sciepub.com
Quick Submission
Boundary Layer Stagnation-Point Flow of Second Grade Fluid over an Exponentially Stretching Sheet
Abdul Rehman1,
, Ghulam Farooq1, Israr Ahmed1, Muhammad Naseer2, Muhammad Zulfiqar3
1Department of Mathematics, University of Balochistan, Quetta, Pakistan
2Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
3Department of Mathematics, GC University, Lahore, Pakistan
Abstract
In this paper, the steady boundary layer stagnation point flow and heat transfer of a second grade fluid over an exponentially stretching sheet is investigated. The solutions are obtained through homotopy analysis method (HAM) and the Keller-box technique. Comparisons of both the solutions are given graphically as well as in tabular form. The effects of second grade parameter β Prandtl number Pr and other important physical parameters are presented through graphs and the salient features are discussed.
Keywords: boundary layer flow, heat transfer, second grade fluid, exponential stretching/shrinking, homotopy analysis method, keller-box technique
Copyright © 2015 Science and Education Publishing. All Rights Reserved.Cite this article:
- Abdul Rehman, Ghulam Farooq, Israr Ahmed, Muhammad Naseer, Muhammad Zulfiqar. Boundary Layer Stagnation-Point Flow of Second Grade Fluid over an Exponentially Stretching Sheet. American Journal of Applied Mathematics and Statistics. Vol. 3, No. 6, 2015, pp 211-219. https://pubs.sciepub.com/ajams/3/6/1
- Rehman, Abdul, et al. "Boundary Layer Stagnation-Point Flow of Second Grade Fluid over an Exponentially Stretching Sheet." American Journal of Applied Mathematics and Statistics 3.6 (2015): 211-219.
- Rehman, A. , Farooq, G. , Ahmed, I. , Naseer, M. , & Zulfiqar, M. (2015). Boundary Layer Stagnation-Point Flow of Second Grade Fluid over an Exponentially Stretching Sheet. American Journal of Applied Mathematics and Statistics, 3(6), 211-219.
- Rehman, Abdul, Ghulam Farooq, Israr Ahmed, Muhammad Naseer, and Muhammad Zulfiqar. "Boundary Layer Stagnation-Point Flow of Second Grade Fluid over an Exponentially Stretching Sheet." American Journal of Applied Mathematics and Statistics 3, no. 6 (2015): 211-219.
| Import into BibTeX | Import into EndNote | Import into RefMan | Import into RefWorks |
At a glance: Figures
1. Introduction
Boundary layer flow due to stretching and stagnation point flows have achieved considerable attention due to its applications in industry and manufacturing processes. A large number of researchers are engaged with this area. Mention may be made to the works of [1-15][1]. In the analysis mentioned above simple stretching and stagnation flow have been used. However some researchers have used the exponential stretching because of its engineering applications. Sanjayanad and Khan [16] and Khan and Sanjayanad [17] have discussed the boundary layer flow of viscoelastic fluid due to exponential stretching sheet with and without heat transfer analysis. Later on, the idea of exponential stretching have further discussed by Nadeem et al [18, 19] for considering different non-Newtonian fluid models. Recently, Wei et al [20] have considered the stagnation point flow over an exponentially stretching/shrinking sheet for the viscous fluids.
Motivated from the above analysis, the aim of the present paper is to discuss the boundary layer flow of second grade fluid over an exponential stretching sheet. To the best of author’s knowledge only a single attempt is available which discussed the exponential stagnation with the exponential stretching when fluid is taken as Newtonian. However, this analysis has not been discussed so far for non-Newtonian fluids. Therefore, in this paper we have discussed both the analytical and numerical solutions of the second grade fluid with exponential stagnation point flow with exponential stretching in the presence of mixed convection heat transfer. It is also worth able to mention here that we have discussed the two strong solution techniques together. The analytical solutions are carried out with the help of homotopy analysis method [21-28][21]. For validity of the solution we have also provided numerical solutions obtained with the of Keller-box technique [29, 30, 31]. The physical features of embedding parameters are discussed through graphs.
2. Formulation
Let us consider a stagnation point flow of an incompressible second grade fluid over a stretching sheet. The stretching and stagnation point is assumed to be of exponential type. The Cartesian coordinates (x, y) are used such that x is along the surface of the sheet, while y is taken normal to it. The related boundary layer equations of second grade fluid in the presence of heat transfer take the following form
 | (1) |
 | (2) |
 | (3) |
Here 
are the velocity components along the 
axes, 
is the fluid density, 
is the second grade parameter, 
is the kinematic viscosity, 
is temperature, 
is the thermal diffusivity, 
is pressure and 
is the free-stream velocity. The corresponding boundary conditions for the problem are
 | (4) |
 | (5) |
where the free-stream velocity 
, the stretching velocity 
, and the surface temperature 
, are defined as
 | (6) |
in which 
and 
are constant velocities, 
is constant temperature and 
is the reference length. Defining the following similarity transformations:
 | (7) |
 | (8) |
With the help of transformations defined in 
and 
, 
is identically satisfied and 
and 
take the form
 | (9) |
 | (10) |
in which 
is the nondimensional second grade fluid parameter and 
is the Prandtl number. The boundary conditions in nondimensional form can be written as
 | (11) |
 | (12) |
where 
.
