2016
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2
2
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Instabilities of Thin Viscous Liquid Film Flowing down a Uniformly Heated Inclined Plane
2
2
Instabilities of a thin viscous film flowing down a uniformly heated plane are investigated in this study. The heating generates a surface tension gradient that induces thermocapillary stresses on the free surface. Thus, the film is not only influenced by gravity and mean surface tension but also the thermocapillary force is acting on the free surface. Moreover, the heat transfer at the free surface plays a crucial role in the evolution of the film. The main objective of this study is to scrutinize the impact of Biot number Bi which describes heat transfer at the free surface on instability mechanism. Using the long wave expansion method, a generalized nonlinear evolution equation of Benney type, including the above mentioned effects, is derived for the development of the free surface. A normal mode approach and the method of multiple scales are used to obtain the linear and weakly nonlinear stability solution for the film flow. The linear stability analysis of the evolution equation shows that the Biot number plays a double role; for Bi < 1 it gives destabilizing effect but for Bi > 1 it produces stabilization. At Bi = 1, the instability is maximum. The weakly nonlinear study reveals that the impact of Marangoni number Mr is very strong on the bifurcation scenario even for its slight variation. This behaviour of the Biot number is the consequence of the fact that the interfacialtemperature is held close to the plane temperature for Bi > 1, thus weakening the Marangoni eﬀect. The weakly nonlinearstudy reveals that the impact of Marangoni number Mr is very strong on the bifurcation scenario even for its slight variation.
1

77
87


Anandamoy
Mukhopadhyay
Vivekananda Mahavidyalaya, Burdwan, W.B., India.
Vivekananda Mahavidyalaya, Burdwan, W.B.,
India
ananda235@email.com


Sanghasri
Mukhopadhyay
South Asian University, Akbar Bhavan, Chanakyapuri, New Delhi110021, India.
South Asian University, Akbar Bhavan, Chanakyapuri
India
sanghasri@mail.com


Asim
Mukhopadhyay
Vivekananda Mahavidyalaya, Burdwan,West Bengal, India
Vivekananda Mahavidyalaya, Burdwan,West Bengal,
India
as1m_m@yahoo.co.in
Thin ﬁlm
Marangoni instability
Biot number
[References ##[1] J. Marra, J. A. M. Huethorst, Physical principles of Marangoni drying, Langmuir, 7, 27482755 (1991). ##[2] S. B. G. M. O’ Brien, On Marangoni drying : nonlinear kinematic waves in a thin film, J.Fluid Mech., 254, 649670 (1993). ##[3] A. Oron, Nonlinear dynamics of thin evaporating liquid films subject to internal heat generation, In Fluid Dynamics at Interfaces (ed. W. shyy and R. Narayanan), Cambridge University Press, (1999). ##[4] A. A. Nepomnyashchy, M. G. Velarde, P. Colinet, Interfacial phenomena and convection, Chapman and Hall, (2002). ##[5] P. Colinet, J. C. Legros, M. G. Velarde, Nonlinear dynamics of surfacetension driven instabilities, Wiley VCH, (2001). ##[6] M. G. Velarde, R. Kh. Zeytounian, Interfacial Phenomena and the Marangoni effect, Springer, (2002). ##[7] D. A. Goussis, R. E. Kelly, Surface wave and thermocapillary instabilities in a liquid film flow, J. Fluid Mech., 223, 2545, (1991) and corrigendum J. Fluid. Mech. 226, 663 (1991). ##[8] S. W. Joo, S. H. Davis, S. G. Bankoff, Longwave instabilities of heated falling films: twodimensional theory of uniform layers, J. Fluid Mech., 230, 117146 (1991). ##[9] D. J. Benny, Long waves on liquid films, J. Math. Phys., 45, 150155 (1966). ##[10] W. Boos, A. Thess, Cascade of structures in longwavelength Marangoni instability, Phys.Fluids, [1], 14841494 (1999). ##[11] S. Kalliadasis, E. A. Demekhin, C. RuyerQuil, M. G. Velarde, Thermocapillary instability and wave formation on a film falling down a uniformly heated plane, J. Fluid Mech., 492, 303338 (2003). ##[12] P. M. J. Trevelyan, S. Kalliadasis, Wave dynamics on a thinliquid film falling down a heated wall, J. Engineering Mathematics, 50, 177208 (2004). ##[13] B. Scheid, C. RuyerQuil, U. Thiele, O. A. Kabov, J. Legros, P. Colinet, Validity domain of the Benney equation including Marangoni effect for closed and open flows, J. Fluid Mech., 527, 303335 (2005). ##[14] C. RuyerQuil, B. Scheid, S. Kalliadasis, M. G. Velarde, R. Kh. Zeytounian, Thermocapillary long waves in a liquid film flow, Part 1 Lowdimensional formulation, J. Fluid Mech., 538, 199222 (2005). ##[15] B. Scheid, C. RuyerQuil, S. Kalliadasis, M. G. Velarde, R. Kh. Zeytounian, Thermocapillary long waves in a liquid film flow, Part 2 Linear stability and nonlinear waves, J. Fluid Mech., 538, 223244 (2005). ##[16] S. Miladinova, S. Slavtchev, G. Lebon, J. C. Legros, Longwave instabilities of nonuniform heated falling films, J. Fluid Mech., 453, 153175 (2002). ##[17] S. Miladinova, D. Staykova, G. Lebon, B. Scheid, Effect of nonuniform wall heating on the threedimensional secondary instability of falling films, Acta Mechanica, 30, 113 (2002). ##[18] A. Mukhopadhyay, A. Mukhopadhyay, Nonlinear stability of viscous film flowing down an inclined plane with linear temperature variation, J. Phys. D: Appl. Phys., 40, 18 (2007). ##[19] Y. L. Yeo, V. R. Craster, O. K. Matar, Marangoni instability of a thin liquid film resting on a locally heated horizontal wall, Physical Review E, 67, 056315 114 (2003). ##[20] S. Saprykin, P. M. J. Trevelyan, R. J. Koopmans, S. Kalliadasis, Free surface thin film flows over uniformly heated topography, Physical Review E, 75, 026306 117 (2007). ##[21] H. C. Chang, Wave evolution on a falling film, Annu. Rev. Fluid Mech., 26, 103136 (1994). ##[22] . L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Springer, (1997). ##[23] B. S. Dandapat, A. Samanta, Bifurcation analysis of first and second order Benney equations for viscoelastic fluid flowing down a vertical plane, J. Phys. D: Appl. Phys., 41, 095501(9PP) (2008).##]
Numerical Study of Entropy Generation for Natural Convection in Cylindrical Cavities
2
2
In this paper, an enhanced computational code was developed using finitevolume method for solving the incompressible natural convection flow within the cylindrical cavities. Grids were generated by an easy method with a view to computer program providing. An explicit integration algorithm was applied to find the steady state condition. Also instead of the conventional algorithms of SIMPLE, SIMPLEM and SIMPLEC, an artificial compressibility technique is applied for coupling the continuity to the momentum equations. The entropy generation, which is a representation of the irreversibility and efficiency loss in engineering heat transfer processes, has been analyzed in detail. The discretization of the diffusion terms were very simplified using the enhanced scheme similar to the flux averaging in the convective term. Additionally an analysis of the entropy generation in a cylindrical enclosure was performed. In order to show the validation of this study, the code was reproduced to solve similar problem of cited paper. Finally, the solutions were extended for the new cases.
1

