Abstract
This article presents a detailed theoretical and computational analysis of alumina and titania-water nanofluid flow from a horizontal stretching sheet. At the boundary of the sheet (wall), velocity slip, thermal slip and Stefan blowing effects are considered. The Pak-Cho viscosity and thermal conductivity model is employed together with the non-homogeneous Buongiorno nanofluid model. The equations for mass, momentum, energy and nanoparticle species conservation are transformed via Lie-group transformations into a dimensionless system. The partial differential boundary value problem is therefore rendered into nonlinear ordinary differential form. With appropriate boundary conditions, the emerging normalized equations are solved with the semi-numerical homotopy analysis method (HAM). To consider entropy generation affects a second law thermodynamic analysis is also carried out. The impact of some physical parameters on the skin friction, Nusselt number, velocity, temperature and entropy generation number (EGM) are represented graphically. This analysis shows that diffusion parameter is a key factor to retards the friction and rate of heat transfer at the surface. Further, temperature of fluid decreases for the higher value of thermal slip parameter. In addition, EGM enhances with nanoparticles ambient concentration and Reynolds number. A numerical validation of HAM results is also included. The computations are relevant to thermodynamic optimization of nano-material processing operations.
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Abbreviations
- C :
-
Nanoparticles concentration (–)
- D B :
-
Brownian diffusion (m2/s)
- D T :
-
Thermophoresis diffusion (m2/s)
- D :
-
Ratio of thermophoresis and Brownian motion parameter
- Ec :
-
Eckert number (–)
- F :
-
Dimensionless stream function (–)
- H :
-
Enthalpy (J)
- K :
-
Thermal conductivity [W/(mK)]
- Nur :
-
Nusselt number (–)
- N 1 :
-
Velocity slip parameter (m)
- N 2 :
-
Thermal slip parameter (m)
- Pr:
-
Prandtl number (–)
- q :
-
Embedding parameter (–)
- R :
-
Gas constant [J/(molK)]
- Re :
-
Reynolds number (–)
- \( \phi \) :
-
Dimensionless concentration (–)
- S g :
-
Volumetric rate of entropy generation [J/(Km3 s)]
- S c :
-
Characteristic entropy [J/(Km3 s)]
- Sc :
-
Schmidt number (–)
- T :
-
Temperature (K)
- u :
-
Velocity (m/s) along x-axis
- v :
-
Velocity (m/s) along y-axis
- \( \rho \) :
-
Density (kg/m3)
- \( \mu \) :
-
Dynamic viscosity (Ns/m2)
- \( \phi \) :
-
Concentration (–)
- \( \psi \) :
-
Stream function (m2/s)
- \( \upsilon \) :
-
Kinematic viscosity (m2/s)
- \( \delta \) :
-
Thermal slip parameter (–)
- \( \rho c \) :
-
Heat capacity [J/(Km3)]
- \( \theta \) :
-
Dimensionless temperature (–)
- \( \chi \) :
-
Diffusive constant (–)
- \( \lambda_{1} \) :
-
Dimensionless velocity slip parameter
- \( \lambda_{2} \) :
-
Dimensionless thermal slip parameter
- \( \eta \) :
-
Similarity variable (–)
- \( \infty \) :
-
Ambient condition
- \( w \) :
-
Condition on surface
- P :
-
Nanoparticles
- nf :
-
Nanofluid
- f :
-
Fluid
References
Crane, L.J.: Flow past a stretching plate. Z. Für Angew. Math. Phys. ZAMP 21(4), 645–647 (1970)
Choi, S.U.S., Eastman, J.A.: Enhancing thermal conductivity of fluids with nanoparticles. Argonne National Lab., IL, ANL/MSD/CP-84938; CONF-951135-29 (1995)
