Skip to main content
SpringerLink
Log in

Micropolar fluid flow over a shrinking sheet

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

An analysis is carried out for the steady two-dimensional flow of a micropolar fluid over a shrinking sheet in its own plane. The shrinking velocity is assumed to vary linearly with the distance from a fixed point on the sheet. The features of the flow and heat transfer characteristics are analyzed and discussed. It is found that the solution exists only if adequate suction through the permeable sheet is introduced. Moreover, stronger suction is necessary for the solution to exist for a micropolar fluid compared to a classical Newtonian fluid. Dual solutions are obtained for certain suction and material parameters.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

a,m :

constants

C f :

skin friction coefficient

f :

dimensionless stream function

h :

dimensionless microrotation

j :

microinertia density

k :

thermal conductivity

s :

suction parameter

K :

material parameter

N :

angular velocity

Pr :

Prandtl number

T :

fluid temperature

T w :

surface temperature

T ∞ :

ambient temperature

u,v :

velocity components in the x and y directions, respectively

U w :

velocity of the shrinking sheet

V w :

transpiration velocity

x,y :

Cartesian coordinates along the sheet and normal to it, respectively

α :

thermal diffusivity

γ :

spin gradient viscosity

η :

similarity variable

θ :

dimensionless temperature

κ :

vortex viscosity

ν :

kinematic viscosity

μ :

dynamic viscosity

ρ :

fluid density

ψ :

stream function

w :

condition at the solid surface

∞:

ambient condition

′:

differentiation with respect to η

References

  1. Xu H, Liao SJ (2009) Laminar flow and heat transfer in the boundary-layer of non-Newtonian fluids over a stretching flat sheet. Comput Math Appl 57:1425–1431

    Article  MATH  MathSciNet  Google Scholar 

  2. Sakiadis BC (1961) Boundary-layer behavior on continuous solid surfaces: I. Boundary-layer equations for two-dimensional and axisymmetric flow. AIChE J 7:26–28

    Article  Google Scholar 

  3. Tsou FK, Sparrow EM, Goldstein RJ (1967) Flow and heat transfer in the boundary layer on a continuous moving surface. Int J Heat Mass Transf 10:219–235

    Article  Google Scholar 

  4. Crane LJ (1970) Flow past a stretching plate. Z Angew Math Phys 21:645–647

    Article  Google Scholar 

  5. Dutta BK, Roy P, Gupta AS (1985) Temperature field in flow over a stretching sheet with uniform heat flux. Int Comm. Heat Mass Transf 12:89–94

    Article  Google Scholar 

  6. Grubka LJ, Bobba KM (1985) Heat transfer characteristics of a continuous stretching surface with variable temperature. ASME J Heat Transf 107:248–250

    Article  Google Scholar 

  7. Chen CK, Char MI (1988) Heat transfer of a continuous stretching surface with suction and blowing. J Math Anal Appl 135:568–580

    Article  MATH  MathSciNet  Google Scholar 

  8. ME Ali (1995) On thermal boundary layer on a power law stretched surface with suction or injection. Int J Heat Fluid Flow 16:280–290

    Article  Google Scholar 

  9. Elbashbeshy EMA (1998) Heat transfer over a stretching surface with variable surface heat flux. J Phys D, Appl Phys 31:1951–1954

    Article  ADS  Google Scholar 

  10. Fang T (2003) Further study on a moving-wall boundary-layer problem with mass transfer. Acta Mech 163:183–188

    Article  MATH  Google Scholar 

  11. Ishak A, Nazar R, Pop I (2007) Boundary layer on a moving wall with suction and injection. Chin Phys Lett 24:2274–2276

    Article  Google Scholar 

  12. Eringen AC (1966) Theory of micropolar fluids. J Math Mech 16:1–18

    MathSciNet  Google Scholar 

  13. Eringen AC (1972) Theory of thermomicro fluids. J Math Anal Appl 38:480–496

    Article  MATH  Google Scholar 

  14. Takhar HS, Agarwal RS, Bhargava R, Jain S (1998) Mixed convection flow of a micropolar fluid over a stretching sheet. Heat Mass Transf 34:213–219

    Article  ADS  Google Scholar 

  15. Xu H, Liao SJ (2005) Series solutions of unsteady magnetohydrodynamic flows of non-Newtonian fluids caused by an impulsively stretching plate. J Non-Newton Fluid Mech 129:46–55

    Article  MATH  Google Scholar 

  16. Mahmoud MAA (2007) Thermal radiation effects on MHD flow of a micropolar fluid over a stretching surface with variable thermal conductivity. Physica A 375:401–410

    Article  ADS  Google Scholar 

  17. Ishak A, Nazar R, Pop I (2008) Mixed convection stagnation point flow of a micropolar fluid towards a stretching sheet. Meccanica 43:411–418

    Article  MATH  Google Scholar 

  18. Pal D, Hiremath PS (2010) Computational modeling of heat transfer over an unsteady stretching surface embedded in a porous medium. Meccanica 45:415–424

    Article  MathSciNet  Google Scholar 

  19. Ishak A (2010) Thermal boundary layer flow over a stretching sheet in a micropolar fluid with radiation effect. Meccanica 45:367–373

    Article  Google Scholar 

  20. Miklavčič M, Wang CY (2006) Viscous flow due to a shrinking sheet. Q Appl Math 64:283–290

    MATH  Google Scholar 

  21. Hayat T, Abbas Z, Sajid M (2007) On the analytic solution of magnetohydrodynamic flow of a second grade fluid over a shrinking sheet. J Appl Mech Trans ASME 74:1165–1171

