Int. J. Appl. Comput. Math DOI 10.1007/s40819-017-0311-y ORIGINAL PAPER
Slip Viscous Flow Over an Exponentially Stretching Porous Sheet with Thermal Convective Boundary Conditions D. Srinivasacharya1 · P. Jagadeeshwar1
© Springer India Pvt. Ltd. 2017
Abstract The aim of the paper is to investigate the boundary layer flow, heat and mass transfer towards the exponentially stretching sheet in a viscous fluid. Similarity transformations are used to convert non-linear governing equations of the flow into non-linear ordinary differential equations. The successive linearization method is used to linearize the resulting ordinary differential equations and then solved by Chebyshev spectral collocation method. A quantitative analysis is made with the previously published results for special cases. The numerical results for the physical parameters on the development of the flow, temperature, concentration, skin friction coefficient, heat and mass transfer are given. It can be concluded from the present analysis that an increase in slip parameter decreases the velocity and increases the temperature and concentration. The heat transfer coefficient is increasing and the velocity, temeperature and concentration are decreasing with the increase in the suction parameter. The temperature and heat transfer coefficient are increasing with the increase in the Biot number. Keywords Exponentially stretching sheet · Suction/injection · Velocity slip and convective boundary conditions · Heat and mass transfer
Introduction The study of flow and heat transfer over an exponentially stretching sheet in a viscous fluid has ever increasing applications in industry as well as in certain technological processes. The examples include the manufacturing process of both metal and polymer sheets, steel rolling, combustion and furnace design, materials processing and temperature measurements. The eminence of such products is determined by both the kinematics of stretching and the simultaneous heating or cooling during the process. For the first time Sakiadis [1,2]
B 1
D. Srinivasacharya
[email protected] Department of Mathematics, National Institute of Technology, Warangal, Telangana State 506004, India
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investigated the Blasius type flow due to a sheet with constant speed from a slit. Later, Crane [3] extended this problem by assuming stretching velocity directly proportional to the distance from the slit. Since then several researchers considered the Newtonian and non-Newtonian fluid flow problems on stretching sheet by considering different effects. The non-adherence of the fluid to a solid boundary is known as velocity slip. The fluid flow with slip flow effects has become the center of attraction of many scientists, engineers and researchers in the areas of technology and industry. A partial slip may occur on a stationary and stretching boundary when the fluid is particulate such as emulsions, suspensions, foams, and polymer solutions. Beavers and Joseph [4] proposed a slip flow boundary condition. Martin and Boyd [5] studied the Blasius boundary layer solution with slip flow conditions. On the other hand, heat transfer under convective boundary conditions plays a vital role in processes such as thermal energy storage, gas turbines, nuclear plants, various propulsion devices for aircraft etc. Yao et al. [6] discussed the flow and heat transfer in a viscous fluid over a stretching/shrinking sheet with convective boundary conditions. They concluded that the convective boundary conditions result in a temperature slip at the wall. Matthews and Hill [7] obtained the solution of the classical laminar boundary layer equations with a linear slip boundary condition. Ishak [8] studied the effect of radiation on magnetohydrodynamic (MHD) boundary layer flow of a viscous fluid over an exponentially stretching sheet. Nadeem et al. [9] analyzed MHD flow of a Casson fluid over an exponentially shrinking sheet. Mukhopadhyay and Gorla [10] studied the effects of partial slip in the presence of thermal radiation towards a porous exponential stretching sheet. Pavithra and Gireesha [11] discussed the heat generation/absorption on dusty fluid over an exponentially stretching surface with viscous dissipation. Srinivasacharya and Ramreddy [12] studied the mixed convection flow along an exponentially stretching sheet in a viscous fluid saturated non-Darcy porous medium with Soret and Dufour effects. Abbas et al. [13] studied the flow and heat transfer of Maxwell fluid over an exponentially continuous moving sheet. Rahman et al. [14] briefed the steady flow and heat transfer characteristics of nanofluids with second order slip using Buongiorno’s model. Hayat et al. [15] studied the heat generation/absorption of three-dimensional stretched flow. Mustafa et al. [16] examined simulations for the Maxwell fluid flow past a convectively heated exponentially stretching sheet with nano-particles. Loganthan and Vimala [17] investigated the effects of MHD, stratification, suction and radiation of a nanofluid over an exponentially stretching sheet. Remus and Marinca [18] considered the steady boundary layer MHD viscous flow and heat transfer over an exponentially porous stretching sheet with radiation effects. Most of the studies reported in the literature are concerned with constant surface velocity and temperature conditions at the boundary. But for many practical applications the surface undergoes convective cooling or heating with velocity slip. Hamad et al. [19] investigated the heat and mass transfer over a moving porous plate with hydrodynamic slip and thermal convective boundary conditions and concentration dependent diffusivity. The aim of the present paper is to analyze the effects of velocity slip on boundary layer flow on an exponentially stretching sheet with suction or injection with convective heat boundary conditions.
