Meccanica (2013) 48:1451–1464 DOI 10.1007/s11012-012-9677-4
MHD flow and heat transfer over a stretching surface with variable thermal conductivity and partial slip Mahantesh M. Nandeppanavar · K. Vajravelu · M. Subhas Abel · M.N. Siddalingappa
Received: 25 March 2012 / Accepted: 26 November 2012 / Published online: 4 January 2013 © Springer Science+Business Media Dordrecht 2012
Abstract In this paper we analyze the flow and heat transfer of an MHD fluid over an impermeable stretching surface with variable thermal conductivity and non-uniform heat source/sink in the presence of partial slip. The governing partial differential equations of the problem are reduced to nonlinear ordinary differential equations by using a similarity transformation. The temperature boundary conditions are assumed to be linear functions of the distance from the origin. Analytical solutions of the energy equations for Prescribed Surface Temperature (PST) and Prescribed Heat Flux (PHF) cases are obtained in terms of a hypergeometric function, without applying the boundary-layer approximation. The effects of the governing parameters on the flow and heat transfer fields are presented through tables and graphs, and they are discussed. Furthermore, the obtained numerical results for the skin friction, wall-temperature gradient and wall temperature
M.M. Nandeppanavar () · M.N. Siddalingappa Department of PG and UG studies in Mathematics, Government College, Gulbarga 585105, Karnataka, India e-mail:
[email protected] K. Vajravelu Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA M. Subhas Abel · M.N. Siddalingappa Department of Mathematics, Gulbarga University, Gulbarga 585106, Karnataka, India
are analyzed and compared with the available results in the literature for special cases. Keywords Partial slip flow · MHD fluid · Variable thermal conductivity · Non-uniform heat source/sink
1 Introduction The flow over a stretching sheet has important industrial applications, for example in a metallurgical processor such as used for drawing of continuous filaments through quiescent fluids, annealing and tinning of copper wires, extrusion process, glass blowing, hot rolling, manufacturing of plastic and rubber sheets, crystal growing, continuous cooling and fiber spinning. During the manufacture of these sheets, the melt issues from a slit and is subsequently stretched to achieve the desired thickness. The final product of the desired characteristics strictly depends on the stretching rate, the rate of cooling, and the process of stretching. In view of these applications the problem under consideration is of fundamental importance. Also, it has applications to other processes like continuous stretching, rolling, and manufacturing of plastic films and artificial fibers; heat treated materials traveling on conveyer belts; paper production and so on. As we know, Sakiadis [1, 2] initiated the study of the boundary-layer flow over a continuous solid surface moving with constant speed. The boundary-layer
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problem considered by Sakiadis differs from the classical boundary-layer problem of Blasius [3], mainly due to the entrainment of the ambient liquid (i.e., the flow with uniform free stream of constant velocity). Here the surface is assumed to be inextensible (uw = 0) whereas most of the physical situations are concerned with extensible surfaces (uw = cx) moving in a cooling liquid. Crane [4], for the first time, considered the boundary-layer behavior over an extensible surface, where he assumed the velocity of the surface to vary linearly with the distance from the slit. Carragher and Crane [5] analyzed the heat transfer due to a continuous stretching sheet. The pioneering work of Crane was subsequently extended by many authors to boundary-layer flows with various velocity and thermal boundary conditions; see, for example, Gupta and Gupta [6], Grubka and Bobba [7], Chen and Char [8], Chiam [9] and references therein. In engineering