Nonlinear Dyn (2008) 51:259–265 DOI 10.1007/s11071-007-9208-3
ORIGINAL ARTICLE
MHD rotating flow of a viscous fluid over a shrinking surface M. Sajid · T. Javed · T. Hayat
Received: 29 September 2006 / Accepted: 5 January 2007 / Published online: 13 February 2007 C Springer Science + Business Media B.V. 2007
Abstract This study is concerned with the magnetohydrodynamic (MHD) rotating boundary layer flow of a viscous fluid caused by the shrinking surface. Homotopy analysis method (HAM) is employed for the analytic solution. The similarity transformations have been used for reducing the partial differential equations into a system of two coupled ordinary differential equations. The series solution of the obtained system is developed and convergence of the results are explicitly given. The effects of the parameters M, s and λ on the velocity fields are presented graphically and discussed. It is worth mentioning here that for the shrinking surface the stable and convergent solutions are possible only for MHD flows. Keywords Rotating flow . Viscous fluid . Shrinking sheet . HAM solution
1 Introduction The boundary layer viscous flow induced by stretching surface moving with a certain velocity in an otherwise M. Sajid ( ) Theoretical Plasma Physics Division, PINSTECH, P.O. Nilore, Islamabad 44000, Pakistan e-mail:
[email protected] T. Javed · T. Hayat Department of Mathematics, Quaid-I-Azam University, 45320, Islamabad 44000, Pakistan
quiescent fluid medium often occurs in several engineering processes. Such flows have promising applications in industries, for example, in the extrusion of a polymer sheet from a die or in the drawing of plastic films. During the manufacture of these sheets, the melt issues from a slit and is subsequently stretched to achieve the desired thickness. The mechanical properties of the final product strictly depends on the stretching and cooling rates in the process. Since the pioneering work of Sakiadis [1, 2] various aspects of boundary layer flow induced by a stretching sheet have been investigated by several workers in the field. Specifically, Crane’s problem [3] for flow of an incompressible viscous fluid past a stretching sheet has become a classic in the literature. It admits an exact analytical solution. Besides, it has produced a galore of associated problems, each incorporating a new effect and still giving an exact solution. The uniqueness of the exact analytical solution presented in [3] is discussed by McLeod and Rajagopal [4]. Gupta and Gupta [5] examined the stretching flow subject to suction or injection. The flow inside a stretching channel or tube has been analyzed by Brady and Acrivos [6] and the flow outside the stretching tube by Wang [7]. In another paper, Wang [8] extended the flow analysis to the three-dimensional axisymmetric stretching surface. The unsteady flows induced by stretching film have been also discussed by Wang [9] and Usha and Sridharan [10]. The boundary layer flow caused by the stretching of a flat surface in a rotating fluid has been studied by Wang [11], Rajeswari and Nath [12] and Nazar et al. [13] for the Newtonian Springer
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fluids. For the non-Newtonian fluid, Kumari et al. [14] discussed the rotating flow of a power-law fluid over a stretching surface. One can find abundant number of articles in the literature regarding different problems for Newtonian and non-Newtonian fluids, with and without heat transfer analysis dealing with the stretching flow problems. However, the investigations regarding the flow problems due to a shrinking sheet are scarcely available in the literature. To the best of our knowledge, only two such attempts [9, 15] are yet available in the literature. In [9], Wang presented unsteady shrinking film solution and in [15], Miklavcic and Wang proved the existence and uniqueness of steady viscous hydrodynamic flow due to a shrinking sheet for a specific value of the suction parameter. The aim of this investigation is to consider the MHD rotating flow of a viscous fluid over a shrinking sheet for the analytic solution using homotopy analysis method (HAM) [16]. The presented HAM solution is valid for all the values of parameters involved in the problem. The exact infinite series is presented and the recurrence formulae are obtained for finding the coefficients of the series. HAM has already been applied successfully for the analytical solution of several other problems [17–35].
u = −ax, u → 0,
v = 0,
v→0
w = −W
at z = 0,
as z → ∞.
