HEAT AND MASS TRANSFER ON MHD NANOFLUID FLOW PAST A VERTICAL POROUS PLATE IN A ROTATING SYSTEM

In this paper, we study the chemical reaction and heat source effects on unsteady MHD free convection heat and mass transfer of a nanofluid flow past a semi-infinite flat plate in a rotating system. The plate is assumed to oscillate in time with steady frequency so that the solutions of the boundary layer are the similar oscillatory type. The innovation of the present work is closed-form analytic solutions are obtained for the momentum, energy and concentration equations. The influence of various parameters entering into the problem in the nanofluid velocity, temperature and concentration distributions, as well as the skin friction coefficient, Nusselt number and Sherwood number are discussed by considering three different types of nanoparticles with the help of graphs and tables. This model finds applications in studying magnetic resonance imaging, data storage, biomedicine and thermal enhancement of energy systems.


INTRODUCTION
The theory of nanofluid is first introduced by Choi (1995) and has been a field of active research area for about two decades.This fluid is a suspension of a nanometer size solid particles and fibres in a convectional base fluid.Commonly used base fluids for this purpose are water, toluene, oil and ethylene glycol mixture etc.The choice of base fluid-particle combination depends on the application for which the nanofluid is intended.The solid particles used for a nanofluid are metallic solids viz.copper, aluminum, silver and gold; nonmetallic solids viz.Silicon (SiO), alumina (Al2O3) etc. and metallic liquid viz.Sodium.In recent years, the concept of a nanofluid has been proposed as a route for enhancing the performance of the heat transfer rates in the liquids.Materials, with sizes of nanometers possess unique physical and chemical properties.They can flow smoothly through microchannels without clogging because they are sufficiently small to behave similar to liquid molecules Khanafer et al. (2003).This fact has attracted much research into the investigation of the heat transfer characteristics in nanofluids.It has been found that the presence of nanoparticles within the fluid can appreciably increase the effective thermal conductivity of the fluid and, as a consequence, enhance the heat transfer characteristics.An excellent collection of articles on this topic can be found in the book Das et al. (2007) and in the review papers (Xu et al., 2013;Trisaksri and Wongwises, 2009;Kakac and Pramuanjaroenkij, 2009;Soleimani et al., 2012;Fakour, 2014;Nield and Kuznetsov, 2014;Das, 2014;Pourmehran, 2015;Ramesh, 2016).
In the recent years, a great progress in a new generation of MHD heat and mass transfer flow of a nanofluid which provide very desirable features in materials processing, energy applications and also medical engineering.Sheikholeslami et al. (2013Sheikholeslami et al. ( , 2014Sheikholeslami et al. ( , 2015) ) have presented the simulation of MHD CuO and Al2O3-water nanofluid flow and convective heat transfer considering Lorentz forces.Turkyilmazoglu (2014) has studied exact analytical solutions for heat and mass transfer of MHD slip flow in nanofluids.Das and Rana (2015) have analyzed the natural convective magneto-nanofluid flow and radiative heat transfer past a moving vertical plate.Satya Narayana et al. (2015) have studied the influence of thermal radiation and heat source on MHD nanofluid past a vertical plate in a rotating system.Ram Reddy et al. (2013) have analyzed the Soret effects on mixed convection flow in a nanofluid by considering convective boundary condition.Very recently, Das et al. (2016) have examined the transient natural convection in a vertical channel filled with nanofluids in the presence of thermal radiation.
The study of heat and mass transfer with chemical reaction in the presence nanofluids is of immense realistic significance to engineers and scientists because of its almost universal incidence in many branches of science and engineering.This phenomenon plays an important role in chemical industry, power and cooling industry for drying, chemical vapor deposition on surfaces, cooling of nuclear reactors and petroleum industries.Venkateswarlu and Satya Narayana (2015) have analysed the chemical reaction and radiation absorption effects on the flow and heat transfer of a nanofluid in a rotating system.Effect of chemical reaction on MHD boundary layer flow and melting heat transfer of Williamson nanofluid in porous medium by Krishnamurthy et al. (2016).Mabood et al. (2016) have studied the MHD stagnation point flow heat and mass transfer of nanofluids in porous medium with radiation, viscous dissipation and chemical reaction.Recently, Mahanthesh et al. (2016) have investigated the heat and mass transfer effects on the mixed convective flow of chemically reacting nanofluid past a moving/stationary vertical plate.Recently, the effect of chemical reaction with heat radiation in presence of nanofluid over a porous vertical stretching surface is investigated by Rosmila Abdul-Kahar et al. (2011) and it is observed that the thermophoresis particle deposition and Brownian diffusion motion have substantial effects on the flow field.Kandasamy et al. (2011) have analyzed the impact of thermophoresis particle deposition and Brownian diffusion motion on nanofluid in the presence of magnetic field and it is predicted that the magnetic strength plays a significant role on the nanoparticles

