Unsteady Boundary Layer Flow over a Stretching Sheet in a Micropolar Fluid
Unsteady boundary layer flow of an incompressible
micropolar fluid over a stretching sheet when the sheet is stretched in
its own plane is studied in this paper. The stretching velocity is
assumed to vary linearly with the distance along the sheet. Two equal
and opposite forces are impulsively applied along the x-axis so that the
sheet is stretched, keeping the origin fixed in a micropolar fluid. The
transformed unsteady boundary layer equations are solved
numerically using the Keller-box method for the whole transient from
the initial state to final steady-state flow. Numerical results are
obtained for the velocity and microrotation distributions as well as the
skin friction coefficient for various values of the material parameter K.
It is found that there is a smooth transition from the small-time
solution to the large-time solution.
[1] L. J. Crane, "Flow past a stretching plane", J. Appl. Math. Phys. (ZAMP),
vol. 21, pp. 645-647, 1970.
[2] E. Magyari, and B. Keller, "Heat and mass transfer in the boundary layers
on an exponentially stretching continuous surface", J. Phys. D: Appl.
Phys., vol. 32, pp. 577-586, 1999.
[3] E. Magyari, and B. Keller, "Exact solutions for self-similar
boundary-layer flows induced by permeable stretching surfaces", Eur. J.
Mech. B-Fluids, vol. 19, pp. 109-122, 2000.
[4] S. J. Liao, and I. Pop, "Explicit analytic solution for similarity boundary
layer equations", Int. J. Heat Mass Transfer, vol. 47, pp. 75-85, 2004.
[5] R. Nazar, N. Amin and I. Pop, "Unsteady boundary layer flow due to a
stretching surface in a rotating fluid", Mech. Res. Comm., vol. 31, pp.
121-128, 2004.
[6] M. Kumari, A. Slaouti, H. S. Takhar, S. Nakamura, and G. Nath, "Unsteady
free convection flow over a continuous moving vertical surface", Acta
Mechanica, vol. 116, pp. 75-82, 1996.
[7] A. Ishak, R. Nazar, and I. Pop, "Unsteady mixed convection boundary layer
flow due to a stretching vertical surface", Arabian J. Sci. Engng., vol. 31,
pp. 165-182, 2006.
[8] I. Pop, and T. Y. Na, "Unsteady flow past a stretching sheet", Mech. Res.
Comm., vol. 23, pp. 413-422, 1996.
[9] C. Y. Wang, G. Du, M. Miklavcic, and C. C. Chang, "Impulsive stretching
of a surface in a viscous fluid", SIAM J. Appl. Math., vol. 57, pp. 1-14,
1997.
[10] A. C. Eringen, "Theory of micropolar fluids", J. Math. Mech., vol. 16, pp.
1-18, 1966.
[11] A. C. Eringen, "Theory of thermomicrofluids", J. Math. Anal. Appl., vol.
38, pp. 480-496, 1972.
[12] A. Ishak, R. Nazar, and I. Pop, "Heat transfer over a stretching surface with
variable surface heat flux in micropolar fluids", Phys. Lett. A, vol. 372, pp.
559-561, 2008.
[13] A. Ishak, R. Nazar, and I. Pop, "Magnetohydrodynamic stagnation point
flow towards a stretching vertical sheet in a micropolar fluid",
Magnetohydrodynamics, vol. 43(1), pp. 83-97, 2007.
[14] T. Cebeci, and P. Bradshaw, Physical and Computational Aspects of
Convective Heat Transfer. New York: Springer, 1984, p. 391.
[1] L. J. Crane, "Flow past a stretching plane", J. Appl. Math. Phys. (ZAMP),
vol. 21, pp. 645-647, 1970.
[2] E. Magyari, and B. Keller, "Heat and mass transfer in the boundary layers
on an exponentially stretching continuous surface", J. Phys. D: Appl.
Phys., vol. 32, pp. 577-586, 1999.
[3] E. Magyari, and B. Keller, "Exact solutions for self-similar
boundary-layer flows induced by permeable stretching surfaces", Eur. J.
Mech. B-Fluids, vol. 19, pp. 109-122, 2000.
[4] S. J. Liao, and I. Pop, "Explicit analytic solution for similarity boundary
layer equations", Int. J. Heat Mass Transfer, vol. 47, pp. 75-85, 2004.
[5] R. Nazar, N. Amin and I. Pop, "Unsteady boundary layer flow due to a
stretching surface in a rotating fluid", Mech. Res. Comm., vol. 31, pp.
121-128, 2004.
[6] M. Kumari, A. Slaouti, H. S. Takhar, S. Nakamura, and G. Nath, "Unsteady
free convection flow over a continuous moving vertical surface", Acta
Mechanica, vol. 116, pp. 75-82, 1996.
[7] A. Ishak, R. Nazar, and I. Pop, "Unsteady mixed convection boundary layer
flow due to a stretching vertical surface", Arabian J. Sci. Engng., vol. 31,
pp. 165-182, 2006.
[8] I. Pop, and T. Y. Na, "Unsteady flow past a stretching sheet", Mech. Res.
Comm., vol. 23, pp. 413-422, 1996.
[9] C. Y. Wang, G. Du, M. Miklavcic, and C. C. Chang, "Impulsive stretching
of a surface in a viscous fluid", SIAM J. Appl. Math., vol. 57, pp. 1-14,
1997.
[10] A. C. Eringen, "Theory of micropolar fluids", J. Math. Mech., vol. 16, pp.
1-18, 1966.
[11] A. C. Eringen, "Theory of thermomicrofluids", J. Math. Anal. Appl., vol.
38, pp. 480-496, 1972.
[12] A. Ishak, R. Nazar, and I. Pop, "Heat transfer over a stretching surface with
variable surface heat flux in micropolar fluids", Phys. Lett. A, vol. 372, pp.
559-561, 2008.
[13] A. Ishak, R. Nazar, and I. Pop, "Magnetohydrodynamic stagnation point
flow towards a stretching vertical sheet in a micropolar fluid",
Magnetohydrodynamics, vol. 43(1), pp. 83-97, 2007.
[14] T. Cebeci, and P. Bradshaw, Physical and Computational Aspects of
Convective Heat Transfer. New York: Springer, 1984, p. 391.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:60430", author = "Roslinda Nazar and Anuar Ishak and Ioan Pop", title = "Unsteady Boundary Layer Flow over a Stretching Sheet in a Micropolar Fluid", abstract = "Unsteady boundary layer flow of an incompressible
micropolar fluid over a stretching sheet when the sheet is stretched in
its own plane is studied in this paper. The stretching velocity is
assumed to vary linearly with the distance along the sheet. Two equal
and opposite forces are impulsively applied along the x-axis so that the
sheet is stretched, keeping the origin fixed in a micropolar fluid. The
transformed unsteady boundary layer equations are solved
numerically using the Keller-box method for the whole transient from
the initial state to final steady-state flow. Numerical results are
obtained for the velocity and microrotation distributions as well as the
skin friction coefficient for various values of the material parameter K.
It is found that there is a smooth transition from the small-time
solution to the large-time solution.", keywords = "Boundary layer, micropolar fluid, stretching surface,unsteady flow.", volume = "2", number = "2", pages = "136-5", }