# Exact Solutions of Rayleigh-Stokes Problem  for Heated Generalized Maxwell Fluid  in a Porous Half-Space

Authors: Changfeng Xue; Junxiang Nie
Journal: Mathematical Problems in Engineering
 (2008)

---

## Abstract

The Rayleigh-Stokes problem for a generalized Maxwell fluid in a porous
half-space with a heated flat plate is investigated. For the description of such a
viscoelastic fluid, a fractional calculus approach in the constitutive relationship
model is used. By using the Fourier sine transform and the fractional Laplace
transform, exact solutions of the velocity and the temperature are obtained.
Some classical results can be regarded as particular cases of our results, such
as the classical solutions of the first problem of Stokes for Newtonian viscous
fluids, Maxwell fluids, and Maxwell fluids in a porous half-space.


---

## Body

## 1. Introduction

Many materials such as clay coatings, drilling mud,
suspensions, certain oils and greases, polymer melts, elastomers, and many
emulsions have been treated as non-Newtonian fluids. It is difficult to suggest
a single model which can exhibit all properties of non-Newtonian fluids as is
done for Newtonian fluids. They cannot be described in a simple model as for
the Newtonian fluids, and there has been much confusion over the classification
of the non-Newtonian fluids. For this reason, many models of constitutive
equations have been proposed. Recently, fractional calculus has encountered
much success in the description of constitutive relations of viscoelastic
fluids. The starting point of the fractional derivative model of viscoelastic
fluids is usually a classical differential equation which is modified by
replacing the time derivative of an integer order with the so-called
Riemann-Liouville fractional calculus operator. This generalization allows one
to define precisely noninteger order integrals or derivatives.
Bagley and  Torvik [1],
Friedrich [2], Huang et al. [3],
He et al. [4], Xu and  Tan [5–7],
Tan [8–12], and
Shen et al. [13, 14] have
sequentially introduced the fractional calculus approach into various rheology
problems. Fractional derivatives have been found to be quite flexible in
describing viscoelastic behaviors.The first problem of Stokes for the flat plate, like
the Rayleigh-Stokes problem for an edge, has received much attention because of
its practical importance [15–18]. This unsteady flow problem examines the
diffusion of vorticity in a half-space filled with a viscous incompressible
fluid that is set to motion when an infinite flat plate suddenly assumes a
constant velocity parallel to itself from the rest. By using similarity
transformation of variables, the exact solution corresponding to a Newtonian
fluid was obtained in an elegant form by Stokes. But for a Maxwell fluid, it is
difficult to obtain such a strict similarity solution. Further, the order of
the equation of motion for such a fluid is higher than that of the
Navier-Stokes equation and thus, in general, one needs conditions in addition
to the usual adherence boundary condition. Friedrich first investigated this
problem and gave a few exact solutions [19].On the other hand, the problems of the fluid through
porous media play an important role in various practical applications. But very
little efforts have so far been made to discuss these problems. Recently, based
on the local volume averaging technique and the balance of forces acting on a
volume element of viscoelastic fluids in porous media, Tan and Masuoka [20–23]
developed a modified Darcy-Brinkman model for viscoelastic fluid flows in
porous media. Based on their works, in this paper, we extend Stokes' first
problem to that of a heated generalized Maxwell fluid with fractional
derivative in a porous half-space. Exact solutions of the velocity and the
temperature are obtained by using the Fourier sine transform and the fractional
Laplace transform. Some previous classical results can be recovered from our
results, such as the classical solutions of the first problem of Stokes for
Newtonian viscous fluids and those for Maxwell fluids.


