Meccanica DOI 10.1007/s11012-012-9688-1

Slip effects on unsteady stagnation-point flow and heat transfer over a shrinking sheet Tapas Ray Mahapatra · Samir Kumar Nandy

Received: 9 December 2011 / Accepted: 17 December 2012 © Springer Science+Business Media Dordrecht 2013

Abstract This paper investigates the unsteady boundary layer stagnation-point flow and heat transfer over a linearly shrinking sheet in the presence of velocity and thermal slips. Similarity solutions for the transformed governing equations are obtained and the reduced equations are then solved numerically using fourth order Runge-Kutta method with shooting technique. The numerical results show that multiple solutions exist for certain range of the ratio of shrinking velocity to the free stream velocity (i.e., α) which again depend on the unsteadiness parameter β and the velocity slip parameter (i.e., δ). An enhancement of the velocity slip parameter δ causes more increment in the existence range of similarity solution. Fluid velocity at a point increases increases with the increase in the value of the velocity slip parameter δ, resulting in a decrease in the temperature field. The effects of the velocity and thermal slip parameters, unsteadiness parameter (β) and the velocity ratio parameter (α) on the velocity and temperature distributions are computed, analyzed and discussed. The reported results are in

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]

good agreement with the available published results in the literature. Keywords Unsteady stagnation-point flow · Heat transfer · Shrinking sheet · Partial slip · Dual solutions

1 Introduction The problem of boundary layer flow of an incompressible viscous fluid near the stagnation-point on a stretching sheet has an important bearing on several technological processes such as the cooling of the metallic plate, nuclear reactor, extrusion of polymers, etc. In all of these cases, a study of the flow field and heat transfer can be of significant importance since the quality of the final product depends to a large extent on the skin friction coefficient and the surface heat transfer rate. In view of these applications, enormous works have been done in various aspects related to the boundary layer flow and heat transfer over a stretching surface (see [1–4]). Stagnation-point flow is a topic of significance in fluid mechanics, in the sense that stagnation-points appear virtually in all flow fields of science and engineering. In some cases, flow is stagnated by a solid wall while in others, a free stagnation-point or a line exists interior to the fluid domain. The pioneering work in this area was carried out by Hiemenz [5] who studied the steady two-dimensional boundary layer flow in the neighborhood of a stagnation-point on an infinite wall.

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The numerical result found by Hiemenz [5] was later improved by Howarth [6]. Following these works, various aspects of stagnation-point flow and heat transfer over a stretching sheet are investigated by many researchers (see [7–11]). All the above mentioned studies deal with a steady flow only. However in some cases, the flow and heat transfer can be unsteady due to a sudden stretching of the flat sheet or by a step change of the temperature or heat flux of the sheet. Andersson et al. [12] presented a new similarity solution for the temperature field by transforming the time dependent thermal energy equation to an ordinary differential equation. Elbashbeshy and Bazid [13] presented an exact similarity solution for unsteady flow and heat transfer over a horizontal stretching surface. The unsteady two dimensional flow of an incompressible viscous fluid in the neighborhood of a stagnation-point over a stretching surface in the presence of time dependent free stream was investigated by Sharma and Singh [14]. Boundary layer flow and heat transfer over an unsteady stretching vertical surface was analyzed by Ishak et al. [15]. Recently, the boundary layer flow due to a shrinking sheet has attracted considerable interest. The shrinking sheet situation occurs, for example, on a rising shrinking balloon. One of the common applications of shrinking sheet problems in engineering and industries is shrinking film. In packaging of bulk products, shrinking film is very useful as it can be unwrapped easily with adequate heat. For the shrinking sheet flow, 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 adequate suction is applied at the sheet. For this type of flow, it is essentially a backward flow as discussed by Goldstein [16]. 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 forward stretching case. Miklavcic and Wang [17] analyzed both twodimensional and axisymmetric viscous flow induced by a shrinking sheet in the presence of uniform suction. The above shrinking sheet problem was then extended to power-law surface velocity by Fang [18]. The boundary layer flow for a moving flat plate with

