2016
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Conjugate Heat Transfer of MHD nonDarcy Mixed Convection Flow of a Nanofluid over a Vertical Slender Hollow Cylinder Embedded in Porous Media
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In this paper, conjugate heat transfer of magneto hydrodynamic mixed convection of nanofluid about a vertical slender hollow cylinder embedded in a porous medium with high porosity have been numerically studied. The Forchheimer’s modification of Darcy’s law was used in representing the nanofluid motion inside the porous media. The governing boundary layer equations were transformed to nondimensional differential equations by taking suitable similarity variables and solved numerically using differential quadrature method (DQM). The interfacial (solidliquid) temperature distribution and the variations of velocity and temperature within boundary layer for different values of governing parameter in presence of uniform magnetic field have been presented and discussed. Our results demonstrate that heat transfer rate can enhance using nanofluid as well as porous medium, while magnetic field has no remarkable effect on the parameter. The computed results were also compared with those available in the existing literature and a good agreement was observed.
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B.
Jafarian
Department of Chemical Engineering, Persian Gulf University, 75168, Boushehr, I. R.Iran
Department of Chemical Engineering, Persian
Iran


M.
Hajipour
Faculty of Sciences, Science and Research Branch, Islamic Azad University, Tehran, I. R. Iran
Faculty of Sciences, Science and Research
Iran


R.
Khademi
Department of Chemical Engineering, University of Sistan and Baluchestan, 98164161, Zahedan, I. R.Iran
Department of Chemical Engineering, University
Iran
ramin.khademi85@gmail.com
Conjugate heat transfer
differential quadrature method (DQM)
Magneto hydrodynamic (MHD)
MHDmixed convection
Nanofluid
[[1] D. A. Nield, A. Bejan, Convection in Porous Media: Springer (2006). ##[2] K. Vafai, Handbook of Porous Media, Second Edition: Taylor & Francis (2010). ##[3] K. Vafai, Porous Media: Applications in Biological Systems and Biotechnology: Taylor & Francis (2010). ##[4] Y D. B. Ingham, I. Pop, Convective Heat Transfer: Mathematical and Computational Modelling of Viscous Fluids and Porous Media: Elsevier Science (2001). ##[5] D. B. Ingham and I. Pop, Transport Phenomena in Porous Media III: Elsevier Science (2005). ##[6] P. Vadász, Emerging Topics in Heat and Mass Transfer in Porous Media: From Bioengineering and Microelectronics to Nanotechnology: Springer Science+Business Media B.V(2008). ##[7] C.L.Chang: Buoyancy and wall conduction effects on forced convection of micropolar fluid flow along a vertical slender hollow circular cylinder, International Journal of Heat and Mass Transfer 49 (2006) 49324942. ##[8] A. Kaya: The effect of conjugate heat transfer on MHD mixed convection about a vertical slender hollow cylinder, Communications in Nonlinear Science and Numerical Simulation 16 (2011) 19051916. ##[9] A. Kaya: Effects of buoyancy and conjugate heat transfer on nonDarcy mixed convection about a vertical slender hollow cylinder embedded in a porous medium with high porosity, International Journal of Heat and Mass Transfer 54 (2011) 818825. ##[10] S. U. S. Choi, Z. G. Zhang, W. Yu, F. E. Lockwood, ,E. A. Grulke: Anomalous thermal conductivityenhancement in nanotube suspensions, Applied Physics Letters 79 (2001) 22522254. ##[11] S. Ahmad, I. Pop: Mixed convection boundary layer flow from a vertical flat plate embedded in a porous medium filled with nanofluids, International Communications in Heat and Mass Transfer37 (2010) 987991. ##[12] A. V. Kuznetsov, D. A. Nield: Effect of Local Thermal Nonequilibrium on the Onset of Convection in a Porous Medium Layer Saturated by a Nanofluid, Transport in Porous Media 83 (2010) 425436. ##[13] A. V. Rosca, N. C. Rosca, T. Grosan, I. Pop: NonDarcy mixed convection from a horizontal plate embedded in a nanofluid saturated porous media, International Communications in Heat and Mass Transfer 39 (2012) 10801085. ##[14] T.Y. Na: Effect of wall conduction on natural convection over a vertical slender hollow circular cylinder, Applied Scientific Research 54 (1995) 3950. ##[15] A. Kaya: Effects of buoyancy and conjugate heat transfer on nonDarcy mixed convection about a vertical slender hollow cylinder embedded in a porous medium with high porosity, International Journal of Heat and Mass Transfer 54 (2011) 818. ##[16] H. Rahideh, P. Malekzadeh, M. R. Golbahar Haghighi: NonFourier Heat Conduction Analysis with TemperatureDependent Thermal Conductivity, ISRN Mechanical Engineering 2011 (2011) 46. ##[17] P. Malekzadeh, H. Rahideh, A. R. Setoodeh: Optimization of nonsymmetric convective–radiative annular fins by differential quadrature method, Energy Conversion and Management48 (2007) 16711677. ##[18] P. Malekzadeh, H. Rahideh: Twodimensional nonlinear transient heat transfer analysis of variable section pin fins, Energy Conversion and Management 50 (2009) 916922. ##[19] H. Rahideh, P. Malekzadeh, M. R. Golbahar Haghighi: Heat conduction analysis of multilayered FGMs considering the finite heat wave speed, Energy Conversion and Management55 (2012) 1419. ##[20] T. Grosan, I. Pop: Axisymmetric mixed convection boundary layer flow past a vertical cylinder in a nanofluid, International Journal of Heat and Mass Transfer54 (2011) 31393145. ##[21] O. Aydın, A. Kaya: Mixed convection of a viscous dissipating fluid about a vertical flat plate, Applied Mathematical Modelling 31 (2007) 843853. ##[22] K. Vafai, C. L. Tien: Boundary and inertia effects on flow and heat transfer in porous media, International Journal of Heat and Mass Transfer 24 (1981)195203. ##[23] K. A. J. AlFarhany: Numerical investigations of heat and mass transfer in a saturated porous cavity with Soret and Dufour effects (2012) 6465. ##[24] A. J. Chamkha: Nonsimilar solutions for heat and mass transfer by hydromagnetic mixed convection flow over a plate in porous media with surface suction or injection, International Journal of Numerical Methods for Heat & Fluid Flow10 (2000) 142  163.##]
Numerical Study of Single Phase/TwoPhase Models for Nanofluid Forced Convection and Pressure Drop in a Turbulence Pipe Flow
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In this paper, the problem of turbulent forced convection flow of water alumina nanofluid in a uniformly heated pipe has been thoroughly investigated. In numerical study, single and twophase models have been used. In singlephase modeling of nanofluid, thermal and flow properties of nanofluid have been considered to be dependent on temperature and volume fraction. Effects of volume fraction and Reynolds number (3000<Re<9000) on convective heat transfer coefficient and pressure drop were investigated for various axial locations of the tube. Numerical results have shown that the inclusion of nanoparticles into the base fluid produced a considerable augmentation of the heat transfer coefficient that increases with an increase of the volume fraction and Reynolds number. Moreover, the increase of volume fraction has no effects on the coefficient of friction, but it decreases with increasing Reynolds number. Comparison of numerical results with experiments shows that the results of single phase analysis is near to the experimental results.
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M.
Esfandiary
Department of Mechanical Engineering, University of Bu Ali Sina, Hamedan, I. R.Iran
Department of Mechanical Engineering, University
Iran


