Meccanica (2012) 47:1325–1335 DOI 10.1007/s11012-011-9516-z
Oblique stagnation-point flow and heat transfer towards a shrinking sheet with thermal radiation Tapas Ray Mahapatra · Samir Kumar Nandy · Anadi Sankar Gupta
Received: 6 June 2011 / Accepted: 9 November 2011 / Published online: 6 December 2011 © Springer Science+Business Media B.V. 2011
Abstract An analysis is made of steady two-dimensional oblique stagnation-point flow and radiative heat transfer of an incompressible viscous fluid towards a shrinking sheet which is shrunk in its own plane with a velocity proportional to the distance from a fixed point. Here the axis of the stagnation flow and that of the shrinking sheet are not aligned. A similarity transformation reduces the Navier-Stokes equations to a set of non-linear ordinary differential equations and are solved numerically using a shooting technique. The analysis of the results obtained shows that multiple solutions exist for a certain range of the ratio of the shrinking velocity to the free stream velocity. The effect of non-alignment for the wall shear stress and the horizontal velocity components are discussed. Streamline patterns are also shown for shrinking at the sheet with aligned and non-aligned cases. It is found that the
T.R. Mahapatra Department of Mathematics, Visva-Bharati, Santiniketan 731 235, India e-mail:
[email protected] S.K. Nandy () Department of Mathematics, A.K.P.C. Mahavidyalaya, Bengai, Hooghly 712 611, India e-mail:
[email protected] A.S. Gupta Department of Mathematics, Indian Institute of Technology, Kharagpur 721 302, India e-mail:
[email protected]
temperature at a point in the fluid decreases with increase in effective Prandtl number (Preff ). The results pertaining to the present study indicate that as Preff increases, the rate of heat transfer also increases. The reported results are in good agreement with the available published work in the literature. Keywords Oblique stagnation-point flow · Shrinking sheet · Dual solutions · Heat transfer · Thermal radiation
1 Introduction In recent years, a great deal of interest has been generated in the boundary layer flow over a stretching sheet in view of its numerous and increasing technological and industrial applications which include the aerodynamic extrusion of plastic sheets, glass fibre and paper production, cooling of metallic sheets or electronic chips and many others. In view of all these cases, the final product of desired characteristics depend on the rate of cooling and the process of stretching. In view of these practical applications, enormous works are done in various aspects of the boundary layer flow and heat transfer over a stretching sheet (see for details [1–5]). On the other hand, Hiemenz [6] first studied the steady two-dimensional boundary-layer flow near the forward stagnation-point on an infinite wall. This solution was later improved by Howarth [7]. Further, various aspects of stagnation flow and heat transfer
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over a stretching sheet are investigated by many authors (see [8–15]). All of these above mentioned studies are related to orthogonal stagnation-point flow over a stretching sheet. Stuart [16] first studied the oblique stagnationpoint flow of an incompressible fluid towards a fixed rigid surface. Later this problem was investigated independently by Tamada [17] and Dorrepaal [18]. Reza and Gupta [19] generalized the problem of an oblique stagnation point flow over a stretching sheet by Chiam [8] to include surface stretching rate different from that of the stagnation flow. In that paper, they have ignored the displacement thickness parameter and the pressure gradient parameter. This was partially rectified in a paper by Lok et al. [20]. However, in that paper, they did not take into account the pressure gradient parameter in the boundary conditions at infinity. This is a serious omission since the pressure gradient parameter is linked to the free stream shear in oblique stagnation point flow (see Drazin and Rilley [21]). Very recently, Reza and Gupta [22] gave a correct solution to the above problem be rectifying the errors in [19] and [20]. Recently, the boundary layer flow due to a shrinking sheet has attracted considerable interest. One of the common applications of shrinking sheet problems in industries and engineering is shrinking film. In packaging of bulk products, shrinking film is very useful as it can be unwrapped easily with adequate heat. Shrinking problem can also be applied to study the capillary effects in small pores, the shrinking-swell behaviour and the hydraulic properties of agricultural clay soils. The associated changes in hydraulic and mechanical properties of such soils will seriously hamper predictions of the flow and transport processes which are essential for agricultural development and environmental strategies. For this flow configuration, the fluid is attracted towards a slot and the flow is quite different from the stretching case. From a physical point of view, vorticity generated at the shrinking sheet is not confined within a boundary layer and a steady flow is not possible unless either a stagnation-point flow towards the sheet is applied or adequate suction is applied at the sheet. This type of shrinking flow is essentially a backward flow as discussed by Goldstein [23]. For a backward flow configuration, the fluid losses memory of the perturbation introduced by the slot. As a result, the flow induced by the shrinking sheet shows quite distinct physical phenomena from the stretching case.
