Startseite Numerical Solutions for Radiative Heat Transfer in Ferrofluid Flow due to a Rotating Disk: Tiwari and Das Model
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Numerical Solutions for Radiative Heat Transfer in Ferrofluid Flow due to a Rotating Disk: Tiwari and Das Model

  • M. Mustafa EMAIL logo , Junaid Ahmad Khan , T. Hayat und A. Alsaedi
Veröffentlicht/Copyright: 11. Januar 2018
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Abstract

In this paper, we explore the von-Kármán infinite disk problem for the situation where ferrofluid resides in the space above the rotating disk. Furthermore, flow field is influenced by axial magnetic field. In this study, we treat water as the base fluid which consists of homogeneous suspensions of Fe3O4 ferromagnetic particles. The main motivation here is to resolve heat transfer problem in the existence of non-linear radiative heat transfer. With the aid of von-Kármán relations, the equations of fluid motion and heat transfer are changed into a set of self-similar differential equations. These equations are dealt by an implicit finite-difference method with high precision. The results reveal that wall heat transfer rate can be improved by increasing solid volume fraction of ferromagnetic particles. Drag coefficient at the disk and heat transfer rate are increased as the strength of Lorentz force is enhanced. Viscous dissipation effect has an important part in improving heart transfer process which is vital in some applications. The results demonstrate that cooling capability of magnetite–water nanofluid is much superior to the conventional coolants. An excellent correlation of present results with the previous published articles is found in the all the cases.

Nomenclature

r,φ,z

Cylindrical coordinate system

u,v,w

velocity components along the r–, φ–, z– directions

B0

uniform magnetic field

T

fluid temperature

R

radius of rotating disk

Ω

angular velocity

M

Hartman number

cp

specific heat capacity

F,G,H

dimensionless function along radial, azimuthal and axial direction

Pr

Prandtl number

Cf

skin friction coefficient

aR

mean absorption coefficient

Nur

local Nusselt number

q''

wall heat flux

k

thermal conductivity

Re

local Reynolds number

Rd

Radiation parameter

Ec

Eckert number

1st order derivative with respect to η

2nd order derivative with respect to η

Greek symbols
v

kinematic viscosity

α

thermal diffusivity

θ

dimensionless temperature

η

similarity variable

σ

electrical conductivity

σ*

Stefan–Boltzman constant

τr,τφ

wall shear stress along r –and φ– direction

ρ

density

ϕ

nanoparticle volume fraction

μ

dynamic viscosity

Ω

angular velocity

θw

temperature ratio parameter

Subscripts
nf

nanofluid

w

condition at the wall

condition at infinity

f

fluid phase

s

solid phase

References

[1] T. Von Kármán, Uberlaminare und turbulentereibung, Z. Angew. Math. Mech. 1 (1921), 233–252.10.1002/zamm.19210010401Suche in Google Scholar

[2] W.G. Cochran, The flow due to a rotating disk, Proc. Camb. Phil. Soc. 30 (1934), 365–375.10.1017/S0305004100012561Suche in Google Scholar

[3] K. Millsaps and K. Pohlhausen, Heat transfer by laminar flow from a rotating disk, J Aeronaut. Sci. 19 (1952), 120–126.10.2514/8.2175Suche in Google Scholar

[4] J.T. Stuart, On the effects of uniform suction on the steady flow due to a rotating disk, Quart. J. Mech. Appl. Math. 7 (1954), 446–457.10.1093/qjmam/7.4.446Suche in Google Scholar

[5] M.G. Roger and G.N. Lance, The rotationally symmetric flow of a viscous fluid in presence of infinite rotating disc, J. Fluid Mech. 7 (1960), 617–631.10.1017/S0022112060000335Suche in Google Scholar

[6] H.A. Attia, Unsteady MHD flow near a rotating porous disk with uniform suction or injection, Fluid Dynam. Res. 23 (1998), 283–290.10.1016/S0169-5983(98)80011-7Suche in Google Scholar

[7] H.A. Attia and A.L. Aboul-Hassan, Effect of Hall current on the unsteady MHD flow due to a rotating disk with uniform suction or injection, Appl. Math. Model. 25 (2001), 1089–1098.10.1016/S0307-904X(01)00033-6Suche in Google Scholar

