# Steady Flow of a Second-Grade Fluid in an Annulus with Porous Walls

Authors: M. Emin Erdoğan; C. Erdem İmrak
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

An exact solution of an incompressible second-grade fluid for flow between two coaxial cylinders with porous walls is given. It is assumed that the inner cylinder is rotating with a constant angular velocity and the outer one is at rest. The solution is expressed in terms of the confluent hypergeometric functions and it is valid for all values of the cross-Reynolds number and the elastic number. The solutions for−2, +∞, and −∞ values of the cross-Reynolds number are obtained and a comparison with those of the Newtonian fluid is given.  Furthermore, the torque exerted by the fluid on the inner cylinder is calculated. It is shown that the moment coefficient depends on the cross-Reynolds number, the elastic number, and the ratio of the radii of the cylinders. The variation of the moment coefficient with these numbers is discussed.


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## Body

## 1. Introduction

The flow considered in this paper is for flow of a second-grade fluid in
an annulus with porous walls. For a Newtonian fluid, the flow in an
annulus has been investigated by Berman [1, 2] and Yuan [3]. The extensions of this flow to
non-Newtonian fluids have been studied by many authors [4–6]. However, the results for non-Newtonian
fluids have been given by using a perturbation method. Therefore, it
is not possible to compare with the exact solution of a Newtonian
fluid for all values of the cross-Reynolds number which is defined
by the suction velocity at the surface of the cylinder, the radius
of the cylinder, and the kinematic viscosity of the fluid. The exact
solution given in this paper provides comparison with the exact
solution of a Newtonian fluid for all values of the cross-Reynolds
number. The non-Newtonian fluid model in this paper is that of the
second-grade fluid [7]. Although
there are some criticisms on the applications of this model, many
papers have been published and a listing of some of them may be
found in the literature. Furthermore, it has been shown by Walters
[8] that for many types of problems
in which the flow is slow enough in the viscoelastic sense, the
results given by Oldroyd’s constitutive equation will be
substantially similar to those of fluids of the second or third
grade. Therefore, it would seem reasonable to use the second- or
third-grade models in carrying out the calculations. This is
particularly so in view of the fact that the calculation is
generally simpler. The constitutive equation for an incompressible
second-grade fluid has three coefficients. There are some
restrictions on these coefficients due to the Clausius-Duhem
inequality and due to the assumption that the Helmhotlz free energy
is minimum in equilibrium. A comprehensive discussion on the
restrictions for these coefficients has been given by Dunn and
Fosdick [9] and Dunn and Rajagopal
[10]. The sign of the material moduli
is the subject of much controversy [11, 12].The no-slip boundary condition is
sufficient for a Newtonian fluid, but may not be sufficient for a
fluid of second grade. Therefore, one needs an additional condition
at boundary. However, if one uses a perturbation expansion in terms
of the coefficient appearing in the higher-order derivative of the
governing equation, the no-slip boundary condition is sufficient. A
critical review on the boundary conditions, the existence, and
uniqueness of the solution has been given by Rajagopal [13]. In order to overcome the difficulty on
the boundary condition, several workers have studied to give an
additional condition. Frater [14] has
studied the problem without using a perturbation expansion in terms
of the coefficient of the higher-order term of the governing
equation. He imposed an additional condition so that the solution
tends to the Newtonian value as the coefficient of the higher
derivative in the equation approaches zero. This can be used
provided the explicit solution or the numerical solution should
exist. In this paper, the solution is obtained in terms of the
confluent hypergeometric functions and has three coefficients: two
of them can be determined by the no-slip boundary condition and the
other can be determined by using the properties of the confluent
hypergeometric functions.The problem considered in this paper is an
extension of a viscous fluid in an annulus with uniform porous walls
to the problem for the flow of a second-grade fluid. The fluid
injection rate at one wall is taken equal to the withdrawal rate at
the outer wall. Further, because of this restriction on the cross
wall, the velocity components do not vary along the axis of the
cylinder. The flow in an annulus with porous walls has been
investigated by many authors [4–6]. In order to obtain the
solution in [4, 6], a perturbation method has been used, therefore, the
results obtained can be applicable for small values of the
cross-Reynolds number. The axial flow in an annulus with porous
walls of a second-grade fluid has been investigated in [15]. It can be shown that the axial and the
azimuthal velocities are superposable. The solution for the flow of
a second-grade fluid between two parallel porous plates has been
given in [16]. The flow of a
second-grade fluid generated by a rotating porous cylinder has been
investigated in [17]. The flow of a
second-grade fluid between eccentric rotating cylinders was examined
in [18]. Assuming that the
eccentricity is small, the secondary flow for a third-grade fluid
between two eccentrically placed cylinders has been investigated in
[19]. The flow of a second-grade fluid
in a porous cylindrical tube was examined for suction or injection
through the surface of the tube, under the assumption that the
second-order parameter is sufficiently small [20]. In this paper, the solution is obtained in terms of
the confluent hypergeometric functions and it is valid for all
values of the cross-Reynolds number and the elastic number. The
solutions for +∞, −∞,
and −2 values of the cross-Reynolds number are
discussed. The solution for +∞ value of the cross-Reynolds number is given in
terms of the modified Bessel functions of the first kind of order
one. The solution for −∞ value of the cross-Reynolds number is given in
terms of the modified Bessel function of the second kind of order
one. The solution for −2 value of the cross-Reynolds number is given in
terms of the exponential integral.The torque exerted by the fluid on the
inner cylinder is calculated. This can be used to obtain the values
of the material coefficients. It is shown that the moment
coefficient depends on the cross-Reynolds number, the elastic
number, and the ratio of the radii of the cylinders. The variation
of the moment coefficient with these numbers is discussed. There are
appreciable differences between that of the Newtonian fluid and that
of the second-grade fluid.


