Meccanica (2013) 48:23–32 DOI 10.1007/s11012-012-9579-5

Stability of dual solutions in stagnation-point flow and heat transfer over a porous shrinking sheet with thermal radiation Tapas Ray Mahapatra · Samir Kumar Nandy

Received: 21 November 2011 / Accepted: 31 July 2012 / Published online: 14 August 2012 © Springer Science+Business Media B.V. 2012

Abstract An analysis is carried out to study the steady two-dimensional stagnation-point flow and heat transfer of an incompressible viscous fluid over a porous shrinking sheet in the presence of thermal radiation. A set of similarity transformations reduce the boundary layer equations to a set of non-linear ordinary differential equations which are solved numerically using fourth order Runge-Kutta method with shooting technique. The analysis of the result obtained shows that as the porosity parameter β increases, the range of region of existence of similarity solution increases. It is also observed that multiple solutions exist for a certain range of the ratio of the shrinking velocity to the free stream velocity (i.e., α) which again depends on β. We then discuss the stability of the unsteady solutions about each steady solution, showing that one steady state solution corresponds to a stable solution whereas the other corresponds to an unstable solution. The stable solution corresponds to the physically relevant solution. Further we obtain numerical results for each solution, which enable us to discuss the features of the respective solutions.

T.R. Mahapatra Department of Mathematics, Visva-Bharati University, 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]

Keywords Stagnation-point flow · Heat transfer · Porous shrinking sheet · Stability analysis · Thermal radiation

1 Introduction The study of boundary layer flow over a stretching sheet is a subject of great interest due to its various applications in designing cooling system which includes liquid metals, MHD generators, accelerators, pumps and flow meters. Furthermore, the continuous surface heat and mass transfer problems are many practical applications in electro-chemistry and polymer processing. Many chemical engineering processes, like metallurgical and polymer extrusion involve cooling of a molten liquid being stretched into a cooling system. The fluid mechanical properties of the penultimate product depend mainly on the process of stretching and on the cooling liquid used. Some polymer fluids, like Polyethylene oxide and Polyisobutylene solution in cetane are generally used as cooling liquid since their flow can be regulated by external magnetic fields in order to improve the quality of the final product. A large number of analytical and numerical studies explaining various aspects of the boundary layer flow over a stretching surface are available. Mention may be made to some interesting works (see [1–5]). Stagnation-point flow is a topic of significance in fluid mechanics, because stagnation-point appears in virtually of all flow fields of science and engineering.

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In some cases, the flow is stagnated by a solid wall, while in others a free stagnation-point or a line exists interior of the fluid domain. The pioneering work in this area was carried out by Hiemenz [6] who studied the steady two-dimensional stagnation-point flow on an infinite wall. Hiemenz’s [6] work was extended by many authors ([7–16]) by considering the effects of heat, mass transfer and magnetic field under various physical conditions. In contrasts, less works are done on the flow over a shrinking sheet where the velocity on the boundary is towards a fixed point. The study of flow over a shrinking surface has important applications in industries and engineering. Shrinking film is one of the common applications of shrinking problems in industries. In packaging of bulk products, shrinking film is very useful as it can be unwrapped easily with adequate heat. Shrinking problems can also be applied to study the capillary effects in small pores, the shrinking-swell behavior and hydraulic properties of agricultural clay soils. The associated changes in hydraulic and mechanical properties of such soils will hamper predictions of the flow and transport processes which are essential for agricultural development and environmental strategies. The shrinking surface situation also occurs, for example, on a rising shrinking balloon. From a physical point of view, there are two situations for which the similarity solutions likely to exist; either an adequate suction is imposed on the boundary or a stagnation flow is added to confine the vorticity within the boundary layer. Miklavcic and Wang [17] studied the flow over a shrinking sheet, which is an exact solution of the NS equations and they reported that the mass suction is required to maintain the flow. The problem of boundary layer flow over a shrinking sheet with different types of surface velocities were investigated recently [18–20]. The shrinking sheet problem was also extended to other fluids [21–23]. The shrinking sheet problem to unsteady conditions with a time dependent wall shrinking velocity was considered by Fang et al. [24] and the effect of magnetic field was analyzed by Merkin and Kumaran [25]. The flow over a permeable shrinking sheet embedded in a porous medium was also studied [26, 27]. The work on the flow over a shrinking sheet was further generalized to include a stagnation flow as the free stream by Wang [28]. The effect of magnetic field on Wang’s [28] problem was analyzed by Lok et al. [29] and the slip effect was considered by Bhattacharyya et al. [30]. Recently, Van

