Nonlinear radiative peristaltic flow of Jeffrey nanofluid with activation energy and modified Darcy’s law

  • T. Hayat
  • Farhat Bibi
  • S. FarooqEmail author
  • A. A. Khan
Technical Paper


The present communication addresses the magnetoperistalsis of Jeffrey nanomaterial in a vertical asymmetric compliant channel walls. Flow modeling is based upon mixed convection, non-Darcy’s resistance, thermal radiation, Brownian motion and thermophoresis, chemical reaction and activation energy. Nonlinear thermal radiation is taken instead of classical linear radiation consideration. Buongiorno model is used for nanofluid analysis. Channel boundaries are associated with no-slip, compliant characteristics and convective heat and mass transfer effects. Lubrication approach is followed and problems are numerically solved. Quantities of interest are analyzed physically. It is observed that velocity enhances for Hall and Darcy parameters. Temperature decreases with radiation parameter. It is found that concentration increases by increasing activation energy parameter.


Jeffrey nanoliquid Convective boundary condition Nonlinear radiative heat flux Energy activation Non-Darcy’s resistance 

List of symbols

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

Velocity components in dimensional form

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

Coordinate axes in dimensional form

\(H_{1} , H_{2}\)

Lower and upper wall shapes

\(h_{1} , h_{2}\)

Wall shapes in wave frame


Wave length


Pressure in


Phase difference


Joule current


Magnetic field


Constant transverse magnetic field

\(a_{1} , b_{1}\)

Amplitudes of upper and lower waves




Cauchy stress tensor


First Rivlin–Ericksen tensor

\(S_{{\tilde{x}\tilde{x}}} ,\,S_{{\tilde{x}\tilde{y}}} ,\,S_{{\tilde{y}\tilde{y}}}\)

Components of extra stress tensor

\(\tau^{\prime }\)

Tension in membrane

\(d^{{^{\prime } }}\)

Viscous damping coefficient


Permeability of porous media


Radiative heat flux

\(\beta_{1}^{{^{\prime } }}\), \(\beta_{2}^{{^{\prime } }}\)

Heat transfer coefficients

\(T_{0}\), \(T_{1}\)

Constant temperatures of walls


Chemical reaction


Thermophoresis diffusion coefficients


Stream function


Reynolds number


Hartmann number

\(E_{1}\), \(E_{2}\), \(E_{3}\)

Compliant wall parameters


Prandt number


Brinkmann number


Brownian motion parameter

\(G_{c}\), \(G_{r}\)

Mass and thermal Grashof numbers


Ratio of relaxation to retardation number


Radiation parameter


Thermophoresis parameter

\(\beta_{1}\), \(\beta_{2}\)

Convective heat Biot numbers

\(\gamma_{1}\), \(\gamma_{2}\)

Convective mass Biot numbers


Activation energy parameter


Dimensionless concentration

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

Velocity components in dimensionless form

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

Coordinate axes in dimensionless form

\(\alpha , \beta\)

Coefficients of thermal and concentration expansion


Gravity vector

\(d_{1}\), \(d_{2}\)

Channel widths

\(\rho_{f}\), \(\rho_{p}\)

Densities of base and nanoparticle


Thermal conductivity


Dynamic viscosity

\(C_{f}\), \(C_{p}\)

Specific heats of base fluid and nanoparticle

\(T\), \(C\)

Temperature and concentration of fluid


Identity tensor


Extra stress tensor


Material derivative


Electric conductivity

\(m^{\prime }\)

Mass per unit area


Pressure on the outside wall


Fluid mean temperature

\(\gamma_{1}^{{^{\prime } }}\), \(\gamma_{2}^{{^{\prime } }}\)

Mass transfer coefficients

\(C_{0}\), \(C_{1}\)

Constant concentration on walls


Activation energy


Brownian motion coefficient


Wave number


Darcy resistance


Hall parameter


Eckert number


Thermophoresis parameter


Stefan–Boltzmann constant


Fitted rate constant

\(a\), \(b\)

