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Numerical Study of Variable Fluid Properties and Magnetic Field on Convectively Heated Inclined Plate Utilizing Nanofluids

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Abstract

The present paper consists of numerical study of the heat transfer process in a two-dimensional steady natural convective flow of a nanofluid over an inclined plate. The considered transport model includes the effect of Brownian motion and thermophoresis and the zero nanoparticles mass flux boundary condition is also employed. The analysis accounts for temperature dependent viscosity, thermal conductivity and magnetic field. The nonlinear governing differential equations are solved numerically using the Finite Element Method. Results for the dimensionless velocity and temperature and nanoparticle volume fraction profiles are displayed graphically delineating the effect of various nanofluid flow characterizing parameters. It is observed that with the increase in thermal conductivity parameter, velocity and temperature in the respective boundary layers is increased. Whereas, magnetic filed have negative effect on fluid velocity and positive effect on temperature profile. Application of such kind of problem is prevalent in electronic packing and glassblowing.

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Abbreviations

k :

Thermal conductivity

\(Nu_x\) :

Local Nusselt number

\(Shr_x\) :

Local Sherwood number

Nur :

Nusselt number

Shrn :

Sherwood number

\(Ra_x\) :

Rayleigh number

Pr :

Prandtl number

Pr :

Prandtl number

g :

Acceleration due to gravity

\(D_T\) :

Thermophoretic diffusion coefficient

\(D_B\) :

Brownian diffusion coefficient

\(\phi \) :

Nanoparticle volume fraction

\(\phi _\infty \) :

Ambient nanoparticle volume fraction

T :

Local fluid temperature

\(T_\infty \) :

Ambient temperature

\(k_\infty \) :

Ambient thermal conductivity of fluid

Nb :

Brownian motion parameter

Nt :

Thermophoresis parameter

Nr :

Buoyancy ratio parameter

Nc :

Convective heat parameter

M :

Magnetic field parameter

\(q_w\) :

Wall heat flux

\(q_{np}\) :

Nanoparticle mass flux

(x, y):

Cartesian co-ordinate

(u, v):

Velocity components along x and y axes

\(\theta \) :

Dimensionless heat transfer

\(c_p\) :

Heat capacity of fluid

\(\alpha _m\) :

Thermal diffusivity of fluid

\(\beta \) :

Volumetric thermal expansion coefficient of the fluid

\((\rho c)_f\) :

Effective heat capacity of the fluid

\((\rho c)_p\) :

Effective heat capacity of the nanoparticle material

\((\rho c)_m\) :

Effective heat capacity of the porous medium

\(\nu \) :

Kinematic viscosity of the fluid

\(\xi \) :

Viscosity parameter

\(\epsilon \) :

Thermal conductivity parameter

\(\rho _p\) :

Nanoparticle mass density

\(\rho _f\) :

Fluid density

\(\rho _{f\infty }\) :

Ambient fluid density

\(\psi \) :

Stream function

\(\mu \) :

Absolute viscosity of the fluid

\(\mu (T)\) :

Temperature dependent viscosity of the fluid

\(\eta \) :

Similarity variable

\(\delta \) :

Acute angle of the plate to the vertical

\(\infty \) :

Condition far away from the plate

f :

Condition for fluid

nf :

Condition for nanofluid

w :

Condition at plate

References

  1. Choi, S.U.S. Eastman, J.A.: Enhancing thermal conductivity of fluids with nanoparticles, ANL/MSD/CP-84938 (1995)

  2. Wen, D., Lin, G., Vafaei, S., Zhang, K.: Review of nanofluids for heat transfer applications. Particuology 7, 141–150 (2009). doi:10.1016/j.partic.2009.01.007

    Article  Google Scholar 

  3. Saidur, R., Leong, K.Y., Mohammadc, H.A.: A review on applications and challenges of nanofluids. Renew. Sustain. Energy Rev. 15, 1646–1668 (2011)

