Skip to main content
SpringerLink
Log in

Impact of third-grade nanofluid flow across a convective surface in the presence of inclined Lorentz force: an approach to entropy optimization

  • Published:
Journal of Thermal Analysis and Calorimetry Aims and scope Submit manuscript

Abstract

The current article deals with heat transfer and entropy generation analysis using Cattaneo–Christov heat flux (CCHF) for a flow of third grade nanofluid toward stretchable sheet with inclined magnetic field effects. Thermal radiation, heat generation/absorption, and convective heating effects are considered. Further entropy generation for fluid friction with inclined magnetic field, heat and mass transfer is calculated. This research is specially investigated for the impact of radiation effects using the CCHF model with entropy generation subject to distinct flow constants. Similarity transformations are used to reduce nonlinear PDE to a set of nonlinear ODE systems. The impact of suitable flow constants on the velocity, entropy generation, Bejan number, concentration, and temperature profiles are discussed. The coefficient of mass gradient and temperature gradient is calculated. Entropy generation and Bejan number profiles show the reverse effect on the larger thermal relaxation time parameter. Moreover, entropy of the system is found to be improving with the radiation constant, Biot number, suction/injection constant, Hartmann number and Brinkman number.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Abbreviations

a :

Stretching rate (s−1)

Bi:

Biot number (–)

Be:

Bejan number (–)

Br:

Brinkman number (–)

B 0 :

Constant magnetic field \( \left( {{\text{kg}}\,{\text{s}}^{ - 2} \,{\text{A}}^{ - 1} } \right) \)

C :

Concentration (kg m−3)

C p :

Specific heat \( \left( {{\text{J}}\,{\text{kg}}^{ - 1} \,{\text{K}}^{ - 1} } \right) \)

C ∞ :

Ambient concentration (kg m−3)

\( C_{\rm w} \) :

Fluid wall concentration (kg m−3)

\( \text{Cf}_{\rm x} \) :

Skin friction coefficient (–)

\( D_{\rm B} \) :

Brownian diffusion coefficient \( \left( {{\text{m}}^{2} \,{\text{s}}^{ - 1} } \right) \)

\( D_{\rm T} \) :

Thermophoretic diffusion coefficient \( \left( {{\text{m}}^{2} \,{\text{s}}^{ - 1} } \right) \)

\( E_{\rm G} \) :

Entropy generation parameter (–)

f(η):

Velocity similarity function (–)

\( f_{\rm w} \) :

Suction/injection parameter (–)

\( h_{\rm f} \) :

Convective heat transfer coefficient \( \left( {{\text{W}}\,{\text{m}}^{ - 1} \,{\text{K}}^{ - 1} } \right) \)

\( K_{1} , K_{2} , K_{3} \) :

Fluid parameters (–)

k :

Thermal conductivity \( \left( {{\text{W}}\,{\text{m}}^{ - 1} \,{\text{K}}^{ - 1} } \right) \)

\( {\mathcal{L}} \) :

Auxiliary linear operator (–)

Le:

Lewis number (–)

M :

Magnetic parameter (–)

\( {\mathcal{N}} \) :

Nonlinear operator (–)

Nb:

Brownian motion parameter (–)

Nt:

Thermophoresis parameter (–)

Nux :

Nusselt number (–)

Pr:

Prandtl number (–)

Q 0 :

Dimensional heat generation/absorption coefficient (–)

\( \hat{q} \) :

Heat flux \( \left( {{\text{W}}\,{\text{m}}^{ - 2} } \right) \)

Rd:

Radiation parameter (–)

Re:

Reynolds number (–)

Shx :

Sherwood number (–)

S :

Heat generation parameter (–)

\( S_{\text{gen}}^{\prime\prime\prime} \) :

Local volumetric entropy generation rate (W m−3 K−1)

\( S_{0}^{\prime\prime\prime} \) :

Characteristic entropy generation rate (W m−3 K−1)

T :