applications, homogeneous or heterogeneous reactions often lead to a significant heat release accompanied by non-isothermal conditions that require the introduction of a heat source/sink term in the energy equation. Cortell [10–12] studied the flow and heat transfer characteristics with linearly and nonlinearly stretching sheets for both Newtonian and non-Newtonian fluids with internal heat generation/absorption and suction/injection. Sarpakaya [13] was the first to study the MHD effects on the flow of a non-Newtonian fluid. Pal and Mondal [14, 15] considered the MHD fluid for their study. The MHD fluids have been considered by many researchers [16–21, 41]. Abel et al. [22] studied the effect of a variable viscosity on the heat transfer of viscoelastic fluid due to stretching sheet. Vajravelu and Rollins [23], and Vajravelu and Nayfeh [24] have studied the effect of a uniform heat source/sink on the heat transfer from a stretching sheet into a cooling liquid. Vajravelu and Cannon [25] studied the fluid flow due to a nonlinear stretching sheet. Bhattacharya and Vajravelu [26] studied the stagnation-point flow over an exponential stretching sheet. Abo-Eldahab and ElAziz [27] investigated heat transfer considering a nonuniform heat source/sink. Recent work of Nandeppanavar et al. [28], Abel et al. [29], and Bataller [30] analyzed the effect of a non-uniform heat source/sink in the case of a viscoelastic liquid flow due to a stretching sheet. The no-slip boundary condition is known as the central tenet of the Navier–Stokes theory. But there
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are situations wherein such a condition is not appropriate. Especially the no-slip condition is inadequate for most non-Newtonian liquids, as some polymer melts often exhibit microscopic wall slip and that in general is governed by a nonlinear and monotone relation between the slip velocity and traction. The liquids exhibiting boundary slip find applications in a technology such as polishing of artificial heart valves and internal cavities. Navier [31] suggested a slip boundary condition in terms of shear stress. Of late, the work of Navier was extended by many authors (see Refs. [32– 40, 43, 44]). Mostafa et al. [45] studied the effect of non-uniform heat source on heat transfer of nonNewtonian power-law fluid over a nonlinear stretching sheet. John and Kumaran [46] studied the heat transfer due to a heat source in MHD unsteady stretching sheet flow. In the above work, except that of Andersson [40], the authors restricted themselves to the boundarylayer approximation, which is valid for small viscosity or high Reynolds number. Andersson [40] found a closed-form solution of the full Navier–Stokes equations for a magnetohydrodynamic flow over a stretching sheet using a similarity transformation, without applying the boundary-layer approximation. Due to ∂2u the vanishing of the transformed term ∂x 2 in the full form of the Navier–Stokes equation in the longitudinal direction, his solutions turned out to be the same as those of Pavlov [41], within the frame work of high Reynolds number boundary-layer theory. Besides, the transverse momentum equation can be employed to find the pressure distribution, which increases along the direction normal to the flow. Since the no boundary-layer approximation is applied in Andersson [40], his solution is exact and valid for all Reynolds numbers. Motivated by the work of Andersson [40] and others, we have attempted here to present an analytical study of the MHD flow and heat transfer due to a stretching sheet in the presence of variable thermal conductivity and non-uniform heat source/sink without applying the boundary-layer approximation. The temperature boundary conditions are assumed to be linear functions of the distance from the origin. Heat transfer solutions are obtained for two different types of boundary heating, namely, the prescribed surface temperature (PST) case and the prescribed heat flux (PHF) case.
