Starting solutions for a viscoelastic fluid with fractional Burgers’ model in an annular pipe

Nonlinear Analysis: Real World Applications 11 (2010) 547–554

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Starting solutions for a viscoelastic fluid with fractional Burgers’ model in an annular pipe S. Hyder Ali Muttaqi Shah a,∗ , Haitao Qi b a

Institute of Mathematics & Computer Science, University of Sindh, Jamshoro, Pakistan

b

School of Mathematics & Statistics, Shandong University at Weihai, Weihai 264209, China

article

info

Article history: Received 24 April 2008 Accepted 12 January 2009 Keywords: Exact solutions Burgers’ fluid Fractional calculus Annular pipe

abstract In this work, we have discussed some simple flows of a viscoelastic fluid with fractional Burgers’ model in an annular pipe. The fractional calculus approach is introduced in the constitutive relationship of a Burgers’ fluid model. Exact analytical solutions are obtained by using Laplace and Weber transforms for two types of flows, namely: Poiseuille flow and Axial Couette flow. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Non-Newtonian fluids, such as paints, grease, oils, liquid polymers, glycerin etc, exhibit some remarkable phenomena which are due to their elastic nature. These fluids retain memory of their past deformations and their behavior is a function of these old deformations. Typical non-Newtonian characteristics include shear-thinning or shear-thickening, stress relaxation, normal stress differences etc. Because of these complexities there are several models of non-Newtonian fluids in the literature. In the category of non-Newtonian fluids, the fluids of differential type have acquired special status as well as much controversy [1]. These fluids cannot describe the influence of relaxation and retardation times. Recently, one of the rate type fluid models proposed by Burgers’ [2] has become very popular amongst researchers. This is because of its success in describing asphalt in geomechanics and food products such as cheese etc. There are numerous examples of the use of Burgers’ model to study asphalt and asphalt mixes [3]. Many of the developments in the theory of viscoelastic flows have been mainly restricted to the formulations of the basics equations and constitutive models. There are many fluids such as second grade, Maxwell, Oldroyd-B etc which lie in the category of the viscoelastic fluid, and some recent attempts regarding the flow of a viscoelastic fluid through porous media are present in [4–8]. In order to describe the viscoelasticity [9,10] the fractional calculus approach is very important. Recently, fractional calculus approach has encountered much success in the description of complex dynamics. In particular, it has been proved to be a valuable tool in handling viscoelastic properties. The starting point of the fractional derivative model of viscoelasticity is usually a classical differential equation which is modified by replacing the time derivatives of an integer order by the so-called Riemann-Liouville fractional calculus operators. This generalization allows one to describe precisely non-integer order derivative [11,12]. So, due to the importance of viscoelasticity, many researchers are engaged in solving the problems of non-Newtonian fluids [13–29] with fractional calculus approach. Recently, Fetecau [30] established the analytical solutions for a non-Newtonian fluid flow in pipe-like domains. Tong and Liu [31] studied the unsteady rotating flows of a non-Newtonian fluid in an annular pipe with Oldroyd-B fluid model. Tong

∗

Corresponding author. E-mail address: [email protected] (S. Hyder Ali Muttaqi Shah).

1468-1218/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2009.01.012

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S. Hyder Ali Muttaqi Shah, H. Qi / Nonlinear Analysis: Real World Applications 11 (2010) 547–554

et al. [32] discussed the flow of Oldroyd-B fluid with fractional derivative in an annular pipe. Khan et al. [33] discussed the starting flows for a fractional Burgers’ fluid between coaxial cylinders. In this paper, our aim is to investigate the unsteady flows of a viscoelastic fluid with fractional Burgers’ fluid model in an annular pipe. The fractional calculus approach is used in the constitutive equation of Burgers’ fluid model. The velocity fields are obtained for two types of flows, namely: (i) flow induced by the constant pressure gradient (Poiseuille flow) and (ii) flow induced by the motion of the outer cylinder while the inner is kept stationary (Axial Couette flow). 2. Governing equations The flow of the incompressible viscoelastic fluid is governed by the following equations dV