89
99


Abdollah
Rezvani
Enter affiliation
Enter affiliation
Iran(Islamic Republic of Iran)
rezvani_61@yahoo.com


Mohammad Sadegh
Valipour
Faculty of mechanical engineering
Faculty of mechanical engineering
Iran
msvalipour@semnan.ac.ir


Mojtaba
Biglari
Faculty of mechanical engineering
Faculty of mechanical engineering
Iran(Islamic Republic of Iran)
mbiglari@semnan.ac.ir
Artificial compressibility
Entropy
Explicit finitevolume method
natural convection
Nuselt number
[References ##[1] M. Salari, A. Rezvani, A. Mohammadtabar, and M. Mohammadtabar, "Numerical Study of Entropy Generation for Natural Convection in Rectangular Cavity with Circular Corners," Heat Transfer Engineering, vol. 36, pp. 186199, 8/ 2014. ##[2] D. K. Edwards and I. Catton, "Prediction of heat transfer by natural convection in closed cylinders heated from below," International Journal of Heat and Mass Transfer, vol. 12, pp. 2330, 1/ 1969. ##[3] A. Horibe, R. Shimoyama, N. Haruki, and A. Sanada, "Experimental study of flow and heat transfer characteristics of natural convection in an enclosure with horizontal parallel heated plates," International Journal of Heat and Mass Transfer, vol. 55, pp. 70727078, 11/ 2012. ##[4] B. M. Ziapour and R. Dehnavi, "A numerical study of the arcroof and the onesided roof enclosures based on the entropy generation minimization," Computers & Mathematics with Applications, vol. 64, pp. 16361648, 9/ 2012. ##[5] B. M. Ziapour and R. Dehnavi, "Finitevolume method for solving the entropy generation due to air natural convection in shaped enclosure with circular corners," Mathematical and Computer Modelling, vol. 54, pp. 12861299, 9/ 2011. ##[6] R. Dehnavi and A. Rezvani, "Numerical investigation of natural convection heat transfer of nanofluids in a Γ shaped cavity," Superlattices and Microstructures, vol. 52, pp. 312325, 8/ 2012. ##[7] I. Rashidi, O. Mahian, G. Lorenzini, C. Biserni, and S. Wongwises, "Natural convection of Al2O3/water nanofluid in a square cavity: Effects of heterogeneous heating," International Journal of Heat and Mass Transfer, vol. 74, pp. 391402, 7/ 2014. ##[8] R. D. C. Oliveski, M. H. Macagnan, and J. B. Copetti, "Entropy generation and natural convection in rectangular cavities," Applied Thermal Engineering, vol. 29, pp. 14171425, 6/ 2009. ##[9] I. Dagtekin, H. F. Oztop, and A. Bahloul, "Entropy generation for natural convection in Γshaped enclosures," International Communications in Heat and Mass Transfer, vol. 34, pp. 502510, 4/ 2007. ##[10] S. K. Pandit and A. Chattopadhyay, "Higher order compact computations of transient natural convection in a deep cavity with porous medium," International Journal of Heat and Mass Transfer, vol. 75, pp. 624636, 8/ 2014. ##[11] Y. Liu, C. Lei, and J. C. Patterson, "Natural convection in a differentially heated cavity with two horizontal adiabatic fins on the sidewalls," International Journal of Heat and Mass Transfer, vol. 72, pp. 2336, 5/ 2014. ##[12] É. Fontana, A. d. Silva, and V. C. Mariani, "Natural convection in a partially open square cavity with internal heat source: An analysis of the opening mass flow," International Journal of Heat and Mass Transfer, vol. 54, pp. 13691386, 3/ 2011. ##[13] J. V. C. Vargas and A. Bejan, "Thermodynamic optimization of the match between two streams with phase change," Energy, vol. 25, pp. 1533, 1/ 2000. ##[14] J. J. Flores, G. Alvarez, and J. P. Xaman, "Thermal performance of a cubic cavity with a solar control coating deposited to a vertical semitransparent wall," Solar Energy, vol. 82, pp. 588601, 7/ 2008. ##[15] L. Chen, Q. Xiao, Z. Xie, and F. Sun, "Constructal entransy dissipation rate minimization for treeshaped assembly of fins," International Journal of Heat and Mass Transfer, vol. 67, pp. 506513, 12/ 2013. ##[16] S. Chen and M. Krafczyk, "Entropy generation in turbulent natural convection due to internal heat generation," International Journal of Thermal Sciences, vol. 48, pp. 19781987, 10/ 2009. ##[17] A. Bradji, "A full analysis of a new second order finite volume approximation based on a loworder scheme using general admissible spatial meshes for the unsteady one dimensional heat equation," Journal of Mathematical Analysis and Applications, vol. 416, pp. 258288, 1/ 2014. ##[18] Z. Zhang, X. Zhang, and J. Yan, "Manifold method coupled velocity and pressure for Navier–Stokes equations and direct numerical solution of unsteady incompressible viscous flow," Computers & Fluids, vol. 39, pp. 13531365, 9/ 2010. ##[19] R. K. Shukla, M. Tatineni, and X. Zhong, "Very highorder compact finite difference schemes on nonuniform grids for incompressible Navier–Stokes equations," Journal of Computational Physics, vol. 224, pp. 10641094, 10/ 2007. ##[20] F. Bassi, A. Crivellini, D. A. Di Pietro, and S. Rebay, "An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible Navier–Stokes equations," Journal of Computational Physics, vol. 218, pp. 794815, 1/ 2006. ##[21] V. Esfahanian and P. Akbarzadeh, "The Jameson’s numerical method for solving the incompressible viscous and inviscid flows by means of artificial compressibility and preconditioning method," Applied Mathematics and Computation, vol. 206, pp. 651661, 12/ 2008. ##[22] A. J. Chorin, "A numerical method for solving incompressible viscous flow problems," Journal of Computational Physics, vol. 2, pp. 1226, 8/ 1967. ##[23] L. Ge and F. Sotiropoulos, "A numerical method for solving the 3D unsteady incompressible Navier–Stokes equations in curvilinear domains with complex immersed boundaries," Journal of Computational Physics, vol. 225, pp. 17821809, 10/ 2007. ##[24] K. T. Yoon, S. Y. Moon, S. A. Garcia, G. W. Heard, and T. J. Chung, "Flowfielddependent mixed explicitimplicit (FDMEI) methods for high and low speed and compressible and incompressible flows," Computer Methods in Applied Mechanics and Engineering, vol. 151, pp. 75104, 1/ 1998. ##[25] S. Yoon and D. Kwak, "Implicit methods for the NavierStokes equations," Computing Systems in Engineering, vol. 1, pp. 535547, 2/ 1990. ##[26] F. Xiao, R. Akoh, and S. Ii, "Unified formulation for compressible and incompressible flows by using multiintegrated moments II: Multidimensional version for compressible and incompressible flows," Journal of Computational Physics, vol. 213, pp. 3156, 3/ 2006. ##[27] S. Kaushik and S. G. Rubin, "Incompressible navierstokes solutions with a new primitive variable solver," Computers & Fluids, vol. 24, pp. 2740, 1/ 1995. ##[28] H. S. Tang and F. Sotiropoulos, "Fractional step artificial compressibility schemes for the unsteady incompressible Navier–Stokes equations," Computers & Fluids, vol. 36, pp. 974986, 6/ 2007. ##[29] A. Shah, L. Yuan, and S. Islam, "Numerical solution of unsteady Navier–Stokes equations on curvilinear meshes," Computers & Mathematics with Applications, vol. 63, pp. 15481556, 6/ 2012. ##[30] A. Shah, L. Yuan, and A. Khan, "Upwind compact finite difference scheme for timeaccurate solution of the incompressible Navier–Stokes equations," Applied Mathematics and Computation, vol. 215, pp. 32013213, 1/ 2010. ##[31] C. Kiris and D. Kwak, "Numerical solution of incompressible Navier–Stokes equations using a fractionalstep approach," Computers & Fluids, vol. 30, pp. 829851, 9/ 2001. ##[32] M. Breuer and D. Hänel, "A dual timestepping method for 3d, viscous, incompressible vortex flows," Computers & Fluids, vol. 22, pp. 467484, 7/ 1993. ##[33] J.g. Lin, Z.h. Xie, and J.t. Zhou, "Application of a threepoint explicit compact difference scheme to the incompressible navierstokes equations," Journal of Hydrodynamics, Ser. B, vol. 18, pp. 151156, 7/ 2006.##]
Flow field and heat transfer in a channel with a permeable wall filled with Al2O3Cu/water micropolar hybrid nanofluid, effects of chemical reaction and magnetic field
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2
In this study, flow field and heat transfer of Al2O3Cu/water micropolar hybrid nanofluid is investigated in a permeable channel using the least square method. The channel is encountered to chemical reaction, and a constant magnetic field is also applied. The bottom wall is hot and coolant fluid is injected into the channel from the top wall. The effects of different parameters such as the Reynolds number, the Hartmann number, microrotation factor and nanoparticles concentration on flow field and heat transfer are examined. The results show that with increasing the Hartmann number and the Reynolds number, the Nusselt and Sherwood numbers increase. Furthermore, when the hybrid nanofluid is applied rather than pure nanofluid, the heat transfer coefficient will increase significantly. It is also observed that in the case of generative chemical reaction, the fluid concentration is more than the case of destructive chemical reaction. Moreover, the Nusselt number and Sherwood number when the micropolar model is used, is less than when it is not considered.
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101
114