## 2. Basic Equations

The well-known
Darcy law states that, in the flow of a Newtonian fluid through a porous
medium, the pressure gradient∇p caused by the friction drag is directly
proportional to the velocity, namely,(2.1)∇p=−μKVD,where VD is the Darcian velocity, μ is the dynamic viscosity of the fluid, and K is the permeability of the porous medium,
respectively. The Darcy law, however, does not consider the boundary effect on
the flow, which unfortunately cannot be ignored in many applications. In 1947,
Brinkman proposed an equation to describe the locally averaged flow in a porous
medium [24]. In the Brinkman equation, the pressure gradient, the divergence of
the viscous stress tensor, and the friction force exerted by the porous medium
satisfy(2.2)−∇p+μe∇2VD−μKVD=0,where μe is the effective dynamic viscosity of the
medium. Although the Brinkman equation takes the boundary effect on the flow
into account, it is justified for and can thus be applied to only steady-state
flows of a Newtonian fluid in porous media. For viscoelastic fluid flows in
porous media, on the other hand, only a few mathematical macroscopic filtration
models have been proposed. In particular, by analogy with the constitutive
equation of a Maxwell fluid [25, 26], the following phenomenological model has
been introduced: (2.3)(1+tmα∂α∂tα)∇p=−μKVD,where tm is the relaxation time of the generalized
Maxwell fluid, α(0<α≤1) is fractional coefficient, and ∂ατ/∂tα=Dtα is Riemann-Liouville fractional derivative,
which is defined as in [27](2.4)Datα[f(t)]=1Γ(k−α)dkdtk∫at(t−τ)k−α−1f(τ)dτ(k−1≤α<k),where Γ(⋅) is the gamma function.Let us now follow the same procedure as used in [22]
to investigate the flow of a generalized Maxwell fluid in a porous half-space.
With the positive y-axis of the rectangular coordinate system pointing
up, a generalized Maxwell fluid flows through the
porous half-space (y>0) above and in contact with a flat plate
occupying the xz-plane. At time t=0, both the fluid and the
plate are at rest. At time  t=0+,
the flat plate suddenly starts to slide in its plane with a constant speed U0.
Under these conditions, no flow occurs in the y- and the z-directions,
and the flow velocity at a given point in the porous half-space depends only on
its y-coordinate and time t, that is, the intrinsic velocity
field takes the form(2.5)V=u(y,t)i,where i represents the unit vector in the x-direction.
The intrinsic velocity V, averaged over the pore space, is related to
the Darcian velocity by VD=ϕV,
where ϕ is the porosity of the porous medium. In this
work, employing a local volume averaging technique [20, 28–30], we will
consider a modified, more rigorous Darcy law for Stokes' first problem with a