Mathematical Formulation Consider a steady, two-dimensional, laminar slip flow of a viscous incompressible fluid towards an exponentially stretching sheet. Choose the x-axis along the surface and the y-axis is perpendicular to the plate so that the fluid occupies the space y > 0. The sheet velocity
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is supposed to vary as an exponential function of the distance x from the slit. The ambient temperature and concentration, respectively, are T∞ and C∞ . It is assumed that the plate is either heated or cooled from left by convection from a fluid of with constant temperature T f with a constant heat transfer coefficient h f . If the left surface of the plate is heated (cooled) by convection from a hot (cold) fluid then T f > T∞ (T f < T∞ ). The species concentration is maintained at a prescribed constant value Cw at the sheet such that Cw > C∞ . The governing equations using the boundary-layer approximations are derived by applying order of magnitude analysis of the terms in the Navier–Stokes equations. Let u and v are the velocity components in x and y-directions, respectively, T is the temperature, C is the concentration, ν is the kinematic viscosity of the fluid, α is the thermal diffusivity and D is the mass diffusivity of the medium. Choosing L as a distance in the flow direction along the sheet and δ as the boundary-layer thickness (δ L), introduce the following normalized quantities x∗ =
x ∗ y u ρv ∗ P ; v∗ = . ; y = ; u∗ = ;P = L δ U∞ μδ P∞
The orders of magnitude of various dimensionless variables are u ∗ ∼ O(1), x ∗ ∼ O(1) and y ∗ ∼ O(δ/L). Using the orders of magnitude of these variables in continuity equations, we get ν ∗ ∼ O(δ ∗ ) 1. Therefore ν ∗ is very small but nonzero otherwise boundary layer concept does not hold. By applying the order of magnitude of the above variables in the momentum equation in the x-direction, we can see that ∂ 2 u/∂ x 2 is much smaller than ∂ 2 u/∂ y 2 , and it can be neglected, i.e., ∂ 2 u/∂ x 2 ∂ 2 u/∂ y 2 . Similar analysis can be extended to momentum equation in y-direction, energy and species equations. Therefore, the boundary layer form of continuity, momentum, energy and concentration equation are (see Refs. [20–24]) ∂u ∂v + ∂x ∂y ∂u ∂u +v u ∂x ∂y ∂T ∂T u +v ∂x ∂y ∂C ∂C +v u ∂x ∂y
=0 ∂ 2u ∂ y2 ∂2T =α 2 ∂y ∂ 2C =D 2 ∂y
=ν
(1) (2) (3) (4)
The boundary conditions are ⎫ ⎪ u = U + N ν ∂u ∂ y , v = −V (x), C = C w , ⎬ ∂T −k ∂ y = h f (T f − T ) at y = 0 ⎪ u → 0, T → T∞ , C → C∞ as y → ∞ ⎭
(5)
x
where k is the thermal conductivity, U = U0 e L is the stretching velocity and U0 is the x reference velocity, V = V0 e 2L is the special velocity at the wall and V0 is the initial strength −x slip factor which changes with x and N0 is the initial of suction, N = N0 e 2L is the velocity x
0 2L value of the velocity slip factor, h f = h U is the heat transfer coefficient and V (x) > 0 2L e represent that case of fluid suction and V (x) < 0 represents the injection. N = 0 corresponds to no slip case.