Mahdi
Mollamahdi
University of kashan
University of kashan
Iran(Islamic Republic of Iran)
mahdimollamahdi@gmail.com


Mahmoud
Abbaszadeh
university of kashan
university of kashan
Iran(Islamic Republic of Iran)
abbaszadeh.mahmoud@gmail.com


Ghanbar Ali
Sheikhzadeh
University of kashan
University of kashan
Iran(Islamic Republic of Iran)
sheikhz@kashanu.ac.ir
Micropolar hybrid nanofluid
magnetic field
Chemical reaction
Channel with a permeable wall
Least square method
[[1]. C. E. Mehmet, B. Elif, “Naturalconvection flow under a magnetic field in an inclined rectangular enclosure heated and cooled on adjacent walls,” Fluid Dynamics Research, 38(8), (2006) pp. 564–590. ##[2]. M. Pirmohammadi, M. Ghassemi, “Effect of magnetic field on convection heat transfer inside a tilted square enclosure,” International Communications in Heat and Mass Transfer, 36(7), (2009) pp. 776–780. ##[3]. H. Nemati, M. Farhadi, K. Sedighi, H.R. Ashorynejad, E. Fattahi, “Magnetic field effects on natural convection flow of nanofluid in a rectangular cavity using the Lattice Boltzmann model,” Scientia Iranica, 19(2), (2012) pp. 303–310. ##[4]. H. Ashorynejad, A. A. Mohamadb, M. Sheikholeslami, “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) pp. 240–250. ##[5]. M. M. Rashidi, N. Vishnu Ganesh, A.K. Abdul Hakeem, B. Ganga, “Buoyancy effect on MHD flow of nanofluid over a stretching sheet in the presence of thermal radiation,” Journal of Molecular Liquids, 198, (2014) pp. 234–238. ##[6]. M. S. Valipour, S. Rashidi, R. Masoodi, “Magnetohydrodynamics flow and heat transfer around a solid cylinder wrapped with a porous ring,” Journal of Heat Transfer, 136(6), (2014) pp. 062601. ##[7]. A. Aghaei, H. Khorasanizadeh, G. Sheikhzadeh, M. Abbaszadeh, “Numerical study of magnetic field on mixed convection and entropy generation of nanofluid in a trapezoidal enclosure,” Journal of Magnetism and Magnetic Materials, 403, (2016) pp. 133–145. ##[8]. S. Rashidi, J. Abolfazli Esfahani, M. S. Valipour, M. Bovand, I. Pop, “Magnetohydrodynamic effects on flow structures and heat transfer over two cylinders wrapped with a porous layer in side,” International Journal of Numerical Methods for Heat & Fluid Flow, 26(5), (2016). ##[9]. G.S. Seth, S.K. Ghosh, “Unsteady hydromagnetic flow in a rotating channel in the presence of inclined magnetic field,” International Journal of Engineering Science, 24(7), (1986) pp. 1183–1193. ##[10]. K. Jha, “Effects of applied magnetic field on transient freeconvective flow in a vertical channel,” Indian Journal of Pure and Applied Mathematics, 29(4), (1998) pp. 441445. ##[11]. O. D. Makinda, P. Y. Mhone, “Heat transfer to MHD oscillatory flow in a channel filled with porous medium,” Romanian Journal of physics, 50(9/10), (2005) pp. 931938. ##[12]. S.K. Parida, S. Panda, M. Acharya, “Magnetohydrodynamic (MHD) flow of a second grade fluid in a channel with porous wall,” Meccanica, 46(5), (2011) pp. 10931102. ##[13]. R. Nouri, D. D. Ganji, M. Hatami, “MHD nanofluid flow analysis in a semiporous channel by a combined series solution method,” Transport Phenomena in Nano and Micro Scales, 1(2), (2013) pp. 124137. ##[14]. M. Fakour, A. Vahabzadeh, D.D. Ganji, “Study of heat transfer and flow of nanofluid in permeable channel in the presence of magnetic field,” Propulsion and Power Research, 4(1), (2015) pp. 50–62. ##[15]. M. Bovand, S. Rashidi, M. Dehghan, A. J. Esfahani, M. S. Valipour, “Control of wake and vortex shedding behind a porous circular obstacle by exerting an external magnetic field,” Journal of Magnetism and Magnetic Materials, 385, (2015) pp. 198206. ##[16]. M. Bovand, S. Rashidi, A. J. Esfahani, R. Masoodi, “Control of wake destructive behavior for different bluff bodies in channel flow by magnetohydrodynamics,” The European Physical Journal Plus, 131(6), (2016) pp. 113. ##[17]. S.U.S. Choi, “Enhancing thermal conductivity of fluids with nanoparticles,” ASMEPublicationsFed, 231, (1995) pp. 99106. ##[18]. C.J. Ho, M.W. Chen, Z.W. Li, “Numerical simulation of natural convection of nanofluid in a square enclosure: Effects due to uncertainties of viscosity and thermal conductivity,” International Journal of Heat and Mass Transfer, 51(17), (2008) pp. 4506–4516. ##[19]. M. Sheikholeslami , M. M. Rashidi, D.D. Ganji, “Numerical investigation of magnetic nanofluid forced convective heat transfer in existence of variable magnetic field using two phase model,” Journal of Molecular Liquids, 212, (2015) pp. 117126. ##[20]. X. Wang, X. Xu, S. Choi, “Thermal conductivity of nanoparticle–fluid mixture,” Journal of Thermophysics and Heat Transfer, 13(4), (1999) pp. 474–480. ##[21]. S. Suresha, K.P. Venkitaraj, P. Selvakumar, M. Chandrasekar, “Synthesis of Al2O3–Cu/water hybrid nanofluids using two step method and its thermo physical properties,” Colloids and Surfaces A: Physicochemical and Engineering Aspects, 388(1), (2011) pp.41– 48. ##[22]. D. Madhesh, R. Parameshwaran, S. Kalaiselvam, “Experimental investigation on convective heat transfer and rheological characteristics of Cu–TiO2 hybrid nanofluids,” Experimental Thermal and Fluid Science, 52, (2014) pp. 104–115. ##[23]. M. Hemmat Esfe, S. Wongwises, A. Naderi, A. Asadi, M. R. Safaei, H. Rostamian, M. Dahari, A. Karimipour, “Thermal conductivity of Cu/TiO2–water/EG hybrid nanofluid: Experimental data and modeling using artificial neural network and correlation,” International Communications in Heat and Mass Transfer, 66, (2015) pp. 100–104. ##[24]. C.J. Ho, J.B. Huang, P.S. Tsai, Y.M. Yang, “Preparation and properties of hybrid waterbased suspension of Al2O3 nanoparticles and MEPCM particles as functional forced convection fluid,” International Communications in Heat and Mass Transfer, 37(5), (2010) pp. 490–494. ##[25]. S. M. Abbasia, A. Rashidib, A. Nematia, K. Arzania, “The