Some exact solutions for fractional generalized Burgers’ fluid in a porous space

$Page 1: Some exact solutions for fractional generalized Burgers’ fluid in a porous space$
Nonlinear Analysis: Real World Applications 9 (2008) 1952–1965www.elsevier.com/locate/na

Some exact solutions for fractional generalized Burgers’ fluid in aporous space

Masood Khan∗, Tasawar HayatDepartment of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan

Received 18 April 2007; accepted 7 June 2007

Abstract

This work is concerned with deriving the equation for describing the magnetohydrodynamic (MHD) flow of a fractional generalizedBurgers’ fluid in a porous space. Modified Darcy’s law has been taken into account. Closed form solutions for velocity are obtainedin three problems. The solutions for Navier–Stokes, second grade, Maxwell, Oldroyd-B and Burgers’ fluids appear as the limitingcases of the obtained solutions. A parametric study of some physical parameters involved in the problems is performed to illustratethe influence of these parameters on the velocity profiles.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Fractional calculus; Generalized Burgers’ fluid; Three problems; Exact solutions

1. Introduction

Considerable attention has been given to the flows of non-Newtonian fluids in the last few years. This is because of theirvarious applications in petroleum drilling, manufacturing of food and paper and other similar activities. Undoubtedly, thefluid motion of non-Newtonian fluids is much more complicated and subtle in comparison with that of the Newtonianfluids. Due to complex behavior, it is very difficult to suggest a single fluid model to exhibit all properties of non-Newtonian fluids. Therefore, several constitutive equations for the non-Newtonian fluid models have been proposed.Amongst these the viscoelastic fluid models have acquired a special status. Extensive literature is available now onthe viscoelastic flows. Some recent attempts in this direction have been made by Tan [13], Tan and Masuoka [14–16],Fetecau and Fetecau [2,3], Fetecau et al. [4] and Hayat et al. [5–7]. But the generalized Burgers’ fluid, which forma subclass of the fluids of the viscoelastic type, have been given a little attention. Moreover, in order to describe theviscoelasticity [11] the fractional calculus approach is very important. The starting point of the fractional derivativemodel of non-Newtonian fluid is usually a classical differential equation, which is modified by replacing the timederivative of an integer order by the so-called Riemann–Liouville fractional calculus operator. Some related attemptsof non-Newtonian flows with fractional calculus approach have been made by Song and Jiang [12], Tan et al. [17–20],Hayat et al. [8], Khan et al. [10], Tong et al. [21,22], and Yin and Zhu [23].

The aim of the present paper is to extend the analysis of Ref. [8] in three directions i.e.: (i) to consider the generalizedBurgers’ fluid; (ii) to include the Hall effects and (iii) to consider the flows in a porous space. The modified Darcy’s

∗ Corresponding author.E-mail address: [email protected] (M. Khan).

1468-1218/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.nonrwa.2007.06.005

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M. Khan, T. Hayat / Nonlinear Analysis: Real World Applications 9 (2008) 1952–1965 1953

law for fractional generalized Burgers’ fluid has been introduced first time in the literature. Exact analytical solutionsof the flows for the three cases are obtained using the Fourier transform for fractional derivatives. The existing resultsfor fractional Maxwell fluid [8] can be recovered by taking �2 = �3 = �4 = M = �1 = 0.

2. Mathematical model

For a fractional generalized Burgers’ fluid, the Cauchy stress tensor is defined as

T = −pI + S, (1)

where the extra stress tensor S is decomposed as follows:(1 + ��

1D̃�t + ��

2D̃2�t

)S = �

(1 + ��

3D̃�t + ��

4D̃2�t

)A1, (2)

D̃�t S = D�

t S + (V · ∇)S − (∇V)S − S(∇V)T, (3)

D̃2�t S = D̃�

t (D̃�t S), (4)

A1 = (∇V) + (∇V)T. (5)

