governing ordinary differential equations of the boundary layer corresponding to a stretching surface have been reported. Power law velocity and temperature distribution were assumed for velocity exponent 3≥m≥?0.41176, ?1.1≥m≥?3, and for temperature exponent 3≥n≥?3. Solutions have been found forn=0 and allm where heat transferred from the stretching surface to the ambient. The direction and amount of heat flow were found to be dependent on the magnitude ofn andm for the same Prandtl number. Nusselt number increases with increasingm andPr for uniform and variable surface temperature however, for uniform surface heat flux it decreases with increasingm for constantPr.     

Flow and heat transfer of nanofluid past stretching/shrinking sheet with partial slip boundary conditions

The boundary layer flow of a nanofluid past a stretching/shrinking sheet with hydrodynamic and thermal slip boundary conditions is studied. Numerical solutions to the governing equations are obtained using a shooting method. The results are found for the skin friction coefficient, the local Nusselt number, and the local Sherwood number as well as the velocity, temperature, and concentration profiles for some values of the velocity slip parameter, thermal slip parameter, stretching/shrinking parameter, thermophoresis parameter, and Brownian motion parameter. The results show that the local Nusselt number, which represents the heat transfer rate, is lower for higher values of thermal slip parameter, thermophoresis parameter, and Brownian motion parameter.     

Radiation effect on Darcy free convection flow along an inclined surface placed in porous media

The paper investigates the effect of radiation on Darcy's buoyancy induced flow of an optically dense viscous incompressible fluid along a heated inclined flat surface maintained at uniform temperature placed in a saturated porous medium with Rosseland diffusion approximation employing the implicit finite difference method together with Keller box elimination technique. Both the streamwise and normal components of the buoyancy force are retained in the momentum equations. The numerical results show that as the buoyancy parameter, ξ, increases the local Nusselt number increases. The results for the locally nonsimilar solutions are compared with the locally similar solutions for small angle of inclination and approximate similar solutions along vertical surface. The effect of the conduction-radiation parameter, R _d , and the surface temperature excess ration, θ _w , on the local Nusselt number, the tangential velocity distribution and the temperature distribution are also shown graphically.     

Thin film flow and heat transfer on an unsteady stretching sheet with internal heating

This research studied the influence of internal heat generation on flow and heat transfer in a thin liquid film on an unsteady stretching sheet. The velocity and temperature fields were solved using the Homotopy Analysis Method (HAM), taking a general surface temperature into consideration. The analytical series solution are presented and the numerical results obtained are tabulated. The effects of unsteadiness parameter, Prandtl number and temperature-dependent parameter in this study are discussed and presented graphically via the velocity and temperature profiles.     

Mixed convection boundary layer flow over a horizontal circular cylinder with Newtonian heating

The steady mixed convection boundary layer flow over a horizontal circular cylinder, generated by Newtonian heating in which the heat transfer from the surface is proportional to the local surface temperature, is considered in this study. The governing boundary layer equations are first transformed into a system of non-dimensional equations via the non-dimensional variables, and then into non-similar equations before they are solved numerically using a numerical scheme known as the Keller-box method. Numerical solutions are obtained for the skin friction coefficient Re ^1/2 C _f and the local wall temperature θ _w (x) as well as the velocity and temperature profiles with two parameters, namely the mixed convection parameter λ and the Prandtl number Pr.     

Mixed convection heat transfer over a non-linear stretching surface with variable fluid properties

This article presents a numerical solution for the steady two-dimensional mixed convection MHD flow of an electrically conducting viscous fluid over a vertical stretching sheet, in its own plane. The stretching velocity and the transverse magnetic field are assumed to vary as a power function of the distance from the origin. The temperature dependent fluid properties, namely, the fluid viscosity and the thermal conductivity are assumed to vary, respectively, as an inverse function of the temperature and a linear function of the temperature. A generalized similarity transformation is introduced to study the influence of temperature dependent fluid properties. The transformed boundary layer equations are solved numerically, using a finite difference scheme known as Keller Box method, for several sets of values of the physical parameters, namely, the stretching parameter, the temperature dependent viscosity parameter, the magnetic parameter, the mixed convection parameter, the temperature dependent thermal conductivity parameter and the Prandtl number. The numerical results thus obtained for the flow and heat transfer characteristics reveal many interesting behaviors. These behaviors warrant further study of the effects of the physical parameters on the flow and heat transfer characteristics. Here it may be noted that, in the case of the classical Navier-Stokes fluid flowing past a horizontal stretching sheet, McLeod and Rajagopal (1987) [42] showed that there exist an unique solution to the problem. This may not be true in the present case. Hence we would like to explore the non-uniqueness of the solution and present the findings in the subsequent paper.     

