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Exact solutions for a viscoelastic fluid with the generalized Oldroyd-B model
Nonlinear Analysis: Real World Applications 10 (2009) 2590–2599
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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
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Exact solutions for a viscoelastic fluid with the generalized Oldroyd-B model S. Hyder Ali Muttaqi Shah a,∗ , M. Khan a , Haitao Qi b a Department of Mathematics, Quaid-i-Azam University 45320, Islamabad, Pakistan b Department of Applied Mathematics, Shandong University at Weihai, Weihai 264209, China
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Article history: Received 4 December 2007 Accepted 10 March 2008 Keywords: Exact solutions Generalized Oldroyd-B fluid Fox H-function Fractional calculus
a b s t r a c t In this paper, the generalized Oldroyd-B model with the fractional calculus approach is used. Exact analytic solutions for the velocity and the stress fields in term of the Fox H-function are obtained by using the discrete Laplace transform for two types of flows of a viscoelastic fluid, namely, (i) flow due to impulsive motion in the presence of a constant pressure gradient and (ii) flow induced by an impulsive pressure gradient. The influence of various parameters of interest on the velocity and shear stress has been shown and discussed through several graphs. Moreover, a comparison between Oldroyd-B and generalized Oldroyd-B fluids is also made. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Since a large class of real fluids does not exhibit a linear relationship between stress and the rate of deformation this is now of great interest to scientists and engineers. This attraction has grown considerably during the past few decades because of their various applications in engineering and industry. These applications ranges from oil and gas well drilling to well completion operations, from industrial processes involving waste fluids, synthetics fibers, foodstuffs, extrusion of molten plastic, cooling of metallic plates in a bath, exotic lubricants, artificial and natural gels and as well as some flows of polymer solutions. The modeling of the equations governing the non-Newtonian fluids gives rise to highly nonlinear differential equations. Generally rheological properties of materials are specified by their so called constitutive equations. The simplest constitutive equation for a fluid is the Newtonian one on which the classical Navier-Stokes’ theory is based. The mechanical behavior of many fluids can be described by this theory, however, there are many rheologically complicated fluids for which this theory is inadequate. For such reasons various models have been proposed. These models are usually classified as fluids of differential, rate and integral types [1–3]. The differential and rate type models are used to describe the response of fluids that have a slight memory such as dilute polymeric solutions, whereas the integral models are used to describe the response of fluids that have considerable memory such as polymer melts. Recently, the Oldroyd-B fluid [4] has acquired a special status amongst the many fluids of the rate type, as it includes as special cases of classical Newtonian and Maxwell fluids. Although computer techniques make the complete integration of the governing equations of motion feasible, the accuracy of the results can only be established by a comparison with an exact solution. Exact solutions corresponding to different flows of Newtonian as well as non-Newtonian fluids are very important for several reasons. In addition to serving as approximations to some specific problems, they also play a very important role, namely, they can be used as tests to verify the numerical or empirical methods that are developed to study complex flow problems. An Oldroyd-B fluid is one particular subclass of non-Newtonian fluids for which one can reasonably hope to obtain an exact solution. ∗ Corresponding author. E-mail address: [email protected] (S. Hyder Ali Muttaqi Shah). 1468-1218/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2008.03.012
S. Hyder Ali Muttaqi Shah et al. / Nonlinear Analysis: Real World Applications 10 (2009) 2590–2599
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Recently, fractional calculus has gained considerable interest due to its applications in different areas of physics and engineering, including complex dynamics. The constitutive equations with fractional derivative have been proved to be a valuable tool to handle viscoelastic properties. In general, these equations are derived from known models via substituting time ordinary derivatives of stress and strain by derivatives of fractional order from which one can define the non-integer order integral and derivatives [5,6]. To date, various researchers [7–25] have introduced the fractional calculus approach into various rheological problems. Fractional derivatives have been found to be quite flexible in describing viscoelastic behavior. Recently, Haitao and Xu [26] have investigated the Stokes’ problem for a viscoelastic fluid with a generalized Oldroyd-B fluid model and Khan et al. [27] has discussed some accelerated flows of generalized Oldroyd-B fluids. No attempt has been made regarding the exact solutions for flows due to constant pressure gradient and impulsive pressure gradient of a generalized Oldroyd-B fluid when the plate is moving with the constant velocity. The velocity and the stress fields of the flows for an Oldroyd-B fluid with fractional derivative model are described by fractional differential equations. The expressions for the velocity and the stress fields are constructed by means of discrete Laplace transform of the sequential fractional derivative. Moreover, the solutions for fractional Maxwell, fractional second grade and generalized Oldroyd-B fluids can be recovered by the current analysis. 2. Governing equations The fundamental equations governing the unsteady motion of an incompressible fluid are div V = 0,
ρ
dV
(1)
= div T,
dt
(2)
where V is the velocity field, ρ the density, T the Cauchy stress tensor and d/dt the material time derivative. The Cauchy stress tensor T for a generalized Oldroyd-B fluid [24,26,27] is ! T = −pI + S,
1 + λα
Dα
Dtα
S = µ 1 + θβ
Dβ
Dtβ
A1 ,
(3)
in which p is the pressure, I the identity tensor, µ the dynamic viscosity, S the extra stress tensor, λ and θ are relaxation and retardation times, respectively, α and β are fractional calculus parameters such that 0 ≤ α ≤ β ≤ 1, and first Rivlin–Ericksen tensor A1 is given by A1 = L + L T ,
L = grad V,
(4)
where T denotes the matrix transpose and Dα S Dtα
= Dαt S + (V.∇ )S − LS − SLT ,
Dβ A1 Dtβ
(5)
β
= Dt A1 + (V.∇ )A1 − LA1 − A1 LT ,
(6)
β
in which Dαt and Dt are the fractional differentiation operators of order α and β with respect to t, respectively and may be defined as [5,6] Z 1 d t f (τ) Dpt [f (t)] = dτ, 0 ≤ p ≤ 1, (7) 0(1 − p) dt 0 (t − τ)p where 0(·) is the Gamma function. For unidirectional flow, we consider the velocity and the stress of the form V = u(y, t)i,
S = S(y, t),
(8)
where u is the x-component of velocity and i is the unit vector in the x-direction. Thus using Eq. (8), the continuity Eq. (1) is satisfied identically and (3)2 and (5), having in mind the initial condition S(y, 0) = 0, yield Sxz = Syy = Syz = Szz = 0 and
(1 + λα Dαt )T (y, t) = µ(1 + θβ Dβt )∂y u(y, t), α α
α
β
(9)
(1 + λ Dt )Sxx − 2λ T ∂y u(y, t) = −2µθ (∂y u) , 2
(10)
where Sxy = T (y, t) is the tangential stress. The balance of linear momentum (2) leads to the following governing equation
∂y T (y, t) − ∂x p = ρ∂t u(y, t), where ρ is the constant density and p the pressure in the flow direction.