(4)
in which a > 0 is the shrinking constant and W > 0 is the suction velocity. Introducing the following similarity transformations u = ax f (η),
v = axg(η), √ a w = − aν f (η), η = z ν
(5)
Equation (1) is identically satisfied and Equations (2) and (3) give f − f 2 + f f + 2λg − M 2 f = 0,
Consider the steady laminar MHD boundary layer flow of a viscous fluid caused by a two-dimensional shrinking surface in a rotating fluid. For mathematical modelling we use the Cartesian coordinate system (x, y, z) with Ω being the angular velocity of the rotating fluid in the z-direction. In addition, a constant magnetic field B0 is applied in the z-direction. The electric field is assumed to be zero. Under the assumption of small magnetic Reynolds number, the boundary layer equations which govern the MHD flow in the absense of pressure gradient are
g − f g + f g − 2λ f − M g = 0, f = s,
f → 0,
f = −1, g→0
g=0
as η → ∞,
(8)
3 Analytic solution For the HAM solution of Equations (6) and (7) subject to conditions (8), we choose the following initial guess approximations of the functions f and g f 0 (η) = s − 1 + e−η ,
g0 (η) = η e−η
(9)
L2 ( f ) = f − f
(10)
and L1 ( f ) = f − f ,
are the auxiliary linear operators satisfying
∂u ∂ 2u ∂u σ B02 +w − 2v = ν 2 − u, ∂x ∂z ∂z ρ
(2)
∂v σ B02 ∂v ∂ 2v u +w + 2u = ν 2 − v, ∂x ∂z ∂z ρ
L1 [C1 + C2 eη + C3 e−η ] = 0, L2 C4 eη + C5 e−η = 0,
(3)
Springer
(7)
at η = 0,
(1)
u
(6)
2
√ where M = σ B02 /ρa, s = W/ aν and λ = /a.
2 Mathematical formulation
∂u ∂w + = 0, ∂x ∂z
where ν = μ/ρ is the kinematic viscosity, σ is the electrical conductivity and u, v and w are the velocity components in x, y and z-directions, respectively. The boundary conditions applicable to the present flow are
where Ci ’s (i = 1, . . . , 5) are arbitrary constants.
(11)
Nonlinear Dyn (2008) 51:259–265
261
3.1 Zeroth-order deformation problems
where
(1 − p)L1 [ f (η, p) − f 0 (η)]
f (η, p) 1 ∂m , f m (η) = m! ∂ p m p=0 1 ∂ m g (η, p) gm (η) = . m! ∂ p m p=0
= pN1 [ f (η, p), g (η, p)],
(12)
(1 − p)L2 [ g (η, p) − g0 (η)] f (η, p), g (η, p)], = pN2 [
f (0, p) = s,
f (0, p) = −1,
g (0, p) = 0,
g (∞, p) = 0,
(13)
f (∞, p) = 0, (14)
f (η, p) ∂3 N1 [ f (η, p), g (η, p)] = ∂η3
L1 [ f m (η) − χm f m−1 (η)] = Rmf (η),
f (η, 1) = f (η),
g (η, 0) = g0 (η),
g (η, 1) = g(η).
f m (η) p , m
Rmf (η) = f m−1 (η) − M 2 f m−1 (η) + 2λgm−1 (η)
gm (η) p ,
+
m−1
(η) f k (η) , f m−1−k (η) f k (η)− f m−1−k
k=0
(22) (η) − M 2 gm−1 (η) − 2λ f m−1 Rmg (η) = gm−1 (η)
(17)
g(η, p) = g0 (η)
m=1
m=1
(21)
(16)
As p increases from 0 to 1, f (η, p), g (η, p) varies from the initial guesses f 0 (η), g0 (η) to the exact solutions f (η), g(η). By Taylor’s theorem and Equation (17), one can write
+
(20)
f m (0) = f m (0) = f m (∞) = gm (0) = gm (∞) = 0,
f (η, 0) = f 0 (η),
m
gm (η).
