Nonlinear convection flow of micropolar liquid: an application of improved Fourier’s expression

  • T. Hayat
  • M. Zubair
  • M. Waqas
  • M. Ayub
  • A. Alsaedi
Technical Paper


Here, improved Fourier’s expression is utilized for heat transfer in nonlinear convection flow of micropolar liquid. The flow is generated due to a stretchable surface with variable thickness. Application of non-Fourier conduction phenomenon illustrates the characteristics of thermal relaxation factor. Temperature-dependent conductivity of liquid is also adopted. The set of partial differential equations governing the flow of micropolar liquid and heat transfer through non-Fourier heat conduction concept is established. The relevant transformations provide the highly nonlinear ordinary differential system. Homotopy theory is employed to acquire convergent solutions for nonlinear differential systems. Coefficient of skin friction is computed and examined for distinct embedded variables. Our presented analysis shows that temperature is decaying for larger thermal relaxation time.


Nonlinear convection flow Micropolar liquid Improved Fourier’s expression Variable thermal conductivity Stagnation point flow 

List of symbols


Velocity components


Space coordinates


Stretching velocity


Free stream velocity


Reference velocities


Micro-rotation variable


Gravitational accelerational


Linear thermal expansion coefficient


Nonlinear thermal expansion coefficient


Spin gradient viscosity




Boundary parameter


Heat flux


Relaxation time of heat flux


Variable thermal conductivity


Thermal conductivity of ambient fluid


Dynamic viscosity


Kinematic viscosity


Fluid density


Vortex viscosity


Specific heat


Dimensional constant


Variable temperature at the sheet


Variable temperature away from the sheet


Small parameter regarding the surface is sufficiently thin


Micropolar parameter


Local buoyancy parameter


Local Grashof number


Nonlinear convection


Prandtl number


Thermal relaxation parameter


Wall thickness parameter


Power law index


Temperature dependent thermal conductivity parameter


Stream function


Dimensionless variable


Dimensionless velocity


Dimensionless micro-rotation velocity


Dimensionless temperature


Skin friction coefficient


Surface shear stress


Local Reynolds number


Initial approximations for velocity


Initial approximations for temperature


Linear operators


Arbitrary constants

f, ℏg, ℏθ

Auxiliary variables


  1. 1.
    Eringen AC (1966) Theory of micropolar fluids. J Math Mech 16:1–18MathSciNetGoogle Scholar
  2. 2.
    Lukaszewicz G (1999) Micropolar fluids: theory and applications. Brikhauser, BaselCrossRefMATHGoogle Scholar
  3. 3.
    Eringen AC (2001) Microcontinuum field theories II: fluent media. Springer, New YorkMATHGoogle Scholar
  4. 4.
    Rashad AM, Abbasbandy S, Chamkha AJ (2014) Mixed convection flow of a micropolar fluid over a continuously moving vertical surface immersed in a thermally and solutally stratified medium with chemical reaction. J Taiwan Inst Chem Eng 45:2163–2169CrossRefGoogle Scholar
  5. 5.
    Sandeep N, Sulochana C (2015) Dual solutions for unsteady mixed convection flow of MHD micropolar fluid over a stretching/shrinking sheet with non-uniform heat source/sink. Eng Sci Technol Int J 18:738–745CrossRefGoogle Scholar
  6. 6.
    Shehzad SA, Waqas M, Alsaedi A, Hayat T (2016) Flow and heat transfer over an unsteady stretching sheet in a micropolar fluid with convective boundary condition. J Appl Fluid Mech 9:1437–1445CrossRefGoogle Scholar
  7. 7.
    Hayat T, Farooq S, Ahmad B, Alsaedi A (2016) Homogeneous-heterogeneous reactions and heat source/sink effects in MHD peristaltic flow of micropolar fluid with Newtonian heating in a curved channel. J Mol Liq 223:469–488CrossRefGoogle Scholar
  8. 8.
    Waqas M, Farooq M, Khan MI, Alsaedi A, Hayat T, Yasmeen T (2016) Magnetohydrodynamic (MHD) mixed convection flow of micropolar liquid due to nonlinear stretched sheet with convective condition. Int J Heat Mass Transfer 102:766–772CrossRefGoogle Scholar
  9. 9.
    Turkyilmazoglu M (2017) Mixed convection flow of magnetohydrodynamic micropolar fluid due to a porous heated/cooled deformable plate: exact solutions. Int J Heat Mass Transfer 106:127–134CrossRefGoogle Scholar
  10. 10.
    Crane L (1970) Flow past a stretching plate. Z Angew Math Phys 21:645–647CrossRefGoogle Scholar