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Introducing the following Similarity variables ⎫ √ x x U0 2L ⎪ 2L f, ⎪ η = y 2ν ⎬ L e , ψ = 2ν LU0 e x x νU 0 u = U0 e L f , v = − 2L e 2L ( f + η f ) ⎪ ⎪ ⎭ T = T∞ + (T f − T∞ )θ (η), C = C∞ + (Cw − C∞ )φ(η)
(6)
into Eqs. (1)–(4), we obtain f
+ f f − 2 f 2 = 0 1 θ + fθ = 0 Pr 1 φ + fφ = 0 Sc
(7) (8) (9)
The corresponding boundary conditions reduce to
f (0) = S, f (0) = 1 + λ f (0), θ (0) = −Bi(1 − θ (0)), φ(0) = 1 f (∞) → 0, θ (∞) → 0, φ(∞) → 0.
(10)
where the prime denotes differentiation with respect to η, Pr = αν is the Prandtl number, 2L Sc = Dν is the Schmidt number, S = V0 νU is the suction/injection parameter according 0 √ νU0 as S > 0 or S < 0 respectively, λ = N0 2L is the velocity slip parameter and Bi = hk ν is the Biot number. Results of practical interest are the wall shear stress, heat and mass transfer rates, respectively, acting on the stretching surface and are given by ∂u ∂T ∂C τω = μ , qw = −k and qm = −D (11) ∂ y y=0 ∂ y y=0 ∂ y y=0 The non-dimensional skin friction C f =
2τω , the local Nusselt ρU 2
number N u x =
xqw k(T f −T∞ ) ,
m and local Sherwood number Sh x = k(Cwxq−C , are given by ∞)
1 −1 −1 L 2L 2L 2 2 f (0) = Rex C f , −θ (0) = Rex N u x and − φ (0) = Rex2 Sh x 2x x x
where Rex =
xU (x) ν
(12)
is the local Reynolds number.
Method of Solution The system of differential Eqs. (7)–(9) along with the boundary conditions (10) are solved using the successive linearization method (SLM: [25–27]). By using this method, the nonlinear governing Eqs. (7)–(9)are reduced to a system of linear differential equations. The Chebyshev pseudo spectral method is then applied to solve these linearized ordinary differential equations. In SLM, the functions f (η), θ (η) and φ(η) are assumed to be expressed as f (η) = f i (η) +
i−1 n=0
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f n (η), θ (η) = θi (η) +
i−1 n=0
θn (η), φ(η) = φi (η) +
i−1 n=0
φn (η)
(13)
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where f i (η), θi (η) and φi (η) (i = 1, 2, 3, . . .) are unknown functions and f n (η), θn (η) and φn (η)(n ≥ 1) are approximations which are obtained by recursively solving the linear part of the system of equations that results from substituting Eq. (13) in the governing Eqs. (7) to (9). We choose the initial approximation f 0 (η), θ0 (η) and φ0 (η) such that they satisfy boundary conditions (10). The subsequent solutions for f i (η), θi (η) and φi (η)(i ≥ 1) are obtained by recursively solving the following linearized system of ordinary differential equations f i + a1,i−1 f i + a2,i−1 f i + a3,i−1 f i = r1,i−1