m=1
(15)
in which p ∈ [0, 1] is the embedding parameter and is the auxiliary nonzero parameter. For p = 0 and p = 1, we respectively have
∞
∞
L2 [gm (η) − χm gm−1 (η)] = Rmg (η),
∂ f (η, p) ∂ g (η, p) g (η, p) + f (η, p) , ∂η ∂η
f (η, p) = f 0 (η) +
f m (η), f (η) = g0 (η)+
3.2 mth-order deformation problems
∂ f (η, p) ∂η
∞
∞
(19)
∂ 2 g (η, p) N2 [ f (η, p), g (η, p)] = ∂η2
−
f (η) = f 0 (η) +
m=1
∂ f (η, p) − M2 + 2λ g (η, p) ∂η 2 ∂ f (η, p) f (η, p) ∂2 − + f (η, p) , ∂η ∂η2
g (η, p) − 2λ − M 2
The convergence of the two series in (18) depends on the auxiliary parameter . Assume that is chosen in such a way that the two series in (18) are convergent at p = 1, then due to Equation (17) we have
(18)
+
m−1
(η) gk (η) , f m−1−k (η)gk (η)− f m−1−k
k=0
(23) in which
0, χm = 1,
m≤1 m>1
.
MATHEMATICA has been used to solve the system of linear nonhomogeneous Equations (20) and (21) up to first few order of approximations and it is found that the Springer
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solution of the problems can be expressed as an infinite series of the form f (η) =
∞
f m (η) = lim
M→∞
m=0
×
M m+1−n
m=n−1
g(η) =
∞
M
0 am,0 +
m=0
M+1
= lim
M→∞
M+1
q
e
M
n=1
m+1−n
m=n−1
Akm,n ηk
0 0 am,0 = χm χm+2 am−1,0 −
m
q
k=0
q
m+1
⎢ ⎣
n=2
q
+
q
m,n (n
⎤ ⎥
q 1)μn,0
−
q=1
−
q ⎦ μn,1
(26) k k am,0 = χm χm+1−k am−1,0 , 0 0 am,1 = χm χm+1 am−1,1 +
+
m+1
1 ≤ k ≤ m + 1,
m
q
q
m,1 μ1,1
0 n m,n μ0n,0
n=2
+
(27)
q=0
m+1−n
q
m,n
q nμn,0
−
q μn,1
ν1,k = q
νn,k =
m
q
(28) q
m,1 μ1,k ,
qm,n =
n=2
Springer
q=0
r =0
p=0
q! , k!(n − 1)q+1−k−r − p n r +1 (n + 1) p+1
q!2q+2−k , k!
q ≥ 0,
n ≥ 2,
0 ≤ k ≤ q + 1,
q+1−k
q ≥ 0,
q! k!(n −
1) p+1 (n
(35)
+ 1)q+1−k− p
(36)
,
m−1
min{n,k+1}
min{q,k+1−i}
q− j
m−1
min{n,k+1}
(39) min{q,k+1−i}
j
q
q− j
bk,i bm−1−k,n−i , qm,n =
q
m,n μn,k ,
m−1
qm,n = q
min{q,k+1−i}
q− j
Bk,i am−1−k,n−i ,
(30)
qm,n νn,0 ,
min{n,k+1}
(40)
k=0 i=max{0,n−m+k} j=max{0,q−m+k+n−i} j
0 ≤ k ≤ m + 1 − n, m+1 m+1−n
q−k q−k−r
j
(29)
A0m,1 = χm χm+1 A0m−1,1 −
1 ≤ k ≤ q + 1,
ck,i am−1−k,n−i ,
q=k
2 ≤ n ≤ m + 1,
, q ≥ 0,
k=0 i=max{0,n−m+k} j=max{0,q−m+k+n−i}
1 ≤ k ≤ m + 1, m+1−n
p=0
k=0 i=max{0,n−m+k} j=max{0,q−m+k+n−i}
,
q=k−1
k k am,n = χm χm+2−n−k am−1,n +
q! k!2q+1−k− p
(33)
0 ≤ k ≤ q + 1 − n, q ≥ 0, n ≥ 2, (37) q q q q
m,n = χm+2−n−q dm−1,n − M 2 bm−1,n +2λAm−1,n q +δm,n − qm,n , q q q qm,n = χm+2−n−q Cm−1,n − M 2 Am−1,n +2λbm−1,n q +qm,n − m,n , (38)
q=1 k k am,1 = χm χm−k+1 am−1,1 +
q+1−k
p=0
q δm,n =
0 ≤ k ≤ m + 1 − n,
0 ≤ k ≤ q + 1 − n,
q
0 μ0n,0 (n − 1) m,n m+1−n
μn,k =
m,1 μ1,1
q=0
⎡
q
qm,n νn,k ,
(34)
, (25)
−
2 ≤ n ≤ m + 1, μ1,k =
m+1−n q=k
(24)
k=0
−nη
(32)
Akm,n = χm χm+2−n−k Akm−1,n +
gm (η)
m=0
q
1 ≤ k ≤ m + 1,
,
k am,n ηk
q
m,1 ν1,k ,
q=k−1
e−nη
n=1
m
Akm,1 = χm χm−k+1 Akm−1,1 +
m−1
min{n,k+1}
(41) min{q,k+1−i}
k=0 i=max{0,n−m+k} j=max{0,q−m+k+n−i}
(31)
j
q− j
Ak,i bm−1−k,n−i ,
(42)