The shear stress on the surfaces 
, the frictional drag coefficient 
the heat flux at the surface 
and the local Nusselt numbers 
in dimensionless form are defined as
 | (13) |
 | (14) |
 | (15) |
 | (16) |
where 
, and 
is the local Reynolds number.
3. Analytical Solution of the Problem
The analytical solution of the above boundary value problem is obtained with the help of HAM. For HAM solution we choose the initial guesses as
 | (17) |
The corresponding auxiliary linear operators are
 | (18) |
They satisfy
 | (19) |
where 
are arbitrary constants. The zeroth-order deformation equations are defined as
 | (20) |
 | (21) |
in which
 | (22) |
 | (23) |
The appropriate boundary conditions for the zeroth order system are
 | (24) |
 | (25) |
Further details of the HAM solution can be found in [21, 22].
4. Numerical Solution
For accuracy of the HAM solution the problem is also solved using the Keller-box technique. For Keller-box scheme the nonlinear system of differential equations is first converted into a first order system using appropriate substitution. This first order system is then approximated by difference equations using central difference. The resulting finite difference system is linearized by applying Newton's method, at the end the obtained linearized system is solved using block-elimination procedure.
5. Results and Discussion
The problem of stagnation point boundary layer flow of a second grade fluid over an exponentially stretching sheet is solved analytically as well as numerically. The analytical solutions of the system of ordinary differential equations 
subject to the boundary conditions 
are obtained through homotopy analysis method (HAM). The convergence of the HAM solution is havily dependent upon the proper selection of 
. To find appropriate values of 
and 
the convergence regions of 
and 
are plotted in Figure 1-Figure 3 for specified combinations of involved parameters. Figure 1 is displaying the convergence regions for 
for different values of second grade parameter 
when stretching parameter 
It is observed that convergence region for viscous fluids 
is 
and convergence region is 
for 
that is with increase in 
convergence region reduced. Figure 2 is graphed for convergence region of 
for different values of 
when second grade parameter 
From Figure 2 it is observed that convergence region for rigid plat case 
is 
while for 
convergence region is 
Figure 3 is sketched to observe convergence region for the temperature gradient for specified vales of stretching parameter, second grade parameter and the Prandtl number. It is noted that convergence region is 
Download as
for different 
drawn at the 20th order-approximation
Download as
for different 
drawn at the 20th order-approximation
Download as
for different 
drawn at the 20th order-approximation
Download as
for 
Download as
for different values of 
Download as
for different values of 
Download as
over 
for different values of 
Download as
over 
for different values of 
Download as
over 
for different values of 
Download as
over 
for different values of 
Download as
and 
against 
To check accuracy of the HAM solution, the problem is also solved with the help of second order implicit finite difference scheme Keller-box technique. Figure 4-Figure 6 are prepared for comparison of both the HAM and Keller-box solutions for specified values of involved parameters. Figure 4 is schemed to spot behavior of the HAM and Keller-box solutions for different values of 
when 
It is noted that both solutions are in excellent contract. Figure 5 is organized to observe behavior of the two solutions for different values of second grade parameter 
when 
It is witnessed that both the solutions are in decent agreement. From Figure 5 it is also noted that an escalation in second grade parameter 
demands a decrease in the velocity profile, this is due to the fact that higher 
resembles to higher tensile stress between fluid layers, that in return corresponds to higher resistance to fluid motion. Figure 6 is drawn to check the behavior of the two solutions for the temperature profile 
for different values of the Prandtl number 
when 
and 
respectively. From Figure 6 it is seen that both the solutions are again in good agreement and that an increase in 
demands a decline in the temperature profile 
and also corresponds to decrease in thermal boundary layer thickness. This is due to the fact that large 
corresponds to low thermal diffusivity, which in return corresponds to less energy transfer ability due to which thermal boundary layer decreases. Figure 7 is designed to observe influence of second grade parameter 
over the velocity profile 
for different values of 
. From Figure 7 it is observed that 
has a dual behavior for 
for different 
. That is, for 
an increase in 
implies decrease in 
, whereas for 
an increase in 
corresponds to a decrease in 
. From Figure 7 it is also noted that 
has greater reliance over 
when 
is away from unity, but when 
is near to unity, dependence of 
over 
is minimal. Figure 8 is plotted to perceive the effects of 
for different values of 
. From Figure 8 it is observed that as stretching parameter 
increases the temperature profile decreases more rapidly with respect to 
as compare to smaller 
Figure 9 is drafted to observe the behavior of 
with respect to 
for different values of second grade parameter 
. From Figure 8 it is observed that decrease in 
with 
is slow for larger values of 
. Figure 10 is included to check the behavior of temperature profile against different 
when 
and 
. It is perceived that 
has a dual behavior against 
for different 
, that is when 
, an increase in 
produces increase in 
, whereas when 
, an increase in 
gives opposite behavior. Figure 11 is affiliated to check behavior of second grade parameter 
over skin-friction coefficient against Reynolds number 
for different values of 
. It is depicted from Figure 11 that with an increase in 
skin-friction coefficient increases. The rate of increase in 
against 
decreases with an increase in 
. It is also noted that skin-friction coefficient 
has strong dependence over 
and behavior of other involved parameters are noticeable only when 
whereas, 
shoots to infinity when 
while 
decays to zero for 
Figure 12 portrays behavior of Nusselt numbers 
for different values of local Reynolds number 
against 
. From Figure 12 an increase is observed in 
with respect to 
.