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Fig. 1 Schematic of a stretching sheet problem, with partial slip condition
2 Mathematical formulation and solution of considered flow
cosity, ρ is the density, and σ is the electric conductivity. The appropriate boundary conditions are ∂u , ∂y p = pw at y = 0, u → 0 as y → ∞. u(x, y) − cx = L
Consider a steady, laminar, two-dimensional flow of an incompressible viscous liquid past a flat impermeable sheet coinciding with the plane y = 0; and the flow being confined to upper half of the plane (y > 0). The flow is generated, due to stretching of the sheet, caused by the simultaneous application of two equal and opposite forces along the x-axis (see Fig. 1). The sheet is stretched, keeping the origin fixed, with a velocity varying linearly with the distance from the slit. Let B0 be the externally applied magnetic field. Since the magnetic Reynolds number is assumed to be small, the induced magnetic field is negligible. We take the x-axis along the surface and the y-axis normal to it. The equations governing the motion of the electrically conducting liquid (for example, Mazola corn oil is one of such electrically conducting liquids) due to a stretching sheet are given by ∂u ∂v + = 0, ∂x ∂y u
(1)
2 ∂u ∂u 1 ∂p ∂ u ∂ 2u σ B0 +v =− +υ u, − + 2 2 ∂x ∂y ρ ∂x ρ ∂x ∂y
u
∂v ∂v 1 ∂p ∂ 2v ∂ 2v , + +v =− +υ ∂x ∂y ρ ∂y ∂x 2 ∂y 2
(2) (3)
where u and v are the velocity components along x and y directions, respectively, υ is the kinematic vis-
v = 0,
Using the new variables c (u, v) (X, Y ) = (x, y), (U, V ) = √ , υ cυ p P= , ρcυ
(4)
(5)
Eqs. (1)–(3) can be written in non-dimensional form as ∂V ∂U + = 0, (6) ∂X ∂Y 2 ∂U ∂U ∂P ∂ U ∂ 2U − Mn U, U +V =− + + ∂X ∂Y ∂X ∂X 2 ∂Y 2 (7) 2 ∂V ∂V ∂P ∂ V ∂ 2V U , (8) +V =− + + ∂X ∂Y ∂Y ∂X 2 ∂Y 2 σ B2
where Mn = ρc0 is the magnetic parameter. The boundary conditions (4) reduce to ∂U , V = 0, ∂Y (9) P = Pw at Y = 0, U → 0 as Y → ∞, where γ = L υc is the slip parameter (L denotes the U −X=γ
pw is the dimensionless presslip length) and Pw = ρcυ sure distribution at the wall (which is very negligible
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and hence ignored in the further discussion). We now introduce the stream function ψ(X, Y ) satisfying the continuity equation (6) as ∂ψ ∂ψ , V =− . (10) ∂Y ∂X Thus we have the following basic equations along with the boundary conditions: U=
∂(ψ, ∂ψ ∂ 3ψ ∂P ∂ψ ∂ 3ψ ∂Y ) − − Mn = 0, + + ∂(X, Y ) ∂X ∂Y ∂Y 3 ∂X 2 ∂Y (11) ∂ 3ψ ∂X 3
+
∂ 3ψ ∂X∂Y 2
+
∂ψ ∂X )
∂(ψ, ∂P + = 0, ∂(X, Y ) ∂Y
∂ 2ψ ∂ψ −X=γ , ∂Y ∂Y 2 P = Pw at Y = 0,
(12)
∂ψ = 0, ∂X (13)
∂ψ → 0 as Y → ∞. ∂Y In order to find the analytical solution to the problem, we introduce the transformation ψ(X, Y ) = Xf (η), and η = Y.
1 P = Pw − h(η) 2
(14)
Using the above transformation the basic equations (11) and (12) can be written as f + ff − f 2 − Mn f = 0, (15) 1 (16) f + ff − h = 0. 2 The corresponding boundary conditions in (13) take the form h(0) = 0, f (0) = 0, f (0) = 1 + γf (0), f (∞) → 0.
Note that the solution (18) satisfies the boundary conditions (first and the third) of (17). It also has to satisfy the remaining boundary condition in (17). Using (18) and the second boundary condition in (17), we get γ As 2 + As − 1 = 0.
(20)
Substituting (19) in (20) we get the following cubic equation for s: γ s 3 + s 2 − γ Mn s − (Mn + 1) = 0.
(21)
Hence, the solution for the momentum equation becomes 2 s − Mn f (η) = 1 − e−sη . (22) s Using (22) in (16) and applying one of boundary conditions in (17), the pressure distribution function is obtained in the form 2 s − Mn h(η) = 2Mn e−sη + s 2 − Mn e−2sη s2
− s 2 + Mn . (23) The parameter s appearing in the solutions (22) and (23) is the minimum positive root of the cubic equation (21). The minimum positive root ensures the realistic behavior of the solution and hence the same is considered for further analysis. Now, the dimensionless velocity components are given by U = Xf (η)
and V = f (η).