effect of functionalisation method on the stability and the thermal conductivity of nanofluid hybrids of carbon nanotubes/gamma alumina,” Ceramics International, 39(4), (2013) pp. 3885–3891. ##[26]. H. Balla, Sh. Abdullah, W. MohdFaizal, R. Zulkifli, K. Sopian, “Numerical Study of the Enhancement of Heat Transfer for Hybrid CuOCu Nanofluids Flowing in a Circular Pipe,” Journal of Oleo Science, 62(7), (2013) pp. 533539. ##[27]. B. Takabi, S. Salehi, “Augmentation of the Heat Transfer Performance of a Sinusoidal Corrugated Enclosure by Employing Hybrid Nanofluid,” Advances in Mechanical Engineering, 6, (2014) pp. 147059. ##[28]. A. C. Eringen, “Theory of micropolar fluids,” Journal of Mathematics and Mechanics, 16, (1966) pp. 116. ##[29]. M. Sheikholeslami, M. Hatami, D.D. Ganji, “Micropolar fluid flow and heat transfer in a permeable channel using analytical method,” Journal of Molecular Liquids, 194, (2014) pp. 3036. ##[30]. S. Mosayebidorcheh, “Analytical investigation of the micropolar flow through a porous channel with changing walls,” Journal of Molecular Liquids, 196, (2014) pp. 113–119. ##[31]. G. C. Bourantas, V.C. Loukopoulos, “MHD naturalconvection flow in an inclined square enclosure filled with a micropolarnanofluid,” International Journal of Heat and Mass Transfer, 79, (2014) pp. 930–944. ##[32]. M. Fakour, A. Vahabzadeh, D.D. Ganji, M. Hatami, “Analytical study of micropolar fluid flow and heat transfer in a channel with permeable walls,” Journal of Molecular Liquids, 204, (2015) pp. 198–204. ##[33]. G.C. Bourantas, V.C. Loukopoulos, “Modeling the natural convective flow of micropolar nanofluids,” International Journal of Heat and Mass Transfer, 68, (2014) pp. 35–41. ##[34]. S.T. Hussain, S. Nadeem, R. U. Haq, “Modelbased analysis of micropolar nanofluid flow over a stretching surface,” The European Physical Journal Plus, 129(8), (2014) pp. 161171. ##[35]. M. Turkyilmazoglu, “A note on micropolar fluid flow and heat transfer over a porous shrinking sheet,” International Journal of Heat and Mass Transfer, 72, (2014) pp. 388–391. ##[36]. E. A. Sameh, M. A. Mansour, A. K. Hussein, S. Sivasankaran, “Mixed convection from a discrete heat source in enclosures with two adjacent moving walls and filled with micropolar nanofluids,” Engineering Science and Technology, an International Journal, 19(1), (2016) pp. 364376. ##[37]. K. J. Sofen, K. M. Laxman, K. M. Swarup, A. J. Chamkha, “Transient buoyancyopposed double diffusive convection of micropolar fluids in a square enclosure,” International Journal of Heat and Mass Transfer, 81, (2015) pp. 681–694. ##[38]. Borrelli, G. Giantesio, M.C. Patria, “Magnetoconvection of a micropolar fluid in a vertical channel,” International Journal of Heat and Mass Transfer, 80, (2015) pp. 614–625. ##[39]. M. Sheikholeslami, M. Hatami, D.D. Ganji, “Analytical investigation of MHD nanofluid flow in a semi porous channel,” Powder Technology, 246, (2013) pp. 327336. ##[40]. M. Hatami, M. Sheikholeslami, D.D. Ganji, “Nanofluid flow and heat transfer in an asymmetric porous channel with expanding or contracting wall,” Journal of Molecular Liquids, 195, (2014) pp. 230239. ##[41]. M. Hatami, M. Sheikholeslami, D.D. Ganji, “Laminar flow and heat transfer of nanofluid between contracting and rotating disks by least square method,” Powder Techonology, 253, (2014) pp. 769779. ##[42]. A. Arefmanesh, M. Mahmoodi, “Effects of uncertainties of viscosity models for Al2O3water nanofluid on mixed convection numerical simulations,” International Journal of Thermal Sciences, 50, (2011) pp. 17061719. ##[43]. S.K. Das, S.U.S. Choi, W. Yu, T. Predeep, “Nanofluids Science and Technology,” John Wiley & sons, (2008). ##[44]. Y. Xuan, W. Roetzel, “Conceptions for heat transfer correlations of nanofluids,” International Journal of Heat and Mass Transfer, 43, (2000) pp. 37013707. ##[45]. AE. Jery, N. Hidouri, M. Magherbi, AB. Brahim, “Effect of an external oriented magnetic field on entropy generation in natural convection,” Entropy, 12(6), (2010) pp. 1391–1417. ##[46]. C. Zhang, L. Zheng, X. Zhang, G. Chen, “MHD flow and radiation heat transfer of nanofluids in porous media with variable surface heat flux and chemical reaction,” Applied Mathematical Modelling, 19(1), (2015) pp. 165181. ##[47]. J.C. Maxwell, “A Treatise on Electricity and magnetism,” second ed., clarendon press, (1881). ##H.C. Brinkman, “The viscosity of concentrated suspensions and solutions,” The Journal of Chemical al Physic, 20(4), (1952) 571571##]
A Comparative Solution of Natural Convection in an Open Cavity using Different Boundary Conditions via Lattice Boltzmann Method
2
2
A Lattice Boltzmann method is applied to demonstrate the comparison results of simulating natural convection in an open end cavity using different hydrodynamic and thermal boundary conditions. The Prandtl number in the present simulation is 0.71, Rayleigh numbers are 104,105 and 106 and viscosities are selected 0.02 and 0.05. OnGrid bounceback method with firstorder accuracy and nonslip method with secondorder accuracy are employed for implementation of hydrodynamic boundary conditions. Moreover, two different thermal boundary conditions (with first and second order of accuracy) are also presented for thermal modelling. The results showed that first and second order boundary conditions (thermal/hydrodynamic) are the same for a twodimensional, single phase, convective heat transfer problem including geometry with straight walls. The obtained results for different hydrodynamic and thermal boundary conditions are useful for the researchers in the field of lattice Boltzmann method in order to implement accurate condition on the boundaries, in different physics.
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115
129


