Two types of fluid motions such as parallel forced flow in the channel and the recirculation flow inside the cavity were found by the authors. Raji and Hasnaoui [7] studied the opposing **mixed** **convection** in a rectangular cavity heated from the side with a constant heat flux using the finite difference method. The problem of steady laminar forced **convection** inside a square cavity with inlet and outlet ports was presented by Saeidi and Khodadadi [8]. Sharif et al. [9] studied the assisting flow of **MHD** **mixed** **convection** inside a ventilated enclosure. Al-Rashed and Badruddin [10].investigated heat transfer in a porous cavity. Billah et al. [11] investigated a simulation of **MHD** combined free and forced **convection** heat transfer enhancement in a double lid-driven blocked enclosure. Kumar and Dalal [12] studied free **convection** around a heated square cylinder kept in an enclosure for the Reynolds number varied as 10 3 to 10 6 and reported that the uniform wall temperature heating is quantitatively different from the uniform wall heat flux heating. The **mixed** **convection** in a ventilated square cavity with a heat conducting horizontal solid circular cylinder was analyzed by Rahman et al. [13]. Chamkha [14] focused an unsteady laminar combined **convection** flow and heat transfer of an electrically conducting and heat generating or absorbing fluid considering the effect of a magnetic field in a vertical lid-driven cavity. Oztop et al. [15] investigated the fluid flow characteristics of combined **convection** in a lid-driven cavity containing a circular body where the cavity was heated at the left wall, cooled at the right wall and other two walls of the enclosure were insulated. A Hydromagnetic **mixed** **convection** problem in a double-lid driven cavity with a heat- generating block have been computed to observe the effect of Prandtl number by Rahman et al. [16]. Gau et al. [17] performed an experimental study on combined free and forced **convection** in a horizontal rectangular channel heated from a side. Mamun et al. [18] numerically studied **mixed** **convection** heat transfer in a bottom heated trapezoidal enclosure along with a moving upper wall and in this work the consequence of Richardson number, aspect ratio as well as rotational angle of the optimum trapezoidal cavity are presented by the authors. Hasanuzzaman et al. [19] analyzed the effects of variables on natural convective heat transfer through V-corrugated vertical plates. Mehmet and Elif [20] investigated natural convective flow in a rectangular enclosure having electrically conducting fluid with a magnetic field to find the inclination effect of the ______________________

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This thesis is focused on the unsteady **MHD** **mixed** **convection** flow of nanofluids inside a vertical channel. Nanoparticles are suspended inside regular fluids where water and ethylene glycol are chosen for this purpose. Nanofluids or ferrofluids are introduced by using several models and equations. Three different driving forces have been considered, which are responsible for inducing the motion into the fluid. These are buoyancy force, external pressure gradient and boundary wall. The first problem emphasized on **MHD** **mixed** **convection** flow of ferrofluids passing through a vertical channel with stationary walls. The second problem focuses on the influence of radiation and heat generation effects on **MHD** **mixed** **convection** flow of nanofluids along a vertical channel. The third problem explores the **MHD** **mixed** **convection** flow of nanofluids in a vertical channel filled with porous medium together with stationary and oscillating boundary conditions. The fourth problem highlights the study of **mixed** **convection** flow of nanofluids in a porous channel filled with permeable walls. Perturbation technique has been used to solve the governing linear partial differential equations. Analytical solutions for velocity and temperature are obtained for all the proposed problems and plotted through various graphs. A computational software namely Mathcad has been used for plotting graphs and for computing tabulated results. Further, the limiting cases of the present results give the published results in literature.

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Abstract: **MHD** **mixed** **convection** flow in a lid driven enclosure with a sinusoidal wavy wall and heated circular body located at the center of the enclosure is studied numerically using finite element analysis. The upper wall is moving with a uniform velocity by unity, and other walls are in no slip condition. The enclosure vertical walls are insulated while the heated circular body inside the enclosure is maintained at a uniform temperature higher than the upper wall and lower wavy surface. Calculations are carried out through solving governing equations for different parameters by using finite element method. The investigation is conducted for various values of Richardson number Ri and Prandtl number Pr. The heat transfer characteristics and flow pattern inside the enclosure are presented in the form of streamlines and isothermal contours. Heat transfer rate in terms of the average Nusselt number and average fluid temperature inside the enclosure are presented for different parameters. The results indicate that the average Nusselt number at the heated surface and average temperature of the fluid inside the enclosure are strongly dependent on the configuration of the system under different geometrical and physical conditions.