Frontiers in Heat and Mass Transfer
Available at www.ThermalFluidsCentral.org in the presence of base fluid.Recently, many researchers Venkateswarlu Satya Narayana (2015); Satya Narayana and Harish Babu (2016); Macha Madhu and Naikoti Kishan (2015); Gireesha et al. (2016) have studied the flow and heat transfer characteristic of Newtonian/non-Newtonian nanofluids over varies geometries.
Motivated by the above reference work and the numerous possible industrial applications of the problem, it is of paramount interest to investigate the effects of chemical reaction and heat source on MHD nanofluid flow in a rotating system with three different types of nanoparticles.It is worth mentioning that the nanofluid model proposed by Buongiorno (2006) is used by many recent studies (Rana and Bharava, 2012;Alsaedi and Hayat, 2012;Hajipour and Molaei Dehkordi, 2012).However, we are following the nanofluid model proposed by Tiwari and Dass (2007), which is being used by many current researches Hamad and Ferdows (2012); Norifiah Bachok (2012); Hamad and Pop (2011) on various flow fields.The layout of the paper as follows.The problem formulation and its solution is presented in Section 2. In Section 3, we present results and discussion.Finally, Section 4 contains the main conclusions.

MATHEMATICAL FORMULATION OF THE PROBLEM
Let us consider unsteady three dimensional flow of a nanofluid past a semi-infinite vertical permeable plate in a rotating system with chemical reaction, thermal diffusion and the thermal radiation in the presence of a uniform transverse magnetic field.It is assumed that there is no applied voltage which implies the absence of an electric field.The flow is assumed to be in the x -direction which is taken along the plate in the upward direction, the Y -axis is normal to it and the z axis along the width of the plate as shown in Fig. 1.Also it is assumed the whole system is rotate with a constant vector Ω about z -axis.The radiation heat flux in x - direction is considered negligible in comparison that the zdirection.Due to semi-infinite plate surface assumption the flow variables are functions of z and time t only.

METHOD OF SOLUTIONS
To find the analytical solutions of the above system of partial differential equations ( 13), ( 14) and ( 18) under the boundary conditions (19) in the neighborhood of the plate, we assume that (see Ganepathy Ref. ( 39)) Invoking the above equations ( 20)-( 22) into the equations ( 13)-( 16) and equating the harmonic and non-harmonic terms and neglecting the higher order terms of 2 ( ) O ε , we obtain the following set of equations: Zeroth order equations are: First order equations are: Where primes denote differentiation with respect to z The corresponding boundary conditions can be written as: Solving the above equations with the boundary conditions ( 29) we obtain the expression for velocity, temperature and concentration as: where The physical quantities of engineering interest are skin-friction coefficient, Nusselt number and Sherwood number are, respectively, defined as: The local skin friction coefficient C f is given by The local Nusselt number Nu is given by The local Sherwood number Sh x is given by 1 2 (Re ) (0) ( ) Where , q and q w m τ are the wall shear stresses or skin friction, the wall heat flux and the wall mass flux from the plate respectively