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Wang [24] conferred the concept of the flow around the shrinking sheet while studying behaviour of liquid film on an unsteady stretching sheet. The existence and uniqueness of the solution of steady viscous flow over a shrinking sheet was studied by Miklavcic and Wang [25]. This problem was then extended to magnetohydrodynamic (MHD) flow by Sajid and Hayat [26]. Fang and Zhang [27] obtained the closed form analytic solution for steady MHD flow over a shrinking sheet subjected to applied suction and they reported greatly different solution behaviour with multiple solution branches compared to the corresponding stretching sheet problem. The boundary layer flow over a continuously shrinking sheet with a power law surface velocity and mass transfer were investigated by Fang [28]. The unsteady viscous flow over a continuously shrinking surface with mass suction was also investigated by Fang et al. [29]. Steady two-dimensional and axisymmetric stagnation point flow with heat transfer over a shrinking sheet was investigated by Wang [30]. Very recently, Mahapatra et al. [31] investigated the MHD stagnation point flow and heat transfer over a shrinking sheet, the flow being permeated by a uniform transverse magnetic field. The existence and uniqueness results for MHD stagnation-point flow over a stretching or shrinking sheet were analyzed by Van Gorder et al. [32]. Note that with an added stagnation flow to contain the vorticity, similarity solution is possible even in the absence of suction at the surface. All of these studies were restricted to orthogonal stagnation point flow over a shrinking sheet. To the best of our knowledge, no investigation is made for oblique stagnation point flow towards a shrinking sheet. In this paper, we investigate the steady twodimensional oblique stagnation point flow and heat transfer of an incompressible viscous fluid over a shrinking sheet. The temperature distribution in the flow is determined when the surface is held at a constant temperature. The influence of effective Prandtl number (Preff ) on the temperature distribution of the fluid is investigated and analyzed with the help of their graphical representations.
2 Flow analysis Consider the steady two-dimensional flow of an incompressible viscous fluid near a non-orthogonal stagnation-point on an elastic surface which is shrunk
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in its own plane with a velocity proportional to the distance from the shrinking (stretching) origin. Two equal and opposite forces are applied along the negative direction of x-axis so that the surface is shrunk keeping the origin fixed. Here the axis of the shrinking sheet and stagnation flow are not, in general, aligned. On the sheet, the velocities are u = c(x + l), v = 0 where c(< 0) is the shrinking rate (stretching rate if c > 0) and x = −l is the location of the shrinking (stretching) origin. We consider the two-dimensional stagnation point flow at infinity given by u = ax and v = −ay where a(> 0) is the strength of the stagnation flow. The velocity components along the x and y directions in the inviscid free stream are U0 = ax + 2b(y − δ1 ),
V0 = −a(y − δ2 ),
(1)
ψ0 = ax(y − δ2 ) + b(y − δ1 ) .
(2)
The boundary conditions at the sheet are u = c(x + l),
v = 0 at y = 0.