mass transfer in a stationary fluid was discussed by Fang et al. [19]. Steady two-dimensional and axisymmetric stagnation-point flow and heat transfer of a viscous fluid towards a shrinking sheet was investigated by Wang [20]. He found that solutions do not exist for larger shrinking rates and non-unique in the twodimensional case. After this pioneering work, the flow field in the neighborhood of a stagnation-point towards a stretching/shrinking sheet has drawn considerable attention and a good amount of literature has been generated on this problem (see [21–26]). Unlike the unsteady stretching sheet problem, little work is done on unsteady boundary layer flow induced by a shrinking sheet. The unsteady viscous flow over a continuously shrinking surface with mass suction was investigated by Fang et al. [27] and showed that multiple solutions exist for certain range of mass suction and unsteadiness parameters. Ali et al. [28] analyzed the unsteady viscous flow over a shrinking sheet with mass transfer in a rotating fluid. The unsteady boundary layer flow of an electrically conducting fluid on a shrinking surface with a constant transverse magnetic field was investigated by Merkin and Kumaran [29]. Unsteady stagnation point flow and heat transfer over a continuously shrinking sheet was analyzed by Fan et al. [30] and the dual solutions in unsteady stagnationpoint flow over a linearly shrinking sheet was considered by Bhattacharyya [31]. Fluid flow in micro-electro-mechanical systems (MEMS) has become an interesting topic because in micro-scale dimensions, the fluid flow behavior deviates significantly from the traditional no-slip flow. Under the micro-scale dimensions, the fluid motion still obeys the Navier-Stokes equations but under slip velocity boundary conditions. For large scale problems with low density, the fluid can be modeled as a rarefied gas and rarefied gas flows with slip boundary conditions are often countered in the micro-scale devices. The non-adherence of the fluid to a solid boundary, known as velocity slip, is a phenomenon observed in certain circumstances. Partial slips occur for fluids with particulate such as emulsions, suspensions, foams and polymer solutions. Fluids exhibiting slip are important in technological applications such as in the polishing of artificial heart valves and internal cavities. With a slip at the wall boundary, the flow behavior and the shear stress in the fluid are quite different from those in the no-slip flows. The slip flows in dif-

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ferent flow configurations were studied in recent years (see [32–36]). The MHD flow under slip conditions over a permeable shrinking surface was solved analytically by Fang et al. [37] and they reported that the velocity slip at the shrinking surface greatly affects the velocity distribution and drag forces on the wall. Very recently, Bhattacharyya et al. [38] studied the effects of partial slip on the boundary layer stagnationpoint flow and heat transfer towards a shrinking surface. Motivated by the above studies, in this paper we investigate the slip effects on an unsteady boundary layer stagnation-point flow and heat transfer towards a shrinking sheet. The slip condition is taken into account in terms of the shear stress. The results pertaining to the present study indicate that the flow and the temperature field are influenced by the unsteadiness, velocity and thermal slip parameters.

2 Flow analysis Consider an unsteady two-dimensional laminar flow of an incompressible viscous fluid in the vicinity of a stagnation-point over a continuously unsteady shrinking sheet. The sheet shrinking velocity is uw (x, t) = cx/(1 − λt), where c is a constant and λ is a parameter showing the unsteadiness of the problem. The free stream velocity is U (x, t) = ax/(1 − λt), where a (> 0) is the strength of the stagnation flow. Here xaxis runs along the shrinking sheet in a direction opposite to sheet motion and y-axis is perpendicular to it. It should be noted that c > 0 and c < 0 correspond to stretching and shrinking cases respectively. Using the boundary layer approximations, the equations for mass, momentum and the temperature are written in the usual notations as ∂u ∂v + = 0, ∂x ∂y

(1)

∂u ∂u ∂U ∂U ∂ 2u ∂u +u +v = +U +ν 2, ∂t ∂x ∂y ∂t ∂x ∂y

(2)

∂T ∂T ∂ 2T ∂T +u +v =κ 2 , ∂t ∂x ∂y ∂y

(3)

where u and v are the velocity components in the x and y directions respectively, ν is the kinematic viscosity, κ is the thermal diffusivity and T is the temperature of the fluid.