Amplitude in dimensionless form


Schmidt number


Dimensionless chemical reaction


Dimensionless temperature


Dimensionless pressure


Retardation time


Mass spectral absorption coefficient


Boltzmann constant


Dimensionless width



  1. 1.
    Choi SU, Eastman JA (1995) Enhancing thermal conductivity of fluids with nanoparticles. Argonne National Lab, LemontGoogle Scholar
  2. 2.
    Hashmi MM, Hayat T, Alsaedi A (2012) On the analytic solutions for squeezing flow of nanofluid between parallel disks. J Nonliner Anal Model Control 17(4):418–430MathSciNetzbMATHGoogle Scholar
  3. 3.
    Mustafa M, Hina S, Hayat T, Alsaedi A (2012) Influence of wall properties on the peristaltic flow of a nanofluid: analytic and numerical solutions. J Heat Mass Transf 55(17–18):4871–4877Google Scholar
  4. 4.
    Abbasi FM, Gul M, Shehzad SA (2018) Hall effects on peristalsis of boron nitride-ethylene glycol nanofluid with temperature dependent thermal conductivity. J Phys E Low-dimens Syst Nanostruct 99:275–284Google Scholar
  5. 5.
    Reddy MG, Reddy KV (2015) Influence of Joule heating on MHD peristaltic flow of a nanofluid with compliant walls. Proc Eng 127:1002–1009Google Scholar
  6. 6.
    Farooq S, Hayat T, Alsaedi A, Ahmad B (2017) Numerically framing the features of second order velocity slip in mixed convective flow of Sisko nanomaterial considering gyrotactic microorganisms. J Heat Mass Transf 112:521–532Google Scholar
  7. 7.
    Latham TW (1966) Fluid motions in a peristaltic pump. M.Sc. Thesis, MIT, Cambridge, MAGoogle Scholar
  8. 8.
    Shapiro AH, Jaffrin MY, Weinberg SL (1969) Peristaltic pumping with long wavelengths at low Reynolds number. J Fluid Mech 37(4):799–825Google Scholar
  9. 9.
    Hayat T, Zahir H, Alsaedi A, Ahmad B (2017) In Peristaltic flow of rotating couple stress fluid in a non-uniform channel. J Res Phys 7:2865–2873Google Scholar
  10. 10.
    Lew HS, Fung YC, Lowenstein CB (1971) Peristaltic carrying and mixing of chyme in the small intestine (an analysis of a mathematical model of peristalsis of the small intestine). J Biomech 4(4):297–315Google Scholar
  11. 11.
    Al-Khafajy DG, Abdulhadi AM (2014) Effects of MHD and wall properties on the peristaltic transport of a Carreau fluid through porous medium. J Adv Phys 6(2):1106–1121Google Scholar
  12. 12.
    Tripathi D, Pandey SK, Das S (2010) Peristaltic flow of viscoelastic fluid with fractional Maxwell model through a channel. J Appl Math Comput 215(10):3645–3654MathSciNetzbMATHGoogle Scholar
  13. 13.
    Mekheimer KS (2008) Peristaltic flow of a couple stress fluid in an annulus: application of an endoscope. J Phys A Stat Mech Appl 387(11):2403–2415Google Scholar
  14. 14.
    Hayat T, Ali N (2006) On mechanism of peristaltic flows for power-law fluids. Phys A 371(2):188–194Google Scholar
  15. 15.
    Mekheimer KS (2004) Peristaltic flow of blood under effect of a magnetic field in a non-uniform channels. J Appl Maths Comput 153(3):763–777MathSciNetzbMATHGoogle Scholar
  16. 16.
    Akram S, Nadeem S (2013) Influence of induced magnetic field and heat transfer on the peristaltic motion of a Jeffrey fluid in an asymmetric channel: closed form solutions. J Magn Magn Mater 328:11–20Google Scholar
  17. 17.
    Hayat T, Farooq S, Mustafa M, Ahmad B (2017) Peristaltic transport of Bingham plastic fluid considering magnetic field, Soret and Dufour effects. J Res Phys 7:2000–2011Google Scholar