The local skin-friction coefficient is given by 1 τw Cf = 1 = −2 √ f (0), 2 Re ρu w 2
(24)
(25)
where τw = ( ∂u ∂y )y=0 is the wall shearing stress and Rex is the local Reynolds number.
(17) The boundary conditions in (17) admit a solution of the form f (η) = A 1 − e−sη , (18) where A and s are constants to be determined. For the above function to be a valid solution it must satisfy Eq. (15). Substituting (18) in (15) and solving for A, we get A=
s 2 − Mn . s
(19)
3 Solution of heat transport equation The energy equation for the present problem is given by ∂T ∂T u +v ∂x ∂y kuw (x) 1 ∂ ∂T 1 = k + ρCP ∂y ∂y ρCP xυ
∗ −sη ∗ + B (T − T∞ ) , (26) × A (Tw − T∞ )e
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where T is the temperature of the liquid, k is the thermal conductivity, Cp is the specific heat at constant pressure, and the second term on the RHS of (26) represents the non-uniform heat source/sink term (see [27]). Further, A∗ and B ∗ are the parameters of the space- and temperature-dependent internal heat generation/absorption. The case A∗ > 0 and B ∗ > 0 corresponds to internal heat generation, while the case A∗ < 0 and B ∗ < 0 corresponds to internal heat absorption. The thermal conductivity is assumed to vary linearly with temperature in the form T − T∞ , (27) k = k∞ 1 + ε T w − T∞ where ε is a small parameter. The solution of Eq. (26) is obtained for two different types of boundary condition, namely (i) the Prescribed Surface Temperature (PST) case and (ii) the Prescribed Heat Flux (PHF) case. We first discuss first the case of PST.
The boundary surface is prescribed by a power-law temperature of general degree (that is, λ = 0, 1, −1, 2, −2, . . .). Here we consider the boundary conditions for PST case as ⎫ λ ⎪ x ⎬ at y = 0, T = Tw = T∞ + A0 (28) L ⎪ as y → ∞, ⎭ T → T∞ where A0 is a positive constant. We define the following non-dimensional temperature variable: T − T∞ Tw − T∞
∂ψ ∂Θ ∂ψ ∂Θ ∂ψ Θ − + ∂Y ∂X ∂X ∂Y ∂Y X 2 ∂ Θ ∂ 2Θ 1 (1 + εΘ) = + Pr ∂X 2 ∂Y 2 2 2 2 + 3εΘ ∂Θ ∂Θ ∂Θ + +ε + ∂X ∂Y X ∂X ∂ψ + B ∗Θ . (30) + (1 + εΘ) A∗ ∂Y The boundary conditions can be written as Θ = 1 at Y = 0, Θ →0
as Y → ∞.
(29)
where
x θ (η) T − T∞ = A0 L and
x . Tw − T∞ = A0 L Here L is the characteristic length, Tw is the wall temperature, T∞ is the temperature of the liquid for away from the sheet, and λ is the variable wall-temperature parameter (in this problem it is taken as linear, i.e., λ = 1).
(31)
Using the similarity transformation (14) along with Θ(X, Y ) = θ (η), Eq. (30) can be written as (1 + εθ )θ + Pr f θ − f θ + εθ 2 + (1 + εθ ) A∗ f + B ∗ θ = 0. (32) The boundary conditions in (31) take the form θ (η) = 1 at η = 0,
3.1 Prescribed Surface Temperature (PST)
Θ=
Using the relations in (10), (27), and (28), Eq. (26) can be written as
θ (η) → 0 as η → ∞.
(33)
Defining the new independent variable −A Pr −sη (34) e , s using (34), Eq. (32) can be written as dθ APr d 2θ −ξ +θ (1 + εθ )ξ 2 + 1 + εθ − s dξ dξ 2 dθ −A∗ B ∗ θ + εξ + 2 = 0. + (1 + εθ ) dξ Pr s ξ (35)
ξ=
The boundary conditions in (33) take the form ⎫ A Pr ⎬ , θ (ξ ) = 1 at ξ = − s ⎭ θ (ξ ) → 0 as ξ → 0.