RESULT AND DISCUSSION
In order to get physical insight into the problem, we have carried out numerical calculations for non-dimensional velocity, temperature and species concentration, skin-friction, and Nusselt number by assigning some specific values to the parameters entering into the problem for three different types of water based nanofluids.For this purpose, Figs.
2-8 have been displayed.Table1 shows the thermo physical properties of water and the Nano elements (Cu, 2 3 Al O and 2 TiO ).To verify the validity and exactness of the present results, we have compared skin friction co efficient and local Nusselt number with those of Satya Narayana et al. [18] and Hamad et al. [38] for various values of Pr.The results of this comparison are given in Table 2 and found them to be in excellent agreement.Therefore, we are confident that the present results are more accurate.This is because, the thermal boundary layer thickness increases with an increase in the thermal radiation.Thus, it is pointed out that, the radiation should be minimized to have the cooling process at a faster rate.It is also observed that the nanofluid velocity and temperature in the case of 2 TiO -water nanofluid is relatively lesser than that of the Cu -water, 2 3 Al O -water nanofluids.This occurrence has a higher conformity with the physical realities.5 shows the influence of magnetic field parameter M on the nanofluid velocity profiles in the boundary layer.Application of magnetic field to an electrically conducting fluid gives rise to a resistive type force called the Lorentz force.This force has the tendency to slow down the motion of the fluid in the boundary layer.Thus the presence of the magnetic field parameter decreases the momentum boundary layers thickness.These results are in good agreement with the results obtained in case of Hamad et al. [38].Table 3 shows that the skin friction coefficient increases with the increase in M where as M have no effect on Sherwood number.5) it is clear that the equation is independent of M. So M has no effect on Sherwood number.This table also displays that the skin friction coefficient and Sherwood number have greater value for Cu than 2 3 Al O and 2 TiO .This is due to the physical properties of fluid and the thermal conductivity of Cu is much higher than that of 2 3 Al O and 2 TiO .This means that the nanofluids will be important in the cooling and heating processes.
Figs. 6-8 have been plotted to find the variation of nanofluid velocity, temperature and concentration profiles for different values of heat generation parameter Q.It is clear that, there is a decrease in the velocity and temperature with increase of Q.This is due to the fact that when heat is absorbed, the buoyancy forces decrease which retard the flow rate and thereby give rise to a decrease in the velocity and temperature profiles while the opposite is true in case of concentration profiles.Figs.9-11 display the effect of volume fraction of the nanoparticles on the nanofluid velocity, temperature and concentration profiles, respectively.It is noticed that the velocity decreases with increase of φ whereas reverse trend is observed in case of temperature and concentration profiles.It is also observed that, with increase of φ , the thermal boundary-layer increases.This agrees with the physical behavior that, when the volume fraction of copper increases, the thermal conductivity increases, and then the thermal boundary-layer thickness increases.We also observe that the nanofluid velocity in the case of Cu-water nanofluid is relatively higher than that of a 2 TiO -water nanofluid.In the same vein we also note that the conductivity of 2 3 Al O is more than that of 2 TiO and the temperature distribution in the 2 3 Al O -water nanofluid is higher than that of the 2 TiO -water nanofluid.It is observed from Fig. 12, the concentration asymptotically decreases from the maximum value 1 to 0 as z → ∞ .This shows that the concentration field is greatest near the plate surface and least in the outer flow region.The same figure further indicates that there is a steady fall in species concentration due to the effect of chemical reaction.This means that the consumption of chemical species leads to fall in the species concentration field.This clearly agrees with the physical laws.The influence of Soret effect on the concentration profiles is illustrated in figure 13.It is observed that, the increase in Sr contributes to decrease in the concentration.

CONCLUSIONS
The following conclusions can be made from the present investigation: (nanofluid) has a dramatic effect on the liquid thermophysical properties such as thermal conductivity.
We hope that the findings of this investigation may be useful in catalysis, biomedicine, magnetic resonance imaging, data storage and environmental remediation.Hence, the subject of nanofluids is of great interest worldwide for basic and applied research.

Fig. 2
Fig.2Velocity profile for various values of F

Fig. 3
Fig. 3 Temperature profile for various values of F

Fig. 4
Fig.4Concentration profile for various values of F Fig.5shows the influence of magnetic field parameter M on the nanofluid velocity profiles in the boundary layer.Application of magnetic field to an electrically conducting fluid gives rise to a resistive type force called the Lorentz force.This force has the tendency to slow down the motion of the fluid in the boundary layer.Thus the presence of the magnetic field parameter decreases the momentum boundary layers thickness.These results are in good agreement with the results obtained in case ofHamad et al. [38].Table3shows that the skin friction coefficient increases with the increase in M where as M have no effect on Sherwood number.

Fig. 5
Fig. 5 Velocity profile for various values of M Also from Eq. (5) it is clear that the equation is independent of M. So M has no effect on Sherwood number.This table also displays that the skin friction coefficient and Sherwood number have greater value for Cu

Fig. 6 Fig. 9
Fig. 6 Velocity profile for various values of Q

Fig. 10
Fig. 10 Temperature profile for various values of φ

Fig. 11
Fig. 11 Concentration profile for various values of φ

Fig. 12
Fig. 12 Concentration profile for various values of Kr

Fig. 13 Fig. 15
Fig. 13 Concentration profile for various values of Sr Figure 14 shows the variation of skin friction coefficient C f with respect to suction parameter S and rotational parameter R. The magnitude of a skin friction coefficient decrease with increasing R. It is also observed that the 2 3 Al O -water nanofluid has a low skin friction coefficient than Cu , 2 TiO -water nanofluids.The coefficient of Nusselt number is plotted against F for different values of Q in figure 15.It is obvious that the Nusselt number increases with Q.It is also observed that Cu -water nanofluid has the highest Nusselt number.

Table 2 :
Hamad and Pop (2011)riction and Nusselt number of the present case with those ofHamad and Pop (2011)and Satya Narayana

Table 3 :
Values of skin friction and Sherwood number for various


The velocity profile decreases with an increase in nanoparticle volume fraction parameter, while the opposite is true in the case of temperature and concentration profiles.Dueto chemical reaction, the concentration of the fluid decreases.This is because the consumption of chemical species leads to a fall in the species concentration field.But it is reverse in case of micropolar fluid [seeRef.40].This remarkable feature of the concentration profiles in our investigation is due to the presence of Soret number and nanoparticles in the flow field. It has been shown that mixing nanoparticles in a liquid