Temperature (K)

T ∞ :

Ambient temperature (K)

\( T_{\rm f} \) :

Convective surface temperature (K)

\( u_{\rm w} \) :

Velocity of the sheet \( \left( {{\text{m}}\,{\text{s}}^{ - 1} } \right) \)

u, v :

Velocity components in (x, y) directions \( \left( {{\text{m}}\,{\text{s}}^{ - 1} } \right) \)

\( v_{\rm w} > 0 \) :

Suction velocity

\( v_{\rm w} < 0 \) :

Injection velocity

x, y :

Cartesian coordinates (m)

\( \chi_{\rm m} \) :

Auxiliary parameter (–)

\( \phi \left( { \eta } \right) \) :

Concentration similarity function (–)

γ :

Dimensionless thermal relaxation time (–)

η :

Similarity parameter (–)

\( \lambda_{\rm T} \) :

Thermal relaxation time

λ :

Dimensionless constant

ν :

Kinematic viscosity \( \left( {{\text{m}}^{2} \,{\text{s}}^{ - 1} } \right) \)

Ω :

Dimensionless temperature difference

\( \theta \left( {\eta } \right) \) :

Temperature similarity function (–)

τ :

Ratio of the effective heat capacity

ρ :

Density \( \left( {{\text{kg}}\,{\text{m}}^{ - 1} } \right) \)

σ :

Electrical conductivity \( \left( {{\text{S}}\,{\text{m}}} \right) \)

ψ :

Stream function \( \left( {{\text{m}}\,{\text{s}}^{ - 1} } \right) \)

ζ :

Dimensionless concentration difference

References

  1. Bejan A. Entropy generation through heat and fluid flow. New York: Wiley; 1982.

    Google Scholar 

  2. Bejan A. Entropy generation minimization. New York: CRC Press; 1995.

    Google Scholar 

  3. Khan MI, Qayyum S, Hayat T, Imran Khan M, Alsaedi A, Khan TA. Entropy generation in radiative motion of tangent hyperbolic nanofluid in presence of activation energy and nonlinear mixed convection. Phys Lett A. 2018;382:2017–26.

    Article  CAS  Google Scholar 

  4. Qayyum S, Khan MI, Hayat T, Alsaedi A, Tamoor M. Entropy generation in dissipative flow of Williamson fluid between two rotating disks. Int J Heat Mass Transf. 2018;127:933–42.

    Article  Google Scholar 

  5. Dalir N, Dehsara M, Nourazar SS. Entropy analysis for magnetohydrodynamic flow and heat transfer of a Jeffrey nanofluid over a stretching sheet. Energy. 2015;79:351–62.

    Article  Google Scholar 

  6. Loganathan K, Rajan S. An entropy approach of Williamson nanofluid flow with Joule heating and zero nanoparticle mass flux. J Therm Anal Calorim. 2020. https://doi.org/10.1007/s10973-020-09414-3.

    Article  Google Scholar 

  7. Hayat T, Khan SA, Khan MI, Alsaedi A. Theoretical investigation of Ree–Eyring nanofluid flow with entropy optimization and Arrhenius activation energy between two rotating disks. Comput Methods Programs Biomed. 2019;177:57–68.

    Article  CAS  Google Scholar 

  8. Rashidi MM, Bagheri S, Momoniat E, Freidoonimehr N. Entropy analysis of convective MHD flow of third grade non-Newtonian fluid over a stretching sheet. Ain Shams Eng J. 2017;8:77–85.

    Article  Google Scholar 

  9. Hayat T, Khan MI, Qayyum S, Alsaedi A. Entropy generation in flow with silver and copper nanoparticles. Colloids Surf A. 2018;539:335–46.

    Article  CAS  Google Scholar 

  10. Mustafa M, Pop I, Naganthran K, Nazar R. Entropy generation analysis for radiative heat transfer to Bödewadt slip flow subject to strong wall suction. Eur J Mech B Fluids. 2018;72:179–88.

