Computers & Mathematics with Applications Volume 62 issue 8 2011 [doi 10.1016_j.camwa.2011.08.025]...

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Computers and Mathematics with Applications 62 (2011) 31233131

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Computers and Mathematics with Applications

journal homepage:www.elsevier.com/locate/camwa

MHD flow and heat transfer of a generalized Burgers fluid due to anexponential accelerating plate with the effect of radiation

Yaqing Liu a,b, Liancun Zheng a,, Xinxin Zhang ba Department of Mathematics and Mechanics, University of Science and Technology Beijing, Beijing 100083, Chinab Mechanical Engineering School, University of Science and Technology Beijing, Beijing 100083, China

a r t i c l e i n f o

Article history:

Received 4 September 2010

Received in revised form 6 May 2011

Accepted 8 August 2011

Keywords:

MHD flow

Generalized Burgers fluid

Fourier sine transform

Laplace transform

Gfunction

a b s t r a c t

This paper presents a research for the MHD flow and heat transfer of an incompressible

generalized Burgers fluid due to an exponential accelerating plate with the effect of

radiation. The fractional calculus approach is used to establish the constitutive relationship

of the viscoelastic fluid. Exact analytic solutions are obtained for the velocity and

temperature fieldsin integral andseries form in terms of the G function by means of Fourier

sine transform and Laplace transform.Moreover, the figures are plotted to show theeffects

of different parameters on the velocity and temperature fields.

2011 Elsevier Ltd. All rights reserved.

1. Introduction

The interest for flow and heat transfer of non-Newtonian fluids has considerably grown in recent years because of theadvance in technological applications. However, it is difficult to suggest a single model which exhibits all properties of non-Newtonian fluids as is done for Newtonian fluids. For this reason, a number of constitutive equations have been proposed.Among them the models of differential type and those of rate type have received much attention [13]. A thermodynamicframework has been put into place to develop a rate type model known as the Burgers model that is used to describethe motion of the earths mantle. This model is also used to characterize diverse viscoelastic materials, such as asphalt ingeomechanics and cheese in food products.

Fractional derivatives have been found to be quite flexible in describing viscoelastic behavior [4]. In general, theconstitutive equations for generalized non-Newtonian fluids are modified from the well known fluid models by replacingthe time derivative of an integer order with the so-called RiemannLiouville fractional calculus operators [58]. A verygood fit of experimental data is achieved when the constitutive equation with fractional derivative is used [9]. Recently, theBurgers fluid models which form a subclass of the viscoelastic type have given attention. Xue [10,11] considered fractionalgeneralized Burgers fluid in a porous half-space. Hyder [12] discussed some unidirectional flows of a viscoelastic fluidbetween two parallel plates with fractional Burgers fluid models. Khan [1315]investigated some fractional Burgers fluidmodels including oscillating flow, accelerated flow and rotating flow. Moreover, MHD flows have widely converged on thedevelopment of energy generation and in astrophysical and geophysical fluid dynamics. Recently, the theory of MHD hasreceived much attention, see[16,17] and references therein. The effect of radiation on the heat and fluid over an unsteadystretching surface is analyzed [1820]. However, there are no attempts to consider the viscoelastic fluids under the effect ofthermal radiation.

Corresponding author. Tel.: +86 10 62332002.E-mail addresses:[email protected],[email protected](L. Zheng).

0898-1221/$ see front matter 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.camwa.2011.08.025
http://dx.doi.org/10.1016/j.camwa.2011.08.025http://www.elsevier.com/locate/camwahttp://www.elsevier.com/locate/camwamailto:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.camwa.2011.08.025http://dx.doi.org/10.1016/j.camwa.2011.08.025mailto:[email protected]:[email protected]://www.elsevier.com/locate/camwahttp://www.elsevier.com/locate/camwahttp://dx.doi.org/10.1016/j.camwa.2011.08.025


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3124 Y. Liu et al. / Computers and Mathematics with Applications 62 (2011) 31233131

Motivated by the above mentioned works, this paper considers the MHD flow of a generalized Burgers fluid due to anexponential accelerating plate. The magnetic field and thermal radiation and their influence on the flow are considered. Aparametric study of some physical parameters involved is performed to illustrate the influence of these parameters.

2. Governing equations

The constitutive equations for an incompressible fractional generalized Burgers fluid are given by

T= pI+S, (1+1Dt+2D2t )S=(1+3Dt)A, (1)whereTis the Cauchy stress tensor, pI denotes the indeterminate spherical stress, S is the extra-stress tensor,A=L+LT isthe first RivlinEricksen tensor, Lis the velocity gradient, , 1, 3 are material constants, known as the viscositycoefficient,

the relaxation and retardation times, respectively, 2is a material constant, andDpt the upper convected fractional derivativedefined by

DptS=DptS+V SLSSLT, D2pt S=Dpt(DptS). (2)

In the above relationsVis the velocity, is the gradient operator, Dt and Dt are based on RiemannLiouvilles definition asdefined in [4]:

Dptf(t)

=

1

(1p)

d

dt

t

0

f( )

