Stability analysis of unsteady MHD stagnation point flow and heat transfer over a shrinking sheet in the presence of viscous dissipation

The unsteady MHD stagnation-point flow and heat transfer over a shrinking sheet was carried out. This study also was conducted in the existence of suction and viscous dissipation. In order to convert the governing partial differential equations to an ordinary differential equation, an appropriate similarity transformation was applied in this study. Then, the resulting equations are worked out by Bvp4c solver in Matlab. The impacts of the parameters involved in this study towards skin friction, Nusselt number, velocity and temperature profile are showed graphically and thoroughly discussed. Remarkably, there were dual solutions present in this study which made us continue deeper in performing the stability analysis. As expected, our study proves that the solution is stable only in the first one while not in the second solution.     

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.     

Thermodynamic method to two dimensional stagnation flow and heat transfer

The purpose of this paper is to develop a variational technique, using the variational principle of non-equilibrium thermodynamics for the solution of laminar stagnation point flow and heat transfer in two dimensions. The governing principle of dissipative processes is formulated for the problem and then using the boundary conditions of the system the elementary trial functions of third degree are assumed for the flow and temperature fields in the vicinity of the wall. The viscous and thermal boundary layer thicknesses, the skin friction and the rate of heat transfer from the wall to the fluid are obtained. It is found that these new approximate results are quite close to the exact known ones. These results are better than those of the Kármán-Pohlhausen technique.The authors are thankful to the referee for his valuable suggestions.     

Three-dimensional magnetohydrodynamics axisymmetric stagnation flow and heat transfer due to an axisymmetric shrinking/stretching sheet with viscous dissipation and heat source/sink

The present work is concerned with the effects of viscous dissipation and heat source/sink on a three-dimensional magnetohydrodynamic boundary layer axisymmetric stagnation flow, and the heat transfer of an electrically conducting fluid over a sheet, which shrinks or stretches axisymmetrically in its own plane where the line of the symmetry of the stagnation flow and that of the shrinking （stretching） sheet are, in general, not aligned. The governing equations are transformed into ordinary differential equations by using suitable similarity transformations and then solved numerically by a shooting technique. This investigation explores the conditions of the non-existence, existence and uniqueness of the solutions of the similar equations numerically. It is noted that the range of the velocity ratio parameter, where the similarity solution exists, is increased with the increase of the value of the magnetic parameter. Furthermore, the study reveals that the non-alignment function affects the shrinking sheet more than the stretching sheet. In addition, the numerical results of the velocity profile, temperature profile, skin-friction coefficient, and rate of heat transfer at the sheet are discussed in detail with different parameters.     

The influence of heat transfer in an MHD second grade fluid film over an unsteady stretching sheet

The flow and heat transfer problem of a second grade fluid film over an unsteady stretching sheet is considered. The fluid is incompressible and electrically conducting in the presence of a uniform applied magnetic field. The series solutions of the governing boundary value problems are obtained by employing homotopy analysis method (HAM). The convergence of the developed solutions is discussed explicitly. The dependence of velocity and temperature profiles on various parameters is shown and discussed through graphs. The values of skin-friction coefficient, Nusselt number and free surface temperature are given in tabular form for various emerging parameters.     

Experimental study of heat transfer characteristics in oscillating fluid flow in tube

In the present work, an experimental investigation of heat transfer enhancement parameters of the oscillating flow heat exchanger under different frequencies, tidal displacement, and heat fluxes is carried out. The effect of different parameters on experimental effective thermal conductivity and convective heat transfer coefficient in cooling region is studied using correlations given by different researchers. A correlation for experimental effective thermal conductivity in terms of S²√ω is derived based on experimental data. This correlation is useful for predicting the optimum value of S²√ω before onset of turbulence.     