A.
Habibzadeh
Department of Mechanical Engineering, Miandoab Branch, Islamic Azad University, Miandoab, I. R. Iran
Department of Mechanical Engineering, Miandoab
Iran
amin.habibzadeh@yahoo.com


H.
Sayehvand
Department of Mechanical Engineering, University of Bu Ali Sina, Hamedan, I. R.Iran
Department of Mechanical Engineering, University
Iran
Heat transfer
Nanofluid
Single Phase/TwoPhase Models
Tube flow
Turbulent Forced convection
[[1] Y. Yang: Characterizations and Convective Heat Transfer Performance of Nanofluids, Ph.D thesis, LehighUniversity, UMI Publisher, Ann Arbor (2011). ##[2] Y. Xuan, Q. Li: Investigation on Convective Heat Transfer and Flow Features of Nanofluids, Journal of Heat Transfer 125 (2003) 151155. ##[3] W. Williams, J Bourgiorno, J Hu: Experimental Investigation of TurbulentConvective Heat Transfer andPressure Loss of Alumina/Water and Zirconia/WaterNanoparticle Colloids (Nanofluid) in Horizontal Tubes, Journal of Heat Transfer 130 (2008) 042412042419. ##[4] U. Rea, T. McKrell, L. Hu, J. Buongiorno: Laminar convective heat transfer and viscous pressure loss of alumina–water and zirconia–water nanofluids, International Journal of Heat and Mass Transfer (2008) 20422048. ##[5] S.M. Fotukian, M. Nasr Esfahany: Experimental study of turbulent convective heat transfer and pressure drop of dilute CuO/water nanofluid inside a circular tube, International Communications in Heat and Mass Transfer 37 (2010) 214–219. ##[6] B. Pak, Y.I. Cho: Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particle, Experimental Heat Transfer 11(1998) 151–170. ##[7] V. Bianco, O. Manca, S. Nardini: Numerical investigation on nanofluids turbulent convection heat transfer inside a circular tube, International Journal of Thermal Sciences, 50 (2011) 341349. ##[8] A. Behzadmehr, M. SaffarAvval, N. Galanis: Prediction of turbulent forced convection of a nanofluid in a tube with uniform heat flux using a two phase approach, International Journal of Heat and Fluid Flow 28 (2007) 211–219. ##[9] R. Lotfi, Y. Saboohi, A.M. Rashidi: Numerical study of forced convective heat transfer of Nanofluids: Comparison of different approaches, International Communications in Heat and Mass Transfer 37 (2010) 74–78. ##[10] W. Yu, D. M. France, E.V. Timofeeva, D. Singh and J.L. Routbort: Convective Heat Transfer of Nanofluids in Turbulent Flow, Argonne NATIONAL LABORATORY, presented at: Carbon Nano Materials and Applications Workshop (2011). ##[11] F. Vahidinia, M. Rahmdel: Turbulent mixed convection of a nanofluid in a horizontal circular tube with nonuniform wall heat flux using twophase approach, Transport Phenomena in Nano and Micro Scales 3 (2015) 106117. ##[12] P. Hanafizadeh, M. Ashjaee, M. Goharkhah, K. Montazeri, M. Akram: The comparative study of single and twophase models for magnetite nanofluid forced convection in a tube, International Communications in Heat and Mass Transfer 65 (2015) 58–70. ##[13] S. Göktepe, K. Atalik, H. Erturk: Comparison of single and twophase models for nanofluid convection at the entrance of a uniformly heated tube, International Journal of Thermal Sciences 80 (2014) 8392. ##[14] I. Behroyan, P. Ganesana, S. Heb, S. Sivasankaran: Turbulent forced convection of Cu–water nanofluid: CFD model comparison, International Communications in Heat and Mass Transfer 67 (2015) 163–172. ##[15] M. Shehnehpour Borazjani, M. Bamdad, R. Hashemi: Numerical investigation on the single phase forced convection heat transfer characteristics of Al2o3 nanofluids in a doubletube counter flow heat exchanger, International Journal of Basic Sciences and Applied Research 3 (2014) 266273. ##[16] R. Raj, N.S. Lakshman, Y. Mukkamala: Single phase flow heat transfer and pressure drop measurements in doubly enhanced tubes, International Journal of Thermal Sciences 88 (2015) 215227. ##[17] F. K. Suguimoto, R. A. Mazza: Experimental analysis of pressure gradients on a liquidliquid two phase flow, IV Journeys in MultiphaseFlows (2015) 2327. ##[18] M. Akbari, N. Galanis, A. Behzadmehr: Comparative assessment of single and twophase models for numerical studies of nanofluid turbulent forced convection, International Journal of Heat and Fluid Flow 37 (2012) 136–146. ##[19] H.K. Versteeg , W. Malalaskera: An introduction to computational fluid dynamic, Longman Group Ltd (1995). ##[20] M. Corcione: Empirical correlating equations for predicting the effective thermal conductivity and dynamic viscosity of nanofluids, Energy Conversion and Management 52 (2011) 789–793. ##[21] C.T. Nguyen, F. Desgranges, G. Roya, N. Galanis, T. Mare´ d, S. Boucher, H. Angue Mintsa: Temperature and particlesize dependent viscosity data for waterbased nanofluids – Hysteresis phenomenon, International Journal of Heat and Fluid Flow 28 (2007) 1492–1506. ##[22] F.P. Incorpra, D. Duit: An introduction to heat transfer.##]
The effect of various conductivity and viscosity models considering Brownian motion on nanofluids mixed convection flow and heat transfer
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In this paper the effect of using various models for conductivity and viscosity considering Brownian motion of nanoparticles is investigated. This study is numerically conducted inside a cavity full of WaterAl2O3 nanofluid at the case of mixed convection heat transfer. The effect of some parameters such as the nanoparticle volume fraction, Rayleigh, Richardson and Reynolds numbers has been examined. The governing equations with specified boundary conditions has been solved using finite volume method. A computer code has been prepared for this purpose. The results are presented in form of stream functions, isotherms, Nusselt number and the flow power with and without the Brownian motion taken into consideration. The results show that for all the applied models the stream functions and isotherm have approximately same patterns and no considerable difference has been observed. In all the studied models when considering the Brownian motion, the average Nusselt number is higher than not taking this effect into account. The models of KooKleinstreuer and LiKleinstreuer give almost same values for the maximum stream function and average Nusselt number. It is also true about the models of VajjhaDas and Xiao et al.
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H. R.
Ehteram
Department of Mechanical Engineering, Mechanical Engineering University of Kashan, Kashan, I. R. Iran
Department of Mechanical Engineering, Mechanical
Iran