(t )p

d , 0

p < 1, (3)

where()is the Gamma function.The motion equation for an incompressible fluid are given by

dV

dt= T+b, (4)

where is the constant density of the fluid,bis the body force field.Assuming a velocity field of the form:

V=u(y, t)i (5)whereu is the velocity and i is the unit vector in the x-direction. Sinceu is a function ofy and t, the stress field will alsodepend upony and t. There we consider the conducting fluid is permeated by an imposed magnetic field B= [0, B0, 0],which acts in the positivey-coordinate. In the low-magnetic Reynolds number approximation [21,22], in which the induced

magnetic field can be ignored, the magnetic body force is represented by B2

0u, where is the electrical conductivity of thefluid. Substituting Eq.(5)into Eqs.(1) 2 ,(4)and taking into account the initial conditions S(y, 0)=St(y, 0)=0, we get therelevant equations

du

dt= p

x+ Sxy

yB20u (6)

(1+1 Dt+2 D2t )Sxy=(1+3 Dt)yu (7)

(1+1 Dt+2 D2t )Sxx2Sxy[1+2 Dt]u

y22

u

yDtSxy= 23

u

y

2. (8)

EliminatingSxybetween Eqs.(7)and(8),in the absence of a pressure gradient in the x-direction, we obtain:

(1+1 Dt+2 D2

t )u(y, t)

t = (1+3 D

t)

2u(y, t)

y2 M(1+1 Dt+2 D2

t )u(y, t) (9)

where=/ is the kinematic viscosity andM= B20/.The fluid is considered to be a gray, absorbing-emitting radiation but non-scattering medium. When the Fouriers law of

heat conduction is considered[2325], the energy equation may be written in the form:

t= kT

Cp

2

y2+

Cp

[u

y

]2 1

Cp

qr

y, (10)

wherekTis the thermal conductivity, Cpis the specific heat of a fluid at constant pressure and qris the radiative heat flux.

3. Statement of the problem and its solution

Consider the generalized Burgers fluid occupying the space above a flat plate. A schematic representation of the physicalmodel and coordinates system is depicted inFig. 1.The plate is aligned with thex-axis aty= 0. Initially the fluid as well


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Y. Liu et al. / Computers and Mathematics with Applications 62 (2011) 31233131 3125

Fig. 1. Schematic representation of the physical model and coordinates system.

as the plate are at rest, and at timet= 0+ the plate slides in its plane with the velocityu0(eat 1), whereu0(>0), aareconstants. Let 0(t) denote the temperature of the plate for t 0, and suppose that the temperature of the fluid at themomentt= 0 is . Due to the shear, the fluid speeds up gradually. Accordingly, the flow is governed by Eqs. (4)(6)andthe associated boundary and initial conditions of the motion equation are

u(y, 0)= u(y, 0)t

= 2u(y, 0)

t2 =0, y> 0. (11)

u(0, t)=u0(eat 1), t>0, (12)

u(y, t),u(y, t)

y0, asy . (13)

The corresponding initial and boundary conditions of the energy equation are:

(y, 0)=, fory > 0, (14) (0, t)=0(t), fort 0, (15)

(y, t), (y, t)

y 0, fory . (16)

3.1. Velocity field

Employing the non-dimensional quantities in Eqs.(9),(11)(13):

u= uu0

, y= u0y

, t= u20t

, 1=1

u20

,

2= 2

u20

2, 3= 3

u20

, M= M

u20, a= a

u20. (17)

Dimensionless motion equations can be given (for brevity the dimensionless mark is omitted here)

(1+1 Dt+2 D2t )u(y, t)

t=(1+3 Dt)

2u(y, t)

y2 M(1+1 Dt+2 D2t )u(y, t) (18)

Initial condition: u(y, 0)= u(y, 0)t

= 2u(y, 0)

t2 =0, y> 0 (19)

Boundary conditions: u(0, t)=eat 1, t>0 (20)

u(y, t),u(y, t)

y0, asy . (21)

In order to solve the above problem, we use the Fourier sine transform [ 26] and Laplace transform for fractionalderivatives. Firstly, multiplying both sides of Eq. (18)by

2/sin(y), integrating then with respect to y from 0 to

and taking account of the corresponding initial and boundary conditions (19)(21),we obtain

(1+1 Dt+2 D2t )us( , t)

t=(1+3 Dt)[

2

(eat 1)2us( , t)] M(1+1 Dt+2 D2t )us( , t) (22)

where the Fourier sine transform us( , t)ofu(y, t)has to satisfy the conditions

us( , 0)= us( , 0)t

= 2

us( , 0)t2

=0, >0. (23)


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Applying the Laplace transform for sequential fractional derivatives to Eq.(22)and using the initial condition(23),we get

us( , s)=

2

1

1

sa 1

s

2(1+3 s )

[(s+M)(1+1 s +2 s2)+2(1+3 s )](24)

whereus( , s)is the Laplace transform ofus( , t)with respect tot. In order to avoid the lengthy procedure of residues andcontour integrals, we rewrite Eq.(24)into the series form

us( , s)=

2

1

1

sa 1s

2

1

1

sa 1s

k=0

(1)k 1(2)k+1(k+1)3

m+

l=

k+

1m,l0

(k+1)!m!l! M

m

n+w+=k

n,w,0

k!n!w!!