(11)
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3. Flow due to impulsive motion in the presence of constant pressure gradient Let us consider the flow problem of an incompressible Oldroyd-B fluid with fractional derivative model occupies plane y ≥ 0 and x-axis is chosen as plane wall. Initially, the system is at rest and at time t = 0+ the fluid is suddenly set in motion
due to a constant pressure gradient and by the motion of the plane wall. The corresponding partial differential equation and the initial and boundary conditions take the form ! 2 t −α ∂u β ∂ u = − 1 + λα A + ν(1 + θβ Dt ) 2 , ∂t 0(1 − α) ∂y
(1 + λα Dαt )
(12)
where A = (1/ρ)(dp/dx) is the constant pressure gradient. u(y, 0) =
(13)
u(0, t) = U,
∂t u(y, 0) = 0, y > 0, t > 0, u(y, t), ∂y u(y, t) → 0, as y → ∞,
(14) (15)
where ν is the kinematic viscosity. To obtain an exact solution of above initial value problem, the Laplace transform is used. Let u(y, s) be the Laplace transform of u(y, t) defined by Z ∞ u(y, s) = u(y, t)e−st dt, s > 0. (16) 0
Taking the Laplace transform of Eqs. (12)–(15) we arrive at d2 u dy2
−
s(1 + λα sα )
A(1 + λα sα ) . u = − ν(1 + θβ sβ ) νs(1 + θβ sβ )
u(0, s) =
U
u(y, s),
∂y u(y, s) → 0,
s
(17)
,
(18) as y → ∞.
(19)
The solution of Eq. (17) satisfying the boundary conditions (18) and (19) is of the following form !1/2 !1/2 U s(1 + λα sα ) A s (1 + λα sα ) . exp −y 1 − exp u(y, s) = − − y s ν 1 + θβ sβ s2 ν 1 + θβ sβ
(20)
As the fluid is moved by the action of the shear stress at the plate, the stress field can be calculated. Now taking the Laplace transform of Eq. (9) and introducing Eq. (20), we get !−1/2 !1/2 s (1 + λα sα ) s (1 + λα sα ) T (y, s) = −U ρ exp −y β β β β ν 1+θ s ν 1+θ s
−
Aρ s
s(1 + λα sα ) ν 1 + θβ sβ
!−1/2
exp −y
s(1 + λα sα ) ν 1 + θβ sβ
!1/2
,
(21)
where T (y, s) is the Laplace transform of T (y, t). In order to avoid the burdensome calculations of residues and contour integrals, we will apply the discrete inverse Laplace transform [5,6] to get the velocity and the stress fields. Now writing Eqs. (20) and (21) in series form as q ∞ (−y λα /νθβ )k X ∞ ∞ (−1)n 0 m − 2k 0 2k + n X U (−1)m X k u(y, s) = +U k! m!λαm n=0 n!θβn 0 k 0 − k s 2 (β−α−1)+mα+nβ+1 m=0 2 2 q ∞ ∞ (−y λα /νθβ )k X ∞ (−1)n 0 m − 2k 0 2k + n X (−1)m X +A , k! m!λαm n=0 n!θβn 0 k 0 − k s 2k (β−α−1)+mα+nβ+2 k=1 m=0 2 2 ∞ ∞ ∞ (−1)n 0 m + 1−2 k 0 k−2 1 + n X (−y)k (λα /νθβ )(k−1)/2 X (−1)m X T (y, s) = −U ρ k! m!λαm n=0 n!θβn 0 1−k 0 k−1 s k−2 1 (β−α−1)+mα+nβ k=0 m=0 2 2 ∞ ∞ ∞ (−1)n 0 m + 1−2 k 0 k−2 1 + n X (−y)k (λα /νθβ )(k−1)/2 X (−1)m X − Aρ . k! m!λαm n=0 n!θβn 0 1−k 0 k−1 s k−2 1 (β−α−1)+mα+nβ+1 k=0 m=0 2 2 s
k =1
(22)
(23)
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Applying the discrete inverse Laplace transform to Eqs. (22) and (23) we get k q k α /νθβ λ − y ∞ ∞ ∞ (−1)n 0 m − 2k 0 2k + n t 2 (β−α−1)+mα+nβ X X (−1)m X u(y, t) = U + U k! m!λαm n=0 n!θβn 0 k 0 − k 0 k (β − α − 1) + mα + nβ + 1 m=0 k=1 2 2 2
+A
∞ X
q
−y λα /νθβ
k
k!