generalized Maxwell fluid in the porous half-space. In fact, in terms of the
pressure gradient, the local volume-averaged balance of linear momentum can be
given by(2.6)ρ∂V∂t=−∇p+∇τ+r,where ρ is the fluid density, τ is the viscous stress tensor, and r is the Darcy resistance, namely, the viscous
damping force caused by the micropore structure of a porous medium, respectively.
Under the assumption that(2.7)τ+tmα[∂ατ∂tα+(V⋅∇)τ−(∇V)Tτ−τ(∇V)]=μ[∇V+(∇V)T],as in [23, 31], next we discuss
the viscoelastic properties of the fluid using the upper convected derivative
generalized Maxwell fluid.It should be pointed out that the classical viscous
Newtonian fluid is just a special case of this model attm=0.
Inserting (2.5) into (2.7) and using the initial condition τ(y,0)=0,
we get τxz=τyy=τyz=τzz=0 and(2.8)τxx+tmα∂ατxx∂tα=2τxy∂u∂y,(2.9)τxy+tmα∂ατxy∂tα=μ∂u∂y.Note that the pressure gradient∇p given in (2.3) can be taken as a measure of the
same resistance to flow in the bulk of the porous medium [26]. Since the Darcy
resistance r in (2.6) is also in effect a measure of the flow resistance offered
by the solid matrix, from (2.3) we get(2.10)(1+tmα∂α∂tα)r=−μKVD.Using (2.9) and (2.10) to eliminate τ and r in (2.6) and recalling that VD=ϕV,
we have the momentum equation of Stokes' first problem for a generalized
Maxwell fluid in the porous half-space(2.11)ρ(1+tmα∂α∂tα)∂u∂t=−(1+tmα∂α∂tα)∂p∂x+μ∂2u∂y2−μϕKu.Ignoring the pressure gradient in thex-direction
finally leads to the equation of motion as follows:(2.12)ρ(1+tmα∂α∂tα)∂u∂t=μ∂2u∂y2−μϕKu.In addition, the initial and the boundary conditions
are(2.13)u(y,0)=0,u(0,t)=U0,respectively, and the natural
condition is(2.14)u(y,t)→0,∂u(y,t)∂y→0asy→∞.We will assume that there is a local thermal
equilibrium. Then,Ts=Tf=T,
where Ts and Tf are the temperatures of the solid and the
fluid phases in porous media, respectively. The thermal energy equation is then
[32, 33](2.15)(ρc)fs∂T∂t=λe∂2T∂y2+QT,where (ρc)fs is the specific heat capacity of the system
formed by both the fluid and the solid phases. More precisely, (ρc)fs=(1−ϕ)ρscs+ϕcfρ,
where ρs is the density of the solid phase, cs is the specific heat capacity of the solid
phase, and cf is the specific heat capacity of the fluid at
a constant pressure, respectively. λe is the equivalent thermal conductivity of the
fluid in the porous half-space, and QT is the contribution due to viscous dissipation
[34, 35](2.16)QT=μϕ2Ku2+μ(∂u∂y)2.Equation (2.16) is a unified form compatible with an
expression derived from the Navier-Stokes equation for a fluid clear of solid
[36, 37].The corresponding initial and boundary conditions are(2.17)T(y,0)=T1,T(0,t)=T0(t).Moreover, the natural conditions are(2.18)T(y,t)→T1,∂T(y,t)∂y→0asy→∞.