MohsenShahrood
Nazari
Mechanical Engineering Dept., Shahrood university
Mechanical Engineering Dept., Shahrood university
Iran(Islamic Republic of Iran)
nazari_me@yahoo.com


MH
Kayhani
University of Shahrood
University of Shahrood
Iran
h_kayhani@shahroodut.ac.ir
Lattice Boltzmann method
open cavity
hydrodynamic/Thermal boundary conditions
order of accuracy
[[1]. Y. L. Chan, C. L. Tien, A numerical study of twodimensional natural convection in square open cavities, Int. J. Heat Mass Transfer, vol. 28, No.3, pp. 65–80, 1985. ##[2]. E. Bilgen, A. Muftuoglu, Natural convection in an open square cavity with slots, Int. Communications. Heat and Mass Transfer, vol. 35, pp. 896–900, 2008. ##[3]. T. Inamuro, M. Yoshino, F. Ogino, A nonslip boundary condition for lattice Boltzmann simulations, Phisics of Fliuds, vol. 7, pp. 2928–2930, 1995. ##[4]. R. S. Maier, R. S. Bernard, D. W. Grunau, Boundary conditions for the lattice Boltzmann method, Phys. Fluids, vol. 8, pp. 1788–1801, 1996. ##[5]. Q. Zou, X. He, On pressure and velocity boundary conditions for the lattice Boltzmann BGK model, Phys. Fluids, vol. 9, pp. 1591–1598, 1997. ##[6]. C. Chang, C. H. Liu, C. A. Lin, Boundary conditions for lattice Boltzmann simulations with complex geometry flows, Computers & Mathematics with Applications, vol. 58, Issue 5, pp. 940–949, 2009. ## [7]. I. Ginzburg, Generic boundary conditions for lattice Boltzmann models and their application to advection and anisotropic dispersion equations, Advances in Water Resources, vol. 28, pp. 1196–1216, 2005. ##[8]. V. Sofonea, R. F. Sekerka, Boundary conditions for the upwind finite difference Lattice ##Boltzmann model: Evidence of slip velocity in microchannel flow, J. Computational Physics, vol. 207, pp. 639–659, 2005. ##[9]. P. A. Skordos, Initial and boundary conditions for the lattice Boltzmann method, Phys, Rev. E, vol. 48, pp. 4823–4842, 1993. ##[10]. 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, Int. J. Thermal Sci, vol. 43, pp. 575–586, 2004. ##[11]. M. A. Gallivan, D. R. Noble, J. G. Georgiadis, R. O. Buckius, An evaluation of bounceback boundary condition for lattice Boltzmann simulations, International journal for numerical methods in fluids, vol.25, pp. 249–263, 1997. ##[12]. ZL. Guo, CG. Zheng, BC. Shi , Nonequilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method, Chinese Phys. vol.4, No.11, pp. 366–374, 2002. ##[13]. D. P. Zeigler, Boundary condition for lattice Boltzmann simulations, Journal of statistical physics, vol.71, Nos.5/6, pp. 1171–1177, 1993. ##[14]. D. R. Noble, S. Chen, J. G. Georgiadis, R. O. Buckius, A consistent hydrodynamics boundary condition for the lattice Boltzmann method, Phys. Fluids, vol. 7, No. 1, pp. 203–209, 1995. ##[15]. S. Chen, D. Martnez, R. Mei, On boundary conditions in lattice Boltzmann methods, Phys. Fluids, vol. 8, No. 1, pp. 2527–2536, 1996. ##[16]. G. H. Tang, W. Q. Tao, Y. L. He, Thermal boundary condition for the thermal lattice Boltzmann equation, Physical Review, E, Vol. 72, pp. 016703.1–016703.6, 2005. ##[171]. H. Huang, T. S. Lee, C. Shu, Thermal curved boundary treatment for the thermal lattice Boltzmann equation, International Journal of Modern Physics C, vol. 17, No.5, pp. 631–643, 2006. ##[18]. L. Zheng, Z. L. Guo, B. C. Shi, Discrete effects on thermal boundary conditions for the thermal lattice Boltzmann method in simulating micro scale gas flows, Europhysics Letters, vol. 82 ,No. 4, pp. 44002, 2008. ##[19]. M. Corcione, Effects of the thermal boundary conditions at the sidewalls upon natural convection in rectangular enclosures heated from below and cooled from above, International. J. Thermal Sci, vol. 42, No. 2, pp. 199–208, 2003. ##[201]. A. D’Orazio, S. Succi, Boundary conditions for thermal lattice Boltzmann simulations, Lecture Notes Comput. Sci, vol. 2657, pp. 977–986, 2003. ##[21]. A. D'Orazio, S. Succi, Simulating twodimensional thermal channel flows by means of a lattice Boltzmann method with new boundary conditions, Future Generation Computer Systems, vol. 20, pp. 935–944, 2004. ##[22]. L. S. Kuo, P. H. Chen, Numerical implementation of thermal boundary conditions in the lattice Boltzmann method, Int. J. Heat and Mass Transfer, vol. 52, pp. 529–532, 2009. ##[23]. A. A. Mohamad, R. Bennacer, M. ElGanaoui, Lattice Boltzmann simulation of natural convection in an open ended cavity, Int. J. Thermal Sci, vol.48, pp. 1870–1875, 2009. ##[24]. H. N. Dixit, V. Babu, Simulation of high Rayleigh number natural convection in a square cavity using the lattice Boltzmann method, Int. J. Heat and Mass Transfer, vol. 49, pp. 727–739, 2006. ##[25]. A. A. Mohamad, R. Bennacer, M. ElGanaoui, Double dispersion, natural convection in an open end cavity simulation via Lattice Boltzmann Method. Int. J. Thermal Sci, vol. 49, pp. 1944–1953, 2010. ##[26]. M. C. Sukop, D. T. Thorne.Jr, Lattice Boltzmann Modeling, SpringerVerlag, Berlin, 2006. ##[27]. S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, Clarendon Press, Oxford, London, 2001. ##[28]. J. Wang, M. Wang, Z. Li, A lattice Boltzmann algorithm for fluid–solid conjugate heat transfer, Int. J. Thermal Sciences, vol. 46, pp. 228–234, 2007. ##[29]. A. A. Mohamad, A. Kuzmin, A critical evaluation of force term in lattice Boltzmann method, natural convection problem. Int. J. Heat and Mass Transfer, vol. 53, pp. 990–996, 2010. ##[30]. A. A. Mohamad, Natural convection in open cavities and slots, Numer. Heat Transfer, vol. 27, pp. 705–716, 1995. ##[31]. J. F. Hinojosa, R. E. Cabanillas, G. Alvarez, C. E. Estrada, Nusslet number for the natural convection and surface thermal radiation in a square tilted open cavity, Int. Comm. Heat Mass Transfer, vol. 32, pp. 1184–1192, 2005.##]
Development of a phase change model for volumeoffluid method in OpenFOAM
2
2
In this present study, volume of fluid method in OpenFOAM open source CFD package will be extended to consider phase change phenomena with modified model due to condensation and boiling processes. This model is suitable for the case in which both unsaturated phase and saturated phase are present and for beginning boiling and condensation process needn't initial interface. Both phases (liquidvapor) are incompressible and immiscible. Interface between two phases is tracked with color function volume of fluid (CFVOF) method. Surface Tension is taken into consideration by Continuous Surface Force (CSF) model. PressureVelocity coupling will be solved with PISO algorithm in the collocated grid. The accuracy of this phasechange model is verified by two evaporation problems (a onedimensional Stefan problem and a twodimensional film boiling problem) and two condensation problem (a onedimensional Stefan problem and Filmwise condensation). The simulation results of this model show good agreement with the classical analytical or numerical results, proving its accuracy and feasibility.
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131
143


