Mathematical Modeling and Theoretical Analysis of Second-Grade Nanomaterial with Entropy Optimization

  • Mujeeb Ur Rahman
  • M. Ijaz KhanEmail author
  • Fazal Haq
  • T. Hayat
Research Paper
Part of the following topical collections:
  1. Physics


A theoretical study of a second-grade nanofluid over a porous medium has been conducted. Stagnation point flow is considered. Effects of nonlinear radiative heat flux, dissipation and Joule heating are considered in the modeling of energy equation. Furthermore, chemical reaction is accounted. The wall is not stationary, but stretching at rate a. Total irreversibility rate is obtained through the second thermodynamics law. Slip mechanism of nanoparticles like Brownian movement and thermophoresis are considered. Suitable transformations lead to ordinary system. Solution development is done through HAM. Effects of pertinent variables are graphically discussed. Skin friction and temperature gradient are examined graphically versus different parameters. It is observed that velocity field decreased versus larger magnetic parameter. Temperature enhances versus rising values of magnetic and radiation variables. Main idea of present flow is listed.


Second-grade nanofluids Chemical reaction Nonlinear radiative heat flux Entropy generation Joule heating and viscous dissipation Nonlinear mixed convection 


  1. Abbas Z, Naveed M, Sajid M (2016) Heat generation effects in the hydromagnetic flow of nanofluid induced by a curved stretching sheet. J Mol Liq 215:756–762CrossRefGoogle Scholar
  2. Ahmad SE, Raizah ZAS, Aly AM (2017) Entropy generation due to mixed convection over vertical permeable cylinders using nanofluids. J King Saud Univ SciGoogle Scholar
  3. Alsaedi A, Awais M, Hayat T (2012) Effects of heat generation/absorption on stagnation point flow of nanofluid over a surface with convective boundary conditions. Nonlinear Simul 17:4210–4223MathSciNetCrossRefzbMATHGoogle Scholar
  4. Buongiorno J (2006) Convective transport in nanofluids. J Heat Transf 128:240–250CrossRefGoogle Scholar
  5. Choi SUS (1995) Enhancing thermal conductivity of fluids with nanoparticles. ASME Fluids Eng Div 231:99–105Google Scholar
  6. Farooq S, Khan MI, Hayat T, Waqas M, Alsaedi A (2019) Theoretical investigation of peristalsis transport in flow of hyperbolic tangent fluid with slip effects and chemical reaction. J Mol Liq 285:314–322CrossRefGoogle Scholar
  7. Gul A, Khan I, Makhanov SS (2018) Entropy generation in a mixed convection Poiseuille flow of molybdenum disulphide Jeffery nanofluid. Results Phys 9:947–954CrossRefGoogle Scholar
  8. Hayat T, Imtiaz M, Alsaedi A (2015) MHD 3D flow of nanofluid in presence of convective conditions. J Mol Liq 212:203–208CrossRefGoogle Scholar
  9. Hayat T, Imtiaz M, Alsaedi A (2016a) Unsteady flow of nanofluid with double stratification and magnetohydrodynamics. Int J Heat Mass Transf 92:100–109CrossRefGoogle Scholar
  10. Hayat T, Khan MI, Farooq M, Alsaedi A, Waqas M, Yasmeen T (2016b) Impact of Cattaneo Christov heat flux model in flow of variable thermal conductivity fluid over a variable thicked surface. Int J Heat Mass Transf 99:702–710CrossRefGoogle Scholar
  11. Hayat T, Khan MI, Farooq M, Yasmeen T, Alsaedi A (2016c) Stagnation point flow with Cattaneo-Christov heat flux and homogeneous–heterogeneous reactions. J Mol Liq 220:49–55CrossRefGoogle Scholar
  12. Hayat T, Khan MI, Farooq M, Gull N, Alsaedi A (2016d) Unsteady three-dimensional mixed convection flow with variable viscosity and thermal conductivity. J Mol Liq 223:1297–1310CrossRefGoogle Scholar