Effects of variable properties on MHD heat and mass transfer flow near a stagnation point towards a stretching sheet in a porous medium with thermal radiation

The effect of variable viscosity and thermal conductivity on steady magnetohydrodynamic(MHD) heat and mass transfer flow of viscous and incompressible fluid near a stagnation point towards a permeable stretching sheet embedded in a porous medium are presented,taking into account thermal radiation and internal heat genberation/absorbtion.The stretching velocity and the ambient fluid velocity are assumed to vary linearly with the distance from the stagnation point.The Rosseland approximation is used to describe the radiative heat flux in the energy equation.The governing fundamental equations are first transformed into a system of ordinary differential equations using a scaling group of transformations and are solved numerically by using the fourth-order Rung-Kutta method with the shooting technique.A comparison with previously published work has been carried out and the results are found to be in good agreement.The results are analyzed for the effect of different physical parameters,such as the variable viscosity and thermal conductivity,the ratio of free stream velocity to stretching velocity,the magnetic field,the porosity,the radiation and suction/injection on the flow,and the heat and mass transfer characteristics.The results indicate that the inclusion of variable viscosity and thermal conductivity into the fluids of light and medium molecular weight is able to change the boundary-layer behavior for all values of the velocity ratio parameter λ except for λ = 1.In addition,the imposition of fluid suction increases both the rate of heat and mass transfer,whereas fluid injection shows the opposite effect.     

Heat and mass transfer on unsteady three-dimensional convective MHD flow past a porous plate moving in a parallel free stream

The unsteady three-dimensional convective flow of a viscous incompressible, electrically conducting fluid over a vertical, insulated, porous surface moving in a parallel free stream has been investigated, which flows belong to a separate class of problem of boundary layer theory. The effects of heat and mass transfer on this unsteady laminar flow past porous surface with transverse sinusoidal suction and oscillatory wall temperature have been analysed and discussed.     

圆管等壁温振荡流动对流换热理论研究

对等壁温边界条件下圆管内层流振荡流动对流换热问题进行理论推导。在随时间做正弦变化的速度入口条件下,经求解圆管内流体振荡时的能量方程,得到振荡流动圆管内温度场分布的解析解表达式,并利用Matlab对表达式进行直观表示。计算结果表明:等壁温边界条件下速度入口脉冲流动可引起温度随时间波动,波动随无量纲振荡频率和速度振荡幅度增加,换热效果加强,且速度振荡幅度和无量纲振荡频率存在协同作用。     

Numerical study of the onset of chemical reaction and heat source on dissipative MHD stagnation point flow of Casson nanofluid over a nonlinear stretching sheet with velocity slip and convective boundary conditions

The magnetohydrodynamic (MHD) stagnation point flow of Casson nanofluid over a nonlinear stretching sheet in the presence of velocity slip and convective boundary condition is examined. In this analysis, various effects such as velocity ratio, viscous dissipation, heat generation/absorption and chemical reaction are accentuated. Possessions of Brownian motion and thermophoresis are also depicted in this study. A uniform magnetic field as well as suction is taken into account. Suitable similarity transformations are availed to convert the governing nonlinear partial differential equations to a system of nonlinear ordinary differential equations and then series solutions are secured using a homotopy analysis method (HAM). Notable accuracy of the present results has been obtained with the earlier results. Impact of distinct parameters on velocity, temperature, concentration, skin friction coefficient,Nusselt number and Sherwood number is canvassed through graphs and tabular forms.     

Thermal radiation and slip effects on MHD stagnation point flow of nanofluid over a stretching sheet

Present model is devoted for the stagnation point flow of nanofluid with magneto-hydrodynamics (MHD) and thermal radiation effects passed over a stretching sheet. Moreover, we have considered the combined effects of velocity and thermal slip. Condition of zero normal flux of nanoparticles at the wall for the stretched flow phenomena is yet to be explored in the literature. Convinced partial differential equations of the model are transformed into the system of coupled nonlinear differential equations and then solved numerically. Graphical results are plotted for velocity, temperature and nanoparticle concentration for various values of emerging parameters. Variation of stream lines, skin friction coefficient, local Nusselt and Sherwood number are displayed along with the effective parameters. Final conclusion has been drawn on the basis of both numerical and graphs results.     

