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Scaling Transformations for Boundary Layer Flow near the Stagnation-Point on a Heated Permeable Stretching Surface in a Porous Medium Saturated with a Nanofluid and Heat Generation/Absorption Effects

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Abstract

In this article, a similarity solution of the steady boundary layer flow near the stagnation-point flow on a permeable stretching sheet in a porous medium saturated with a nanofluid and in the presence of internal heat generation/absorption is theoretically studied. The governing partial differential equations with the corresponding boundary conditions are reduced to a set of ordinary differential equations with the appropriate boundary conditions via Lie-group analysis. Copper (Cu) with water as its base fluid has been considered and representative results have been obtained for the nanoparticle volume fraction parameter \({\phi}\) in the range \({0\leq \phi \leq 0.2}\) with the Prandtl number of Pr = 6.8 for the water working fluid. Velocity and temperature profiles as well as the skin friction coefficient and the local Nusselt number are determined numerically. The influence of pertinent parameters such as nanofluid volume fraction parameter, the ratio of free stream velocity and stretching velocity parameter, the permeability parameter, suction/blowing parameter, and heat source/sink parameter on the flow and heat transfer characteristics is discussed. Comparisons with published results are also presented. It is shown that the inclusion of a nanoparticle into the base fluid of this problem is capable to change the flow pattern.

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Abbreviations

a, c:

Constants

C f :

Skin friction coefficient

C p :

Specific heat at constant pressure

G :

Invariant group of transformations

k nf :

Effective thermal conductivity of the nanofluid

k f :

Thermal conductivity of the fluid

K :

Permeability of the porous medium

K 1 :

Permeability parameter

Nu :

Local Nusselt number

Pr :

Prandtl number

q w :

Wall heat flux

Q 0 :

Heat generation/absorption coefficient

Re x :

Local Reynolds number

S :

Suction/injection parameter

T :

Temperature of the nanofluid

T w :

Surface temperature

T ∞ :

Temperature of the ambient nanofluid fluid

\({\bar{{u}}, \bar{{v}}}\) :

Velocity components along \({\bar{{x}}}\) and \({\bar{{y}}}\) directions

u, v:

Dimensionless velocity components

\({\bar{{u}}_w (\bar{{x}})}\) :

Stretching velocity

\({\bar{{U}}}\) :

Free stream velocity of the nanofluid

U :

Dimensionless free stream velocity of the nanofluid

\({\bar{{v}}_w}\) :

Mass flux velocity

\({\bar{{x}}, \bar{{y}}}\) :

Cartesian coordinates along the surface and normal to it

x, y:

Dimensionless coordinates

α nf :

Effective thermal diffusivity of the nanofluid

α f :

Thermal diffusivity of the fluid

α i :

Constants in (14)

\({\phi}\) :

Solid volume fraction of the nanoparticles

η :

Similarity variable

λ:

Heat generation/absorption parameter

μ nf :

Effective dynamic viscosity of the nanofluid

μ f :

Dynamic viscosity of the fluid

ν f :

Kinematic viscosity of the fluid

θ :

Dimensionless temperature

ρ nf :

Effective density of the nanofluid

ψ :

Stream function

nf:

Nanofluid

f:

Fluid

s:

Solid

*:

G Group variables

References

  • Abel M.S., Sanjayanand E., Nandeppanavar M.M.: Viscoelastic MHD flow and heat transfer over a stretching sheet with viscous and ohmic dissipations. Commun. Nonlinear Sci. Numer. Simul. 13, 1808–1821 (2008)

    Article  Google Scholar 

  • Abu-Nada E.: Application of nanofluids for heat transfer enhancement of separated flows encountered in a backward facing step. Int. J. Heat Fluid Flow 29, 242–249 (2008)

    Article  Google Scholar 

  • Abu-Nada E., Oztop H.F.: Effects of inclination angle on natural convection in enclosures filled with Cu-water nanofluid. Int. J. Heat Fluid Flow 30, 669–678 (2009)

    Article  Google Scholar 

  • Ahmad S., Pop I.: Mixed convection boundary layer flow from a vertical flat plate embedded in a porous medium filled with nanofluids. Int. Commun. Heat Mass Transf. 37, 987–991 (2010)

    Article  Google Scholar 

  • Aminossadati S.M., Ghasemi B.: Natural convection cooling of a localized heat source at the bottom of a nanofluid-filled enclosure. Eur. J. Mech. B/Fluids 28, 630–640 (2009)

