# Decay of Potential Vortex and Diffusion of Temperature in a Generalized Oldroyd-B Fluid through a Porous Medium

Authors: Xiaoyi Guo
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

Based on a modified Darcy law, the decay of potential vortex and diffusion of temperature in a generalized Oldroyd-B fluid with fractional derivatives through a porous medium is studied. Exact solutions of the velocity and temperature fields are obtained in terms of the generalized Mittag-Leffler function by using the Hankel transform and discrete Laplace transform of the sequential fractional derivatives. One of the solutions is the sum of the Newtonian solutions and the non-Newtonian contributions. As limiting cases of the present solutions, the corresponding solutions of the fractional Maxwell fluid and classical Maxwell fluids are given. The influences of the fractional parameters, material parameters, and the porous space on the decay of the vortex are interpreted by graphical results.


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## Body

## 1. Introduction

In recent years, non-Newtonian fluids that cannot be described by the classical Navier-Stokes equations increasingly play an important role in many industries and technological applications. For the well-known fact that all non-Newtonian fluids are so complicated that they cannot be described by a single constitutive equation, various models of non-Newtonian fluid have been suggested in the literature. Amongst those models, the rate type fluid models, by the pioneering work of Frohlick and Sack [1] and Oldroyd [2], have received much more attention. As one of the rate type models, the Oldroyd-B model [3–6] has succeeded in describing important responses of some polymeric liquids.Additionally, fractional calculus has been successful in the description of the viscoelastic properties of non-Newtonian fluids. The starting point of the fractional derivative model of a non-Newtonian fluid is usually a classical differential equation, which is modified by replacing the time derivative of an integer order by the fractional calculus operators. These fractional models have been proved to be valuable tools in characterizing the properties of polymeric solutions and melts [7–10].The studies on vortex flow of viscoelastic fluid and heat flow for viscoelastic fluids are motivated by many practical applications, such as petroleum, chemical, and food industries and bioengineering. The corresponding problem for Newtonian fluids has been investigated by Zierep [11], by means of the similarity transformation, and for second grade fluids it was studied by Fetecau et al. [12]. Later, the results were extended to the generalized second grade fluids [13], fractional Maxwell model [14], Oldroyd-B fluids [15], and generalized Oldroyd-B fluids [16].Moreover, special attention is given to the flow of viscoelastic fluid through porous media, due to the demands of such diverse areas as biorheology, geophysics, and chemical and petroleum industries [17–19]. In these references, the classical Darcy law was used to describe the effects of the pores on the velocity of a viscoelastic fluid.In this paper, we consider vortex velocity and temperature fields for a generalized Oldroyd-B fluid with fractional derivatives through porous medium. Based on a modified Darcy law for a viscoelastic fluid, exact solutions are obtained in terms of generalized Mittag-Leffler function by using the Hankel transform and discrete Laplace transform of the sequential fractional derivatives.