Download as
against 
Table 1-Table 2 are prepared to compare the HAM and Keller-box solutions for velocity and temperature gradients at the surface of stretching sheet. From Table 1-Table 2 it is obvious that both the solutions are compatible. From Table 1 it is also predicted that wall shear stress at the surface 
decreases with 
, when 
, while 
increases with 
, when 
. Whereas with respect to second grade parameter 
wall shear stress 
decreases. Table 2 contains values for temperature gradient at surface of stretching sheet. From Table 2 it is noted that with an increase in 
and 
the local heat flux decreases, whereas with an increase in 
the heat transfer at the wall decrease.
Table 1. Comparison of behavior of wall shear stress at the surface τw, against different parameters. The tabulated values are the corresponding absolute values of f''(0)
)
References
| [1] | M. Y. Malik, M. Naseer, S. Nadeem, Abdul Rehman, The boundary layer flow of Casson nanofluid over a vertical exponentially stretching cylinder, Appl. NanoSci. | ||
 In article | |||
| [2] | M. Y. Malik, M. Naseer, Abdul Rehman, Numerical study of convective heat transfer on the Power Law fluid over a vertical exponentially stretching cylinder, Applied and Computational Mathematics, 4(5), (2015) 346-350. | ||
| In article | |||
| [3] | Abdul Rehman, R. Baza, S. Achakzai, S. Iqbal, M. Naseer, Boundary layer flow and heat transfer of micropolar fluid over a vertical exponentially stretched cylinder, App. Comp. Math. 4(6), (2015), 424-430. | ||
| In article | |||
| [4] | Y. Y. Lok, N. Amin, I. Pop, Non-orthogonal stagnation point flowtowards a stretching sheet, Int. J. Non-Linear Mech. 41 (2006) 622-627. | ||
| In article | View Article | ||
| [5] | S. Nadeem, A. Hussain, M. Khan, HAM solutions for boundary layer flow in the region of the stagnation point towards a stretching sheet, Commun. Nonlinear Sci. Numer. Simulat. 15 (2010) 475-481. | ||
| In article | View Article | ||
| [6] | M. Naseer, M. Y. Malik, S. Nadeem, Abdul Rehman, The boundary layer flow of hyperbolic tangent fluid over a vertical exponentially stretching cylinder, Alexandria Eng. J. 53(3) (2014), 747-750. | ||
| In article | View Article | ||
| [7] | A. Ishak, K. Jafar, R. Nazar, I. Pop, MHD stagnation point flow towards a stretching sheet, Physica A 388 (2009) 3377-3383. | ||
| In article | View Article | ||
| [8] | Abdul Rehman, S. Nadeem, M. Y. Malik, Stagnation flow of couple stress nanofluid over an exponentially stretching sheet through a porous medium, J. Power Tech. 93(2) (2013) 122-132. | ||
| In article | |||
| [9] | T. C. Chiam, Heat transfer with variable conductivity in a stagnation-point flow towards a stretching sheet, Int. Comm. Heat Mass Tran. 23 (1996) 239-248. | ||
| In article | View Article | ||
| [10] | R. A. V. Gorder, K. Vajravelu, Hydromagnetic stagnation point flow of a second grade fluid over a stretching sheet, Mech. Res. Commun. 37 (2010) 113-118. | ||
| In article | View Article | ||
| [11] | Abdul Rehman, S. Nadeem, M. Y. Malik, Boundary layer stagnation-point flow of a third grade fluid over an exponentially stretching sheet, Braz. J. Che. Eng. 30(3) (2013) 611-618. | ||
| In article | View Article | ||
| [12] | Abdul Rehman, S. Nadeem, Heat transfer analysis of the boundary layer flow over a vertical exponentially stretching cylinder, Global J. Sci. Fron. Res. 13(11) (2013) 73-85. | ||
| In article | |||
| [13] | T. Fan, H. Xu, I. Pop, Unsteady stagnation flow and heat transfer towards a shrinking sheet, Int. Commun. Heat Mass Tran. 37 (2010) 1440-1446. | ||