A. A.
Abbasian Arani
Department of Mechanical Engineering, Mechanical Engineering University of Kashan, Kashan, I. R. Iran
Department of Mechanical Engineering, Mechanical
Iran


G. A.
Sheikhzadeh
Department of Mechanical Engineering, Mechanical Engineering University of Kashan, Kashan, I. R. Iran
Department of Mechanical Engineering, Mechanical
Iran
sheikhz@kashanu.ac.ir


A.
Aghaei
Department of Mechanical Engineering, Mechanical Engineering University of Kashan, Kashan, I. R. Iran
Department of Mechanical Engineering, Mechanical
Iran


A. R.
Malihi
Department of Mechanical Engineering, Mechanical Engineering University of Kashan, Kashan, I. R. Iran
Department of Mechanical Engineering, Mechanical
Iran
alirezaaghaei21@gmail.com
Brownian motion
Mixed convection
Nanofluid
Numerical study
Variable properties
[[1] M.A. Mansour, R.A. Mohamed, M.M. AbdElaziz, S.E. Ahmed: Numerical simulation of mixed convection flows in a square liddriven cavity partially heated from below using nanofluid, International Communications in Heat and Mass Transfer 37 (2010) 1504–1512. ##[2] B. Ghasemi, S.M. Aminossadati: Mixed convection in a liddriven triangular enclosure filled with nanofluids, International Communications in Heat and Mass Transfer 37 (2010) 1142–1148. ##[3] G.A. Sheikhzadeh, M. Ebrahim Qomi, N. Hajialigol, A. Fattahi: Numerical study of mixed convection flows in a liddriven enclosure filled With nanofluid using variable properties, International Results in Physics 2 (2012) 513. ##[4] I. Pishkar, B. Ghasemi: Cooling enhancement of two fins in a horizontal channel by nanofluid mixed convection, International Journal of Thermal Sciences 59 (2012) 141151. ##[5] A.J. Chamkhaa, E. AbuNada: Mixed convection flow in single and doublelid driven square cavities filled with water–Al2O3 nanofluid: Effect of viscosity models, European Journal of Mechanics B/Fluids 36 (2012) 8296. ##[6] A.A. Abbasian Arani, S. Mazrouei Sebdani, M. Mahmoodi, A. Ardeshiri, M. Aliakbari: Numerical study of mixed convection flow in a liddriven cavity with sinusoidal heating on side walls using nanofluid, Super lattices and Micro structures 51 (2012) 893911. ##[7] B. Ghasemi, S.M. Aminossadati: Brownian motion of nanoparticles in a triangular enclosure with natural convection, International Journal of Thermal Sciences 49 (2010) 931–940. ##[8] H.A. Pakravan, M. Yaghoubi: Combined thermophoresis, Brownian motion and Dufour effec ts on natural convection of nano fluids, International Journal of Thermal Sciences 50 (2011) 394 – 402. ##[9] X. Wang, D. Li, H. Jiao: Heat Transfer Enhancement of CuOwater Nanouids Considering Brownian Motion of Nanoparticles in a Singular Cavity, Journal of Information & Computational Science 9 (2012) 12231235. ##[10] Z. Haddad, E. AbuNada, H.F. Oztop, A. Mataoui: Natural convection in nanofluids: Are the thermophoresis and Brownian motion effects significant in nano fluid heat transfer enhancement?, International Journal of Thermal Sciences 57 (2012) 152 162. ##[11] H.R. Seyf, B. Nikaaein: Analysis of Brownian motion and particle size effects on the thermal behavior and cooling performance of microchannel heat sinks, International Journal of Thermal Sciences 58 (2012) 3644. ##[12] A.A. Abbasian Arani, A. Aghaei, H. Ehteram: Numerical investigation of Brownian motion effect on nanofluid mixed convection in enclosure with a hot central heat source, Journal of Modeling in Engineering Semnan University (2013). ##[13] E. AbuNada, Z. Masoud, A. Hijazi: Natural convection heat transfer enhancement in horizontal concentric annuli using nanofluids, International Communications in Heat and Mass Transfer 35(2008) 657 – 665. ##[14] A. Bijan: Convection heat transfer, Willey, NewYork, Third edition (1984). ##[15] K. Khanafer, K. Vafai: A critical synthesis of thermo physical characteristics of nanofluids, International Journal of Heat and Mass Transfer 54 (2011) 44104428. ##[16] H.C. Brinkman: The viscosity of concentrated suspensions and solution, The Journal of Chemical Physics 20 (1952) 571–581. ##[17] J. Maxwell: A Treatise on Electricity and Magnetism, second ed., Oxford University Press, Cambridge, UK (1904). ##[18] J. Koo, C. Kleinstreuer: A new thermal conductivity model for nanofluids, Journal of Nanoparticle Research 6 (2004) 577–588. ##[19] R.S. Vajjha, D.K. Das: Experimental determination of thermal conductivity of three nanoﬂuids and development of new correlations, International Journal of Heat and Mass Transfer 52 (2009) 4675–4682. ##[20] J. Li, C. Kleinstreuer: Thermal performance of nanoﬂuid ﬂow in microchannels, International Journal of Heat and Fluid Flow 29 (2008) 1221–1232. ##[21] B. Xiao, Y. Yang, L. Chen: Developing a novel form of thermal conductivity of nanofluids with Brownian motion effect by means of fractal geometry, Powder Technology 23 (2013) 409–414. ##[22] M. Hemmat Esfe, S. Saedodin, O. Mahian, S, Wongwises: Thermal conductivity of Al2O3/water nanofluids, Journal of Thermal Analysis and Calorimetry 117(2014) 675–681.##]
A numerical investigation of γAl2O3water nanofluids heat transfer and pressure drop in a shell and tube heat exchanger
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The effect of γAl2O3 nanoparticles on heat transfer rate, baffle spacing and pressure drop in the shell side of small shell and tube heat exchangers was investigated numerically under turbulent regime. γAl2O3water nanofluids and pure water were used in the shell side and the tube side of heat exchangers, respectively. Since the properties of γAl2O3water nanofluids were variable, they were defined using the user define function. The results revealed that heat transfer and pressure drop were increased with mass flow rate as well as baffle numbers. Adding nanoparticles to the based fluid did not have a significant effect on pressure drop in the shell side. The best heat transfer performance of heat exchangers was for γAl2O3water 1 vol.% and higher nanoparticles concentration was not suitable. The suitable baffle spacing was 43.4% of the shell diameter, showing a good agreement with BellDelaware method.
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P.
Shahmohammadi
Department of Chemical Engineering, Quchan University of Advanced Technology, 6733594771, Quchan, I.R. Iran
Department of Chemical Engineering, Quchan
Iran