In above equations, V=(u, v, w) is the velocity field, −pI the indeterminate spherical stress, � the dynamic viscosity,A1 the first Rivlin–Ericksen tensor and �, � are fractional calculus parameters such that 0��1. For � > � therelaxation fraction is increasing, which is generally not responsible [9] and requires that ��. In Eqs. (3) and (4),D�

t (=��t ) indicates the fractional derivative of order � and is defined by

D�t [f (t)] = 1

�(1 − �)

d

dt

∫ t

0

f (t)

(t − z)�dz (0 < � < 1), (6)

where �(·) is a Gamma function. It should be noted that the model (2) reduces to that of generalized Burgers’ fluidmodel when �=�=1. It is also important to mention that the model (2) includes many non-Newtonian fluid models asthe special cases. For example when �4 = 0, it gives fractional Burgers’ fluid model and for �4 = 0, � = � = 1 we haveBurgers’ fluid model. It reduces to Oldroyd-B, Maxwell, second grade and Navier–Stokes fluid models, respectively,when �2 = �4 = 0, �2 = �3 = �4 = 0, �1 = �2 = �4 = 0 and �1 = �2 = �3 = �4 = 0.

The basic equations which govern the magnetohydrodynamic (MHD) incompressible flow in a porous space are

�dVdt

= −∇p + div S + J × B + R, (7)

div V = 0, (8)

div B = 0, curl B = �mJ, curl E = −�B�t

, (9)

in which � is the fluid density, J the current density, B the total magnetic field so that B = B0 + b, B0 and b theapplied and induced magnetic fields, respectively, R the Darcy’s resistance for the generalized Burgers’ fluid in porousmedium, �m the magnetic permeability and E the total electric field current. In the present analysis, it is assumed thatno applied and polarization voltage exist so that E = 0. This then corresponds to the case where no energy is added orextracted from the fluid by the electric field. The induced magnetic field is further neglected compared with appliedmagnetic field. If the Hall term is retained in the generalized Ohm’s law, then the current density J is given by [1]

J + ee

B0(J × B0) = �

[E + V × B + 1

ene∇p

], (10)

in which e is the cyclotron frequency of electron, e the electron collision time, � the electrical conductivity of thefluid, e the electron charge, ne the number density of electrons and pe the electron pressure. Note that ion-slip andthermoelectric effects have not been taken into account in Eq. (10). Also ee ≈ O(1) and ii>1 (i and i are thecyclotron frequency and collision time for ions, respectively).

1954 M. Khan, T. Hayat / Nonlinear Analysis: Real World Applications 9 (2008) 1952–1965

We shall consider the velocity and the stress of the form

V = u(y, t)i, S = S(y, t)i, (11)

where u is the x-component of velocity V and i the unit vector in the x-direction.Upon making use of Eqs. (11), Eq. (8) is satisfied identically and Eqs. (2)–(5) after using the initial condition

S(y, 0) = 0, yield(1 + ��

1��

�t�+ ��

2�2�

�t2�

)Sxy = �

(1 + ��

3��

�t�+ ��

4�2�

�t2�

)�u

�y, (12)

(1 + ��

1��

�t�+ ��

2�2�

�t2�

)Sxx − 2��

1�u

�ySxy − 2��

2

[��

�t�

(�u

�ySxy

)+ �u

�y

��

�t�Sxy

]

= −2��3

(�u

�y

)2

− 6��4�u

�y

��

�t�

(�u

�y

)(13)

and Syy = Syz = Szz = Sxz = 0.The pressure drop and the velocity of a generalized Burgers’ fluid in a porous space are related according to(

1 + ��1

��

�t�+ ��

2�2�

�t2�

)∇p = −�

k

(1 + ��

3��

�t�+ ��

4�2�

�t2�

)VD, (14)

where k is the permeability of the porous medium, VD the Darcian velocity, which is related to the usual (i.e. volumeaverage over a volume element consisting of fluid only in the pores) the velocity V by VD =�1V and �1 is the porosity.Since the pressure gradient in Eq. (14) can also be interpreted as measure of the resistance to the flow in the bulk of theporous space and R is a measure of the flow resistance offered by the solid matrix. Therefore R can be inferred fromEq. (14) to give(