  18. 18.
    Reddy MG, Makinde OD (2016) Magnetohydrodynamic peristaltic transport of Jeffrey nanofluid in an asymmetric channel. J Mol Liq 223:1242–1248Google Scholar
  19. 19.
    Hayat T, Javed M, Ali N (2008) MHD peristaltic transport of a Jeffery fluid in a channel with compliant walls and porous space. J Transp Porous Media 74(3):259–274MathSciNetGoogle Scholar
  20. 20.
    Tripathi D (2013) Study of transient peristaltic heat flow through a finite porous channel. J Math Comput Model 57(5–6):1270–1283MathSciNetGoogle Scholar
  21. 21.
    Eldesoky IM, Mousa AA (2010) Peristaltic flow of a compressible non-Newtonian Maxwellian fluid through porous medium in a tube. J Biomath 3(02):255–275MathSciNetzbMATHGoogle Scholar
  22. 22.
    Mekheimer KS, Komy SR, Abdelsalam SI (2013) Simultaneous effects of magnetic field and space porosity on compressible Maxwell fluid transport induced by a surface acoustic wave in a microchannel. J Chin Phys B 22(12):124702Google Scholar
  23. 23.
    Vajravelu K, Sreenadh S, Lakshminarayana P (2011) The influence of heat transfer on peristaltic transport of a Jeffrey fluid in a vertical porous stratum. J Commun Nonliner Sci Numer Simul 16(8):3107–3125MathSciNetzbMATHGoogle Scholar
  24. 24.
    Asghar S, Hussain Q, Hayat T, Alsaedi A (2015) Peristaltic flow of a reactive viscous fluid through a porous saturated channel and convective cooling conditions. J Appl Mech Tech Phys 56(4):580–589MathSciNetzbMATHGoogle Scholar
  25. 25.
    Ramesh K (2016) Influence of heat and mass transfer on peristaltic flow of a couple stress fluid through porous medium in the presence of inclined magnetic field in an inclined asymmetric channel. J Mol Liq 219:256–271Google Scholar
  26. 26.
    Bhatti MM, Zeeshan A, Ijaz N, Ellahi R (2017) Heat transfer and inclined magnetic field analysis on peristaltically induced motion of small particles. J Braz Soc Mech Sci Eng 39(9):3259–3267Google Scholar
  27. 27.
    Bhatti MM, Zeeshan A, Ellahi R, Shit GC (2018) Mathematical modeling of heat and mass transfer effects on MHD peristaltic propulsion of two-phase flow through a Darcy-Brinkman-Forchheimer porous medium. J Adv Powd Tech 29(5):1189–1197Google Scholar
  28. 28.
    Bhatti MM, Zeeshan A, Ellahi R (2017) Simultaneous effects of coagulation and variable magnetic field on peristaltically induced motion of Jeffrey nanofluid containing gyrotactic microorganism. J Microvasc Res 110:32–42Google Scholar
  29. 29.
    Ijaz N, Zeeshan A, Bhatti MM, Ellahi R (2018) Analytical study on liquid-solid particles interaction in the presence of heat and mass transfer through a wavy channel. J Mol Liq 250:80–87Google Scholar
  30. 30.
    Bhatti MM, Zeeshan A, Ellahi R, Ijaz N (2017) Heat and mass transfer of two-phase flow with Electric double layer effects induced due to peristaltic propulsion in the presence of transverse magnetic field. J Mol Liq 230:237–246Google Scholar
  31. 31.
    Hsiao KL (2017) To promote radiation electrical MHD activation energy thermal extrusion manufacturing system efficiency by using Carreau-Nanofluid with parameters control method. J Energy 130:486–499Google Scholar
  32. 32.
    Anuradha S, Yegammai M (2017) MHD radiative boundary layer flow of nanofluid past a vertical plate with effects of binary chemical reaction and activation energy. J Pure Appl Math 13:6377–6392Google Scholar