H.
Beiki
Department of Chemical Engineering, Quchan University of Advanced Technology, 6733594771, Quchan, I.R. Iran
Department of Chemical Engineering, Quchan
Iran
hbeiki@qiet.ac.ir
Baffle spacing
CFD models
γAl2O3water nanofluids
Shell and tube heat exchanger
User define function (UDF)
[[1] S. Kim, H. Chung, H. Jeong, B. Lee, B. Ochirkhuyag, J. Lee, H. Choi: The study of heat transfer for nanofluid with carbon nano particle in an exhaust gas recirculation (EGR) cooler, Heat and Mass Transfer 49 (2013) 10511055. ##[2] R. Lotfi, A. M. Rashidi, A. Amrollahi: Experimental study on the heat transfer enhancement of MWNTwater nanofluid in a shell and tube heat exchanger, International Communications in Heat and Mass Transfer 39 (2012) 108111. ##[3] K. Y. Leong, R. Saidur, T. M. Mahlia, Y. H. Yau: Modeling of shell and tube heat recovery exchanger operated with nanofluid based coolants, International Journal of Heat and Mass Transfer 55 (2012) 808816. ##[4] L. Godson, K. Deepak, C. Enoch, B. Jefferson, B. Raja: Heat transfer characteristics of silver/water nanofluids in a shell and tube heat exchanger. Archives of Civil and Mechanical Engineering 14 (2014) 489496. ##[5] R. Aghayari, H. Maddah, F. Ashori, A. Hakiminejad, M. Aghili: Effect of nanoparticles on heat transfer in mini doublepipe heat exchangers in turbulent flow, Heat and Mass Transfer 51 (2014) 301306. ##[6] D. Madhesh, S. Kalaiselvam: Experimental study on the heat transfer and flow properties of Ag–ethylene glycol nanofluid as a coolant, Heat and Mass Transfer 50 (2014) 15971607. ##[7] A. Hussein, R. A. Bakar, K. Kadirgama, K. V. Sharma: Heat transfer augmentation of a car radiator using nanofluids. Heat and Mass Transfer 50 (2014) 15531561. ##[8] J. Cao, Y. Ding, C. Ma: Aqueous Al2O3 nanofluids: the important factors impacting convective heat transfer Heat and Mass Transfer 50 (2014) 16391648. ##[9] Y. S. Son , J. Y. Shin: Performance of a shellandtube heat exchanger with spiral baffle plates, KSME International Journal 15 (2001) 15551562. ##[10] B. K. Sonage, P. Mohanan: Heat transfer and pressure drop characteristic of zinc–water nanofluid, Heat and Mass Transfer 51 (2014) 521527. ##[11] C. Pang, J. W. Lee, Y. T. Kang: Review on combined heat and mass transfer characteristics in nanofluids, International Journal of Thermal Sciences 87 (2015) 4967. ##[12] D. Q. Kern: Process heat transfer, McGrawHill, New York (1950). ##[13] R. W. Serth, T. G. Lestina: Process Heat Transfer, Second ed, Academic Press, Boston (2014). ##[14] K. J. Bell: Delaware method for shellside design, In: Kakaç S, Berger A, Mayinger F (eds) Heat exchangers: thermalhydraulic fundamentals and design. Hemisphere (1981) 581618. ##[15] M. Kahani, S. Zeinali Heris, S. M. Mousavi: Experimental investigation of TiO2/water nanofluid laminar forced convective heat transfer through helical coiled tube, Heat and Mass Transfer 50 (2014) 15631573. ##[16] X. Wang, A. S. Mujumdar: Heat transfer characteristics of nanofluids: a review, International Journal of Thermal Sciences 46 (2007) 119. ##[17] G. Huminic, A. Huminic: A: Heat transfer characteristics in double tube helical heat exchangers using nanofluids, International Journal of Heat and Mass Transfer 54 (2011) 42804287. ##[18] H. A. Mohammed, H. A. Hasan, M. A. Wahid: Heat transfer enhancement of nanofluids in a double pipe heat exchanger with louvered strip inserts, International Communications in Heat and Mass Transfer 40 (2013) 3646. ##[19] B. H. Chun, H. U. Kang, S. H. Kim: Effect of alumina nanoparticles in the fluid on heat transfer in doublepipe heat exchanger system, Korean Journal of Chemical Engineering 25 (2008) 966971. ##[20] M. Akhtari, M. Haghshenasfard, M. R. Talaie: Numerical and Experimental Investigation of Heat Transfer of αAl2O3/Water Nanofluid in Double Pipe and Shell and Tube Heat Exchangers, Numerical Heat Transfer, Part A: Applications 63 (2013) 941958. ##[21] B. Farajollahi, S. G. Etemad, M. Hojjat: Heat transfer of nanofluids in a shell and tube heat exchanger, International Journal of Heat and Mass Transfer 53 (2010) 1217. ##[22] M. M. Elias, I. M. Shahrul, I. M. Mahbubul, R. Saidur, N. A. Rahim: Effect of different nanoparticle shapes on shell and tube heat exchanger using different baffle angles and operated with nanofluid, International Journal of Heat and Mass Transfer 