MHD FREE CONVECTION FLOW WITH FRACTIONAL HEAT CONDUCTION ...

MAGNETOHYDRODYNAMICS Vol. 48 (2012), No. 4, pp. 587–606

MHD FREE CONVECTION FLOW WITHFRACTIONAL HEAT CONDUCTION LAW

M.A. Ezzat 1, A.A. El-Bary 2

1 Department of Mathematics, Faculty of Education, Alexandria, Egypt2 Arab Academy for Science and Technology, P.O. Box 1029, Alexandria, Egypt

e-Mail: [email protected]

A new mathematical model of MHD theory has been constructed in the context of anew consideration of heat conduction with a time-fractional derivative of the order α(0 < α ≤ 1) and a time-fractional integral of the order υ (0 < υ ≤ 2). This modelis applied to MHD free convection flow of a viscous conducting fluid past an infinitesurface with heat sources. Laplace transforms and state-space techniques [1] are usedto obtain the general solution for any set of boundary conditions. According to thenumerical results and their graphs, a conclusion about the new theory has been made.Some comparisons are shown in figures to estimate the effects of the fractional orderparameters α, υ on all studied fields for different theories.

Nomenclature.

ρ densityx, y, z space coordinatest timeT temperatureH magnetic field intensity vectorB magnetic induction vectorE electric field intensity vectorJ conduction electric density vectorq heat flux vectorH0 constant component of magnetic fieldHw temperature of the plane surfaceH∞ temperature of the fluid away from the plane surfaceT0 = Tw − T∞ reference temperatureβ0 coefficient of volume expansionCp specific heat at constant pressurec concentrationγ diffusion conductivityGr Grashof numberPr Prandtl numberκ thermal conductivityσ0 electrical conductivityμ0 magnetic permeabilityμ viscosityλ = μ/ρ kinematics viscosityQ intensity of the applied heat sourceα fractional derivative parameterυ fractional integral parameterτ0 relaxation timeM = λσ0B

20/(ρU2) magnetic number

587

M.A.Ezzat, A.A.El-Bary

Introduction. The flow of fluids over boundaries has many applications suchas boundary-layer control. The study of unsteady boundary layers owes its impor-tance to the fact that all boundary layers, which occur in real life are, in a sense,unsteady. In recent years, the requirements of modern technology have stimulatedinteresting fluid flow studies, which involve the intersection of several phenomena.One such study is related to the effects of free convection flow through a porousmedium, which plays an important role in agriculture, engineering, petroleum in-dustries, and in heat transfer. The flow in the boundary layer on a continuoussemi-infinite sheet moving steadily through on other wise quiescent fluid environ-ment was first studied theoretically by Sakiadis [2] and experimentally verified byTsou et al. [3].

The effects of a magnetic field on the laminar flow of an incompressible electri-cally conducting fluid are important problems that have been studied frequentlyby many researchers in relation to MHD power generation and boundary-layercontrol. Pathak and Choudhary [4] considered the MHD behaviour of boundarylayer flow along a flat plate with a pressure gradient. Ezzat and Abd-Elall [5]studied the flow of hydromagnetic fluctuating boundary layer through a porousmedium saturated with a viscoelastic fluid bounded by an infinite non-magneticvertical plate in the presence of a transverse magnetic field.

The study of free convection flow of an electrically conducting fluid past aninfinite surface plane under the influence of a magnetic field has attracted theinterest of many authors [6-11], and increasing attention is being devoted to theinteraction between the magnetic field and the fluid-flow field in a thermofluidowing to its numerous applications in science. Usually, in these investigations,the heat equation under consideration is taken as a heat equation based on theclassical Fourier law not generalized. This attitude is justified in many situationssince the solutions obtained, using any of these equations, quantitatively differ alittle. However, when short-time effects are considered, the full-generalized systemof equations has to be used, so a great deal of accuracy is not lost.

In the literature concerning thermal effects in continuum mechanics, severalparabolic and hyperbolic theories describing heat conduction have been developed.The hyperbolic theories are also called theories of second sound, and there the flowof heat is modelled with a finite propagation speed, in contrast to the classicalmodel based on the Fourier law leading to infinite propagation speed of heat sig-nals. A review of these theories is presented in the articles by Chandrasekharaiah[12], Ezzat and El-Bary [13].

The modification of the heat conduction equation from diffusive to a wavetype may be affected either by a microscopic consideration of the heat transportphenomenon or in a phenomenological way by modifying the classical Fourier lawof heat conduction. The first is due to Cattaneo [14], who derived a wave-typeheat equation by postulating a new law of heat conduction to replace the classicalFourier law. Puri and Kythe [15] investigated the effects of using the Maxwell—Cattaneo model in Stokes second problem for a viscous fluid, and they note thatthe non-dimensional thermal relaxation time defined as τ0 = C Pr, where C andPr are the Cattaneo and the Prandtl number, respectively, is of the order (10)−2.

Among the few works devoted to applications of the Cattaneo theory to MHD,we can refer to the works of Ezzat et al. [16–18] and Ezzat [19–21].

Differential equations of fractional order have been the focus of many studiesdue to their frequent appearance in various applications in fluid mechanics, vis-coelasticity, biology, physics and engineering. The most important advantage offractional differential equations in these and other applications is their non-localproperty. It is well known that the integer order differential operator is a local

588

MHD free convection flow with fractional heat conduction law

operator, but the fractional order differential operator is non-local. This meansthat the next state of a system depends not only upon its current state, but alsoupon all of its historical states. This is more realistic and it is one reason whyfractional calculus has become popular [22–24].

Although the tools of fractional calculus have been available and applicableto various fields of study, the investigation of the theory of fractional differentialequations has been started quite recently [22]. The differential equations, involvingRiemann–Liouville differential operators of the fractional order 0 < α < 1, appearto be important in modelling several physical phenomena [25] and, therefore, seemto deserve an independent study of their theory along with to the well-knowntheory of ordinary differential equations.