    Article  Google Scholar 

  11. Qayyum S, Hayat T, Khan MI, Imran Khan M, Alsaedi A. Optimization of entropy generation and dissipative nonlinear radiative Von Karman’s swirling flow with Soret and Dufour effects. J Mol Liq. 2018;262:261–74.

    Article  CAS  Google Scholar 

  12. López A, Ibáñez G, Pantoja J, Moreira J, Lastres O. Entropy generation analysis of MHD nanofluid flow in a porous vertical microchannel with nonlinear thermal radiation, slip flow and convective-radiative boundary conditions. Int J Heat Mass Transf. 2017;107:982–94.

    Article  CAS  Google Scholar 

  13. Khan MI, Hayat T, Waqas M, Khan MI, Alsaedi A. Entropy generation minimization (EGM) in nonlinear mixed convective flow of nanomaterial with Joule heating and slip condition. J Mol Liq. 2018;256:108–20.

    Article  CAS  Google Scholar 

  14. Qing J, Bhatti MM, Abbas MA, Rashidi MM, Ali MES. Entropy generation on MHD Casson nanofluid flow over a porous stretchy/shrinking surface. Entropy. 2016;18:e18040123.

    Article  CAS  Google Scholar 

  15. Hayat T, Khan MI, Khan TA, Imran Khan M, Ahmad S, Alsaedi A. Entropy generation in Darcy-Forchheimer bidirectional flow of water-based carbon nanotubes with convective boundary conditions. J Mol Liq. 2018;265:629–38.

    Article  CAS  Google Scholar 

  16. Khan MI, Kumar A, Hayat T, Waqas M, Singh R. Entropy generation in flow of Carreau nanofluid. J Mol Liq. 2019;278:677–87.

    Article  CAS  Google Scholar 

  17. Loganathan K, Sivasankaran S, Bhuvaneshwari M, Rajan S. Second-order slip, cross-diffusion and chemical reaction effects on magneto-convection of Oldroyd-B liquid using Cattaneo–Christov heat flux with convective heating. J Therm Anal Calorim. 2019;136:401–9.

    Article  CAS  Google Scholar 

  18. Imtiaz M, Alsaedi A, Shafiq A, Hayat T. Impact of chemical reaction on third grade fluid flow with Cattaneo–Christov heat flux. J Mol Liq. 2017;229:501–7.

    Article  CAS  Google Scholar 

  19. Khan MI, Waqas M, Hayat T, Alsaedi A. Colloidal study of Casson fluid with homogeneous heterogeneous reactions. J Colloid Interface Sci. 2017. https://doi.org/10.1016/j.jcis.2017.03.024.

    Article  PubMed  Google Scholar 

  20. Fourier JBJ. Theorie Analytique De La Chaleur. Paris: Chez Firmin Didot; 1822.

    Google Scholar 

  21. Cattaneo C. Sulla conduzionedelcalore. Atti del Seminario Matematico e Fisico dell Universita di Modena e Reggio Emilia. 1948;3:83–101.

    Google Scholar 

  22. 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 

  23. Hayat T, Khan MI, Farooq M, Yasmeen T, Alsaedi A. Stagnation point flow with Cattaneo–Christov heat flux and homogeneous-heterogeneous reactions. J Mol Liq. 2016;220:49–55.

    Article  CAS  Google Scholar 

  24. Bhattacharyya A, Seth GS, Kumar R, Chamkha AJ. Simulation of Cattaneo–Christov heat flux on the flow of single and multi-walled carbon nanotubes between two stretchable coaxial rotating disks. J Therm Anal Calorim. 2019. https://doi.org/10.1007/s10973-019-08644-4.

    Article  Google Scholar 

  25. Waqas M, Hayat T, Farooq M, Shehzad SA, Alsaedi A. Cattaneo–Christov heat flux model for flow of variable thermal conductivity generalized Burgers fluid. J Mol Liq. 2016;220:642–8.