Mohammad
Bahreini
Faculty of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Iran
Faculty of Mechanical Engineering, Babol
Iran(Islamic Republic of Iran)
m.bahreini1990@gmail.com


Abbas
Ramiar
Faculty of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Iran
Faculty of Mechanical Engineering, Babol
Iran(Islamic Republic of Iran)
aramiar@nit.ac.ir


Ali Akbar
Ranjbar
Faculty of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Iran
Faculty of Mechanical Engineering, Babol
Iran(Islamic Republic of Iran)
ranjbar@nit.ac.ir
Phase change model
Volumeoffluid
Boiling
Condensation
OpenFOAM
[References ##[1] D.L. Youngs, Timedependent multimaterial flow with large fluid distortion, Numerical methods for fluid dynamics, 24 (1982) 273285. ##[2] C.W. Hirt, B.D. Nichols, Volume of fluid (VOF) method for the dynamics of free boundaries, Journal of Computational Physics, 39 (1981) 201225. ##[3] M. Huang, L. Wu, B. Chen, A piecewise linear interfacecapturing volumeoffluid method based on unstructured grids, Numerical Heat Transfer, Part B: Fundamentals, 61 (2012) 412437. ##[4] S. Osher, R.P. Fedkiw, Level set methods: an overview and some recent results, Journal of Computational Physics, 169 (2001) 463502. ##[5] P. Wang, H. Sun, P.Y. Wong, H. Fukuda, T. Ando, Modeling of dropletbased processing for the production of highperformance particulate materials using the level set method, Numerical Heat Transfer, Part A: Applications, 61 (2012) 401416. ##[6] M. Sussman, P. Smereka, S. Osher, A level set approach for computing solutions to incompressible twophase flow, Journal of Computational physics, 114 (1994) 146159. ##[7] S.W. Welch, J. Wilson, A volume of fluid based method for fluid flows with phase change, Journal of computational physics, 160 (2000) 662682. ##[8] M.W. Akhtar, S.J. Kleis, Boiling flow simulations on adaptive octree grids, International Journal of Multiphase Flow, 53 (2013) 8899. ##[9] S. Hardt, F. Wondra, Evaporation model for interfacial flows based on a continuumfield representation of the source terms, Journal of Computational Physics, 227 (2008) 58715895. ##[10] R.W. Schrage, A theoretical study of interphase mass transfer, Columbia University Press, 1953. ##[11] D. Juric, G. Tryggvason, Computations of boiling flows, International Journal of Multiphase Flow, 24 (1998) 387410. ##[12] B.A. Nichita, J.R. Thome, A level set method and a heat transfer model implemented into FLUENT for modeling of microscale two phase flows, in: AVT178 Specialists" Meeting on System Level Thermal Management for Enhanced Platform Efficiency, 2010. ##[13] W. Mao, Numerical Simulation of Vapor–liquid Phase Change Heat Transfer and Micromixing in Microfluidic Systems, Master's thesis, GuangZhou Institute of Energy Conversion Chinese Academy of Sciences, China, (2009). ##[14] D. Guo, D. Sun, Z. Li, W. Tao, Phase change heat transfer simulation for boiling bubbles arising from a vapor film by the VOSET method, Numerical Heat Transfer, Part A: Applications, 59 (2011) 857881. ##[15] H.G. Weller, G. Tabor, H. Jasak, C. Fureby, A tensorial approach to computational continuum mechanics using objectoriented techniques, Computers in physics, 12 (1998) 620631. ##[16] J.U. Brackbill, D.B. Kothe, C. Zemach, A Continuum Method for Modeling Surface Tension, J. Comput. Phys., 100 (1992) 335354. ##[17] J. Klostermann, K. Schaake, R. Schwarze, Numerical simulation of a single rising bubble by VOF with surface compression, International Journal for Numerical Methods in Fluids, 71 (2013) 960982. ##[18] A. Albadawi, D. Donoghue, A. Robinson, D. Murray, Y. Delauré, Influence of surface tension implementation in volume of fluid and coupled volume of fluid with level set methods for bubble growth and detachment, International Journal of Multiphase Flow, 53 (2013) 1128. ##[19] N. Samkhaniani, A. Gharehbaghi, Z. Ahmadi, Numerical simulation of reaction injection molding with polyurethane foam, Journal of Cellular Plastics, 49 (2013) 405421. ##[20] W.H. Lee, Pressure iteration scheme for twophase flow modeling, IN" MULTIPHASE TRANSPORT: FUNDAMENTALS, REACTOR SAFETY, APPLICATIONS". (1980) 407432. ##[21] M. Bahreini, A. Ramiar, A.A. Ranjbar, Numerical simulation of bubble behavior in subcooled flow boiling under velocity and temperature gradient, Nuclear Engineering and Design, 293 (2015) 238248. ##[22] M. Bahreini, A. Ramiar, A. Ranjbar, Numerical simulation of subcooled flow boiling under conjugate heat transfer and microgravity condition in a vertical mini channel, Applied Thermal Engineering, 113 (2017) 170185. ##[23] H.G. Weller, G. TaboraI, H. Jasak, C. Fureby, A tensorial approach to computational continuum mechanics using objectoriented techniques, Comput. Phys., 12 (1998) 620632. ##[24] C. Rhie, W. Chow, Numerical study of the turbulent flow past an airfoil with trailing edge separation, AIAA journal, 21 (1983) 15251532. ##[25] H. Jasak, Error analysis and estimation for the finite volume method with applications to fluid flows, in: Imperial College, Vol. PhD Thesis, University of London, 1996. ##[26] H. Weller, A new approach to VOFbased interface capturing methods for incompressible and compressible flow, OpenCFD Ltd., Report TR/HGW/04, (2008). ##[27] B. Van Leer, Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a secondorder scheme, Journal of Computational Physics, 14 (1974) 361370. ##[28] M. Darwish, F. Moukalled, TVD schemes for unstructured grids, International Journal of Heat and Mass Transfer, 46 (2003) 599611. ##[29] B. Shu, F. Dammel, P. Stephan, Phase change model for twophase fluid flow based on the volume of fluid method, ICHMT DIGITAL LIBRARY ONLINE, 13 (2008). ##[30] G. Son, V. Dhir, Numerical simulation of film boiling near critical pressures with a level set method, Journal of Heat Transfer, 120 (1998) 183192. ##[31] P. Berenson, Filmboiling heat transfer from a horizontal surface, Journal of Heat Transfer, 83 (1961) 351356. ##[32] A. Gavrish, V.G. Rifert, Investigation of transition mode phenomenon from dropwise to filmwise steam condensation, Heat Transfer Research, 30 (1999). ##[33] V. Semenov, N. Nikitin, Condensation heat transfer on noncircular pipes in stationary vapor, Heat Transfer Research, 39 (2008). ##[34] Z. Liu, B. Sunden, J. Yuan, VOF modeling and analysis of filmwise condensation between vertical parallel plates, Heat Transfer Research, 43 (2012). ##[35] S.M. Ghiaasiaan, Convective heat and mass transfer, Cambridge University Press, 2011. ##[36] Z. Liu, B. Sunden, J. Yuan, Numerical Modelling of Condensation of Multiple Bubbles in Subcooled Flow Boiling With VOF Method, in: ASME 2013 Heat Transfer Summer Conference collocated with the ASME 2013 7th International Conference on Energy Sustainability and the ASME 2013 11th International Conference on Fuel Cell Science, Engineering and Technology, American Society of Mechanical Engineers, 2013, pp. V001T003A014V001T003A014. ##[37] C. Aghanajafi, K. Hesampour, Heat transfer analysis of a condensate flow by VOF method, Journal of Fusion Energy, 25 (2006) 219223. ##[38] R. Marek, J. Straub, Analysis of the evaporation coefficient and the condensation coefficient of water, International Journal of Heat and Mass Transfer, 44 (2001) 3953.##]
A study of a Stefan problem governed with space–time fractional derivatives
2
2
This paper presents a fractional mathematical model of a onedimensional phasechange problem (Stefan problem) with a variable latentheat (a power function of position). This model includes space–time fractional derivatives in the Caputo sense and timedependent surfaceheat flux. An approximate solution of this model is obtained by using the optimal homotopy asymptotic method to find the solutions of temperature distribution in the domain 0 ≤x≤s(t) and interface’s tracking or location. The results thus obtained are compared with existing exact solutions for the case of the integer order derivative at some particular values of the governing parameters. The dependency of movement of the interface on certain parameters is also studied.
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145
151