ρ

dt

= −∇ p + div S,

(1)

∇ · V = 0,

(2)

where ρ is the fluid density, V the velocity vector, p the pressure and S the extra-stress tensor. The constitutive equation for fractional Burgers’ fluid model is



β

β

1 + λα1 e Dαt + λα2 e D2t α S = µ 1 + λ3 e Dt






A,

(3)

where A = L + LT is the first Rivlin–Ericksen tensor with L the velocity gradient, µ the dynamic viscosity, λ1 and λ3 (<λ1 ) are the relaxation and retardation times, respectively, λ2 a new material constant of Burgers’ fluid, α and β are fractional calculus parameters such that 0 ≤ α ≤ β ≤ 1 and e Dαt is the upper convected time derivative defined by


e Dαt S = Dαt S + (V · ∇) S − LS − SLT and e D2t α S = e Dαt e Dαt S
where Dαt =∂tα is the fractional differential operator, which is defined as Z t 1 d Dαt [f (t )] = (t − τ )−α f (τ ) dτ , (0 ≤ α ≤ 1) 0 (1 − α) dt 0

(4)

(5)

with 0 (·) as the Gamma function. For the flow under consideration we have the velocity and the stress fields in the following form V (r , t ) = v(r , t )ez ,

S = S(r , t ),

(6)

where ez is the unit vector along the z-direction. The velocity field (6) automatically satisfies the incompressibility condition. Substituting Eq. (6) into Eq. (3) which together with the initial condition S(r , 0) = 0, yields Srr = Sθ z = Sr θ = Sθ θ = 0 and

∂α ∂ 2α + λα2 2α α ∂t ∂t

∂β ∂tβ

∂v , ∂r      2 ∂α ∂ 2α ∂α ∂v ∂v β 1 + λα1 α + λα2 2α Szz − 2Srz λα1 + λα2 α = −2µλ3 . ∂t ∂t ∂t ∂r ∂r 

1 + λα1





β

Srz = µ 1 + λ3



(7)

(8)

At the same time, the balance of linear momentum (1) in the presence of applied pressure gradient gives

ρ

∂v dp 1 ∂ =− + (rSrz ) . ∂t dz r ∂r

(9)

Eliminating Srz between Eqs. (7) and (9), we obtain the governing equation in the form



1 + λα1

∂α ∂ 2α + λα2 2α α ∂t ∂t



∂v 1 =− ∂t ρ



1 + λα1

∂α ∂ 2α + λα2 2α α ∂t ∂t



dp dz

  2 β β ∂ +ν 1 + λ3 ∂∂t β + ∂r2

1 ∂ r ∂r



v (r , t ) ,

(10)

where ν = µ/ρ is the kinematic viscosity of the fluid. 3. First problem Consider that the Burgers’ fluid is initially at rest between two infinitely long coaxial cylinders of radii R0 and R1 (>R0 ). At time t = 0+ the flow is generated due to the constant pressure gradient −Aρ = ∂ p/∂ z that acts on the liquid in the z-direction. So the flow is governed by Eq. (10) and its associated initial and boundary conditions become

S. Hyder Ali Muttaqi Shah, H. Qi / Nonlinear Analysis: Real World Applications 11 (2010) 547–554

549

v (r , 0) = ∂t v (r , 0) = ∂tt v (r , 0) = 0 for R0 ≤ r ≤ R1 , v (R0 , t ) = v (R1 , t ) = 0 for t > 0.