(14)
b1,i−1 f i + θi + b2,i−1 θi = r2,i−1
(15)
c1,i−1 f i + φi + c2,i−1 φi = r3,i−1
(16)
where the coefficients ⎫ i−1 ⎪ θn , b2,i−1 = Pr fn ⎪ ⎪ ⎪ ⎪ n=0 n=0 n=0 n=0 n=0 ⎪
2 ⎪ ⎪ ⎬ i−1 i−1 i−1 i−1 i−1 i−1 c1,i−1 = Sc φn , c2,i−1 = Sc f n , r1,i−1 = − fn − fn fn + 2 fn ⎪ ⎪ n=0 n=0 n=0 n=0 n=0 n=0 ⎪ ⎪ ⎪ i−1 i−1 i−1 i−1 i−1 i−1 ⎪ ⎪ ⎪ r2,i−1 = − θn − Pr fn θn , r3,i−1 = − φn − Sc fn φn ⎭ a1,i−1 =
i−1
f n , a2,i−1 = −4
n=0
i−1
n=0
f n , a3,i−1 =
i−1
n=0
f n , b1,i−1 = Pr
n=0
i−1
n=0
n=0
(17) The boundary associated conditions are
f i (0) = λ f i (0) − f i (0) = f i (∞) = θi (0) − Biθi (0) = θi (∞) = φi (0) = φi (∞) = 0. (18) Hence, the approximate solutions for f (η), θ (η) and φ(η) are obtained as f (η) ≈
M
f n (η), θ (η) ≈
n=0
M n=0
θn (η), φ(η) ≈
M
φn (η)
(19)
n=0
where M is the order of SLM approximation. The linearized Eqs. (14) to (16) are solved using the Chebyshev spectral collocation method [28]. The functions that are to be determined are approximated by the Chebyshev interpolating polynomials in such a manner that they are collocated at the Gauss–Lobatto points defined as πk ξk = cos , k = 0, 1, 2, . . . , N (20) N where N +1 is the number of collocation points used. To apply this method the domain under consideration [0, ∞) is transformed to [−1, 1] by the transformation ξ = 2η L − 1, −1 ≤ ξ ≤ 1 and the problem is solved for [0, L] instead of [0, ∞). Here L is a parameter used to recover the conditions at infinity. The functions f i , θi and φi are expressed at the collocation points by f i (ξ ) ≈
N k=0
f i (ξk )Tk (ξk ), θi (ξ ) ≈
N k=0
θi (ξk )Tk (ξk ), φi (ξ ) ≈
N
φi (ξk )Tk (ξk )
(21)
k=0
where Tk is the k th Chebyshev polynomial defined as Tk (ξ ) = cos(k cos−1 (ξ )).
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The derivatives of the unknown functions at the collocation points are represented as d a θi d a φi d a fi a a = D f (ξ ), = D θ (ξ ), = Dakj φi (ξk ) i k i k kj kj dηa dηa dηa N
N
N
k=0
k=0
k=0
(22)
where a is the order of differentiation and D = L2 D and D being the Chebyshev spectral differentiation matrix. Substituting Eqs. (21)–(22) into the Eqs. (14)–(18) leads to the following system of algebraic equations Ai−1 Xi = Ri−1 , (23) The boundary conditions reduce to f i (ξ N ) = N
N
D0k f i (ξk ) =
k=0
N k=0
(λD2N k − D N k ) f i (ξk ) = 0
⎫ ⎪ ⎪ ⎬
⎪ ⎭ D N k θi (ξk ) − Biθi (ξ N ) = θi (ξ0 ) = φi (ξ0 ) = φi (ξ N ) = 0 ⎪