Non-linear Oberbeck-electroconvection in a poorly conducting fluid through a vertical channel in the presence of an electric field

Non-linear Oberbeck-electroconvection (OBEC) in a poorly electrically conducting fluid through a vertical channel, when the walls are held at different temperatures with temperature difference perpendicular to gravity, is studied using the modified Navier stokes equation in the presence of both induced and an applied electric field. Both analytical and numerical solutions for the non-linear coupled equations governing the motion are obtained and found that analytical solutions agree well with numerical solutions for values of the buoyancy parameter N<1. It is shown that OBEC can be controlled by maintaining the temperature difference either in the same direction or opposing the potential difference with a suitable value of electric number W. The effect of W on velocity, temperature, rate of heat transfer, skin friction and mass flow rate are computed and the results are depicted graphically. We found that analytical results agree well with numerical results for small values of N. We also found that an increase in W accelerates the flow and hence increases linearly the skin friction and mass flow rate.     

On stagnation point flow of Sisko fluid over a stretching sheet

The steady two-dimensional stagnation-point flow, represented by Sisko fluid constitutive model, over a stretching sheet is investigated theoretically. Using suitable similarity transformations, the governing boundary-layer equations are transformed into the self-similar non-linear ordinary differential equation. The transformed equation is then solved using a very efficient analytic technique namely the homotopy analysis method (HAM) and the HAM solutions are validated by the exact analytic solutions obtain in certain special cases. The influence of the power-law index (n), the material parameter (A) and the velocity ratio parameter (d/c) on the flow characteristics is studied and presented through several graphs. In addition, the local skin friction coefficient for several values of these parameters is tabulated and examined. The similarity solutions for both the Newtonian and the power-law fluids are presented as special cases of the analysis. The results obtained reveal that, in comparison with the Newtonian and the power-law fluids, the velocity profiles of the Sisko fluid are much faster (slower), for d/c<1 (d/c>1), respectively.     

Oblique stagnation-point flow of an incompressible visco-elastic fluid towards a stretching surface

An analysis is made of the steady two-dimensional stagnation-point flow of an incompressible viscoelastic fluid over a flat deformable surface when the surface is stretched in its own plane with a velocity cx, where x is the distance from the stagnation-point and c is a positive constant. It is shown that for a viscoelastic fluid of short memory (obeying Walters’ B model), a boundary layer is formed when the stretching velocity of the surface is less than ax, where ax+2by is the inviscid free-stream velocity and y is the distance normal to the plate, a and b being constants and the velocity at a point increases with increase in the elasticity of the fluid. On the other hand an inverted boundary layer is formed when the surface stretching velocity exceeds ax and the velocity decreases with increase in the elasticity of the fluid. A novel result of the analysis is that the flow near the stretching surface is that corresponding to an inviscid stagnation-point flow when a=c. Temperature distribution in the boundary layer is found in three cases, namely: (i) the sheet with constant surface temperature (CST); (ii) the sheet with variable surface temperature (VST) and (iii) the sheet with prescribed quadratic power law surface heat flux (PHF) for various values of non-dimensional parameters. It is found that in all the three cases when a/c>1, temperature at a point decreases with increase in the elasticity of the fluid and when a/c<1, temperature at a point increases with increase in the elasticity of the fluid. Further temperature at a point decreases with increase in the radiation parameter and wall temperature parameter.     

Investigation of nanofluid flow in a vertical channel considering polynomial boundary conditions by Akbari-Ganji's method

In this research, a vertical channel containing a laminar and fully developed nanofluid flow is investigated. The channel surface's boundary conditions for temperature and volume fraction functions are considered qth-order polynomials. The equations related to this problem have been extracted and then solved by the AGM and validated through the Runge-Kutta numerical method and another similar study. In the study, the effect of parameters, including Grashof number, Brownian motion parameter, etc., on the motion, velocity, temperature, and volume fraction of nanofluids have been analyzed. The results demonstrate that increasing the Gr number by 100% will increase the velocity profile function by 78% and decrease the temperature and fraction profiles by 20.87% and 120.75%. Moreover, rising the Brownian motion parameter in five different sizes (0.1, 0.2, 0.3, 0.4, and 0.5) causes lesser velocity, about 24.3% at first and 4.35% at the last level, and a maximum 52.86% increase for temperature and a 24.32% rise for Ψ occurs when N_b rises from 0.1 to 0.2. For all N_t values, at least 55.44%, 18.69%, for F(η), and Ω(η), and 20.23% rise for Ψ(η) function is observed. Furthermore, enlarging the N_r parameter from 0.25 to 0.1 leads F(η) to rise by 199.7%, fluid dimensionless temperature, and dimensional volume fraction to decrease by 18% and 92.3%. In the end, a greater value of q means a more powerful energy source, amplifying all velocity, temperature, and volume fraction functions. The main novelty of this research is the combined convection qth-order polynomials boundary condition applied to the channel walls. Moreover, The AMG semi-analytical method is used as a novel method to solve the governing equations.