k=1
(−1)n 0 m −
∞ ∞ X (−1)m X
m!λαm n=0 n!θβn 0
m=0
k
2
2
0
k
0 − 2k 0
k
2 k
2
+ n t 2 (β−α−1)+mα+nβ+1
k
,
(24)
(β − α − 1) + mα + nβ + 2
and k−1 ∞ ∞ ∞ (−1)n 0 m + 1−2 k 0 k−2 1 + n t 2 (β−α−1)+mα+nβ−1 X (−1)m X (−y)k (λα /νθβ )(k−1)/2 X T (y, t) = −U ρ k! m!λαm n=0 n!θβn 0 1−k 0 k−1 0 k−1 (β − α − 1) + mα + nβ m=0 k=0 2 2 2 k− 1 ∞ ∞ ∞ (−1)n 0 m + 1−2 k 0 k−2 1 + n t 2 (β−α−1)+mα+nβ X (−y)k (λα /νθβ )(k−1)/2 X (−1)m X . − Aρ k! m!λαm n=0 n!θβn 0 1−k 0 k−1 0 k−1 (β − α − 1) + mα + nβ + 1 k=0 m=0 2 2 2
In terms of the Fox H-function, Eqs. (24) and (25) take the simpler forms q 1− k ,1 , 1−m+ k ,0) ∞ ∞ (−y λα /νθβ )k X 2 X (−1)m k (β−α−1)+mα 1,2 tβ ( 2 ) ( u(y, t) = U + U t2 H2,4 α m β k! m!λ θ k k m=0 k=1
(0,1),(1− 2 ,0),(1+ 2 ,0),( 2k (α+1−β)−mα,β)
+A
∞ X k=1
q
−y λα /νθβ k!
k ∞ X (−1)m
m!λαm
m=0
t
− 2k (α+1−β)+mα+1 1,2
H2,4
(25)
(1− k ,1),(1−m+ k ,0) 2 2 tβ θβ k k
,
(26)
(0,1),(1− 2 ,0),(1+ 2 ,0),( 2k (α+1−β)−mα−1,β)
and T (y, t) = −U ρ
∞ ∞ X (−y)k (λα /νθβ )(k−1)/2 X (−1)m
k!
k=0
m=0
m!λαm
t
k−1
2
(β−α−1)+mα−1
( 3−k ,1),( k+1 −m,0) 2 tβ 2 β θ k+1 3−k
× H21,,42 − Aρ
(0,1),(
2
,0),(
2
,0),( k−2 1 (1+α−β)−mα+1,β)
∞ ∞ X (−y)k (λα /νθβ )(k−1)/2 X (−1)m
m!λαm m=0
k!
k=0
( 3−k ,1),( k−1 −m,0) 2 tβ 2 β θ k+1 3−k
t
k− 1
2
(β−α−1)+mα
× H21,,42
(0,1),(
2
,0),(
2
,0),( k−2 1 (1+α−β)−mα,β)
,
(27)
in which Hpm,,qn (z) is the Fox H-function. Now for λ → 0, the above expressions (26) and (27) reduce to that of Khan et al. [22] and making A = (1/ρ)(dp/dx) = 0 the Eqs. (26) and (27) reduce to that of Haitao et al. [26]. In obtaining Eqs. (26) and (27), the following property of the Fox H-function is used ∞ h i X 0(a1 + A1 k) . . . 0 ap + Ap k k (1−a1 ,A1 ),...,(1−ap ,Ap ) 1,p x . Hp,q+1 −x|(0,1),(1−b ,B ),...,(1−bq ,Bq ) = (28) 1 1 k=0
k!0(b1 + B1 k) . . . 0 bq + Bq k
4. Flow induced by an impulsive pressure gradient Let us consider the flow problem of a generalized Oldroyd-B fluid bounded by an infinite plane wall at y = 0. Initially, the system is at rest and at time t = 0+ the fluid suddenly set in motion due to a impulsive pressure gradient and by the motion of the plane wall. The corresponding partial differential equation and the initial and boundary conditions take the form 2 ∂u β ∂ u = −K (1 + λα Dαt )δ(t) + ν(1 + θβ Dt ) 2 , ∂t ∂y where K = k/ρ and K δ(t) = (1/ρ)(dp/dx) is the unit impulsive pressure gradient and Dαt δ(t) = t−α−1 /0(−α).
(1 + λα Dαt )
(29)
u(y, 0) =
(30)
∂t u(y, 0) = 0, y > 0, u(0, t) = U, t > 0, u(y, t), ∂y u(y, t) → 0, as y → ∞,
(31) (32)
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Fig. 1. Velocity profiles U (y, t) versus y for different values of λ keeping θ, α, β, A and t fixed (constant pressure gradient).
Applying the discrete Laplace transform to Eqs. (29)–(32) and solving the resulting ordinary differential equation we find the solution of the form ( " ( 1/2 ) 1/2 )# U s(1 + λα sα ) s(1 + λα sα ) K u(y, s) = exp −y 1 − exp − y − , (33) s ν(1 + θβ sβ ) s ν(1 + θβ sβ ) and the corresponding stress will be of the form T (y, s) = −ρ(U + K )
s(1 + λα sα )
−1/2
(
exp −y
ν(1 + θβ sβ )
s(1 + λα sα )
1/2 )
ν(1 + θβ sβ )
.