  13. Hayat T, Tamoor M, Khan MI, Alsaedi A (2016e) Numerical simulation for nonlinear radiative flow by convective cylinder. Results Phys 6:1031–1035CrossRefGoogle Scholar
  14. Hayat T, Khan MI, Waqas M, Alsaedi A (2017a) On Cattaneo–Christov heat flux in the flow of variable thermal conductivity Eyring–Powell fluid. Results Phys 7:446–450CrossRefGoogle Scholar
  15. Hayat T, Qayyum S, Khan MI, Alsaedi A (2017b) Modern developments about statistical declaration and probable error for skin friction and Nusselt number with copper and silver nanoparticles. Chin J Phys 55:2501–2513CrossRefGoogle Scholar
  16. Hayat T, Qayyum S, Khan MI, Alsaedi A (2017c) Current progresses about probable error and statistical declaration for radiative two phase flow using Ag-H2O and Cu-H2O nanomaterials. Int J Hydrogen Energy 42:29107–29120CrossRefGoogle Scholar
  17. Hayat T, Khan MI, Waqas M, Alsaedi A (2017d) Newtonian heating effect in nanofluid flow by a permeable cylinder. Results Phys 7:256–262CrossRefGoogle Scholar
  18. Hayat T, Khan MWA, Alsaedi A, Khan MI (2017e) Squeezing flow of second grade liquid subject to non-Fourier heat flux and heat generation/absorption. Colloid Polym Sci 295:967–975CrossRefGoogle Scholar
  19. Hayat T, Khan MI, Qayyum S, Alsaedi A (2018a) Entropy generation in flow with silver and copper nanoparticles. Colloid Surf A Physicochem Eng Asp 539:335–346CrossRefGoogle Scholar
  20. Hayat T, Khan MI, Qayyum S, Alseadi A, Khan MI (2018b) New thermodynamics of entropy generation minimization with nonlinear thermal radiation and nanomaterials. Phys Lett A 382:749–760MathSciNetCrossRefGoogle Scholar
  21. Hayat T, Khan SA, Khan MI, Alsaedi A (2019a) Theoretical investigation of Ree–Eyring nanofluid flow with entropy optimization and Arrhenius activation energy between two rotating disks. Comput Methods Prog Biomed 177:57–68CrossRefGoogle Scholar
  22. Hayat T, Rashid M, Khan MI, Alsaedi A (2019b) Physical aspects of MHD nonlinear radiative heat flux in flow of thixotropic nanomaterial. Iran J Sci Technol Trans A Sci 1–12Google Scholar
  23. Hsiao KL (2016) Stagnation electrical MHD nanofluid mixed convection with slip boundary on a stretching sheet. Appl Therm Eng 98:850–861CrossRefGoogle Scholar
  24. Hsiao KL (2017a) Micropolar nanofluid flow with MHD and viscous dissipation effects towards a stretching sheet with multimedia feature. Int J Heat Mass Transf 112:983–990CrossRefGoogle Scholar
  25. Hsiao KL (2017b) To promote radiation electrical MHD activation energy thermal extrusion manufacturing system efficiency by using Carreau-Nanofluid with parameters control method. Energy 130:486–499CrossRefGoogle Scholar
  26. Hsiao KL (2017c) Combined electrical MHD heat transfer thermal extrusion system using Maxwell fluid with radiative and viscous dissipation effects. Appl Therm Eng 112:1281–1288CrossRefGoogle Scholar
  27. Imtiaz M, Hayat T, Hussain M, Shehzad SA, Chen GQ, Ahmad B (2014) Mixed convection flow of nanofluid with Newtonian heating. Eur Phys J Plus 129:97CrossRefGoogle Scholar
  28. Khan MI, Yasmeen T, Khan MI, Farooq M, Wakeel M (2016) Research progress in the development of natural gas as fuel for road vehicles: a bibliographic review (1991–2016). Renew Sustain Energy Rev 66:702–741CrossRefGoogle Scholar
  29. Khan MI, Waqas M, Hayat T, Alsaedi A (2017a) A comparative study of Casson fluid with homogeneous-heterogeneous reactions. J Colloid Interface Sci 498:85–90CrossRefGoogle Scholar
  30. Khan MI, Waqas M, Hayat T, Alsaedi A, Khan MI (2017b) Significance of nonlinear radiation in mixed convection flow of magneto Walter-B nanoliquid. Int J Hydrogen Energy 42:26408–26416CrossRefGoogle Scholar