(3)
The boundary conditions at infinity are u → U0 (x, y)
and v → V0 (x, y)
as y → ∞, (4)
where U0 and V0 are given by (1). Stream function in the boundary layer is assumed in the form ψ = ξ F (η) + W (η), (5) ν where a 1/2 a 1/2 ξ= x and η = y, (6) ν ν and ν is the kinematic viscosity. The velocity components u and v along x and y directions are given by u=
∂ψ ∂y
and v = −
∂ψ . ∂x
(8)
V = −F (η),
(9)
where U = u/(aν)1/2 and V = v/(aν)1/2 . Substituting (8) and (9) into the Navier-Stokes equations we get a 3/2 1 ∂p 2 = ν2 ξ(F − F F − F ) − ρ ∂x ν +(F W − F W − W ) , (10) 3/2 1 ∂p a − = ν2 F F + F , (11) ρ ∂y ν where a prime denotes differentiation with respect to η. Eliminating pressure p(x, y) between (10) and (11), and equating the coefficients of ξ 0 and ξ 1 , we get upon integration F − F F − F = C1 , 2
respectively, where a and b are constants. Here δ1 is the parameter which controls the horizontal pressure gradient that produces the shear flow and δ2 is the displacement thickness that arises due to boundary layer on the stretching (shrinking) sheet. It should be noted that the flow field given by (1) may be viewed as the composition of an orthogonal stagnation-point flow together with a horizontal shear flow. The corresponding stream function ψ0 for this present velocity distribution is 2
U = ξ F (η) + W (η),
(12)
and F W − F W − W = C2 ,
(13)
where C1 and C2 are the constants of integration. Using (8) and (9), the no-slip conditions (3) become F (0) = 0,
F (0) = α,
(14)
W (0) = αL,
(15)
and W (0) = 0,
where α = c/a and L(= ( aν )1/2 l) is the dimensionless distance from the stretching (shrinking) origin. Further from (1), (8) and (9), the boundary conditions for F (η) and W (η) at infinity are F (η) = 1,
F (η) = η − d2
as η → ∞,
and W (η) = 2β(η − d1 )
as η → ∞,
This gives the dimensionless velocity components U and V from (5)–(7) as
(17)
where β = b/a, d1 (= ( aν )1/2 δ1 ) is the dimensionless pressure gradient parameter due to the free stream shear flow and d2 (= ( aν )1/2 δ2 ) is the dimensionless displacement thickness parameter. Using the boundary conditions (16) and (17) in (12) and (13) we get C1 = 1
and C2 = 2β(d2 − d1 ).
(18)
Hence (12) and (13) become F − F F − F = 1, 2
(7)
(16)
(19)
and F W − F W − W = 2β(d2 − d1 ).
(20)
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Here the displacement thickness d2 is obtained by integrating (19) numerically by using the boundary conditions (14) and (16). Substituting (19) and (20) into (10) and (11), we get the dimensionless pressure distribution P (ξ, η) after integration as 1 ξ2 −P = F 2 + F + + 2β(d2 − d1 )ξ + constant, 2 2 (21)
1+
4R θ + PrF θ = 0, 3
(28)
where Pr(= μcp /K) is the Prandtl number and 3 /KK ) is the thermal radiation parameter. R(= 4σ T∞ 1 Introducing the effective Prandtl number Preff =
Pr , 1 + 4R/3
(29)
equation (28) can be converted into
where P (ξ, η) = p(x, y)/ρaν. The dimensionless wall shear stress τ is given as
θ + Preff F θ = 0.
τ = ξ F (0) + W (0).
The boundary conditions for θ are obtained from (26) and (27) as
(22)
θ (0) = 1 and θ (∞) = 0. 3 Heat transfer To determine the temperature distribution in the above flow field, we solve the following energy equation using boundary layer approximation, neglecting viscous dissipation, u
∂T ∂T K ∂ 2T 1 ∂qr +v = , − ∂x ∂y ρcp ∂y 2 ρcp ∂y
(23)
where K, ρ, cp denote the thermal conductivity, fluid density, the specific heat at constant pressure, respectively and qr is the radiative heat flux. Using Rosseland’s approximation for radiation (Brewster [33]), we can write 4σ ∂T 4 , qr = − 3K1 ∂y
(24)
where σ is the Stefan-Boltzmann constant and K1 is the mean absorption coefficient. Assuming the temperature difference within the flow is such that T 4 may be expanded in a Taylor series about T∞ and neglecting 3 T − 3T 4 . Hence higher order terms we get T 4 ≈ 4T∞ ∞ (23) becomes u
3 ∂ 2T ∂T K ∂ 2T ∂T 16σ T∞ +v = + . ∂x ∂y ρcp ∂y 2 3K1 ρcp ∂y 2
(25)
The appropriate boundary conditions are T = Tw
at y = 0
and T = T∞
as y → ∞, (26)
where Tw and T∞ are constants with Tw > T∞ . We introduce the dimensionless temperature θ as θ (η) =
T − T∞ . T w − T∞
Substituting (8), (9) and (27) into (25), we get
(27)
(30)
(31)
Equation (30) shows that the temperature actually does not depend on the Prandtl number Pr and the thermal radiation parameter R separately, but depends only on a combination of them which is the effective Prandtl number (Preff ) given by (29). The effect of thermal radiation in the linearized Rosseland approximation on the heat transfer characteristics of various boundary layer flows is discussed in some details in the reference [34].
