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Finite element and Chebyshev finitedifference methods for micropolar flowpast a stretching surface with heat andmass transferMoncef Aouadi aa Rustaq Faculty of Education, Department of Mathematics andComputer Science, Rustaq, 329, P.O. Box 10, Sultanate of OmanVersion of record first published: 31 Dec 2007.
To cite this article: Moncef Aouadi (2008): Finite element and Chebyshev finite difference methodsfor micropolar flow past a stretching surface with heat and mass transfer, International Journal ofComputer Mathematics, 85:1, 105-122
To link to this article: http://dx.doi.org/10.1080/00207160701374392
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International Journal of Computer MathematicsVol. 85, No. 1, January 2008, 105–122
Finite element and Chebyshev finite difference methods formicropolar flow past a stretching surface with heat and
mass transfer
MONCEF AOUADI*
Rustaq Faculty of Education, Department of Mathematics and Computer Science,Rustaq 329, P.O. Box 10, Sultanate of Oman
(Received 25 March 2006; revised version received 10 March 2007; accepted 22 March 2007)
This paper presents a numerical study of a micropolar fluid flowing past a stretching surface involvingboth heat and mass transfer with Ohmic heating and viscous dissipation. A similarity transformation isemployed to change the governing equations into nonlinear coupled higher-order ordinary differentialequations. These equations are solved numerically using a finite element method with linear shapefunctions and a Chebyshev finite difference method that involves higher-order polynomials. Numericalresults of both methods have been compared with the closed-form solution of a particular case whenthe material parameter is taken to be zero. Good agreement between the numerical results of bothmethods, together with excellent agreement with the closed-form solution, ensures the reliability ofusing linear shape functions in the finite element method. The effect of the parameters governing theproblem on the velocity, microrotation, temperature and concentration functions has been studied fordifferent boundary conditions.
Keywords: Micropolar fluid; Finite element method; Chebyshev finite difference method
AMS Subject Classifications: 76M10; 65Z05; 65N30
1. Introduction
Eringen [1, 2] introduced micropolar fluid theories in order to describe some physical systemsthat do not satisfy the Navier–Stokes equations. To explain the kinematics of such mediatwo new variables should be added to the velocity. These variables are the spin, responsiblefor microrotation, and the microinertia tensor, which accounts for the atoms and moleculeinside the macroscopic fluid particle. Micropolar fluids are able to describe the behaviourof colloidal solutions, suspensions, liquid crystals, animal blood, etc. Eringen [3] extendedmicropolar fluid theory and developed the theory of thermomicropolar fluids.
The study of magneto-hydrodynamic flow for an electrically conducting fluid past a heatedand stretched surface has attracted the interest of many researchers in view of its importantapplications in many engineering problems such as plasma studies, the petroleum industry,
*Email: [email protected]
International Journal of Computer MathematicsISSN 0020-7160 print/ISSN 1029-0265 online © 2008 Taylor & Francis
http://www.tandf.co.uk/journalsDOI: 10.1080/00207160701374392
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106 M. Aouadi
MHD power generators, cooling of nuclear reactors, boundary layer control in aerodynamics,and crystal growth. Until recently, this study has been largely concerned with the flow andheat transfer characteristics in various physical situations [4–8]. Attention has also been givento the problem of MHD convection flow past a heated vertical plate in a porous medium[9, 10].
Recently, several researchers have focused their attention on the problem of combined heatand mass transfer in MHD free convection flow due to the fact that natural convection inducedby the simultaneous action of buoyancy forces resulting from thermal and mass diffusionis of considerable interest in nature and in many industrial applications. The monograph byGebhart et al. [11] provided an overview of early studies concerning the natural convectionboundary-layer flow due to simultaneous heat and mass transfer over heated surfaces withvarious geometries. Elbashbeshy [12] studied heat and mass transfer along a vertical platein the presence of a magnetic field, neglecting the effects of viscous dissipation and Ohmicheating. El-Hakiem et al. [13] presented an analysis of MHD free convection heat transfer ofan electrically-conducting micropolar fluid past a semi-infinite plate, including the effects ofviscous and Joule heating. Ganesan and Rani [14] considered the problem of unsteady MHDfree convection flow past a vertical cylinder with heat and mass transfer. Aboeldahab andElbarbary [15] took into account the Hall current effect on MHD free-convection heat andmass transfer over a vertical surface. Chamkha and Khaled [16] investigated the problem ofcoupled heat and mass transfer by hydromagnetic free convection, neglecting both viscousdissipation and Ohmic heating effects.