Fractional calculus has been used successfully to modify many existing modelsof physical processes. One can state that the whole theory of fractional deriva-tives and integrals was established in the second half of the 19th century. The firstapplication of fractional derivatives was given by Abel, who applied fractional cal-culus in the solution of an integral equation, which arises in the formulation of thetautochrone problem. The generalization of the concept of derivative and integralto a non-integer order has been subjected to several approaches, and some variousalternative definitions of fractional derivatives appeared [26–30]. In the last fewyears, fractional calculus was applied successfully in various areas to modify manyexisting models of physical processes, e.g., chemistry, biology, modelling and iden-tification, electronics, wave propagation and viscoelasticity [31–35]. One can referto Padlubny [23] for a survey of applications of fractional calculus. Some applica-tions of fractional calculus to various problems of viscoelastic fluid mechanics arereviewed in the literature [36–40]. In most of these investigations, the effect of thethermal state in the fluid is not considered.

Recently, a considerable research effort has been initiated to study anomalousdiffusion, which is characterized by the time fractional diffusion–wave equation byKimmich [41]:

ρc = γIυ∇2c, 0 < υ ≤ 2, (1)

where the notion Iυ is the Riemann–Liouville fractional integral introduced asa natural generalization of the well-known n-fold repeated integral written in aconvolution-type form as [42]

Iυf(t) =1

Γ(υ)

∫ t

0

(t − ξ)υ−1f(ξ)dξ

I0f(t) = f(t)

⎫⎪⎬⎪⎭ 0 < υ ≤ 0, (2)

where Γ(υ) is the gamma function.According to Kimmich [41], Eq. (1) describes different cases of diffusion, where

0 < υ < 0 corresponds to weak diffusion (sub-diffusion), υ = 1 corresponds tonormal diffusion, 0 < υ < 2 corresponds to strong diffusion (super diffusion),and υ = 2 corresponds to ballistic diffusion. It should be noted that the term“diffusion” is often used in a more generalized sense including various transportphenomena. Eq. (1) is a mathematical model of a wide range of important physicalphenomena, for example, the sub-diffusive transport occurs in widely differentsystems ranging from dielectrics and semiconductors through polymers to fractals,glasses, porous, and random media. Super diffusion is comparatively rare and hasbeen observed in porous glasses, polymer chain, biological systems, transport oforganic molecules and atomic clusters on surface [43]. One might expect theanomalous heat conduction in media, where anomalous diffusion is observed.

589


Fujita [44, 45] considered the constitutive equation for the heat flux in theform

q = −κIυ−1∇T, 1 < υ ≤ 2, (3)

Povstenko [43] used the Caputo heat wave equation to define the fractionalheat conduction equation in the form

q = −κIυ−1∇T, 0 < υ ≤ 2, (4)

Cattaneo [14] introduced a law of heat conduction to replace the classicalFourier law in the form (C model)

q + τ0∂q∂t

= −κ∇T. (5)

Sherief et al. [46] introduced a formula of heat conduction taking into accountconsideration (7)

q + τ0∂αq∂tα

= −κ∇T, 0 < α ≤ 1, (6)

where

∂α

∂tαf(y, t) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

f(y, t) − f(y, 0) α → 0,

Iα−1 ∂f(y, t)∂t

0 < α < 1,

∂f(y, t)∂t

α = 1,

(7)

and proved a uniqueness theorem and derived a reciprocity relation and a varia-tional principle.

In the limit as tends to unity, Eq. (6) reduces to the well-known Cattaneo lawused by Lord and Shulman [47] to derive the equation of the generalized theoryof thermoelasticity with one relaxation time. It is known from Lebon et al. [48]and Jou et al. [49] that the classical entropy derived using this law, instead ofbeing monotonically increasing, behaves in an oscillatory way. Strictly speaking,this result is not incompatible with the Clausius formulation of the second law,which states that the entropy of the final equilibrium state must be higher thanthe entropy of the initial equilibrium state. However, the non-monotonic behaviorof entropy is in contradiction to the local equilibrium formulation of the secondlaw, which requires that the entropy production must be positive everywhere atany time (Lebon et al. [48]). During the last two decades, this has become thesubject of many research papers and resulted in the introduction of what nowadaysis known as extended irreversible thermodynamics. A review can be found in [49].

Youssef [50] introduced another formula of heat conduction taking into ac-count considerations (3)–(5)

q + τ0∂q∂t

= −κIυ−1∇T, 0 < υ ≤ 2, (8)

and its uniqueness has been proved.The main objectives of this work is firstly to work out a mathematical model

of MHD boundary layer equations for a conducting thermofluid in the context ofa new consideration of heat conduction with time fractional orders and thermalrelaxation time, and, secondly, to derive governing decoupled equations in theframe of the boundary layer theory applied to free convection flow of fluid pastan infinite plate with a heat source distribution in the presence of a transversemagnetic field normal to the flow.

590


The solution is obtained using a state space approach [1]. With this approach,the governing equations are written in the matrix form using a state vector thatconsists of Laplace transforms in time of the temperature, velocity component andtheir gradients. Their integration subjected to zero initial conditions is carried outby means of a matrix exponential method. Influence functions in the Laplace trans-form domain are explicitly developed. The Laplace transforms are obtained usingthe complex inversion formula of the transform together with Fourier expansiontechniques [51]. The effects of physical parameters on various momentum and heattransfer characteristics are discussed in detail and represented graphically.