70 (2014) 289297. ##[23] A. Ghozatloo, A. Rashidi, M. ShariatyNiassar: Convective heat transfer enhancement of graphene nanofluids in shell and tube heat exchanger, Experimental Thermal and Fluid Science 53 (2014) 136141. ##[24] J. Taborek: Thermal and hydraulic design of heat exchangers, In: Hewitt GF (ed) Handbook of Heat Exchanger Design 3 (2002). ##[25] R. Mukherjee: Use doublesegmental baffles in the shellandtube heat exchangers, Chem Eng Progress 88 (1992) 47–52. ##[26] M. SaffarAvval, E. Damangir: A general correlation for determining optimum baffle spacing for all types of shell and tube exchangers, International Journal of Heat and Mass Transfer 38 (1995) 25012506. ##[27] B. Khalifeh Soltan, M. SaffarAvval, E. Damangir: Minimizing capital and operating costs of shell and tube condensers using optimum baffle spacing, Applied Thermal Engineering 24 (2004) 28012810. ##[28] D. Eryener: Thermoeconomic optimization of baffle spacing for shell and tube heat exchangers, Energy Conversion and Management 47(2006) 14781489. ##[29] E. Ozden, I. Tari: Shell side CFD analysis of a small shellandtube heat exchanger, Energy Conversion and Management 51 (2010) 10041014. ##[30] R. Mukherjee: Use doublesegmental baffles in the shellandtube heat exchanger, chem Eng progress88(1992) 47–52. ##[31] A. Sasmito, J. Kurnia, A. Mujumdar: Numerical evaluation of laminar heat transfer enhancement in nanofluid flow in coiled square tubes, Nanoscale Research Letters 6 (2011) 114. ##[32] A. K. Tiwari, P. Ghosh, J. Sarkar, H. Dahiya, J. Parekh: Numerical investigation of heat transfer and fluid flow in plate heat exchanger using nanofluids, International Journal of Thermal Sciences 85 (2014) 93103. ##[33] Y. Xuan, W. Roetzel: Conceptions for heat transfer correlation of nanofluids, International Journal of Heat and Mass Transfer 43 (2000) 37013707. ##[34] B. C. Pak, Y. I. Cho: Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles, Experimental Heat Transfer 11 (1998) 151170. ##[35] A. A. Minea: Uncertainties in modeling thermal conductivity of laminar forced convection heat transfer with water alumina nanofluids, International Journal of Heat and Mass Transfer 68 (2014) 7884. ##[36] M. Keshavarz Moraveji, M. Hejazian: Modeling of turbulent forced convective heat transfer and friction factor in a tube for Fe3O4 magnetic nanofluid with computational fluid dynamics, International Communications in Heat and Mass Transfer 39 (2012) 12931296. ##[37] A. Kamyar, R. Saidur, M. Hasanuzzaman: Application of Computational Fluid Dynamics (CFD) for nanofluids, International Journal of Heat and Mass Transfer 55 (2012) 41044115. ##[38] O. Kaya: Numerical study of turbulent flow and heat transfer of Al2O3–water mixture in a square duct with uniform heat flux, Heat and Mass Transfer 49 (2013) 15491563. ##[39] R. R. T. Karuppa, G. Srikanth: Shell side numerical analysis of a shell and tube heat exchanger considering the effects of baffle inclination angle on fluid flow using CFD, Thermal Science 16 (2012) 11651174. ##[40] G. Towler, R. Sinnott: Chemical engineering design: principles, practice and economics of plant and process design. ButterworthHeinemann (2008). ##[41] B. Sahin, G. G. Gültekin, E. Manay, S. Karagoz: Experimental investigation of heat transfer and pressure drop characteristics of Al2O3–water nanofluid, Experimental Thermal and Fluid Science 50 (2013) 2128.##]
Lattice Boltzmann simulation of EGM and solid particle trajectory due to conjugate natural convection
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2
The purpose of this paper is to investigate the EGM method and the behavior of a solid particle suspended in a twodimensional rectangular cavity due to conjugate natural convection. A thermal lattice Boltzmann BGK model is implemented to simulate the two dimensional natural convection and the particle phase was modeled using the Lagrangian–Lagrangian approach where the solid particles are treated as points moving in the computational domain as a result of the fluid motion. Entropy generation due to heat transfer irreversibility, isotherms, streamlines and Nusselt numbers were obtained and discussed. Total entropy generations in various cases are also reported and optimum case is presented based on minimum entropy generation.
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43