Rajeev
.
Indian Institute of Technology(BHU)
Indian Institute of Technology(BHU)
India
rajeev.apm@itbhu.ac.in


M.
Kushwaha
IIT (BHU), Varanasi
IIT (BHU), Varanasi
India
kushwahamohansingh76@gmail.com


Abhishek
Singh
IIT (BHU), VARANASI
IIT (BHU), VARANASI
India
aksingh.iitbhu@gmail.com
Optimal homotopy asymptotic method
Stefan problem
moving interface
fractional derivatives
[[1] A.S. Chaves, A fractional diffusion equation to describe Lévy flights. Phys. Lett. A, 239, 13–16 (1998). ##[2] D.A. Benson, S.W. Wheatcraft, M. M. Meerschaert, The fractionalorder governing equation of Lévy motion, Water Resources Res., 36, 1413–1423 (2000). ##[3] Y. Aoki, M. Sen, S. Paolucci, Approximation of transient temperatures in complex geometries using fractional derivatives, Heat Transfer, 44, 771–777 (2008). ##[4] H. Jiang, F. Liu, I. Turner, K. Burrage, Analytical solutions for the multiterm timespace CaputoRiesz fractional advectiondiffusion equations on a finite domain, Journal of Mathematical Analysis and Applications, 389, 11171127 (2012). ##[5] Zˇ. Tomovskia, T. Sandev, R. Metzler, J. Dubbeldam, Generalized space–time fractional diffusion equation with composite fractional time derivative, Physica A, 391, 2527–42 (2012). ##[6] X.C. Li, M. Y. Xu, S.W. Wang, Analytical solutions to the moving boundary problems with time–space fractional derivatives in drug release devices, J Phys A: Math. Theor., 40, 12131–12141 (2007). ##[7] X.C. Li, M.Y. Xu, S.W. Wang, Scaleinvariant solutions to partial differential equations of fractional order with a moving boundary condition, J Phys A: Math. Theor., 41, 155202 (2008). ##[8] J. Liu, M. Xu, Some exact solutions to Stefan problems with fractional differential equations, J. Math Anal. Appl., 351, 536542 (2009). ##[9] C. J. Vogl, M. J. Miksis, S. H. Davis, Moving boundary problems governed by anomalous diffusion.Proc. R. Soc. A,468, 33483369 (2012). ##[10] S. Das, R. Kumar, P.K. Gupta, Analytical approximate solution of space–time fractional diffusion equation with a moving boundary condition, Z. Naturforsch. A 66 a, 281–288 (2011). ##[11] V.R. Voller, An exact solution of a limit case Stefan problem governed by a fractional diffusion equation, Int. J. Heat Mass Transfer., 53, 56225625 (2010). ##[12] Y. Zhou, Y. Wang, W. Bu, Exact solution for a Stefan problem with latent heat a power function of position, International Journal of Heat and Mass Transfer, 69, 451–454 (2014). ##[13] L. Xicheng, M. Xu, X. Jiang, Homotopy perturbation method to timefractional diffusion equation with a moving boundary condition, Applied Mathematics and Computation, 208, 434–439 (2009). ##[14] Rajeev, M.S. Kushwaha, Homotopy perturbation method for a limit case Stefan problem governed by fractional diffusion equation, Appl. Math Model, 37, 35893599 (2013). ##[15] S. Das, Rajeev, Solution of fractional diffusion equation with a moving boundary condition by variational iteration method and Adomian decomposition method, Z Naturforsch . 65a, 793799 (2010). ##[16] R. Grzymkowski, D. Słota, Onephase inverse Stefan problem solved by Adomian decomposition method, Comput Math Appl., 51, 3340 (2006). ##[17] Rajeev, M. S. Kushwaha, A. Kumar, An approximate solution to a moving boundary problem with space–time fractional derivative in fluviodeltaic sedimentation process, Ain Shams Engineering Journal, 4, 889–895 (2013). ##[18] V. Marinca, N. Herisanu, Application of homotopy Asymptotic method for solving nonlinear equations arising in heat transfer, Int. Comm. Heat Mass Transfer, 35 , 710–715 (2008). ##[19] N. Herisanu, V. Marinca, Accurate analytical solutions to oscillators with discontinuities and fractionalpower restoring force by means of the optimal homotopy asymptotic method. Comput. Math. Appl. 60, 1607–1615 (2010). ##[20] V. Marinca, N. Herisanu, Determination of periodic solutions for the motion of a particle on a rotating parabola by means of the optimal homotopy asymptotic method, J. Sound Vib. 329, 1450–1459 (2010). ##[21] S. Iqbal, M. Idrees, A.M. Siddiqui, A.R. Ansari, Some solutions of the linear and nonlinear Klein–Gordon equations using the optimal homotopy asymptotic method, Appl. Math. Comput., 216, 2898–2909 (2010). ##[22] S. Iqbal, A. Javed, Application of optimal homotopy asymptotic method for the analytic solution of singular Lane–Emden type equation, Appl. Math. Comput., 217, 7753–7761 (2011). ##[23] M. S. Hashmi, N. Khan, S. Iqbal, Optimal homotopy asymptotic method for solving nonlinear Fredholm integral equations of second kind, Applied Mathematics and Computation, 218, 10982–10989 (2012). ##[24] M. Ghoreishi , A.I.B. Md. Ismail , A.K. Alomari ,A.S.Bataineh, The comparison between Homotopy Analysis Method and Optimal Homotopy Asymptotic Method for nonlinear agestructured population Models, Commun Nonlinear Sci. Numer. Simulat., 17, 1163–1177 (2012). ##[25] S. Dinarvand, R. Hosseini, Optimal homotopy asymptotic method for convective–radiative cooling of a lumped system, and convective straight fin with temperaturedependent thermal conductivity, Afrika Matematika., 24, 103116 (2013).##]
Analytical and Numerical Studies on Hydromagnetic Flow of Boungiorno Model Nanofluid over a Vertical Plate
2
2
MHD boundary layer flow of two phase model nanofluid over a vertical plate is investigated both analytically and numerically. A system of governing nonlinear partial differential equations is converted into a set of nonlinear ordinary differential equations by suitable similarity transformations and then solved analytically using homotopy analysis method and numerically by the fourth order RungeKutta method along with shooting iteration technique. The effects of magnetic parameter, Prandtl number, Lewis number, buoyancyratio parameter, Brownian motion parameter and thermophoresis parameter on the velocity profile, temperature profile and concentration profile of the nanofluid are discussed graphically. The values of reduced local Nusselt number and reduced local sherwood number are tabulated and discussed. It is noted that the Brownian motion and thermophoresis parameters enhance the velocity distribution and the temperature distribution, but it suppress the concentration distribution. Furthermore, comparisons have been made with bench mark solutions for a special case and obtained a very good agreement..
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153
164