    Article  Google Scholar 

  22. Sajid M, Javed T, Hayat T (2008) MHD rotating flow of a viscous fluid over a shrinking surface. Nonlinear Dyn 51:259–265

    Article  MATH  Google Scholar 

  23. Fang T, Liang W, Lee CF (2008) A new solution branch for the Blasius equation: a shrinking sheet problem. Comput Math Appl 56:3088–3095

    Article  MATH  MathSciNet  Google Scholar 

  24. Fang T (2008) Boundary layer flow over a shrinking sheet with power-law velocity. Int J Heat Mass Transf 51:5838–5843

    Article  MATH  Google Scholar 

  25. Fang T, Zhang J, Yao S (2009) Viscous flow over an unsteady shrinking sheet with mass transfer. Chin Phys Lett 26:014703

    Article  ADS  Google Scholar 

  26. Fang T, Yao S, Zhang J, Aziz A (2010) Viscous flow over a shrinking sheet with a second order slip flow model. Commun Nonlinear Sci Numer Simul 15:1831–1842

    Article  MATH  ADS  MathSciNet  Google Scholar 

  27. Fang T, Zhang J (2010) Thermal boundary layers over a shrinking sheet: an analytical solution. Acta Mech 209:325–343

    Article  MATH  Google Scholar 

  28. Fang T, Zhang J (2009) Closed-form exact solutions of MHD viscous flow over a shrinking sheet. Commun Nonlinear Sci Numer Simul 14:2853–2857

    Article  MATH  ADS  Google Scholar 

  29. Nadeem S, Abbasbandy S, Hussain M (2009) Series solutions of boundary layer flow of a micropolar fluid near the stagnation point towards a shrinking sheet. Z Naturforsch 64a:575–582

    Google Scholar 

  30. Nadeem S, Hussain A, Malik MY, Hayat T (2009) Series solutions for the stagnation flow of a second-grade fluid over a shrinking sheet. Appl Math Mech 30:1255–1262

    Article  MATH  MathSciNet  Google Scholar 

  31. Nadeem S, Hussain A (2009) MHD flow of a viscous fluid on a nonlinear porous shrinking sheet with homotopy analysis method. Appl Math Mech 30:1569–1578

    Article  MATH  MathSciNet  Google Scholar 

  32. Nadeem S, Faraz N (2010) Thin film flow of a second grade fluid over a stretching/shrinking sheet with variable temperature-dependent viscosity. Chin Phys Lett 27:034704

    Article  ADS  Google Scholar 

  33. Nadeem S, Hussain A, Khan M (2010) Stagnation flow of a Jeffrey fluid over a shrinking sheet. Z Naturforsch 65a:540–548

    Google Scholar 

  34. Nadeem S, Akbar NS, Malik MY (2010) Exact and numerical solutions of a micropolar fluid in a vertical annulus. Numer Methods Partial Differ Equ 26:1660–1674

    MATH  MathSciNet  Google Scholar 

  35. Ishak A, Lok YY, Pop I (2010) Stagnation-point flow over a shrinking sheet in a micropolar fluid. Chem Eng Commun 197:1417–1427

    Article  Google Scholar 

  36. Bachok N, Ishak A, Pop I (2010) Melting heat transfer in boundary layer stagnation-point flow towards a stretching/shrinking sheet. Phys Lett A 374:4075–4079

    Article  ADS  Google Scholar 

  37. Hayat T, Abbas Z, Ali N (2008) MHD flow and mass transfer of a upper-convected Maxwell fluid past a porous shrinking sheet with chemical reaction species. Phys Lett A 372:4698–4704

    Article  MATH  ADS  Google Scholar 

  38. Nadeem S, Awais M (2008) Thin film flow of an unsteady shrinking sheet through porous medium with variable viscosity. Phys Lett A 372:4965–4972

    Article  MATH  ADS  Google Scholar 

  39. Ishak A, Nazar R, Pop I (2006) Boundary-layer flow of a micropolar fluid on a continuous moving or fixed surface. Can J Phys 84:399–410

    Article  ADS  Google Scholar 

  40. Ahmadi G (1976) Self-similar solution of incompressible micropolar boundary layer flow over a semi-infinite plate. Int J Eng Sci 14:639–646

    Article  MATH  Google Scholar 

  41. Cebeci T, Bradshaw P (1988) Physical and computational aspects of convective heat transfer. Springer, New York

    Book  MATH  Google Scholar 

  42. Ishak A, Yacob NA, Bachok N (2010) Radiation effects on the thermal boundary layer flow over a moving plate with convective boundary condition. Meccanica. doi:10.1007/s11012-010-9338-4

    Google Scholar 

  43. Ishak A (2009) Mixed convection boundary layer flow over a horizontal plate with thermal radiation. Heat Mass Transf 46:147–151

    Article  ADS  Google Scholar 

  44. Ishak A (2009) Radiation effects on the flow and heat transfer over a moving plate in a parallel stream. Chin Phys Lett 26:034701

    Article  Google Scholar 

  45. Ishak A, Nazar R, Bachok N, Pop I (2010) Melting heat transfer in steady laminar flow over a moving surface. Heat Mass Transf 46:463–468

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA Pahang, 26400, Bandar Jengka, Pahang, Malaysia

    Nor Azizah Yacob

  2. School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600, UKM Bangi, Selangor, Malaysia

    Anuar Ishak

Authors
  1. Nor Azizah Yacob
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anuar Ishak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anuar Ishak.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yacob, N.A., Ishak, A. Micropolar fluid flow over a shrinking sheet. Meccanica 47, 293–299 (2012). https://doi.org/10.1007/s11012-011-9439-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-011-9439-8

Keywords

Search

Navigation