  31. Khan MI, Tamoor M, Hayat T, Alsaedi A (2017c) MHD Boundary layer thermal slip flow by nonlinearly stretching cylinder with suction/blowing and radiation. Results Phys 7:1207–1211CrossRefGoogle Scholar
  32. Khan MI, Waqas M, Hayat T, Khan MI, Alsaedi A (2017d) Numerical simulation of nonlinear thermal radiation and homogeneous-heterogeneous reactions in convective flow by a variable thicked surface. J Mol Liq 246:259–267CrossRefGoogle Scholar
  33. Khan MWA, Khan MI, Hayat T, Alsaedi A (2018a) Entropy generation minimization (EGM) of nanofluid flow by a thin moving needle with nonlinear thermal radiation. Phys B Condens Matter 534:113–119CrossRefGoogle Scholar
  34. Khan MI, Qayyum S, Hayat T, Khan MI, Alsaedi A (2018b) Entropy generation in radiative motion of tangent hyperbolic nanofluid in presence of activation energy and nonlinear mixed convection. Phys Lett A 382:2017–2026MathSciNetCrossRefGoogle Scholar
  35. Khan MI, Sumaira S, Hayat T, Waqas M, Khan MI, Alsaedi A (2018c) Entropy generation minimization and binary chemical reaction with Arrhenius activation energy in MHD radiative flow of nanomaterial. J Mol Liq 259:274–283CrossRefGoogle Scholar
  36. Khan MI, Hayat T, Waqas M, Khan MI, Alsaedi A (2018d) Entropy generation minimization (EGM) in nonlinear mixed convective flow of nanomaterial with Joule heating and slip condition. J Mol Liq 256:108–120CrossRefGoogle Scholar
  37. Khan MI, Ullah S, Hayat T, Khan MI, Alsaedi A (2018e) Entropy generation minimization (EGM) for convection nanomaterial flow with nonlinear radiative heat flux. J Mol Liq 260:279–291CrossRefGoogle Scholar
  38. Khan MI, Javed S, Hayat T, Waqas M, Alsaedi A (2019a) Entropy optimization in cubic autocatalysis chemical reactive flow of Williamson fluid subjected to viscous dissipation and uniform magnetic field. J Cent South Univ 26:1218–1232CrossRefGoogle Scholar
  39. Khan MI, Khan SA, Hayat T, Javed MF, Alsaedi A (2019b) Entropy generation in radiative flow of Ree-Eyring fluid due to due rotating disks. Int J Numer Methods Heat Fluid Flow 29:2057–2079CrossRefGoogle Scholar
  40. Khan MI, Rashid M, Hayat T, Khan NB, Alsaedi A (2019c) Physical aspects of Darcy–Forchheimer bidirectional flow in carbon nanotubes (SWCNTs and MWCNTs). Int J Numer Methods Heat Fluid Flow 29:2032–2056CrossRefGoogle Scholar
  41. Khan NB, Ibrahim ZB, Ali MA, Jameel M, Khan MI, Javanmardi A, Oyejobi DO (2019d) Numerical simulation of flow with large eddy simulation at Re = 3900: a study on the accuracy of statistical quantities. Int J Numer Methods Heat Fluid Flow. Google Scholar
  42. Khan MI, Khan SA, Hayat T, Alsaedi A (2019e) Entropy optimization in magnetohydrodynamic flow of third-grade nanofluid with viscous dissipation and chemical reaction. Iran J Sci Technol Trans A Sci 1–11Google Scholar
  43. Khan MI, Shah F, Hayat T, Alsaedi A (2019f) Transportation of CNTs based nanomaterial flow confined between two coaxially rotating disks with entropy generation. Phys A Stat Mech Appl 527:121154MathSciNetCrossRefGoogle Scholar
  44. Khan MI, Hayat T, Shah F, Rahman MU, Haq F (2019g) Physical aspects of CNTs and induced magnetic flux in stagnation point flow with quartic chemical reaction. Int J Heat Mass Transf 135:561–568CrossRefGoogle Scholar
  45. Kurnia JC, Sasmito AP (2017) Heat transfer performance and entropy generation of helical square tubes with various curvature radiuses. Energy Procedia 142:4064–4069CrossRefGoogle Scholar
  46. Liao SJ (2012) Homotopy analysis method in nonlinear differential equations. Springer, BerlinCrossRefzbMATHGoogle Scholar