The boundary conditions with partial slip for the velocity and the temperature are given by ∂u , v = 0, ∂y ∂T at y = 0, T = Tw (x, t) + M1 ∂y ax u → U (x, t) = , 1 − λt T → T∞ as y → ∞,

u = uw (x, t) + L1 ν

(4)

(5)

where L1 = L(1 − λt)1/2 is the velocity slip factor and M1 = M(1 − λt)1/2 is the thermal slip factor, L and M are respectively the initial values of velocity and thermal slip factors. Note that both the essential slip factors L1 and M1 change with time and their dimensions are (velocity)−1 and length respectively. The wall temperature Tw (x, t) is given by 2 −3/2 (see [12]), where Tw (x, t) = T∞ + T0 ax 2ν (1 − λt) T∞ is the constant free stream temperature and T0 is a reference temperature such that 0 ≤ T0 ≤ Tw . Note that the expressions for uw (x, t), Tw (x, t) are valid for time t < λ−1 . The governing equations (1)–(3) subject to the boundary conditions (4) and (5) can be expressed in a simpler form by introducing the following transformations:  1/2 aν T − T∞ ψ= xF (η), θ= , (6) 1 − λt Tw − T∞ where the similarity variable η is defined as  1/2 a η= y, ν(1 − λt)

(7)

and ψ is the stream function defined in the usual way as u = ∂ψ/∂y and v = −∂ψ/∂x. In view of these relations, Eqs. (2) and (3) become   η 2 F  + F F  − F  + β 1 − F  − F  + 1 = 0, (8) 2  β θ  + Pr F θ  − 2 Pr F  θ − Pr ηθ  + 3θ = 0, (9) 2 where β (= λ/a) is the unsteadiness parameter and Pr (= ν/κ) is the Prandtl number. The boundary conditions (4) and (5) become F (0) = 0,

F  (0) = α + δF  (0),

θ (0) = 1 + γ θ  (0),

(10)

F  (∞) = 1,

(11)

θ (∞) = 0,

Meccanica Table 1 Comparison of the values of F  (0) (with δ = 0 and β = 0) for shrinking sheet with different values of α (< 0) Present work

α

Wang [20]

Fan et al. [30]

first sol

second sol

first sol

second sol

first sol

second sol

−0.25

1.402242

–

1.40224

–

1.402321

–

−0.50

1.495672

–

1.49567

–

1.496050

–

−0.75

1.489296

–

1.48930

–

1.491277

–

−1.00

1.328819

0.0

1.32882

0.0

1.33019

–

−1.10

1.186680

0.049229

–

−

–

–

−1.15

1.082232

0.116702

1.08223

0.116702

–

–

−1.20

0.932470

0.233648

–

−

–

–

−1.246

0.584374

0.554215

0.55430

–

–

–

where δ = L(aν)1/2 is the dimensionless velocity slip parameter, γ = M(a/ν)1/2 is the dimensionless thermal slip parameter, and α (= c/a) is the velocity ratio parameter.

3 Numerical solution Equations (8) and (9) along with the boundary conditions (10) and (11) form a two point boundary value problem. These equations are solved numerically using shooting technique by converting them into an initial value problem (IVP). To do this, we first transform the non-linear differential equations (8) and (9) to a system of first order differential equations as: y1 = y2 ,

y2 = y3 ,   η  2 y3 = y2 − y1 y3 + β y2 + y3 − 1 − 1, 2

(12)

y4 = y5 ,   β y5 = Pr (ηy5 + 3y4 ) + 2y2 y4 − y1 y5 , 2

(13)

where y1 = F (η), y2 = F  (η), y3 = F  (η), y4 = θ (η) and y5 = θ  (η) and a prime denotes differentiation with respect to the independent variable η. The boundary conditions (10) and (11) become y1 = 0,

y2 = α + δy3 ,

y4 = 1 + γ y5 y2 → 1,

at η = 0, y4 → 0 as η → ∞.