J.
Alinejad
Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, I. R. Iran
Department of Mechanical Engineering, Sari
Iran
alinejad_javad@iausari.ac.ir


J. A.
Esfahani
Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad 917751111, I. R. Iran
Department of Mechanical Engineering, Ferdowsi
Iran
Conjugate convection
Entropy generation
Lagrangian–Lagrangian (L–L)
Lattice Boltzmann model
particle trajectory
[[1] J.W. Dally, P. Lall, J.C. Suhling: Mechanical Design of Electronic Systems. Knoxville, TN USA: College House Enterprises, LLS (2008). ##[2] E. Samadiani, Y. Joshi, F. Mistree:The thermal design of a next generation data center: a conceptual exposition, J. Electron 130 (2008) 1104 −1112. ##[3] Y.K. Kim, K.H. Lee, H.R. Kim: Cold neutron source at KAERI, Korea, J. Nuclear Engng. and Design 238 (2008)1664−1669. ##[4] S. Karthikeyan, T. Sundararajan, U.S.P. Shet, P. Selvaraj: Effect of turbulent natural convection on sodium pool combustion in the steam generator building of a fast breeder reactor, J. Nuclear Engng. and Design 239(2009)2992−3002. ##[5] I. Rodriguez, J. Castro, C.D. PerezSegarra, A. Oli va: L Unsteady numerical simulation of the cooling process of vertical storage tanks under laminar natural convection, Inter. J. Thermal Sci. 48(2009)708−721. ##[6] W. Lin, S.W. Armfield: Direct simulation of natural convection cooling in a vertical circular cylinder, Int. J. Heat Mass Transfer 42 (1999) 4117−4130. ##[7] V. Kurian, M.N. Varma, A. Kannan: Numerical studies on laminar natural convection inside inclined cylinders of unity aspect ratio, Int. J. Heat Mass Transfer 52 (2009) 822−838. ##[8] G. V. Kuznetsov, M. A. Sheremet: show all 2 hide:Twodimensional problem of natural convection in a rectangular domain with local heating and heatconducting boundaries of finite thickness, fluid dynamics 41(2006) 881890. ##[9] S.G. Cherkasov: Natural convection and temperat ure stratification in a cryogenic fuel tank in microgravity, Fluid Dynamics 29 (1994) 710−716. ##[10] V.I. Polezhaev and S.G. Cherkasov, Unsteady thermal convection in a cylindrical vessel heated from the side, Fluid Dynamics 18 (1983) 620−629. ##[11] S.G. Cherkasov: Natural convection in a vertical cylindrical vessel with heat supplied to its side and free surfaces, Fluid Dynamics 19 (1984) 902−906. ##[12] L.A. Moiseeva, S.G. Cherkasov: Mathematical modeling of natural convection in a vertical cylindrical tank with alternatingsign heat flux distribution on the wall, Fluid Dynamics 31(1996)218−223. ##[13] S.G. Martyushev, M.A. Sheremet: Mathematical Modeling of the Laminar Regime of Conjugate Convective Heat Transfer in an Enclosure with an Energy Source Under SurfaceRadiation Conditions,” Journal of Engineering Physics and Thermophysics 86 (2013) 110–119. ##[14] M.A. Sheremet: Unsteady conjugate thermo gravitational convection in a cylindrical region with local energy source, Thermophysics and Aeromechanics 18 (2011) 447458. ##[15] J. A. Esfahani, J. Alinejad: Entropy generation of conjugate natural convection in enclosures: the Lattice Boltzmann Method, Journal of Thermophysics and Heat Transfer 27 (2013) 498505. ##[16] A. C. Baytas: Entropy generation for natural convection in an inclined porous cavity, Int. J. Heat Mass Transfer 43 (2000) 2089–2099. ##[17] G. Naterer: Transition criteria for entropy reduct  ion of convective heat transfer from micropatterned surfaces, J. Thermophysics and Heat Transfer 22(2008)271280. ##[18] A. B. S. Alquaity, S. A. AlDini, B. S. Yilbas: Entropy generation in microchannel flow with presence of nanosized phase change particles, J. Thermophysics and Heat Transfer 26(2012)134140. ##[19] A. Vosoogh, A.R. Falahat: Effect of nanofluid on entropy generation and pumping power in coiled Tube, J. Thermophysics and Heat Transfer 26(2012)141146. ##[20] S. Chakraborty, D. Chatterjee: An enthalpybased hybrid latticeBoltzmann method for modelling solidliquid phase transition in the presence of convective transport , J. Fluid Mechanics 592 (2007) 155176. ##[21] D. Chatterjee , S. Amiroudine : Lattice kinetic simulation of non isothermal magnet ohydrodynamics, Physical Review E 81(2010) 16. ##[22] Z. Guo, T.S. Zhao: Lattice Boltzmann model for incompressible ﬂows through porous media. Physical Review E 66 (2002) 304–312. ##[23] A. D’Orazio, M. Corcione, G.P Celata: Application to natural convection enclosed ﬂows of a lattice Boltzmann BGK model coupled with a general purpose thermal boundary condition. International Journal of Thermal Sciences 43 (2004) 575–586. ##[24] C. Shu, Y. Peng, Y.T. Chew: Simulation of natural convection in a square cavity by Taylor series expansion and least squaresbased lattice Boltzmann method. International Journal of Modern Physics 13 (2002) 1399–1414. ##[25] B. Chopard, P. O. Luthi: Lattice Boltzmann computations and applications to physics. Theoret. Comput. Phys 217 (1999) 115–130. ##[26] R. R. Nourgaliev, T. N. Dinh, T. G. Theofanous, D. Joseph: The lattice Boltzmann equation method: theoretical interpretation, numerics and implications. Int. J. Multiph. Flow 29 (2003) 117–169. ##[27] D. Yu, R. Mei, L. S. Luo, W. Shyy: Viscous flow computations with the method of lattice Boltzmann equation. Progr. Aerospace. Sci 39 (2003) 329–367. ##[28] A. A. Mohammad: Applied Lattice Boltzmann Method for Transport Phenomena Momentum Heat Mass Transfer. University of Calgary Press, Calgary (2007). ##[29] D. M. Aghajani, M. Farhadi, K. Sedighi: Effect of heater location on heat transfer and entropy generation in the cavity using the lattice Boltzmann method. Heat Transfer Research 40 (2009) 521–536. ##[30] A. Mezrhab, M. Jami, C. Abid, M. Bouzidi, P. Lallemand: Lattice Boltzmann modeling of natural convection in an inclined square enclosure with partitions attached to its cold wall. Int. J. Heat Fluid Flow 27 (2006) 456–465. ##[31] X. He, L. S. Luo: Lattice Boltzmann model for the incompressible Navier–Stokes equations. J. Stat. Phys 88 (1997) 927–944. ##[32] N. Thürey, U. Rüde: Stable free surface flows with the lattice Boltzmann method on adaptively coarsened grids. Comput. Vis. Sci12 (2009) 247–263. ##[33] Y. Varol, H. F.Oztop, A. Koca: Entropy generation due to conjugate natural convection in enclosures bounded by vertical solid walls with different thicknesses, International Communications in Heat and Mass Transfer 35 (2008) 648–656.##]
MHD Boundary Layer Flow of a Nanofluid over an Exponentially Permeable Stretching Sheet with radiation and heat Source/Sink
2
2
The problem of steady Magnetohydrodynamic boundary layer flow of an electrically conducting nanofluid due to an exponentially permeable stretching sheet with heat source/sink in presence of thermal radiation is numerically investigated. The effect of transverse Brownian motion and thermophoresis on heat transfer and nano particle volume fraction considered. The governing partial differential equations of mass, momentum, energy and nanoparticle volume fraction equations are reduced to ordinary differential equations by using suitable similarity transformation. These equations are solved numerically using an implicit finite difference scheme, for some values of flow parameters such as Magnetic parameter (M), Wall mass transfer parameter(S), Prandtl number(Pr), Lewis number (Le), Thermophoresis parameter (Nt), Brownian motion parameter(Nb), Radiation parameter (R). The numerical values presented graphically and analized for velocity, temperature and nanoparticle volume fraction.
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44
51