## 2. Basic Equations

We begin by recalling the definition of the constitutive relationship of an incompressible fluid of Oldroyd-B type (also called the Jeffreys model). The extra stress tensorT is given by the constitutive equation [15]:
(1)T=-pI+S,S+λDSDt=μ[1+λtDDt]A,
where -pI denotes the indeterminate spherical stress due to the constraint of incompressibility, S is the extra stress tensor, μ is the viscosity of the fluid, and λ and λt are constant relaxation and retardation times, respectively. It is assumed that λ≥λt≥0. A=L+LT is the first Rivlin-Ericksen tensor, L=∇V is the velocity gradient, V is the velocity vector, and D/Dt is defined as follows:
(2)DSDt=∂S∂t+(V·∇)S-S(∇V)-(∇V)TS,DADt=∂A∂t+(V·∇)A-A(∇V)-(∇V)TA.
Due to the symmetry of the considered circular motion for an Oldroyd-B fluid, the velocity field, in a system of cylindrical coordinates (r,θ,z), can be assumed as vr=0,  vθ=ω(r,t), and  vz=0. This flow field automatically satisfies the constraint of incompressibility. We suppose the initial distribution of the velocity is that of a potential of circulation Γ0 and the flow is symmetrical with respect to the axis through a porous medium. Having in mind the initial condition S(r,0)=0, we can obtain the constitutive relationship:
(3)(1+λ∂t)Srθ=μ(1+λt∂t)r∂∂r(ωr),
for the Oldroyd-B fluid performing this flow.A direct approach to create fractional rheological constitutive equations is to replace the regular time derivatives of ordinary equation by fractional time derivatives of noninteger order. Therefore, according to the constitutive relationship of the classical Oldroyd-B fluid, the constitutive relationship of the generalized Oldroyd-B is given by(4)(1+λDtα)Srθ=μ(1+λtDtβ)r∂∂r(ωr),
where Dtα and Dtβ are fractional derivatives of order α and β with respect to t, respectively, and are defined as [20]
(5)Dtpf(t)=1Γ(1-p)∫0t(t-τ)-pf′(τ)dτ,0≤p≤1.The new material constantsλ and λt in (4) have the dimensions of tα and tβ, respectively. Some of the papers use λα and λβ instead of λ and λt. However, for the sake of simplicity, we keep the same notations as in the ordinary case. This model includes the ordinary Oldroyd-B fluid as a special case for α=β=1  and includes the fractional Maxwell and classical Maxwell fluid for λt→0,  β=1 and for λt→0,  α=β=1, respectively.It is well known that in the flow of viscous Newtonian fluid at a low speed through a porous medium the pressure drop caused by the frictional drag is directly proportional to velocity, which is the Darcy law. Recently many modified Darcy’s laws have been presented for viscous Newtonian fluid flows in porous media. However, for viscoelastic fluid in porous media, only a few mathematical macroscopic filtration models were proposed. By analogy with Oldroyd-B fluid’s constitutive relationship, the following phenomenological model, which relates pressure drop and velocity for a viscoelastic fluid in an unbounded porous medium, has been introduced [17]:
(6)(1+λ∂∂t)∇p=-μϕK(1+λt∂∂t)V,
where K is the permeability and ϕ is the porosity of the porous medium. Equation (6) can be simplified to Darcy’s law when λ=λt=0. In the same way, by analogy with generalized Oldroyd-B fluid’s constitutive relationship equation (4), we introduce a modified phenomenological Darcy law with fractional derivatives as follows:
(7)(1+λDtα)∇p=-μϕK(1+λtDtβ)V.In consideration of the balance of forces acting on a volume element of fluid, the local volume average balance of linear momentum can be given by(8)ρdVdt=-∇p+div⁡S+r,
where d/dt is the material time derivative, ρ denotes the density, and r is Darcy’s resistance for a generalized Oldroyd-B fluid in the porous medium and can be inferred from (7) to satisfy the following equation [17]:
(9)(1+λDtα)r=-μϕK(1+λtDtβ)V.Then we obtain the momentum equation through a porous medium in the form:(10)ρ(1+λDtα)∂ω(r,t)∂t=(1+λDtα)1r2∂∂r(r2Srθ)-μϕK(1+λtDtβ)ω(r,t),
where the pressure gradient in the θ direction is ignored.Substituting (4) into (10), we obtain
(11)(1+λDtα)∂ω∂t=ν(1+λtDtβ)(∂2∂r2+1r∂∂r-1r2)ω-νϕK(1+λtDtβ)ω,
in which ν=μ/ρ is the kinematic viscosity. The initial and natural conditions are
(12)ω(r,0)=Γ02πr,∂ω∂t(r,0)=0,ω(r,t),∂ω(r,t)∂r⟶0,asr⟶∞,t>0.Further, we consider that there exists temperature field during the vortex flow in a generalized Oldroyd-B fluid. Its initial distribution and natural conditions are similarly assumed to be in the same form of previous studies [13, 18] with
(13)θ(r,0)=θ02πr,(14)θ(r,t),∂θ(r,t)∂r⟶0,asr⟶∞,t>0.The energy equation, when the Fourier law of heat conduction is applied, may be written in the form [13, 18]:
(15)∂θ(r,t)∂t=kρc(∂2∂r2+∂r∂r)θ(r,t)+νc[∂ω(r,t)∂r-ω(r,t)r]2+h(r,t)c,
where h(r,t) is the radiant heating which is neglected in this paper, c is the specific heat, and k is the conductivity (be assumed to be constant).


