Neural Computing and Applications

, Volume 31, Supplement 1, pp 597–605 | Cite as

On non-Fourier flux in nonlinear stretching flow of hyperbolic tangent material

  • M. WaqasEmail author
  • Gulnaz Bashir
  • T. Hayat
  • A. Alsaedi
Original Article


The present study explores the features of hyperbolic tangent material due to a nonlinear stretched sheet with variable sheet thickness. Non-Fourier flux theory is implemented for the development of energy expression. Such consideration accounts for the contribution by thermal relaxation. The resulting nonlinear differential system has been determined for the convergent series expressions of velocity and temperature. The solutions are demonstrated and analyzed through plots. Presented results indicate that velocity decays via larger material power law index and Weissenberg number. Temperature is the decreasing function of Prandtl number and thermal relaxation time.


Variable sheet thickness Non-Fourier flux Hyperbolic tangent fluid 


u , v

Velocity components


Dynamic viscosity


Kinematic viscosity


Fluid density


Heat flux


Relaxation time of heat flux


Variable thermal conductivity


Thermal conductivity of ambient fluid

x , y

Space coordinates


Stretching velocity


Specific heat

a , b

Dimensional constants


Wall temperature


Ambient temperature


Surface shear stress


Temperature of fluid


Velocity field


Extra stress tensor


Cauchy stress tensor


Stream function


Wall thickness parameter


Material constant


Material power law index


Power law index


Prandtl number


Thermal relaxation parameter


Weissenberg number


Small parameter regarding the surface is sufficiently thin


Temperature dependent thermal conductivity parameter


Skin friction coefficient


Local Reynolds number


Dimensionless velocity


Dimensionless temperature


Dimensionless space variable


First Rivlin-Ericksen tensor


Zero shear rate viscosity


Infinite shear rate viscosity


Compliance with ethical standards

Conflict of interest

The authors declare they have no conflict of interest.