  47. Marinca B, Marinca V (2018) Some exact solutions for MHD flow and heat transfer to modified second grade fluid with variable thermal conductivity in the presence of thermal radiation and heat generation/absorption. Comput Math Appl 76:1515–1524MathSciNetCrossRefGoogle Scholar
  48. Nandy SK, Mahapatra TR (2013) Effects of slip and heat generation/absorption on MHD stagnation flow of nanofluid past a stretching/shrinking surface with convective boundary conditions. Int J Heat Mass Transf 64:1091–1100CrossRefGoogle Scholar
  49. Qayyum S, Khan MI, Hayat T, Alsaedi A (2017) A framework for nonlinear thermal radiation and homogeneous–heterogeneous reactions flow based on silver-water and copper-water nanoparticles: a numerical model for probable error. Results Phys 7:1907–1914CrossRefGoogle Scholar
  50. Tamoor M, Waqas M, Khan MI, Alsaedi A, Hayat T (2017) Magnetohydrodynamic flow of Casson fluid over a stretching cylinder. Results Phys 7:498–502CrossRefGoogle Scholar
  51. Tiwari RK, Das MK (2007) Heat transfer augmentation in a two-sided lid driven differentially heated square cavity utilizing nanofluids. Int J Heat Mass Transf 50:2002–2018CrossRefzbMATHGoogle Scholar
  52. Turkyilmazoglu M (2016) Determination of the correct range of physical parameters in the approximate analytical solutions of nonlinear equations using the Adomian decomposition method. Mediterr J Math 13:4019–4037MathSciNetCrossRefzbMATHGoogle Scholar
  53. Turkyilmazoglu M (2018) Convergence accelerating in the homotopy analysis method: a new approach. Adv Appl Math Mech 2018:925–947MathSciNetCrossRefGoogle Scholar
  54. Turkyilmazoglu M (2019a) Accelerating the convergence of decomposition method of Adomian. J Comput Sci 31:54–59MathSciNetCrossRefGoogle Scholar
  55. Turkyilmazoglu M (2019b) Free and circular jets cooled by single phase nanofluids. Eur J Mech B Fluid 76:1–6MathSciNetCrossRefzbMATHGoogle Scholar
  56. Turkyilmazoglu M (2019c) MHD natural convection in saturated porous media with heat generation/absorption and thermal radiation. Arch Mech 71:49–64Google Scholar
  57. Turkyilmazoglu M (2019d) Latitudinally deforming rotating sphere. Appl Math Model 71:1–11MathSciNetCrossRefGoogle Scholar
  58. Waqas M, Khan MI, Hayat T, Alsaedi A (2017a) Stratified flow of an Oldroyd-B nanoliquid with heat generation. Results Phys 7:2489–2496CrossRefGoogle Scholar
  59. Waqas M, Khan MI, Hayat T, Alsaedi A, Khan MI (2017b) Nonlinear thermal radiation in flow induced by a slendering surface accounting thermophoresis and Brownian diffusion. Eur Phys J Plus 132:280CrossRefGoogle Scholar
  60. Yih KA (1999) Free convection effect on MHD coupled heat and mass transfer of a moving permeable vertical surface. Int Commun Heat Mass Transf 26:95–104CrossRefGoogle Scholar
  61. Ziaei-Rad M, Saeedan M, Afshari E (2016) Simulation and prediction of MHD dissipative nanofluid flow on a permeable stretching surface using artificial neural network. Appl Therm Eng 99:373–382CrossRefGoogle Scholar

Copyright information

© Shiraz University 2019

Authors and Affiliations

  • Mujeeb Ur Rahman
    • 1
  • M. Ijaz Khan
    • 2
    Email author
  • Fazal Haq
    • 1
    • 3
  • T. Hayat
    • 2
    • 4
  1. 1.Department of MathematicsKarakoram International University, Gilgit-BaltistanGilgitPakistan
  2. 2.Department of MathematicsQuaid-I-Azam UniversityIslamabadPakistan
  3. 3.Karakoram International UniversityHunzaPakistan
  4. 4.Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia

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