(34)
Adopting a similar procedure as above, the expressions for the velocity and the stress fields in case of impulsive pressure gradient are given by q k u(y, t) = U + (U + K )
∞ X
−y λα /νθβ k!
k=1
× H21,,42
∞ X (−1)m m=0
m!λαm
(1− k ,1),(1−m+ k ,0) 2 2 tβ θβ k k
t− 2 (1+α−β)+mα k
,
(35)
(0,1),(1− 2 ,0),(1+ 2 ,0),( 2k (1+α−β)−mα,β)
and T (y, t) = −ρ(U + K )
∞ ∞ X (−y)k (λα /νθβ )(k−1)/2 X (−1)m
k!
k=0
m=0
(1+ 1−k ,1),(1−m+ k−1 ,0) 2 2 tβ β θ k+1 3−k k−1
m!λαm
t
k−1
(0,1),(
2
,0),(
2
,0),(
2
(β−α−1)+mα−1
× H21,,42
2
(1+α−β)−mα+1,β)
.
(36)
Making λ → 0 in Eqs. (35) and (36), the solutions reduce to that of Khan et al. [22] for the case of flow induced by an impulsive pressure gradient. 5. Numerical results and discussion This section displays the graphical results of the flows considered in this paper for different parameters of interest. A comparison between an Oldroyd-B (panel a) and generalized Oldroyd-B (panel b) fluids is also made graphically. For the sake of simplicity all the diagrams are plotted by assuming U = ν = ρ = 1. Fig. 1 shows the behavior of material parameterλ. As expected, an increase in λ decreases the velocity profiles for Oldroyd-B as well as generalized Oldroyd-B fluids. From Fig. 2, an increase in material parameter θ shows quite the opposite effect to that of λ for both Oldroyd-B and generalized Oldroyd-B fluids. In Fig. 3, the variations of non-integer fractional parameters of interest, α and β is shown. By increasing α, the velocity profile decreases for Oldroyd-B as well as generalized Oldroyd-B fluids, however, an increase in β shows an opposite trend for both the fluids. Further, it is noted that the velocity profiles for a generalized Oldroyd-B fluid are greater in magnitude than those of an Oldroyd-B fluid. Fig. 4 is shows the variations of flow velocity for different values of time. With increasing
S. Hyder Ali Muttaqi Shah et al. / Nonlinear Analysis: Real World Applications 10 (2009) 2590–2599
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Fig. 2. Velocity profiles U (y, t) versus y for different values of θ keeping λ, α, β, A and t fixed (constant pressure gradient).
Fig. 3. Velocity profiles U (y, t) versus y for different values of α and β keeping λ, θ, A and t fixed (constant pressure gradient).
Fig. 4. Velocity profiles U (y, t) versus y for different values of t keeping λ, α, β, θ and A fixed (constant pressure gradient).
time the flow velocity increases for both the fluids. In Fig. 5, the effects of the pressure gradient A are shown for both the fluids. From this figure, we can see that for a favorable pressure gradient (A < 0), the velocity is positive and increases by
2596
S. Hyder Ali Muttaqi Shah et al. / Nonlinear Analysis: Real World Applications 10 (2009) 2590–2599
Fig. 5. Velocity profiles U (y, t) versus y for different values of A keeping λ, α, β, θ and t fixed (constant pressure gradient).
Fig. 6. Velocity profiles T (y, t) versus y for different values of t keeping λ, α, β, θ and A fixed (constant pressure gradient).
Fig. 7. Velocity profiles U (y, t) versus y for different values of λ keeping θ, α, β, K and t fixed (impulsive pressure gradient).
increasing favorable pressure gradient. For adverse pressure gradient (A > 0), the velocity may either be all positive or a combination of a positive and negative regime, depending on the value of the adverse pressure gradient. Moreover, with an
S. Hyder Ali Muttaqi Shah et al. / Nonlinear Analysis: Real World Applications 10 (2009) 2590–2599
Fig. 8. Velocity profiles U (y, t) versus y for different values of θ keeping λ, α, β, K and t fixed (impulsive pressure gradient).
Fig. 9. Velocity profiles U (y, t) versus y for different values of α and β keeping λ, θ, K and t fixed (impulsive pressure gradient).
Fig. 10. Velocity profiles U (y, t) versus y for different values of t keeping λ, α, β, θ and K fixed (impulsive pressure gradient).
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Fig. 11. Velocity profiles U (y, t) versus y for different values of K keeping λ, α, β, θ and t fixed (impulsive pressure gradient).
Fig. 12. Velocity profiles T (y, t) versus y for different values of t keeping λ, α, β, θ and K fixed (impulsive pressure gradient).
increase of adverse pressure gradient it decreases. Further, the curvature of the profiles depends on the amplitude of the pressure gradient. Fig. 6 shows the variations of shear stress for increasing time for Oldroyd-B and generalized Oldroyd-B fluids. It is noted that the maximum stress occurs near the wall and decreases far away from the wall. Figs. 7–12 are prepared to show the variations of velocity and stress fields for flow induced by an impulsive pressure gradient. Qualitatively, the observations in this case are similar to those of previous case, however, the velocity and the stress profiles are not similar quantitatively. Acknowledgement The authors would like to express their deepest gratitude to the reviewer for his constructive comments and suggestions regarding an earlier version of this manuscript. References [1] J.E. Dunn, K.R. Rajagopal, Fluids of differential type-critical review and thermodynamic analysis, Int. J. Eng. Sci. 33 (1995) 689–729. [2] K.R. Rajagopal, Mechanics of non-Newtonian fluids, in: Recent Developments in Theoretical Fluids Mechanics, in: Pitman Research Notes in Mathematics, 291, Longman, New York, 1993, pp. 129–162. [3] J.E. Dunn, K.R. Rajagopal, Thermodynamics, stability and boundedness of fluids of complexity 2 and fluids of second grade, Arch. Ration. Mech. Anal. 56 (1974) 191–252. [4] J.G. Oldroyd, On the formulation of rheological equations of state, Proc. R. Soc. Lond. Ser. A 200 (1950) 523–541. [5] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [6] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Press, Singapore, 2002.