## 3. Exact Solutions

Employing the nondimensional quantities(3.1)u*=uU0,y*=yU0v,t*=tU02v,T*=T−T1T0(0)−T1,the dimensionless motion and
energy equations can be written as (for brevity the dimensionless marks “*” are omitted here)(3.2)(1+γ∂α∂tα)∂u∂t=∂2u∂y2−β2u,(3.3)u(y,0)=0,(3.4)u(0,t)=1,(3.5)u(y,t)→0,∂u(y,t)∂y→0asy→∞,(3.6)∂T∂t=ξ∂2T∂y2+Qt(y,t),(3.7)T(y,0)=0,(3.8)T(0,t)=f(t),(3.9)T(y,t)→0,∂T(y,t)∂y→0asy→∞.where γ=(tmU02/v)α, β=v/(U0K/ϕ), ξ=De/v, f(t)=(T0(t)−T1)/(T0(0)−T1), De=λe/(ρc)fs. v=μ/ρ is the kinematic viscosity. The dimensionless
viscous dissipation, namely, Qt(y,t), has the following form:(3.10)Qt(y,t)=ϕ2μ2(ρc)fsK(T0(0)−T1)ρu2+U02ρ(ρc)fs(T0(0)−T1)(∂u∂y)2.


### 3.1. Solution of the Velocity Field

Multiplying both sides of (3.2) and (3.3) by integrating
with respect to y from 0 to ∞,
and then using the boundary conditions of (3.4) and (3.5), in fact we have(3.11)∂U(ζ,t)∂t+γ∂αU(ζ,t)∂tα=−ζ2U(ζ,t)+ζ−β2U(ζ,t),(3.12)U(ζ,0)=0,where U(ζ,t) denotes the Fourier sine transform of U(y,t) with respect to y. In order to obtain an exact solution of (3.11)
subject to the initial condition of (3.12), the fractional Laplace transform is
used. Let(3.13)U¯(ζ,s)=L{U(ζ,t)}=∫0∞e−stU(ζ,t)dtbe the image function of U(ζ,t),
where s is the transform parameter. Using Laplace
transform principle of sequential fractional derivatives [27], we
get(3.14)U¯(ζ,s)=ζs[γsα+1+s+(ζ2+β2)].In order to avoid the burdensome
calculations of residues and contour integrals, we apply the discrete inverse
Laplace transform method here. First, we rewrite (3.14) in a series form(3.15)U¯(ζ,s)=ζζ2+β2∑k=0∞(−1)k(ζ2+β2γ)k+1s−k−2(sα+γ−1)k+1.Then, applying the inversion
formulae term by term for the Laplace transform, (3.15) becomes(3.16)U(ζ,t)=ζζ2+β2∑k=0∞(−1)kk!(ζ2+β2γ)k+1t(k+1)(α+1)Eα,α+k+2(k)(−tαγ),in which(3.17)Eα,β(t)=∑k=0∞tkΓ(αk+β)denotes the generalized
Mittag-Leffler function. Here, we use the following
property of the generalized Mittag-Leffler function's inverse Laplace transform
[27](3.18)L−1[n!sλ−μ(sλ∓c)n+1]=tλn+μ−1Eλ,μ(n)(±ctλ)(Re(s)>|c|1/λ).Inverting (3.18) by the Fourier
sine inverse transform, we obtain an exact solution to the velocity field as
follows:(3.19)u(y,t)=2π∫0∞ζsin(ζy)ζ2+β2∑k=0∞(−1)kk!(ζ2+β2γ)k+1t(k+1)(α+1)Eα,α+k+2(k)(−tαγ)dζ.In a special case, whenα=1,
corresponding to Stokes' first problem for a Maxwell fluid in a porous
half-space, (3.14) can be written as(3.20)U¯(ζ,s)=ζs[γs2+s+(ζ2+β2)].Correspondingly, (3.19) can be
written as(3.21)u(y,t)=2π∫0∞ζsin(ζy)ζ2+β2∑k=0∞(−1)kk!(ζ2+β2γ)k+1t2(k+1)E1,k+3(k)(−tγ)dζ,which is just the dimensionless
velocity solution of a Maxwell fluid in a porous half-space obtained in another
form by Tan [22].In a special case, whenα=1 and β=0,
we have(3.22)u(y,t)=2π∫0∞sin(ζy)ζ∑k=0∞(−1)kk!(ζ2γ)k+1t2(k+1)E1,k+3(k)(−tγ)dζ,which is just the dimensionless
velocity solution of a clear Maxwell fluid obtained in another form by Jordan et al.
[38].In another special case, whenα=1, β=0, and γ=0,
corresponding to the classical version of Stokes' first problem for a Newtonian
fluid, (3.14) can be written as(3.23)U¯(ζ,s)=ζs(s+ζ2).Correspondingly, (3.19) can be
written as(3.24)u(y,t)=1−2π∫0∞sin(ζy)ζexp(−ζ2t)dζ.Equation (3.24) can be simplified
further as follows:(3.25)u(y,t)=1−erf(y2t),where erf(x) is the error function of Gauss. This is just
the well-known result of Stokes' first problem for a clear Newtonian fluid.