As mentioned above, neither viscous dissipation nor Ohmic heating effects were consideredin previous studies of the problem of coupled heat and mass transfer in MHD free convection.However, it is more reasonable to include these two effects in order to explore momentum,heat and mass transfer characteristics in the thermal boundary layer with an externally appliedmagnetic field.
Keeping this view in mind, very recently Chen [17] has taken into consideration the viscousdissipation term and the Ohmic heating effect on the combined heat and mass transfer inMHD free convection flow adjacent to an impermeable vertical surface with uniform walltemperature and concentration. In order to obtain more insight into the heat and mass transferbehaviour of MHD flow, it is necessary to treat this kind of problem in a more general situation.Therefore, the present analysis is the generalization of the recent work of Chen [17], i.e. itattempts to investigate the problem of coupled heat and mass transfer in MHD micropolarflow past a stretching surface, including the effects of viscous dissipation and Ohmic heating,the non-isothermal boundary conditions of prescribed surface temperature, and concentrationdistributions for different boundary conditions.
Most researchers in the field of fluid mechanics try to obtain similarity solutions by intro-ducing a general similarity transformation with unknown parameters into the differentialequation, obtaining in this way an algebraic system. Then the solution of this system, if itexists, determines the values of the unknown parameters. Sparrow et al. [18] studied thecauses of non-similarities. Three different cases were considered: non-similarity caused by(a) spatial variations in the freestream velocity, (b) surface mass transfer, and (c) transversecurvature. These cases are not present in our problem considered here. Hence, we apply theprocedure of the symmetry transformation, which transforms the governing equations intoordinary differential equations.
Methods available for solving the resulting ordinary differential equations are the Chebyshevfinite difference method, the finite element method, the finite difference method, the fourth-order Runge–Kutta shouting method, and others.
The normal procedure for handing the solutions of differential equations using the finitedifference method is to express the derivatives of a function in terms of its values. The numbers
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Numerical methods for micropolar flow 107
of these terms are two and three for the central finite difference of the first and second-order derivatives, respectively. The Chebyshev finite difference method is more accurate incomparison to the above mentioned methods, because the approximation of the derivatives isdefined over the whole domain. However the finite difference method produces a second-orderaccurate derivative with the error decreasing as 1/L2 (L being the number of grid points). Thiserror from the global method decreases exponentially [19].
It is interesting instead to solve the nonlinear equations of the flow of a micropolar fluidwith the help of the Chebyshev finite difference method (ChFDM) and compare the resultswith those of the finite element method (FEM) based on linear shape functions [20]. To ensurethe reliability of the proposed numerical methods, both numerical results are compared withthe analytical solution obtained by neglecting the vortex viscosity.
Good agreement between the numerical results of both methods, together with excellentagreement with the closed-form solution, enables one to use linear shape functions in the finiteelement method.
2. Mathematical analysis
Let us consider a steady, laminar, free convection flow of an incompressible electrically con-ducting micropolar fluid caused by a moving surface coinciding with the plane y = 0, the flowbeing confined in the region y > 0. Two equal and opposite forces are introduced along thex-axis so that the surface is stretched keeping the origin fixed. The magnetic field B0 is appliedperpendicular to the stretching sheet and the effect of induced magnetic field is neglected sincethe magnetic Reynolds number is assumed to be small. We further assume that the impressedelectric field is zero and the Hall effect is neglected. The component of velocity varies lin-early along the x-axis, i.e. u(x, 0) = Bx where B(>0) is an arbitrary constant. The governingequations of the flow in two dimensions are as follows.