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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
Short communication
Exact solutions for a viscoelastic fluid with the generalized Oldroyd-B model S. Hyder Ali Muttaqi Shah a,∗ , M. Khan a , Haitao Qi b a Department of Mathematics, Quaid-i-Azam University 45320, Islamabad, Pakistan b Department of Applied Mathematics, Shandong University at Weihai, Weihai 264209, China
article
info
Article history: Received 4 December 2007 Accepted 10 March 2008 Keywords: Exact solutions Generalized Oldroyd-B fluid Fox H-function Fractional calculus
a b s t r a c t In this paper, the generalized Oldroyd-B model with the fractional calculus approach is used. Exact analytic solutions for the velocity and the stress fields in term of the Fox H-function are obtained by using the discrete Laplace transform for two types of flows of a viscoelastic fluid, namely, (i) flow due to impulsive motion in the presence of a constant pressure gradient and (ii) flow induced by an impulsive pressure gradient. The influence of various parameters of interest on the velocity and shear stress has been shown and discussed through several graphs. Moreover, a comparison between Oldroyd-B and generalized Oldroyd-B fluids is also made. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Since a large class of real fluids does not exhibit a linear relationship between stress and the rate of deformation this is now of great interest to scientists and engineers. This attraction has grown considerably during the past few decades because of their various applications in engineering and industry. These applications ranges from oil and gas well drilling to well completion operations, from industrial processes involving waste fluids, synthetics fibers, foodstuffs, extrusion of molten plastic, cooling of metallic plates in a bath, exotic lubricants, artificial and natural gels and as well as some flows of polymer solutions. The modeling of the equations governing the non-Newtonian fluids gives rise to highly nonlinear differential equations. Generally rheological properties of materials are specified by their so called constitutive equations. The simplest constitutive equation for a fluid is the Newtonian one on which the classical Navier-Stokes’ theory is based. The mechanical behavior of many fluids can be described by this theory, however, there are many rheologically complicated fluids for which this theory is inadequate. For such reasons various models have been proposed. These models are usually classified as fluids of differential, rate and integral types [1–3]. The differential and rate type models are used to describe the response of fluids that have a slight memory such as dilute polymeric solutions, whereas the integral models are used to describe the response of fluids that have considerable memory such as polymer melts. Recently, the Oldroyd-B fluid [4] has acquired a special status amongst the many fluids of the rate type, as it includes as special cases of classical Newtonian and Maxwell fluids. Although computer techniques make the complete integration of the governing equations of motion feasible, the accuracy of the results can only be established by a comparison with an exact solution. Exact solutions corresponding to different flows of Newtonian as well as non-Newtonian fluids are very important for several reasons. In addition to serving as approximations to some specific problems, they also play a very important role, namely, they can be used as tests to verify the numerical or empirical methods that are developed to study complex flow problems. An Oldroyd-B fluid is one particular subclass of non-Newtonian fluids for which one can reasonably hope to obtain an exact solution. ∗ Corresponding author. E-mail address: [email protected] (S. Hyder Ali Muttaqi Shah). 1468-1218/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2008.03.012
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Recently, fractional calculus has gained considerable interest due to its applications in different areas of physics and engineering, including complex dynamics. The constitutive equations with fractional derivative have been proved to be a valuable tool to handle viscoelastic properties. In general, these equations are derived from known models via substituting time ordinary derivatives of stress and strain by derivatives of fractional order from which one can define the non-integer order integral and derivatives [5,6]. To date, various researchers [7–25] have introduced the fractional calculus approach into various rheological problems. Fractional derivatives have been found to be quite flexible in describing viscoelastic behavior. Recently, Haitao and Xu [26] have investigated the Stokes’ problem for a viscoelastic fluid with a generalized Oldroyd-B fluid model and Khan et al. [27] has discussed some accelerated flows of generalized Oldroyd-B fluids. No attempt has been made regarding the exact solutions for flows due to constant pressure gradient and impulsive pressure gradient of a generalized Oldroyd-B fluid when the plate is moving with the constant velocity. The velocity and the stress fields of the flows for an Oldroyd-B fluid with fractional derivative model are described by fractional differential equations. The expressions for the velocity and the stress fields are constructed by means of discrete Laplace transform of the sequential fractional derivative. Moreover, the solutions for fractional Maxwell, fractional second grade and generalized Oldroyd-B fluids can be recovered by the current analysis. 2. Governing equations The fundamental equations governing the unsteady motion of an incompressible fluid are div V = 0,
ρ
dV
(1)
= div T,
dt
(2)
where V is the velocity field, ρ the density, T the Cauchy stress tensor and d/dt the material time derivative. The Cauchy stress tensor T for a generalized Oldroyd-B fluid [24,26,27] is ! T = −pI + S,
1 + λα
Dα
Dtα
S = µ 1 + θβ
Dβ
Dtβ
A1 ,
(3)
in which p is the pressure, I the identity tensor, µ the dynamic viscosity, S the extra stress tensor, λ and θ are relaxation and retardation times, respectively, α and β are fractional calculus parameters such that 0 ≤ α ≤ β ≤ 1, and first Rivlin–Ericksen tensor A1 is given by A1 = L + L T ,
L = grad V,
(4)
where T denotes the matrix transpose and Dα S Dtα
= Dαt S + (V.∇ )S − LS − SLT ,
Dβ A1 Dtβ
(5)
β
= Dt A1 + (V.∇ )A1 − LA1 − A1 LT ,
(6)
β
in which Dαt and Dt are the fractional differentiation operators of order α and β with respect to t, respectively and may be defined as [5,6] Z 1 d t f (τ) Dpt [f (t)] = dτ, 0 ≤ p ≤ 1, (7) 0(1 − p) dt 0 (t − τ)p where 0(·) is the Gamma function. For unidirectional flow, we consider the velocity and the stress of the form V = u(y, t)i,
S = S(y, t),
(8)
where u is the x-component of velocity and i is the unit vector in the x-direction. Thus using Eq. (8), the continuity Eq. (1) is satisfied identically and (3)2 and (5), having in mind the initial condition S(y, 0) = 0, yield Sxz = Syy = Syz = Szz = 0 and
(1 + λα Dαt )T (y, t) = µ(1 + θβ Dβt )∂y u(y, t), α α
α
β
(9)
(1 + λ Dt )Sxx − 2λ T ∂y u(y, t) = −2µθ (∂y u) , 2
(10)
where Sxy = T (y, t) is the tangential stress. The balance of linear momentum (2) leads to the following governing equation
∂y T (y, t) − ∂x p = ρ∂t u(y, t), where ρ is the constant density and p the pressure in the flow direction.