(36)
We employ the perturbation method to solve Eq. (35) with boundary conditions (36). That is, let us assume θ (ξ ) = θ0 (ξ ) + εθ1 (ξ ) + ε 2 θ2 (ξ ) + · · · .
(37)
Using (37) in (35) and equating the coefficients of like powers of ε on both sides, we get the following sequence of boundary value problems:
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O ε0 : ξ
d 2θ
0 dξ 2
+ 1−
A∗ = , Pr θ0 (ξ ) = 1
B∗
+ c12 ξ 2r c0 + c1 ξ + c2 ξ 2 + · · · ,
A Pr dθ0 −ξ + 1 + 2 θ0 s dξ s ξ
where (38)
⎫ A Pr ⎬ , at ξ = − s ⎭ as ξ → 0,
(39)
θ0 (ξ ) → 0 O ε1 : d 2 θ1 dθ1 B∗ A Pr ξ − ξ + 1 + θ1 + 1 − s dξ dξ 2 s2ξ A∗ B∗ 2 dθ0 2 θ0 − 2 θ0 − ξ = Pr dξ s ξ dθ0 d 2 θ0 − ξ θ0 2 , dξ dξ ⎫ A Pr ⎬ , θ1 (ξ ) = 0 at ξ = − s ⎭ θ1 (ξ ) → 0 as ξ → 0 − θ0
where a+b , r= 2 A Pr a= , s
1−a+
B∗ s2
k=0
a0 = a2 = (40)
(41)
b0 = b1 = b2 = ,c0 =
(42)
ξ,
⎪ ⎪ b= ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ B∗ ⎪ ) ⎪ ⎪ s2 ⎭ . c1 = r (−a) M[r − 1, b + 1, −a]
(43)
a2
Here the value of b should be positive for realistic so∗ lution and there is no solution if a 2 > 4B (as b is s2 imaginary). The solution of (38) subject to the boundary conditions (39) in terms of Kummer’s function (see Abramowitz and Stegun [42]) is obtained as θ1 (ξ ) = c2 ξ r F [r − 1, b + 1, ξ ] + ξ 2 a0 + a1 ξ + a2 ξ 2 + · · · + c1 ξ r+1 b0 + b1 ξ + b2 ξ 2 + · · ·
k=0
ck (−a)k , a0
4 − 2a +
B∗ s2
(9 − 3a +
a0
a1 =
,
2 a0 B∗ )(16 − 4a s2
9 − 3a + +
b0 (r
+ 1)2
(r
+ 2)2
− a(r + 1) + b1 + r b0 − a(r + 2) +
b2 + (r + 1)b1 (r + 3)2 − a(r + 3) +
B∗ ) s2
B∗ s2
,
B∗ s2
,
B∗ s2
,
B∗ s2
,
and so on
,
and so on
c0 (2r)2
c1 = c2 =
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
4B ∗ − 2 , s ∗ 1 + ( APra /1 − a +
∞ k=0
θ0 (ξ ) = c1 ξ r F [r − 1, b + 1, ξ ] +
−[a 2 s1 + c1 (−a)r+1 s2 + c12 (−a)2r s3 ] , (−a)r F [r − 1, b + 1, −a] ∞ ∞ ak (−a)k , s2 = bk (−a)k , s1 = c2 =
s3 =
and so on. The solution of (36) subject to the boundary conditions (37) in terms of Kummer’s function is obtained as A∗ Pr
(44)
− a(2r) +
B∗ s2
c1 + (2r − 1) c0 (2r + 1)2 − a(2r + 1) + c2 + (2r) c1 (2r + 2)2 − a(2r + 2) +
B∗ s2
,
B∗ s2
,
and so on
The local heat flux can be expressed as c ∂T Bxθ (−a) qw = −k = −k ∂y y=0 υ
(45)
where θ (−a) = θ0 (−a) + θ1 (−a), and the Nusselt number in this PST case is given by qw = −θ (−a) . (46) Nu = T w − T∞ 3.2 Prescribed Heat Flux (PHF) The boundary conditions in this case are given as ⎫ x λ ∂T −k = qw = D( ) at y = 0, ⎬ ∂y L (47) ⎭ T → T∞ as y → ∞ where D is a positive constant.
