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Investigation of generalized Fick’s and Fourier’s laws in the second-grade fluid flow

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Abstract

The second-grade fluid flow due to a rotating porous stretchable disk is modeled and analyzed. A porous medium is characterized by the Darcy relation. The heat and mass transport are characterized through Cattaneo-Christov double diffusions. The thermal and solutal stratifications at the surface are also accounted. The relevant nonlinear ordinary differential systems after using appropriate transformations are solved for the solutions with the homotopy analysis method (HAM). The effects of various involved variables on the temperature, velocity, concentration, skin friction, mass transfer rate, and heat transfer rate are discussed through graphs. From the obtained results, decreasing tendencies for the radial, axial, and tangential velocities are observed. Temperature is a decreasing function of the Reynolds number, thermal relaxation parameter, and Prandtl number. Moreover, the mass diffusivity decreases with the Schmidt number.

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References

  1. KARMAN, T. V. Uber laminare und turbulente reibung. Journal of Applied Mathematics and Mechanics, 1, 233–252 (1921)

    MATH  Google Scholar 

  2. REDDY, P. S., SREEDEVI, P., and CHAMKHA, A. J. MHD boundary layer flow, heat and mass transfer analysis over a rotating disk through porous medium saturated by Cu-water and Ag-water nanofluid with chemical reaction. Powder Technology, 307, 46–55 (2017)

    Article  Google Scholar 

  3. SHEIKHOLESLAMI, M., HATAMI, M., and GANJI, D. D. Numerical investigation of nanofluid spraying on an inclined rotating disk for cooling process. Journal of Molecular Liquids, 211, 577–583 (2015)

    Article  Google Scholar 

  4. TURKYILMAZOGLU, M. Nanofluid flow and heat transfer due to a rotating disk. Computer and Fluids, 94, 139–146 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. HAYAT, T., JAVED, M., IMTIAZ, M., and ALSAEDI, A. Convective flow of Jeffrey nanofluid due to two stretchable rotating disks. Journal of Molecular Liquids, 240, 291–302 (2017)

    Article  Google Scholar 

  6. IMTIAZ, M., HAYAT, T., ALSAEDI, A., and AHMAD, B. Convective flow of carbon nanotubes between rotating stretchable disks with thermal radiation effects. International Journal of Heat and Mass Transfer, 101, 948–957 (2016)

    Article  Google Scholar 

  7. HAYAT, T., QAYYUM, S., KHAN, M. I., and ALSAEDI, A. Current progresses about probable error and statistical declaration for relitive two phase flow using Ag-H2O and Cu-H2O nanoma-terials. International Journal of Hydrogen Energy, 42, 29107–29120 (2017)

    Article  Google Scholar 

  8. TAKHAR, H. S., BHARGAVA, R., AGRAWAL, R. S., and BALAJI, A. V. S. Finite element solution of micropolar fluid flow and heat transfer between two porous discs. International Journal of Engineering Science, 38, 1907–1922 (2000)

    Article  MATH  Google Scholar 

  9. KHAN, M. I., WAQAS, M., HAYAT, T., and ALSAEDI, A. A comparative study o. Casson fluid with homogeneous-heterogeneous reactions. Journal of Colloid and Interface Science, 498, 85–90 (2017)

    Article  Google Scholar 

  10. KHAN, M. I., WAQAS, M., HAYAT, T., KHAN, M. I., and ALSAEDI, A. Behavior of stratifica-tion phenomenon in flow of Maxwell nanomaterial with motile gyrotactic microorganisms in the presence of magnetic field. International Journal of Mechanical Sciences, 132, 426–434 (2017)

    Article  Google Scholar 

  11. KHAN, M. I., WAQAS, M., HAYAT, T., ALSAEDI, A., and KHAN, M. I. Significance of nonlinear radiation in mixed convection flow of magneto Walter-B nanoliquid. International Journal of Hydrogen Energy, 42, 26408–26416 (2017)

    Article  Google Scholar 

  12. AZIZ, A., ALSAEDI, A., MUHAMMAD, T., and HAYAT, T. Numerical study for heat genera-tion/absorption in flow of nanofluid by a rotating disk. Results in Physics, 8, 785–792 (2018)

    Article  Google Scholar 

  13. HAYAT, T., HAIDER, F., MUHAMMAD, T., and ALSAEDI, A. On Darcy-Forchheimer flow of carbon nanotubes due to a rotating disk. International Journal of Heat and Mass Transfer, 112, 248–254 (2017)

