Unsteady MHD flow and heat transfer near stagnation point over a stretching/shrinking sheet in porous medium filled with a nanofluid

In this article, the unsteady magnetohydrodynamic （MHD） stagnation point flow and heat transfer of a nanofluid over a stretching/shrinking sheet is investigated numerically. The similarity solution is used to reduce the governing system of partial differential equations to a set of nonlinear ordinary differential equations which are then solved numerically using the fourth-order Runge-Kutta method with shooting technique. The ambient fluid velocity, stretching/shrinking velocity of sheet, and the wall temperature are assumed to vary linearly with the distance from the stagnation point. To investigate the influence of various pertinent parameters, graphical results for the local Nusselt number, the skin friction coefficient, velocity profile, and temperature profile are presented for different values of the governing parameters for three types of nanoparticles, namely copper, alumina, and titania in the water-based fluid. It is found that the dual solution exists for the decelerating flow. Numerical results show that the extent of the dual solution domain increases with the increases of velocity ratio, magnetic parameter, and permeability parameter whereas it remains constant as the value of solid volume fraction of nanoparticles changes. Also, it is found that permeability parameter has a greater effect on the flow and heat transfer of a nanofluid than the magnetic parameter.

Boundary Layer Flow over a Continuously Moving Thin Needle in a Parallel Free Stream

We investigate the boundary-layer flow on a moving isothermal thin needle parallel to a moving stream. The governing equations are solved numerically by a finite-difference method. Dual solutions are found to exist when the needle and the free stream move in the opposite directions.

Boundary Layer on a Moving Wall with Suction and Injection

We investigate the boundary-layer flow on a moving permeable plate parallel to a moving stream. The governing equations are solved numerically by a finite-difference method. Dual solutions are found to exist when the plate and the free stream move in the opposite directions.

Boundary layer flow and heat transfer of a micropolar fluid near the stagnation point on a stretching vertical surface with prescribed skin friction

The steady laminar mixed convection boundary layer flow and heat transfer of a micropolar fluid near the stagnation point on a stretched vertical surface with prescribed skin friction were considered.The governing partial differential equations were transformed into a system of ordinary differential equations,which were then solved numerically using the shooting method.Results for the stretching velocity,the local Nusselt number,the temperature,and the velocity profiles are presented for various values of the mixed convection parameter λ and material parameter K when the Prandtl number is equal to 1.Both assisting（heated plate） and opposing（cooled plate） flow regions are considered.It is found that dual solutions exist for both assisting and opposing flows.

Similarity solutions for the mixed convection flow over a vertical plate with thermal radiation

The steady laminar boundary layer flow adjacent to a vertical plate with prescribed surface temperature immersed in an incompressible viscous fluid,where the effect of thermal radiation was taken into consideration,was investigated.The governing partial differential equations were transformed into a system of ordinary differential equations using similarity transformation,before being solved numerically by the shooting method.Both assisting and opposing buoyant flows were considered.It is found that dual solutions exist for both cases. Moreover,numerical results show that the heat transfer rate at the surface decreases in the presence of the radiation effect.

Stagnation Point Flow Over a Permeable Stretching/Shrinking Sheet with Chemical Reaction and Heat Source/Sink

The present study considers the magnetohydrodynamic(MHD)stagnation point flow with chemical reaction effect over a permeable stretching/shrinking sheet.The partial differential equations are reduced to a set of ordinary differential equations using a similarity transformation.The transformed equations are then solved numerically by employing the bvp4c function available in the MATLAB software.The numerical results illustrate the effects of several parameters on the skin friction coefficient,local Nusselt number and the local Sherwood number.Dual solutions are obtained for a certain range of parameters.The temporal stability analysis is carried out to determine which one of these solutions is stable and thus physically reliable in a long run.