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Let T − T∞ g= Tw − T∞
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(48)
where
g(ξ ) = g0 (ξ ) + εg1 (ξ ) + ε 2 g2 (ξ ) + · · · .
λ x g(η) T − T∞ = D L and Tw − T∞ = D
(55)
Using (55) in (53) and equating the coefficients of like powers of ε on both sides we get the following sequence of boundary value problems:
λ x L
O ε0 :
(as in the case of PHT, also here λ = 1 is considered). Using Eqs. (10), (25), (47), and (48), Eq. (26) and the conditions in (47) can be written as ∂ψ ∂g ∂ψ g ∂ψ ∂g − + ∂Y ∂X ∂X ∂Y ∂Y X 2 ∂ g 1 ∂ 2g (1 + εg) = + Pr ∂X 2 ∂Y 2 2 2 + 3εg ∂g ∂g 2 ∂g + +ε + ∂X ∂Y X ∂X ∂ψ + B ∗g , + (1 + εg) A∗ (49) ∂Y ⎫ −1 ∂g = at Y = 0, ⎬ ∂Y 1 + εg (50) ⎭ g→0 as Y → ∞. Using the transformation (14) along with g(X, Y ) = g(η), Eq. (49) and the conditions in (50) take the form (1 + εg)g Pr f g − f g + εg 2 + (1 + εg) A∗ f + B ∗ g = 0, (51) dg −1 = at η = 0, dη 1 + εg(η) (52) g(η) → 0 as η → ∞. Using the new independent variable ξ = −As Pr e−sη , Eq. (51) can be written as dg APr d 2g −ξ +g (1 + εg)ξ 2 + 1 + εg − s dξ dξ 2 dg −A∗ B ∗ g + εξ + (1 + εg) + 2 = 0. dξ Pr s ξ (53) The boundary conditions (52) become ⎫ −1 A Pr ⎬ dg = at ξ = − , dξ A Pr[1 + g(ξ )] s ⎭ φ(ξ ) → 0 as ξ → 0.
Now, we employ the perturbation method to solve Eq. (53) with the boundary conditions (54). The solution is assumed as
dg0 B∗ A Pr d 2 g0 −ξ + 1 + 2 g0 + 1− ξ s dξ dξ 2 s ξ A∗ , Pr dg0 −1 = dξ A Pr g0 (ξ ) → 0 =
(57)
O ε1 : ξ
dg1 B∗ A Pr d 2 g1 − ξ + 1 + g1 + 1 − s dξ dξ 2 s2ξ dg0 2 A∗ B∗ g0 − 2 g02 − ξ = , Pr dξ s ξ
dg0 d 2 g0 − ξg0 2 dξ dξ ⎫ A Pr ⎬ dg0 dg1 = −g0 at ξ = − , dξ dξ s ⎭ g1 (ξ ) → 0 as ξ → 0 − g0
(58)
(59)
and so on. The solution of (56) subject to the boundary a condition (57) in terms of Kummer’s function is obtained as g0 (ξ ) = c3 ξ r M[r − 1, b + 1, ξ ] +
A∗ Pr
1−a+
B∗ s2
(60)
ξ,
where ∗
c3 = (54)
(56) ⎫ A Pr ⎬ at ξ = − , s ⎭ as ξ → 0.
∗
−[ A1Pr +( APra /1 − a+ B2 )] s
r−1 (−a)r {( b+1 )M[r,b + 2,−a]− ar M[r−1,b + 1,−a]}
and the other constants have their usual definitions as given earlier in the case of PST.