1 + ��1

��

�t�+ ��

2�2�

�t2�

)R = −��1

k

(1 + ��

3��

�t�+ ��

4�2�

�t2�

)V. (15)

From Eqs. (1)–(5), (7) and (9)–(15) we obtain

�

(1 + ��

1��

�t�+ ��

2�2�

�t2�

)�u

�t+(

1 + ��1

��

�t�+ ��

2�2�

�t2�

)�p

�x

= �

(1 + ��

3��

�t�+ ��

4�2�

�t2�

)�2u

�y2− �B2

0

1 − i�

(1 + ��

1��

�t�+ ��

2�2�

�t2�

)u

− ��1

k

(1 + ��

3��

�t�+ ��

4�2�

�t2�

)u, (16)

where � = ee is the Hall parameter.

3. Flow due to a rigid plate

Let us consider an incompressible generalized Burgers’ fluid bounded by a plate at y = 0. At time t = 0+, the fluidis set to motion due to the periodic plate oscillation of the form f (t) with period T0. Because of the shear effects, thefluid over the plate is gradually disturbed. The problem which governs the flow is(

1 + ��1

��

�t�+ ��

2�2�

�t2�

)�u

�t= �

(1 + ��

3��

�t�+ ��

4�2�

�t2�

)�2u

�y2− �B2

0

�(1 − i�)

(1 + ��

1��

�t�+ ��

2�2�

�t2�

)u

− ��1

k

(1 + ��

3��

�t�+ ��

4�2�

�t2�

)u, (17)


u(0, t) = Uf (t), t > 0, (18)

u → 0 as y → ∞, (19)

where

f (t) =∞∑

k=−∞akeik0t

and the Fourier series coefficients {ak} are

ak = 1

T0

∫T0

f (t)e−ik0t

with non-zero fundamental frequency 0 = 2 /T0.Defining the following non-dimensional quantities

u∗ = u

U, y∗ = y

(�/U), t∗ = t

(�/U2), ∗

0 = 0

(U2/�),

�∗1,3 = �1,3

(�/U2), �∗

2,4 = �2,4

(�/U2)2, M2 = �B2

0

(�2U2/�),

1

K= �1

(kU2/�2)(20)

and then suppressing the asterisks, we have(1 + ��

1��

�t�+ ��

2�2�

�t2�

)�u

�t=(

1 + ��3

��

�t�+ ��

4�2�

�t2�

)�2u

�y2− M2

1 − i�

(1 + ��

1��

�t�+ ��

2�2�

�t2�

)u

− 1

K

(1 + ��

3��

�t�+ ��

4�2�

�t2�

)u, (21)

u(0, t) =∞∑

k=−∞akeik0t , t > 0, (22)

u → 0 as y → ∞. (23)

For the solution of above problem, we define the temporal Fourier transform pair as

�(y, ) =∫ ∞

−∞u(y, t)e−it dt , (24)

u(y, t) = 1

2

∫ ∞

−∞�(y, )eit d, (25)

where is the temporal frequency. The Fourier transform for the fractional derivative is defined by [9]∫ ∞

−∞D

�t [u(y, t)]e−it dt = (i)��(y, ), (26)

∫ ∞

−∞D

2�t [u(y, t)]e−it dt = (i)2��(y, ), (27)

in which

(i)� = ||�ei� /2 sign = ||�(

cos�

2+ i sign sin

�

2

),

(i)2� = ||2�(cos � + i sign sin � )

and sign is the signum function.


Taking Fourier transform and then solving the resulting problem we have

�(y, ) = 2 ∞∑

k=−∞ak�( − k0)e

−(mk+ink)y , (28)

where �(·) is the dirac delta function.From Eqs. (25) and (28) we arrive at

u(y, t) =∞∑

k=−∞ake−mky+i(k0t−nky), (29)

where

mk =⎡⎢⎣√

L2r + L2

i + Lr

2

⎤⎥⎦

1/2

, nk =⎡⎢⎣√

L2r + L2

i − Lr

2

⎤⎥⎦

1/2

,

Lr = 1

K+ M2(ac + bd) − (bc − ad){M2� + k0(1 + �2)}

(1 + �2)(c2 + d2),

Li = M2(bc − ad) + (ac + bd){M2� + k0(1 + �2)}(1 + �2)(c2 + d2)

,

a = 1 + ��1|k0|� cos

�

2+ ��

2|k0|2� cos � ,

c = 1 + ��3 |k0|� cos

�

2+ ��

4 |k0|2� cos � ,

b = sign k0

[��

1|k0|� sin�

2+ ��

2|k0|2� sin � ]

,

d = sign k0

[��

3 |k0|� sin�

2+ ��

4 |k0|2� sin �

].