(11)
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3. Flow due to impulsive motion in the presence of constant pressure gradient Let us consider the flow problem of an incompressible Oldroyd-B fluid with fractional derivative model occupies plane y ≥ 0 and x-axis is chosen as plane wall. Initially, the system is at rest and at time t = 0+ the fluid is suddenly set in motion
due to a constant pressure gradient and by the motion of the plane wall. The corresponding partial differential equation and the initial and boundary conditions take the form ! 2 t −α ∂u β ∂ u = − 1 + λα A + ν(1 + θβ Dt ) 2 , ∂t 0(1 − α) ∂y
(1 + λα Dαt )
(12)
where A = (1/ρ)(dp/dx) is the constant pressure gradient. u(y, 0) =
(13)
u(0, t) = U,
∂t u(y, 0) = 0, y > 0, t > 0, u(y, t), ∂y u(y, t) → 0, as y → ∞,
(14) (15)
where ν is the kinematic viscosity. To obtain an exact solution of above initial value problem, the Laplace transform is used. Let u(y, s) be the Laplace transform of u(y, t) defined by Z ∞ u(y, s) = u(y, t)e−st dt, s > 0. (16) 0
Taking the Laplace transform of Eqs. (12)–(15) we arrive at d2 u dy2
−
s(1 + λα sα )
A(1 + λα sα ) . u = − ν(1 + θβ sβ ) νs(1 + θβ sβ )
u(0, s) =
U
u(y, s),
∂y u(y, s) → 0,
s
(17)
,
(18) as y → ∞.
(19)
The solution of Eq. (17) satisfying the boundary conditions (18) and (19) is of the following form !1/2 !1/2 U s(1 + λα sα ) A s (1 + λα sα ) . exp −y 1 − exp u(y, s) = − − y s ν 1 + θβ sβ s2 ν 1 + θβ sβ
(20)
As the fluid is moved by the action of the shear stress at the plate, the stress field can be calculated. Now taking the Laplace transform of Eq. (9) and introducing Eq. (20), we get !−1/2 !1/2 s (1 + λα sα ) s (1 + λα sα ) T (y, s) = −U ρ exp −y β β β β ν 1+θ s ν 1+θ s
−
Aρ s
s(1 + λα sα ) ν 1 + θβ sβ
!−1/2
exp −y
s(1 + λα sα ) ν 1 + θβ sβ
!1/2
,
(21)
where T (y, s) is the Laplace transform of T (y, t). In order to avoid the burdensome calculations of residues and contour integrals, we will apply the discrete inverse Laplace transform [5,6] to get the velocity and the stress fields. Now writing Eqs. (20) and (21) in series form as q ∞ (−y λα /νθβ )k X ∞ ∞ (−1)n 0 m − 2k 0 2k + n X U (−1)m X k u(y, s) = +U k! m!λαm n=0 n!θβn 0 k 0 − k s 2 (β−α−1)+mα+nβ+1 m=0 2 2 q ∞ ∞ (−y λα /νθβ )k X ∞ (−1)n 0 m − 2k 0 2k + n X (−1)m X +A , k! m!λαm n=0 n!θβn 0 k 0 − k s 2k (β−α−1)+mα+nβ+2 k=1 m=0 2 2 ∞ ∞ ∞ (−1)n 0 m + 1−2 k 0 k−2 1 + n X (−y)k (λα /νθβ )(k−1)/2 X (−1)m X T (y, s) = −U ρ k! m!λαm n=0 n!θβn 0 1−k 0 k−1 s k−2 1 (β−α−1)+mα+nβ k=0 m=0 2 2 ∞ ∞ ∞ (−1)n 0 m + 1−2 k 0 k−2 1 + n X (−y)k (λα /νθβ )(k−1)/2 X (−1)m X − Aρ . k! m!λαm n=0 n!θβn 0 1−k 0 k−1 s k−2 1 (β−α−1)+mα+nβ+1 k=0 m=0 2 2 s
k =1
(22)
(23)
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Applying the discrete inverse Laplace transform to Eqs. (22) and (23) we get k q k α /νθβ λ − y ∞ ∞ ∞ (−1)n 0 m − 2k 0 2k + n t 2 (β−α−1)+mα+nβ X X (−1)m X u(y, t) = U + U k! m!λαm n=0 n!θβn 0 k 0 − k 0 k (β − α − 1) + mα + nβ + 1 m=0 k=1 2 2 2
+A
∞ X
q
−y λα /νθβ
k
k!