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This study numerically investigates the steady and unsteady **MHD** **mixed** **convection** flow of Casson and Casson nanofluid over a nonlinearly stretching sheet and moving wedge in the presence of thermal radiation, chemical reaction, slip and convective boundary conditions. The highly nonlinear coupled partial differential equations are transformed into nonlinear ordinary differential equation with the help of suitable transformations and then solved using Keller-box method. The obtained results are displayed graphically and discussed in detail. In order to validate the present method, numerical results for skin friction and Nusselt number are compared with the previously published results. Following are the objectives of the present study:

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Many investigators have studied the effect of magnetic field on the fluid flow and heat transfer in conductive fluids problems. Ozoe [1, 2] studied the ability of magnetic field to manipulate heat and fluid flow in metallurgical industries. Reynolds and prandtl numbers effects on **MHD** **mixed** **convection** in a lid-driven cavity along with joule heating and a centered heat conducting circular block investigated by Rahman et al. [3]. Oztop et al. [4] studied buoyancy induced flow in magneto hydrodynamic-fluid filled enclosure with sinusoidally varying temperature boundary conditions. They found that increasing Hartmann number decreases heat transfer rate within the enclosure. Salam et al. [5] numerically studied steady laminar natural **convection** of air flow in a sinusoidal corrugated enclosure containing a tilted hot thin plate located symmetrically at its centreline. Rahman et al. [6] studied the effects of magnetic field on **mixed** **convection** flow in a horizontal channel with a heated square cavity at the bottom wall numerically. Their investigation was carried out using Galerkin weighted residual finite element technique. It is found that magnetic field is mostly effective inside the cavity. Moreover, a numerical study on conduction-combined forced and natural **convection** heat transfer flow of air in a differentially heated lid-driven parallelogram-shaped enclosure divided by a solid partition has been conducted extensively by Chamkha et al. [7]. Rahman et al. [8, 9] also numerically studied **mixed** **convection** flow inside a ventilated cavity with different heat conducting solid bodies. They found that the characteristics of the heat transfer and fluid flow are influenced by the interaction

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The magnetohydrodynamic (**MHD**) **mixed** **convection** heat transfer in a non-Newtonian Powell-Erying fluid flow due to an exponentially shrinking porous sheet is investigated. Both assisting and opposing flows are considered. After use of the suitable transformations, the governing equations become non-similar ODEs. Numerical computations of resulting equations are obtained by very efficient shooting method for several values of involved parameters. The results exhibit that dual non-similar solutions can be found only when some amount of fluid mass is sucked from the flow field through the porous sheet. Many important results on the effect of external magnetic field on **mixed** convective flow of Powell-Erying fluid have been explored. Where, it is found that dual non-similar solutions exist for the opposing flow, while for the assisting flow, solution is unique and for steady Powell-Erying fluid flow stronger mass suction is required compared to the Newtonian fluid flow.

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Diﬀerent from the ﬂow induced by a stretching horizontal sheet, the eﬀect of **mixed** con- vection due to a buoyancy force could not be neglected for the vertical sheet. There has been increasing interest in studying the problem of **MHD** with **mixed** **convection** bound- ary layer ﬂow and heat transfer characteristics over a stretching vertical surface [–]. Very recently, Ali et al. [] studied the **MHD** **mixed** **convection** stagnation-point ﬂow and heat transfer of an incompressible viscous ﬂuid over a vertical stretching sheet, and the **MHD** boundary layer ﬂow over a vertical stretching/shrinking sheet in a nano-ﬂuid was investigated by Makinde et al. [] and Das et al. [].