1. Heat transfer equation with fractional derivative and integral.The conventional thermal MHD is based on the principles of the classical theoryof heat conductivity, in particular, on the classical Fourier law, which relates theheat flux vector q to the temperature gradient [43]

q = −κIυ−1∇T, 0 < υ ≤ 2, (9)

where V = {u, v, w} is the fluid vector velocity.The energy equation in terms of the heat conduction vector q reads

ρCp =DDt

T = −∇ · q + Φ, (10)

where ρ is the density of the fluid, and Cp is the specific heat at constant pressure,and

Φ = μ

[2

(∂u

∂x

)2

+ 2(

∂v

∂y

)2

+ 2(

∂w

∂z

)2

+

(∂w

∂y+

∂v

∂z

)2

+(

∂u

∂z+

∂w

∂x

)2

+(

∂v

∂x+

∂u

∂y

)2]

, (11)

is the internal heat due to viscous stresses, and the material operator D/Dt isdefined as

DDt

=∂

∂t+ (V · ∇). (12)

During the past three decades, non-classical thermofluid theories, in which theFourier law (9) and heat equation (10) are replaced by more general equations,have been formulated. The first well-known generalization of such type [14]

q + τ0∂q∂t

= −κ∇T (13)

leads to the hyperbolic-type heat transport equation[∂

∂t+ (V · ∇)

] [T + τ0

∂T

∂t

]=

κ

ρCp∇2T +

1ρCp

(Φ + τ0

∂Φ∂t

). (14)

In this paper, we consider the theory developed by taking a new fractionalTaylor’s series of the time-fractional order α [52] to expand q(x, t+τ0) and retainingterms up to the order α in the thermal relaxation time τ0. One then obtains anon-Fourier formula of heat conduction including the current density effect, inwhich the evolution equation contains a fractional order derivative with respect totime. That is [53–55]

q(x, t + τ0) = q(x, t) +τα0

α!∂αq∂tα

, 0 < α ≤ 1. (15)

591


From the mathematical viewpoint, the Fourier law (9) in the theory of generalizedfractional heat conduction is given by

q(x, t) +τα0

α!∂αq∂tα

= −κIυ−1∇T, 0 < α ≤ 1, 0 < υ ≤ 2. (16)

Taking the partial time derivative of the fraction order α of Eq. (10) yields[56]

ρCpDDt

(∂αT

∂tα

)= −∇ ·

(∂αq∂tα

)+

∂αΦ∂tα

, 0 < α ≤ 1. (17)

By multiplying Eq. (17) by τα0 /α! and adding to Eq. (10), we have

ρCpDDt

(T +

τα0

α!∂αT

∂tα

)= −∇ ·

(q +

τα0

α!∂αq∂tα

)+

(Φ +

τα0

α!∂αΦ∂tα

),

0 < α ≤ 1, 0 < υ ≤ 2.

(18)

Substituting from Eq. (16) yields

ρCpDDt

(T +

τα0

α!∂αT

∂tα

)= κIυ−1∇2T +

(Φ +

τα0

α!∂αΦ∂tα

),

0 < α ≤ 1, 0 < υ ≤ 2.

(19)

Taking into account the definition of D/Dt from Eq. (12), we arrive at[∂

∂t+ (V · ∇)

] [T +

τα0

α!∂αT

∂tα

]=

κ

ρCpIυ−1∇2T +

1ρCp

(Φ +

τα0

α!∂αΦ∂tα

),

0 < α ≤ 1, 0 < υ ≤ 2.

(20)

Eq. (20) is a generalized energy equation with fractional derivatives and inte-grals taking into account the relaxation time τ0. Some theories of heat conductionlaw follow as limit cases for different values of the parameters α, υ and τ0.

2. Limiting cases.(i) In the theory of hydrodynamic free convection flow: the heat equation (20)

in the limiting case and is transformed by Ezzat in [9].(ii) In the theory of MHD free convection flow: (1) The heat equation (20) in

the limiting case and is transformed by Ezzat et al. in [10] and [11]. (2) The heatequation (20) in the limiting case and is transformed by Ezzat in [19–21].

(iii) In the theory of fractional MHD boundary layer flow: the heat equation(20) in the limiting case and is transformed by Ezzat in [53–55].

3. Free convection flow with heat source distribution. Let us considerthe laminar flow of an incompressible conducting fluid past a non-conducting half-space y ≥ 0, taking the positive y-axis of the Cartesian coordinate system inthe upward direction and the fluid flows through the half-space y > 0 above andin contact with the plane surface occupying the xz-plane. A constant magneticfield of strength H0 acts in the z-direction. The electric current induced dueto the motion of the fluid caused by the buoyancy forces does not distort theapplied magnetic field. The previous assumption is reasonably true if the magneticReynolds number of the flow (Rm = UL0σ0μ0) is assumed to be small, which is thecase in many aerodynamic applications, where rather low velocities and electricalconductivities are involved. All the fluid properties are assumed constant exceptthat the influence of the density variation with temperature is considered onlyin the body force term. The influence of the density variations in other terms

592


of the momentum and the energy equations and the variations of the expansioncoefficient with temperature are considered negligible. Under these conditions,no flow occurs in the y- and z-directions, and all the considered functions at agiven point in the half-space depend only on its y-coordinate and time t, andV = (u, 0, 0) is the vector velocity of the fluid. This is the well-known Boussinesqapproximation.

With the above assumptions, we have:1 – the magnetic induction has one non-vanishing component

Bz = μ0H0 = B0 (const);

2 – the Lorentz force F = J ∧ B has one component in the x-direction

Fx = −σ0B20u; (21)

3 – the constitutive relation

ρ∞ − ρ = ρβ0(T − T∞);

4 – the equation of motion

∂u

∂t= gβ0(T − T∞) + λ

∂2u

∂y2− σ0B

20

ρu; (22)

5 – the generalized energy equation with fractional orders

ρCp∂

∂t

(T +

τα0

α!∂αT

∂tα

)= κIυ−1 ∂2T

∂t2+ Q +

τα0

α!∂αQ

∂tα, 0 < α ≤ 1, 0 < υ ≤ 2. (23)

Let us introduce the following non-dimensional variables:

y∗ =U

λy, t∗ =

U2

λt, τ∗

0 =U2

λτ0, u∗ =

u

U, pr =

Cpμ

κ, M =

λσ0B20

ρU2,

q∗ =λ

κT0Uq, Θ =

T − T∞T0

, Q∗ =λ2

κU2T0Q, Gr =

λβ0g(T − T∞)U3

.