  1. 1.
    Cattaneo C (1948) Sulla conduzione del calore, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 3:83–101Google Scholar
  2. 2.
    Christov CI (2009) On frame indifferent formulation of the Maxwell-Cattaneo model of finite speed heat conduction. Mech Res Commun 36:481–486MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Straughan B (2010) Thermal convection with the Cattaneo-Christov model. Int J Heat Mass Transf 53:95–98CrossRefzbMATHGoogle Scholar
  4. 4.
    Tibullo V, Zampoli V (2011) A uniqueness result for the Cattaneo-Christov heat conduction model applied to incompressible fluids. Mech Res Commun 38:77–99CrossRefzbMATHGoogle Scholar
  5. 5.
    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 38:87–93MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hayat T, Khan MI, Farooq M, Alsaedi A, Khan MI (2017) Thermally stratified stretching flow with Cattaneo-Christov heat flux. Int J Heat Mass Transf 106:289–294CrossRefGoogle Scholar
  7. 7.
    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
  8. 8.
    Hayat T, Zubair M, Ayub M, Waqas M, Alsaedi A (2016) Stagnation point flow towards nonlinear stretching surface with Cattaneo-Christov heat flux. Eur Phys J Plus 131:355CrossRefGoogle Scholar
  9. 9.
    Kreiss HO, Nagy GB, OEO O, Reula A (1997) Global existence and exponential decay for hyperbolic dissipative relativistic fluid theories. J Math Phys 38:5272–5279MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jyothi B, Rao PK (2013) Influence of magnetic field on hyperbolic tangent fluid through a porous medium in a planar channel with peristalsis. Int J Mathematical archive 4:171–182Google Scholar
  11. 11.
    Kothandapani M Prakash J (2014) Influence of heat source, thermal radiation and inclined magnetic field on peristaltic flow of hyperbolic tangent nanofluid in a tapered asymmetric channel. IEEE Trans Nanobioscience DOI: 10.1109/TNB.2363673.
  12. 12.
    Akbar NS, Nadeem S, Haq RU, Khan ZH (2013) Numerical solutions of Magnetohydrodynamic boundary layer flow of tangent hyperbolic fluid towards a stretching sheet. Indian J Phys 87:1121–1124CrossRefGoogle Scholar
  13. 13.
    Hayat T, Qayyum S, Alsaedi A, Waqas M (2016) Radiative flow of a tangent hyperbolic fluid with convective conditions and chemical reaction. Eur Phys J Plus 131:422CrossRefGoogle Scholar
  14. 14.
    Salahuddin T, Malik MY, Hussain A, Bilal S, Awais M (2015) Effects of transverse magnetic field with variable thermal conductivity on tangent hyperbolic fluid with exponentially varying viscosity. AIP Adv 5:127103CrossRefGoogle Scholar
  15. 15.
    Khan MI Hayat T Waqas M Alsaedi A (2017) Outcome for chemically reactive aspect in flow of tangent hyperbolic material. J Mol Liq DOI:  10.1016/j.molliq.2017.01.016.
  16. 16.
    Hayat T, Waqas M, Alsaedi A, Bashir G, Alzahrani F (2017) Magnetohydrodynamic (MHD) stretched flow of tangent hyperbolic nanoliquid with variable thickness. J Mol Liq 229:178–184CrossRefGoogle Scholar
  17. 17.
    Sakiadis BC (1961) Boundary-layer behavior on continuous solid surface: I. Boundary-layer equations for two-dimensional and axisymmetric flow. AICHE J 7:26–28CrossRefGoogle Scholar
  18. 18.
    Crane LJ (1970) Flow past a stretching plane. J Appl Math Phys 21:645–647Google Scholar
  19. 19.
    Turkyilmazoglu M (2016) Equivalences and correspondences between the deforming body induced flow and heat in two-three dimensions. Physics Fluids 28:043102CrossRefGoogle Scholar
  20. 20.
    Ramzan M, Bilal M, Chung JD, Farooq U (2016) Mixed convective flow of Maxwell nanofluid past a porous vertical stretched surface—an optimal solution. Results Physics 6:1072–1079CrossRefGoogle Scholar
  21. 21.
    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
  22. 22.
    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 Surfaces A: Physicochem Eng Aspects 518:263–272CrossRefGoogle Scholar
  23. 23.
    Hayat T, Anwar MS, Farooq M, Alsaedi A (2015) Mixed convection flow of viscoelastic fluid by a stretching cylinder with heat transfer. PLoS One 10:e0118815CrossRefGoogle Scholar
  24. 24.
    Pal D, Chatterjee S Soret and Dufour effects on MHD convective heat and mass transfer of a power-law fluid over an inclined plate with variable thermal conductivity in a porous medium, Appl Math Comput 219: 7556–7574.Google Scholar
  25. 25.
    Vajravelu K, Prasad KV, Ng C (2013) Unsteady convective boundary layer flow of a viscous fluid at a vertical surface with variable fluid properties. Nonlinear Anal Real World Appl 14:455–464MathSciNetCrossRefGoogle Scholar
  26. 26.
    Hayat T, Waqas M, Shehzad SA, Alsaedi A (2016) Mixed convection flow of viscoelastic nanofluid by a cylinder with variable thermal conductivity and heat source/sink. Int J Numer Methods Heat Fluid Flow 26:214–234MathSciNetCrossRefzbMATHGoogle 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 Transf 99:702–710CrossRefGoogle Scholar
  28. 28.
    Umavathi JC, Sheremet MA, Mohiuddin S (2016) Combined effect of variable viscosity and thermal conductivity on mixed convection flow of a viscous fluid in a vertical channel in the presence of first order chemical reaction. Eur J Mech B/Fluids 58:98–108MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Salawu SO, Dada MS (2016) Radiative heat transfer of variable viscosity and thermal conductivity effects on inclined magnetic field with dissipation in a non-Darcy medium. J Nigerian Math Soc 35:93–106MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    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
  31. 31.
    Animasaun IL (2015) Effects of thermophoresis, variable viscosity and thermal conductivity on free convective heat and mass transfer of non-darcian MHD dissipative Casson fluid flow with suction and image order of chemical reaction. J Nigerian Math Soc 34:11–31MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Animasaun IL, Sandeep N (2016) Buoyancy induced model for the flow of 36 nm alumina-water nanofluid along upper horizontal surface of a paraboloid of revolution with variable thermal conductivity and viscosity. Powder Techn 301:858–867CrossRefGoogle Scholar
  33. 33.
    Turkyilmazoglu M (2010) An optimal analytic approximate solution for the limit cycle of Duffing-van der Pol equation. J Appl Mech Trans ASME 78:021005CrossRefGoogle Scholar
  34. 34.
    Turkyilmazoglu M (2012) Solution of Thomas-Fermi equation with a convergent approach. Commun Nonlin Sci Numer Simul 17:4097–4103MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Zheng L, Zhang C, Zhang X, Zhang J (2013) Flow and radiation heat transfer of a nanofluid over a stretching sheet with velocity slip and temperature jump in porous medium. J Frankl Inst 350:990–1007MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Turkyilmazoglu M (2016) An effective approach for evaluation of the optimal convergence control parameter in the homotopy analysis method. Filomat 30:1633–1650MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Hayat T, Khan MI, Waqas M, Alsaedi A (2017) Newtonian heating effect in nanofluid flow by a permeable cylinder. Res Physics 7:256–262Google Scholar
  38. 38.
    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 Transf 102:1123–1129CrossRefGoogle Scholar
  39. 39.
    Sui J, Zheng L, Zhang X, Chen G (2015) Mixed convection heat transfer in power law fluids over a moving conveyor along an inclined plate. Int. J Heat Mass Transf 85:1023–1033CrossRefGoogle Scholar
  40. 40.
    Khan WA, Khan M, Alshomrani AS (2016) Impact of chemical processes on 3D Burgers fluid utilizing Cattaneo-Christov double-diffusion: applications of non-Fourier’s heat and non-Fick's mass flux models. J Mol Liq 223:1039–1047CrossRefGoogle Scholar
  41. 41.
    Hayat T, Ullah I, Muhammad T, Alsaedi A (2016) Magnetohydrodynamic (MHD) three-dimensional flow of second grade nanofluid by a convectively heated exponentially stretching surface. J Mol Liq 220:1004–1012CrossRefGoogle Scholar
  42. 42.
    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 Transf 102:766–772CrossRefGoogle Scholar
  43. 43.
    Hayat T, Hussain Z, Alsaedi A, Mustafa M (2017) Nanofluid flow through a porous space with convective conditions and heterogeneous--homogeneous reactions. J Taiwan Inst Chem Eng 70:119–126CrossRefGoogle Scholar
  44. 44.
    Turkyilmazoglu M (2016) Determination of the correct range of physical parameters in the approximate analytical solutions of nonlinear equations using the Adomian decomposition method. Medit J Math 13:4019–4037MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Makinde OD, Aziz A (2011) Boundary layer flow of nanofluid past a stretching sheet with a convective boundary condition. Int J Therm Sci 50:1326–1332CrossRefGoogle Scholar

Copyright information

© The Natural Computing Applications Forum 2017

Authors and Affiliations

  • M. Waqas
    • 1
    Email author
  • Gulnaz Bashir
    • 1
  • T. Hayat
    • 1
    • 2
  • A. Alsaedi
    • 2
  1. 1.Department of MathematicsQuaid-I-Azam UnisversityIslamabadPakistan
  2. 2.Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia

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