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Apr 28, 2017
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Nov 1, 2016
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Abstract

The aim of the present study is to analyze numerically the steady boundary layer flow and heat transfer characteristics of Casson fluid with variable temperature and viscous dissipation past a permeable shrinking sheet with second order slip velocity. Using appropriate similarity transformations, the basic nonlinear partial differential equations have been transformed into ordinary differential equations. These equations have been solved numerically for different values of the governing parameters namely: shrinking parametersuction parameterCasson parameterfirst order slip parametersecond order slip parameter  Prandtl number  and the Eckert number  using the bvp4c function from MATLAB. A stability analysis has also been performed. Numerical results have been obtained for the reduced skin-friction, heat transfer and the velocity and temperature profiles. The results indicate that dual solutions exist for the shrinking surface for certain values of the parameter space. The stability analysis indicates that the lower solution branch is unstable, while the upper solution branch is stable and physically realizable. In addition, it is shown that for a viscous fluida very good agreement exists between the present numerical results and those reported in the open literature. The present results are original and new for the boundary-layer flow and heat transfer past a shrinking sheet in a Casson fluid. Therefore, this study has importance for researchers working in the area of non-Newtonian fluids, in order for them to become familiar with the flow behavior and properties of such fluids.

 

 

Keywords

Boundary layer Casson fluid Dual solutions Shrinking surface Second-order slip Stability analysis.