(24)

Eqs. (22) and (23) are reduce to non-dimensional equations

∂u

∂t= GrΘ +

∂2u

∂y2− Mu, (25)

pr

(∂

∂t+

τα0

α!∂α+1

∂tα+1

)Θ = Iυ−1 ∂2Θ

∂y2+ Q +

τα0

α!∂αQ

∂tα. (26)

From now on, we shall consider a heat source of the form Q = Q0δ(y)H(t), whereQ0 is a constant.

To simplify the algebra, only problems with zero initial conditions are consid-ered. Applying the Laplace transform defined by the formulas [57]

L {g(t)} = g(s) =∫ ∞

0

e−stg(t)dt,

L {Dng(t)} = snL {g(t)}L {Ing(t)} =

1sn

L {g(t)}

⎫⎬⎭ s > 0,

(27)

593


at both sides of Eqs. (25) and (26) and writing the resulting equations in thematrix form result in

ddy

⎧⎪⎪⎨⎪⎪⎩

Θ(y, s)u(y, s)Θ′(y, s)u′(y, s)

⎫⎪⎪⎬⎪⎪⎭ =

⎧⎪⎪⎨⎪⎪⎩

0 0 1 00 0 0 1C 0 0 0

−Gr s + M 0 0

⎫⎪⎪⎬⎪⎪⎭

⎧⎪⎪⎨⎪⎪⎩

Θ(y, s)u(y, s)Θ′(y, s)u′(y, s)

⎫⎪⎪⎬⎪⎪⎭ − Q0βδ(y)

⎧⎪⎪⎨⎪⎪⎩

0010

⎫⎪⎪⎬⎪⎪⎭ (28)

where

C = prsυ

(1 +

τα0

α!sα

)and β = sυ−1

(1 +

τα0

α!sα

).

Eq. (28) can be written in a constricted form as

G′(y, s) = AG(y, s) + B(y, s), (29)

where G(y, s) denotes the state vector in the transform domain, whose componentsconsist of transformed temperature, velocity and induced magnetic field as well asof their gradients.

In order to solve system (29), we need first to find the form of the matrixexp{A(s)y}. The characteristic equation of the matrix A(s)y has the form

k4 − (s + M + C)k2 + (s + M)C = 0, (30)

where k is a characteristic root. The Cayley–Hamilton theorem states that thematrix A satisfies its own characteristic equation in the matrix sense. Therefore,it follows that

A4 − (s + M + C)A2 + (s + M)CI = 0, (31)

Eq. (31) shows that A4 and all higher powers of A can be expressed in termsof A3, A2, A and I, the unit matrix of order 4. The matrix exponential can nowbe presented in the form

exp{Ay} = a0(y, s)I + a1(y, s)A + a2(y, s)A2 + a3(y, s)A3. (32)

The scalar coefficients of Eq. (32) are now evaluated by replacing the matrixA by its characteristic roots ±k1 and ±k1, which are the roots of the biquadraticEq. (30), such that

k21 + k2

2 = C + s + M, (33a)

k21k2

2 = C(s + M). (33b)

This leads to a system of equations

exp{±k1 · y} = a0 ± a1k1 + a2k21 ± a3k

31 , (34a)

exp{±k2 · y} = a0 ± a1k2 + a2k22 ± a3k

32 . (34b)

By solving the system of linear Eqs. (34), we can determine a0–a3 (see Ap-pendix A). By substituting the parameters a0–a3 into Eq. (32), by computing A2

and A3 using Eqs. (33a) and (33b), after some lengthy algebraic manipulations,one can obtain

exp{Ay} = L (y, s) = {�ij(y, s)}, i, j = 1, 2, 3, 4 (35)

whose elements �ij(y, s) are given in Appendix B.

594


In the actual physical problem, the space is divided into two regions, accord-ingly, as y ≥ 0 or y < 0; inside the region 0 ≤ y < ∞ the positive exponentialterms, not bounded at infinity, must be suppressed. Thus, for y ≥ 0, we mustreplace each sh(ky) by −(1/2) exp(−ky)and each ch(ky) by (1/2) exp(−ky). Inthe region y ≤ 0, the negative exponentials are suppressed instead.

We shall now proceed to the solution of the problem for the region y ≥ 0.The solution for the other region is obtained by replacing each y by y. The formalsolution of Eq. (29) can be expressed as

G(y, s) = exp{A(y, s)y}[G(0, s) +

∫ y

0

exp{A(s)z}B(z, s)dz

]. (36)

By evaluating the integral in Eq. (36) using the integral properties of the Diracdelta function, we obtain

G(y, s) = L (y, s)[G(0, s) + ξ(s)

]. (37)

where

ξ(s) = −Q0β

4s

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

1k1

Grk1k2(k1 + k2)

10

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

.

Eq. (37) expresses the solution of the problem in the Laplace transform do-main in terms of the vector ξ(s), representing the applied heat source, and vectorG(0, s), representing the conditions at the plane source of heat. In order to eval-uate the components of this vector, we note first that, due to the symmetry ofthe problem, the temperature is symmetric about y, while the velocity is anti-symmetric. It thus follows that

u(0, t) = 0 or u(0, s) = 0. (38)

The Gauss divergence theorem will be used now to obtain the thermal con-dition at the plane source. We consider a short cylinder of unit base, whose axisis perpendicular to the plane source of heat and whose bases lie on its oppositesides.