Continuity equation:
∂u
∂x+ ∂v
∂y= 0. (1)
Momentum equation:
u∂u
∂x+ v
∂v
∂y=
(ν + κ
ρ
)∂2u
∂y2+ κ
ρ
∂N
∂y− σB2
0
ρu. (2)
Angular momentum equation:
u∂N
∂x+ v
∂N
∂y= γ
Jρ
∂2N
∂y2− κ
Jρ
(2N + ∂u
∂y
). (3)
Energy equation:
u∂T
∂x+ v
∂T
∂y= k
ρcp
∂2T
∂y2+ μ + κ
ρcp
(∂u
∂y
)2
+ σB20
ρcpu. (4)
Concentration equation:
u∂C
∂x+ v
∂C
∂y= D
∂2C
∂y2. (5)
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108 M. Aouadi
The appropriate physical boundary conditions are given by
u(x, y) = Bx, v(x, y) = 0, T = Tw = T∞ + A(x
l
)2,
(6)N(x, y) = −s
∂u
∂y, C(x, y) = Cw, at y = 0,
u(x, y) −→ 0, T (x, y) −→ T∞, N(x, y) −→ 0, C(x, y) −→ C∞, as y −→ ∞,
(7)
where x and y are the coordinate directions, u, v, N, T and C are the fluid velocity componentsin the x- and y-directions, the components of microrotation, temperature and concentration,respectively; ν, κ and ρ are the kinematic viscosity, the vortex viscosity and the fluid density,respectively; μ, σ and k are the dynamic viscosity, the electric conductivity and the thermalconductivity, respectively; cp, Tw and T∞ are the specific heat at constant pressure, the platetemperature and the fluid free-stream temperature, respectively; Cw and C∞ are the speciesconcentration at the surface and the concentration of the fluid far way from the surface,respectively; D is the coefficient of mass diffusivity.
The second term on the right-hand side of equation (4) represents viscous dissipation andthe last term indicates the Ohmic heating effect. Here γ is assumed to be given by [21]:
γ =(μ + κ
2
)J, (8)
and we take J = ν/B as a reference length. Equation (4) is invoked to allow equations (1)–(3)to predict the correct behaviour in the limiting case when microstructure effects becomenegligible, and the microrotation N reduces to the angular velocity.
A linear relationship between the microrotation function N and the surface shear stress(∂u/∂y) is chosen for investigating the effect of different surface conditions on the microro-tation. Here s is the boundary parameter and varies from 0 to 1. The first boundary condition(s = 0) is a generalization of the no-slip condition, which requires that the fluid particlesclosest to a solid boundary stick to it—neither translating nor rotating. The second boundarycondition, i.e. microrotation, is equal to the fluid vorticity at the boundary (s �= 0), mean-ing that, in the neighbourhood of a rigid boundary, the effect of microstructure is negligiblesince the suspended particles cannot get closer to the boundary than their radius. Hence, inthe neighbourhood of the boundary, the only rotation is due to fluid shear, and therefore thegyration vector must be equal to the fluid vorticity [22].
By using the following similarity transformations:
u = Bxf ′(η), v = −√Bνf (η), θ(η) = T − T∞
Tw − T∞,
(9)N = √
B3lνxg(η), φ = C − C∞Cw − C∞
, η = √Blνy,
and substituting equation (2) into equations (1)–(7), we obtain the following similarityequations and boundary conditions:
(1 + R)f ′′′ + ff ′′ − (f ′)2 + Rg′ − Mf ′ = 0, (10)(1 + R
2
)g′′ + fg′ − f ′g − R(2g + f ′′) = 0, (11)
θ ′′ + Pr f θ ′ − 2 Pr f ′θ + (1 + R) Pr Ec(f ′′)2 + Pr EcMf ′2 = 0, (12)
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Numerical methods for micropolar flow 109
φ′′ + Scf φ′ = 0, (13)
f (0) = 0, f ′(0) = 1, g(0) = −sf ′′(0), θ(0) = 1, φ(0) = 1, (14)
f ′(∞) = 0, g(∞) = 0, θ(∞) = 0, φ(∞) = 0, (15)
where the primes denote differentiation with respect to η, R = κ/μ is a material parameter,Pr = ρcpν/k is the Prandtl number, Ec = (Bx)2/cp(Tw − T∞) is the Eckert number, Sc =ν/D is the Schmidt number, and M = σB2
0/(ρB) is the magnetic parameter.Assuming the material parameter to be zero, i.e. R = 0, equation (10), together with the
boundary conditions f (0) = 0, f ′(0) = 1, f ′(∞) = 0, has an exact solution in the form
f (η) = 1
m(1 − e−mη), where m = √
1 + M. (16)
To obtain the solution of equation (11), one introduces a new variable ξ as
ξ = −e−mη
m2, (17)
and substitutes equation (17) into equation (11) yielding
ξd2g
dξ 2+
(1 − 1
m2− ξ
)dg
dξ+ g = 0. (18)
The corresponding boundary conditions are
g
(−1
m2
)= sm, g(0) = 0. (19)
The exact solution of equation (18) satisfying equation (19) in terms of Kummer’s function[23] is
g(ξ) = sm(−m2ξ)1/m2M
(1
m2− 1,
1
m2+ 1, ξ
)
M
(1
m2− 1,
1
m2+ 1,
−1
m2
) , (20)
where Kummer’s function M is defined by
M(a0, b0, z) = 1 +∞∑
n=1
(a0)nzn
(b0)nn! ,
where
(a0)n = a0(a0 + 1)(a0 + 2) · · · (a0 + n − 1),
(b0)n = b0(b0 + 1)(b0 + 2) · · · (b0 + n − 1).