Eastman, J.A., Choi, U.S., Li, S., Thompson, L.J., Lee, S.: Enhanced thermal conductivity through the development of nanofluids, vol. 457. Fall Meet. Mater. Res. Soc. MRS, Boston (1996)
Aybar, H.Ş., Sharifpur, M., Azizian, M.R., Mehrabi, M., Meyer, J.P.: A review of thermal conductivity models for nanofluids. Heat Transf. Eng. 36(13), 1085–1110 (2015)
Rana, P., Shukla, N., Gupta, Y., Pop, I.: Analytical prediction of multiple solutions for MHD Jeffery-Hamel flow and heat transfer utilizing KKL nanofluid model. Phys. Lett. A 383, 176–185 (2018)
Sheikholeslami, M., Rokni, H.B.: CVFEM for effect of Lorentz forces on nanofluid flow in a porous complex shaped enclosure by means of non-equilibrium model. J. Mol. Liq. 254, 446–462 (2018)
Sheikholeslami, M., Darzi, M., Sadoughi, M.K.: Heat transfer improvement and pressure drop during condensation of refrigerant-based nanofluid; an experimental procedure. Int. J. Heat Mass Transf. 122, 643–650 (2018)
Sheikholeslami, M.: Numerical investigation for CuO-H2O nanofluid flow in a porous channel with magnetic field using mesoscopic method. J. Mol. Liq. 249, 739–746 (2018)
Sheikholeslami, M., Rokni, H.B.: Numerical simulation for impact of Coulomb force on nanofluid heat transfer in a porous enclosure in presence of thermal radiation. Int. J. Heat Mass Transf. 118, 823–831 (2018)
Buongiorno, J.: Convective transport in nanofluids. ASME J. Heat Transf. 128(3), 240–250 (2006)
Dhanai, R., Rana, P., Kumar, L.: Critical values in slip flow and heat transfer analysis of non-Newtonian nanofluid utilizing heat source/sink and variable magnetic field: multiple solutions. J. Taiwan Inst. Chem. Eng. 58, 155–164 (2016)
Dhanai, R., Rana, P., Kumar, L.: MHD mixed convection nanofluid flow and heat transfer over an inclined cylinder due to velocity and thermal slip effects: Buongiorno’s model. Powder Technol. 288, 140–150 (2016)
Kuznetsov, A.V., Nield, D.A.: Natural convective boundary-layer flow of a nanofluid past a vertical plate: a revised model. Int. J. Therm. Sci. 77, 126–129 (2014)
Rana, P., Bhargava, R., Bég, O.A.: Numerical solution for mixed convection boundary layer flow of a nanofluid along an inclined plate embedded in a porous medium. Comput. Math. Appl. 64(9), 2816–2832 (2012)
Rana, P., Bhargava, R., Bég, O.A.: Finite element modeling of conjugate mixed convection flow of Al2O3–water nanofluid from an inclined slender hollow cylinder. Phys. Scr. 87(5), 1–15 (2013)
Rashidi, M.M., Freidoonimehr, N., Hosseini, A., Bég, O.A., Hung, T.-K.: Homotopy simulation of nanofluid dynamics from a non-linearly stretching isothermal permeable sheet with transpiration. Meccanica 49(2), 469–482 (2014)
Yang, C., Li, W., Nakayama, A.: Convective heat transfer of nanofluids in a concentric annulus. Int. J. Therm. Sci. 71, 249–257 (2013)
Malvandi, A., Moshizi, S.A., Soltani, E.G., Ganji, D.D.: Modified Buongiorno’s model for fully developed mixed convection flow of nanofluids in a vertical annular pipe. Comput. Fluids 89, 124–132 (2014)
Rana, P., Dhanai, R., Kumar, L.: MHD slip flow and heat transfer of Al2O3–water nanofluid over a horizontal shrinking cylinder using Buongiorno’s model: effect of nanolayer and nanoparticle diameter. Adv. Powder Technol. 28(7), 1727–1738 (2017)
Yoshimura, A., Prud’homme, R.K.: Wall slip corrections for Couette and parallel disk viscometers. J. Rheol. 32(1), 53–67 (1988)
Klein, S., Nellis, G.: Heat Transfer. Cambridge University Press, Cambridge (2008)