(24)
k=0
In Eq. (23), Ai−1 is a (3N + 3) × (3N + 3) square matrix and Xi and Ri−1 are (3N + 3) × 1 column vectors ⎤ ⎤ ⎡ ⎡ ⎤ ⎡ A11 A12 A13 Fi r1,i−1 Ai−1 = ⎣ A21 A22 A23 ⎦ , Xi = ⎣ i ⎦ , Ri−1 = ⎣ r2,i−1 ⎦ (25) A31 A32 A33 i r3,i−1 where
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ T ⎪ i = [θi (ξ0 ), θi (ξ1 ), θi (ξ2 ), . . . , θi (ξ N −1 ), θi (ξ N )] , ⎪ ⎪ ⎪ ⎪ ⎪ T ⎪ i = [φi (ξ0 ), φi (ξ1 ), φi (ξ2 ), . . . , φi (ξ N −1 ), φi (ξ N )] , ⎪ ⎪ ⎪ ⎪ ⎪ r1,i−1 = [r1,i−1 (ξ0 ), r1,i−1 (ξ1 ), r1,i−1 (ξ2 ), . . . , r1,i−1 (ξ N −1 ), r1,i−1 (ξ N )], ⎪ ⎪ ⎬ Fi = [ f i (ξ0 ), f i (ξ1 ), f i (ξ2 ), . . . , f i (ξ N −1 ), f i (ξ N )]T ,
r2,i−1 = [r2,i−1 (ξ0 ), r2,i−1 (ξ1 ), r2,i−1 (ξ2 ), . . . , r2,i−1 (ξ N −1 ), r2,i−1 (ξ N )], ⎪ ⎪ ⎪ ⎪ r3,i−1 = [r3,i−1 (ξ0 ), r3,i−1 (ξ1 ), r3,i−1 (ξ2 ), . . . , r3,i−1 (ξ N −1 ), r3,i−1 (ξ N )], ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 2 ⎪ A11 = D + a1,i−1 D + a2,i−1 D + a3,i−1 , A12 = 0, A13 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ A21 = b1,i−1 , A22 = D + b2,i−1 D, A23 = 0, A31 = c1,i−1 , A32 = 0 ⎪ ⎭ and A33 = D2 + c2,i−1 D.
(26)
Here ak,i−1 (k = 1, 2, 3), bk,i−1 and ck,i−1 (k = 1, 2) are diagonal matrices of size (N + 1) × (N + 1), rk,i−1 (k = 1, 2, 3) are column matrices of size (N + 1)×1 and 0 represents (N + 1) × (N + 1) a null matrix. After modifying the matrix system (23) to incorporate boundary conditions (24), the solution is obtained as −1 Xi = Ai−1 Ri−1 (27)
Results and Discussions In order to validate accuracy and reliability of the method used, the results for particular values of S, λ and Bi large is compared with the results obtained by Magyari and Keller [29] and found to be in good agreement, as shown in Table 1.
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Int. J. Appl. Comput. Math Table 1 Comparative analysis of the Nusselt number calculated by the present method for S = 0, λ = 0 and Bi → ∞
Nusselt number −θ (0) Pr
Magyari and Keller [29]
Present
0.5
0.330493
0.33053741121
1
0.549643
0.54964316819
3
1.122188
1.12208592275
5
1.521243
1.52123757437
8
1.991847
1.99183597409
10
2.257429
2.25742181796
Fig. 1 Effect of slip parameter on velocity, temperature and concentration profiles for S = 0.5 and Bi = 0.5
Convergence of the method to the numerical results is achieved at the fourth order of SLM approximation. All the numerical calculations are done by taking Pr = 0.71, Sc = 0.22, N = 100, L = 20, S = 0.5, Bi = 0.5 and λ = 1.0 unless otherwise mentioned.