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Gorder et al. [31] analyzed the existence and uniqueness results over the semi-infinite interval [0, ∞) for a class of nonlinear third order ordinary differential equations arise in the stagnation-point flow of a hydromagnetic fluid over a stretching or shrinking sheet. Over the past several decades, a considerable number of studies have shown the existence of multiple solutions of boundary layer flows driven by moving surfaces with or without external pressure gradients. The method of finding dual solutions and analyzing stability is of practical importance to those interested in engineering analysis, as it provides one with a way to determine whether a steady state solution is physically meaningful. Hence such study is very much useful for the determination of the treatment of the fluid flow problems with multiple solutions. The motion induced by a flat plate moving at constant velocity beneath a uniform mainstream for both finite and semiinfinite plates were considered by Kemp and Acrivos [32] and dual solutions were found when the plate advances towards the oncoming stream. Later, Riley and Weidman [33] provided an asymptotic analysis for the lower branch solution behavior near the terminal point where the shear stress and plate velocity simultaneously tend to zero. Merrill et al. [34] investigated the unsteady mixed convection stagnation-point flow on a vertical surface in a fluid saturated porous medium and they found that dual solutions exist for some values of the buoyancy parameter. By implementing a stability analysis (Merkin [35]), Merrill et al. [34] showed that the lower branch solutions are unstable and thus they are not asymptotically available solutions. The simultaneous effects of normal transpiration through and tangential movement of a semi-infinite plate on selfsimilar boundary layer flow beneath a uniform free stream were considered by Weidman et al. [36]. Later, Paullet and Weidman [11] analyzed the flow behavior in the neighborhood of a stagnation-point towards a stretching sheet. In that paper, they gave numerical evidence that a second solution exists for some values of the ratio of the stagnation flow strain rate to the stretch rate of the sheet. A linear stability analysis reveals that solutions along with upper branch are linearly stable whilst those on the lower branch are linearly unstable. Very recently, a stability analysis of dual solutions in MHD stagnation-point flow over a stretching sheet was analyzed by Mahapatra et al. [37]. Motivated by the above mentioned investigations and applications, in this paper, we investigate the

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25

u

ν ∂u dU ∂ 2u ∂u (U − u), (2) +v = U (x) +ν 2 + ∂x ∂y dx K1 ∂y

u

∂T ∂T K ∂ 2T 1 ∂qr +v = . − ∂x ∂y ρcp ∂y 2 ρcp ∂y

(3)

Here u and v are the velocity components in the x and y directions respectively, K1 is the permeability of the porous medium, μ is the coefficient of fluid viscosity, ρ is the fluid density, ν (= μ/ρ) is the kinematic viscosity, K is the thermal conductivity, cp is the specific heat at constant pressure and qr is the radiative heat flux. In Eq. (2), U (x) stands for the stagnation-point velocity in the inviscid free stream and T is the temperature of the fluid. The boundary conditions for the velocity and the temperature are given by u = cx, Fig. 1 A sketch of the physical model and coordinate system

boundary layer flow and heat transfer in the neighborhood of a stagnation-point over a porous shrinking sheet with the presence of thermal radiation. The focus of the present endeavor is twofold. First we study the dual solutions of the flow phenomenon numerically. Second, the stability of the dual solutions based on linear disturbances of the steady similarity solutions is determined.

2 Flow analysis Consider the steady two-dimensional stagnation-point flow of an incompressible viscous fluid towards a porous surface which is shrunk in its own plane with a velocity proportional to the distance from the stretching/shrinking origin. Here shrinking of the sheet is along the negative direction of x-axis. On the sheet, the velocity components are u = cx, v = 0, where c (< 0) is the shrinking rate (stretching rate if c > 0). The velocity components at infinity are given by u = ax, v = −ay, where a (> 0) is the strength of the stagnation flow. The flow configuration is shown in Fig. 1. Using the boundary layer approximations, the equations for mass, momentum and energy are written in the usual notations as ∂u ∂v + = 0, ∂x ∂y

(1)

v = 0,

u → U (x) = ax,

T = Tw T → T∞

at y = 0, as y → ∞.