    Article  CAS  Google Scholar 

  26. Hayat T, Khan MI, Farooq M, Alsaedi A, Imran Khan M. Thermally stratified stretching flow with Cattaneo–Christov heat flux. Int J Heat Mass Transf. 2017;106:289–94.

    Article  CAS  Google Scholar 

  27. Abbasi FM, Shehzad SA. Heat transfer analysis for three-dimensional flow of Maxwell fluid with temperature dependent thermal conductivity: application of Cattaneo–Christov heat flux model. J Mol Liq. 2016;220:848–54.

    Article  CAS  Google Scholar 

  28. Liu L, Zheng L, Liu F, Zhang X. Anomalous convection diffusion and wave coupling transport of cells on comb frame with fractional Cattaneo–Christov flux. Commun Nonlinear Sci Numer Simul. 2016;38:45–58.

    Article  Google Scholar 

  29. Salahuddin T, Malik MY, Hussain A, Bilal S, Awais M. MHD flow of Cattaneo–Christov heat flux model for Williamson fluid over a stretching sheet with variable thickness using numerical approach. J Magn Magn Mater. 2016;401:991–7.

    Article  CAS  Google Scholar 

  30. Hayat T, Khan MI, Farooq M, Alsaedi A, Waqas M, Yasmeen T. Impact of Cattaneo–Christov heat flux model in flow of variable thermal conductivity fluid over a variable thicked surface. Int J Heat Mass Transf. 2016;99:702–10.

    Article  Google Scholar 

  31. Choi S. Enhancing thermal conductivity of fluids with nanoparticles. ASME Int Mech Eng Expo. 1995;66:99–105.

    Google Scholar 

  32. Wang N, Maleki A, Nazari MA, Tlili I, Shadloo MS. Thermal conductivity modeling of nanofluids contain MgO particles by employing different approaches. Symmetry. 2020;12:206.

    Article  CAS  Google Scholar 

  33. Ahmadi MH, Ahmadi M-A, Maleki A, Pourfayaz F, Bid M, Açıkkalp E. Exergetic sustainability evaluation and multi-objective optimization of performance of an irreversible nanoscale Stirling refrigeration cycle operating with Maxwell–Boltzmann gas. Renew Sustain Energy Rev. 2017;78:80–92.

    Article  CAS  Google Scholar 

  34. Hayat T, Qayyum S, Khan MI, Alsaedi A. Current progresses about probable error and statistical declaration for radiative two-phase flow using AgeH2O and CueH2O nanomaterials, 2017. https://doi.org/10.1016/j.ijhydene.2017.09.124.

  35. Ramezanizadeh M, Nazari MA, Ahmadi MH, Lorenzini G, Pop I. A review on the applications of intelligence methods in predicting thermal conductivity of nanofluids. J Therm Anal Calorim. 2019. https://doi.org/10.1007/s10973-019-08154-3.

    Article  Google Scholar 

  36. Hayat T, Khan MI, Waqas M, Alsaedi A, Farooq M. Numerical simulation for melting heat transfer and radiation effects in stagnation point flow of carbon–water nanofluid. Comput Methods Appl Mech Eng. 2016. https://doi.org/10.1016/j.cma.2016.11.033.

    Article  Google Scholar 

  37. Maleki A, Elahi M, El Haj Assad M, Nazari MA, Shadloo MS, Nabipour N. Thermal conductivity modeling of nanofluids with ZnO particles by using approaches based on artificial neural network and MARS. J Therm Anal Calorim. 2020. https://doi.org/10.1007/s10973-020-09373-9.

    Article  Google Scholar 

  38. Shahrestani MI, Maleki A, Shadloo MS, Tlili I. Numerical investigation of forced convective heat transfer and performance evaluation criterion of Al2O3/water nanofluid flow inside an axisymmetric microchannel. Symmetry. 2020;12:120.