The solution of equation (18) in terms of the variable η is
g(η) = sme−η/m
M
(1
m2− 1,
1
m2+ 1,
−e−mη
m2
)
M
(1
m2− 1,
1
m2+ 1,
−1
m2
) . (21)
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110 M. Aouadi
The dimensionless wall microrotation gradient g′(0) is obtained as
g′(0) = s1 − m2
1 + m2
⎡⎢⎢⎣
M
(1
m2,
1
m2+ 2,
−1
m2
)
M
(1
m2− 1,
1
m2+ 1,
−1
m2
) − 1
⎤⎥⎥⎦ . (22)
We proceed now to solve equation (12) by substituting
ξ = −re−mη, r = Pr
m2, (23)
into equation (12) to obtain
ξd2θ
dξ 2+ (1 − r − ξ)
dθ
dξ+ 2θ = − Pr Ec
(1 + M
m2
)ξ
r2, (24)
and
θ(−r) = 1 and θ(0−) = 0. (25)
Equation (24) can be further transformed into the standard confluent hypergeometric functionor Kummer’s equation. Making use of the boundary conditions (25) we obtain
θ(ξ) = (1 + C)
(ξ
−r
)rM(r − 2, r + 1, ξ)
M(r − 2, r + 1, −r)− C
(ξ
−r
)2
, (26)
where
C = Ec Pr(1 + M/m2)
2(2 − r).
The solution (26) is, in terms of η,
θ(η) = (1 + C)e−mrη M(r − 2, r + 1, −re−mη)
M(r − 2, r + 1, −r)− Ce−2mη, (27)
and the dimensionless surface temperature gradient θ ′(0) at the wall is
θ ′(0) = (1 + C)mr
[(r − 2
r + 1
)M(r − 2, r + 1, −r)
M(r − 2, r + 1, −r)− 1
]+ 2Cm. (28)
To solve equation (13), we proceed as above by substituting
ξ = −Sc
m2e−mη (29)
into equation (13), yielding
ξd2φ
dξ 2+
(1 − Sc
m2− ξ
)dφ
dξ= 0. (30)
The corresponding boundary conditions are
φ
(−Sc
m2
)= 1, φ(0) = 0. (31)
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Numerical methods for micropolar flow 111
The exact solution of equation (30) satisfying equation (31) is given by
φ(ξ) =(−m2
Scξ
)Sc/m2 M
(Sc
m2,
Sc
m2+ 1, ξ
)
M
(Scm2 ,
Sc
m2+ 1,
−Sc
m2
) . (32)
The solution of equation (30) in terms of the variable η is
φ(η) = e−Scη/m
M
(Sc
m2,
Sc
m2+ 1,
−Sc e−mη
m2
)
M
(Sc
m2,
Sc
m2+ 1,
−Sc
m2
) . (33)
The dimensionless wall concentration gradient φ′(0) is obtained as
φ′(0) = Sc
m
⎡⎢⎢⎣ Sc
Sc + m2
M
(Sc
m2+ 1,
Sc
m2+ 2,
−Sc
m2
)
M
(Sc
m2,
Sc
m2+ 1,
−Sc
m2
) − 1
⎤⎥⎥⎦ . (34)
3. Numerical analysis
3.1 Finite element method
Finite element method has been used for solving linear as well as nonlinear differentialequations. The steps involved in the finite-element analysis of a problem are as follows.
(1) Finite-element discretization. The given domain is divided into a finite number of sub-domains. This is called the discretisation of the domain. Each sub-domain is called anelement. The collection of elements is called a finite-element mesh.
(2) Generation of element equations.(i) A typical element is isolated from the mesh and the variational formulation of the
given problem over the typical element is constructed.(ii) An approximate solution of the variational problem is assumed. On substituting in
(i) above, the element equations are obtained.(iii) The element matrix, also called the stiffness matrix, is constructed by using the
element interpolation functions.(3) Assembly of element equations. The algebraic equations so obtained are assembled by
imposing the inter-element continuity conditions. This yields a large number of algebraicequations known as the global finite element model, which governs the whole domain.
(4) Imposition of boundary conditions. The essential and natural boundary conditions areimposed on the assembled equations.