It is worth emphasizing to note that Eq. (29) is the solution to general periodic plate oscillation. The flow fields due tooscillations in the specific cases can be obtained by inserting the corresponding coefficients into Eq. (29). For example,the flow fields due to the following oscillations:

Oscillations Fourier coefficients

f (t) ak

(i) ei0t a1 = 1 and ak = 0 (k �= 1),(ii) cos 0t a1 = a−1 = 1

2 and ak = 0, otherwise,(ii) sin 0t a1 = −a−1 = 1

2iand ak = 0, otherwise,

(iv)

[1, |t | < T10, T1 < |t | < T0/2

]a0 = 2T1/T0, ak = sin(k0T1)/k ,

for all k �= 0,

(v)∞∑

k=−∞�(t − kT 0) ak = 1/T0 for all k


Table 1A comparison of velocity profiles (flow due to a rigid plate) for various kinds of fluids when M = K = 1 and 0 = t = y = 0.5

Type of fluid Rheological parameters u for � = 0, u for � = 0, u for � = 2, u for � = 2,� = � = 1 � = 0.1,� = 0.5 � = � = 1 � = 0.1,� = 0.5

Newtonian �i = 0 for i = 1 − 4 0.473331 0.473331 0.557548 0.557548Second grade �1 = �2 = �4 = 0, �3 = 1 0.478712 0.476142 0.546511 0.552371Maxwell �2 = �3 = �4 = 0, �1 = 10 0.300458 0.391521 0.532301 0.517897Oldroyd-B �2 = �4 = 0, �1 = 10, �3 = 1 0.309188 0.394349 0.492535 0.512882Burgers’ �1 = 10, �2 = 2, �3 = 1, �4 = 0 0.314115 0.350336 0.506013 0.486670G. Burgers’ �1 = 10, �2 = 2, �3 = 1, �4 = 1.5 0.252047 0.310428 0.421612 0.471846

Table 2A comparison of velocity profiles (periodic flow between two plates) for various kinds of fluids when M = K = 1 and 0 = t = y = 0.5



are given by

u1(y, t) = e−m1y+i(0t−n1y), (30)

u2(y, t) = 12 [e−m1y+i(0t−n1y) + e−m−1y−i(0t+n−1y)], (31)

u3(y, t) = 1

2i[e−m1y+i(0t−n1y) − e−m−1y−i(0t+n−1y)], (32)

u4(y, t) = 1

∞∑k=−∞

sin(k0T1)

ke−mky+i(k0t−nky), k �= 0 (33)

u5(y, t) = 1

T0

∞∑k=−∞

e−mky+i(k0t−nky). (34)

4. Periodic flow between two plates

Here the fluid lies between two infinite parallel plates distant d apart. The plate at y = 0 has velocity Uf (t) for t > 0and the plate at y = d is fixed. The resulting boundary-value problem consists of Eqs. (17) and (18) and the followingboundary condition:

u(d, t) = 0. (35)

Introducing

u∗ = u

U, y∗ = y

d, t∗ = t

(d2/�), ∗

0 = 0

(�/d2),

�∗1,3 = �1,3

(d2/�), �∗

2,4 = �2,4

(d2/�)2, M2 = �B2

0

(�/d2),

1

K= �1

(k/d2)


Table 3A comparison of velocity profiles (time-periodic plane Poiseuille flow) for various kinds of fluids when M = K = 1, Q = −1 and 0 = t = y = 0.5



Fig. 1. Profiles of velocity u(y, t) (flow due to a rigid plate) for various values of permeability parameter K when �1 =10, �3 =1, M =1,0 = t =0.5,� = 0.1,� = 0.5 and � = 0 are fixed: (a) F. Oldroyd-B fluid (�2 = �4 = 0); (b) F.G. Burgers’ fluid (�2 = 2, �4 = 1.5).


