Unsteady Laminar Boundary Layer Forced Flow in the Region of the Stagnation Point on a Stretching Flat Sheet
This paper analyses the unsteady, two-dimensional
stagnation point flow of an incompressible viscous fluid over a flat
sheet when the flow is started impulsively from rest and at the same
time, the sheet is suddenly stretched in its own plane with a velocity
proportional to the distance from the stagnation point. The partial
differential equations governing the laminar boundary layer forced
convection flow are non-dimensionalised using semi-similar
transformations and then solved numerically using an implicit finitedifference
scheme known as the Keller-box method. Results
pertaining to the flow and heat transfer characteristics are computed
for all dimensionless time, uniformly valid in the whole spatial region
without any numerical difficulties. Analytical solutions are also
obtained for both small and large times, respectively representing the
initial unsteady and final steady state flow and heat transfer.
Numerical results indicate that the velocity ratio parameter is found
to have a significant effect on skin friction and heat transfer rate at
the surface. Furthermore, it is exposed that there is a smooth
transition from the initial unsteady state flow (small time solution) to
the final steady state (large time solution).
[1] B.C. Sakiadis, "Boundary-layer behavior on continues solid surfaces: II.
The boundary-layer on continuous flat surface," AIChE J. vol. 7, 1961,
pp. 221-225.
[2] L. Crane, "Flow past a stretching plate". Z. Angew Math Phys (ZAMP),
vol.21, 1970, pp.645-657.
[3] I. Vleggaar, "Laminar boundary layer behavior on continuous
accelerating surfaces", Chem, Eng. Sci., vol. 32, 1977, pp.1517-1525.
[4] C. Y. Wang, "Exact solutions of the steady-state Navier-Stokes
equations", Ann. Rev. Fluid Mech. vol.23, 1991, 159-177.
[5] J. F. Brady and A. Acrivos, "Steady flow in a channel or tube with an
accelerating surface velocity: An exact solution to the Navier-Stokes
equations with reverse flow", J. Fluid Mech. Vol.112, 1981, pp.127-150.
[6] W. H. H. Banks, "Similarity solutions of the boundary-layer equations
for a stretching wall", J. Mec. Theoret. Appl. vol.2, 1983, pp.375-392.
[7] I Pop and T.Y. Na "Unsteady flow past a stretching sheet". Mech.
Research Communications. vol.23, 1996, pp. 413-422.
[8] N. Nazar, N. Amin, I. Pop, "Unsteady boundary-layer flow due to a
streching surface in a rotating fluid. Mech. Res. Commun.vol.31, 2004,
pp. 413-422.
[9] C.Y.Wang, G.Du, M.Mikilavi and C.C.Chang, "Impulsive stretching of
a surface in a viscous fluid-,SIAM J. Appl. Math., vol.57, 1997, pp.1-14.
[10] S. Awang Kechil and I. Hashim, "Series solution for unsteady boundary
layer flows due to impulsively stretching plate". Chinese Physics
Letters, vol.24, 2007, pp.139-142.
[11] J. C.Williams and Rhyne, "Boundary layer development on a wedge
impulsively sent into motion. SIAM J. Appl.Math. vol. 38, 1980, pp.
215-224.
[12] H.B. Keller, "A new difference scheme for parabolic problems". In
J.Bramble(Ed.). Numerical solutions of partial differential equations. vol
II. Academic Press, New York, 1970.
[13] T. Cebeci and P.Bradshaw, "Physical and Computational aspects of
Convective Heat Transfer, Springer, New York, 1988.
[14] T.C.Chiam, "Stagnation-point flow towards a stretching plate", J. Phys.
Soc. Jpn.,vol.63,1994, pp.2443-2444.
[15] T.R.Mahapatra and A.S.Gupta, "Heat transfer in stagnation point flow
towards a stretching sheet". Heat and Mass Transfer, vol.38, 2002, pp.
517-522.
[1] B.C. Sakiadis, "Boundary-layer behavior on continues solid surfaces: II.
