
Research Article


Unsteady Stagnation Point Flow and Heat Transfer over a Stretching/shrinking Sheet


N.M.A. Nik Long,
M. Suali,
A. Ishak,
N. Bachok
and
N.M. Arifin



ABSTRACT

This study investigates the problem of unsteady stagnation point flow and heat transfer over a stretching/shrinking sheet. The governing partial differential equations are converted into a system of nonlinear ordinary differential equations using a similarity transformation, before being solved numerically. Both stretching and shrinking cases are considered. It is found that dual solutions exist for the shrinking case while for the stretching case, the solution is unique. Moreover, it is found that the heat transfer rate at the surface increases as the stretching/shrinking parameter as well as the unsteadiness parameter increases.





Received: July 30, 2011;
Accepted: October 14, 2011;
Published: December 12, 2011


INTRODUCTION
The study of flow and heat transfer over a stretching/shrinking sheet is an
important problem in many engineering processes because it has many applications
in industries such as extrusion of plastic sheets, wire drawing, hot rolling
and glass fiber production. Sakiadis (1961a, b)
investigated the boundary layer flow on a continuously moving surface with a
constant velocity. Later, this work was verified experimentally by Tsou
et al. (1967). Following Sakiadis (1961a,
b), Tsou et al. (1967)
and Crane (1970) studied the flow over a linearly stretching
sheet immersed in an ambient fluid and obtained an exact solution to the NavierStokes
equation. Gupta and Gupta (1977) extended the work of
Crane (1970) by investigating the effect of mass transfer
on a stretching sheet with suction or blowing. On the other hand, the stretching
boundary problem Crane (1970) was extended by Wang
(1984) to a threedimensional flow. Mahapatra and Gupta
(2003) considered the stagnation flow over a stretching surface and then
this problem was extended to oblique stagnation flow by Lok
et al. (2006). Many authors such as Carragher
and Crane (1982), Elbashbesy and Bazid (2000), Magyari
and Keller (1999, 2000), Magyari
et al. (2001), Liao and Pop (2004) and Nazar
et al. (2004) investigated the stretching sheet problem with different
aspects, such as uniform heat flux, permeability of the surface and unsteadiness
flow and heat transfer.
Different from the stretching case, only a few works have been done on the
flow induced by a shrinking sheet. Miklavcic and Wang (2006)
investigated the flow over a shrinking sheet and found that the flow characteristics
are different from that of the stretching case. Fang (2008)
investigated the flow induced by a shrinking sheet with a powerlaw velocity
and reported the existence of multiple solutions for certain range of the mass
transfer parameter. The nonuniqueness solution of the shrinking sheet problem
was also reported by Fang et al. (2008), when
they solved the Blasius equation for the shrinking sheet. The flow characteristics
induced by a shrinking sheet was also investigated by Hayat
et al. (2007) and Sajid et al. (2008)
and the solutions were obtained using the homotopy analysis method. The stagnation
flow towards a shrinking sheet was considered by Wang (2008)
where the existence of dual solutions for a certain range of the shrinking parameter
was reported. He found that solutions do not exist for larger shrinking rates
and may be nonunique in the twodimensional case. This problem was then extended
to a micropolar fluid by Ishak et al. (2010).
Different from the stretching case, solutions do not exist for a shrinking impermeable
sheet in an otherwise still fluid, since vorticity could not be confined in
the boundary layer. However, with an added stagnation flow to contain the vorticity,
similarity solutions may exist.
Different from the abovementioned investigations, the present paper considers
the problem of an unsteady twodimensional stagnationpoint flow and heat transfer
over a stretching/shrinking sheet immersed in an incompressible viscous fluid.
To the best of our knowledge, this problem has not been studied before.
MATHEMATICAL FORMULATION
Consider an unsteady stagnation point flow over a stretching/shrinking sheet
immersed in an incompressible viscous fluid of ambient temperature T_{∞}.
It is assumed that the free stream velocity is in the form, U_{∞}
(x,t) = ax/ (1λt) the sheet is stretched with velocity U_{w}(x,
t) = bx/ (1λt) and the surface temperature is T_{W} (x, t) = T
_{ ∞}+cx/(1λt). The xaxis runs along the sheet while the
yaxis is measured normal to it. With these assumptions along with the boundarylayer
approximations and neglecting the viscous dissipation, the governing equations
are (Fang et al., 2011):
with the boundary conditions: where, u and v are the velocity components in the x and y directions, respectively, v is the kinematic viscosity, α the thermal diffusivity and T is the fluid temperature. In order that Eq. 13 reduce to similarity equations, we introduce the following similarity transformation: where, η is the similarity variable and ψ is the stream function defined as u = , ∂ψ/∂y and v = ∂ψ/∂x, which identically satisfies the continuity Eq. 1. Substituting (5) into Eq. 2 and 3 yield the following nonlinear ordinary differential equations: subject to the boundary conditions: where, primes denote the differentiation with respect to η, Pr = v/α is the Prandtl number, A = λ/a is the unsteadiness parameter, ε = b/a is the stretching/shrinking parameter with ε>0 for stretching and ε<0 for shrinking. The physical quantities of interest are the skin friction coefficient C_{f} and the local Nusselt number Nu_{x}, which are defined as: where, the surface shear stress τ_{w} and the surface heat flux q_{w} are given by: with μ and κ being the dynamic viscosity and the thermal conductivity, respectively. Using the similarity variables (Eq. 5), we obtain: where Re_{x} = U_{∞}x/v is the local Reynolds number. RESULTS AND DISCUSSION
Equations 6 and 7 were solved numerically
using a shooting method. The results are given to carry out a parametric study
showing the influence of the nondimensional parameters, namely the unsteadiness
parameter A and the stretching/shrinking parameter ε, while the Prandtl
number Pr is fixed at Pr = 0.7 (such as air), to conserve space. For the validation
of the numerical results obtained, the case A = 0 (steady state flow) has also
been considered and compared with those of Wang (2008)
and Ishak et al. (2010). The quantitative comparisons
are shown in Table 1 and found to be in a favorable agreement.
Figure 1 shows the variation of the skin friction coefficient
in terms of f” (0) as a function of ε for various values of A.