A.K. Abdul
Hakeem
Assistant Professor
Department of Mathematics
Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore, Tamil Nadu
Assistant Professor
Department of Mathematics
Sr
India
abdulhakeem6@gmail.com


B.
Ganga
Department of Mathematics,Providence College for Women, Coonoor  643 104, INDIA
Department of Mathematics,Providence College
India
gangabhose@gmail.com


S. Mohamed
Yusuff Ansari
Department of Mathematics, Jamal Mohamed College, Trichy  6420 020, INDIA
Department of Mathematics, Jamal Mohamed
India
yusuffsaitu@yahoo.in


N.Vishnu
Ganesh
of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts & Science, Coimbatore  641 020, INDIA.
of Mathematics, Sri Ramakrishna Mission Vidyalaya
India
nvishnuganeshmath@gmail.com
Homotopy Analysis Method
MHD
Nanofluid
RungeKutta method
Vertical plate
[[1] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, {in: D A. Siginer, H.P. Wang (Eds.), Developments and Applications of NonNewtonian Flows, ASME FED, 231/MD (66)}, 66, 99105, (1995). ##[2] J. Boungiorno, et al., A benchmark study of thermalconductivity of nanofluids,J. Appl. Phys., 106paper 094312,(2009). [3] J. Buongiorno, Convective transport in nanofluids, J. Heat Transfer, 128 (2006) 240250. ##[4] P. Rana,R. Bhargava , Flow and heat transfer of a nanofluid over a nonlinearly stretching sheet: a numerical study. Commun. Nonlinear Sci. Numer. Simulat., 17, 212226,(2012). ## [5] M.A.A Hamad, Analytical solution of natural convection flow of a nanofluid over a linearly stretching sheet in the presence of magnetic field, Int. Comm. Heat Mass Transfer., 38, 487492, (2011). ##[6] A.J. Chamkha and A.M. Aly, MHD Free Convection Flow of a Nanofluid past a Vertical Plate in the Presence of Heat Generation or Absorption Effects. Chem. Eng, Commun., 198, 425441, (2011). ##[7] A.J. Chamkha, A.M. Rashad, and E. AlMeshaiei, “Melting Effect on Unsteady Hydromagnetic Flow of a Nanofluid Past a Stretching Sheet.” Int. J. of Chem. Reactor Eng., 9:A113, (2011). ##[8] A.J. Chamkha, A.M. Aly, H. AlMudhaf, Laminar MHD Mixed Convection Flow of a Nanofluid along a Stretching Permeable Surface in the Presence of Heat Generation or Absorption Effects, International Journal of Microscale and Nanoscale Thermal and Fluid Transport Phenomena, 2, 5170, (2011). ##[9] R.S.R. Gorla and A.J. Chamkha, Natural Convective Boundary Layer Flow over a Horizontal Plate Embedded in a Porous Medium Saturated with a NonNewtonian Nanofluid.” International Journal of Microscale and Nanoscale Thermal and Fluid Transport Phenomena, 2, 211 227, (2011). ##[10] R.S.R. Gorla, A.J. Chamkha and A. Rashad, Mixed Convective Boundary Layer Flow over a Vertical Wedge Embedded in a Porous Medium Saturated with a Nanofluid: Natural Convection Dominated Regime. Nanoscale Research Letters, 6 (207), 19, (2011). ##[11] N. Vishnu Ganesh, B. Ganga, A.K. Abdul Hakeem, Lie symmetry group analysis of magnetic field effects on free convective flow of a nanofluid over a semi infinite stretching sheet, J. Egyptian Math. Soc., 22, 304310,(2014). ##[12] N. Vishnu Ganesh, A. K. Abdul Hakeem , R. Jayaprakash , and B. Ganga, break Analytical and Numerical Studies on Hydromagnetic Flow of Water Based Metal Nanofuids Over a Stretching Sheet with Thermal Radiation Effect, J. Nanofluids , 3, 154161, (2014). ##[13] M. Govindaraju, N. Vishnu Ganesh, B. Ganga, A.K. Abdul Hakeem, Entropy generation analysis of magneto hydrodynamic flow of a nanofluid over a stretching sheet., J. Egyptian Math. Soc.,429434(2014). ##[14] M.M. Rashidi, N.Vishnu Ganesh , A.K. Abdul Hakeem and B. Ganga, Buoyancy Effect on MHD Flow of Nanofluid over a Stretching Sheet in the Presence of Thermal Radiation, J. Mol. liq., 234238, (2014). ##[15] A.K. Abdul Hakeem, N.Vishnu Ganesh, B.Ganga, Magnetic field effect on second order slip flow of nanofluid over a stretching/shrinking sheet with thermal radiation effect, J. Magn. Magn. Mater., 381, 243257 (2015). ##[16] A.V. Kuznetsov, D.A. Nield, Natural convective boundary layer flow of a nanofluid past a vertical plate. Int. J. Therm. Sci., 49, 243247, (2010). ##[17] A.V. Kuznetsov, D.A. Nield, Doublediffusive natural convective boundarylayer flow of a nanofluid past a vertical plate. Int. J. Therm. Sci., 50, 712717, (2011). ##[18] W.A. Khan, I. Pop, Boundarylayer flow of a nanofluid past a stretching sheet, Int. J. Heat Mass Transfer, 53,24772483, (2010). ##[19] W.A. Khan, A. Aziz, Natural convection flow of a nanofluid over a vertical plate with uniform surface heat flux, Int. J. Therm. Sci., 50(7), 12071214, (2011). ##[20] R.S.R. Gorla, A. Chamkha, Natural convective boundary layer flow over a horizontal plate embedded in a porous medium saturated with a nanofluid. J. 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