  11. 11.
    Turkyilmazoglu M (2015) An analytical treatment for the exact solutions of MHD flow and heat over two–three dimensional deforming bodies. Int J Heat Mass Transfer 90:781–789CrossRefGoogle Scholar
  12. 12.
    Turkyilmazoglu M (2016) Flow of a micropolar fluid due to a porous stretching sheet and heat transfer. Int J Non-Linear Mech 83:59–64CrossRefGoogle Scholar
  13. 13.
    Waqas M, Hayat T, Farooq M, Shehzad SA, Alsaedi A (2016) Cattaneo–Christov heat flux model for flow of variable thermal conductivity generalized Burgers fluid. J Mol Liq 220:642–648CrossRefGoogle Scholar
  14. 14.
    Turkyilmazoglu M (2016) Equivalences and correspondences between the deforming body induced flow and heat in two-three dimensions. Phys Fluids 28:043102CrossRefGoogle Scholar
  15. 15.
    Babu MJ, Sandeep N (2016) 3D MHD slip flow of a nanofluid over a slendering stretching sheet with thermophoresis and Brownian motion effects. J Mol Liq 222:1003–1009CrossRefGoogle Scholar
  16. 16.
    Khan MI, Waqas M, Hayat T, Alsaedi A (2017) A comparative study of Casson fluid with homogeneous–heterogeneous reactions. J Colloid Interface Sci 498:85–90CrossRefGoogle Scholar
  17. 17.
    Khan WA, Irfan M, Khan M, Alshomrani AS, Alzahrani AK, Alghamdi MS (2017) Impact of chemical processes on magneto nanoparticle for the generalized Burgers fluid. J Mol Liq 234:201–208CrossRefGoogle Scholar
  18. 18.
    Hayat T, Khan MI, Waqas M, Alsaedi A, Farooq M (2017) Numerical simulation for melting heat transfer and radiation effects in stagnation point flow of carbon-water nanofluid. Comp Methods Appl Mech Eng 315:1011–1024MathSciNetCrossRefGoogle Scholar
  19. 19.
    Bilal S, Rehman KU, Malik MY, Hussain A, Khan M (2017) Effects of temperature dependent conductivity and absorptive/generative heat transfer on MHD three dimensional flow of Williamson fluid due to bidirectional non-linear stretching surface. Results Phys 7:204–212CrossRefGoogle Scholar
  20. 20.
    Hayat T, Waqas M, Khan MI, Alsaedi A (2017) Impacts of constructive and destructive chemical reactions in magnetohydrodynamic (MHD) flow of Jeffrey liquid due to nonlinear radially stretched surface. J Mol Liq 225:302–310CrossRefGoogle Scholar
  21. 21.
    Devi SPA, Prakash M (2015) Temperature dependent viscosity and thermal conductivity effects on hydromagnetic flow over a slendering stretching sheet. J Niger Math Soc 34:318–330MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Hayat T, Hussain Z, Alsaedi A, Ahmad B (2017) Numerical study for slip flow of carbon–water nanofluids. Computer Methods Appl Mech Eng 319:366–378MathSciNetCrossRefGoogle Scholar
  23. 23.
    Khan MI, Hayat T, Waqas M, Alsaedi A (2017) Outcome for chemically reactive aspect in flow of tangent hyperbolic material. J Mol Liq 230:143–151CrossRefGoogle Scholar
  24. 24.
    Hayat T, Khan MI, Waqas M, Alsaedi A (2017) Mathematical modeling of non-Newtonian fluid with chemical aspects: a new formulation and results by numerical technique. Colloids Surf A Physicochem Eng Asp 518:263–272CrossRefGoogle Scholar
  25. 25.
    Imran Khan M, Hayat T, Ijaz Khan M, Alsaedi A (2017) A modified homogeneous-heterogeneous reactions for MHD stagnation flow with viscous dissipation and Joule heating. Int J Heat Mass Transfer 113:310–317CrossRefGoogle Scholar
  26. 26.
    Hayat T, Ullah I, Alsaedi A, Farooq M (2017) MHD flow of Powell–Eyring nanofluid over a non-linear stretching sheet with variable thickness. Results Phys 7:189–196CrossRefGoogle Scholar
  27. 27.
    Hayat T, Khan MI, Farooq M, Alsaedi A, Waqas M, Yasmeen T (2016) Impact of Cattaneo–Christov heat flux model in flow of variable thermal conductivity fluid over a variable thicked surface. Int J Heat Mass Transfer 99:702–710CrossRefGoogle Scholar
  28. 28.
    Ijaz Khan M, Hayat T, Waqas M, Imran Khan M, Alsaedi A (2017) Impact of heat generation/absorption and homogeneous–heterogeneous reactions on flow of Maxwell fluid. J Mol Liq 233:465–470CrossRefGoogle Scholar
  29. 29.
    Hayat T, Qayyum S, Imtiaz M, Alsaedi A (2017) Radiative flow due to stretchable rotating disk with variable thickness. Results Phys 7:156–165CrossRefGoogle Scholar
  30. 30.
    Hayat T, Qayyum S, Alsaedi A, Ahmad B (2017) Magnetohydrodynamic (MHD) nonlinear convective flow of Walters-B nanofluid over a nonlinear stretching sheet with variable thickness. Int J Heat Mass Transfer 110:506–514CrossRefGoogle Scholar
  31. 31.
