The application of non-Fourier and Fick's laws to the flow of temperature-dependent thermal conductivity generalized Newtonian liquids: A 3D computational study

Document Type : Article


1 Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan.

2 Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan.; Department of Mathematics, Shaheed Benazir Bhutto University, Sheringal Upper Dir 18000, Pakistan.

3 Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan.;Department of Mathematics and Statistics, Hazara University, Mansehra 21300, Pakistan.

4 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia.


The current article presents a numerical study for local solutions of Sisko liquid flow by bidirectional stretched surface. Additionally, the analysis of relaxation times for the heat and mass transfer mechanisms are made by utilizing modified heat flux and mass diffusion models. These improved relations are the generalized form of Fourier's and Fick's laws in which the time space upper-convected derivative are employed to portray the heat conduction and mass diffusion mechanisms. Appropriate transformations lead to a strongly nonlinear differential system of equations which is then solved numerically by employing the applications of bvp4c package in Matlab software. Another numerical method namely shooting technique with RK45 Fehlberg and Newton-Raphson method is utilized to authenticate the results. Graphical illustrations demonstrating the impacts of sundry physical parameters with required discussion highlighting their physical effect are also a part of this exploration. It is perceived that the temperature and concentration of Sisko liquid are diminishing functions of relaxation times for the heat and mass transfer mechanisms. It is also fascinating to perceive that the temperature and concentration of Sisko liquid are higher in case of classical form in comparison with an improved constitutive relations.