Fang, T., Jing, W.: Flow, heat and species transfer over a stretching plate considering coupled Stefan blowing effects from species transfer. Commun. Nonlinear Sci. Numer. Simul. 19(9), 3086–3097 (2014)
Uddin, M.J., Kabir, M.N., Bég, O.A.: Computational investigation of Stefan blowing and multiple-slip effects on buoyancy-driven bioconvection nanofluid flow with microorganisms. Int. J. Heat Mass Transf. 95, 116–130 (2016)
Latiff, N.A., Uddin, M.J., Ismail, A.I.M.: Stefan blowing effect on bioconvective flow of nanofluid over a solid rotating stretchable disk. Propuls. Power Res. 5(4), 267–278 (2016)
Rana, P., Shukla, N., Beg, O.A., Kadir, A., Singh, B.: Unsteady electromagnetic radiative nanofluid stagnation-point flow from a stretching sheet with chemically reactive nanoparticles, Stefan blowing effect and entropy generation. Proc. Inst. Mech. Eng. Part N J. Nanomater. Nanoeng. Nanosyst. 232, 69–82 (2018)
Bejan, A.: Method of entropy generation minimization, or modeling and optimization based on combined heat transfer and thermodynamics. Rev. Générale Therm. 35(418), 637–646 (1996)
Abolbashari, M.H., Freidoonimehr, N., Nazari, F., Rashidi, M.M.: Entropy analysis for an unsteady MHD flow past a stretching permeable surface in nano-fluid. Powder Technol. 267, 256–267 (2014)
Butt, A.S., Munawar, S., Ali, A., Mehmood, A.: Entropy generation in the Blasius flow under thermal radiation. Phys. Scr. 85(3), 035008 (2012)
Qing, J., Bhatti, M., Abbas, M., Rashidi, M., Ali, M.: Entropy generation on MHD Casson nanofluid flow over a porous stretching/shrinking surface. Entropy 18(4), 1–14 (2016)
Bhatti, M., Abbas, T., Rashidi, M., Ali, M.: Numerical simulation of entropy generation with thermal radiation on MHD Carreau nanofluid towards a shrinking sheet. Entropy 18(6), 200 (2016)
Aïboud, S., Saouli, S.: Second law analysis of viscoelastic fluid over a stretching sheet subject to a transverse magnetic field with heat and mass transfer. Entropy 12(8), 1867–1884 (2010)
Bhatti, M., et al.: Entropy generation on MHD Eyring–Powell nanofluid through a permeable stretching surface. Entropy 18(6), 224 (2016)
Liao, S.J.: Homotopy Analysis Method in Nonlinear Differential Equations. Higher Education Press, Beijing (2012)
Mabood, F., Khan, W.A., Ismail, A.I.M.: MHD flow over exponential radiating stretching sheet using homotopy analysis method. J. King Saud Univ. Eng. Sci. 29(1), 68–74 (2017)
Abdallah, I.A.: Homotopy analytical solution of MHD fluid flow and heat transfer problem. Appl. Math. Inf. Sci. 3(2), 223–233 (2009)
Uddin, M.J., Alginahi, Y., Bég, O.A., Kabir, M.N.: Numerical solutions for gyrotactic bioconvection in nanofluid-saturated porous media with Stefan blowing and multiple slip effects. Comput. Math. Appl. 72(10), 2562–2581 (2016)
Pak, B.C., Cho, Y.I.: Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles. Exp. Heat Transf. 11(2), 151–170 (1998)
Liao, S.J.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman and Hall/CRC Press, London/Boca Ratton (2003)
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Rana, P., Shukla, N., Bég, O.A. et al. Lie Group Analysis of Nanofluid Slip Flow with Stefan Blowing Effect via Modified Buongiorno’s Model: Entropy Generation Analysis. Differ Equ Dyn Syst 29, 193–210 (2021). https://doi.org/10.1007/s12591-019-00456-0
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DOI: https://doi.org/10.1007/s12591-019-00456-0