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Cheng P., Minkowycz W.J.: Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike. J. Geophys. Res. 82, 2040–2044 (1977)

    Article  Google Scholar 

  • Congedo, P.M., Collura, S., Congedo, P.M.: Modeling and analysis of natural convection heat transfer in nanofluids. In: Proceedings of ASME Summer Heat Transfer Conference, vol. 3, pp. 569–579 (2009)

  • Cortell R.: Flow and heat transfer of a fluid through a porous medium over a stretching surface with internal heat generation/absorption and suction/blowing. Fluid Dyn. Res. 37, 231–245 (2005)

    Article  Google Scholar 

  • Das S.K., Choi S.U.S., Yu W., Pradeep T.: Nanofluids: Science and Technology. Wiley, New Jersey (2007)

    Book  Google Scholar 

  • Elbashbeshy E.M.A., Basid M.A.A.: Heat transfer in a porous medium over a stretching surface with internal heat generation and suction or injection. Appl. Math. Comput. 158, 799–807 (2004)

    Article  Google Scholar 

  • Ghasemi B., Aminossadati S.M.: Natural convection heat transfer in an inclined enclosure filled with a water-CuO nanofluid. Numer. Heat Transf. Part A Appl. 55, 807–823 (2009)

    Article  Google Scholar 

  • Ghasemi B., Aminossadati S.M.: Periodic natural convection in a nanofluid-filled enclosure with oscillating heat flux. Int. J. Therm. Sci. 49, 1–9 (2010)

    Article  Google Scholar 

  • Ho, C.J., Chen, M.W., Li, Z.W.: Effect of natural convection heat transfer of nanofluid in an enclosure due to uncertainties of viscosity and thermal conductivity. In: Proceedings of ASME/JSME Thermal Engineering. Summer Heat Transfer Conference—HT, vol. 1, pp. 833–841 (2007)

  • Ho C.J., Chen M.W., Li Z.W.: Numerical simulation of natural convenction of nanofluid in a square enclosure: Effects due to uncertainties of viscosity and thermal conductivity. Int. J. Heat Mass Transf. 51, 4506–4516 (2008)

    Article  Google Scholar 

  • Ibrahim F.S., Mansour M.A., Hamad M.A.A.: Lie-group analysis of radiation and magnetic field effects on free convection and mass transfer flow past a semi-infinite vertical flat plate. Elect. J. Diff. Equ. 2005, 1–17 (2005)

    Google Scholar 

  • Ingham, D.B., Pop, I. (eds): Transport Phenomena in Porous Media III. Elsevier, Oxford (2005)

    Google Scholar 

  • Ishak A., Nazar R., Pop I.: Magnaetohydrodynamic stagnation point flow towards a stretching vertical sheet. Magnetohydrodynamics 42, 17–30 (2006)

    Google Scholar 

  • Kakaç S., Pramuanjaroenkij A.: Review of convective heat transfer enhancement with nanofluids. Int. J. Heat Mass Transf. 52, 3187–3196 (2009)

    Article  Google Scholar 

  • Kandasamy R., Muhaimin I.: Scaling transformation for the effect of temperature-dependent fluid viscosity with thermophoresis particle deposition on MHD-free convection heat and mass transfer over a porous stretching surface. Transp. Porous Media 84, 549–568 (2010)

    Article  Google Scholar 

  • Khan W.A., Pop I.: Boundary-layer flow of a nanofluid past a stretching sheet. Int. J. Heat Mass Transf. 53, 2477–2483 (2010)

    Article  Google Scholar 

  • Lakshmisha K.N., Venkateswaran S., Nath G.: Three-dimensional unsteady flow with heat and mass transfer over a continuous stretching surface. ASME J. Heat Transf. 110, 590–595 (1988)

    Article  Google Scholar 

  • Layek G.C., Mukhopadhyay S., Samad S.A.: Heat and mass transfer analysis for boundary layer stagnation-point flow towards a heated porous stretching sheet with heat absorption/generation and suction/blowing. Int. Commun. Heat Mass Transf. 34, 347–356 (2007)

    Article  Google Scholar 

  • Mahapatra T.R., Gupta A.S.: Heat transfer in stagnation-point flow towards a stretching sheet. Heat Mass Transf. 38, 517–521 (2002)