Taking limits as the height of the cylinder tends to zero and noting thatthere is no heat flux through the lateral surface; upon using the symmetry of thetemperature field, we get

q(0, t) =Q0

2H(t) or q(0, s) =

Q0

2s. (39)

Using the Fourier law of heat conduction in the non-dimensional form, namely,

q = − s1−υ

1 +τα0

α!sα

Θ′ (40)

we derive the condition

Θ′(0, s) = −βQ0

2s. (41)

595


Eqs. (38) and (41) give two components of the vector G(0, s). In order to obtainthe remaining two components, we substitute y = 0 at both sides of Eq. (37) toderive a system of linear equations, whose solution gives

Θ(0, s) =βQ0

2sk1, (42)

u′(0, s) =GrβQ0

2sk1(k1 + k2). (43)

By substituting the values from Eqs. (38), (42) and (43) into the right-handside of Eq. (37) and performing the necessary matrix operations, we have

Θ(y, s) =βQ0

2sk1e±k1y, (44)

u′(y, s) = ± GrβQ0

2sk1(k21 − k2

2)[e±k1y − e±k2y

]. (45)

In the above equations, the upper (plus) sign indicates the solution in the regiony < 0, while the lower (minus) sign indicates the region y ≥ 0, respectively.

4. Numerical inversion of the Laplace transforms. To invert the Laplacetransforms in the above equations, we shall use a numerical technique based onthe Fourier expansion of functions by Honig and Hirdes [51].

Let g(s) be the Laplace transform of a given function g(t). The inversionformula of the Laplace transforms states that

g(t) =1

2πi

d+i∞∫d−i∞

estg(s) ds,

where d is an arbitrary positive constant greater than all the real parts of thesingularities of g(s). Taking s = d + iy, we get

g(t) =edt

2π

+∞∫−∞

e−ity g(d + iy) dy.

This integral can be approximated by

g(t) =edt

2π

+∞∑k=−∞

e−iktΔy g(d + ikΔy)Δy.

Taking Δy = π/t1, we obtain

g(t) =edt

t1

[12g(d) + �e

{+∞∑k=1

e−iktπ/t1 g(d + ikπ/t1)

}].

For numerical purposes, it is approximated by the function

gN (t) =edt

t1

[12g(d) + �e

{N∑

k=1

e−iktπ/t1 g(d + ikπ/t1)

}], (46)

where N is a sufficiently large integer chosen such that

edt

t1�e

{eiNtπ/t1 g(d + iNπ/t1)

}< η,

596


where η is a persecuted small positive number, which corresponds to the degree ofaccuracy to be achieved.

Formula (46) is the numerical inversion formula valid for 0 ≤ t ≤ 2t1 [51]. Inparticular, we choose t = t1 to get

gN (t) =edt

t

[12g(d) + �e

{N∑

k=1

(−1)kg(d + ikπ/t)

}]. (47)

5. Numerical results and discussion. The study of the effect of the frac-tional orders α and υ on the fluid flow over the boundaries with the heat sourcedistribution in the presence of a magnetic field has been carried out in the pre-ceding sections. The computations were performed for t = 1 and Q0 = 1.0. Thenumerical technique outlined above was used to obtain the components of temper-ature Θ and velocity u. The results are shown in Figs. 1–9. The graphs show thecurves predicted by different values of the fractional orders α within a wide range0 < α ≤ 1 and υ within a wide range 0 < υ ≤ 2. Comparisons are made with theresults predicted by some previous theories.

Hence, we conclude with the following emphases:• In all figures, it is noted that the fractional orders α and υ have a significant

effect on all fields.• It is noted that all the waves reach the steady state depending on the value

of the fractional orders α and υ.• Fig. 1 illustrates the Fujita theory [44, 45]. Note from this figure that the

temperature has only two distributions: one in the wide range 1 < υ ≤ 2, which isa continuous function, and the other, when υ = 2, is a continuous function exceptat the points y = ±1.5.

• In the frames of Povstenko theory [43], the temperature has three distribu-tions in the wide range 0 < υ ≤ 2. We learn from Fig. 2 that the temperature is acontinuous function in the wide range 0 < υ < 1, and the solution of Fujita theorycan be obtained from the Povstenko theory as a special case.

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

-4 -3 -2 -1 0 1 2 3 4 5

Tem

per

atu

reΘ

Distance y

τ0 = 0.0, pr = 0.71, Gr = 4.0

1 < υ ≤ 2

υ = 2.0

Fig. 1. A plot of the temperature against the distance in the frames of Fujita theory.

597


0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

-5 -4 -3 -2 -1 0 1 2 3 4 5

Tem

per

atu

reΘ

Distance y

τ0 = 0.0, pr = 0.71, Gr = 4.01 < υ < 1

1 ≤ υ < 2

υ = 2.0

Fig. 2. A plot of the temperature against the distance in the frames of Povstenkotheory.

0

0,1

0,2

0,3

0,4

0,5

0,6

-3 -2 -1 0 1 2 3

Tem

per

atu

reΘ

Distance y

τ0 = 0.0, pr = 0.71, Gr = 4.0

0 < υ < 1

α = 1.0, 0.7, 0.5, 0.3, 0.1

Fig. 3. A plot of the temperature against the distance in the frames of the new theoryin the wide range 0 < υ < 1.

• In the frames of the new theory, Figs. 3 and 4 display the temperature dis-tribution for a wide range of y (−3 ≤ y ≤ 3) at different values of the differentialfractional order 0 < α ≤ 1 in the limiting value of 0 < υ < 1 (weak conductivity)and 1 ≤ υ < 2 (super conductivity) [41]. We note that the increase of the param-eter α value causes a decrease in temperature and the temperature increment isa continuous function that means that the particles easily transport the heat toother particles, and this makes the decreasing rate of temperature greater thanthe other ones.

• Fig. 5 presents the temperature distribution in the limiting case υ = 2 (bal-listic conductivity) with different values of the fractional order α (α = 1, 0.7, 0.5, 0.3).We stress that the temperature increment is a continuous function at all pointsexcept at y = ±1.35.

598


0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

-3 -2 -1 0 1 2 3

Tem

per

atu

reΘ

Distance y

τ0 = 0.0, pr = 0.71, Gr = 4.0

1 ≤ υ < 2

α = 1.0, 0.7, 0.5, 0.3, 0.1

Fig. 4. A plot of the temperature against the distance in the frames of the new theoryin the wide range 1 ≤ υ < 2.