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Fig. 2 Effect of suction parameter on velocity, temperature and concentration profiles for λ = 1.0 and Bi = 0.5
The effect of slip parameter on the velocity, temperature and concentration profiles is shown in Figs. 1a–c. From Fig. 1a, it is observed that velocity is decreasing with increases in the slip parameter. When the velocity slip happens, the flow velocity neighboring the sheet is not equal to the stretching velocity of the sheet. With the increase in λ, slip velocity increases and consequently the fluid velocity reduce. This is because under the slip condition, the pulling of the stretching sheet can be only partly transmitted to the fluid. The effect of slip parameter on the temperature and concentration profiles is depicted through the Fig. 1b, c. It is seen that both the temperature and concentrations are increasing with an increasing in slip parameter. Figures 2a–c depict the velocity, temperature and concentration profiles for various values of suction/injection parameter. It is observed from Fig. 2a that velocity is decreasing with an increase in the value of the suction parameter, whereas fluid velocity is increasing with an increase in the injection parameter. From Fig. 2b, it is observed that there is a decrease in temperature in the thermal boundary layer resulting in thinning of thermal boundary layer thickness in the case of suction and the reverse trend is observed for injection. It is noticed from
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Fig. 2c that the concentration is decreasing with increase in the suction parameter, whereas the trend is reversed in the case of injection. It is known that applying the wall suction have the propensity to lessen both the momentum and thermal boundary layer thickness. This reasons decreases in both the velocity and temperature profiles. However, the exact contradictory behavior is produced by the imposition of the wall injection. The influence of the Biot number on the temperature profile is plotted in Fig. 3. It is witnessed that temperature within the boundary layer increases with the increase of the Biot number. As Bi → ∞, the boundary condition (10) reduce to θ (0) → 1 which is the case of constant wall temperature. The stronger convection leads to the higher surface temperatures, which appreciably increases the temperature and the thermal boundary layer thickness. As the effect of the Biot number on the non-dimensional velocity and concentration is almost negligible, the corresponding graphs are not presented. Figure 4 exhibits the nature of skin-friction coefficient ( f (0)) with suction/injection parameter S for different values of velocity slip (λ = 0.0, 0.5, 1.0, 2.0). It is found that the magnitude of the skin-friction coefficient increases with S whereas it decreases with increasing slip parameter. Physically, when slip occurs; the slipping fluid shows a decrease in the surface skin-friction between the fluid and the stretching sheet because not all the pulling force of the stretching sheet can be transmitted to the fluid. The behavior of heat transfer coefficient (Nusselt number θ (0)) with suction/injection parameter S for different values of velocity slip is presented in Fig. 5. It is clear that the magnitude of the heat transfer increases with increasing values of suction /injection parameter. It decreases with increasing values of velocity slip. The effect of the Biot number on the heat transfer coefficient is shown in Fig. 6. It is observed that the heat transfer coefficient is increasing with the increase in the value of Biot number. An increase in the convection at the sheet results in an increase in the thermal penetration depth. Hence, the rate of heat transfer at the sheet therefore increases with an increase in the values of Bi
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Fig. 5 Variation of Nusselt number with suction/injection parameter for different values of λ
The influence of velocity slip on the mass transfer coefficient is represented in Fig. 7. It is depicted from this figure that the mass transfer coefficient is increasing with a rise in the value of suction/injection parameter. The mass transfer coefficient is decreasing with the increase in the value of slip parameter. An increase in the slip parameter has the tendency to reduce the friction forces which reduces the mass transfer coefficient.
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Fig. 7 Variation of Sherwood number with suction/injection parameter for different values of λ
Conclusion The present study gives numerical solutions for steady boundary layer flow and heat and mass transfer over an exponentially stretching sheet in presence of slip at the boundary and convective boundary conditions. The governing equations are transformed to a set of nonlinear ODE’s by using similarity transformations and then linearized by using SLM technique
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and are solved by using the Chebyshev spectral collocation method. The main findings are listed as follows. • The velocity is decreasing as the slip and suction parameters increase. • The temperature is increasing with the increase in the value of slip parameter and Biot number. It is decreasing with an increase in the value of suction parameter. • An increase in slip parameter increases the concentration, whereas raise in the suction parameter decreases the concentration. • The skin-friction coefficient increases with an increase in suction parameter and decreases with slip parameter. • The heat transfer coefficient increases with an increase in the value of suction parameter and the Biot number, whereas it decreases with an increase in the value of slip parameter. • As the suction parameter increases the Sherwood number increase and it decreases with an increase in the velocity slip.
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