    Article  Google Scholar 

  4. Wong, K.V., Leon, O.D.: Applications of nanofluids: current and future, Hindawi publishing corporation. Adv. Mech. Eng. 2, 11 (2010). doi:10.1155/2010/519659. [Article ID 519659]

    Google Scholar 

  5. Bejan, A.: Convection Heat Transfer. Wiley, NY (1984)

    MATH  Google Scholar 

  6. Chen, C.H.: Heat and mass transfer in MHD flow by natural convection from a permeable, inclined surface with variable wall temperature and concentration. Acta Mech. 172(3–4), 219–235 (2004)

    Article  MATH  Google Scholar 

  7. Aziz, A., Khan, W.A.: Natural convective boundary layer flow of a nanofluid past a convectively heated vertical plate. Int. J. Thermal Sci. 52, 83–90 (2012)

    Article  Google Scholar 

  8. Nield, D.A., Kuznetsov, A.V.: The Cheng–Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid. Int. J. Heat Mass Transf. 52, 5792–5795 (2009)

    Article  MATH  Google Scholar 

  9. Buongiorno, J.: Convective transport in nanofluids. J. Heat Transf. 128(3), 240–250 (2006)

    Article  Google Scholar 

  10. Choi, S.U.S.: Enhancing thermal conductivity of fluids with nanoparticles in developments and applications of non-Newtonian flows. In: Siginer, D.A., Wang, H.P. (eds) vol. 231, pp. 99–105. ASME, New York (1995)

  11. Nield, D.A., Kuznetsov, A.V.: The Cheng–Minkowycz problem for the double-diffusive natural convective boundary-layer flow in a porous medium saturated with a nanofluid. Int. J. Heat Mass Transf. 54, 374–378 (2011)

    Article  MATH  Google Scholar 

  12. Uddin, M.J., Bég, O.A., Aziz, A., Ismail, A.I.M.: Group analysis of free convection flow of a magnetic nanofluid with chemical reaction, Hindawi publishing corporation. Math. Probl. Eng. 2015, 11 (2015). doi:10.1155/2015/621503. [Article ID 621503]

    Article  Google Scholar 

  13. Narahari, M., Akilu, S., Jaafar, A.: Free convection flow of a nanofluid past an isothermal inclined plate. Appl. Mech. Mater. 390, 129–133 (2013). doi:10.4028/www.scientific.net/AMM.390.129. [ISSN: 1662–7482]

    Article  Google Scholar 

  14. Bachok, N., Ishak, A., Pop, I.: Boundary layer flow and heat transfer with variable fluid properties on a moving flat plate in a parallel free stream, Hindawi publishing corporation. J. Appl. Math. 2012, 10 (2012). doi:10.1155/2012/372623. [Article ID 372623]

    Article  MATH  Google Scholar 

  15. Rahman, M.M., Aziz, A., Lawatia, M.A.: Heat transfer in micropolar fluid along an inclined permeable plate with variable fluid properties. Int. J. Thermal Sci. 49, 993–1002 (2010)

    Article  Google Scholar 

  16. Palani, G., Kirubavathi, J.D., Kim, K.Y.: Free convection on an inclined plate with variable viscosity and thermal diffusivity. Thermophys. Aeromech. 21(1), 65 (2014)

    Article  Google Scholar 

  17. Pal, D., Mondal, H.: Effects of temperature-dependent viscosity and variable thermal conductivity on MHD non-Darcy mixed convective diffusion of species over a stretching sheet. J. Egypt. Math. Soc. 22, 123–133 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  18. Anjali, S.P., Prakash, M.: Temperature dependent viscosity and thermal conductivity effects on hydromagnetic flow over a slendering stretching sheet. J. Niger. Math. Soc. 34, 318–330 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  19. Bhuvaneswari, M., Ganesan, P.B., Sivasankaran, S., Viswanathan, K.K.: Effect of variable fluid properties on natural convection of nanofluids in a cavity with linearly varying wall temperature, Hindawi publishing corporation. Math. Probl. Eng. 2015, 13 (2015). doi:10.1155/2015/391786. [Article ID 391786]