(11) (12)

Employing the non-dimensional quantities

v∗ =

v
, AR20 /ν

r∗ =

r R0

,

t∗ =

νt

, 2

R0

λ∗1,3 =

νλ1,3 R20

,

λ∗2 =

ν 2 λ2 R40

and

b=

R1 R0

,

(13)

we find Eqs. (10)–(12) in dimensionless form, after dropping ‘‘∗’’ becomes



1 + λα1

∂α ∂ 2α + λα2 2α α ∂t ∂t



    β   2 ∂v t −α t −2α ∂ 1 ∂ β ∂ = 1 + λα1 + λα2 + 1 + λ3 β + v, ∂t 0 (1 − α) 0 (1 − 2α) ∂t ∂r2 r ∂r

v (r , 0) = ∂t v (r , 0) = ∂tt v (r , 0) = 0 for 1 ≤ r ≤ b, v (1, t ) = v (b, t ) = 0 for t > 0.

(14) (15) (16)

To find the solution for Eq. (14) subject to the initial and boundary conditions (15) and (16), we first make the change of unknown function, by using

v (r , t ) = V (r ) + u (r , t ) , (17)

2 2 2 where V (r ) = b − r /4 + b − 1 ln (r /b) /4 ln b is the steady solution. The corresponding transient problem will take the form



1 + λα1

2α ∂α α ∂ + λ 2 ∂tα ∂ t 2α



    β   2 ∂u ∂ 1 ∂ β t −β β ∂ α t −2α α t −α + λ − λ = 1 + λ3 β + u + λ 2 0 (1−2α) 1 0 (1−α) 3 0 (1−β) , ∂t ∂t ∂r2 r ∂r

u (r , 0) = −V (r ) ,

∂t u (r , 0) = ∂tt u (r , 0) = 0, u (1, t ) = u (b, t ) = 0, t > 0.

(18) (19) (20)

To obtain the exact analytical solution of the above problem (18)–(20), we first apply Weber transform with respect to r, defined as follows uh (ρi , t ) =

b

Z

ru (r , t ) H (ρi , r ) dr ,

(21)

1

and its inverse is u (r , t ) =

∞ X

uh (ρi , t )

H (ρi , r ) N (ρi )

i=1

,

(22)

where H (ρi , r ) = J0 (ρi r ) Y0 (ρi ) − J0 (ρi ) Y0 (ρi r ) , ρi is the positive root of H (ρi , b) and 1 N (ρi )

=

π2 2

J02

ρi2 J02 (ρi b) , (ρi ) − J02 (ρi b)

(23)

where J0 (·) and Y0 (·) are the Bessel functions of the first and second kinds of order zero. Now applying Weber transform to Eqs. (18) and (19), we get



1 + λα1

∂α ∂ 2α + λα2 2α α ∂t ∂t

uh (ρi , 0) = −Vh ,



 β  ∂ uh β ∂ = −ρi2 1 + λ3 β uh ∂t  ∂ t −α  2F (ρi ) t t −2α t −β β α α + λ + λ − λ , 1 2 3 0 (1 − α) 0 (1 − 2α) 0 (1 − β) π ρi2

∂t uh (ρi , 0) = ∂tt uh (ρi , 0) = 0,

(24) (25)

in which F (ρi ) =

bπ ρi 2

[J1 (ρi b) Y0 (ρi ) − J0 (ρi ) Y1 (ρi b)] − 1

b

Z

rV (r ) H (ρi , r ) dr .

and Vh = 1

Let uh (ρi , s) be the Laplace transform of uh (ρi , t ) defined by uh (ρi , s) =

∞

Z

uh (ρi , t ) e−st dt , 0

s > 0.

(26)

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S. Hyder Ali Muttaqi Shah, H. Qi / Nonlinear Analysis: Real World Applications 11 (2010) 547–554

Now applying Laplace transform to Eq. (24), having in mind the initial condition (25), we get

uh (ρi , s) =



2

1−

πρ

2 i

J0 (ρi ) J0 (ρi b)

β





A∗ (ρi , s) +

2 λα1 + λα2 sα − λ3 sβ−α



F (ρi ) β



π ρi2 s1−α s + λα1 sα+1 + λα2 s2α+1 + λ3 ρi2 sβ + ρi2

,

(27)

where A∗ (ρi , s) =

1 + λα1 sα + λα2 s2α β



ρi2 s + λα1 sα+1 + λα2 s2α+1 + λ3 ρi2 sβ + ρi2

.