(14) (15)

To solve Eqs. (12) and (13) as an IVP, we require values for y3 (0) (i.e., F  (0)) and y5 (0) (i.e., θ  (0)). But no such values are given at the boundary. So the values of y3 (0) and y5 (0) are guessed in order to initiate

the integration scheme. Now the values of y2 and y4 as η → ∞ are replaced by y2 = 1 and y4 = 0 at a finite value η (= η∞ ) to be determined later. Now we integrate the system of first order equations (12) and (13) by using a fourth-order Runge-Kutta method up to the end-point η = η∞ . The computed values of y2 and y4 at η = η∞ are then compared with y2 = 1 and y4 = 0 at η = η∞ . The absolute differences between these values should be as small as possible (to six decimal accuracy). 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 and y4 attain asymptotically their values 1 and 0 respectively.

4 Results and discussion In order to validate the method and to judge the accuracy of the present analysis, comparison of the skin friction coefficient F  (0) for steady two-dimensional stagnation-point flow towards a shrinking sheet in the absence of slip with available results of Wang [20] and Fan et al. [30] are made (see Table 1) and found in excellent agreement. For the current work, we assume a decelerating shrinking sheet with β ≤ 0. Figures 1 and 2 show the variation of skin friction coefficient F  (0) with α < 0 (shrinking sheet) and α > 0 (stretching sheet) for the unsteadiness parameter β = 0, −0.3, −0.5 in the absence of slip and in the presence of slip at the boundary, respectively. It is observed that the solution for a particular value of β exists upto a critical value α = αc (< 0), beyond which the boundary layer separates from the sheet and the solution based on the

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Fig. 1 Trajectories of skin friction coefficient F  (0) for different values of the unsteadiness parameter β under no-slip boundary condition (i.e., δ = 0)

Fig. 2 Trajectories of skin friction coefficient F  (0) for different values of the unsteadiness parameter β under slip boundary condition δ = 0.25

boundary layer approximations are not possible. Figures 1 and 2 show the values of the critical or turning point αc for different values of β. It is observed that as the unsteadiness parameter β increases, the solution domain expands. Also enhancement of δ causes more increment in the existence range of similarity solution. From a physical point of view this follows from the fact that as slip increases, the vorticity generated by the shrinking velocity is slightly reduced and hence with the same straining velocity of the stagnation flow, that vorticity remains confined within the boundary layer for larger shrinking velocity also and consequently the steady solution is possible for some large values of α. Dual solutions are observed for different values of β. In the dual solutions range, we identify the first solution as the solution with higher values of F  (0) and second solution as the lesser values of F  (0) for a fixed value of α. Further it is observed from these fig-

Fig. 3 Velocity profiles F  (η) for several values of β with α = −1.1 under the slip and no-slip boundary conditions for the first solution branch

Fig. 4 Velocity profiles F  (η) for several values of β with α = −1.1 under the slip and no-slip boundary conditions for the second solution branch

ures that for the shrinking case, the values of F  (0) increases as |α| increases and these values reach the maximum before decreasing to zero. As β increases, F  (0) is also increases. In Figs. 3 and 4, horizontal velocity profiles F  (η) are shown for different values of the unsteadiness parameter β in the absence and also presence of slip at the boundary for the first and second solution branches, respectively. It is seen from Fig. 3 that the velocity F  (η) along the sheet decreases with the decrease of unsteadiness parameter β and this implies an accompanying reduction of the thickness of the momentum boundary layer. Figure 3 also depicts that for a fixed value of β, velocity component increases with the increase of velocity slip parameter. Totally opposite behavior is observed for second solution branch (see Fig. 4).