Main Subjects

1.Liao, S.J. and Pop, I. Explicit analytic solution for similarity boundary layer equations", Int. J. Heat Mass Transf., 47(1), pp. 75-85 (2004).
2. Alam, M.S. and Ahammad, M.U. E_ects of variable chemical reaction and variable electric conductivity on free convective heat and mass transfer ow along an inclined stretching sheet with variable heat and mass uxes under the inuence of Dufour and Soret effects", Nonlinear Anal. Model. Control, 16, pp. 1{16 (2011). M. Khan et al./Scientia Iranica, Transactions C: Chemistry and ... 26 (2019) 1516{1528 1527
3. Isaa, S.S.P.M., Ari_n, N.M., Nazarc, R., Bachok, N., Ali, F.M., and Pop, I. MHD mixed convection boundary layer ow of a Casson uid bounded by permeable shrinking sheet with exponential variation", Scientia Iranica B, 24(2), pp. 637-647 (2017). 4. Ramly, N.A., Sivasankaran, S., and Noor, N.F.M. Zero and nonzero normal uxes of thermal radiative boundary layer ow of nano uid over a radially stretched surface", Scientia Iranica B, 24(6), pp. 2895- 2903 (2017). 5. Khan, Y. Magnetohydrodynamic ow of linear viscoelastic uid model above a shrinking/stretching sheet: A series solution", Scientia Iranica B, 24(5), pp. 2466- 2472 (2017). 6. Niranjan, H., Sivasankaran, S., and Bhuvaneswari, M. Chemical reaction, Soret and Dufour e_ects on MHD mixed convection stagnation point ow with radiation and slip condition", Scientia Iranica B, 24(2), pp. 698- 706 (2017). 7. Hussain, Q., Asghar, S., and Alsaedi, A. Heat transfer analysis in peristaltic slip ow with Hall and ion-slip currents", Scientia Iranica C, 23(6), pp. 2771-2783 (2016). 8. Heydari, M.M. Investigation of uid ow and heat transfer of compressible ow in a constricted microchannel", Scientia Iranica B, 23(5), pp. 2144-2153 (2016). 9. Cattaneo, C. Sulla conduzione Del Calore", Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia, 3, pp. 83-101 (1948). 10. Christov, C.I. On frame indi_erent formulation of the Maxwell-Cattaneo model of _nite speed heat conduction", Mech. Res. Commun., 36, pp. 481-486 (2009). 11. Ciarletta, M. and Straughan, B. Uniqueness and structural stability for the Cattaneo-Christov equations", Mech. Res. Commun., 37, pp. 445-447 (2010). 12. Hayat, T., Imtiaz, M., Alsaedi, A., and Almezal, S. On Cattaneo-Christov heat ux in MHD ow of Oldroyd-B uid with homogeneous-heterogeneous reactions", J. Mol. Liq., 401, pp. 296-303 (2016). 13. Khan, M. and Khan, W.A. Three-dimensional ow and heat transfer to Burgers uid using Cattaneo- Christov heat ux model", J. Mol. Liq., 221,pp. 651- 657 (2016). 14. Waqas, M., Hayat, T., Farooq, M., Shehzad, S.A., and Alsaedi, A. Cattaneo-Christov heat ux model for ow of variable thermal conductivity generalized Burgers uid", J. Mol. Liq., 220, pp. 642-648 (2016). 15. Sui, J., Zheng, L., and Zhang, X. Boundary layer heat and mass transfer with Cattaneo-Christov doubledi _usion in upper-convected Maxwell nanouid past a stretching sheet with slip velocity", Int. J. Therm. Sci., 104, pp. 461-468 (2016). 16. Liu, L., Zheng, L., Liu, F., and Zhang, X. Anomalous convection di_usion and wave coupling transport of cells on comb frame with fractional Cattaneo-Christov ux", Commun. Nonlinear Sci. Numer. Simulat., 38, pp. 45-58 (2016). 17. Khan, W.A., Khan, M., and Alshomrani, A.S. Impact of chemical processes on 3D Burgers uid utilizing Cattaneo-Christov double-di_usion: Applications of non-Fourier's heat and non-Fick's mass ux models", J. Mol. Liq., 223, pp. 1039-1047 (2016). 18. Malik, M.Y., Khan, M., Salahuddin, T., and Khan, I. Variable viscosity and MHD ow in Casson uid with Cattaneo-Christov heat ux model: Using Keller box method", Engi. Sci. Tech., Int. J., 19(4), pp. 1985- 1992 (2016). 19. Muhammad, N., Nadeem, S., and Mustafa, T. Squeezed ow of a nanouid with Cattaneo-Christov heat and mass uxes", Res. Phys., 7, pp. 862-869 (2017). 20. Khan, M., Shahid, A., Malik, M.Y., and Salahuddin, T. Thermal and concentration di_usion in Je_ery nanouid ow over an inclined stretching sheet: A generalized Fourier's and Fick's perspective", J. Mol. Liq., 251, pp. 7-14 (2018). 21. Munir, A., Shahzad, A., and Khan, M. Convective ow of Sisko uid over a bidirectional stretching surface", PLOS ONE, 10(6), e0130342 (2015). 22. Khan, M., Malik, R., Munir, A., and Khan, W.A. Flow and heat transfer to Sisko nanouid over a nonlinear stretching sheet", PLOS ONE, 10(5), e0125683 (2015). 23. Malik, R., Khan, M., Munir, A., and Khan, W.A. Flow and heat transfer in Sisko uid with convective boundary condition", PLOS ONE, 9(10), e107989 (2014). 24. Khan, W.A., Khan, M., Alshomrani, A.S., and Ahmad, L. Numerical investigation of generalized Fourier's and Fick's laws for Sisko uid ow", J. Mol. Liq., 224, pp. 1016-1021 (2016). 25. Khan, M., Ahmad, L., and Khan, W.A. Numerically framing the impact of radiation on magnetonanoparticles for 3D Sisko uid ow", J. Braz. Soc. Mech. Sci. Eng., 39(11), pp. 4475-4487 (2017). 26. Awais, M., Malik, M.Y., Bilal, S., Salahuddin, T., and Hussain, A. Magnetohydrodynamic (MHD) ow of Sisko uid near the axisymmetric stagnation point towards a stretching cylinder", Res. Phys., 7, pp. 49-56 (2017). 27. Hussain, A., Malik, M.Y., Salahuddin, T., Bilal, S., and Awais, M. Combined e_ects of viscous dissipation and Joule heating on MHD Sisko nanouid over a stretching cylinder", J. Mol. Liq., 231, pp. 341-352 (2017). 28. Hussain, A., Malik, M.Y., Bilal, S., Awais, M., and Salahuddin, T. Computational analysis of magnetohydrodynamic Sisko uid ow over a stretching cylinder in the presence of viscous dissipation and temperature dependent thermal conductivity", Res. Phys., 7, pp. 139-146 (2017). 29. Fourier, J.B.J., TTh_eorie Analytique De La Chaleur, Paris (1822).