MHD Flow Towards a Permeable Surface with Prescribed Wall Heat Flux

The steady magnetohydrodynamic （MHD） mixed convection flow towards a vertical permeable surface with prescribed heat flux is investigated. The governing partial differential equations are transformed into a system of ordinary differential equations, which is then solved numerically by a finite-difference method. The features of the flow and heat transfer characteristics for different values of the governing parameters are analysed and discussed. Both assisting and opposing flows are considered. It is found that dual solutions exist for the assisting flow, besides the solutions usually reported in the literature for the opposing flow.

Heat transfer for boundary layers with cross flow

An analysis is presented to study the dual nature of solutions for the forced convective boundary layer flow and heat transfer in a cross flow with viscous dissipation terms in the energy equation. The governing equations are transformed into a set of three self-similar ordinary differential equations by similarity transformations. These equations are solved numerically using the very efficient shooting method. This study reveals that the dual solutions of the transformed similarity equations for velocity and temperature distributions exist for certain values of the moving parameter, Prandtl number, and Eckert numbers. The reverse heat flux is observed for larger Eckert numbers; that is, heat absorption at the wall occurs.

Magnetohydrodynamic flow over a moving plate in a parallel stream with an induced magnetic field

A viscous boundary layer flow of an electrically-conducting fluid over a moving flat plate in a parallel stream with a constant magnetic field applied outside the boundary layer parallel to the plate was investigated. The goveming system of partial differential equations was transformed to ordinary differential equations using a similarity transformation. The similarity equations were then solved numerically using a finite-difference scheme known as the Keller-box method. Numerical results of the skin friction coefficient, velocity profiles, and the induced magnetic field profiles were obtained for some values of the moving parameter, magnetic parameter, and reciprocal magnetic Prandtl number. The results indicate that dual solutions exist when the plate and the fluid move in the opposite directions up to a critical value of the moving parameter, whose value depends on the value of the magnetic parameter.

Hybrid Nanofluid Flow with Homogeneous-Heterogeneous Reactions

This study examines the stagnation point flow over a stretching/shrinking sheet in a hybrid nanofluid with homogeneous-heterogeneous reactions.The hybrid nanofluid consists of copper(Cu)and alumina(Al2O3)nanoparticles which are added into water to form Cu-Al2O3/water hybrid nanofluid.The similarity equations are obtained using a similarity transformation.Then,the function bvp4c in MATLAB is utilised to obtain the numerical results.The dual solutions are found for limited values of the stretching/shrinking parameter.Also,the turning point arises in the shrinking region(λ<0).Besides,the presence of hybrid nanoparticles enhances the heat transfer rate,skin friction coefficient,and the concentration gradient.In addition,the concentration gradient is intensified with the heterogeneous reaction but the effect is opposite for the homogeneous reaction.Furthermore,the velocity and the concentration increase,whereas the temperature decreases for higher compositions of hybrid nanoparticles.Moreover,the concentration decreases for larger values of homogeneous and heterogeneous reactions.It is consistent with the fact that higher reaction rate cause a reduction in the rate of diffusion.However,the velocity and the temperature are not affected by these parameters.From these observations,it can be concluded that the effect of the homogeneous and heterogeneous reactions is dominant on the concentration profiles.Two solutions are obtained for a single value of parameter.The temporal stability analysis shows that only one of these solutions is stable and thus physically reliable over time.

Boundary layer flow over a moving surface in a nanofluid with suction or injection

An analysis is performed to study the heat transfer characteristics of steady two-dimensional boundary layer flow past a moving permeable flat plate in a nanofluid. The effects of uniform suction and injection on the flow field and heat transfer characteristics are numerically studied by using an implicit finite difference method. It is found that dual solutions exist when the plate and the free stream move in the opposite directions. The results indicate that suction delays the boundary layer separation, while injection accelerates it.