(28)

Now rewriting Eq. (28) into the series form, we have

A∗ (ρi , s) =

1

−

sρi2

∞ X (−1)m m=0

m+1) λα( 2

j,k,l≥0

X

j+k+l=m

 1 j!k!l!

β

sδ + λ3 sβ+δ

βj

ρi2(j+k) λ3



m! ,
m+1 sα + λα1 /λα2

(29)

where δ = l + β j − α m − m − α − 1. And its inverse Laplace transform will take the form A (ρi , t ) =

1

∗

ρi2

−

∞ X (−1)m m=0

m+1) λα( 2

"  α   α # β X ρ 2(j+k) λβ j λ1 α λ3 (m) λ (m) i 3 α(m+1)−δ−1 t Eα,α−δ − α t + β Eα,α−δ−β − 1α t α j ! k ! l ! λ t λ2 2 j+k+l=m j,k,l≥0

(30)

and similarly, we can obtain

L−1

 

 

1

 s1−α s + λα sα+1 + λα s2α+1 + λβ ρ 2 sβ + ρ 2  i 1 2 3 i 



=

∞ X (−1)m m=0

λ2α(m+1)

 α  X ρ 2(j+k) λβ j λ i 3 α m−δ (m) t Eα,1−δ − 1α t α (31) j ! k ! l ! λ2 j+k+l=m j,k,l≥0

and L−1

 

 

1

   s1−2α s + λα sα+1 + λα s2α+1 + λβ ρ 2 sβ + ρ 2  i 1 2 3 i

=

∞ X (−1)m

 α  X ρ 2(j+k) λβ j λ i 3 α m−α−δ (m) t Eα,1−δ−α − α1 t α , j!k!l! λ2 j+k+l=m j,k,l≥0

(32)

m+1) λα( 2 P∞ n where Eλ,µ (z ) = n=0 z /0 (λn + µ) is the Mittag-Leffler function and to obtain Eqs. (30)–(32) the following property of m=0

inverse Laplace transforms is used

( L

−1

k!sλ−µ sλ ∓ c


k+1

)


(k) = t λk+µ−1 Eλ,µ ±ct λ ,

Re (s) > |c |1/λ .




(33)

Now taking the inverse Laplace and Weber transforms of Eq. (27) and using Eqs. (31) and (32) into the resulting one, we get

v (r , t ) = V (r ) + π

∞ X

J02 (ρi b) H (ρi , r )

( 1−

J0 (ρi )

" 1−

∞ X (−1)m

j,k,l≥0

X

λ2α(m+1) j+k+l=m   α )# βj β λ3 ρi t λα1 α λ3 (m) λ (m) × Eα,α−δ − α t + β Eα,α−δ−β − 1α t α j!k!l! λ2 t λ2 (  α  β j β 2 j + k + 1 j , k , l ≥ 0 ( ) ∞ X λ3 (m) λ t α m−δ (−1)m X λ3 ρi + F (ρi ) − E − 1α t α α(m+1) β−α α,α−δ−β+1 j ! k ! l ! t λ λ 2 j+k+l=m m=0 2 ))  α    λ λα (m) λα (m) + λα1 Eα, − α1 t α + α2 Eα, − α1 t α . 1−δ 1−δ−α λ2 t λ2  ρi J02 (ρi ) − J02 (ρi b) i =1 (  2(j+k+1) α(m+1)−δ−1  2

J0 (ρi b)

m=0

(34)

S. Hyder Ali Muttaqi Shah, H. Qi / Nonlinear Analysis: Real World Applications 11 (2010) 547–554

(a) Oldroyd-B fluid (λ2 = 0).

551

(b) Burgers’ fluid (λ2 = 1.5).