    Article  Google Scholar 

  20. James, M., Mureithi, E.W., Kuznetsov, D.: Effects of variable viscosity of nanofluid flow over a permeable wedge embedded in saturated porous medium with chemical reaction and thermal radiation. Int. J. Adv. Appl. Math. Mech. 2(3), 101–118 (2015). [ISSN: 2347–2529]

    MATH  MathSciNet  Google Scholar 

  21. Animasaun I.L., Oyem A.O.: Effect of variable viscosity, Dufour, Soret and thermal conductivity on free convective heat and mass transfer of non-Darcian flow past porous flat surface. Am. J. Comput. Math. 2014(4) 357–365 (2014) http://www.scirp.org/journal/ajcm; doi:10.4236/ajcm.2014.44030

  22. Reddy, J.N.: An Introduction to the Finite Element Method. McGraw-Hill, New York (1984)

    MATH  Google Scholar 

  23. Goyal, M., Bhargava, R.: Numerical study of thermodiffusion effects on boundary layer flow of nanofluids over a power law stretching sheet. Microfluid. Nanofluidics 17(3), 591–604 (2014)

    Article  Google Scholar 

  24. Rahman, M.M., Uddin, M.J., Aziz, A.: Effects of variable electric conductivity and non-uniform heat source (or sink) on convective micropolar fluid flow along an inclined flat plate with surface heat flux. Int. J. Thermal Sci. 48, 2331–2340 (2009)

    Article  Google Scholar 

  25. Srinivasacharya, D., Kumar, P.V.: Mixed convection over an inclined Wavy surface in a nanofluid saturated non-Darcy porous medium with radiation effect, Hindawi publishing corporation. Int. J. Chem. Eng. 2015, 15 (2015). doi:10.1155/2015/927508. [Article ID 927508]

    Article  Google Scholar 

  26. Gorla, S.R.S., Chamkha, A.: Natural convective boundary layer flow over a horizontal plate embedded in a porous medium saturated with a nanofluid. J. Mod. Phys. 2, 62–71 (2011). doi:10.4236/jmp.2011.22011

    Article  Google Scholar 

  27. Yadav, D., Lee, J., Cho, H.H.: Brinkman convection induced by purely internal heating in a rotating porous medium layer saturated by a nanofluid. Powd. Technol. (2015). doi:10.1016/j.powtec.2015.08.048

    Google Scholar 

  28. Yadav, D., Kim, C., Lee, J., Cho, H.H.: Influence of magnetic field on the onset of nanofluid convection induced by purely internal heating. Comput. Fluids (2015). doi:10.1016/j.compfluid.2015.07.024

    MathSciNet  Google Scholar 

  29. Behseresht, A., Noghrehabadi, A., Ghalambaz, M.: Natural-convection heat and mass transfer from a vertical cone in porous media filled with nanofluids using the practical ranges of nanofluids thermo-physical properties. Chem. Eng. Res. Des. 92, 447–452 (2014)

    Article  Google Scholar 

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Acknowledgements

First author would like to thank the Ministry of Human Resource Development, Government of India for its financial support and reviewers for their helpful comments, which helped in improving manuscript.

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  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667, India

    Sarita Nandal & Rama Bhargava

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  1. Sarita Nandal
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  2. Rama Bhargava
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Correspondence to Rama Bhargava.

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Nandal, S., Bhargava, R. Numerical Study of Variable Fluid Properties and Magnetic Field on Convectively Heated Inclined Plate Utilizing Nanofluids. Int. J. Appl. Comput. Math 3, 3305–3320 (2017). https://doi.org/10.1007/s40819-016-0301-5

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