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Fig. 5 Vertical velocity profiles F (η) for several values of β with α = −1.1 under slip and no-slip boundary conditions for the first solution branch

Fig. 7 Variation of F  (η) with η for several values of α with β = −0.25 under slip and no-slip boundary conditions

Fig. 6 Vertical velocity profiles F (η) for several values of β with α = −1.1 under slip and no-slip boundary conditions for the second solution branch

Fig. 8 Shear stress profiles F  (η) for several values of β with α = −1.1 under the slip and no-slip boundary conditions for the first solution branch

Figures 5 and 6 show the variation of the vertical velocity component F (η) with η for several values of β for the same parameter values as in Figs. 3 and 4 respectively. 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 α (< 0), the velocity profiles F (η) exhibit reverse flow. This is expected because the directions of shrinking and stagnation-point flow are opposite. Figures 5 and 6 show that as β increases, the region of reverse cellular flow decreases for the first solution and increases for the second solution. With the increasing values of velocity slip parameter δ, the region of reverse flow decreases for the first solution and increases for the second solution. Figure 7 displays the variation of F  (η) with η for different values of α (< 0) with a fixed value of β (= −0.25) in the absence and also presence of slip

at the boundary. It is observed that as |α| increases, |F  (η)| decreases and also as velocity slip increases, |F  (η)| increases. Figure 8 exhibits the shear stress for variable values of the slip parameter δ. It is clear that shear stress decreases with increasing slip parameter. Next we discuss the effects of slip parameters on the temperature profiles θ (η). Figure 9 is a graphical representation of temperature profiles θ (η) for different values of the velocity slip parameter δ and thermal slip parameter γ . As the thermal slip parameter γ increases, less heat is transferred from the sheet to the fluid and consequently the temperature profiles θ (η) decreases. To observe the influence of Prandtl number Pr on the temperature profile θ (η), Fig. 10 is displayed. From the definition of Pr, it is obvious that an increase in Pr decreases the thermal conductivity which tends to decrease the temperature and the thermal boundary layer thickness. This observation is

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Fig. 9 Temperature profiles θ(η) for several values of β with α = −0.8 and Pr = 0.71 under slip and no-slip boundary conditions

Fig. 10 Temperature profiles θ(η) for several values of Pr with α = −0.8 and β = −0.1 under slip and no-slip boundary conditions

easily visualized in Fig. 10. The rate of heat transfer −θ  (0) with the velocity slip parameter δ for several values of unsteadiness parameter β is shown in Fig. 11. It is observed that the rate of heat transfer increases with the increase of δ and β. Figure 12 depicts the nature of the rate of heat transfer −θ  (0) with the thermal slip parameter γ for different values of β. The figure indicates that −θ  (0) decreases with the thermal slip parameter γ while increases with the unsteadiness parameter β.

Fig. 11 Variation of −θ  (0) with velocity slip parameter δ for several values of the unsteadiness parameter β

Fig. 12 Variation of −θ  (0) with thermal slip parameter γ for several values of the unsteadiness parameter β

thermal slip conditions at the boundary. The existence and duality of solutions are displayed in (α, F  (0)) parameter space for different values of the unsteadiness parameter β. Due to the increase of velocity slip parameter δ, the range of α for which the similarity solution exists, increases. It is noticed that the reverse flow occurs near the sheet due to shrinking sheet effect. The effects of the slip parameter, velocity ratio parameter and the unsteadiness parameter on the velocity and temperature profiles are computed and analyzed. Quite different flow behavior is observed with multiple solution branches for certain parameter domain.

5 Conclusion We have treated the unsteady boundary layer flow and heat transfer about a stagnation-point towards a shrinking sheet in the presence of both velocity and

Acknowledgements We thank the reviewers for their useful comments and suggestions that led to definite improvement in the paper. The work of one of the authors (Tapas Ray Mahapatra) is supported under SAP (DRS PHASE II) program of UGC, New Delhi, India.

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