N.
Kishan
Department of Mathematics, Osmania University, Hyderabad Telangana, India
Department of Mathematics, Osmania University,
Iran


C.
Kalyani
Department of Mathematics, R.B.V.R.R. women’s college, Hyderabad Telangana, India
Department of Mathematics, R.B.V.R.R. women’
Iran
kpskalyani_1996@yahoo.com


M.
Chenna Krishna Reddy
Department of Mathematics, Osmania University, Hyderabad Telangana, India
Department of Mathematics, Osmania University,
Iran
Keller box
MHD
Nanofluid
Stretching permeable sheet
thermal radiation
[[1]B.C.Sakiadas: Boundary layer behaviour on continous solid surface, AICHEJ 7 (1961) 21 28. ##[2]L. J. Crane: Flow past a stretching plate, Zeitschrift fur Angewandte Mathematik und Physik ZAMP 21(1970) 645–647. ##[3]F.K. TSOU, E.M sparrow, V. Goldstein: Flow and heat transfer in the boundary layer flow on a continous moving surface, Int.Journal of Heat and Mass Transfer 10 (1967) 219235. ##[4]E. M. A. Elbashbeshy: Heat transfer over an exponentially stretching continuous surface with suction, Archives of Mechanics 53 (2001) 643–651. ##[5]E. Magyari, B. Keller: Heat and mass transfer in the boundary layers on an exponentially stretching Continuous surface, Journal of Physics D 32(1999) 577–585. ##[6]M.K. Partha, P. V. S. N. Murthy, G. P. Rajasekhar: Effect of viscous dissipation on the mixed convection heat transfer from an exponentially stretching surface, Heat and Mass Transfer 41(2005) 360–366. ##[7]A. Ishak: MHD boundary layer flow due to an exponentially stretching sheet with radiation effect,SainsMalaysiana 40 (2011) 391–395. ##[8]K. Bhattacharyya: Boundary layer flow and heat transfer over an exponentially shrinking sheet, Chinese Physics Letters 28, Article ID 074701 (2011). ## [9]N. Kishan, P. Kavitha: MHD NonNewtonian Power Law Fluid Flow and Heat Transfer Past a NonLinear Stretching Surface with Thermal Radiation and Viscous Dissipation, International Journal of Applied Mathematics 29 (2014)1318 1332 . ## [10]K. Bhattacharyya , I. Pop: MHD boundary layer flow due to an exponentially shrinking sheet,Magnetohydrodynamics 47 (2011) 337–344. ##[11]R.K. Gupta, T. Sridhar: Viscoelastic effects in nonNewtonian flow through porous media, Rheol Acta 24 (1985) 14851. ##[12]S. Abel, PH. Veena: Viscoelastic fluid flow and heat transfer in a porous medium over a stretching sheet, Int.J NonLinear Mech 33 (1998) 5318. ##[13]R. Sharma: Effects of Viscous elastics fluid flow and heat source on unsteady boundary layer flow and heat transfer past stretching surface embedded in a porous medium using element free Galerkin method, Appl. Math Comput 219 (2012) 976 87. ##[14]S. U. S. Choi: Enhancing thermal conductivity of fluids with nanoparticles, in Proceedings of the ASME International Mechanical Engineering Congress and Exposition 66 (1995) 99–105. ##[15]H. Masuda, A. Ebata, K. Teramae, N. Hishinuma: Alteration of thermal conductivity and viscosity of liquid by dispersing ultrafine particles, Netsu Bussei 7 (1993) 227–233,. ##[16]J. Buongiorno: Convective transport in nanofluids, Journal of Heat Transfer 128 (2006) 240–250. ##[17]D. A. Nield, A. V. Kuznetsov: The ChengMinkowycz problemfor natural convective boundarylayer flow in a porous medium saturated by a nanofluid, International Journal of Heat and Mass Transfer 52 (2009) 5792–5795. ##[18]A. V. Kuznetsov, D. A. Nield: Natural convective boundarylayer flow of a nanofluid past a vertical plate, International Journal ofThermal Sciences 49 (2010) 243–247. ##[19]Rosmila et. As theorectically studied the problem of steady boundary layer flow of a nanofluid past a porous stretching surface with variable stream conditions and chemical reaction (2011). ##[20]N.C. Rosc et.al, T. Grosan,I. Pop: stagnation point flow and mass transfer past a permeable stretching /shrinking sheet in a nano fluid, Sains Malaysiana 41(2012) 12711279. ##[21]W. A. Khan, I. Pop, Boundarylayer flow of a nanofluid past a stretching sheet, International Journal of Heat and Mass Transfer53 (2010) 2477–2483. ##[22]D. A. Nield, A. V. Kuznetsov: The ChengMinkowycz problemfor natural convective boundarylayer flow in a porous medium saturated by a nanofluid,International Journal of Heat and Mass Transfer 52 (2009) 5792–5795. ##[23]O.D.Makinde, A. Aziz: Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition, The International Journal ofThermal Sciences 50 (2011) 1326–1332. ##[24]R. Kandasamy, P. Loganathan, P. Puvi Arasu: Scaling group transformation for MHD boundarylayer flow of a nanofluid past a vertical stretching surface in the presence of suction/injection, Nuclear Engineering and Design 241 (2011) 2053–2059. ##[25]P. Rana, R. Bhargava: Flow and heat transfer of a nanofluid over a nonlinearly stretching sheet: a numerical study, Communications in Nonlinear Science and Numerical Simulation 17(2012) 212–226. ##[26]F. M. Hady, F. S. Ibrahim, S. M. AbdelGaied, M. R. Eid: Radiation effect on viscous flow of a nanofluid and heat transfer over a nonlinearly stretching sheet, Nanoscale ResearchLetters 7 article 229 (2012). ##[27]H. naw, N. Kishan: MHD boundary layer flow and heat transfer of nanofluid past a nonlinearly with stretching / shrinking sheet with thermal radiation Journal Nanofluids 3 (2014) 19. ##[28]C. Srinivas Reddy, N. Kishan, B. Chandra Shekar: MHD Boundary Layer Flow and Heat Transfer of a Nanofluid Over Shrinking Sheet with Mass Suction and Chemical Reaction, Journal of Nanofluids 4 (4) 518527. ##[29]K. Bhattacharyya, G. C. Layek: MHD boundary layer flow of nanofluid over an exponentially stretching prmeable sheet, Physics R.J (2014) Article ID 592536. ##]
Investigation of two phase unsteady nanofluid flow and heat transfer between moving parallel plates in the presence of the magnetic field using GM
2
2
In this paper, unsteady two phase simulation of nanofluid flow and heat transfer between moving parallel plates, in presence of the magnetic field is studied. The significant effects of thermophoresis and Brownian motion have been contained in the model of nanofluid flow. The three governing equations are solved simultaneously via Galerkin method. Comparison with other works indicates that this method is very applicable to solve these problems. The semi analytical analysis is accomplished for different governing parameters in the equations e.g. the squeeze number, Eckert number and Hartmann number. The results showed that skin friction coefficient value increases with increasing Hartmann number and squeeze number in a constant Reynolds number. Also, it is shown that the Nusselt number is an incrementing function of Hartmann number while Eckert number is a reducing function of squeeze number. This type of results can help the engineers to make better and researchers to investigate faster and easier.
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52
58


