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This paper presents a numerical analysis of a steady three-dimensional fluid flow and heat transfer towards a permeable shrinking sheet. The governing nonlinear partial differential equations are transformed into a system of ordinary differential equations by a similarity transformation, which are then solved numerically by a shooting method. The effects of the governing parameters on the skin friction and heat transfer from the surface of the shrinking surface are illustrated graphically. It is found that dual solutions exist for the shrinking case. A comparison with known results from the open literature has been done and it is shown to be in excellent agreement. [ABSTRACT FROM AUTHOR]

The problem of steady laminar magnetohydrodynamic (MHD) mixed convection stagnation-point flow of an incompressible viscous fluid over a vertical stretching 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 externally 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. [ABSTRACT FROM AUTHOR]

Steady mixed convection boundary layer flow from an isothermal horizontal circular cylinder embedded in a porous medium filled with a nanofluid has been studied for both cases of a heated and cooled cylinder. The resulting system of nonlinear partial differential equations is solved numerically using an implicit finite-difference scheme. The solutions for the flow and heat transfer characteristics are evaluated numerically for various values of the governing parameters, namely the nanoparticle volume fraction φ and the mixed convection parameter λ. Three different types of nanoparticles are considered, namely Cu, AlO and TiO. It is found that for each particular nanoparticle, as the nanoparticle volume fraction φ increases, the magnitude of the skin friction coefficient decreases, and this leads to an increase in the value of the mixed convection parameter λ which first produces no separation. On the other hand, it is also found that of all the three types of nanoparticles considered, for any fixed values of φ and λ, the nanoparticle Cu gives the largest values of the skin friction coefficient followed by TiO and AlO. Finally, it is worth mentioning that heating the cylinder (λ > 0) delays separation of the boundary layer and if the cylinder is hot enough (large values of λ > 0), then it is suppressed completely. On the other hand, cooling the cylinder (λ < 0) brings the boundary layer separation point nearer to the lower stagnation point and for a sufficiently cold cylinder (large values of λ < 0) there will not be a boundary layer on the cylinder. [ABSTRACT FROM AUTHOR]

In this paper, the mathematical model of free convection boundary layer flow on a solid sphere with Newtonian heating, in which the heat transfer from the surface is proportional to the local surface temperature, is considered. The transformed boundary layer equations are solved numerically using an efficient numerical scheme known as the Keller-box method. Numerical solutions are obtained for the local wall temperature, the local skin friction coefficient, as well as the velocity and temperature profiles. The features of the flow and heat transfer characteristics for different values of the Prandtl number Pr and conjugate parameter γ are analyzed and discussed. [ABSTRACT FROM AUTHOR]

Abstract: The effects of suction and injection on steady laminar mixed convection boundary layer flow over a permeable horizontal flat plate in a viscous and incompressible fluid is investigated in this paper. The similarity solutions of the governing boundary layer equations are obtained for some values of the suction and injection parameter f 0, the constant exponent n of the wall temperature as well as the mixed convection parameter λ. The resulting system of nonlinear ordinary differential equations is solved numerically for both assisting and opposing flow regimes using a finite-difference scheme known as the Keller-box method. Numerical results for the reduced skin friction coefficient, the reduced local Nusselt number, and the velocity and temperature profiles are obtained for various values of the parameters considered. Dual solutions are found to exist for the opposing flow. [Copyright &y& Elsevier]

Abstract: The steady boundary layer flow and heat transfer over a stretching sheet with Newtonian heating in which the heat transfer from the surface is proportional to the local surface temperature, is considered in this study. The transformed governing nonlinear boundary layer equations are solved numerically by a finite-difference method. Numerical solutions are obtained for the heat transfer from the stretching sheet and the wall temperature for a large range of values of the Prandtl number Pr. The Newtonian heating is controlled by a dimensionless conjugate parameter, which varies between zero (insulated wall) and infinity (wall temperature remains constant). The important findings in this study are the variation of the surface temperature and heat flux from the stretching surface with the conjugate parameter and Prandtl number. It is found that these parameters have essential effects on the heat transfer characteristics. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the problem of unsteady uniform flow across a stretching surface in an arbitrary direction is studied theoretically, where the unsteadiness is caused by the impulsive motion of the stretching surface. Numerical results of the governing partial differential equations are obtained using an implicit finite-difference scheme for the whole transient from the early or initial unsteady-state flow to the final steady-state flow. The early unsteady-state flow is solved analytically. The numerical solution obtained for the reduced skin friction coefficient is compared with previously reported results and the results for velocity profiles, h and g profiles are also presented in this paper. It is found that there is a smooth transition from the small-time solution (initial unsteady flow) to the large-time solution (final steady-state flow). [Copyright &y& Elsevier]