(4) (5)

We introduce the similarity transformations √ u = axF  (η), v = − aνF (η), θ (η) =

T − T∞ , T w − T∞

(6)

where η is the dimensionless similarity variable and is defined as  1/2 a η= y, (7) ν and a prime denotes differentiation with respect to the similarity variable η. With these values of u and v (defined by Eq. (6)), we find that the continuity equation (1) is identically satisfied. Using Rosseland’s approximation for radiation (Brewster [38]), we can write qr = −

4σ ∂T 4 , 3K ∗ ∂y

(8)

where σ is the Stefan-Boltzmann constant and K ∗ 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 . higher order terms we get T 4 ≈ 4T∞ ∞ In view of these relations, Eqs. (2) and (3) reduce to   F  + F F  − F  2 + β 1 − F  + 1 = 0, (9)

26

 1+

Meccanica (2013) 48:23–32

 4R  θ + PrF θ  = 0, 3

(10)

where β (= ν/aK1 ) is the permeability parameter of the porous medium, Pr (= μcp /K) is the Prandtl 3 /KK ∗ ) is the thermal radinumber and R (= 4σ T∞ ation parameter. The boundary conditions for F (η) and θ (η) are 

F (0) = 0,

F (0) = α,

θ (0) = 1,

θ (∞) = 0,



F (∞) = 1,

(11) (12)

where α (= c/a) is the velocity ratio parameter. For this two-dimensional flow, the dimensionless stream function ψ ∗ can be defined as ψ ∗ (ξ, η) =

ψ = ξ F (η), ν

Eq. (10) together with the boundary conditions (12) using the same technique to get the numerical values of θ (η).

3 Stability analysis Numerically it is seen that for a certain range of α, there exist two branches of solutions for different values of the porosity parameter β. So we have to test the stability of the dual solutions. To do this, we consider the unsteady form of Eq. (2) as ∂u ∂u dU ∂ 2u ν ∂u +u +v = U (x) +ν 2 + (U − u), ∂t ∂x ∂y dx K1 ∂y

(13)

(17)

where ψ is the dimensional stream function and ξ = (a/ν)1/2 x is the dimensionless distance along the plate. The physical quantities of interest are the skin friction coefficient CF and the local Nusselt number Nux , which are defined by

together with the equation of continuity (1). The unsteady similarity solution can be taken in the form √ u = axf  (η, τ ); v = − νaf (η, τ ), (18)

CF =

τw , ρu2w /2

Nux =

xqw , K(Tw − T∞ )

(14)

where the skin-friction τw and the heat transfer from the plate qw are given by     ∂u ∂T τw = μ , qw = −K . (15) ∂y y=0 ∂y y=0

where η is the same as defined in (7) and τ (= at) is the dimensionless time. Substituting (18) into (17) we get   f  + ff  − f  2 + 1 + β 1 − f  − fτ = 0, (19) where the subscript τ denotes partial derivative with respect to τ . The boundary conditions are f (0, τ ) = 0,

f  (0, τ ) = α,

(20)

Using Eqs. (6) and (7) we have 1 1/2 CF Rex = F  (0), 2

f  (∞, τ ) = 1.

Nux /Rex = −θ  (0), (16) 1/2

where Rex is the local Reynolds number. The coupled ordinary differential equations (9) and (10) subject to the boundary conditions (11) and (12) are solved numerically by an efficient shooting technique for different values of the physical parameters. To do this, we first transform the non-linear differential equation (9) to a system three first order differential equations, which are solved by means of a standard fourth order Runge-Kutta integration technique. Then a Newton iteration procedure is employed to assure quadratic convergence of the iterations required to satisfy the outer boundary condition F  (∞) = 1. Once the numerical values of F (η) are known, we can solve

Stability of the dual solutions is determined by adopting the stability analysis of Merkin [35] and we put f (η, τ ) = F (η) + e−γ τ g(η),

(21)

where F (η) satisfies the steady state boundary value problem (2) and γ is the growth (or decay) rate of the disturbance. Here g(η) and all its derivatives are assumed small compared with the steady solution F (η) and its derivatives. Such an assumption is made because we are studying the linear stability analysis of the basic flow F (η) so that the disturbance g(η) is small. Substitution of (21) into (19) and (20) and linearizing, we get   (22) g  + F g  + γ − 2F  − β g  + F  g = 0,

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27

Table 1 Comparison of the values of F  (0) (with β = 0) for stretching sheet with different values of α α

Present study

Wang [28]

Mahapatra and Gupta [8]*

Lok et al. [12]*

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

– 0.71329

0

0.71334

0

2.0

−1.887308

−1.88731

5.0

−10.264751

−10.26475

–

−1.8874

−1.88733

−10.265

−10.2648

* Results of [8] and [12] are adjusted using our normalization Table 2 Comparison of the values of F  (0) for the shrinking sheet when β = 0 α

Wang [28] first solution

Present work second solution

first solution

second solution

−0.25

1.40224

–

1.402242

–

−0.50

1.49567

–

1.495672

–

−0.75

1.48930

–

1.489296

– 0.0

−1.00

1.32882

0.0

1.328819

−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

and g(0) = 0,

g  (0) = 0,

g  (∞) = 0.