Nonlinear Dyn (2008) 51:259–265
263 k k+1 k = (k + 1)cm,n − ncm,n , dm,n
(45)
k k Bm,n = (k + 1)Ak+1 m,n − n Am,n ,
(46)
k k+1 k Cm,n = (k + 1)Bm,n − n Bm,n ,
(47)
0 a0,0 = s − 1,
0 a0,1 = 1,
A10,1 = 1.
(48)
For the detailed procedure of deriving the above relations the reader is referred to [18]. 4 Convergence of the analytic solution Fig. 1 -curves for 25th order of approximation k k+1 k bm,n = (k + 1)am,n − nam,n ,
(43)
k k+1 k cm,n = (k + 1)bm,n − nbm,n ,
(44)
The explicit, analytic expressions given by Equations (24) and (25) contain the auxiliary parameter . As pointed out by Liao [16] this parameter plays a vital role in finding the convergence region and rate of approximation for the homotopy analysis method. For
Fig. 2 Variation of f and g by increasing suction parameter s
Fig. 3 Variation of f and g by increasing Hartman number M Springer
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Nonlinear Dyn (2008) 51:259–265
Fig. 4 Variation of f and g by increasing rotation parameter λ
this purpose the -curves are plotted for the 25th order of approximations for both f and g in Fig. 1. Figure 1 clearly indicates that the admissible values of the parameter are −0.45 ≤ < −0.1. Our calculations indicate that the series given by Equations (24) and (25) converge in the whole region of η when = −0.25.
graphically and the effect of the emerging parameters are discussed. In comparison to the stretching sheet problem it is found that the results in the case of hydrodynamic flow are not stable for the shrinking surface and only MHD flows are meaningful in the case of shrinking surface.
5 Results and discussion The graphs for the function f (η) and g(η) are drawn against η for different values of the parameters M, s and λ. In all cases Fig. 2(a) displays the function f and (b) displays the function g. It is depicted from Fig. 2(a) that the velocity f increases and boundary layer thickness decreases by increasing the suction parameter s. The effect of s on the velocity g is similar to that of f but in this case, boundary layer thickness increases as shown in Fig. 2(b). Figure 3 elucidates that the effect of the Hartman number is similar to that of the suction parameter. The effect of rotation parameter is quite opposite when compared with suction parameter and Hartman number (Fig. 4). 6 Concluding remarks In this paper, the MHD rotating flow of a viscous fluid due to a shrinking surface is investigated. The series solution is developed for the governing-coupled nonlinear ordinary differential equations using homotopy analysis method. The explicit form of the infinite series is found and the recurrence formulae are constructed for the coefficients of the series. The convergence of the results is shown explicitly. The results are presented Springer
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