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The solution of (58) subject to the boundary conditions (59) is obtained in terms of Kummer’s function (see Abramowitz and Stegun [42]) as a0 + a1 ξ g1 (ξ ) = c4 ξ r F [r − 1, b + 1, ξ ] + ξ 2 ( + a2 ξ 2 + · · ·) + c3 ξ r+1 (b0 + b1 ξ + b2 ξ 2 + · · ·) + c32 ξ 2r ( c0 + c1 ξ + c2 ξ 2 + · · ·),
(61)
where g0 (−a) 2 A Pr −[t1 +c3 t2 +c3 t3 ] c4 = r−1 r r (−a) {( b+1 )F [r,b+2,−a]− a M[r−1,b+1,−a]} ∞ t1 = ak (k + 2)(−a)k+1 , k=0 ∞ t2 = bk (k + r + 1)(−a)k+r , k=0 ∞ t3 = ck (k + 2r)(−a)k+2r−1 k=0
, Fig. 2 Effect of slip parameter γ on axial velocity profiles with Mn = 1.0
and other constants have their usual definitions as given in PST case. The temperature at the sheet is given by D c xg(−a), (62) Tw = T∞ + k υ where g(−a) = g0 (−a) + g1 (−a). The Nusselt number in this PHF case is given by qw = −g (−a) Nu = (63) T w − T∞ where g (−a) = g0 (−a) + εg1 (−a).
4 Results and discussion An analysis has been carried out to study the MHD slip flow and heat transfer due an impermeable stretching sheet in the presence of a non-uniform heat source/sink. The basic equations governing the flow are highly nonlinear partial differential equations. These equations are transformed into a set of nonlinear ordinary differential equations using similarity transformations without the boundary-layer approximation. The regular perturbation technique is used to obtain the solution for the energy equation in terms of the confluent hypergeometric function. The effects
Fig. 3 Effect of magnetic parameter Mn on axial velocity profiles with γ = 1.0
of slip parameter γ , magnetic parameter Mn, Prandtl number Pr, non-uniform source/sink parameters A∗ and B ∗ , and variable thermal conductivity parameter ε on the dynamics are presented graphically in Figs. 2 through 9. Figure 2 demonstrates the effects of the slip parameter γ on the axial velocity f (η). It is readily seen that γ has strong effect on the flow field. In fact, the amount of slip 1 − f (0) increases monotonically with γ from the no-slip situation for γ = 0 and towards full slip as γ → ∞. The latter limiting case implies that the frictional resistance between the cooling liquid and the stretching sheet is eliminated, and the stretching
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Fig. 4 Effect of magnetic parameter Mn on the temperature profiles
of the sheet does no longer impose any motion of the cooling liquid. The effect of the magnetic parameter Mn on the axial velocity f (η) is shown in Fig. 3. From this figure, it is evident that increasing values of the magnetic parameter Mn opposes the motion of the liquid. That is, the Lorentz force offers resistance to the fluid flow. This results in thinning of the momentum boundary layer. Figures 4(a), (b) project the effects of the magnetic parameter Mn on the temperature field in the PST and PHF cases, respectively. These plots highlight the fact that increasing values of Mn result in increasing the thermal boundary-layer thickness; this is true in both PST and PHF cases.
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Fig. 5 Effect of Prandtl number Pr on the temperature profiles
The effect of the Prandtl number Pr on heat transfer can be analyzed from Figs. 5(a), (b) in the PST and PHF cases, respectively. These graphs reveal that an increase in Prandtl number results in a decrease of the temperature distribution in both cases under consideration. That is, an increase in Prandtl number indicates reduction in the rate of thermal diffusion. The effect of the slip parameter γ on the temperature field is depicted in Figs. 6(a), (b) in the cases of PST and PHF, respectively. From these plots it is observed that increasing values of the slip parameter γ results in thickening of the thermal boundary layer. The effect of the variable thermal conductivity parameter ε on the temperature field is shown in
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Fig. 6 Effect of slip parameter γ on temperature profiles
Figs. 7(a), (b) in the cases of PST and PHF, respectively. Increasing values of the variable thermal conductivity parameter ε results in thickening of the thermal boundary layer: This fact suggests that a liquid whose conductivity does not vary with temperature will be a better coolant. Figures 8(a), (b) depict the effects of the spacedependent heat source/sink parameter A∗ in the cases of PST and PHF, respectively. It is observed that the energy is generated in the thermal boundary layer, causing the temperature profiles (both in PST and PHF) to increase with the increasing values of A∗ > 0, whereas in the case of A∗ < 0 the energy is being absorbed in the boundary layer; as a result the temperature falls considerably with decreasing values of A∗ .