(5) Solution of assembled equations. The assembled equations can be solved by using anyof the numerical techniques, viz. Gaussian elimination, LU Decomposition method, etc.The details of the method used here can be obtained from [20, 22, 24, 25].
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112 M. Aouadi
To solve the differential equations (10)–(13), with the boundary conditions (14),(15), weassume
f ′ = h. (35)
The system of equations (10)–(13) then reduces to
(1 + R)h′′ + f h′ − h2 + Rg′ − Mh = 0, (36)(1 + R
2
)g′′ − R(2g + h′) + fg′ − hg = 0, (37)
θ ′′ + Pr f θ ′ − 2 Pr hθ + (1 + R) Pr Ec(h′)2 + Pr EcMh2 = 0, (38)
φ′′ + Scf φ′ = 0. (39)
The corresponding boundary conditions now become
f (0) = 0, h(0) = 1, g(0) = −sh′(0), θ(0) = 1, φ(0) = 1, (40)
h(∞) = 0, g(∞) = 0, θ(∞) = 0, φ(∞) = 0. (41)
3.1.1 Variational formulation. The variational form associated with equations (35)–(39)over a typical two-node line element (ηe, ηe+1) is given by
∫ ηe+1
ηe
℘1(f ′ − h
)dη = 0, (42)
∫ ηe+1
ηe
℘2[(1 + R)h′′ + f h′ − h2 + Rg′ − Mh
]dη = 0, (43)
∫ ηe+1
ηe
℘3
[(1 + R
2
)g′′ − R(2g + h′) + fg′ − hg
]dη = 0, (44)
∫ ηe+1
ηe
℘4[θ ′′ + Pr f θ ′ − 2 Pr hθ + (1 + R) Pr Ec(h′)2 + Pr EcMh2
]dη = 0, (45)
∫ ηe+1
ηe
℘5(φ′′ + Scf φ′) dη = 0. (46)
The functions ℘1, . . . , ℘5 are arbitrary test functions which can be considered as the variationsof f , h, g, θ and φ, respectively.
3.1.2 Finite element formulation. By using finite element approximations of the form
f =2∑
j=1
fjψ(e)j , h =
2∑j=1
hjψ(e)j , g =
2∑j=1
gjψ(e)j , θ =
2∑j=1
θjψ(e)j , φ =
2∑j=1
φjψ(e)j ,
(47)with
℘1 = ℘2 = ℘3 = ℘4 = ℘5 = ψi (i = 1, 2), (48)
where ψi are the shape functions for a typical element (ηe, ηe+1) and are defined as
ψ(e)1 = ηe+1 − η
ηe+1 − ηe
, ψ(e)2 = η − ηe
ηe+1 − ηe
, ηe ≤ η ≤ ηe+1, (49)
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Numerical methods for micropolar flow 113
equations (42)–(46) may be transformed as follows:⎡⎢⎢⎢⎢⎣
[K11] [K12] 0 0 00 [K22] [K23] 0 00 [K32] [K33] 0 00 [K42] 0 [K44] 00 0 0 0 [K55]
⎤⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎢⎣
{f }{h}{g}{θ}{φ}
⎤⎥⎥⎥⎥⎦ =
⎡⎢⎢⎢⎢⎣
{r1}{r2}{r3}{r4}{r5}
⎤⎥⎥⎥⎥⎦ .
Matrices [Kmn] and [rm] (m, n = 1, . . . , 5) are given by
K11ij =
∫ ηe+1
ηe
ψi
dψj
dηdη, K12
ij = −∫ ηe+1
ηe
ψiψj dη,
K22ij =
∫ ηe+1
ηe
[−(1 + R)
dψi
dη
dψj
dη+ f̄ ψi
dψj
dη− ψih̄ψj − Mψiψj
]dη,
K23ij =
∫ ηe+1
ηe
Rψi
dψj
dηdη,
K32ij = −
∫ ηe+1
ηe
Rψi
dψj
dηdη,
K33ij =
∫ ηe+1
ηe
[−
(1 + R
2
)dψi
dη
dψj
dη− 2Rψiψj + ψif̄
dψj
dη− ψih̄ψj
]dη,
K42ij =
∫ ηe+1
ηe
Pr Ec
[(1 + R)ψi
dh̄
dη
dψj
dη+ Mψih̄ψj
]dη,
K44ij =
∫ ηe+1
ηe
[−dψi
dη
dψj
dη+ Pr ψif̄
dψj
dη− 2 Pr ψih̄ψj
]dη,
K55ij =
∫ ηe+1
ηe
(−dψi
dη
dψj
dη+ Sc ψif̄
dψj
dη
)dη,
r1i = 0, r2
i = −(1 + R)
(ψi
dh
dη
)ηe+1
ηe
, r3i = −
(1 + R
2
) (ψi
dg
dη
)ηe+1
ηe
,
r4i = −
(ψi
dθ
dη
)ηe+1
ηe
, r5i = −
(ψi
dφ
dη
)ηe+1
ηe
,
where
f̄ =2∑
i=1
f̄iψi, h̄ =2∑
i=1
h̄iψi.