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micro-electronic devices, the flow is influenced by magnetic fields. Hence, study on natural or **mixed** **convection** fluid flow and heat transfer in the enclosures in the presence of magnetic forces is important in such cases. There are a large number of studies on both **MHD** **mixed** and natural **convection** heat transfer in conventional fluid-filled enclosures. Groson et al. [13] have numerically studied the steady **MHD** free **convection** in a rectangular cavity filled with a fluid-saturated porous medium with internal heat generation. Rahman et al. [14] performed a numerical investigation into conjugate effect of joule heating and magnetic force, acting normal to the left vertical wall of an obstructed lid-driven cavity saturated with an electrically conducting fluid. Sivasankaran et al. [15] carried out a numerical study on **mixed** **convection** in a square cavity of sinusoidal boundary temperatures at the sidewalls in the presence of magnetic field. The results of a numerical study on **mixed** **convection** of **MHD** flow with joule effect in two-dimensional lid-driven cavity with corner heater were reported by Oztop et al. [16]. Revnic et al. [17] conducted a numerical investigation into the effects of an inclined magnetic field and heat generation on unsteady free **convection** within a square cavity filled with a fluid-saturated medium. The effects of moving lid direction on **MHD**-**mixed** **convection** in a cavity with linearly heated bottom wall were analyzed by Al- Salem et al. [1]. In spite of a large number of papers about natural and **mixed** **convection** of conventional fluids in cavities with magnetic field effects which is mentioned before, there is a serious lack of information about **MHD** natural or **mixed** **convection** of nanofluids in enclosures. Ghasemi et al. [18] examined the natural **convection** in an enclosure filled with water-Al 2 O 3 nanofluid, influenced by a magnetic

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In the case of a fluid through a porous medium, Thermal and Solutal transport are of a great interest from both theory and application point of view. Heat transfer in the case of homogeneous fluid-saturated porous media has been studied with relation to different applications like dynamics of hot underground springs, terrestrial heat flow through aquifer, hot fluid and ignition front displacements in reservoir engineering, heat exchange between soil and atmosphere, flow of moisture through porous industrial materials, and heat exchanges with fluidized beds. Mass transfer in isothermal conditions has been studied with applications to problems of mixing of fresh and salt water in aquifers, miscible displacements in oil reservoirs, spreading of solutes in fluidized beds and crystal washers, salt leaching in soils, etc. An integral approach to the heat and mass transfer by natural **convection** from vertical plates with variable wall temperature and concentration in porous media saturated with an electrically conducting fluid in the presence of transverse magnetic field has been studied by [1]. The unsteady free convective **MHD** flow and mass transfer of a viscous, incompressible, electrically conducting fluid past an infinite vertical, non-conducting porous plate with variable temperature was analyzed by [2]. Natural **convection** from a permeable sphere embedded in a variable porosity porous medium due to thermal dispersion was analyzed by [3]. Dual solutions in **mixed** **convection** flow near a stagnation point on a vertical porous plate were investigated by [4]. Combined heat and mass transfer by natural **convection** under boundary layer approximations has been studied by [5], [6]. Coupled heat and mass transfer by **mixed** **convection** in Darcian fluid-saturated porous medium has been analyzed by [7]. The problem of thermal dispersion and radiation effects on non-Darcy natural **convection** in a fluid saturated porous medium were studied by [8]. Thermal radiation heat transfer effects on the Rayleigh of gray viscous fluids under the effect of a transverse magnetic field have been investigated by [9]. The steady two-dimensional stagnation-point flow of an

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Consider the unsteady **mixed** **convection** flow of an elec- trical conducting nanofluid over a permeable stretching sheet surface in the presence of external magnetic and electric fields, thermal radiation, heat generation/absorption, chemical reaction, viscous dissipation and Joule heating. The coordi- nate system is selected in such a manner that the x-axis is measured along the stretching sheet and the y-axis is nor- mal to it and the flow is occupied above the surface sheet y > 0 displayed in Figure 1. Two equal and opposite forces

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[3] Chand K., Kumar R., Sharma S., “Hydro magnetic oscillatory flow through a porous medium bounded by two vertical porous plates with heat source and soret effect”, Advances in Applied Science Research, 2012, 3, 2169-2178. [4] Rabin N. Barik., et.al, “chemical reaction effect on **MHD** oscillatory flow through a porous medium bounded by two vertical porous plates with heat source and soret effect”. Journal of applied analysis and computation, 2013, 3, 4, 307-321

Convective flows often driven by temperature and concentration differences has been the objective of extensive research due to their vast applications in nature and engineering problems. The situations are often encountered in nature includes photo-synthetic mechanism, evaporation and vaporization of mist and fog, while the engineering application includes the chemical reaction in a reactor chamber consisting of rectangular ducts, chemical vapor deposition on surfaces and cooling of electronic equipment. Magneto hydrodynamics flows has applications in meteorology, solar physics, cosmic fluid dynamics, astrophysics, geophysics and in the motion of earth’s core. From the technological point of view, **MHD** free **convection** flows have significant applications in the field of stellar and planetary magnetospheres, aeronautics, chemical engineering and electronics. Due to several important applications, such flows has been studied by several authors. Heat and mass transfer on flow past a vertical plate have been studied by several authors; viz. Somess [1], Soundalgekar and Ganesan [2], Khair and Bejan [3] and Lin and Wu [4] in numerous ways to include various physical aspects. In addition to the above note worthy contributions have been made by Shercliff [5] Ferraro and Plumption [6] Cramer and Pai [7] and Elbashbeshy [8].