Fig. 1: 
Skin friction coefficient f ” (0) as a function of ε
for different values of A 

Fig. 2: 
Local Nusselt number θ as a function of ε for
different value of A when Pr = 0.7 
Table 1: 
Values of f” for different values of ε 


Fig. 3: 
Velocity profiles f ’ (η) when A = 0.01 

Fig. 4: 
Temperature profiles θ (η) when A = 0.01 and Pr
= 0.7 
It is seen that the range of ε for which the solution exists increases
as A increases. Thus, the solution domain is widen for the unsteady flow. For
a particular value of A, the solution exists up to a critical value of ε,
which depends on A. Based on our computations, ε_{c} = 1.2465,
1.2536, 1.3118 and 1.4520 for ε = , 0.01, 0.1 and 0.3, respectively.
For a particular value of ε, the skin friction coefficient is higher for
higher values of A. This results in increasing manner of the local Nusselt number
which represents the heat transfer rate at the surface, as presented in Fig.
2. Moreover, Fig. 2 shows that the heat transfer rate
at the surface increases as the stretching/shrinking parameter ε increases.
Figure 1 shows that the lower solution branch is attracted
to (ε f”(0) = (1,0)), while the local Nusselt number is negative and
unbounded as ε1. As discussed by previous authors (Merkin,
1994; Weidman et al., 2006; Harris
et al., 2009; Postelnicu and Pop, 2011),
we expect that the lower solution branch is unstable and not physically realizable.
Although, such solutions are deprived of physical significance they are nevertheless
of interest as the differential equations are concerned. Similar equations may
reappear in other situations where the corresponding solutions have more realistic
meaning (Ridha, 1996).
Figure 3 and 4 show the velocity and temperature profiles for selected values of parameters respectively. It is seen that there are two different profiles for particular values of ε (as shown in the figures) which support the existence of dual solutions presented in Fig. 1 and 2. Moreover, the velocity and temperature profiles satisfy the far field boundary conditions (8) asymptotically, which support the validity of the numerical results obtained. CONCLUSIONS A numerical study was performed to investigate the flow and heat transfer characteristics of unsteady twodimensional stagnation point flow over a stretching/shrinking sheet. The similarity transformation reduced the partial differential equations into a system of nonlinear ordinary differential equations, which was solved numerically by a shooting method. The effects of the unsteadiness parameter A and the stretching/shrinking parameter ε were obtained and discussed. Both stretching and shrinking cases were considered. It was found that dual solutions exist for the shrinking case while for the stretching case, the solution is unique. The unsteady parameter A widen the range of ε for which the solution exists. For the upper branch solution, which we expect to be the physically relevant solution, the heat transfer rate at the surface increases as A as well as ε well increases. ACKNOWLEDGMENT This study was supported by a research grant (Project Code: UKMGUP2011202) from the Universiti Kebangsaan Malaysia.