0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

0,16

-5 -4 -3 -2 -1 0 1 2 3 4 5

Tem

per

atu

reΘ

Distance y

τ0 = 0.0, pr = 0.71, Gr = 4.0

υ = 2.0

α = 1.0, 0.7, 0.5, 0.3

Fig. 5. A plot of the temperature against the distance in the frames of the new theoryfor υ = 2.0.

• Fig. 6 shows the temperature distribution according to the new theory, whichis compared with some previous theories. The temperature rises to the maximumvalue at the point y ≈ 0.682 for the classical theory result (τ0 = 0, υ = 1.0) [9],y ≈ 0.487 for the Cattaneo theory result (τ0 = 0.02, α = 1.0, υ = 1.0) [14], andy ≈ 0.570 for the theory of Sherief et al. result (τ0 = 0.02, α = 0.5, υ = 1.0) [46],while at the point y ≈ 0.6 for the new theory result (τ0 = 0.02, α = 0.5, υ = 1.5).

• In Fig. 7, the velocity component has been affected by the value of thefractional order α. We note that the velocity component is a continuous functionat all points and increases with a very great rate until the maximum and minimumvalues at y ≈ ±0.75, respectively.

• Fig. 8 display the velocity distribution according to the new theory for allpossible values of the integral fractional order υ, and it is emphasized that all thewaves reach the steady state depending on the value υ.

599


0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

-4 -3 -2 -1 0 1 2 3 4

Tem

per

atu

reΘ

Distance y

pr = 0.71, Gr = 4.0

τ0 = 0.0, υ = 1.0 [9]

τ0 = 0.02, α = υ = 1.0 [14]

τ0 = 0.02, α = 0.5,υ = 1.0 [46]

τ0 = 0.02, α = 0.5,υ = 1.5 [New model]

Fig. 6. A plot of the temperature against the distance for different theories.

-0,1

-0,08

-0,06

-0,04

-0,02

0

0,02

0,04

0,06

0,08

0,1

-5 -4 -3 -2 -1 0 1 2 3 4 5

Vel

oci

tyu

Distance y

Gr = 4.0, M = 1.0, α = 0.5α = 1.0, 0.7, 0.5, 0.3, 0.1

Fig. 7. A plot of the velocity against the distance for different values of the fractionalorder α.

• Fig. 9 presents the velocity distribution according to the new theory (τ0 =0.02, α = 0.5, υ = 1.5), which is compared with the classical case (τ0 = 0.0,υ = 1.0) [9] at two values of the magnetic number M. We learn from this figurethat the magnetic field acts to decrease the velocity component. This is mainlydue to the fact that the magnetic field corresponds to a term signifying a positiveforce, which tends to accelerate the fluid particles.

6. Conclusion. The main goal of this work is to introduce a new mathe-matical model for the Fourier law of heat conduction with time-fractional orders.According to this new theory, we have to construct a new classification for ma-terials according to their fractional parameters α and υ, where these parametersbecome new indicators of their ability to conduct heat in a conducting medium.

600


-0,15

-0,1

-0,05

0

0,05

0,1

0,15

-5 -4 -3 -2 -1 0 1 2 3 4 5

Vel

oci

tyu

Distance y

Gr = 4.0, M = 1.0, α = 0.5

1 < υ < 1

1 ≤ υ < 2

υ = 2.0

Fig. 8. A plot of the velocity against the distance for different values of the fractionalorder υ.

-0,1

-0,08

-0,06

-0,04

-0,02

0

0,02

0,04

0,06

0,08

0,1

-4 -3 -2 -1 0 1 2 3 4

Vel

oci

tyu

Distance y

Gr = 1.0, pr = 1.0, Q0 = 1.0

τ = 0.0, υ = 1.0 [9]

τ = 0.02, α = 0.5

υ = 1.5 [New model]

M = 10.0 M = 1.0

Fig. 9. A plot of the velocity against the distance for two theories.

The result provides a motivation to investigate conducting fluids as a new class ofapplicable thermofluids.

In this work, the method of direct integration by means of the matrix ex-ponential, which is a standard approach in modern control theory and has beendeveloped in detail in, e.g. [1], is introduced in the field of MHD and applied to freeconvection problems. This method gives exact solutions in the Laplace transformdomain without any assumed restrictions on either the applied magnetic field orthe velocity and temperature distributions.

The method used in the present work is applicable to a wide range of problems.It can be applied to problems that are described by linear system equations [53–55].

601


The same approach has been used quite successfully when dealing with problemsin generalized magneto-thermoelasticity theory [57-62].

REFERENCES

[1] M.A.Ezzat. State space approach to solids and fluids. Can. J. Phys. Rev.,vol. 86 (2008), pp. 1241-1250.

[2] B.C. Sakiadis. Boundary layer behaviour on continuous solid surface.A. I. Ch. E. J., vol. 17 (1961) ,pp. 2628, pp. 221–225.

[3] F.Tsou, E.M. Sparrow and R.Goldstein. Flow and heat transfer inthe boundary layer on a continuous moving surface. Int. J. Heat and MassTransfer, vol. 10 (1967), pp. 219–235.

[4] R.P.Pathak and R.C.Choudhary. Unsteady hydromagnetic boundarylayer on infinite plate in the presence of transverse magnetic field. Rev.Roum. Phys., vol. 48 (1983), pp. 405–412.

[5] M.A.Ezzat and M.Z.Abd–Elall. State space approach to viscoelasticfluid flow of hydromagnetic fluctuating boundary layer through a porousmedium. ZAMM , vol. 77 (1997), pp. 197–207.

[6] V.M. Soundalgekar and D.D.Haldavnekar. MHD free convective flowin a vertical channel. Acta Mech., vol. 16 (1973), pp. 77–91.

[7] V.M. Soundalgekar. Free convection effects on steady MHD flow past avertical porous plate. J. Fluid Mech., vol. 66 (1974), pp. 541–551.