Tiwari-Das nanofluid model for magnetohydrodynamics(MHD)natural-convective flow of a nanofluid adjacent to a spinning down-pointing vertical cone

In this article,the natural-convective flow of an electrically conducting nanofluid adjacent to a spinning down-pointing vertical cone in the presence of transverse magnetic field is studied.The mathematical model has been formulated based on Tiwari-Das nanofluid model.Three different types of water-based nanofluid with copper,aluminum oxide(alumina)and titanium dioxide(titania)as nanoparticles are considered in this investigation.Two cases of heat transfer analysis are discussed.These are:(i)the spinning cone with prescribed surface temperature and(ii)the spinning cone with prescribed surface heat flux.Using appropriate transformations,the system of partial differential equations is transformed into an ordinary differential system of three equations,which is solved numerically using the fourth-order Runge-Kutta method with shooting technique.The current solution demonstrates very good agreement with those of the previously published studies in the especial cases.The effects of the three key thermophysical parameters governing the flow;the nanoparticle volume fraction,the magnetic parameter and the spin parameter on dimensionless velocity and temperature distributions,skin friction coefficient,Nusselt number and entropy generation number are presented graphically and discussed in details.Our results demonstrate that,the enhancement of heat transfer is a function of particle concentration,small fraction of metallic particles leading to significant changes in all three quantities of skin friction coefficient,local Nusselt number and entropy generation number.The results illustrate that selecting alumina and copper as the nanoparticle leads to the minimum and maximum amounts of skin friction coefficient value,and also copper and titania nanoparticles have the largest and lowest local Nusselt number.Moreover,it is observed that the magnetic parameter has a decreasing effect on both skin friction coefficient and local Nusselt number and an increasing effect on entropy generation number.In addition,our computation shows that all three quantities of skin friction coefficient,local Nusselt number and entropy generation number are the increasing functions of spin parameter.Finally,this simulation represents the feasibility of using magnetic rotating body drives in novel nuclear space propulsion engines and this model has important applications in heat transfer enhancement in renewable energy systems and industrial thermal management.

通过具有速度滑移和辐射效应的可渗透拉伸/收缩Riga板的驻点流动和热传导（英文）

目的:1.通过分析Riga板的抽吸效应来控制流体运动和减少摩擦力和压力阻力;2.利用磁场的速度滑移效应来控制流体的流速;3.基于辐射原理控制热传导并减小阻力。创新点:1.本研究可应用于核电厂、飞机、潜艇以及卫星等设施中推进装置的设计;2.本研究可用于防止边界层分离以减少湍流的产生。方法:1.构建基于偏微分方程的数理模型;2.利用相似变换法将偏微分方程简化为常微分方程;3.利用Matlab内置求解器bvp4c对常微分方程组进行数值求解;4.基于求解结果讨论稳定性。结论:1.对于拉伸/收缩两种情形的Riga板问题都存在对偶解;2.数值求解结果显示表面摩擦系数和表面传热率均会随着吸力的增大而增大,而随拉伸/收缩参数?的增大而减小;3.上支解的努塞尔数增大而下支解减小;4.辐射会提高边界层内的温度,而增强滑移效应则会提高流速同时降低边界层温度;5.只有上支解是长期稳定的。

Natural convective heat transfer and nanofluid flow in a cavity with top wavywall and corner heater

A numerical analysis of natural convection of nanofluid in a wavy-walled enclosure with an isothermal comer heater has been carried out. The cavity is heated from the left bottom comer and cooled from the top wavy wall while the rest walls are adiaba- tic. Mathematical model has been formulated using the single-phase nanofluid approach. Main efforts have been focused on the effects of the dimensionless time, Rayleigh number, undulation number, nanoparticle volume fraction and length of comer heaters on the fluid flow and heat transfer inside the cavity. Numerical results have been presented in the form of streamlines, isotherms, velocity and temperature profiles, local and average Nusselt numbers. It has been found that nanoparticle volume fraction essentially affects both fluid flow and heat transfer while undulation number changes significantly only the heat transfer rate.