The system of equations after assembly of the elements is nonlinear, therefore an iterativescheme is used to solve it. The system is linearized by incorporating the functions f̄ and h̄,which are assumed to be known.
4. Chebyshev finite difference method
This approach requires the definition of grid points and it is applied to satisfy the differentialequations and the boundary conditions at these grid points. It can be regarded as a non-uniform finite difference scheme. The derivatives of the function f (x) at a point xj is a linear
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114 M. Aouadi
combination of the values of the function f (x) at the Gauss–Lobatto points xk = cos(kπ/L),where k = 0, 1, 2, . . . , L, and j is an integer 0 ≤ j ≤ L [26]. The application of the methodto boundary-value problems leads to algebraic systems. The method permits the applicationof an iterative method in order to solve the algebraic systems.
4.1 Chebyshev finite difference method for derivative calculation
The derivatives of the function f (x) at the points xk are given by [26]:
f (n)(xk) =L∑
j=0
d(n)k,j f (xj ), n = 1, 2, 3, (50)
where
d(1)k,j = 4γj
L
L∑n=0
n−1∑l=0
nγn
cl
Tn(xj )Tl(xk), (n + l) odd, k, j = 0, . . . , L, (51)
d(2)k,j = 2γj
L
L∑n=0
n−2∑l=0
nγn
cl
(n2 − l2)Tn(xj )Tl(xk), (n + l) even, k, j = 0, . . . , L, (52)
d(3)k,j = 4γj
L
L∑n=0
n−2∑l=0
l−1∑i=0
nγnl
clci
(n2 − l2)Tn(xj )Tl(xk), (n + l) even, (i + l) odd, (53)
k, j = 0, . . . , L, and γ0 = γL = 1/2, γj = 1 for j = 1, . . . , L − 1.
4.2 Chebyshev finite difference approximation for the governing equations
The domain is 0 ≤ η ≤ η∞, where η∞ is the edge of the boundary layer. Using the algebraicmapping
ξ = 2η
η∞− 1, (54)
the domain [0, η∞] is mapped into the computational domain [−1, 1] and the equations(10)–(15) are transformed into the following equations:
(1 + R)f ′′′ + η∞2
ff ′′ − η∞2
(f ′)2 + R(η∞
2
)2g′ − M
(η∞2
)2f ′ = 0, (55)
(1 + R
2
)g′′ − 2R
(η∞2
)2g − Rf ′′ + η∞
2fg′ − η∞
2f ′g = 0, (56)
θ ′′ + Prη∞2
f θ ′ − Pr η∞f ′θ + Pr Ec(1 + R)
(2
η∞
)2
(f ′′)2 + Pr EcM(f ′)2 = 0, (57)
φ′′ + Scη∞2
f φ′ = 0 (58)
f (−1) = 0, f ′(−1) = η∞2
, g(−1) = −s(η∞
2
)2, θ(−1) = 1, φ(−1) = 1, (59)
f ′(1) = 0, g(1) = 0, θ(1) = 0, φ(1) = 0, (60)
where the primes denote differentiation with respect to ξ . Thus by applying the ChFD approx-imation to the equations (55)–(60), we obtain the following Chebyshev finite difference
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Numerical methods for micropolar flow 115
equations:
(1 + R)
L∑j=0
d(3)k,j f (ξj ) + η∞
2f (ξk)
L∑j=0
d(2)k,j f (ξj ) − η∞
2
⎛⎝ L∑
j=0
d(1)k,j f (ξj )
⎞⎠
2
+(η∞
2
)2R
L∑j=0
d(1)k,j g(ξj ) − M
(η∞2
)2 L∑j=0
d(1)k,j f (ξj ) = 0, k = 2, 3, . . . , L − 1, (61)
(1 + R
2
) L∑j=0
d(2)k,j g(ξj ) − 2R
(η∞2
)2g(ξk) − R
L∑j=0
d(2)k,j f (ξj )
+ η∞2
f (ξk)
L∑j=0
d(1)k,j g(ξj ) − η∞
2g(ξk)
L∑j=0
d(1)k,j f (ξj ) = 0, k = 1, 2, . . . , L − 1, (62)
L∑j=0
d(2)k,j θ(ξj ) + η∞
2Pr f (ξj )
L∑j=0
d(1)k,j θ(ξj ) − η∞ Pr θ(ξj )
L∑j=0
d(1)k,j f (ξj )
+(
2
η∞
)2
Pr Ec(1 + R)
⎛⎝ L∑
j=0
d(2)k,j f (ξj )
⎞⎠
2
+ Pr EcM
⎛⎝ L∑
j=0
d(1)k,j f (ξj )
⎞⎠
2
= 0, (63)
k = 1, 2, . . . , L − 1.