[8] A.A.Raptis, C.P.Perdikis, and J.G.Tjivanidis. Effects of free convec-tion currents on the flow of an electrically conducting fluid past an accel-erated vertical infinite plate with variable suction. ZAMM , vol. 61 (1981),pp. 341–342.

[9] M.A.Ezzat. State space approach to unsteady free convection flow througha porous medium. Appl. Math. Comput., vol. 94 (1994), pp. 191–205.

[10] M.A.Ezzat, M.Z.Abd–Elall, O. Shaker and F.Barakat. State spaceformulation to viscoelastic fluid flow of magnetohydrodynamic free convec-tion through a porous medium. Acta Mech., vol. 119 (1996), pp. 147–164.

[11] M.A.Ezzat, A.A.El–Bary and M.I.Morsey. Space approach to thehydromagnetic flow of a dusty fluid through a porous medium. Comput.Math. Applic., vol. 59 (2010), pp. 2868–2879.

[12] D.S.Chandrasekharaiah. Hyperbolic thermoelasticity: A review of re-cent literature. Appl. Mech. Rev., vol. 51(1998), pp.705–729.

[13] M.A.Ezzat and A.A.El–Bary. On three models of magneto–hydrodynamic free convection flow. Can. J. Phys., vol. 87 (2009), 12131226.

[14] C.Cattaneo. Sur une forme de lequation de la Chaleur eliminant le para-doxe dune propagation instantane. C. R. Acad. Sci., vol. 247 (1958), pp. 431–433.

[15] P.Puri and P.K.Kythe. Non–classical thermal effects in Stokes secondproblem. Acta Mech., vol. 112 (1995), pp. 1–9.

602


[16] M.A.Ezzat, M.I.Othman and K.A.Helmy. A problem of a micropolarmagnetohydrodynamic boundary layer flow. Can. J. Phys., vol. 77 (1999),pp. 813–827.

[17] M.A.Ezzat, A.A. Samaan and A.A.El–Bary. State space formulationfor boundary layer magnetohydrodynamic free convection flow with one re-laxation time. Can. J. Phys., vol. 80 (2002), pp. 1157–1174.

[18] M.A.Ezzat, A.A.El–Bary and S.M.Ezzat. Combined heat and masstransfer for unsteady MHD flow of perfect conducting micropolar fluid withthermal relaxation. Energ. Conver. Manag., vol. 52 (2011), pp. 934–945.

[19] M.A.Ezzat. Free convection effects on perfectly conducting fluid. Int. J.Enging. Sci., vol. 39 (2001), pp. 799–819.

[20] M.A.Ezzat. Free convection flow of conducting micropolar fluid with ther-mal relaxation including heat sources. J. Appl. Math., vol. 4 (2004), pp. 271–292.

[21] M.A.Ezzat. Free convection effects on extracellular fluid in the presence ofa transverse magnetic field. Appl. Math. Comput., vol. 151 (2004), pp. 455–482.

[22] M.Caputo. Linear models of dissipation whose Q is almost frequency–independent II. Geophys. J. Roy. Astro. Soc., vol.13 (1967), pp. 529–539.

[23] I. Podlubny. Fractional Differential Equations (Academic Press, NewYork, 1999).

[24] F.Mainardi. Fractional calculus: Some basic problems in continuum andstatistical mechanics. In: A. Carpinteri, F. Mainardi. Fractals and Frac-tional Calculus in Continuum Mechanics (Springer–Verlag, New York, 1997,pp. 291–348).

[25] V.Kiryakova. Generalized fractional calculus and applications. In: PitmanRes. Notes Math. Ser., vol. 301 (Longman-Wiley, New York, 1994).

[26] K.S.Miller and B.Ross. An Introduction to the Fractional Integrals andDerivatives – Theory and Applications (John Wiley & Sons Inc, New York,1993).

[27] S.G. Samko, A.A.Kilbas and O.I.Marichev. Fractional Integrals andDerivatives – Theory and Applications (Gordon and Breach, Longhorne, PA,1993).

[28] K.B.Oldham and J. Spanier. The Fractional Calculus (Academic Press,New York, 1974).

[29] R.Gorenflo and F.Mainardi. Fractional Calculus: Integral and Differ-ential Equations of Fractional Orders, Fractals and Fractional Calculus inContinuum Mechanics (Springer, Wien, 1997).

[30] R.Hilfer. Applications of Fraction Calculus in Physics (World Scientific,Singapore, 2000).

[31] M.Caputo and F.Mainardi. Linear model of dissipation in anelasticsolids. Rivis ta el Nuovo Cimento, vol. 1 (1971), pp.161– 198.

603


[32] M.Caputo. Vibrations on an infinite viscoelastic layer with a dissipativememory. J. Acous. Soc. Amer., vol. 56 (1974), pp. 897–904.

[33] R.L. Bagley and P.J.Torvik. A theoretical basis for the application offractional calculus to viscoelasticity. J. Rheol., vol. 27 (1983), pp. 201–210.

[34] R.C.Koeller. Applications of fractional calculus to the theory of viscoelas-ticity. Trans. ASME J. Appl. Mech., vol. 51 (1984), pp. 299–307.

[35] Yu.A.Rossikhin and M.V. Shitikova. Applications of fractional calculusto dynamic problems of linear and nonlinear heredity mechanics of solids.Appl. Mech. Rev., vol. 50 (1997), pp. 15–67.

[36] M.Khan, A.Anjum, C. Fetecau and Q.Haitao. Exact solutions forsome oscillating motions of a fractional Burgers’ fluid. Math. Compu. Model.,vol. 51 (2010), pp. 682–692.

[37] S.Hyder, Q.Haitao. Starting solutions for a viscoelastic fluid with frac-tional Burgers’ model in an annular pipe. Nonlinear Analys., vol. 11 (2010),pp. 547–554.