Fig. 2. Profiles of velocity u(y, t) (flow due to a rigid plate) for various values of permeability parameter K when�1 = 10, �3 = 1, M = 1,0 = t = 0.5, � = 0.1,� = 0.5 and � = 2 are fixed: (a) F. Oldroyd-B fluid (�2 = �4 = 0); (b) F.G. Burgers’fluid (�2 = 2, �4 = 1.5).

the problem here is governed by Eqs. (21), (22) and

u(1, t) = 0. (36)

The solutions for flow fields are

u(y, t) =∞∑

k=−∞ak

sinh �k(1 − y)

sinh �k

eik0t , (37)


Fig. 3. Profiles of velocity u(y, t) (periodic flow between two plates) for various values of permeability parameter K when�1 = 10, �3 = 1, M = 1,0 = t = 0.5, � = 0.1,� = 0.5 and � = 0 are fixed: (a) F. Oldroyd-B fluid (�2 = �4 = 0); (b) F.G. Burgers’ fluid(�2 = 2, �4 = 1.5).

u1(y, t) = sinh �1(1 − y)

sinh �1ei0t , (38)

u2(y, t) = 1

2

[sinh �1(1 − y)

sinh �1ei0t + sinh �−1(1 − y)

sinh �−1e−i0t

], (39)

u3(y, t) = 1

2i

[sinh �1(1 − y)

sinh �1ei0t − sinh �−1(1 − y)

sinh �−1e−i0t

], (40)


Fig. 4. Profiles of velocity u(y, t) (periodic flow between two plates) for various values of permeability parameter K when�1 = 10, �3 = 1, M = 1,0 = t = 0.5, � = 0.1,� = 0.5 and � = 2 are fixed: (a) F. Oldroyd-B fluid (�2 = �4 = 0); (b) F.G. Burgers’ fluid(�2 = 2, �4 = 1.5).

u4(y, t) = 1

∞∑k=−∞

sin k0T1

k

sinh �k(1 − y)

sinh �k

eik0t , k �= 0, (41)

u5(y, t) = 1

T0

∞∑k=−∞

sinh �k(1 − y)

sinh �k

eik0t , (42)

where

�2k = (mk + ink)

2.


Fig. 5. Profiles of velocity u(y, t) (time-periodic plane Poiseuille flow) for various values of permeability parameter K when�1 = 10, �3 = 1, M = 1,0 = t = 0.5, � = 0.1,� = 0.5 and � = 0 are fixed: (a) F. Oldroyd-B fluid (�2 = �4 = 0); (b) F.G. Burgers’ fluid(�2 = 2, �4 = 1.5).

5. Time-periodic plane Poiseuille flow

This section deals with the flow induced by an oscillating pressure gradient between the two fixed plates, of the form

�p

�x= �Q0ei0t . (43)

The flow is governed by Eq. (16) and the conditions

u(±d, t) = 0. (44)


Fig. 6. Profiles of velocity u(y, t) (time-periodic plane Poiseuille flow) for various values of permeability parameter K when�1 = 10, �3 = 1, M = 1,0 = t = 0.5, � = 0.1,� = 0.5 and � = 2 are fixed: (a) F. Oldroyd-B fluid (�2 = �4 = 0); (b) F.G. Burgers’ fluid(�2 = 2, �4 = 1.5).

The solution is of the following form:

u(y, t) = Q0

[1 + (i0)

��1 + (i0)

2��2

1 + (i0)��

3 + (i0)2��

4

(cosh �∗y − cosh �∗

�∗2cosh �∗

)ei0t

], (45)

with

�∗ = �k|k=1.