    Fourier J B J (1822) Théorie Analytique De La Chaleur, ParisGoogle Scholar
  32. 32.
    Catteneo C (1958) A form of heat conduction equation which eliminates the paradox of instantaneous propagation. Comput Rendus 247:431–433Google Scholar
  33. 33.
    Vernotte P (1958) Les paradoxes de la theorie continue de L’equation de lachaleur. Comput Rendus 246:3154–3155MathSciNetMATHGoogle Scholar
  34. 34.
    Christov CI (2009) On frame indifferent formulation of the Maxwell-Cattaneo model of finite speed heat conduction. Mech Res Commun 36:481–486MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Straughan B (2011) Thermal convection with the Cattaneo–Christov model. Int J Heat Mass Transfer 53:95–98CrossRefMATHGoogle Scholar
  36. 36.
    Tibullo V, Zampoli V (2011) A uniqueness result for the Cattaneo–Christov heat conduction model applied to incompressible fluids. Mech Res Commun 38:77–99CrossRefMATHGoogle Scholar
  37. 37.
    Han S, Zheng L, Li C, Zhang X (2014) Coupled flow and heat transfer in viscoelastic fluid with Cattaneo–Christov heat flux model. Appl Math Lett 38:87–93MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Hayat T, Waqas M, Shehzad SA, Alsaedi A (2016) On 2D stratified flow of an Oldroyd-B fluid with chemical reaction: an application of non-Fourier heat flux theory. J Mol Liq 223:566–571CrossRefGoogle Scholar
  39. 39.
    Khan WA, Khan M, Alshomrani AS, Ahmad L (2016) Numerical investigation of generalized Fourier’s and Fick’s laws for Sisko fluid flow. J Mol Liq 224:1016–1021CrossRefGoogle Scholar
  40. 40.
    Nadeem S, Muhammad N (2016) Impact of stratification and Cattaneo–Christov heat flux in the flow saturated with porous medium. J Mol Liq 224:423–430CrossRefGoogle Scholar
  41. 41.
    Hayat T, Khan MI, Waqas M, Alsaedi A (2017) On Cattaneo–Christov heat flux in the flow of variable thermal conductivity Eyring–Powell fluid. Results Phys 7:446–450CrossRefGoogle Scholar
  42. 42.
    Khan M, Ahmad L, Khan WA, Alshomrani AS, Alzahrani AK, Alghamdi MS (2017) A 3D Sisko fluid flow with Cattaneo–Christov heat flux model and heterogeneous-homogeneous reactions: a numerical study. J Mol Liq 238:19–26CrossRefGoogle Scholar
  43. 43.
    Zhang Y, Chen B, Li D (2017) Non-Fourier effect of laser-mediated thermal behaviors in bio-tissues: a numerical study by the dual-phase-lag model. Int J Heat Mass Transfer 108:1428–1438CrossRefGoogle Scholar
  44. 44.
    Turkyilmazoglu M (2016) An effective approach for evaluation of the optimal convergence control parameter in the homotopy analysis method. Filomat 30:1633–1650MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Shehzad SA, Hayat T, Alsaedi A, Chen B (2016) A useful model for solar radiation. Energy Ecol Environ 1:30–38CrossRefGoogle Scholar
  46. 46.
    Hayat T, Waqas M, Khan MI, Alsaedi A (2016) Analysis of thixotropic nanomaterial in a doubly stratified medium considering magnetic field effects. Int J Heat Mass Transfer 102:1123–1129CrossRefGoogle Scholar
  47. 47.
    Ellahi R, Hassan M, Zeeshan A (2016) Aggregation effects on water base Al2O3-nanofluid over permeable wedge in mixed convection. Asia-Pac J Chem Eng 11:179–186CrossRefGoogle Scholar
  48. 48.
    Hayat T, Zubair M, Waqas M, Alsaedi A, Ayub M (2017) Application of non-Fourier heat flux theory in thermally stratified flow of second grade liquid with variable properties. Chin J Phys 55:230–241CrossRefGoogle Scholar
  49. 49.
    Khan WA, Alshomrani AS, Khan M (2016) Assessment on characteristics of heterogeneous-homogenous processes in three-dimensional flow of Burgers fluid. Results Phys 6:772–779CrossRefGoogle Scholar
  50. 50.
    Muhammad T, Alsaedi A, Shehzad SA, Hayat T (2017) A revised model for Darcy–Forchheimer flow of Maxwell nanofluid subject to convective boundary condition. Chin J Phys. Google Scholar
  51. 51.
    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 13:4019–4037MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Mahapatra TR, Gupta AS (2002) Heat transfer in stagnation-point flow towards a stretching sheet. Heat Mass Transfer 38:517–521CrossRefGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  • T. Hayat
    • 1
    • 2
  • M. Zubair
    • 1
  • M. Waqas
    • 1
  • M. Ayub
    • 1
  • A. Alsaedi
    • 2
  1. 1.Department of MathematicsQuaid-i-Azam UniversityIslamabadPakistan
  2. 2.Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia

Personalised recommendations