A THEORETICAL ANALYSIS is made for the steady two-dimensional post-stagnation-point flow of an incompressible viscous fluid over a stretching vertical sheet in its own plane. The stretching velocity, the free stream velocity and the surface temperature are assumed to vary linearly with the distance from the stagnation point. The governing partial differential equations are transformed into a coupled system of ordinary differential equations, which is then solved numerically by a finite-difference method. Results are presented in terms of the skin friction coefficient and local Nusselt number, along with a selection of velocity and temperature profiles. It was shown that for both cases of a fixed surface (ε = 0) and a stretching surface (ε ≠ 0), dual solutions exist for the assisting flow (positive values of the buoyancy parameter λ), besides that usually reported in the literature for the opposing flow (λ < 0). It was also found that for the assisting flow, a solution exists for all values of λ (> 0), while for the opposing flow, a solution exists only if the magnitude of the buoyancy parameter is small. [ABSTRACT FROM AUTHOR]

The steady laminar MHD boundary-layer flow past a wedge immersed in an incompressible micropolar fluid in the presence of a variable magnetic field is investigated. The governing partial differential equations are transformed to the ordinary differential equations using similarity variables, and then solved numerically using a finite-difference scheme known as the Keller-box method. Numerical results show that the micropolar fluids display drag reduction and consequently reduce the heat transfer rate at the surface, compared to the Newtonian fluids. The opposite trends are observed for the effects of the magnetic field on the fluid flow and heat transfer characteristics. [ABSTRACT FROM AUTHOR]

Unsteady mixed convection boundary layer flow adjacent to a vertical, stretching surface in a viscous and incompressible quiescent fluid is studied in this paper. The stretching velocity and the surface temperature are assumed to vary linearly with the distance along the surface. Two equal and opposite forces are impulsively applied along the x-axis so that the wall is stretched, keeping the origin fixed in a viscous fluid of constant ambient temperature T∞. The transformed unsteady boundary layer equations are solved numerically using the Keller-box method for the whole transient from t = 0 (initial state) to t → ∞ (final steady-state flow). The equations for the early unsteady flow (0 < t ⪡ 1) are also solved analytically. It is found that there is a smooth transition from the small-time solution to the large-time solution. Results for the local skin friction coefficient and the local Nusselt number, temperature profiles as well as velocity profiles are presented for different values of the governing parameters, namely, the buoyancy parameter, λ (= constant), and Prandt1 number, Pr. [ABSTRACT FROM AUTHOR]

The laminar mixed convection boundary layer flow of an incompressible micropolar fluid past a horizontal circular cylinder with a constant surface heat flux qw has been studied in both cases of a heated and cooled cylinder. The transformed conservation equations of the non-similar boundary layers are solved numerically using a very efficient finite-difference method known as the Keller-box scheme. The solutions for the flow and heat transfer characteristics are evaluated numerically for different parameters, such as the mixed convection parameter λ, the material parameter K (vortex viscosity parameter) and the Prandtl number Pr. It is found that heating the cylinder delays separation of the boundary layer and can, if the cylinder is warm enough, suppress it completely. Cooling the cylinder, on the other side, brings the separation point nearer to the lower stagnation point and for sufficiently cold cylinder there will not be a boundary layer on the cylinder. [ABSTRACT FROM AUTHOR]

Abstract: The problem of steady laminar mixed convection boundary layer flow of an incompressible viscous fluid along vertical thin needles for both assisting and opposing flow cases is considered in this paper. The transformed boundary layer equations are solved numerically using an implicit finite-difference scheme known as the Keller-box method. Numerical computations are carried out for various values of the dimensionless parameters of the problem, namely the mixed convection parameter λ and the parameter a representing the needle size, with Prandtl number, Pr =0.7. It has been found that the flow and heat transfer characteristics are significantly influenced by these parameters. [Copyright &y& Elsevier]