[8] H.A. Attia, Steady flow over a rotating disk in porous medium with heat transfer, Nonlinear Anal. Modell. Control 14 (2009), 21–26.10.15388/NA.2009.14.1.14527Suche in Google Scholar

[9] N. Bachok, A. Ishak and I. Pop, Flow and heat transfer over a rotating porous disk in a nanofluid, Physica B 406 (2011), 1767–1772.10.1016/j.physb.2011.02.024Suche in Google Scholar

[10] M.M. Rashidi, S.A. Mohimanian Pour, T. Hayat and S. Obaidat, Analytic approximate solutions for steady flow over a rotating disk in porous medium with heat transfer by homotopy analysis method, Comp. Fluids 54 (2012), 1–9.10.1016/j.compfluid.2011.08.001Suche in Google Scholar

[11] M. Turkyilmazoglu, MHD fluid flow and heat transfer due to a shrinking rotating disk, Comp. Fluids 90 (2014), 51–56.10.1016/j.compfluid.2013.11.005Suche in Google Scholar

[12] M. Turkyilmazoglu, Nanofluid flow and heat transfer due to a rotating disk, Comp. Fluids 94 (2014), 139–146.10.1016/j.compfluid.2014.02.009Suche in Google Scholar

[13] J.A. Khan, M. Mustafa, T. Hayat and A. Alsaedi, A revised model to study the MHD nanofluid flow and heat transfer due to rotating disk: Numerical solutions, Neural Comput. Appl. (2016), doi: 10.1007/s00521-016-2743-4.Suche in Google Scholar

[14] A.J. Hunt, Small particle heat exchangers, Lawrence Berkeley Lab Report Number LBL-7841.10.2172/6070780Suche in Google Scholar

[15] J. Buongiorno and L.W. Hu, Nanofluid heat transfer enhancement for nuclear reactor application, Proceedings of the ASME 2009 2nd Micro/Nanoscale Heat & Mass Transfer International Conference, MNHMT, 2009. DOI: 10.1115/MNHMT2009-18062.Suche in Google Scholar

[16] G. Huminic and A. Huminic, Application of nanofluids in heat exchangers: A review, Renew. Sust. Ener. Rev. 16 (2012), 5625–5638.10.1016/j.rser.2012.05.023Suche in Google Scholar

[17] J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transf. 128 (2006), 240–250.10.1115/1.2150834Suche in Google Scholar

[18] R.K. Tiwari and M.K. Das, Heat transfer augmentation in a two-sided lid driven differentially heated square cavity utilizing nanofluids, Int. J. Heat Mass. Transf. 50 (2007), 2002–2018.10.1016/j.ijheatmasstransfer.2006.09.034Suche in Google Scholar

[19] A.V. Kuznetsov and D.A. Nield, Natural convective boundary-layer flow of a nanofluid past a vertical plate, Int. J. Therm. Sci. 49 (2010), 243–247.10.1016/j.ijthermalsci.2009.07.015Suche in Google Scholar

[20] D.A. Nield and A.V. Kuznetsov, The Cheng–Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid, Int. J. Heat Mass Transf. 52 (2009), 5792–5795.10.1016/j.ijheatmasstransfer.2009.07.024Suche in Google Scholar

[21] W.A. Khan and I. Pop, Boundary-layer flow of a nanofluid past a stretching sheet, Int. J. Heat Mass Transf. 53 (2010), 2477–2483.10.1016/j.ijheatmasstransfer.2010.01.032Suche in Google Scholar

[22] M. Mustafa, T. Hayat, I. Pop, S. Asghar and S. Obaidat, Stagnation-point flow of a nanofluid towards a stretching sheet, Int. J. Heat Mass Transf. 54 (2011), 5588–5594.10.1016/j.ijheatmasstransfer.2011.07.021Suche in Google Scholar