## 3. Velocity Field

Let us introduce dimensionless variablesω*=ων/Γ02,  r*=rΓ0/ν,  and t*=tΓ02/ν and dimensionless parameters λ*=λ(Γ02/ν)α and  λt*=λt(Γ02/ν)β. Then (11) and (12) can be changed into dimensionless equations as follows (for brevity the dimensionless mark “*” is omitted here):
(16)(1+λDtα)∂ω∂t=(1+λtDtβ)(∂2∂r2+1r∂∂r-1r2)ω-η2(1+λtDtβ)ω,(17)ω(r,0)=12πr,∂ω∂t(r,0)=0,(18)ω(r,t),∂ω(r,t)∂r⟶0,asr⟶∞,t>0,
where η=1/Re  (Re=Γ0K/ϕ/ν is the Reynolds number in porous media).In order to get the exact solution of above equations, we introduce the Hankel transform of ordern and its inverse transform as follows [21]:
(19)f-(ξ)=Hn[f(r);ξ]=∫0∞rJn(ξr)f(r)dr,f(r)=Hn-1[f-(ξ);r]=∫0∞ξJn(ξr)f-(ξ)dξ,
where Jn(ξr) is the first kind of the Bessel function of order n. Applying the Hankel transform of order one (n=1) to (16) and (17) and having in mind condition (18), we can obtain
(20)(1+λDtα)∂ω-∂t+(ξ2+η2)(1+λtDtβ)ω-=0,ω-(ξ,0)=12πξ,∂ω-∂t(ξ,0)=0.The Laplace transform is defined as follows:(21)ω-~(ξ,s)=L[ω-(ξ,t);s]=∫0∞e-stω-(ξ,t)dt.Then, applying the Laplace transform to (17) and considering the formula [19]:
(22)L[Dtqf(t);s]=sqf~(s)-∑k=0n-1sq-k-1f(k)(0),n-1<q≤n,
we get
(23)ω-~(ξ,s)=12πξλsα+1+(ξ2+η2)λtsβ-1λsα+1+s+(ξ2+η2)λtsβ+ξ2+η2.In order to avoid the burdensome calculations of residues and contour integrals, we apply the discrete inverse Laplace transform method to giveω-(ξ,t). Firstly, we rewrite (23) as a series form
(24)ω-~(ξ,s)=12πξ[∑m=0k(km)λtmsβm-k-2(sα+λ-1)k+11s-∑k=0∞(-1)k(ξ2+η2λ)k+1×∑m=0k(km)λtmsβm-k-2(sα+λ-1)k+1].Then, applying the inverse Laplace transform term by term to (24) and considering the following property of the generalized Mittag-Leffler function [20]:
(25)L-1[n!sλ-μ(sλ∓c)n+1]=tλn+μ-1Eλ,μ(n)(±ctλ),Re(s)>|c|1/λ,
we have
(26)ω-(ξ,t)=12πξ[1-∑k=0∞(-1)k(ξ2+η2)k+1k!λk+1×∑m=0k(km)λtmtαk+α-βm+k+1×Eα,α-βm+k+2(k)(-λ-1tα)∑k=0∞(-1)k(ξ2+η2)k+1k!λk+1].Finally, applying the inverse Hankel transform to (26), we obtain the exact solution to the velocity field
(27)ω(r,t)=12π∫0∞J1(ξr)dξ-12π∫0∞∑k=0∞∑m=0k(-1)k(ξ2+η2)k+1λtmtαk+α-βm+k+1m!(k-m)!λk+1×Eα,α-βm+k+2(k)(-λ-1tα)J1(ξr)dξ.In addition, we can also rewrite (23) as another form of series as follows:
(28)ω-~(ξ,s)=12πξ1s+ξ2+η2+ξ2+η22πξ1s+ξ2+η2×∑k=0∞∑m=0k(-1)k(ξ2+η2)kλtmk!λk+1m!(k-m)!×[λsα+βm-(k+1)(sα+λ-1)k+1+λt(ξ2+η2)sβ+βm-(k+2)(sα+λ-1)k+1].Applying the discrete inverse Laplace transform to the above equation and considering (25), we get
(29)ω-(ξ,t)=12πξe-t(ξ2+η2)+ξ2+η22πξ∫0te-(t-s)(ξ2+η2)F(ξ,s)ds,
where
(30)F(ξ,t)=∑k=0∞∑m=0k(-1)k(ξ2+η2)kλtmtαk+k-βmλkm!(k-m)!×Eα,k+1-βm(k)(-λ-1tα)+∑k=0∞∑m=0k(-1)k(ξ2+η2)k+1λtm+1tαk+k+α+1-β-βmλk+1m!(k-m)!×Eα,α+k+2-β-βm(k)(-λ-1tα).Then, applying the inverse Hankel transform to (29), we obtain the equivalent solution of vortex velocity as follows:
(31)ω(r,t)=ωNP(r,t)+12π∫0∞∫0t(ξ2+η2)e-(t-s)(ξ2+η2)F(ξ,s)J1(ξr)dsdξ,
where (cf. [19, equation (41)])
(32)ωNP(r,t)=12π∫0∞e-t(ξ2+η2)J1(ξr)dξ
is the velocity field corresponding to a Newtonian fluid performing the same motion through a porous medium.