Fig. 7 Effect of variable thermal conductivity ε on temperature profiles
The effect of the temperature-dependent heat source/sink parameter B ∗ on temperature field is shown in Figs. 9(a), (b) in the cases of PST and PHF, respectively. These graphs illustrate that energy is released for increasing values of B ∗ > 0, which causes the temperature to increase in both the PST and the PHF case, whereas energy is absorbed for decreasing values of B ∗ < 0, resulting in a significant temperature drop in the boundary layer. In Table 1 we present the values of the skin-friction coefficient for different values of γ with Mn = 0. In this table, we compare our results (for special cases) with those obtained by Andersson [31] and Sahoo [37]. From this table we see that there is very good
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Fig. 9 Effect of temperature-dependent heat source/sink parameter B ∗ on temperature Fig. 8 Effect of space-dependent heat source/sink parameter A∗ on temperature profiles
agreement between our results and the ones available in the literature. In Table 2 we present the values of Nu for different values of γ , Mn, Pr, A∗ , B ∗ , and ε. From this table it is evident that an increase in all the parameters except the Prandtl number decreases the wall-temperature gradient.
1. The individual and collective effect of increasing values of the parameters Mn, γ , ε, A∗ , and B ∗ (except Pr) is to increase the thermal boundary-layer thickness. 2. Non-uniform heat sinks can be opted for in order to reach effective cooling of the stretching sheet. 3. Liquids with a negligibly varying thermal conductivity property with temperature gradient must be opted for as cooling liquids.
5 Conclusions The analytical solutions for the complete governing equations of momentum and heat transfer in the case of MHD slip flow are obtained. Some specific conclusions derived from the study are presented now.
Acknowledgements The authors are thankful to the reviewers’ comments and suggestion for better improvement of the quality of paper. This work is supported by University Grants Commission, New-Delhi, India under Major Research Project [F.No.39-59/2010 (SR)].
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Meccanica (2013) 48:1451–1464 Table 1 Skin-friction coefficient for various values of γ when Mn = 0
γ
Cf (Skin friction) Andersson [36]
Sahoo [43]
Wang [44]
Present results
0.0
−1.0000
−1.001154
−1.00000
1.000000
0.1
−0.8721
−0.871447
–
0.87208
0.2
−0.7764
−0.774933
–
0.77638
0.5
−0.5912
−0.589195
–
0.59120
1
−0.4302
−0.428450
−0.430
0.43016
2
−0.2840
−0.282893
−0.284
0.28398
5
−0.1448
−0.144430
−0.145
0.14484
10
−0.0812
−0.081091
–
0.08124
20
−0.0438
−0.043748
−0.0438
0.04379
50
−0.0186
−0.018600
–
0.01860
Table 2 Nusselt number for PST and PHF cases for different values of governing parameters γ
Mn
Pr
A∗
B∗
Values of
ε
θ (0) (PST) 0.0
1.0
1.0
−0.01
−0.01
0.1
φ (0) (PHF)
0.836973
0.866429
1.0
0.518719
0.755687
5.0
0.271043
0.497638
1.0
1.0
1.0
1.0
1.0
1.0
1.0
−0.01
−0.01
0.1
0.518719
0.755687
2.0
0.410482
0.656622
3.0
0.344737
0.577458
1.0
1.0
1.0
1.0
1.0
−0.01
−0.01
0.1
0.518719
0.755687
2.0
0.832306
0.877669
5.0
1.478850
0.963913
0.912443
0.896506
1.0
1.0
1.0
−0.5
−0.01
0.1
0.0
0.502514
0.739313
0.3
0.385212
0.501422
−0.01
−0.01
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