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Effects of nonlinear thermal radiation and activation energy on modified second-grade nanofluid with Cattaneo–Christov expressions

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Abstract

The aim of the current contribution is to exploit the effects of nonlinear thermal radiation and activation energy on the flow of modified second-grade fluid with the utilization of nanoparticles. The induced flow has been considered by linear movement of stretched surface which is considered to be porous. Famous modified second-grade fluid enables one to capture the important rheological features of shear thinning and thickening. Following the modern aspects of heat and mass transportations, we have employed the theories of Cattaneo–Christov heat flux and generalized Fick’s relations. The additional features of thermal radiation are also utilized in the energy equations with nonlinear expressions. Further, the impact of activation energy is also considered in the current continuation which makes the study quite versatile. While operating the suitable variables, the constituted problem has been distracted in a dimensionless form. The solution procedure has been followed with the implementation of the famous shooting technique with desirable accuracy. The involved engineering parameters are explained graphically with interesting physical consequences. It is noted that velocity distribution declined with a stretching ratio constant and combined parameter. An improved nanoparticles temperature distribution is observed with increases in stretching parameter and Brownian motion constant. The study also reports that the presence of activation energy can be more useful to enhance reaction processes. Based on the obtained scientific computations, it is claimed that the reported results can play a useful role in manufacturing processes and improvement in energy and heat resources.

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Abbreviations

\(\left( {u,v} \right)\) :

Velocity components, m s−1

\(\left( {x,y} \right)\) :

Cartesian coordinates, m

\(\nu\) :

Kinematic viscosity, m2 s−1

\(\rho_{\text{p}}\) :

Fluid particles density, kg m−3

\(\left( {\rho c} \right)_{\text{f}}\) :

Specific heat coefficient of fluid, J kg−1 K−1

\(T_{\infty }\) :

Ambient fluid temperature, K

\(\left( {\rho c} \right)_{\text{p}}\) :

Specific heat coefficient of nanoparticles, J kg−1 K−1

\(\sigma_{\text{e}}\) :

Stefan–Boltzmann constant, W m−2K−4

\(c\) :

Stretching rate

\(\alpha_{1}\) :

Material parameter

T :

Temperature of nanofluid, K

\(k^{ * }\) :

Permeability of porous medium

\(\alpha_{\text{f}}\) :

Liquid thermal diffusivity

\(k\) :

Generalized second-grade parameter

\({\text{Rd}}\) :

Radiation parameter

\(N_{\text{b}}\) :

Brownian motion constraint

\({\text{Le}}\) :

Lewis number

\(\theta\) :

Dimensionless temperature

\({\text{Nu}}_{\text{x}}\) :

Local Nusselt number

\(B_{0}\) :

Uniform magnetic field kg S2 A1

\(\mu\) :

Dynamic viscosity, Pas

\(\rho_{\text{f}}\) :

Base fluid density, kg m−3

\(\sigma_{1}\) :

Electrical conductivity, 1 Ω m−1

\(T_{\text{w}}\) :

Surface temperature, K

\(D_{\text{B}}\) :

Brownian coefficient, kg m−1 s−1

\(D_{\text{T}}\) :

Coefficient of thermophoretic force, kg m−1 s−1 K−1

\(k\) :

Mean absorption coefficient, 1 m−1

\(C_{\text{w}}\) :

Surface concentration

\(m\) :

Power law index

\(\sigma_{1}\) :

Electrical conductivity

\(\varphi\) :

Porous medium

\(k_{\text{a}}\) :

Thermal conductivity

\(\Pr\) :

Prandtl number

\(Q\) :

Heat absorption/generation parameter

\(N_{\text{t}}\) :

Thermophoresis parameter

\(\tau_{\text{w}}\) :

Wall shear stress

\(\phi\) :

Dimensionless concentration

References

  1. Rivlin RS, Ericksen JL. Stress deformation relations for isotropic materials. J Ration Mech Anal. 1955;4:323–425.

    Google Scholar 

  2. Schowalter WR. The application of boundary-layer theory to power-law pseudoplastic fluids: similar solutions. AIChE J. 1960;6:24–8.

    Article  CAS  Google Scholar 

  3. Aksoy Y, Pakdemirli M, Khalique CM. Boundary layer equations and stretching sheet solutions for the modified second grade fluid. Int J Eng Sci. 2007;45:829–41.

    Article  CAS  Google Scholar 

  4. Khan M, ur Rahman M. Flow and heat transfer to modified second grade fluid over a non-linear stretching sheet. AIP Adv. 2015;5:087157.

    Article  Google Scholar 

  5. ur Rahman M, Khan M, Manzur M. Boundary layer flow and heat transfer of a modified second grade nanofluid with new mass flux condition. Results Phys. 2018;10:594–600.

    Article  Google Scholar 

  6. Waqas H, Khan SU, Hassan M, Bhatti MM, Imran M. Analysis for bioconvection flow of modified second grade fluid containing gyrotactic microorganisms and nanoparticles. J Mol Liq. 2019;291(1):111231.