(23)

Now the homogeneous linear equation (22) subject to the homogeneous boundary conditions (23) constitutes an eigenvalue problem with γ as the eigenvalue. Solutions of (22) and (23) give an infinite set of eigenvalues γ1 < γ2 < γ3 < · · · . If the smallest eigenvalue γ1 is negative, then there is an initial growth of disturbances and the flow is unstable. On the other hand, when γ1 is positive, there is an initial decay and the flow is stable.

4 Results and discussion In order to validate the method used in this study and to judge the accuracy of the present analysis, we compare our results for the value of the skin friction coefficient F  (0) (taking β = 0) for the stretching/shrinking

sheet with those of Wang [28], Mahapatra and Gupta [8] and Lok et al. [12]. These comparisons are shown in Tables 1 and 2. A good agreement is observed between these results. This lends confidence in the numerical results to be reported subsequently. Figure 2 shows the variation of the skin friction coefficient F  (0) with α < 0 (shrinking sheet) and α > 0 (stretching sheet) for different values of the porosity parameter β. The figure shows that it is possible to obtain dual solutions of the similarity equations (9)–(12) when the sheet velocity and free stream velocity act in opposite directions (i.e., α < 0). This observation is in agreement with the result reported in Ref. [28]. It is observed that the similarity solution for a fixed value of β exists up to a critical value α = αc (< 0), beyond which the boundary layer separates from the sheet and the solution based on the boundary layer approximations is not possible. The coordinates of the points (αc , F  (0)), where the upper branch solution (designated as first solution) meets the lower branch solution (designated as second solution) for corresponding

28

Fig. 2 Wall shear stress F  (0) with α for different values of the porosity parameter β

Fig. 3 Variation of −θ  (0) with α for different values of the porosity parameter β

values of β are shown in Fig. 2. The figure also reveals that as β increases, the range of α where similarity solution exists progressively increases. Also the values of |F  (0)| increases as |α| increases and these values reach the maximum before decreasing to zero. It is to be noted that for α = 1, F (η) = η is the solution of Eq. (9) subject to the boundary conditions (11) and consequently F  (0) = 0 for all β. The variations of the local Nusselt number −θ  (0) with different values of α and β are depicted in Fig. 3. From the figure, the dual nature of the temperature in the range of α as stated above is observed. The coordinates of the points (αc , −θ  (0)) for corresponding values of β are also shown in this figure. The figure reveals that the values of −θ  (0) are higher for higher values of β. Thus the porosity parameter β reduces the heat transfer rate at the surface. Figure 4 shows the variation of the smallest eigenvalue γ1 with α for several values of the porosity pa-

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Fig. 4 Plot of lowest eigenvalues γ1 as a function of α for different values of the porosity parameter β (solid line: first branch, dotted line: second branch)

rameter β. In this figure the smallest eigenvalue γ1 for the first solution is plotted by solid line and that for the second solution is plotted by dotted line. Since for the first solution, γ1 is real and positive, it follows that the first solution is linearly stable. Also for a given value of α, γ1 increases with increase in β. Thus we may conclude that for these stable solutions, disturbances decay more quickly with increase in β. Since for the second solution, γ1 is real and negative, it is clear that the second solution is linearly unstable. For this solution γ1 decreases with increase in β (for a fixed α) and hence the solution becomes more unstable with increasing β. The novel result that emerges from the analysis is that the first solution branch is stable and the second solution branch is unstable. Also the stable solution corresponds to the physically meaningful solution for the above flow, while the unstable solution is not physically meaningful. That is to say, though two solutions exist mathematically, only the stable solution is physically meaningful and can be realized physically. The influence of the porosity parameter β on the horizontal velocity F  (η), vertical velocity F (η) and the temperature θ (η) are displayed in Figs. 5, 6, 7 for both the solution branches. In these figures the dual nature of the velocity and temperature profiles are observed. From Fig. 5, it is observed that magnitude of the horizontal velocity increases as β increases in first solution and decreases in second solution. Figure 6 shows the variation of the vertical velocity components F (η) with η for different values of β. It is interesting to note that F (η) is initially decreasing with values being negative and for large η it starts to increase and ultimately it becomes positive. Hence for

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Fig. 5 Variation of F  (η) with η for several values of the porosity parameter β with α = −1.5 (solid line → first solution branch, dashed line → second solution branch)

Fig. 7 Variation of θ(η) with η for several values of the porosity parameter β with α = −1.5, Pr = 0.71 and R = 1.0 for (solid line → first solution branch, dashed line → second solution branch)

Fig. 6 Variation of F (η) with η for several values of the porosity parameter β with α = −1.5 (solid line → first solution branch, dashed line → second solution branch)