Fig. 1. Velocity profiles v (r , t ) /Vm for different values of α when t = 0.1, λ1 = 2, λ3 = 1, β = 0.8 (solid line α = 0.1, dash line α = 0.3, dot line α = 0.5, dash-dot line α = 0.7).

(a) Oldroyd-B fluid (λ2 = 0).

(b) Burgers’ fluid (λ2 = 1.5).

Fig. 2. Velocity profiles v (r , t ) /Vm for different values of α when t = 2, λ1 = 2, λ3 = 1, β = 0.8 (solid line α = 0.1, dash line α = 0.3, dot line α = 0.5, dash-dot line α = 0.7).

4. Second problem For this problem, suppose that the fluid is between two infinitely long coaxial cylinders. Initially the system is at rest and at time t = 0+ the outer cylinder moves with velocity V while the inner cylinder is kept stationary. The associate non-dimensional governing equation in the absence of pressure gradient will take the form



1 + λα1

2α ∂α α ∂ + λ 2 ∂tα ∂ t 2α



  β   2 ∂v ∂ 1 ∂ β ∂ = 1 + λ3 β + v, ∂t ∂t ∂r2 r ∂r

(35)

and its associated initial and boundary conditions will be

v (r , 0) = ∂t v (r , 0) = ∂tt v (r , 0) = 0 for 1 ≤ r ≤ b, v (1, t ) = 0, v (b, t ) = 1, for t > 0.

(36) (37)

For this problem, we have introduced the quantity v ∗ = v/V in place of v ∗ = v/ AR20 /ν in Eq. (13) while all the rest of the quantities are kept the same. Making the change of unknown function




v (r , t ) = U (r ) + u (r , t )

(38)

where U (r ) = ln r / ln b, we attain the next transient problem i.e.,



1 + λα1

∂α ∂ 2α + λα2 2α α ∂t ∂t

u (r , 0) = −U (r ) , u (1, t ) = u (b, t ) = 0,



  β   2 ∂u ∂ 1 ∂ β ∂ = 1 + λ3 β + u, ∂t ∂t ∂r2 r ∂r

∂t u (r , 0) = ∂tt u (r , 0) = 0 for 1 ≤ r ≤ b, for t > 0.

(39) (40) (41)

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S. Hyder Ali Muttaqi Shah, H. Qi / Nonlinear Analysis: Real World Applications 11 (2010) 547–554

(a) Oldroyd-B fluid (λ2 = 0).

(b) Burgers’ fluid (λ2 = 1.5).

Fig. 3. Velocity profiles v (r , t ) /Vm for different values of β when t = 0.1, λ1 = 2, λ3 = 1, α = 0.2 (solid line β = 0.3, dash line β = 0.5, dot line β = 0.7, dash-dot line β = 0.9).

(a) Oldroyd-B fluid (λ2 = 0).

(b) Burgers’ fluid (λ2 = 1.5).

Fig. 4. Velocity profiles v (r , t ) /Vm for different values of β when t = 2, λ1 = 2, λ3 = 1, α = 0.2 (solid line β = 0.3, dash line β = 0.5, dot line β = 0.7, dash-dot line β = 0.9).

(a) t = 0.1.

(b) t = 2.

Fig. 5. Velocity profiles v (r , t ) /Vm for different values of λ2 when λ1 = 5, λ3 = 1, α = 0.2, β = 0.6.

Using Weber and Laplace transforms to Eq. (39), having in mind Eqs. (40) and (41), we get uh (ρi , s) = −

2 J0 (ρi )

π J0 (ρi b)

A∗ (ρi , s)

where A∗ (ρi , s) is given by Eq. (28).

(42)

S. Hyder Ali Muttaqi Shah, H. Qi / Nonlinear Analysis: Real World Applications 11 (2010) 547–554

(a) Newtonian fluid.

(b) Second grade fluid.

(c) Oldroyd-B fluid.

(d) Burgers’ fluid.