## 4. Temperature Field

Equations (13)~(15) can be nondimensionalized by introducing dimensionless variables θ*=θν/θ0Γ0,  ω*=ων/Γ02,  r*=rΓ0/ν,  and  t*=tΓ02/ν. The dimensionless equation with the initial and natural boundary conditions can be given as follows (for brevity, the dimensionless mark “*” is omitted here):
(33)∂θ(r,t)∂t=β1(∂2∂r2+∂r∂r)θ(r,t)+η1f(r,t),(34)θ(r,0)=12πr,(35)θ(r,t),∂θ(r,t)∂r⟶0,asr⟶∞,t>0,
where β1=k/μc, η1=Γ03/cνθ0, and
(36)f(r,t)=[∂ω(r,t)∂r-ω(r,t)r]2.Making the Hankel transform of zero order(n=0) with respect to r in (33) and (34), and having in mind the condition (35), we obtain
(37)∂θ-(ξ,t)∂t+β1ξ2θ-(ξ,t)=η1f-(ξ,t),(38)θ(r,0)=12πξ,
in which f-(ξ,t) is given by
(39)f-(ξ,t)=14π2×∫0∞rJ0(ξr)×{∑k=0∞∑m=0k(-1)k(ξ2+η2)k+1λtmtαk+α-βm+k+1m!(k-m)!λk+1∫0∞xJ2(xr)×[1-∑k=0∞∑m=0k(((-1)k(ξ2+η2)k+1×λtmtαk+α-βm+k+1(-1)k(ξ2+η2)k+1)×(m!(k-m)!λk+1)-1((-1)k(ξ2+η2)k+1)×Eα,α-βm+k+2(k)×(-λ-1tα)∑k=0∞∑m=0k(-1)k(ξ2+η2)k+1λtmtαk+α-βm+k+1m!(k-m)!λk+1]dx}2dr.To obtain the equation above, we substituted (27) into (36) and used the following property of the Bessel function:
(40)xJ1′(x)=J1(x)-xJ2(x).Solving (37) with (38), we have
(41)θ-(ξ,t)=e-β1ξ2t[12πξ+η1∫0tf-(ξ,τ)eβ1ξ2τdτ].Then, applying the inverse Hankel transform of zero order to (41), we get the exact solution to the temperature field
(42)θ(r,t)=∫0∞J0(ξr)e-β1ξ2t[12π+η1ξ∫0tf-(ξ,τ)eβ1ξ2τdτ]dξ,
in which f-(ξ,t) is given by (39).