L∑j=0
d(2)k,jφ(ξj ) + η∞
2Scf (ξj )
L∑j=0
d(1)k,jφ(ξj ) = 0, k = 1, 2, . . . , L − 1. (64)
The ChFD approximation for the derivative boundary conditions f ′(1) = 0, f ′(−1) = η∞/2are formed by
L∑j=0
d(1)N,j f (ξj ) = 0,
L∑j=0
d(1)0,j f (ξj ) = η∞
2. (65)
The system of nonlinear equations which contains 4L − 2 equations for the unknownsf (ξj ), j = 1, 2, 3, . . . , L, and g(ξj ), θ(ξj ), φ(ξj ), j = 1, 2, 3, . . . , L − 1, is solved byNewton’s method.
5. Results and discussion
The velocity, microrotation and temperature functions obtained using FEM and ChFDM areshown in figures 1–3, taking constant values of the Prandtl number (Pr = 0.7), the Schmidtnumber (Sc = 0.22), and the Eckert number (Ec = 0.2). The effects of other important para-meters, namely the magnetic field parameter M , the material parameter R and the surfaceparameter s, have also been studied for these functions.
Figures 1(a), 2(a) and 3(a) illustrate the variation of velocity, microrotation and temperaturefunctions for different M and R and for s = 0.As expected, f ′ and g decrease while θ increaseswith increasing magnetic field parameter M . First, as M increases, the Lorenz force, whichopposes the flow, also increases and leads to enhanced deceleration of the flow. Secondly, it isobserved that the Ohmic heating effect due to the electromagnetic work produces an increase
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116 M. Aouadi
Figure 1. Velocity distribution for (a) s = 0; (b) s = 0.5; (c) s = 1.
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Numerical methods for micropolar flow 117
Figure 2. Microrotation distribution for (a) s = 0; (b) s = 0.5; (c) s = 1.
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118 M. Aouadi
Figure 3. Temperature distribution for (a) s = 0; (b) s = 0.5; (c) s = 1.
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Numerical methods for micropolar flow 119
in the fluid temperature, and thus a decrease in the surface temperature gradient. Also, it isfound that the effect of viscous heating leads to an increase in the temperature; this effectis more pronounced in the presence of the magnetic field. The profiles of f ′, g and θ areshown in figures 1(a), 2(a) and 3(a), respectively, for various values of the material parameterR. With increasing material parameter R, it is shown that f ′ and g increase as expected butθ decreases. Of course, when the viscosity of the fluid decreases the angular velocity of theadditive increases.
Figures 1(b), 2(b) and 3(b) illustrate the variation of velocity, microrotation and temperaturefunctions for the surface parameter s = 0.5. The microrotation exhibits behaviour opposite tothat for s = 0, as can be seen by comparing figure 2(a) with figure 2(b), whereas the velocityand temperature distribution profiles remain the same.
For s = 1, the magnetic field has the same effect on the three functions as for s = 0, as canbe seen in figures 1(c), 2(c) and 3(c), but the material parameter has the opposite effect.
It is clear, from these figures, that the velocity decreases with an increase in the parameters. Near the boundary it remains positive whereas away from it becomes negative and thusretards the flow. The velocity values corresponding to the no-slip condition are maximum.The microrotation distribution continuously increases with increasing s; however, for s = 0(i.e. the no-slip condition) the profile is different from that for non-zero s. As expected,the microrotation effects are more dominant near the wall. As s increases, temperature alsoincreases—which is physically acceptable. The parameter s can thus be used effectively tocontrol velocity as well as temperature.