k=1
(−1)n 0 m −
∞ ∞ X (−1)m X
m!λαm n=0 n!θβn 0
m=0
k
2
2
0
k
0 − 2k 0
k
2 k
2
+ n t 2 (β−α−1)+mα+nβ+1
k
,
(24)
(β − α − 1) + mα + nβ + 2
and k−1 ∞ ∞ ∞ (−1)n 0 m + 1−2 k 0 k−2 1 + n t 2 (β−α−1)+mα+nβ−1 X (−1)m X (−y)k (λα /νθβ )(k−1)/2 X T (y, t) = −U ρ k! m!λαm n=0 n!θβn 0 1−k 0 k−1 0 k−1 (β − α − 1) + mα + nβ m=0 k=0 2 2 2 k− 1 ∞ ∞ ∞ (−1)n 0 m + 1−2 k 0 k−2 1 + n t 2 (β−α−1)+mα+nβ X (−y)k (λα /νθβ )(k−1)/2 X (−1)m X . − Aρ k! m!λαm n=0 n!θβn 0 1−k 0 k−1 0 k−1 (β − α − 1) + mα + nβ + 1 k=0 m=0 2 2 2
In terms of the Fox H-function, Eqs. (24) and (25) take the simpler forms q 1− k ,1 , 1−m+ k ,0) ∞ ∞ (−y λα /νθβ )k X 2 X (−1)m k (β−α−1)+mα 1,2 tβ ( 2 ) ( u(y, t) = U + U t2 H2,4 α m β k! m!λ θ k k m=0 k=1
(0,1),(1− 2 ,0),(1+ 2 ,0),( 2k (α+1−β)−mα,β)
+A
∞ X k=1
q
−y λα /νθβ k!
k ∞ X (−1)m
m!λαm
m=0
t
− 2k (α+1−β)+mα+1 1,2
H2,4
(25)
(1− k ,1),(1−m+ k ,0) 2 2 tβ θβ k k
,
(26)
(0,1),(1− 2 ,0),(1+ 2 ,0),( 2k (α+1−β)−mα−1,β)
and T (y, t) = −U ρ
∞ ∞ X (−y)k (λα /νθβ )(k−1)/2 X (−1)m
k!
k=0
m=0
m!λαm
t
k−1
2
(β−α−1)+mα−1
( 3−k ,1),( k+1 −m,0) 2 tβ 2 β θ k+1 3−k
× H21,,42 − Aρ
(0,1),(
2
,0),(
2
,0),( k−2 1 (1+α−β)−mα+1,β)
∞ ∞ X (−y)k (λα /νθβ )(k−1)/2 X (−1)m
m!λαm m=0
k!
k=0
( 3−k ,1),( k−1 −m,0) 2 tβ 2 β θ k+1 3−k
t
k− 1
2
(β−α−1)+mα
× H21,,42
(0,1),(
2
,0),(
2
,0),( k−2 1 (1+α−β)−mα,β)
,
(27)
in which Hpm,,qn (z) is the Fox H-function. Now for λ → 0, the above expressions (26) and (27) reduce to that of Khan et al. [22] and making A = (1/ρ)(dp/dx) = 0 the Eqs. (26) and (27) reduce to that of Haitao et al. [26]. In obtaining Eqs. (26) and (27), the following property of the Fox H-function is used ∞ h i X 0(a1 + A1 k) . . . 0 ap + Ap k k (1−a1 ,A1 ),...,(1−ap ,Ap ) 1,p x . Hp,q+1 −x|(0,1),(1−b ,B ),...,(1−bq ,Bq ) = (28) 1 1 k=0
k!0(b1 + B1 k) . . . 0 bq + Bq k
4. Flow induced by an impulsive pressure gradient Let us consider the flow problem of a generalized Oldroyd-B fluid bounded by an infinite plane wall at y = 0. Initially, the system is at rest and at time t = 0+ the fluid suddenly set in motion due to a impulsive pressure gradient and by the motion of the plane wall. The corresponding partial differential equation and the initial and boundary conditions take the form 2 ∂u β ∂ u = −K (1 + λα Dαt )δ(t) + ν(1 + θβ Dt ) 2 , ∂t ∂y where K = k/ρ and K δ(t) = (1/ρ)(dp/dx) is the unit impulsive pressure gradient and Dαt δ(t) = t−α−1 /0(−α).
(1 + λα Dαt )
(29)
u(y, 0) =
(30)
∂t u(y, 0) = 0, y > 0, u(0, t) = U, t > 0, u(y, t), ∂y u(y, t) → 0, as y → ∞,
(31) (32)
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Fig. 1. Velocity profiles U (y, t) versus y for different values of λ keeping θ, α, β, A and t fixed (constant pressure gradient).
Applying the discrete Laplace transform to Eqs. (29)–(32) and solving the resulting ordinary differential equation we find the solution of the form ( " ( 1/2 ) 1/2 )# U s(1 + λα sα ) s(1 + λα sα ) K u(y, s) = exp −y 1 − exp − y − , (33) s ν(1 + θβ sβ ) s ν(1 + θβ sβ ) and the corresponding stress will be of the form T (y, s) = −ρ(U + K )
s(1 + λα sα )
−1/2
(
exp −y
ν(1 + θβ sβ )
s(1 + λα sα )
1/2 )
ν(1 + θβ sβ )
.