n1

w2 (1+1 s +2 s2)

s

(3 +s )k+1

. (25)

In which= n+2w+l. Taking the discrete Laplace transform method, we obtain

us( , t)=

2

1

(eat 1)

2

1

k=0

(1)k 12(k+1)(k+1)3

m+l=k+1m,l0

(k+1)!m!l! M

m

n+w+=kn,w,0

k!n!w!!

n1

w2

t

0

(ea(t ) 1){G,,k+1(3 , )+1 G,+,k+1(3 , )+2 G,+2,k+1(3 , )}d (26)

where

G,a,b(d, t)=

l=0

(b)lt(l+b)a1

(l+1)((l+b)a) (d)l, (27)

(b)l= b(b+1) (b+l1)is the Pochhammer polynomial[27]. In obtaining Eq.(26),the following property of the Gfunction is used:

Ga,b,c(p, t)=L1

sb

(sa p)c

; Re(acb) >0, Re(s) >0, p

sa


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Fig. 2. Velocity fields for different values ofandMwhen keeping other parameters fixed.

Fig. 3. Velocity fields for different values of and Mwhen keeping other parameters fixed.

Eqs.(32),(14)(16)can reduce to dimensionless equations as follows (for brevity the dimensionless mark are omittedhere):

(y, t)

t= 1

Pr

[3NR+4

3NR

] 2 (y, t)

y2 +

[u(y, t)

y

]2

, (34)

whereNR= kkT

43. Lettingg(y, t)=

u(y,t)

y

2, k0= 3NR3NR+4 , Eq.(34)can be rewritten as

(y, t)

t= 1

k0Pr

2 (y, t)

y2 +g(y, t). (35)

The corresponding initial and boundary conditions become:

(y, 0)=0, fory > 0, (36) (0, t)=f(t), fort 0, (37)

(y, t)0, (y, t)y

0, fory . (38)


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Fig. 4. Velocity fields for different values oftwhen keeping other parameters fixed.

Fig. 5. Temperature profiles for different values oftwhenf(t)=1.

where f(t)= (0(t))/(0(0)). In order to obtain the exact solution to Eqs. (35)(38),using the Fourier sinetransform on Eq.(35),we obtain

ds( , t)

dt+ 1

k0Pr2s( , t)=

2

k0Prf(t)+gs( , t) (39)

s( , 0)=0, (40)where s( , t) andgs( , t) denote theFourier sine transform of(y, t) andg(y, t) with respect toy, respectively. The solutionof the ordinary differential equation(39)subject to the initial condition(40)is given by

s( , t)=e2t/(k0Pr)

t0

2

k0Prf( )+gs( , )

e

2 /(k0Pr)d . (41)

Inverting Eq.(41)by means of the Fourier sine transform, we get

(y, t)=2

0sin(y)e

2

t/(k0Pr) t

0

2

k0Prf( )+gs( , ) e2 /(k0Pr)dd . (42)


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Fig. 6. Temperature profiles for different values oftwhenf(t)=t.

Fig. 7. Temperature profiles for different values of Pr when keeping other parameters fixed.

4. Results and conclusions

The purpose of this paper is to provide exact solutions for the unsteady MHD flow of a generalized Burgers fluid due toan exponential accelerating plate. The exact solutions for the velocity field u(y, t)and temperature field (y, t)in terms oftheGfunction are obtained by using the Fourier sine transform and Laplace transform. Moreover, some figures are plottedto show the behavior of some different emerging parameters of interest involved in the velocity field and temperaturefield.

Figs. 23show the velocity changes with the fractional parameters and the magnetic field parameter. It is clearly seenthat the smaller the, the more slowly the velocity decays. However, one can see that an increase in material parameter has quite the opposite effect to that of. Moreover, the magnetic body force is favorable to the velocity decays. The velocityin the case of a magnetohydrodynamic fluid is less when compared with hydrodynamic fluid. This is due to the fact thatapplied transverse magnetic field produces a drag in the form of Lorentz force thereby decreasing the magnitude of velocity.Fig. 4is the velocity profile u vs. the time t. With the increasing the values oft, the velocity rapidly speeds up.Figs. 56demonstrate the influence oftfor the temperature field. As it was to be expected, it clearly shows that the non-Newtonian

effects are stronger at larger values oft. The greater the value oft, the higher the temperature. The temperature profilesare similar with different values oft, but not similar quantitatively.Fig. 7is the graph of temperature distribution vs. the


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Fig. 8. Temperature profiles for different values ofNRwhen keeping other parameters fixed.

Prandtl number Pr. It is clear that there is a fall in temperature with increasing the Prandtl number.Fig. 8depicts the effectof varyingNRfor the temperature field. The results show a marked decrease in the temperature distributions with increaseinNR.

Acknowledgments

The work is supported by the National Natural Science Foundations of China (No. 50936003, 51076012) and the openProject of State Key Lab. for Adv. Metals and Materials (2009Z-02) USTB.

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