Fig. 8 Variation of F  (η) with η for several values of the shrinking rate parameter α with β = 0.3 (solid line → first solution branch, dashed line → second solution branch)

all α (< 0), the velocity profiles F (η) exhibit reverse flow. This is expected because the shrinking velocity and stagnation-point velocity act in opposite direction. Figure 6 reveals that as β increases, the region of reverse cellular flow decreases for the first solution and increases for the second solution. The temperature profiles θ (η) for selected values of Pr, α and R with different values of β are presented in Fig. 7 and show that the far field boundary condition θ (∞) = 0 is approached asymptotically. It is observed that as β increases, temperature at a point decreases for the first solution and reverse effect is observed for the second solution. The boundary layer thickness for the second solution is higher compared to the first solution. This is consistent with the results that the first branch solutions are stable whilst the second branch solutions are not.

Figures 8, 9, 10 indicate that the effects of the shrinking rate parameter α (< 0) on F  (η), F (η) and θ (η) respectively. Figure 8 depicts the influence of α (< 0) on F  (η). In this figure, the dual velocity profiles F  (η) reveal that |F  (η)| decreases as |α| increases for the first solution branch. But for the second solution branch, |F  (η)| decreases as |α| decreases except in a small neighborhood near the sheet. From Fig. 9, it is observed that the region of flow reversal increases as |α| increases in first solution and decreases in second solution. Figure 10 gives an idea about the dual temperature profile θ (η) for several values of α. The value of the temperature profile θ (η) at a point increases with increase in α for first solution, while for second solution it decreases. Figure 11 shows the streamline patterns for the stagnation-point flow over a shrinking sheet for dif-

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Fig. 9 Variation of F (η) with η for several values of the shrinking rate parameter α with β = 0.3 (solid line → first solution branch, dashed line → second solution branch) Fig. 11 Streamlines for the first solution branch for α = −1.5, solid line → β = 0.3, dashed line → β = 0.4

Fig. 10 Variation of θ(η) with η for several values of α(< 0) with β = 0.3, Pr = 0.71 and R = 1.0 (solid line → first solution branch, dashed line → second solution branch)

ferent values of the porosity parameter β in the case of first solution branch. The figure indicates that there is a flow reversal. The region of flow reversal near the shrinking sheet decreases as β increases. This is consistent with the result for the first solution branch as shown in Fig. 6. The streamline patterns for different values of α (< 0) with a fixed value of β (= 0.5) for the first solution branch are shown in Fig. 12. The figure reveals that as |α| increases, the region of reverse cellular flow also increases and is consistent with the result as obtained in Fig. 9. The influence of the Prandtl number Pr on temperature distribution is depicted in Fig. 13. It is observed that the increase in Pr causes the decrease in thermal boundary layer thickness. From a physical point of view, if Pr increases, the thermal diffusivity decreases and this phenomenon leads to the decreasing of energy ability that reduces the thermal boundary layer.

Fig. 12 Streamlines for the first solution branch for β = 0.5, solid line → α = −1.5, dashed line → α = −1.6

Figure 14 illustrates the effect of radiation parameter R on the temperature profiles θ (η). It is seen that the temperature profiles and the thermal boundary layer thickness increase with an increase of R. It is to be noted that the radiation parameter R and Prandtl number Pr have no influence on the flow field, which is clear from Eqs. (9)–(12).

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ison to the stretching sheet, the shrinking sheet has some interesting characteristics. Acknowledgements We appreciate the comments of the reviewers, which have led to improvement of the manuscript. The work of one of the authors (T.R.M.) is supported under SAP (DRS PHASE II) program of UGC, New Delhi, India.

References

Fig. 13 Variation of θ(η) with η for several values of Prandtl number Pr with α = −1.5, β = 0.3 and R = 1.0

Fig. 14 Variation of θ(η) with η for several values of radiation parameter R with α = −1.2, β = 0.5 and Pr = 0.71

5 Conclusion In summery, the boundary layer stagnation-point flow and heat transfer adjacent to a porous shrinking sheet in the presence of thermal radiation have been investigated. Greatly different solution behavior with multiple solution branches has been found compared with the corresponding stretching sheet problem. A linear stability analysis reveals that one solution is stable whilst the other is unstable and the stable solution is the physically meaningful solution. The numerical result reveals that with an increase in the porosity parameter β, the range of α where similarity solution exists, increases. A region of reverse cellular flow occurs near the shrinking sheet. It has been observed that the boundary layer thickness for the first solution is always smaller than that of the second solution for both velocity and temperature distributions. So in compar-

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