(34)
Adopting a similar procedure as above, the expressions for the velocity and the stress fields in case of impulsive pressure gradient are given by q k u(y, t) = U + (U + K )
∞ X
−y λα /νθβ k!
k=1
× H21,,42
∞ X (−1)m m=0
m!λαm
(1− k ,1),(1−m+ k ,0) 2 2 tβ θβ k k
t− 2 (1+α−β)+mα k
,
(35)
(0,1),(1− 2 ,0),(1+ 2 ,0),( 2k (1+α−β)−mα,β)
and T (y, t) = −ρ(U + K )
∞ ∞ X (−y)k (λα /νθβ )(k−1)/2 X (−1)m
k!
k=0
m=0
(1+ 1−k ,1),(1−m+ k−1 ,0) 2 2 tβ β θ k+1 3−k k−1
m!λαm
t
k−1
(0,1),(
2
,0),(
2
,0),(
2
(β−α−1)+mα−1
× H21,,42
2
(1+α−β)−mα+1,β)
.
(36)
Making λ → 0 in Eqs. (35) and (36), the solutions reduce to that of Khan et al. [22] for the case of flow induced by an impulsive pressure gradient. 5. Numerical results and discussion This section displays the graphical results of the flows considered in this paper for different parameters of interest. A comparison between an Oldroyd-B (panel a) and generalized Oldroyd-B (panel b) fluids is also made graphically. For the sake of simplicity all the diagrams are plotted by assuming U = ν = ρ = 1. Fig. 1 shows the behavior of material parameterλ. As expected, an increase in λ decreases the velocity profiles for Oldroyd-B as well as generalized Oldroyd-B fluids. From Fig. 2, an increase in material parameter θ shows quite the opposite effect to that of λ for both Oldroyd-B and generalized Oldroyd-B fluids. In Fig. 3, the variations of non-integer fractional parameters of interest, α and β is shown. By increasing α, the velocity profile decreases for Oldroyd-B as well as generalized Oldroyd-B fluids, however, an increase in β shows an opposite trend for both the fluids. Further, it is noted that the velocity profiles for a generalized Oldroyd-B fluid are greater in magnitude than those of an Oldroyd-B fluid. Fig. 4 is shows the variations of flow velocity for different values of time. With increasing
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Fig. 2. Velocity profiles U (y, t) versus y for different values of θ keeping λ, α, β, A and t fixed (constant pressure gradient).
Fig. 3. Velocity profiles U (y, t) versus y for different values of α and β keeping λ, θ, A and t fixed (constant pressure gradient).
Fig. 4. Velocity profiles U (y, t) versus y for different values of t keeping λ, α, β, θ and A fixed (constant pressure gradient).
time the flow velocity increases for both the fluids. In Fig. 5, the effects of the pressure gradient A are shown for both the fluids. From this figure, we can see that for a favorable pressure gradient (A < 0), the velocity is positive and increases by
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Fig. 5. Velocity profiles U (y, t) versus y for different values of A keeping λ, α, β, θ and t fixed (constant pressure gradient).
Fig. 6. Velocity profiles T (y, t) versus y for different values of t keeping λ, α, β, θ and A fixed (constant pressure gradient).
Fig. 7. Velocity profiles U (y, t) versus y for different values of λ keeping θ, α, β, K and t fixed (impulsive pressure gradient).
increasing favorable pressure gradient. For adverse pressure gradient (A > 0), the velocity may either be all positive or a combination of a positive and negative regime, depending on the value of the adverse pressure gradient. Moreover, with an
S. Hyder Ali Muttaqi Shah et al. / Nonlinear Analysis: Real World Applications 10 (2009) 2590–2599
Fig. 8. Velocity profiles U (y, t) versus y for different values of θ keeping λ, α, β, K and t fixed (impulsive pressure gradient).
Fig. 9. Velocity profiles U (y, t) versus y for different values of α and β keeping λ, θ, K and t fixed (impulsive pressure gradient).
Fig. 10. Velocity profiles U (y, t) versus y for different values of t keeping λ, α, β, θ and K fixed (impulsive pressure gradient).
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Fig. 11. Velocity profiles U (y, t) versus y for different values of K keeping λ, α, β, θ and t fixed (impulsive pressure gradient).
Fig. 12. Velocity profiles T (y, t) versus y for different values of t keeping λ, α, β, θ and K fixed (impulsive pressure gradient).
increase of adverse pressure gradient it decreases. Further, the curvature of the profiles depends on the amplitude of the pressure gradient. Fig. 6 shows the variations of shear stress for increasing time for Oldroyd-B and generalized Oldroyd-B fluids. It is noted that the maximum stress occurs near the wall and decreases far away from the wall. Figs. 7–12 are prepared to show the variations of velocity and stress fields for flow induced by an impulsive pressure gradient. Qualitatively, the observations in this case are similar to those of previous case, however, the velocity and the stress profiles are not similar quantitatively. Acknowledgement The authors would like to express their deepest gratitude to the reviewer for his constructive comments and suggestions regarding an earlier version of this manuscript. References [1] J.E. Dunn, K.R. Rajagopal, Fluids of differential type-critical review and thermodynamic analysis, Int. J. Eng. Sci. 33 (1995) 689–729. [2] K.R. Rajagopal, Mechanics of non-Newtonian fluids, in: Recent Developments in Theoretical Fluids Mechanics, in: Pitman Research Notes in Mathematics, 291, Longman, New York, 1993, pp. 129–162. [3] J.E. Dunn, K.R. Rajagopal, Thermodynamics, stability and boundedness of fluids of complexity 2 and fluids of second grade, Arch. Ration. Mech. Anal. 56 (1974) 191–252. [4] J.G. Oldroyd, On the formulation of rheological equations of state, Proc. R. Soc. Lond. Ser. A 200 (1950) 523–541. [5] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [6] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Press, Singapore, 2002.
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