4 Numerical solution In the absence of an analytic solution of a problem, a numerical solution is indeed an obvious and natural choice. Thus the governing momentum equations (19) and (20) along with the boundary conditions (14)–(17) and the thermal equation (30) together with boundary conditions (31) are solved numerically using shooting technique by converting it into an initial value problem. To do this, we first transform the non-linear differential equation (19) to a system of three first order differential equations as: y2 = y3 , y1 = y2 , 2 y3 = y2 − y1 y3 − 1,
(32)
where y1 = F (η), y2 = F (η), y3 = F (η) and a prime denotes differentiation with respect to η. The boundary conditions (14) and (16) become y1 = 0, y2 = α at η = 0, y2 → 1 as η → ∞.
(33)
For a given α, the values of y1 and y2 are known at the starting point η = 0. Now the value of y2 as η → ∞ is
Meccanica (2012) 47:1325–1335
replaced by y2 = 1 at a finite value η(= η∞ ) to be determined later. The value of y3 at η = 0 is guessed in order to initiate the integration scheme. Starting from the given values of y1 and y2 at η = 0 and the guessed value of y3 at η = 0, we integrate the system of first order equations (32) by using a fourth-order RungeKutta method up to the end-point η = η∞ . The computed value of y2 at η = η∞ is then compared with y2 = 1 at η = η∞ . The absolute difference between these two values should be as small as possible. To this end we use a Newton-Raphson iteration procedure to assure quadratic convergence of the iterations. The value of η∞ is then increased till y2 attains the value unity asymptotically. Using the numerical values of F (η) from the solutions of (14), (16) and (19), (20) along with the boundary conditions (15) and (17) are solved numerically using the same method as described above to obtain W (η). Once the numerical values of F (η) are known, we can solve (30) together with the boundary conditions (31) using the same technique to get the numerical values of θ (η).
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Fig. 1 Trajectories of F (0) and W (0) for different values of β with d1 = 0.5 and L = 0.5
5 Results and discussions In order to validate the method used in this study and to judge the accuracy of the present analysis, comparisons with the available literature of Wang [30], Mahapatra and Gupta [9], Lok et al. [20] for skin friction coefficients F (0) and W (0) for orthogonal stagnation point flow towards a stretching/shrinking sheet are made and found excellent agreement, as shown in Tables 1–3 (taking β = 0). Therefore we are confident that the present results are accurate. From a physical point of view, vorticity generated at the shrinking sheet is not confined within a boundary layer and a steady solution is not possible unless sufficient stagnation flow is added to the free stream. Even with an added stagnation flow, a steady solution is possible only when the ratio of shrinking velocity and free stream velocity becomes less than a certain numerical value. Our numerical solution reveals that when α < −1.24658077, there is no similarity solution although multiple solutions exist for −1.24658077 ≤ α ≤ −1.0 and unique solution exists for α > −1. Figure 1 shows the trajectories of the components of the skin friction coefficient F (0) and W (0) for different values of β(= b/a) with d1 = 0.5
Fig. 2 Variation of F (η) with η for different values of α(< 0)
and L = 0.5 (i.e., non-aligned). It should be noted that increase in β results in increase in the shearing motion in turn leads to increased obliquity of the flow towards the sheet. It also to be noted that the values of F (0) do not depend on β and L (see (19) along with boundary conditions (14) and (16)). The figure reveals that for shrinking at the sheet, F (0) ≥ 0 but W (0) may be negative. In case of orthogonal stagnation flow, W (0) becomes negative beyond α < −1. But interestingly, for oblique stagnation flow W (0) > 0 for all α(< 0) where the similarity solution exists. Figure 2 shows the variation of the vertical velocity component F (η) with η for different values of α(< 0). It is interesting to note that for shrinking at the sheet, the function F (η) is initially negative, showing regions of reverse cellular flow. This is due to the fact that the shrinking velocity and free stream velocity act in opposite directions. As |α| increases, the region of reverse cellular flow gradually increases.