553

Fig. 6. Velocity profiles v (r , t ) /Vm for different constitutive models (λ1 = 2, λ2 = 1.5, λ3 = 1, α = 0.2, β = 0.6).

Now, adopting the similar procedure as in previous section, the expression for the velocity field is given by

( β j 2 j+k+1) j,k,l≥0 ∞ ∞ X X J0 (ρi ) J0 (ρi b) H (ρi , r ) (−1)m X λ3 ρi ( v (r , t ) = −π 1 − α(m+1) ln b j!k!l! J02 (ρi ) − J02 (ρi b) j+k+l=m i=1 m=0 λ2 #) "  α   α  β λ3 (m) λ λ1 α (m) α(m+1)−δ−1 + β Eα,α−δ−β − 1α t α . × t Eα,α−δ − α t λ2 t λ2 ln r

(43)

5. Numerical results and discussion This section displays the graphical illustration of the velocity profiles for the first problem (Poiseuille flow) for different types of fluid models. Special attention has been given to two types of fluid models, namely: fractional Oldroyd-B and fractional Burgers’ fluid models.  For the  sake of simplicity, all the diagrams are plotted by taking b = 2 and Vm = 1 4



b2 −

b 2 −1 2 ln b




+ b2 − 1 ln

1 b

q

b2 −1 2 ln b

/4 ln b in order to unit the velocity. Graphically, in all the figures the symmetric

parabolic flow profiles are obtained for both fractional Oldroyd-B and fractional Burgers’ fluid models. We interpret these results with respect to the variation of emerging parameters of interest. Figs. 1–4 are established to show the behavior of non-integers fractional parameters α and β for small as well as for large time. It is observed in Fig. 1 that for small time t = 0.1 the increase in fractional parameter α will increase the velocity profiles for both the fluid models. While quite the opposite effects are observed for large time t = 2 i.e., the increase in fractional parameter α will decrease the velocity profiles for both fractional Oldroyd-B as well as for fractional Burgers’ fluid models. However, in Figs. 2 and 3 the increase in β shows an opposite trend for both the fluid models. Fig. 5 is plotted to demonstrate the effects of new material parameter λ2 . It is observed that the velocity is an increasing function of new material parameter λ2 of the fractional Burgers’ fluid. Moreover, it is observed that velocity profiles for large time t = 2 are larger than those of initial time t = 0.1.

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S. Hyder Ali Muttaqi Shah, H. Qi / Nonlinear Analysis: Real World Applications 11 (2010) 547–554

Fig. 6 displays the graphical illustration of time t on the velocity profiles for different constitutive models, namely: Newtonian, fractional second grade, fractional Oldroyd-B and fractional Burgers’ fluid models. It is observed that the phenomenon of velocity overshooting is happening here for both Oldroyd-B and Burgers’ fluid model. But for the Newtonian and second grade fluid model, the over shooting phenomenon is not happening. Thus the time t and different emerging parameters α, β and λ2 show the strong effects on the velocity field. 6. Conclusion In this work, we have discussed some starting flows of a viscoelastic fluid in an annular pipe with fractional Burgers’ fluid model. Fractional calculus approach is used in the constitutive model of the ordinary Burgers’ fluid model. Exact analytical solutions are obtained by using Laplace and Weber transforms for two types of flows, namely: (i) flow due to constant pressure gradient and, (ii) flow due to the constant velocity of the outer cylinder. It has been shown that the fractional constitutive relationship model is more flexible than the conventional model in describing the properties of viscoelastic fluids. Moreover, it is shown graphically that different emerging parameters of interest have strong effects on the velocity field. The phenomenon of velocity overshooting is also observed, as observed in [34] for Oldroyd-B and Burgers’ fluid model. But for the Newtonian and second grade fluid model, this overshooting phenomenon does not happen [34]. Conclusively, some non-traditional effects on the velocity profiles are observed for the fractional parameters α and β . Acknowledgments The authors are grateful to the reviewers for their useful comments. References [1] J.E. Dunn, K.R. 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