## 5. Special Cases and Numerical Results

Whenη=0, (27) and (31) can be simplified to the dimensionless solutions of vortex velocity field for generalized Oldroyd-B fluids in nonporous cases. Equivalent expression for velocity in terms of G function is obtained by Fetecau et al. [16]. And when α=1,  β=1, the above results can be simplified to the solution for ordinary Oldroyd-B fluid through porous medium.Makingλt→0,  β=1 to (27) and (42), we have vortex velocity of the fractional Maxwell fluid through porous medium as follows:
(43)ω(r,t)=12π∫0∞J1(ξr)dξ-12π∫0∞∑k=0∞(-1)k(ξ2+η2)k+1tαk+α+k+1k!λk+1×Eα,α+k+2(k)(-λ-1tα)J1(ξr)dξ=12π∫0∞J1(ξr)dξ-12π∫0∞∑k=0∞∑n=0∞(-1)k+n(ξ2+η2)k+1(k+n)!tαk+αn+α+k+1k!λk+n+1Γ(αk+αn+α+k+2)×J1(ξr)dξ
and the temperature field of the fluid as
(44)θ(r,t)=∫0∞J0(ξr)e-β1ξ2t[12π+η1ξ∫0tf-(ξ,τ)eβ1ξ2τdτ]dξ,
in which f-(ξ,t) is given by
(45)f-(ξ,t)=14π2∫0∞rJ0(ξr)×{+2))-1)∑k=0∞∑n=0∞((αk+αn+α+k+2))-1+2))-1((-1)k+n(ξ2+η2)k+1]dx∫0∞xJ2(xr)×[1-∑k=0∞∑n=0∞((αk+αn+α+k+2))-1+2))-1((-1)k+n(ξ2+η2)k+1×(k+n)!×tαk+αn+α+k+1(-1)k+n(ξ2+η2)k+1)×(k!λk+n+1Γ×(αk+αn+α+k+2)k!λk+n+1)-1((-1)k+n(ξ2+η2)k+1)∑k=0∞∑n=0∞((αk+αn+α+k+2))-1+2))-1((-1)k+n(ξ2+η2)k+1]dx}2dr.Also when makingλt→0,  β=1 to (31), we get the solutions
(46)ω(r,t)=ωNP(r,t)+12π∫0∞∫0t∑k=0∞∑n=0∞(-1λ)k+n×(ξ2+η2)k+1sαk+αn+kn!k!Γ(αk+αn+k+1)×e-(t-s)(ξ2+η2)J1(ξr)dsdξ,
for the fractional Maxwell fluid through porous medium. When η→0, (46) is the same result as that obtained by Fetecau et al. [16].Further makingα=1 to (43), (44), and (45), we, respectively, have the velocity field
(47)ω(r,t)=12π∫0∞J1(ξr)dξ-12π∫0∞∑k=0∞∑n=0∞(-1)k+n(ξ2+η2)k+1(k+n)!t2k+n+2k!λk+n+1Γ(2k+n+3)×J1(ξr)dξ
and the temperature field
(48)θ(r,t)=∫0∞J0(ξr)e-β1ξ2t[12π+η1ξ∫0tf-(ξ,τ)eβ1ξ2τdτ]dξ,
where
(49)f-(ξ,t)=14π2∫0∞rJ0(ξr)×{+3)k!λk+n+1Γ)-1((-1)k+n(ξ2+η2)k+1)∑k=0∞∑n=0∞(+3))-1((-1)k+n(ξ2+η2)k+1]dx∫0∞xJ2(xr)×[1-∑k=0∞∑n=0∞(+3))-1((-1)k+n(ξ2+η2)k+1×(k+n)!t2k+n+2(-1)k+n(ξ2+η2)k+1)×(k!λk+n+1Γ×(2k+n+3)k!λk+n+1Γ)-1((-1)k+n(ξ2+η2)k+1)∑k=0∞∑n=0∞(+3))-1((-1)k+n(ξ2+η2)k+1]dx∫0∞xJ2(xr)}2dr,
for the classical Maxwell fluid through porous medium.Whenλ→0, (23) reduces to
(50)ω-~(ξ,s)=12πξ1+(ξ2+η2)λtsβ-1s+(ξ2+η2)λtsβ+ξ2+η2.Separately applying the inverse Laplace and inverse Hankel transform tos and ξ in the above equation, we have the solution of vortex velocity field
(51)ω(r,t)=12π∫0∞∑k=0∞(-t)k(ξ2+η2)kk!×E1-β,βk+1(k)(t1-β-λt(ξ2+η2)×t1-β)J1(ξr)dξ×12π∫0∞∑k=0∞(-t)k(ξ2+η2)k+1λtt1-βk!×E1-β,βk-β+2(k)(t1-β-λt(ξ2+η2)×t1-β)J1(ξr)dξ,
which is the same as the result obtained by Shen et al. [13], when η→0, and temperature field is
(52)θ(r,t)=∫0∞J0(ξr)e-β1ξ2t[12π+η1ξ∫0tf-(ξ,τ)eβ1ξ2τdτ]dξ,
where
(53)f-(ξ,t)=14π2∫0∞rJ0(ξr)×{(-λt(ξ2+η2)t1-β)∑k=0∞(-t)k(ξ2+η2)kk!]d∫0∞xJ2(xr)×[∑k=0∞(-t)k(ξ2+η2)kk!×E1-β,βk+1(k)(-λt(ξ2+η2)t1-β)+∑k=0∞(-t)k(ξ2+η2)k+1λtt1-βk!×E1-β,βk-β+2(k)(t1-β-λt(ξ2+η2)×t1-β)∑k=0∞(-t)k(ξ2+η2)kk!]dx∫0∞xJ2(xr)}2dr
for the generalized second grade fluid through a porous medium.Makingλt→0,  β=1 to (51), (52), and (53), we find the velocity
(54)ω(r,t)=12π∫0∞J1(ξr)∑k=0∞(-t)k(ξ2+η2)ktkk!dξ=12π∫0∞J1(ξr)e-(ξ2+η2)tdξ
and the temperature field
(55)θ(r,t)=∫0∞J0(ξr)e-β1ξ2t×[e-(ξ2+η2)tdx∫0∞xJ2(xr)}2dreβ1ξ2τdτ12π+η1ξ×∫0t14π2∫0∞rJ0(ξr)×{∫0∞xJ2(xr)×e-(ξ2+η2)tdx∫0∞xJ2(xr)}2dreβ1ξ2τdτ]dξ,