[23] M. Mustafa, M.A. Farooq, T. Hayat and A. Alsaedi, Numerical and series solutions for stagnation-point flow of nanofluid over an exponentially stretching sheet, PLoS ONE 8 (2013), doi: 10.1371/journal.pone.0061859.Suche in Google Scholar PubMed PubMed Central

[24] M. Mustafa, T. Hayat and A. Alsaedi, Unsteady boundary layer flow of nanofluid past an impulsively stretching sheet, J. Mech. 29 (2013), 423–432.10.1017/jmech.2013.9Suche in Google Scholar

[25] O.D. Makinde, W.A. Khan and Z.H. Khan, Buoyancy effects on MHD stagnation point flow and heat transfer of a nanofluid past a convectively heated stretching/shrinking sheet, Int. J. Heat Mass Transf. 62 (2013), 526–533.10.1016/j.ijheatmasstransfer.2013.03.049Suche in Google Scholar

[26] A.V. Kuznetsov and D.A. Nield, Natural convective boundary-layer flow of a nanofluid past a vertical plate: A revised model, Int. J. Therm. Sci. 77 (2014), 126–129.10.1016/j.ijthermalsci.2013.10.007Suche in Google Scholar

[27] A. Mushtaq, M. Mustafa, T. Hayat and A. Alsaedi, Nonlinear radiative heat transfer in the flow of nanofluid due to solar energy: A numerical study, J. Taiwan Inst. Chem. Eng. 45 (2014), 1176–1183.10.1016/j.jtice.2013.11.008Suche in Google Scholar

[28] M.M. Rashidi, N. Freidoonimehr, A. Hosseini, O.A. Bég and T.K. Hung, Homotopy simulation of nanofluid dynamics from a non-linearly stretching isothermal permeable sheet with transpiration, Meccan 49 (2014), 469–482.10.1007/s11012-013-9805-9Suche in Google Scholar

[29] M.M. Rashidi, S. Abelman and N. Freidoonimehr, Entropy generation in steady MHD flow due to a rotating porous disk in a nanofluid, Int. J. Heat Mass Transf. 62 (2013), 515–525.10.1016/j.ijheatmasstransfer.2013.03.004Suche in Google Scholar

[30] M. Mustafa, T. Hayat and A. Alsaedi, Boundary layer flow of a nanofluid over an exponentially stretching sheet with convective boundary conditions, Int. J. Num. Meth. Heat Fluid Flow 23 (2013), 945–959.10.1108/HFF-09-2011-0179Suche in Google Scholar

[31] M. Turkyilmazoglu and I. Pop, Heat and mass transfer of unsteady natural convection flow of some nanofluids past a vertical infinite flat plate with radiation effect, Int. J. Heat Mass Transf. 59 (2013), 167–171.10.1016/j.ijheatmasstransfer.2012.12.009Suche in Google Scholar

[32] M. Sheikholeslami and M. Gorji-Bandpy, Free convection of ferrofluid in a cavity heated from below in the presence of an external magnetic field, Powder Technol. 256 (2014), 490–498.10.1016/j.powtec.2014.01.079Suche in Google Scholar

[33] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji and S. Soleimani, Thermal management for free convection of nanofluid using two phase model, J. Mol. Liq. 194 (2014), 179–187.10.1016/j.molliq.2014.01.022Suche in Google Scholar

[34] M.A. Sheremet and I. Pop, Free convection in a triangular cavity filled with a porous medium saturated by a nanofluid: Buongiorno’s mathematical model, Int. J. Num. Meth. Heat Fluid Flow 25 (2015), 1138–1161.10.1108/HFF-06-2014-0181Suche in Google Scholar

[35] S. Dinarvand, R. Hosseini and I. Pop, Unsteady convective heat and mass transfer of a nanofluid in Howarth’s stagnation point by Buongiorno’s model, Int. J. Num. Meth. Heat Fluid Flow 25 (2015), 1176–1197.10.1108/HFF-04-2014-0095Suche in Google Scholar

[36] M. Mustafa and A. Mushtaq, Model for natural convective flow of viscoelastic nanofluid past an isothermal vertical plate, Eur. Phys. J. Plus 130 (2015), doi: 10.1140/epjp/i2015-15178-1.Suche in Google Scholar