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Meccanica (2012) 47:1325–1335 Table 1 Comparison of the values of F (0) (with β = 0) for stretching sheet with different values of α α
Present study
Wang [30]
Mahapatra and Gupta [9]*
Lok et al. [20]*
0.0
1.232588
1.232588
–
–
0.1
1.146560
1.146560
–
–
0.2
1.051131
1.051130
–
0.5
0.713296
0.713300
1.0
0
0
– 0.71329
0.71334
0
2.0
−1.887308
−1.88731
5.0
−10.264751
−10.26475
* Results
of [9] and [20] are adjusted using our normalization
–
−1.8874
−1.88733
−10.265
−10.2648
Table 2 Comparison of the values of F (0) for the shrinking sheet when β = 0 α
Wang [30]
Present work
first solution
second solution
first solution
second solution
−0.25
1.40224
–
1.402242
–
−0.50
1.49567
–
1.495672
–
−0.75
1.48930
–
1.489296
–
−1.00
1.32882
0.0
1.328819
0.0
−1.10
–
–
1.186680
0.049229
−1.15
1.08223
0.116702
1.082232
0.116702
−1.20
–
–
0.932470
0.233648
−1.2465
0.55430
–
0.584374
0.554215
Table 3 Comparison of the values of W (0) for the shrinking sheet when β = 0 α
Wang [30]
Present work
first solution
second solution
first solution
second solution
−0.25
−0.66857
–
−0.668575
–
−0.50
−0.50145
–
−0.501451
–
−0.75
−0.29376
–
−0.293760
–
−1.00 −1.10 −1.15 −1.20 −1.2465
0.0 – 0.297995 – 0.99904
–
0.0
–
–
0.176959
4.265787
0.276345
0.297990
2.763441
–
0.471858
1.883152
–
0.947612
0.999190
The variation of W (η) with η is displayed in Fig. 3 for both stretching (α > 0) and shrinking (α < 0) at the sheet for fixed values of β and d1 . It is noticed that the effect of W (η) is small for stretching sheet but large in case of a shrinking sheet. The variations of the horizontal velocity component U (ξ, η) with η
(for a fixed value of ξ ) are displayed in Figs. 4–7 for different values of the physical parameters. Figure 4 displays the variation of U (ξ, η) with η for different values of α(< 0) in aligned case and Fig. 5 shows the same for non-aligned case. It is found that as |α| increases, |U (ξ, η)| decreases and this effect is more
Meccanica (2012) 47:1325–1335
Fig. 3 Variation of W (η) with η for different values of α at a fixed value of β = 0.5 and d1 = 0.5 (aligned)
Fig. 4 Variation of U (ξ, η) with η for different values of α(< 0) at a fixed value of ξ = 0.5, d1 = 0.5 and β = 0.05 (aligned)
pronounced for non-aligned case. Figure 6 shows the variation of U (ξ, η) with η at a fixed value of ξ for several values of β with fixed values of the other parameters. It can be seen that velocity at a point increases with increase in β which is consistent with the fact that increase in free stream shear increases the velocity near the surface. It may be observed from the figure that the flow displays boundary layer behaviour even for large value of shear and the slope of the boundary layer is asymptotically of order 2β. Figure 7 displays the variation of U (ξ, η) with η for several values of the displacement thickness d1 with fixed values of the other parameters. It is seen that as d1 increases, U (ξ, η) decreases. Figures 8–10 show the streamlines pattern in case of two different values of α(< 0) keeping the other parameters fixed for aligned (L = 0), non-aligned with positive L(= 1.2) and non-aligned with negative
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Fig. 5 Variation of U (ξ, η) with η for different values of α(< 0) at a fixed value of ξ = 0.5, d1 = 0.5 and β = 0.05 (non-aligned)
Fig. 6 Variation of U (ξ, η) with η for different values of β(> 0) at a fixed value of ξ = 0.5, α = −0.5 and d1 = 0.5 (non-aligned)
L(= −0.25), respectively. These figures reveal that as |α| increases, the region of flow reversal increases. This is consistent with the result previously obtained viz., the region of flow reversal increases with increase in |α|. Figures 11 and 12 show the streamline patterns for aligned and non-aligned cases for different values of β. These figures show how the streamlines change with the change of β in aligned and non-aligned cases. It is to be noted that for the stagnation-point flow over a shrinking sheet, there is a region near the sheet where flow reversal takes place. This is happened because the directions of stagnation-point velocity and shrinking velocity are opposite. But for such flow over a stretching sheet, these two directions are same and consequently no flow reversal region is found (see Wang [30]).