Entropy analysis on unsteady MHD biviscosity nanofluid flow with convective heat transfer in a permeable radiative stretchable rotating disk

We examine the entropy analysis in three-dimensional hydromagnetic flow and convective heat transport of a biviscosity nanofluid over a rotating porous disk with a time-dependent stretching rate in the direction of the radius of the circular disk. We also examine the influence of thermal radiation and viscous dissipation due to nanoparticles and applied magnetic field. We invoked suitable self-similar transformations to covert the modeled coupled nonlinear PDEs into a set of nonlinear ODEs. The transformed system of equations is then worked out numerically by a well-known shooting technique and the fourth-order Runge–Kutta–Fehlberge method. The rotating phenomenon yields an additional parameter known as a rotation parameter, which controls the disk’s rotation. The study shows that the fluid motion is accelerated along the radial and cross-radial directions with an increase in the rotation of the disk. The skin-friction and the heat transfer rate at the disk strongly depend on the rotation of the disk, permeability of the porous medium, thermal radiation, and nanoparticle size. The Bejan number quantifies the entropy production of the system. It has a considerable impact on the magnetic field, rotation of the disk, thermal radiation, and Biot number. The efficient performance of the system is possible by a suitable choice of the physical parameters discussed in this article.     

MHD mixed convection flow and heat transfer in a porous medium

The effects of a steady two-dimensional laminar MHD mixed convection flow and heat transfer against a heated vertical semi-infinite permeable surface in a porous medium are discussed. The coupled nonlinear partial differential equations describing the conservation of mass, momentum, and energy are solved by a perturbation technique. The results are presented to illustrate the influence of Hartmann number (M), Prandtl number (Pr), permeability parameter (K _p ), suction/blowing parameter (f _w ), heat generation/absorption coefficient (?), and mixed convection or buoyancy parameter (γ). The effects of different parameters on the velocity and temperature as well as the skin friction and wall heat transfer are discussed with the help of figures.     

Brownian motion and thermophoresis effects on unsteady stagnation point flow of Eyring-Powell nanofluid: a Galerkin approach

This article concerns the analysis of an unsteady stagnation point flow of Eyring-Powell nanofluid over a stretching sheet. The influence of thermophoresis and Brownian motion is also considered in transport equations. The nonlinear ODE set is obtained from the governing nonlinear equations via suitable transformations. The numerical experiments are performed using the Galerkin scheme. A tabular form comparison analysis of outcomes attained via the Galerkin approach and numerical scheme (RK-4) is available to show the credibility of the Galerkin method. The numerical exploration is carried out for various governing parameters, namely, Brownian motion, steadiness, thermophoresis, stretching ratio, velocity slip, concentration slip, thermal slip, and fluid parameters, and Hartmann, Prandtl and Schmidt numbers. The velocity of fluid enhances with an increase in fluid and magnetic parameters for the case of opposing, but the behavior is reversed for assisting cases. The Brownian motion and thermophoresis parameters cause an increase in temperature for both cases (assisting and opposing). The Brownian motion parameter provides a drop-in concentration while an increase is noticed for the thermophoresis parameter. All the outcomes and the behavior of emerging parameters are illustrated graphically. The comparison analysis and graphical plots endorse the appropriateness of the Galerkin method. It is concluded that said method could be extended to other problems of a complex nature.     