    Article  CAS  Google Scholar 

  7. Fourier JBJ (1822) Théorie Analytique De La Chaleur. Paris.

  8. Cattaneo C. Sulla conduzioned elcalore. Atti Semin Mat Fis Univ Modena Reggio Emilia. 1948;3:83–101.

    Google Scholar 

  9. Christov CI. On frame indifferent formulation of the Maxwell–Cattaneo model of finite-speed heat conduction. Mech Res Commun. 2009;36:481–6.

    Article  Google Scholar 

  10. Khan M. On Cattaneo–Christov heat flux model for Carreau fluid flow over a slendering sheet. Results Phys. 2017;7:310–9.

    Article  Google Scholar 

  11. Hayat T, Farooq M, Alsaedi A, Al-Solamy F. Impact of Cattaneo–Christov heat flux in the flow over a stretching sheet with variable thickness. AIP Adv. 2015;5:087159.

    Article  Google Scholar 

  12. Ullah KS, Ali N, Hayat T, Abbas Z. Heat transfer analysis based on Cattaneo–Christov heat flux model and convective boundary conditions for flow over an oscillatory stretching surface. Therm Sci. 2019;23(2A):443–55.

    Google Scholar 

  13. Mustafa M. Cattaneo–Christov heat flux model for rotating flow and heat transfer of upper-convected Maxwell fluid. AIP Adv. 2015;5:047109.

    Article  Google Scholar 

  14. Hayat T, Imtiaz M, Alsaedi A, Almezal S. On Cattaneo–Christov heat flux in MHD flow of Oldroyd-B fluid with homogeneous–heterogeneous reactions. J Magn Magn Mater. 2016;401(1):296–303.

    Article  CAS  Google Scholar 

  15. Khan SU, Shehzad SA, Ali N, Bashir MN. Some generalized results for Maxwell fluid flow over porous oscillatory surface with modified Fourier and Fick’s theories. J Braz Soc Mech Sci Eng. 2018;40:474.

    Article  Google Scholar 

  16. Ahmad I, Faisal M, Javed T. Bi-directional stretched nanofluid flow with Cattaneo–Christov double diffusion. Results Phys. 2019;15:102581.

    Article  Google Scholar 

  17. Choi SUS. Enhancing thermal conductivity of fluids with nanoparticles. In: Proceedings of the ASME international mechanical engineering congress and exposition, FED-vol. 231/MD-vol. 66; 1995, p. 99–105.

  18. Buongiorno J. Convective transport in nanofluids. J Heat Transf. 2006;128:240–50.

    Article  Google Scholar 

  19. Khan WA, Pop I. Boundary-layer flow of a nanofluid past a stretching sheet. Int J Heat Mass Transf. 2010;53:2477–83.

    Article  CAS  Google Scholar 

  20. Bhatti MM, Rashidi MM. Effects of thermo-diffusion and thermal radiation on Williamson nanofluid over a porous shrinking/stretching sheet. J Mol Liq. 2016;21:567–73.

    Article  Google Scholar 

  21. Sheikholeslami M, Rashidi MM, Ganji DD. Effect of non-uniform magnetic field on forced convection heat transfer of Fe3O4 water nanofluid. Comput Methods Appl Mech Eng. 2015;294:299–312.

    Article  Google Scholar 

  22. Rashidi MM, Freidoonimehr N, Hosseini A, Beg OA, Hung TK. Homotopy simulation of nanofluid dynamics from a non-linearly stretching isothermal permeable sheet with transpiration. Meccanica. 2014;49:469–82.

    Article  Google Scholar 

  23. Hayat T, Muhammad K, Farooq M, Alsaedi A. Melting heat transfer in stagnation point flow of carbon nanotubes towards variable thickness surface. AIP Adv. 2016;6(1):015214.

    Article  Google Scholar 

  24. Muhammad T, Hayat T, Shehzad SA, Alsaedi A. Viscous dissipation and Joule heating effects in MHD 3D flow with heat and mass fluxes. Results Phys. 2018;8:365–71.

    Article  Google Scholar 

  25. Hsiao K-L. Stagnation electrical MHD nanofluid mixed convection with slip boundary on a stretching sheet. Appl Therm Eng. 2016;98(5):850–61.

    Article  CAS  Google Scholar 

  26. Khan M, Malik R, Munir A, Khan WA. Flow and heat transfer to Sisko nanofluid over a nonlinear stretching sheet. PLoS ONE. 2015;10(5):e0125683.

    Article  Google Scholar 

  27. Turkyilmazoglu M. Flow of nanofluid plane wall jet and heat transfer. Eur J Mech B Fluids. 2016;59:18–24.

    Article  Google Scholar 

  28. Mahanthesh B, Gireesha BJ, Gorla RSR. Nonlinear radiative heat transfer in MHD three-dimensional flow of water based nanofluid over a non-linearly stretching sheet with convective boundary condition. J Niger Math Soc. 2016;35:178–98.