6. Results and discussion

This section displays the graphical illustration of velocity profiles for the flows analyzed in this investigation.Emphasis has been given to examine the difference between the velocity profiles for six fluid models: Newtonian fluid(�i = 0 for i = 1 − 4), second grade fluid (�1 = �2 = �4 = 0, �3 �= 0), Maxwell fluid (�2 = �3 = �4 = 0, �1 �= 0),Oldroyd-B fluid (�2 = �4 = 0, �1 �= 0, �3 �= 0), Burgers’ fluid (�i �= 0 for i = 1 − 3, �4 = 0) and generalized Burgers’fluid (�i �= 0 for i = 1 − 4) for both cases i.e.: (a) when � = � = 1 and (b) for fractional model when 0 < � < � < 1.A comparison for various kind of fluids is also given in the form of tables when the oscillation is of the type cos 0t .Graphs are included only for fractional Oldroyd-B and fractional generalized Burgers’ fluids. The effects of variousemerging parameters especially permeability parameter K, Hall parameter � and the respective rheological parameters�2 and �4 of the Burgers’ and generalized Burgers’ fluid on the velocity profiles have been investigated (Tables 1–3).

Figs. 1 and 2 are prepared for flow due to a rigid plate, Figs. 3 and 4 for periodic flow between two plates whenthe oscillation is of the type cos 0t . Figs. 5 and 6 for time-periodic plane Poiseuille flow when the pressure gradientis also of the form cos 0t . The influence of the permeability of the porous medium K on the flows is illustrated inFigs. 1–6 for fractional Oldroyd-B fluid (panel a) and fractional generalized Burgers’ fluid (panel b) in the absence aswell as in the presence of Hall parameter � by keeping the values of other parameters fixed. As anticipated, the increaseof the permeability of the porous medium reduces the drag force and hence causes the flow velocity to increase inall the three cases. It is further observed that when the flow is driven by the oscillation of the boundary, the velocityprofiles for Oldroyd-B fluid are larger than those for Burgers’ fluid. However, the situation is quite reverse when theflow is driven by the adverse pressure gradient. For such situation the Oldroyd-B fluid has smaller velocity profileswhen compared with Burgers’ fluid. Further, some differences among Newtonian, second grade Maxwell, Oldroyd-B,Burgers’ and generalized Burgers’ fluids can also be observed from the given tables. Moreover, it can be expected fromthe governing equation (16) that increasing the magnitude of the magnetic field yields an effect opposite to that of thepermeability.

These figures also elucidate the influence of the Hall parameter � on the velocity profiles with fixed values of otherparameters. As expected, the velocity increases by increasing � for examined fluids. This has the effect of reducingeffective conductivity by increasing �. In fact the magnetic damping force on velocity decreases.

7. Concluding remarks

In this article we have investigated some unidirectional flows of a magnetohydrodynamic (MHD) fluid througha porous space. A theoretical model is developed for the analysis concerning the flows of a fractional generalizedBurgers’ fluid. By using the constitutive equations for a generalized Burger’ fluid, in the literature, the governing timedependent equation is modelled by introducing modified Darcy’s law. Exact analytical solutions are obtained for threeflow situations of MHD generalized Burgers’ fluid in a porous space. The analysis for the analytic solutions is carriedout using fractional calculus approach. A comparison of the present results is made with previously published resultsand have found in excellent agreement. The main conclusions are outlined as follows:

• The velocity profiles in the presence of Hall parameter differ substantially when compared with the results of noHall current.

• Predictions of present fluid model are in very good agreement with the results of Maxwell fluid [8] by taking�2 = �3 = �4 = M = �1 = 0.

• In Oldroyd-B and generalized Burgers’ fluids, the velocity profile is an increasing function of the Hall parameter.However, decreases monotonically by increasing Hartmann number.

• It is important to note that in the absence of pressure gradient, the velocity profiles induced by the oscillatingboundary are greater for Oldroyd-B fluid when compared with generalized Burger’ fluid.

• The velocity profiles show similar characteristics in an Oldroyd-B and generalized Burgers’ fluids.

Acknowledgment

This work is supported by the Quaid-i-Azam University Research Fund (URF scheme).


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