Numerical study of flow and heat transfer from a torus placed in a uniform flow

Forced convection heat transfer characteristics of a torus (maintained at a constant temperature) immersed in a streaming fluid normal to the plane of the torus are studied numerically. The governing equations, namely, continuity, momentum and thermal energy in toroidal coordinate system, are solved using a finite difference method over ranges of parameters (aspect ratio of torus, 1.4 ≤ Ar ≤ 20; Reynolds number, 20 ≤ Re ≤ 40; Prandtl number, 0.7 ≤ Pr ≤ 10). Over the ranges of parameters considered herein, the nature of flow is assumed to be steady. In particular, numerical results elucidating the influence of Reynolds number, Prandtl number and aspect ratio on the isotherm patterns, local and average Nusselt numbers for the constant temperature (on the surface of the torus) boundary condition. As expected, at large aspect ratio the flow pattern and heat transfer are similar to the case of flow and heat transfer over a single circular cylinder.     

A numerical study of MHD generalized Couette flow and heat transfer with variable viscosity and electrical conductivity

The steady flow and heat transfer of an electrically conducting fluid with variable viscosity and electrical conductivity between two parallel plates in the presence of a transverse magnetic field is investigated. It is assumed that the flow is driven by combined action of axial pressure gradient and uniform motion of the upper plate. The governing nonlinear equations of momentum and energy transport are solved numerically using a shooting iteration technique together with a sixth-order Runge-Kutta integration algorithm. Solutions are presented in graphical form and given in terms of fluid velocity, fluid temperature, skin friction and heat transfer rate for various parametric values. Our results reveal that the combined effect of magnetic field, viscosity, exponents of variable properties, various fluid and heat transfer dimensionless quantities and the electrical conductivity variation, have significant impact on the hydromagnetic and electrical properties of the fluid.     

MHD flow and heat transfer over a radially stretching/shrinking disk

A steady magnetohydrodynamic (MHD) flow past a radially stretching or shrinking disk is investigated. The governing partial differential equations are transformed into a set of ordinary (similarity) differential equations by a similarity transformation. These equations along with the corresponding boundary conditions are solved numerically using the boundary value problem solver (bvp4c) in Matlab. The effects of magnetic field and suction on the shear stress and the heat transfer are analyzed and discussed. It is found that both parameters affect more in the shrinking region. The increase in the magnetic parameter results in the increase of the skin friction coefficient but decrease in the local Nusselt number.The skin friction coefficient and the local Nusselt number increase as suction increases.     

Numerical study of heat transfer enhancement of counter nanofluids flow in rectangular microchannel heat exchanger

This paper reports a numerical analysis of the performance of a counter-flow rectangular shaped microchannel heat exchanger (MCHE) using nanofluids as the working fluids. Finite volume method was used to solve the three-dimensional steady, laminar developing flow and conjugate heat transfer in aluminum MCHE. The nanofluids used were Ag, Al₂O₃, CuO, SiO₂, and TiO₂ and the performance was compared with water. The thermal, flow fields and performance of the MCHE were analyzed using different nanofluids, different Reynolds numbers and different nanoparticle concentrations. Temperature profile, heat transfer coefficient, pressure profile, and wall shear stress were obtained from the simulations and the performance was discussed in terms of heat transfer rate, pumping power, effectiveness, and performance index. Results indicated enhanced performance with the usage of nanofluids, and slight penalty in pressure drop. The increase in Reynolds number caused an increase in the heat transfer rate and a decrease in the overall bulk temperature of the cold fluid. The increase in nanoparticle concentration also yielded better performance at the expense of increased pressure drop.     

Series solutions of stagnation slip flow and heat transfer by the homotopy analysis method

An analytical approximation for the similarity solutions of the two- and three-dimensional stagnation slip flow and heat transfer is obtained by using the homotopy analysis method. This method is a series expansion method, but it is different from the perturbation technique, because it is independent of small physical parameters at all. Instead, it is based on a continuous mapping in topology so that it is applicable for not only weakly but also strongly nonlinear flow phenomena. Convergent [m,m] homotopy Padé approximants are obtained and compared with the numerical results and the asymptotic approximations. It is found that the homotopy Padé approximants agree well with the numerical results. The effects of the slip length ℓ and the thermal slip constant β on the heat transfer characteristics are investigated and discussed. Supported by the National Natural Science Foundation of China (Grant No. 10872129)