    Article  Google Scholar 

  29. Wakif A, Boulahia Z, Mishra SR, Rashidi MM, Sehaqui R. Influence of a uniform transverse magnetic field on the thermo-hydrodynamic stability in water-based nanofluids with metallic nanoparticles using the generalized Buongiorno’s mathematical model. Eur Phys J Plus. 2018;133:181. https://doi.org/10.1140/epjp/i2018-12037-7.

    Article  CAS  Google Scholar 

  30. Wakif A, Boulahia Z, Ali F, Eid MR, Sehaqui R. Numerical analysis of the unsteady natural convection MHD Couette nanofluid flow in the presence of thermal radiation using single and two-phase nanofluid models for Cu–water nanofluids. Int J Appl Comput Math. 2018;4:81. https://doi.org/10.1007/s40819-018-0513-y.

    Article  Google Scholar 

  31. Wakif A, Boulahia Z, Sehaqui R. Numerical analysis of the onset of longitudinal convective rolls in a porous medium saturated by an electrically conducting nanofluid in the presence of an external magnetic field. Results Phys. 2017;7:2134–52.

    Article  Google Scholar 

  32. Wakif A, Boulahia Z, Sehaqui R. Numerical study of the onset of convection in a Newtonian nanofluid layer with spatially uniform and non uniform internal heating. J Nanofluids. 2017;6(1):136–48.

    Article  Google Scholar 

  33. Wakif A, Boulahia Z, Sehaqui R. A semi-analytical analysis of electro-thermo-hydrodynamic stability in dielectric nanofluids using Buongiorno’s mathematical model together with more realistic boundary conditions. Results Phys. 2018;9:1438–54.

    Article  Google Scholar 

  34. Wakif A, Boulahia Z, Amine A, Animasaun IL, Afridi MI, Qasimd M, Sehaqui R. Magneto-convection of alumina-water nanofluid within thin horizontal layers using the revised generalized Buongiorno’s model. Front Heat Mass Transf (FHMT). 2019;12:3.

    CAS  Google Scholar 

  35. Qing J, Bhatti MM, Abbas MA, Rashidi MM, Ali ME. Entropy generation on MHD Casson nanofluid flow over a porous stretching/shrinking surface. Entropy. 2016;18:123. https://doi.org/10.3390/e18040123.

    Article  CAS  Google Scholar 

  36. Waqas H, Imran M, Khan SU, Shehzad SA, Meraj MA. Slip flow of Maxwell viscoelasticity-based micropolar nano particles with porous medium: a numerical study. Appl Math Mech (Engl Ed). 2019;40(9):1255–68.

    Article  Google Scholar 

  37. Patel HS, Meher R. Simulation of counter-current imbibition phenomenon in a double phase flow through fracture porous medium with capillary pressure. Ain Shams Eng J. 2018;9(4):2163–9.

    Article  Google Scholar 

  38. Waqas H, Khan SU, Shehzad SA, Imran M. Significance of nonlinear radiative flow of micropolar nanoparticles over porous surface with gyrotactic microorganism, activation energy and Nield’s condition. Heat Transf Asian Res. 2019;48(7):3230–56.

    Article  Google Scholar 

  39. Ali N, Khan SU, Abbas Z, Sajid M. Soret and Dufour effects on hydromagnetic flow of viscoelastic fluid over porous oscillatory stretching sheet with thermal radiation. J Braz Soc Mech Sci Eng. 2016;38:2533–46.

    Article  CAS  Google Scholar 

  40. Yuan Y, Xu K, Zhao K. Numerical analysis of transport in porous media to reduce aerodynamic noise past a circular cylinder by application of porous foam. J Therm Anal Calorim. 2019. https://doi.org/10.1007/s10973-019-08619-5.

    Article  Google Scholar 

  41. Moradi A, Toghraie D, Isfahani AH, Hosseinian A. An experimental study on MWCNT–water nanofluids flow and heat transfer in double-pipe heat exchanger using porous media. J Therm Anal Calorim. 2019;137:1797–807.

    Article  CAS  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, COMSATS University Islamabad, Sahiwal, 57000, Pakistan

    Sami Ullah Khan

  2. Department for Management of Science and Technology Development, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Iskander Tlili

  3. Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Iskander Tlili

  4. Department of Mathematics, Government College University, Faisalabad, Pakistan

    Hassan Waqas & Muhammad Imran

Authors
  1. Sami Ullah Khan
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  2. Iskander Tlili
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  4. Muhammad Imran
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Correspondence to Iskander Tlili.

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Khan, S.U., Tlili, I., Waqas, H. et al. Effects of nonlinear thermal radiation and activation energy on modified second-grade nanofluid with Cattaneo–Christov expressions. J Therm Anal Calorim 143, 1175–1186 (2021). https://doi.org/10.1007/s10973-020-09392-6

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