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Fig. 7 Variation of U (ξ, η) with η for different values of d1 (> 0) at a fixed value of ξ = 0.5, α = −0.5 and β = 0.5 (non-aligned)
Fig. 9 Streamlines of oblique stagnation-point flow for two different values of α with d1 = 0.5, β = 0.3 and L = 1.2 (non-aligned) (solid line → α = −0.5, dashed line → α = −0.8)
Fig. 8 Streamlines of oblique stagnation-point flow for two different values of α with d1 = 0.5, β = 0.3 and L = 0 (aligned) (solid line → α = −0.5, dashed line → α = −0.8)
The temperature distribution given by θ (η) is shown in Fig. 13 for several values of α(< 0) with fixed values of the parameters. It is seen that temperature at a point increases with increase in |α|. The influence of effective Prandtl number (Preff ) on temperature distribution is enlightened in Fig. 14. It is to be noted that Preff is nothing but a simple rescaling of the Prandtl number Pr by a factor involving the radiation parameter R. The figure reveals that an increase in Preff causes the decrease in temperature as well as in thermal boundary layer thickness. From a physical
Fig. 10 Streamlines of oblique stagnation-point flow for two different values of α with d1 = 0.5, β = 0.3 and L = −0.25 (non-aligned) (solid line → α = −0.5, dashed line → α = −0.8)
point of view, if Preff increases, the thermal diffusivity decreases and this phenomenon leads to decrease the thermal boundary layer thickness. The variation of the temperature gradient (−θ (0)) at the sheet with several values of α for different val-
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Fig. 13 Variation of θ(η) with η for several values of α(< 0) with several values of effective Prandtl number Preff = 0.35
Fig. 11 Streamlines of oblique stagnation-point flow for two different values of β with d1 = 0.2, α = −0.5 and L = 0 (aligned) (solid line → β = 0.0, dashed line → β = 0.3)
Fig. 14 Variation of θ(η) with η for several values of effective Prandtl number (Preff ) with α = −0.8
Fig. 12 Streamlines of oblique stagnation-point flow for two different values of β with d1 = 0.2, α = −0.5 and L = 0.5 (non-aligned) (solid line → β = 0.0, dashed line → β = 0.3)
ues of the effective Prandtl number (Preff ) is shown in Fig. 15. It is seen that −θ (0) is positive and this is consistent with the fact that in the absence of viscous dissipation, heat flows from the surface to the fluid as long as Tw > T∞ . The figure reveals that for a fixed
Fig. 15 Variation of −θ (0) with α for several values of effective Prandtl number (Preff )
value of α, −θ (0) increases with increasing Preff beyond a certain value of α.
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6 Conclusions The present paper deals with the analysis of twodimensional oblique stagnation-point flow and heat transfer of an incompressible viscous fluid towards a shrinking surface. In comparison with the stretching sheet, the shrinking sheet has some interesting result. For the stretching sheet, studied previously, similarity solution exists for all stretching velocity. But for the shrinking sheet, similarity solution may or may not exists. Even if it exists it may or may not unique depending on the ratio of the shrinking velocity and free stream velocity. A region of reverse flow occurs near the shrinking surface but such a situation does not appear in the stretching case. The effect of non-alignment is also studied for both stretching and shrinking sheet. It can be concluded that the effect of non-alignment is small for the stretching case but large for the shrinking case. Acknowledgements The authors thank the reviewers for their time and interest as well as constructive suggestions and comments for improving the paper. The work of one of the authors (T.R.M) is supported under SAP (DRS PHASE II) program of UGC, New Delhi, India. One of the authors (A.S.G) acknowledges the financial support of Indian National Science Academy for carrying out this work.
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