    Article  CAS  Google Scholar 

  39. Ramezanizadeh M, Ahmadi MA, Ahmadi MH, Nazari MA. Rigorous smart model for predicting dynamic viscosity of Al2O3/water Nanofluid. J Therm Anal Calorim. 2018. https://doi.org/10.1007/s10973-018-7916-1.

    Article  Google Scholar 

  40. Khan SU, Rauf A, Shehzad SA, Abbas Z, Javed T. Study of bioconvection flow in Oldroyd-B nanofluid with motile organisms and effective Prandtl approach. Phys A. 2019;527:121179.

    Article  CAS  Google Scholar 

  41. Komeilibirjandi A, Raffiee AH, Maleki A, Nazari MA, Shadloo MS. Thermal conductivity prediction of nanofluids containing CuO nanoparticles by using correlation and artificial neural network. J Therm Anal Calorim. 2019. https://doi.org/10.1007/s10973-019-08838-w.1.

    Article  Google Scholar 

  42. Hayat T, Khan MI, Qayyum S, Alsaedi A. Modern developments about statistical declaration and probable error for skin friction and Nusselt number with copper and silver nanoparticles. Chin J Phys. 2017;55:2501–13.

    Article  CAS  Google Scholar 

  43. Farooq M, Javed M, Khan MI, Anjum A, Hayat T. Melting heat transfer and double stratification in stagnation flow of viscous nanofluid. Results Phys. 2017;7:2296–301.

    Article  Google Scholar 

  44. Waqas M, Farooq M, Khan MI, Alsaedi A, Hayat T, Yasmeen T. Magnetohydrodynamic (MHD) mixed convection flow of micropolar liquid due to nonlinear stretched sheet with convective condition. Int. J. Heat Mass Transf. 2016;102:766–72.

    Article  Google Scholar 

  45. Hayat T, Shah F, Khan MI, Alsaedi A. Framing the performance of heat absorption/generation and thermal radiation in chemically reactive Darcy–Forchheimer flow. Results Phys. 2017;7:3390–5.

    Article  Google Scholar 

  46. Rivlin RS, Ericksen J. Stress–deformation relations for isotropic materials. Collected Papers of RS Rivlin. Berlin: Springer; 1997; pp. 911–1013.

  47. Fosdick R, Rajagopal K, editors. Thermodynamics and stability of fluids of third grade. In: Proceedings of the royal society of London A: mathematical, physical and engineering sciences. The Royal Society; 1980.

  48. Pakdemirli M. The boundary layer equations of third-grade fluids. Int J Non-Linear Mech. 1992;27(5):785–93.

    Article  Google Scholar 

  49. Makinde OD, Aziz A. Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition. Int J Therm Sci. 2011;50:1326–32.

    Article  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Erode Arts and Science College, Erode, Tamilnadu, 638009, India

    K. Loganathan & S. Rajan

  2. Department of Mathematics, Faculty of Engineering, Karpagam Academy of Higher Education, Coimbatore, Tamilnadu, 641021, India

    K. Loganathan

  3. Department of Mathematics, Sasuri College of Engineering, Tirupur, Tamilnadu, 638056, India

    K. Mohana

  4. Department of Mechanical Engineering, Hindustan College of Engineering and Technology, Coimbatore, Tamilnadu, 641032, India

    M. Mohanraj

  5. Department of Physics, Faculty of Engineering, Karpagam Academy of Higher Education, Coimbatore, Tamilnadu, 641021, India

    P. Sakthivel

Authors
  1. K. Loganathan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Mohana
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Mohanraj
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. P. Sakthivel
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. S. Rajan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. Loganathan.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Loganathan, K., Mohana, K., Mohanraj, M. et al. Impact of third-grade nanofluid flow across a convective surface in the presence of inclined Lorentz force: an approach to entropy optimization. J Therm Anal Calorim 144, 1935–1947 (2021). https://doi.org/10.1007/s10973-020-09751-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10973-020-09751-3

Keywords

Search

Navigation