which correspond to the results for a Newtonian fluid performing the same motion through a porous medium.In addition, the influences of fractional orders and material parameters on the decay of vortex velocity are also analyzed in graphical form for generalized Oldroyd-B fluids through porous medium. In Figure1, the velocity field corresponding to (27) is drawn against r for different values of t. The influences of the fractional parameters QUOTEαα and β on the decay of the vortex are showed in Figures 2 and 3. The influences of the material parameters λ and λt on the decay of the vortex are showed in Figures 4 and 5, respectively. In order to compare the vortex velocities for fractional Oldroyd-B, fractional Maxwell, and Newtonian fluids performing the same motion in porous and nonporous media, we draw Figures 6 and 7, respectively.Figure 1
Profiles of velocityω(r,t) versus r for different values of t, for λ=6,  λt=3.Figure 2
Profiles of velocityω(r,t) versus r for different values of β, for λ=6,  λt=3.Figure 3
Profiles of velocityω(r,t) versus r for different values of α, for λ=6,  λt=3.Figure 4
Profiles of velocityω(r,t) versus r for different values of λt, for α=0.5,  β=0.8.Figure 5
Profiles of velocityω(r,t) versus r for different values of λ, for α=0.5,  β=0.8.Figure 6
Profiles of velocityω(r,t) versus r, in a porous medium, for fractional Oldroyd-B fluids (curve ωFOP), fractional Maxwell fluids (curve ωFMP), and Newtonian fluids (curve ωNP).Figure 7
Profiles of velocityω(r,t) versus r, in a nonporous medium, for fractional Oldroyd-B fluids (curve ωFO), fractional Maxwell fluids (curve ωFM), and Newtonian fluids (curve ωN).


## 6. Conclusions

In this paper, by analogy to constitutive relationship of the generalized Oldroyd-B fluid, we introduce a modified phenomenological Darcy law with fractional derivative. Based on this model, the decay of a potential vortex and diffusion of temperature in a generalized Oldroyd-B fluid with fractional derivatives through a porous medium is studied. Using the Laplace transform and Hankel transform, we provide the exact solutions of the velocity and temperature fields. Particularly, we give the solution of vortex velocity of two equivalent forms, one of which is the sum of the Newtonian solutions and the non-Newtonian contributions. And the results for decay of a potential vortex in Oldroyd-B, generalized Maxwell, and Maxwell fluids filling the porous space all can be deduced as the special cases. We also discuss the velocity and temperature fields of the generalized second grade fluid through a porous medium performing the same motion, from which the Newtonian solutions can be easily deduced.The graphs of the velocity field indicate that the potential velocity decays in time and space and the strength of the potential vortex is directly proportional to parametersλ and λt and is inversely proportional to α and β. The vortexes of fractional Oldroyd-B fluid and fractional Maxwell fluid are much stronger than those of Newtonian one, no matter in porous or nonporous cases. The vortex in a nonporous medium is stronger than that in a porous medium.



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