| In article | View Article | ||
| [14] | S. Nadeem, Abdul Rehman, Axisymmetric stagnation flow of a nanofluid in a moving cylinder, Comp. Math. Mod. 24(2) (2013) 293-306. | ||
| In article | View Article | ||
| [15] | S. Nadeem, Abdul Rehman, Mohamed Ali, The boundary layer flow and heat transfer of a nanofluid over a vertical slender cylinder, J. NanoEngineering and NanoSystems (2012) 1-9. | ||
| In article | PubMed | ||
| [16] | E. Sanjayanad, S. K. Khan, On heat and mass transfer in a viscoelastic boundary layer flow over an exponentially stretching sheet, Int. J. Thermal Sci. 45 (2006) 819-828. | ||
| In article | View Article | ||
| [17] | S. K. Khan, E. Sanjayanad,.Viscoelastic boundary layer flow and heat transfer over an exponential stretching sheet, Int. J. Heat Mass Tran. 48 (2005) 1534-1542. | ||
| In article | View Article | ||
| [18] | S. Nadeem, S. Zaheer, T. Fang, Effects of thermal radiation on the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface,, Numer. Algor. 57 (2011) 187-205. | ||
| In article | View Article | ||
| [19] | S. Nadeem, T. Hayat,M. Y. Malik, S. A. Rajput, Thermal radiation effects on the flow by an exponentially stretching surface: a series solution, Z. Na.. 65 (2010) 495-503 | ||
| In article | View Article | ||
| [20] | S. Wei. Wong, M. Abu O. Awang, A. Ishak, Stagnation-Point Flow over an Exponentially Shrinking/Stretching Sheet, ZNa. 66a (2011) 705-711. | ||
| In article | |||
| [21] | Sj. Liao, On the homotopy analysis method for nonlinear problems, App. Math. Comp. 147 (2004) 499-513. | ||
| In article | View Article | ||
| [22] | Sj. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC Press, Boca Raton, 2003 | ||
| In article | View Article | ||
| [23] | S. Nadeem, Noreen Sher Akbar, Influence of heat and mass transfer on the peristaltic flow of a Johnson Segalman fluid in a vertical asymmetric channel with induced MHD, J. Taiwan Inst. Che. Eng. 42 (2011) 58-66. | ||
| In article | View Article | ||
| [24] | S. Nadeem, Noreen Sher Akbar, Effects of heat and mass transfer peristaltic flow of Williamson fluid in a vertical annulus, Meccanica. | ||
| In article | |||
| [25] | Abdul Rehman, S. Nadeem, Mixed convection heat transfer in micropolar nanofluid over a vertical slender cylinder, Chin. Phy. Lett. 29 (12) (2012) 124701-5. | ||
| In article | View Article | ||
| [26] | S. Nadeem, Abdul Rehman, Changhoon Lee, Jinho Lee, Boundary layer flow of second grade fluid in a cylinder with heat transfer, Mathematical Problems in Engineering, Volume 2012, Article ID 640289. | ||
| In article | |||
| [27] | L. Zou, Z. Zong, G. H. Dong, Generalizing homotopy analysis method to solve Lotka--Volterra equation, Comp Math App 56 (2008) 2289-2293. | ||
| In article | View Article | ||
| [28] | S. Nadeem, Abdul Rehman, K. Vajravelu, Jinho Lee, Changhoon Lee, Axisymmetric stagnation flow of a micropolar nanofluid in a moving cylinder, Mathematical Problems in Engineering, Volume 2012, Article ID 378259. | ||
| In article | |||
| [29] | H. B. Keller, T. Cebeci, Accurate numerical methods for boundary-layer flows. II: Two-dimensional flows, AIAA Journal 10 (1972) 1193-1199. | ||
| In article | View Article | ||
| [30] | M. Abramowitz, Stegun IA (1965) Handbook of mathematical functions, Dover, New York. | ||
| In article | PubMed | ||
| [31] | F. Aman, A. Ishak, Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet with prescribed surface heat flux, Heat Mass Transfer, 46 (2010) 61-620. | ||
| In article | View Article | ||
CiteULike
Mendeley
StumbleUpon
Delicious
Facebook
Twitter
LinkedIn