    Article  Google Scholar 

  14. HAYAT, T., MUHAMMAD, T., SHEHZAD, S. A., and ALSAEDI, A. On magnetohydrodynamic flow of nanofluid due to a rotating disk with slip effect: a numerical study. Computer Methods in Applied Mechanics and Engineering, 315, 467–477 (2017)

    Article  MathSciNet  Google Scholar 

  15. SRINIVASACHARYA, D. and SURENDER, O. Effect of double stratification on mixed convection boundary layer flow of a nanofluid past a vertical plate in a porous medium. Applied Nanoscience, 5, 29–38 (2015)

    Article  Google Scholar 

  16. HAYAT, T., WAQAS, M., KHAN, M. I., and ALSAEDI, A. Analysis of thixotropic nanomaterial in a doubly stratified medium considering magnetic field effects. International Journal of Heat and Mass Transfer, 102, 1123–1129 (2016)

    Article  Google Scholar 

  17. BABU, M. J. and SANDEEP, N. UCM flow across a melting surface in the presence of double stratification and cross-diffusion effects. Journal of Molecular Liquids, 232, 27–35 (2017)

    Article  Google Scholar 

  18. NADEEM, S. and MUHAMMAD, N. Impact of stratification and Cattaneo-Christov heat flux in the flow saturated with porous medium. Journal of Molecular Liquids, 224, 423–430 (2016)

    Article  Google Scholar 

  19. WAQAS, M., KHAN, M. I., HAYAT, T., and ALSAEDI, A. Stratified flow of an Oldroyd-B nanoliquid with heat generation. Results in Physics, 7, 2489–2496 (2017)

    Article  Google Scholar 

  20. FOURIER, J. B. J. Théorie Analytique De La Chaleur, Cambridge University Press, Cambridge (1822)

    MATH  Google Scholar 

  21. CATTANEO, C. Sulla Conduzione Del Calore, Springer, Berlin Heidelberg (1948)

    MATH  Google Scholar 

  22. CHRISTOV, C. I. On frame indifferent formulation of the Maxwell-Cattaneo model of finite-speed heat conduction. Mechanics Research Communications, 36, 481–486 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. HAYAT, T., FAROOQ, T., ALSAEDI, A., and AL-SOLAMY, F. Impact of Cattaneo-Christov heat flux in the flow over a stretching sheet with variable thickness. AIP Advances, 5, 087159 (2015)

    Article  Google Scholar 

  24. HAYAT, T., QAYYUM, S., IMTIAZ, M., and ALSAEDI, A. Impact of Cattaneo-Christov heat flux in Jeffrey fluid flow with homogeneous-heterogeneous reactions. PloS One, 11, e0148662(2016)

    Google Scholar 

  25. SHEHZAD, S. A., ABBASI, F. M., HAYAT, T., and AHMAD, B. Cattaneo-Christov heat flux model for third-grade fluid flow towards exponentially stretching sheet. Applied Mathematics and Mechanics (English Edition), 37(6), 761–768 (2016) https://doi.org/10.1007/s10483-016-2088-6

    Article  MathSciNet  Google Scholar 

  26. HAYAT, T., AYUB, T., MUHAMMAD, T., and ALSAEDI, A. Flow of variable thermal conduc-tivity Oldroyd-B fluid with generalized Fourier’s and Fick’s laws. Journal of Molecular Liquids, 234, 9–17 (2017)

    Article  Google Scholar 

  27. HAYAT, T., KHAN, M. I., FAROOQ, M., ALSAEDI, A., WAQAS, M., and YASMEEN, T. Impact of Cattaneo-Christov heat flux model in flow of variable thermal conductivity fluid over a variable thicked surface. International Journal of Heat and Mass Transfer, 99, 702–710 (2016)

    Article  Google Scholar 

  28. MERAJ, M. A., SHEHZAD, S. A., HAYAT, T., ABBASI, F. M., and ALSAEDI, A. Darcy-Forchheimer flow of variable conductivity Jeffrey liquid with Cattaneo-Christov heat flux theory. Applied Mathematics and Mechanics (English Edition), 38(4), 557–566 (2017) https://doi.org/10.1007/s10483-017-2188-6

    Article  MathSciNet  Google Scholar 

  29. RIVLIN, R. S. and ERICKSEN, J. L. Stress deformation relations for isotropic materials. Journal of Rational Mechanics and Analysis, 4, 323–425 (1955)