N.
Hedayati
Babol University of Technology, Department of Mechanical Engineering, Babol, I. R.Iran
Babol University of Technology, Department
Iran
nima.hedayati883@gmail.com


A.
Ramiar
Babol University of Technology, Department of Mechanical Engineering, Babol, I. R.Iran
Babol University of Technology, Department
Iran
Brownian
Galerkin method (GM)
Eckert number
Hartmann number
Nanofluid
Thermophoresis
[[1] M.Sheikholeslami , H.R.Ashorynejad, D.D. Ganji, A. Kolahdooz: Investigation of Rotating MHD Viscous Flow and Heat Transfer between Stretching and Porous Surfaces Using Analytical Method. Hindawi Publishing Corporation Mathematical Problems in Engineering (2011) , http://dx.doi.org/10.1155/2011/ 258734 (Article ID 258734, 17 pages). ##[2] M. Sheikholeslami, H.R. Ashorynejad, D.D. Ganji, A. Yıldırım: Homotopy perturbation method for threedimensional problem of condensation film on inclined rotating disk, Scientia Iranica B 19 (2012) 437–442. ##[3] D.D.Ganji, H.B.Rokni, M.G.Sfahani, S.S.Ganji. Approximate traveling wave solutions for coupled shallow water, Advances in Engineering Software 41 (2010) 956–961. ##[4] M. Keimanesh, M.M.Rashidi, A.J. Chamkha, R.Jafari: Study of a third grade non Newtonian fluid flow between two parallel plates using the multistep differential transform method, Computers and Mathematics with Applications 62 (2011) 2871–2891 ##[5] M.Hatami, Kh.Hosseinzadeh, G. Domairry , M.T. Behnamfar: Numerical study of MHD twophase Couette flow analysis for fluidparticle suspension between moving parallel plates, Journal of the Taiwan Institute of Chemical Engineers 45 (2014) 22382245 ##[6] K.Khanafer, K.Vafai, M.Lightstone: Buoyancydriven heat transfer enhancement in a twodimensional enclosure utilizing nanofluids, International Journal of Heat and Mass Transfer 46 (2003) 3639–3653. ##[7] E.AbuNada ,Z. Masoud , A.Hijazi: Natural convection heat transfer enhancement in horizontal concentric annuli using nanofluids, International Communications in Heat and Mass Transfer 35 (2008) 657–665. ##[8] M.M. Rashidi , S.Abelman, N. Freidooni Mehr: Entropy generation in steady MHD flow due to a rotating porous disk in a nanofluid, International Journal of Heat and Mass Transfer 62 (2013) 515–525. ##[9] M .Sheikholeslami ,Sh. Abelman, D.D.Ganji: Numerical simulation of MHD nanofluid flow and heat transfer considering viscous dissipation, International Journal of Heat and Mass Transfer 79 (2014) 212–222. ##[10] M.Sheikholeslami, M.GorjiBandpy ,D.D Ganji: MHD free convection in an eccentric semiannulus filled with nanofluid, Journal of the Taiwan Institute of Chemical Engineers 45(2014)1204–16. ##[11] M.Sheikholeslami, M.GorjiBandpy, D.D.Ganji, S.Soleimani: MHD natural convection in a nanofluid filled inclined enclosure with sinusoidal wall using CVFEM, Neural Comput & Applic 24 (2014) 873–882. ##[12] A.Malvandi ,D.D. Ganji: Brownian motion and thermophoresis effects on slip flow of alumina/water nanofluid inside a circular microchannel in the presence of a magnetic field, International Journal of Thermal Sciences 84 (2014) 196206. ## [13] M. Hatami , D.D.Ganji: Heat transfer and nanofluid flow in suction and blowing process between parallel disks in presence of variable magnetic, Field Journal of Molecular Liquids190 (2014) 159–168. ##[14] H.R. Ashorynejad, A.A. Mohamad, M.Sheikholesla mi: Magnetic field effects on natural convection flow of a nanofluid in a horizontal cylindrical annulus using Lattice Boltzmann method, International Journal of Thermal Sciences 64 (2013) 240250. ##[15] D.A.Nield, A.V.Kuznetsov: Thermal instability in a porous medium layer saturated by a nanofluid, International Journal of Heat and Mass Transfer 52 (2009) 5796–5801. ##[16] W.A. Khan, I. Pop: Boundarylayer flow of a nanofluid past a stretching sheet, International Journal of Heat and Mass Transfer 53 (2010) 2477–2483. ##[17] M.Sheikholeslami, M.GorjiBandpy, D.D.Ganji, S.Soleimani: Thermal management for free convection of nanofluid using two phase model, Journal of Molecular Liquids 194(2014) 179–87. ##[18] M .Mahmood, S. Asghar, M.A.Hossain: Squeezed flow and heat transfer over a porous surface for viscous fluid, Heat Mass Transfer 44 (2007) 165–173. ##[19] G.Domairry, A .Aziz: Approximate analysis of MHD squeeze flow between two parallel disks with suction or injection by homotopy perturbation method, Hindawi Publishing Corporation Mathematical Problems in Engineering (2009) (Article ID 603916, 19 pages) http://dx. doi.org /10.1155/2009/603916. ##[20] M.Mustafa, T.Hayat, S.Obaidat: On heat and mass transfer in the unsteady squeezing flow between parallel plates, Meccanica 47(2012)1581–1589.##]