### 3.2. Solution of Temperature Field

In order to determine the temperature distribution in
the porous half-space, we multiply both sides of (3.6) and (3.7) by sin(sy) and then integrate with respect to y from 0 to ∞.
Using (3.8) and (3.9), we get(3.26)∂T¯(s,t)∂t+ξs2T¯(s,t)=ξsf(t)+g(s,t),(3.27)T¯(s,0)=0,where T¯(s,t) and g(s,t) denote the Fourier sine transforms of T(y,t) and Qt(y,t) with respect to y, respectively. Since the solution of velocity u(y,t) is obtained by (3.21), g(s,t) is a known function. The solution of the
ordinary differential equation (3.26) subject to the initial condition
(3.27) can
be easily obtained as(3.28)T¯(s,t)=exp(−ξs2t)∫0t(ξsf(τ)+g(s,τ))exp(ξs2t)dτ.Inverting (3.28) by the Fourier
sine inverse transform, and integrating by parts the term containing f(τ),
we get(3.29)T(y,t)=f(0)(1−erf(y2ξt))+∫0tf′(τ)(1−erf(y2ξ(t−τ)))dτ+2π∫0∞∫0tg(s,t)exp(−ξs2(t−τ))sin(sy)dτds.In a special case, whenU0=0,
the fluid is at rest, and the temperature distribution is the same for both a
Maxwell fluid and a Newtonian one. Then, (3.29) takes the form(3.30)T(y,t)=f(0)(1−erf(y2ξt))+∫0tf′(τ)(1−erf(y2ξ(t−τ)))dτ.From (3.30), we easily obtain that(3.31)T(y,t)→T0(∞)ast→∞.If, moreover, the plate is kept
at a constant temperature, (3.31) reduces to a well-known form,(3.32)T(y,t)=f(0)(1−erf(y2ξt)).


## 3.1. Solution of the Velocity Field

Multiplying both sides of (3.2) and (3.3) by integrating
with respect to y from 0 to ∞,
and then using the boundary conditions of (3.4) and (3.5), in fact we have(3.11)∂U(ζ,t)∂t+γ∂αU(ζ,t)∂tα=−ζ2U(ζ,t)+ζ−β2U(ζ,t),(3.12)U(ζ,0)=0,where U(ζ,t) denotes the Fourier sine transform of U(y,t) with respect to y. In order to obtain an exact solution of (3.11)
subject to the initial condition of (3.12), the fractional Laplace transform is
used. Let(3.13)U¯(ζ,s)=L{U(ζ,t)}=∫0∞e−stU(ζ,t)dtbe the image function of U(ζ,t),
where s is the transform parameter. Using Laplace
transform principle of sequential fractional derivatives [27], we
get(3.14)U¯(ζ,s)=ζs[γsα+1+s+(ζ2+β2)].In order to avoid the burdensome
calculations of residues and contour integrals, we apply the discrete inverse
Laplace transform method here. First, we rewrite (3.14) in a series form(3.15)U¯(ζ,s)=ζζ2+β2∑k=0∞(−1)k(ζ2+β2γ)k+1s−k−2(sα+γ−1)k+1.Then, applying the inversion
formulae term by term for the Laplace transform, (3.15) becomes(3.16)U(ζ,t)=ζζ2+β2∑k=0∞(−1)kk!(ζ2+β2γ)k+1t(k+1)(α+1)Eα,α+k+2(k)(−tαγ),in which(3.17)Eα,β(t)=∑k=0∞tkΓ(αk+β)denotes the generalized
Mittag-Leffler function. Here, we use the following
property of the generalized Mittag-Leffler function's inverse Laplace transform
[27](3.18)L−1[n!sλ−μ(sλ∓c)n+1]=tλn+μ−1Eλ,μ(n)(±ctλ)(Re(s)>|c|1/λ).Inverting (3.18) by the Fourier
sine inverse transform, we obtain an exact solution to the velocity field as
follows:(3.19)u(y,t)=2π∫0∞ζsin(ζy)ζ2+β2∑k=0∞(−1)kk!(ζ2+β2γ)k+1t(k+1)(α+1)Eα,α+k+2(k)(−tαγ)dζ.In a special case, whenα=1,
corresponding to Stokes' first problem for a Maxwell fluid in a porous
half-space, (3.14) can be written as(3.20)U¯(ζ,s)=ζs[γs2+s+(ζ2+β2)].Correspondingly, (3.19) can be
written as(3.21)u(y,t)=2π∫0∞ζsin(ζy)ζ2+β2∑k=0∞(−1)kk!(ζ2+β2γ)k+1t2(k+1)E1,k+3(k)(−tγ)dζ,which is just the dimensionless
velocity solution of a Maxwell fluid in a porous half-space obtained in another
form by Tan [22].In a special case, whenα=1 and β=0,
we have(3.22)u(y,t)=2π∫0∞sin(ζy)ζ∑k=0∞(−1)kk!(ζ2γ)k+1t2(k+1)E1,k+3(k)(−tγ)dζ,which is just the dimensionless
velocity solution of a clear Maxwell fluid obtained in another form by Jordan et al.
[38].In another special case, whenα=1, β=0, and γ=0,
corresponding to the classical version of Stokes' first problem for a Newtonian
fluid, (3.14) can be written as(3.23)U¯(ζ,s)=ζs(s+ζ2).Correspondingly, (3.19) can be
written as(3.24)u(y,t)=1−2π∫0∞sin(ζy)ζexp(−ζ2t)dζ.Equation (3.24) can be simplified
further as follows:(3.25)u(y,t)=1−erf(y2t),where erf(x) is the error function of Gauss. This is just
the well-known result of Stokes' first problem for a clear Newtonian fluid.