    Article  Google Scholar 

  • Muhaimin I., Kandasamy R., Hashim I.: Scalimg transformation for the effect of chemical reaction on free convection heat and mass transfer in the presence of variable stream conditions. Chem. Eng. Res. Des. 88, 1320–1328 (2010)

    Article  Google Scholar 

  • Mukhopadhyay S., Layek G.C., Samad S.A.: Study of MHD boundary layer flow over a heated stretching sheet with variable viscosity. Int. J. Heat Mass Transf. 48, 4460–4466 (2005)

    Article  Google Scholar 

  • Muthtamilselvan M., Kandaswamy P., Lee J.: Heat transfer enhancement of copper-water nanofluids in a lid-driven enclosure. Commun. Nonlinear Sci. Numer. Simul. 15, 1501–1510 (2010)

    Article  Google Scholar 

  • Nazar R., Amin N., Filip D., Pop I.: Unsteady boundary layer flow in the region of the stagnation-point on a stretching sheet. Int. J. Eng. Sci. 42, 1241–1253 (2004)

    Article  Google Scholar 

  • Nield D.A., Bejan A.: Convection in Porous Media, 3rd edn. Springer, New York (2006)

    Google Scholar 

  • 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  Google Scholar 

  • Oztop H.F., Abu-Nada E.: Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids. Int. J. Heat Fluid Flow 29, 1326–1336 (2008)

    Article  Google Scholar 

  • Pal D.: Heat and mass transfer in stagnation-point flow towards a stretching surface in the presence of buoyancy force and thermal radiation. Meccanica 44, 145–158 (2009)

    Article  Google Scholar 

  • Prasad K.V., Vajravelu K., Datti P.S.: The effects of variable fluid properties on the hydro-magnetic flow and heat transfer over a non-linearly stretching sheet. Int. J. Therm. Sci. 49, 603–610 (2010a)

    Article  Google Scholar 

  • Prasad K.V., Pal D., Umesh V., Rao N.S.P.: The effect of variable viscosity on MHD viscoelastic fluid flow and heat transfer over a stretching sheet. Commun. Nonlinear Sci. Numer. Simul. 15, 331–344 (2010b)

    Article  Google Scholar 

  • Raptis A., Perdikis C.: Viscous flow over a non-linearly stretching sheet in the presence of chemical reaction and magnetic field. Int. J. Non-Linear Mech. 41, 527–529 (2006)

    Article  Google Scholar 

  • Tiwari R.K., Das M.K.: Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int. J. Heat Mass Transf. 50, 2002–2018 (2007)

    Article  Google Scholar 

  • Vadasz, P. (ed.): Emerging Topics in Heat and Mass Transfer in Porous Media. Springer, New York (2008)

    Google Scholar 

  • Vafai, K. (ed.): Handbook of Porous Media, 2nd edn. Taylor & Francis, New York (2005)

    Google Scholar 

  • Wang C.Y.: Free convection on a vertical stretching surface. J. Appl. Math. Mech. (ZAMM) 69, 418–420 (1989)

    Google Scholar 

  • Wang X.-Q., Mujumdar A.S.: Heat transfer characteristics of nanofluids: a review. Int. J. Therm. Sci. 46, 1–19 (2007)

    Article  Google Scholar 

  • Wang X.-Q., Mujumdar A.S.: A review on nanofluids—Part I: theoretical and numerical investigations. Braz. J. Chem. Eng. 25, 613–630 (2008a)

    Google Scholar 

  • Wang X.-Q., Mujumdar A.S.: A review on nanofluids—Part II: experiments and applications. Braz. J. Chem. Eng. 25, 631–648 (2008b)

    Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, Faculty of Science, Assiut University, Assiut, 71516, Egypt

    M. A. A. Hamad

  2. School of Mathematical Sciences, Universti Sains Malaysia, Penang, 11800, Malaysia

    M. A. A. Hamad

  3. Faculty of Mathematics, University of Cluj, CP 253, Cluj, 3400, Romania

    I. Pop

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  1. M. A. A. Hamad
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  2. I. Pop
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Correspondence to I. Pop.

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Hamad, M.A.A., Pop, I. Scaling Transformations for Boundary Layer Flow near the Stagnation-Point on a Heated Permeable Stretching Surface in a Porous Medium Saturated with a Nanofluid and Heat Generation/Absorption Effects. Transp Porous Med 87, 25–39 (2011). https://doi.org/10.1007/s11242-010-9683-8

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