Physical assessments on variable thermal conductivity and heat generation/absorption in cross magneto-flow model

  • Faisal SultanEmail author
  • Waqar Azeem Khan
  • Muhammad Shahzad
  • Mehboob AliEmail author
  • Zakir Hussain


The present work focused here is a mixed convective MHD flow of cross fluid in the presence of heat generation/absorption and variable thermal conductivity over a bidirectional stretchable sheet. To elaborate the mechanism of heat transfer is analyzed in view of non-Fourier heat flux based upon Cattaneo–Christov theory. The influence of a simple isothermal model of homogeneous–heterogeneous reactions is further used for solute concentration. As a result, the relevant Buongiorno fluid model is utilized in mathematical modeling and then it is simplified through lubrication technique. By using appropriate transformations, the raised PDEs initially converted to ODEs. Convergent solutions of ODEs are obtained by the implementation of the numerical procedure bvp4c technique. However, the velocity, temperature and concentration profiles have been sketched by distinct physical flow parameter. Drag coefficients and heat transport are also computed numerically. Our results reveal that temperature profile has an inverse relation between the relaxation parameter and variable thermal conductivity.


Cross fluid model Cattaneo–Christov heat flux Variable thermal conductivity Heat generation/absorption Homogeneous–heterogeneous reactions 

List of symbols

\(T_{\infty }\)

Ambient temperature of fluid


Concentrations of chemical species \(P,Q\)

\(D_{\text{P}} ,\;D_{\text{Q}}\)

Diffusion coefficient of species \(P\) and \(Q\)


Magnetic field strength


Power-law index

\(k_{\text{m}} ,\;K_{\text{s}}\)

Rate coefficient of homogeneous/heterogeneous reactions


Space coordinates

\(U_{\text{w}} \left( {x,t} \right),\;V_{\text{w}} \left( {y,t} \right)\)

Stretching velocities


Temperature of fluid

\(K\left( T \right)\)

Variable thermal conductivity


Velocity components


Dimensionless velocities


Skin friction


Schmidt number


Magnetic parameter


Prandtl number


Strength coefficient homogenous reaction


Local Reynolds number

\({\text{We}}_{1} ,\;{\text{We}}_{2}\)

Weissenberg numbers


Local Nusselt number

Greek alphabets


Heat transfer coefficient

\(\left( {\rho c} \right)_{\text{f}}\)