The boundary-layer on continuous flat surface," AIChE J. vol. 7, 1961,
pp. 221-225.
[2] L. Crane, "Flow past a stretching plate". Z. Angew Math Phys (ZAMP),
vol.21, 1970, pp.645-657.
[3] I. Vleggaar, "Laminar boundary layer behavior on continuous
accelerating surfaces", Chem, Eng. Sci., vol. 32, 1977, pp.1517-1525.
[4] C. Y. Wang, "Exact solutions of the steady-state Navier-Stokes
equations", Ann. Rev. Fluid Mech. vol.23, 1991, 159-177.
[5] J. F. Brady and A. Acrivos, "Steady flow in a channel or tube with an
accelerating surface velocity: An exact solution to the Navier-Stokes
equations with reverse flow", J. Fluid Mech. Vol.112, 1981, pp.127-150.
[6] W. H. H. Banks, "Similarity solutions of the boundary-layer equations
for a stretching wall", J. Mec. Theoret. Appl. vol.2, 1983, pp.375-392.
[7] I Pop and T.Y. Na "Unsteady flow past a stretching sheet". Mech.
Research Communications. vol.23, 1996, pp. 413-422.
[8] N. Nazar, N. Amin, I. Pop, "Unsteady boundary-layer flow due to a
streching surface in a rotating fluid. Mech. Res. Commun.vol.31, 2004,
pp. 413-422.
[9] C.Y.Wang, G.Du, M.Mikilavi and C.C.Chang, "Impulsive stretching of
a surface in a viscous fluid-,SIAM J. Appl. Math., vol.57, 1997, pp.1-14.
[10] S. Awang Kechil and I. Hashim, "Series solution for unsteady boundary
layer flows due to impulsively stretching plate". Chinese Physics
Letters, vol.24, 2007, pp.139-142.
[11] J. C.Williams and Rhyne, "Boundary layer development on a wedge
impulsively sent into motion. SIAM J. Appl.Math. vol. 38, 1980, pp.
215-224.
[12] H.B. Keller, "A new difference scheme for parabolic problems". In
J.Bramble(Ed.). Numerical solutions of partial differential equations. vol
II. Academic Press, New York, 1970.
[13] T. Cebeci and P.Bradshaw, "Physical and Computational aspects of
Convective Heat Transfer, Springer, New York, 1988.
[14] T.C.Chiam, "Stagnation-point flow towards a stretching plate", J. Phys.
Soc. Jpn.,vol.63,1994, pp.2443-2444.
[15] T.R.Mahapatra and A.S.Gupta, "Heat transfer in stagnation point flow
towards a stretching sheet". Heat and Mass Transfer, vol.38, 2002, pp.
517-522.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:57399", author = "A. T. Eswara", title = "Unsteady Laminar Boundary Layer Forced Flow in the Region of the Stagnation Point on a Stretching Flat Sheet", abstract = "This paper analyses the unsteady, two-dimensional
stagnation point flow of an incompressible viscous fluid over a flat
sheet when the flow is started impulsively from rest and at the same
time, the sheet is suddenly stretched in its own plane with a velocity
proportional to the distance from the stagnation point. The partial
differential equations governing the laminar boundary layer forced
convection flow are non-dimensionalised using semi-similar
transformations and then solved numerically using an implicit finitedifference
scheme known as the Keller-box method. Results
pertaining to the flow and heat transfer characteristics are computed
for all dimensionless time, uniformly valid in the whole spatial region
without any numerical difficulties. Analytical solutions are also
obtained for both small and large times, respectively representing the
initial unsteady and final steady state flow and heat transfer.
Numerical results indicate that the velocity ratio parameter is found
to have a significant effect on skin friction and heat transfer rate at
the surface. Furthermore, it is exposed that there is a smooth
transition from the initial unsteady state flow (small time solution) to
the final steady state (large time solution).", keywords = "Forced flow, Keller-box method, Stagnation point, Stretching flat sheet, Unsteady laminar boundary layer, Velocity ratio parameter.", volume = "7", number = "6", pages = "993-6", }