    MathSciNet  MATH  Google Scholar 

  30. WAQAS, M., KHAN, M. I., HAYAT, T., ALSAEDI, A., and KHAN, M. I. Nonlinear thermal ra-diation in flow induced by a slendering surface accounting thermophoresis and Brownian diffusion. The European Physical Journal Plus, 132, 280 (2017)

    Article  Google Scholar 

  31. KHAN, M. I., WAQAS, M., HAYAT, T., and ALSAEDI, A. Chemically reactive flow of micropolar fluid accounting viscous dissipation and Joule heating. Results in Physics, 7, 3706–3715 (2017)

    Article  Google Scholar 

  32. HAYAT, T., KHAN, M. I., FAROOQ, M., YASMEEN, T., and ALSAEDI, A. Water-carbon nanofluid flow with variable heat flux by a thin needle. Journal of Molecular Liquids, 224, 786–791 (2016)

    Article  Google Scholar 

  33. KHAN, M. I., HAYAT, T., KHAN, M. I., and ALSAEDI, A. A modified homogeneous-heterogeneous reactions for MHD stagnation flow with viscous dissipation an. Joule heating. International Journal of Heat and Mass Transfer, 113, 310–317 (2017)

    Article  Google Scholar 

  34. HAYAT, T., WAQAS, M., KHAN, M. I., and ALSAEDI, A. Analysis of thixotropic nanomaterial in a doubly stratified medium considering magnetic field effects. International Journal of Heat and Mass Transfer, 102, 1123–1129 (2016)

    Article  Google Scholar 

  35. ZHAO, Q., XU, H., TAO, L., RAEES, A., and SUN, Q. Three-dimensional free bio-convection of nanofluid near stagnation point on general curved isothermal surface. Applied Mathematics and Mechanics (English Edition), 37(4), 417–432 (2016) https://doi.org/10.1007/s10483-016-2046-9

    Article  MathSciNet  MATH  Google Scholar 

  36. HAYAT, T., MAKHDOOM, S., AWAIS, SALEEM, S., and RASHIDI, M. M. Axisymmetric Powell-Eyring fluid flow with convective boundary condition: optimal analysis. Applied Mathematics and Mechanics (English Edition), 37(7), 919–928 (2016) https://doi.org/10.1007/s10483-016-2093-9

    Article  MathSciNet  Google Scholar 

  37. HAYAT, T., AHMAD, S., KHAN, M. I., ALSAEDI, A., and WAQAS, M. Investigation of second-grade fluid through temperature dependent thermal conductivity and non-Fourier heat flux. Results in Physics, 9, 871–878 (2018)

    Article  Google Scholar 

  38. ZHU, J., WANG, S., ZHENG, L., and ZHANG, X. Heat transfer of nanofluids considering nanopar-ticle migration and second-order slip velocity. Applied Mathematics and Mechanics (English Edition ), 38(1), 125–136 (2017) https://doi.org/10.1007/s10483-017-2155-6

    Article  MathSciNet  Google Scholar 

  39. HAYAT, T., AHMAD, S., KHAN, M. I., and ALSAEDI, A. Modeling and analyzing flow of third grade nanofluid due to rotating stretchable disk with chemical reaction and heat source. Physica B: Condensed Matter, 537, 116–126 (2018)

    Article  Google Scholar 

  40. HAYAT, T., QAYYUM, S., ALSAEDI, A., and AHMAD, B. Significant consequences of heat generation/absorption and homogeneous-heterogeneous reactions in second-grade fluid due to rotating disk. Results in Physics, 8, 223–230 (2018)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Quaid-i-Azam University 45320, Islamabad, 44000, Pakistan

    T. Hayat, S. Ahmad & M. I. Khan

  2. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    T. Hayat & A. Alsaedi

Authors
  1. T. Hayat
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  2. S. Ahmad
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  3. M. I. Khan
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  4. A. Alsaedi
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Correspondence to M. I. Khan.

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Project supported by the Natural Science and Engineering Research Council (NSERC) of Canada (No.NSERC-RGPIN204992)

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Hayat, T., Ahmad, S., Khan, M.I. et al. Investigation of generalized Fick’s and Fourier’s laws in the second-grade fluid flow. Appl. Math. Mech.-Engl. Ed. 39, 1617–1630 (2018). https://doi.org/10.1007/s10483-018-2390-6

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  • DOI: https://doi.org/10.1007/s10483-018-2390-6

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