[37] T. Hayat, T. Muhammad, S.A. Shehzad and A. Alsaedi, On magnetohydrodynamic flow of nanofluid due to a rotating disk with slip effect: A numerical study, Comput. Meth. Appl. Mech. Eng. 315 (2017), 467–477.10.1016/j.cma.2016.11.002Suche in Google Scholar

[38] T. Hayat, T. Muhammad, S.A. Shehzad and A. Alsaedi, An analytical solution for magnetohydrodynamic Oldroyd-B nanofluid flow induced by a stretching sheet with heat generation/absorption, Int. J. Therm. Sci. 111 (2017), 274–288.10.1016/j.ijthermalsci.2016.08.009Suche in Google Scholar

[39] T. Hayat, F. Haider, T. Muhammad and A. Alsaedi, On Darcy-Forchheimer flow of viscoelastic nanofluids: A comparative study, J. Mol. Liq. 233 (2017), 278–287.10.1016/j.molliq.2017.03.035Suche in Google Scholar

[40] T. Hayat, A. Aziz, T. Muhammad and A. Alsaedi, On magnetohydrodynamic three-dimensional flow of nanofluid over a convectively heated nonlinear stretching surface, Int. J. Heat Mass Transf. 100 (2016), 566–572.10.1016/j.ijheatmasstransfer.2016.04.113Suche in Google Scholar

[41] T. Hayat, T. Muhammad, A. Alsaedi and M.S. Alhuthali, Magnetohydrodynamic three-dimensional flow of viscoelastic nanofluid in the presence of nonlinear thermal radiation, J. Magn. Magn. Mater. 385 (2015), 222–229.10.1016/j.jmmm.2015.02.046Suche in Google Scholar

[42] N.S. Bondareva, M.A. Sheremet and I. Pop, Magnetic field effect on the unsteady natural convection in a right-angle trapezoidal cavity filled with a nanofluid: Buongiorno’s mathematical model, Int. J. Numer. Meth. Heat Fluid Flow 25 (2015), 1924–1946.10.1108/HFF-07-2014-0236Suche in Google Scholar

[43] M.A. Sheremet, I. Pop and N.C. Roşca, Magnetic field effect on the unsteady natural convection in a wavy-walled cavity filled with a nanofluid: Buongiorno’s mathematical model, J. Taiwan Inst. Chem. Eng. 61 (2016), 211–222.10.1016/j.jtice.2015.12.015Suche in Google Scholar

[44] M.A. Sheremet, H.F. Oztop and I. Pop, MHD natural convection in an inclined wavy cavity with corner heater filled with a nanofluid, J. Magn. Magn. Mater. 416 (2016), 37–47.10.1016/j.jmmm.2016.04.061Suche in Google Scholar

[45] T. Cebeci and P. Bradshaw, Physical and computational aspects of convective heat transfer, Springer-Verlag, New York, 1988. (Chapter 13).10.1007/978-1-4612-3918-5Suche in Google Scholar

[46] H.C. Brinkman, The viscosity of concentrated suspensions and solutions, J. Chem. Phys. 20 (1952), 571–581.10.1063/1.1700493Suche in Google Scholar

[47] K. Khanafer, K. Vafai and M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, Int. J. Heat Mass. Transf. 46 (2003), 3639–3653.10.1016/S0017-9310(03)00156-XSuche in Google Scholar

[48] J.C. Maxwell, A treatise on electricity and magnetism, 3rd Edition, Oxford, Clarendon Press, 1904.Suche in Google Scholar

[49] N. Kelson and A. Desseaux, Note on porous rotating disk flow, Anziam J. 42 (2000), 837–855.10.21914/anziamj.v42i0.624Suche in Google Scholar

[50] M. Turkyilmazoglu, Determination of the correct range of physical parameters in the approximate analytical solutions of nonlinear equations using the Adomian decomposition method, Mediterranean J. Math. 13 (2016), 4019–4037.10.1007/s00009-016-0730-8Suche in Google Scholar

Received: 2015-12-26
Accepted: 2017-12-5
Published Online: 2018-1-11
Published in Print: 2018-2-23

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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