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References

  1. Sparrow, E.M. and Abraham, J.P. Universal solutions for the streamwise variation of the temperature of a moving sheet in the presence of a moving fluid. Int. J. Heat Mass Trans., 2005, 48, 3047-3056.
  2. Abraham, J.P. and Sparrow, E.M. Friction drag resulting from the simultaneous imposed motions of a free stream and its bounding surface. Int. J. Heat Fluid Flow, 2005, 26, 289-295.
  3. Crane, L.J. Flow past a stretching plate. J. Appl. Math. Physics (ZAMP), 1970, 21, 645–647.
  4. Skelland, A.H.P. Non-Newtonian Flow and Heat Transfer, John Wiley & Sons, New York, 1966.
  5. Denn, M.M. Boundary-layer flows for a class of elastic fluids. Chem. Eng. Sci., 1967, 22, 395–405.
  6. Rajagopal, K.R., Gupta, A.S. and Wineman, A.S. On a boundary layer theory for non-Newtonian fluids. Appl. Sci. Eng. Lett., 1980, 18, 875–883.
  7. Bird, R.B., Armstrong, R.C. and Hassager, O. Dynamics of Polymer Liquids (2nd edition), Wiley, New York, 1987.
  8. Slattery, J.C. Advanced Transport Phenomena, Cambridge University Press, Cambridge, 1999.
  9. Rivlin, R.S. and Ericksen, J.L. Stress deformation relations for isotropic materials. J. Rational Mech. Anal., 1955, 4, 323–425.
  10. Oldroyd, J.G. On the formulation of rheological equations of state. Proc. R. Soc. London A, 1950, 200, 523–541.
  11. Hayat, T., Shafiq, A. and Alsaedi A. Effect of Joule heating and thermal radiation in flow of third grade fluid over radiative surface. PLoS ONE, 2015, 9(1), e83153, doi:10.1371/journal.pone.0083153.
  12. Fredrickson, A.G. Principles and Applications of Rheology. Prentice-Hall, Englewood Cliffs, New Jersey, 1964.
  13. Eldabe, N.T.M. and Salwa, M.G.E. Pulsatile magnetohydrodynamic viscoelastic flow through a channel bounded by two permeable parallel plates. J. Phys. Soc. Japan,1995, 64, 4163-4174.
  14. Boyd, J., Buick, J.M. and Green, S. Analysis of the Casson and Carreau-Yasuda non-Newtonian blood models in steady and oscillatory flow using the lattice Boltzmann method. Phys. Fluids, 2007, 19, 93-103.
  15. Mernone, A.V., Mazumdar, J.N. and Lucas, S.K. A mathematical study of peristaltic transport of a Casson fluid. Math. Comp. Model., 2002, 35, 895–912.
  16. Mustafa, M., Hayat, T., Pop, I. and Aziz, A. Unsteady boundary layer flow of a Casson fluid due to an impulsively started moving flat plate. Heat Transfer Asian Res., 2011, 40, 563–576.
  17. Pramanik, S. Casson fluid flow and heat transfer past an exponentially porous stretching surfcae in presence of thermal radiation. Ain Shams Eng. J., 2014, 5, 205-212.
  18. Nakamura, M. and Sawada, T. Numerical study on the flow of a non-Newtonian fluid through an axisymmetric stenosis. ASME J. Biomechanical Eng., 1988, 110, 137–143.
  19. Wu, L. A slip model for rarefied gas flows at arbitrary Knudsen number. Appl. Phys. Lett., 2008, 93, 253103 (3 pages).
  20. Fang, T., Yao, S., Zhang, J. and Aziz, A. Viscous flow over a shrinking sheet with a second order slip flow model. Commun. Nonlinear Sci. Numer. Simulat., 2010, 15, 1831–1842.
  21. Mahmood, T., Shah, S.M. and Abbas, G. Magnetohydrodynamic viscous flow over a shrinking sheet with second order slip flow model. Heat Trsnf. Res., 2015, doi:10.1615/HeatTransRes.2015007512.
  22. Fukui, S. and Kaneko, R. A database for interpolation of Poiseuille flow rates for high Knudsen number lubrication problems. ASME J. Tribol., 1990, 112, 78–83.
  23. Weidman, P.D., Kubittschek, D.G. and Davis, A.M.J. The effect of transpiration on self-similar boundary layer flow over moving surfaces. Int. J. Eng. Sci., 2006, 44, 730-737.
  24. Roşca, A.V. and Pop, I. Flow and heat transfer over a vertical permeable stretching/shrinking sheet with a second order slip. Int. J. Heat Mass Trans., 2013, 60, 355–364.
  25. Roşca, N.C. and Pop, I. Mixed convection stagnation point flow past a vertical flat plate with a second order slip: heat flux case. Int. J. Heat Mass Trans., 2013, 65, 102-109.
  26. Rahman, M.M., Roşca, A.V. and Pop, I. Boundary layer flow of a nanofluid past a permeable exponentially shrinking/stretching surface with second order slip using Buongiorno’s model. Int. J. Heat Mass Trans., 2014, 77, 1133-1143.
  27. Rahman, M.M. and Pop, I. Mixed convection boundary layer stagnation-point flow of a Jeffrey fluid past a permeable vertical flat plate. Zeitschrift fuer Naturforschung A (ZNA), 2014, 69(12), 687-696.
  28. Rahman, M.M., Roşca, A.V. and Pop, I. Boundary layer flow of a nanofluid past a permeable exponentially shrinking surface with convective boundary condition using Buongiorno’s model. Int. J. Num. Methods Heat & Fluid Flow, 2015, 25(2), 299-319.
  29. Harris, S.D., Ingham, D.B. and Pop, I. Mixed convection boundary-layer flow near the stagnation point on a vertical surface in a porous medium: Brinkman model with slip. Trans. Porous Media, 2009, 77, 267-285.
  30. Shampine, L.F., Gladwell, I. and Thompson, S. Solving ODEs with MATLAB. Cambridge University Press, Cambridge, 2003.
  31. Shampine, L.F., Reichelt, M.W. and Kierzenka, J. Solving boundary value problems for ordinary differential equations in MATLAB with bvp4c, 2010. <http://www.mathworks.com/bvp_tutorial>.
  32. Wang, C.Y. Flow due to a stretching boundary with partial slip-an exact solution of the Navier-Stokes equations. Chem. Eng. Sci., 2002, 57, 3745-3747.
  33. Wang, C.Y. Analysis of viscous flow due to a stretching sheet with surface slip and suction. Nonlinear Anal. Real World Appl., 2009, 10(1), 375-380.
  34. Anderson, H.I. Slip flow past a stretching surface. Acta Mech., 2002, 158, 121-125.
  35. Sahoo, B. and Do, Y. Effects of slip on sheet-driven flow and heat transfer of third grade fluid past a stretching sheet. Int. Commun. Heat Mass Transfer, 2010, 37, 1064-1071.
  36. Miklavčič, M. and Wang, C.Y. Viscous flow due to a shrinking sheet. Q. Appl. Math., 2006, 64, 283-290.