REFERENCES 
Carragher, P. and L.J. Crane, 1982. Heat transfer on a continuous stretching sheet. ZAMM J. Applied Math. Mech., 62: 564565. CrossRef 
Crane, L.J., 1970. Flow past a stretching plate. Zeitschrift Angewandte Mathematik Physik, 21: 645647. CrossRef  Direct Link 
Elbashbesy, E.M.A. and M.A.A. Bazid, 2000. The effect of temperaturedependent viscosity on heat transfer over a continuous moving surface. J. Phys. D: Applied Phys., 33: 27162721. CrossRef  Direct Link 
Fang, T., 2008. Boundary layer flow over a shrinking sheet with powerlaw velocity. Int. J. Heat Mass Transfer, 51: 58385843. CrossRef  Direct Link 
Fang, T., C.F. Lee and J. Zhang, 2011. The boundary layers of an unsteady incompressible stagnationpoint flow with mass transfer. Int. J. NonLinear Mech., 46: 942948. CrossRef  Direct Link 
Fang, T., W. Liang and C.F. Lee, 2008. A new solution branch for the Blasius equation: A shrinking sheet problem. Comp. Math. Appl., 56: 30883095. CrossRef  Direct Link 
Gupta, P.S. and A.S. Gupta, 1977. Heat and mass transfer on a stretching sheet with suction or blowing. Can. J. Chem. Eng., 55: 744746. CrossRef  Direct Link 
Harris, S.D., D.B. Ingham and I. Pop, 2009. Impulsive FalknerSkan flow with constant wall heat flux: Revisited. Int. J. Numer. Methods Heat Fluid Flow, 19: 10081037. CrossRef  Direct Link 
Hayat, T., Z. Abbas and M. Sajid, 2007. On the analytic solution of magnetohydrodynamic flow of a second grade fluid over a shrinking sheet. J. Applied Mech. Trans. ASME, 74: 11651171. CrossRef  Direct Link 
Ishak, A., Y.Y. Lok and I. Pop, 2010. Stagnationpoint flow over a shrinking sheet in a micropolar fluid. Chem. Eng. Commun., 197: 14171427. CrossRef  Direct Link 
Liao, S.J. and I. Pop, 2004. On explicit analytic solutions of boundarylayer equations about flows in porous medium or for a stretching wall. Int. J. Heat Mass Transfer, 47: 7585.
Lok, Y.Y., N. Amin and I. Pop, 2006. Nonorthogonal stagnation point flow towards a stretching sheet. Int. J. Nonlinear Mech., 41: 622627.
Magyari, E. and B. Keller, 1999. Heat and mass transfer in the boundary layers on an exponentially stretching continuous surface. J. Phys. D: Applied Phys., 32: 557585. CrossRef  Direct Link 
Magyari, E. and B. Keller, 2000. Exact solutions for selfsimilar boundarylayer flows induced by permeable stretching surface. Eur. J. Mech. B/Fluids, 19: 109122. CrossRef  Direct Link 
Magyari, E., M.E. Ali and B. Keller, 2001. Heat and mass transfer characteristics of the self similar boundary layer flow induced by continuous surface stretched with rapidly decreasing velocities. Heat Mass Transfer, 38: 6574. Direct Link 
Mahapatra, T.R. and A.S. Gupta, 2003. Stagnationpoint flow towards a stretching surface. Can. J. Chem. Eng., 81: 258263. CrossRef  Direct Link 
Merkin, J.H., 1994. A note on the similarity equations arising in free convection boundary layers with blowing and suction. Zeitschrift Angewandte Mathematik Physik, 45: 258274. CrossRef  Direct Link 
Miklavcic, M. and C.Y. Wang, 2006. Viscous flow due to a shrinking sheet. Quart. Applied Math., 64: 283290. Direct Link 
Nazar, R., N. Amin, D. Filip and I. Pop, 2004. Unsteady boundary layer flow in the reagion of the stagnation point on a stretching sheet. Int. J. Eng. Sci., 42: 12411253. CrossRef 
Postelnicu, A. and I. Pop, 2011. FalknerSkan boundary layer flow of a powerlaw fluid past a stretching wedge. Applied Math. Comput., 217: 43594368. CrossRef  Direct Link 
Ridha, A., 1996. Aiding flows nonunique similarity solutions of mixedconvection boundarylayer equations. J. Applied Math. Phys., 47: 341352. CrossRef  Direct Link 
Sajid, M., T. Hayat and T. Javed, 2008. MHD rotating flow of a viscous fluid over a shrinking surface. Nonlinear Dyn., 51: 259265. CrossRef  Direct Link 
Sakiadis, B.C., 1961. Boundarylayer behavior on continuous solid surfaces: I. Boundarylayer equations for twodimensional and axisymmetric flow. AIChE J., 7: 2628. CrossRef  Direct Link 
Sakiadis, B.C., 1961. Boundarylayer behavior on continuous solid surfaces: II. The boundary layer on a continuous flat surface. AIChE J., 7: 221225. CrossRef  Direct Link 
Tsou, F.K., E.M. Sparrow and R.J. Goldstain, 1967. Flow and heat transfer in the boundary layer on a continuous moving surface. Int. J. Heat Mass Transfer, 10: 219235. CrossRef  Direct Link 
Wang, C.Y., 1984. The three dimensional flow due to a stretching flat surface. Phys. Fluid, 27: 19151917. CrossRef 
Wang, C.Y., 2008. Stagnation flow towards a shrinking sheet. Int. J. NonLinear Mech., 43: 377382. CrossRef  Direct Link 
Weidman, P.D., D.G. Kubitschek and A.M.J. Davis, 2006. The effect of transpiration on selfsimilar boundary layer flow over moving surfaces. Int. J. Eng. Sci., 44: 730737. CrossRef  Direct Link 