[38] G.Haitao and J.Hui. Unsteady helical flows of a generalized Oldroyd-Bfluid with fractional derivative. Nonlinear Analys., vol. 10 (2010), pp.2700–2708.

[39] A.Saadatmandi and M.Dehghan. A new operational matrix for solvingfractional order differential equations. Comp. Math. Applic., vol. 59 (2010),pp.1326–1336.

[40] Z.Odibat and S.Momani. The variational iteration method: An efficientscheme for handling fractional partial differential equations in fluid mechan-ics. Comput. Math. Appl., vol. 58 (2009), pp. 2199–2208.

[41] R.Kimmich. Strange kinetics, porous media, and NMR . J. Chem. Phys.,vol. 284 (2002), pp. 243285.

[42] F.Mainardi and R.Gorenflo. On Mittag–Lettler-type function in frac-tional evolution processes. J. Comput. Appl. Math., vol. 118 (2000), pp. 283–299.

[43] Y.Z.Povstenko. Fractional heat conduction equation and associated ther-mal stresses. J. Therm. Stress., vol. 28 (2005), pp. 83–102.

[44] Y.Fujita. Integrodifferential equation which interpolates the heat equationand wave equation (I). Osaka J. Math., vol. 27(1990), pp. 309321.

[45] Y.Fujita. Integrodifferential equation which interpolates the heat equationand wave equation (II). Osaka J. Math., vol. 27 (1990), pp. 797804.

[46] H.Sherief, A. El-Said, A.AbdEl-Latief. Fractional order theory ofthermoelasticity. Int. J. Solid Struct., vol. 47 (2010), pp. 269–275.

[47] H.Lord, Y. Shulman. A generalized dynamical theory of thermoelasticity.J. Mech. Phys. Solids , vol. 15 (1967), pp. 299–309.

[48] G.Lebon, D. Jou and J. Casas-Vazquez. Understanding Non-Equilib-rium Thermodynamics: Foundations, Applications, Frontiers (Springer Ver-lag, Berlin, Heidelberg, 2008).

604


[49] D. Jou, J.Casas-Vazquez and G. Lebon. Extended irreversible thermo-dynamics. Rep. Prog. Phys., vol. 51 (1988), pp. 1105–1179.

[50] H.Youssef. Theory of fractional order generalized thermoelasticity. J.Heat Transfer , vol. 132 (2010), pp. 1–7.

[51] G.Honig and U.Hirdes. A method for the numerical inversion of theLaplace transform. J. Comp. Appl. Math., vol. 10 (1984), pp. 113–132.

[52] G. Jumarie. Derivation and solutions of some fractional Black–Scholesequations in coarse-grained space and time. Application to Merton’s optimalportfolio. Comput. Math. Applic., vol. 59 (2010), pp. 1142–1164.

[53] M.A.Ezzat. Thermoelectric MHD non-Newtonian fluid with fractionalderivative heat transfer. Physica B , vol. 405 (2010), pp. 4188–4194.

[54] M.A.Ezzat. Thermoelectric MHD with modified Fourier’s law. Int. J.Therm. Sci., vol. 50 (2011), pp. 449–455.

[55] M.A.Ezzat. State space approach to thermoelectric fluid with fractionalorder heat transfer. Heat Mass Trans., vol. 48 (2012), pp. 71–82.

[56] V.E.Tarasov. Fractional vector calculus and fractional Maxwell’s equa-tions. Annals Phys., vol. 323 (2008), pp. 2756–2778.

[57] M.A.Ezzat. Magneto-thermoelasticity with thermoelectric properties andfractional derivative heat transfer. Physica B , vol. 406 (2011), pp. 30–35.

[58] M.A.Ezzat, M.I.Othman and A.S.El-Karamany. State space ap-proach to generalized thermo-viscoelasticity with two relaxation times. J.Therm. Stress., vol. 25 (2002), pp. 295–316.

[59] M.A.Ezzat and A.S.El-Karamany. Magnetothermoelasticity with tworelaxation times in conducting medium with variable electrical and thermalconductivity. Appl. Math. Comput., vol. 142 (2003), pp. 449–467.

[60] M.A.Ezzat. Fundamental solution in generalized magneto-thermoelasticitywith two relaxation times for perfect conductor cylindrical region. Int. J.Eng. Sci., vol. 42 (2004), pp. 455–482.

[61] M.A.Ezzat. The relaxation effects of the volume properties of electricallyconducting viscoelastic material. Mater. Sci. Eng. B , vol. 130 (2006), pp. 11–23.

[62] M.A.Ezzat and A.S.El-Karamany. Fractional order heat conductionlaw in magneto-thermoelasticity involving two temperatures. ZAMP , vol. 62(2011), pp.937–952.

Appendix A.

a0 =k21ch(k2y) − k2

2ch(k1y)k21 − k2

2

, a1 =k31sh(k2y) − k3

2sh(k1y)k1k2(k2

1 − k22)

,

a2 =ch(k1y) − ch(k2y)

k21 − k2

2

, a3 =k2sh(k1y) − k1sh(k2y)

k1k2(k21 − k2

2)

(48)

605


Appendix B: The elements of the matrix L (s, y).

�11 = �11 = ch(k1y), �22 = �44 = ch(k2y),

�12 = �14 = �32 = �34 = 0, �13 = k−11 sh(k1y),

�21 = �43 =Gr(ch(k2y) − ch(k1y))

k21 − k2

2

, �23 =Gr(k1sh(k2y) − k2sh(k1y))

k1k2(k21 − k2

2),

�24 = k−12 sh(k2y), �31 = k1sh(k1y),

�41 =Gr(sh(k2y) − sh(k1y))

k21 − k2

2

, �42 = k2ch(k2y).

Received 06.12.2012

606

MHD FREE CONVECTION FLOW WITH FRACTIONAL HEAT CONDUCTION ...

Documents

Transcript of MHD FREE CONVECTION FLOW WITH FRACTIONAL HEAT CONDUCTION ...