Heat capacity of fluid


Kinematics viscosity


Ratio of diffusion coefficient


Stefan–Boltzmann constant


Thermal conductivity


Thermal conductivity time of the heat discussion


Time material constant


The ratio of stretching rates parameter


Biot number


Specific heat at constant pressure


Dimensionless concentration


Relaxation time of heat flux




Variable thermal conductivity parameter


Thermal relaxation parameter


Dimensionless temperature


Heat generation/absorption


Dimensionless variable



  1. 1.
    Reddya JVR, Sugunamma V, Sandeep N, Sulochana C. Influence of chemical reaction, radiation and rotation on MHD nanofluid flow past a permeable flat plate in porous medium. J Niger Math Soc. 2016;35:48–65.CrossRefGoogle Scholar
  2. 2.
    Khan MI, Hayat T, Waqas M, Alsaedi A. Outcome for chemically reactive aspect in flow of tangent hyperbolic material. J Mol Liq. 2017;230:143–51.CrossRefGoogle Scholar
  3. 3.
    Sulochana C, Ashwinkumar GP, Sandeep N. Effect of frictional heating on mixed convection flow of chemically reacting radiative Casson nanofluid over an inclined porous plate. Alex Eng J. 2018;57:2573–2584.CrossRefGoogle Scholar
  4. 4.
    Sultan F, Shahzad M, Ali M, Khan WA. The reaction routes comparison with respect to slow invariant manifold and equilibrium points. AIP Adv. 2019;9:015212. Scholar
  5. 5.
    Animasaun IL, Raju CSK, Sandeep N. Unequal diffusivities case of homogeneous–heterogeneous reactions within viscoelastic fluid flow in the presence of induced magnetic-field and nonlinear thermal radiation. Alex Eng J. 2016;55:1595–606. Scholar
  6. 6.
    Tanveer T, Hayat A Alsaedi, Ahmad B. Mixed convective peristaltic flow of Sisko fluid in curved channel with homogeneous–heterogeneous reaction effects. J Mol Liq. 2017;233:131–8.CrossRefGoogle Scholar
  7. 7.
    Hayat T, Saif RS, Ellahi R, Muhammad T, Ahmad B. Numerical study for Darcy–Forchheimer flow due to a curved stretching surface with Cattaneo–Christov heat flux and homogeneous–heterogeneous reactions. Results Phys. 2017;7:2886–92.CrossRefGoogle Scholar
  8. 8.
    Hayat T, Ayub S, Alsaedi A. Homogeneous–heterogeneous reactions in curved channel with porous medium. Results Phys. 2018;9:1455–61.CrossRefGoogle Scholar
  9. 9.
    Shahzad M, Sultan F, Haq I, Ali M, Khan WA. C-matrix and invariants in chemical kinetics: a mathematical concept. Pramana J Phys. 2019;92:64. Scholar
  10. 10.
    Shahzad M, Sultan F, Shah SIA, Ali M, Khan HA, Khan WA. Physical assessments on chemically reacting species and reduction schemes for the approximation of invariant manifolds. J Mol Liq. 2019;285:237–43.CrossRefGoogle Scholar
  11. 11.
    Shahzad M, Sultan F, Ali M, Khan WA, Irfan M. Slow invariant manifold assessments in multi-route reaction mechanism. J Mol Liq. 2019;284:265–70.CrossRefGoogle Scholar
  12. 12.
    Sultan F, Khan WA, Ali M, Shahzad M, Khan F, Waqas M. Slow invariant manifolds and its approximation in a multi-route reaction mechanism: a case study of iodized H2/O mechanism. J Mol Liq. 2019;288:111048.CrossRefGoogle Scholar
  13. 13.
    Bahiraei M, Alighardashi M. Investigating non-Newtonian nanofluid flow in a narrow annulus based on second law of thermodynamics. J Mol Liq. 2016;219:117–27. Scholar
  14. 14.
    Bahiraei M, Khosravi R, Heshmatian S. Assessment and optimization of hydrothermal characteristics for a non-Newtonian nanofluid flow within miniaturized concentric-tube heat exchanger considering designer’s viewpoint. Appl Therm Eng. 2017;123:266–76. Scholar
  15. 15.
    Khan WA, Sultan F, Ali M, Shahzad M, Khan M, Irfan M. Consequences of activation energy and binary chemical reaction for 3D flow of cross-nanofuid with radiative heat transfer. J Braz Soc Mech Sci Eng. 2019;41:4. Scholar
  16. 16.
    Bahiraei M, Mazaheri N, Alighardashi M. Development of chaotic advection in laminar flow of a non-Newtonian nanofluid: a novel application for efficient use of energy. Appl Therm Eng. 2017;124:1213–23. Scholar
  17. 17.
    Bahiraei M, Gharagozloo K, Alighardashi M, Mazaheri N. CFD simulation of irreversibilities for laminar flow of a power-law nanofluid within a mini channel with chaotic perturbations: an innovative energy-efficient approach. Energy Convers Manag. 2017;144:374–87. Scholar
  18. 18.
    Bhatti MM, Zeeshan A, Ellahi R. Heat transfer with thermal radiation on MHD particle–fluid suspension induced by metachronal wave. Pramana J Phys. 2017;89:48.CrossRefGoogle Scholar
  19. 19.
    Sheikholeslami M. Numerical approach for MHD Al2O3-water nanofluid transportation inside a permeable medium using innovative computer method. Comput Methods Appl Mech Eng. 2019;344:306–18.CrossRefGoogle Scholar
  20. 20.
    Sheikholeslami M, Gorji-Bandpy M, Ganji DD. MHD free convection in an eccentric semi-annulus filled with nanofluid. J Taiwan Inst Chem Eng. 2014;45(4):1204–16.CrossRefGoogle Scholar
  21. 21.
    Sheikholeslami M, Kataria HR, Mittal AS. Effect of thermal diffusion and heat-generation on MHD nanofluid flow past an oscillating vertical plate through porous medium. J Mol Liq. 2018;257:12–25.CrossRefGoogle Scholar
  22. 22.
    Khan WA, Ali M, Sultan F, Shahzad M, Khan M, Irfan M. Numerical interpretation of autocatalysis chemical reaction for nonlinear radiative 3D flow of cross magnetofluid. Pramana J Phys. 2019;92:16. Scholar
  23. 23.
    Shehzad SA, Hayat T, Alsaed A, Meraj MA. Cattaneo–Christov heat and mass flux model for 3D hydrodynamic flow of chemically reactive Maxwell liquid. Appl Math Mech Engl Ed. 2017;38(10):1347–56.CrossRefGoogle Scholar
  24. 24.
    Muhammad S, Ali G, Shah SIA, Irfan M, Khan WA, Ali M, Sultan F. Numerical treatment of activation energy for the three-dimensional flow of a cross magnetonanoliquid with variable conductivity. Pramana J Phys. 2019;93:40. Scholar
  25. 25.
    Megahed AM. Variable fluid properties and variable heat flux effects on the flow and heat transfer in a non-newtonian maxwell fluid over an unsteady stretching sheet with slip velocity. Chin Phys B. 2013;22:094701.CrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Department of MathematicsHazara UniversityMansehraPakistan
  2. 2.School of Mathematics and StatisticsBeijing Institute of TechnologyBeijingChina
  3. 3.Departement of MathematicsCOMSATS Institute of Information TechnologyAbbottabadPakistan

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