  33. 33.
    Khan MI, Hayat T, Khan MI, Alsaedi A (2018) Activation energy impact in nonlinear radiative stagnation point flow of Cross nanofluid. Int Commun Heat Mass Transf 91:216–224Google Scholar
  34. 34.
    Hayat T, Farooq S, Ahmad B, Alsaedi A (2018) Consequences of variable thermal conductivity and activation energy on peristalsis in curved configuration. J Mol Liq 263:258–267Google Scholar
  35. 35.
    Noreen S, Saleem M (2016) Soret and Dufour effects on the MHD peristaltic flow in a porous medium with thermal radiation and chemical reaction. J Heat Transf Res 47(1):1–28Google Scholar
  36. 36.
    Ayub S, Hayat T, Asghar S, Ahmad B (2017) Thermal radiation impact in mixed convective peristaltic flow of third grade nanofluid. J Res Phys 7:3687–3695Google Scholar
  37. 37.
    Kothandapani M, Prakash J (2015) Influence of heat source, thermal radiation, and inclined magnetic field on peristaltic flow of a hyperbolic tangent nanofluid in a tapered asymmetric channel. J IEEE Trans Nanobiosci 14(4):385–392Google Scholar
  38. 38.
    Hayat T, Shafique M, Tanveer A, Alsaedi A (2016) Radiative peristaltic flow of Jeffrey nanofluid with slip conditions and Joule heating. J PLoS ONE 11(2):e0148002Google Scholar
  39. 39.
    Hussain Q, Latif T, Alvi N, Asghar S (2018) Nonlinear radiative peristaltic flow of hydromagnetic fluid through porous medium. J Res Phys 9:121–134Google Scholar
  40. 40.
    Bhatti MM, Zeeshan A, Ijaz N, Bég OA, Kadir A (2017) Mathematical modelling of nonlinear thermal radiation effects on EMHD peristaltic pumping of viscoelastic dusty fluid through a porous medium duct. J Eng Sci Tech 20(3):1129–1139Google Scholar
  41. 41.
    Bhatti MM, Zeeshan A, Ellahi R (2016) Study of heat transfer with nonlinear thermal radiation on sinusoidal motion of magnetic solid particles in a dusty fluid. J Theor Appl Mech 46(3):75–94MathSciNetGoogle Scholar
  42. 42.
    Hayat T, Yasmin H, Ahmad B, Chen B (2014) Simultaneous effects of convective conditions and nanoparticles on peristaltic motion. J Mol Liq 193:74–82Google Scholar
  43. 43.
    Farooq S, Hayat T, Ahmad B, Alsaedi A (2018) MHD flow of Eyring-Powell liquid in convectively curved configuration. J Braz Soc Mech Sci Eng 40(3):159Google Scholar
  44. 44.
    Sayed HM, Aly EH, Vajravelu K (2016) Influence of slip and convective boundary conditions on peristaltic transport of non-Newtonian nanofluids in an inclined asymmetric channel. Alex Eng J 55(3):2209–2220Google Scholar
  45. 45.
    Hayat T, Farooq S, Alsaedi A, Ahmad B (2016) Hall and radial magnetic field effects on radiative peristaltic flow of Carreau-Yasuda fluid in a channel with convective heat and mass transfer. J Magn Magn Mater 412:207–216Google Scholar
  46. 46.
    Mittra TK, Prasad SN (1973) On the influence of wall properties and Poiseuille flow in peristalsis. J Biomech 6(6):681–693Google Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  • T. Hayat
    • 1
    • 2
  • Farhat Bibi
    • 3
  • S. Farooq
    • 1
    Email author
  • A. A. Khan
    • 3
  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
  3. 3.Department of Mathematics and StatisticsInternational Islamic UniversityIslamabadPakistan

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