## 3.2. Solution of Temperature Field

In order to determine the temperature distribution in
the porous half-space, we multiply both sides of (3.6) and (3.7) by sin(sy) and then integrate with respect to y from 0 to ∞.
Using (3.8) and (3.9), we get(3.26)∂T¯(s,t)∂t+ξs2T¯(s,t)=ξsf(t)+g(s,t),(3.27)T¯(s,0)=0,where T¯(s,t) and g(s,t) denote the Fourier sine transforms of T(y,t) and Qt(y,t) with respect to y, respectively. Since the solution of velocity u(y,t) is obtained by (3.21), g(s,t) is a known function. The solution of the
ordinary differential equation (3.26) subject to the initial condition
(3.27) can
be easily obtained as(3.28)T¯(s,t)=exp(−ξs2t)∫0t(ξsf(τ)+g(s,τ))exp(ξs2t)dτ.Inverting (3.28) by the Fourier
sine inverse transform, and integrating by parts the term containing f(τ),
we get(3.29)T(y,t)=f(0)(1−erf(y2ξt))+∫0tf′(τ)(1−erf(y2ξ(t−τ)))dτ+2π∫0∞∫0tg(s,t)exp(−ξs2(t−τ))sin(sy)dτds.In a special case, whenU0=0,
the fluid is at rest, and the temperature distribution is the same for both a
Maxwell fluid and a Newtonian one. Then, (3.29) takes the form(3.30)T(y,t)=f(0)(1−erf(y2ξt))+∫0tf′(τ)(1−erf(y2ξ(t−τ)))dτ.From (3.30), we easily obtain that(3.31)T(y,t)→T0(∞)ast→∞.If, moreover, the plate is kept
at a constant temperature, (3.31) reduces to a well-known form,(3.32)T(y,t)=f(0)(1−erf(y2ξt)).


## 4. Conclusion

In this work, we have presented some results about the
generalized Maxwell fluid in a porous half-space on a heated flat plate. An
exact solution of the velocity field was obtained by using the Fourier sine
transform and the fractional Laplace transform. The temperature distribution in
the generalized Maxwell fluid in the porous half-space subject to a linear flow
on a heated flat plate was determined. Some classical results can be considered
as particular cases of our results, such as the solutions corresponding to a
Newtonian viscous fluid, a clear Maxwell fluid, and a Maxwell fluid in a porous
half-space. They can be easily obtained from the results presented in this
paper by lettingα=1, β=0, γ=0; α=1, β=0, and α=1,
respectively. It has also been shown that the fractional constitutive
relationship model is more flexible than the conventional model in describing
the properties of viscoelastic fluids.



---

*Source: 641431-2008-05-25.xml*