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In this work, we show a comparative study of two different kinds of **convection** heat transfer enhancement techniques carried out with an aim of better thermal/energy/power management. We explored and presented the recent articles in the field of passive heat transfer enhancement using extra surface to kill the thermal boundary layer as well as the **MHD** **mixed** **convection** in a lid-driven cavity. Hereby, we are confident enough that this review article will help in planning for future design and development of heat exchangers in plethora of applications.

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A theoretical analysis for magnetohydrodynamic (**MHD**) **mixed** **convection** of non- Newtonian tangent hyperbolic nanofluid flow with suspension dust particles along a vertical stretching sheet is carried out. The current model comprises of non-linear partial differential equations expressing conservation of total mass, momentum, and thermal energy for two-phase tangent hyperbolic nanofluid phase and dust particle phase. Primitive similarity formulation is given to mutate the dimensional boundary layer flow field equations into a proper nonlinear ordinary differential system then Runge-Kutta-Fehlberg method (RKF45 method) is applied. Distinct pertinent parameter impact on the fluid or particle velocity, temperature, concentration, and skin friction coefficient is illustrated. Analysis of the obtained computations shows that the flow field is affected appreciably by the existence of suspension dust particles. It is concluded that an increment in the mass concentration of dust particles leads to depreciate the velocity distributions of the nanofluid and dust phases. The numerical computations has been validated with earlier published contributions for a special cases.

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The effects of Hall current and radiation on **MHD** **mixed** **convection** in a rotating vertical channel temperature in the presence of a uniform transverse magnetic field have been investigated. It is found that the Hall parameter m accelerates the primary velocity u 1 as well as the magnitude of the secondary velocity v 1 . An increase in radiation parameter R leads to fall in the primary velocity u 1 as well as the magnitude of the secondary velocity v 1 . Further, the amplitudes and tangent of phases of the shear stresses due to the primary and the secondary flows at the plates are significantly affected by characteristic parameters. The amplitudes and tangent of phases of the rate of heat transfer at the plates increases with an increase in radiation parameter R .

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2. M. Waqas, M. Farooq, M. Ijaz Khan, A. Alsaedi, T. Hayat and T. Yasmeen, Magnetohydrodynamic (**MHD**) **mixed** **convection** flow of micropolar liquid due to nonlinear stretched sheet with convective condition, International Journal of Heat and Mass Transfer 102 (2016) 766-772.

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flow inside a square cavity. Fan et al. [20] investigated **mixed** **convection** heat transfer in a horizontal channel filled with nanofluids. Tiwari and Das [21] and Sheikhza- deh et al. [22] analyzed laminar **mixed** **convection** flow of a nanofluid in two-sided lid-driven enclosures. Further, magnetic field in nanofluids has its numerous applications such as in the polymer industry and metal- lurgy where hydromagnetic techniques are being used. Nadeem and Saleem [23] examined the unsteady flow of a rotating **MHD** nanofluid in a rotating cone in the pres- ence of magnetic field. Al-Salem et al. [24] investigated **MHD** **mixed** **convection** flow in a linearly heated cavity. The effects of variable viscosity and variable thermal conductivity on the **MHD** flow and heat transfer over a non-linear stretching sheet was investigated by Prasad et al. [25]. The problem of Darcy Forchheimer **mixed** **convection** heat and mass transfer in fluid-saturated por- ous media in the presence of thermophoresis was pre- sented by Rami et al. [26]. Effect of radiation and magnetic field on the **mixed** **convection** stagnation-point flow over a vertical stretching sheet in a porous medium bounded by a stretching vertical plate was presented by Hayat et al. [27]. Few other studies on **mixed** **convection** and nanofluids are given in [28–38].

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