In figure 4, the temperature profiles θ are plotted for several values of Prandtl number Prstarting from Pr = 0.7 (which represents air at 20◦). This figure shows that θ decreases withincreasing Pr. This is due to the fact that the thermal boundary layer thickness decreases withincreasing Pr. Figure 4 also displays the temperature profiles θ for selected values of Eckertnumber Ec. It is obvious that θ increases with an increase in Ec. The effect of the Schmidtnumber Sc on the concentration distribution φ is shown in figure 5 (the values of Sc are chosenso that they represent the diffusing chemical species of most common interest in air, such as
Figure 4. Temperature distribution for different Ec and Pr.
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120 M. Aouadi
Figure 5. Effect of Schmidt number on concentration.
H2, H2O, NH3 and CO2, for which the values of Sc are 0.22, 0.6, 0.78 and 0.94, respectively).From this figure it is clear that φ decreases as Sc increases.
Tables 1 and 2 present a comparison between the numerical results for −f ′′(0), −g′(0)
and −θ ′(0) using FEM and ChFDM, and the analytical results for the special case R = 0for various values of M . In tables 1 and 2 we show that −f ′′(0) and −g′(0) increase as M
increases, whereas the values of −θ ′(0) and −φ′(0) decrease for increasing values of M .Tables 1 and 2 show that the values calculated by FEM and ChFDM are in excellent
agreement with those obtained from the exact solutions given by equations (16), (22), (28)and (34).
Table 1. Values of −f ′′(0) for various values of M , R = 0, Sc = 0.22, Ec = 0.02, Pr = 0.7, s = 1.
−f ′′(0) −g′(0)
M FEM ChFDM equation (16) FEM ChFDM equation (22)
0 1.000483 1.000483 1.000000 1.000483 1.000483 1.0000000.5 1.224742 1.224712 1.224776 1.152392 1.152356 1.1523611 1.414121 1.412166 1.142130 1.261628 1.261668 1.2616661.5 1.581141 1.581139 1.581138 1.344946 1.344967 1.3449302 1.732057 1.732051 1.732050 1.410726 1.410746 1.410784
Table 2. Values of −θ ′(0) and −φ′(0) for various values of M , R = 0, Sc = 0.22, Ec = 0.02, Pr = 0.7, s = 1.
−θ ′(0) −φ′(0)
M FEM ChFDM equation (28) FEM ChFDM equation (34)
0 1.064153 1.064234 1.064288 0.183578 0.183545 0.1834120.5 0.996628 0.996642 0.996632 0.157972 0.157949 0.1579761 0.940419 0.940473 0.940490 0.140844 0.140867 0.1408511.5 0.892573 0.892538 0.892505 0.128322 0.128382 0.1283122 0.850795 0.850738 0.850732 0.118620 0.118633 0.118619
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Numerical methods for micropolar flow 121
6. Conclusions
Owing to the complicated nature of the governing equations for magneto-micropolar fluidflowing past a stretching surface with Ohmic heating and viscous dissipation involving heatand mass transfer, work done in this field by means of the finite element method or theChebyshev finite difference method are unfortunately limited in number. Numerical resultsfrom both methods have been compared with those obtained by the closed-form solution of theparticular case R = 0. The finite element method and the Chebyshev finite difference methodprovide a quite successfully approach to dealing with such problems, and give numericalsolutions without any assumed restrictions on the actual physical quantities that appear in thegoverning equations.
According to the description presented in the previous section, we come to the followingconclusions.
(1) The numerical results indicate that the effect of the magnetic field M and the materialparameter R on the microrotation depend on the boundary conditions. For the no-slipcondition (s = 0), an increase in the magnetic parameter M gives a decrease in the valuesof the velocity and microrotation, but an increase in the values of the temperature. Thematerial parameter R has an effect opposite to that of the magnetic field on the threefunctions. For the rigid condition (s �= 0), M and R have the same effect on the velocityand temperature, but an opposite effect on the microrotation as compared to the no-slipcondition (s = 0).
(2) The velocity decreases with increasing s, whereas the microrotation and the temperaturebehave in precisely the opposite way to increasing s.
(3) The temperature decreases as the Prandtl number increases, but increases as the Eckertnumber increases. The concentration decreases as the Schmidt number increases.
(4) Good agreement between the numerical results of both methods, together with excellentagreement with the closed-form solution, ensures the reliability of using linear shapefunctions in the finite element method.
(5) The methods presented in this paper should prove useful for researchers working on thedevelopment of computational methods in applied mathematics and engineering sciences.
Acknowledgements
The author would like to thank the reviewers for their valuable comments, which thoroughlyimproved the paper.
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