## 2. Basic Equations

The
equation of motion for a fluid in the absence of body forces is(2.1)ρDuDt=∇.σ, where ρ is the density of the fluid, u is the velocity, σ is the stress tensor, and D/Dt represents the material derivative. The
continuity equation for the velocity is
(2.2)∇.u=0. Equations (2.1)
and (2.2) can be applied to all types of incompressible Newtonian
and non-Newtonian fluids. The stress depends on the local properties
of the fluid. The relation between the stress and the local
properties of the fluid is called constitutive equation. The
constitutive equation for an incompressible second-grade fluid is in
the following form [7]:
(2.3)σ=−pI+μA1+α1A2+α2A12, where μ, α1, and α2 are material constants, and An represents the Rivlin-Eriksen tensor defined
as [21]
(2.4)A0=I,A1=∇u+(∇u)T,A2=(∂∂t+u.∇)A1+A1.(∇u)+(∇u)T.A1, where t is time, p is pressure, and I is the identity tensor. The Clausius-Duhem
inequality and the condition that Helmholtz free energy is minimum
in equilibrium provide the following restrictions [9, 10]:
(2.5)μ≥0,α1+α2=0,α1≥0. A comprehensive
discussion on the restriction for μ, α1, and α2 can be found in the work by Dunn and Rajagopal
[10]. The sign of the material moduli α1
and  α2 is the subject of much controversy [11, 12].
The experiments have not confirmed these restriction on α1
and
α2.
Thus, the conclusion is that the fluids which have been tested are
not fluids of second grade and characterized by a different
constitutive structure.Fully developed laminar flow of an
incompressible fluid of second grade in an annulus with porous walls
is considered. The cylindrical polar coordinates are used. The radii
of the porous cylinders area1 and b1(a1<b1).
The rate of fluid withdrawal at one wall of the annulus is assumed
to be equal to the rate of injection of fluid at the other wall, and
that these rates are independent of axial position in annulus. The
condition that the suction rate at wall is equal to the injection
rate at the outer is satisfied if
(2.6)b1ub=a1ua, where ua and ub are the radial velocities at walls of smaller
and larger pipes, respectively. Equation (2.6) provides that the
velocity is independent of the axial position in annulus. The
velocity field is assumed to be in the following form:
(2.7)ur=αr,uθ=u(r),uz=0where ur,uθ,uz
are components of the velocity in cylindrical polar coordinates,
α is positive for injection at the inner
cylinder and negative for suction at the inner one. The velocity
field given by (2.7) is not a special case of the flow that is given
in [22]. The velocity field in [22] is a superposition of the stagnation
point flow and the rotating flow. Equation (2.2) is satisfied
identically by the velocity given by (2.7). Inserting the velocity
given by (2.7) into the expression of the stress, the components of
the stress tensor, in cylindrical polar coordinates, can be written
in the following forms:
(2.8)σrr=−p−2αμr2+α1[8α2r4+2u′2−2(u2r)′]+α2[4α2r4+(u′−ur)2],σrθ=μ(u′−ur)+α1αr(u′′+u′r−ur2),σrz=0,σθθ=−p+2μαr2+α2[4α2r4+(u′−ur)2],σθz=0,σzz=−p,where σrθ=σθr, σrz=σzr, σθz=σzθ;
the primes denote differentiation with respect to r.
Inserting the stress components and the velocity given by (2.7) into
(2.1), one obtains
(2.9)α1α(u′′′r+2u′′r2−u′r3+ur4)+μ(u′′+u′r−ur2)−ραr(u′+ur)=0 and the boundary
conditions are
(2.10)u(a1)=Ωa1,u(b1)=0, where a1 is the radius of the inner cylinder and b1 is the radius of the outer cylinder. The
solution to (2.9) has three coefficients: two of them can be
determined by the no-slip condition (2.10) and the other can be
determined by the confluent hypergeometric functions. It is expected
that when the viscoelastic parameter goes to zero, the solution
should give the Newtonian case [23],
leading to entirely two different kinds of flow behavior.
Similar discussions can be found in [24–26].


## 3. Solution of the Problem

After
some manipulations, (2.9) takes the form
(3.1)xy′′+(1−x)y′+R2y=0, where (3.2)u=g(r)r,g′=rb12y(x),x=−ξ22εR,ξ=rb1,R=αv,ε=α1/ρb12,where prime
denotes the differentiation with respect to r, y(x) is the new function, x is the new variable, R is the cross-Reynolds number, and ε is the elastic number and the boundary
conditions are
(3.3)g(a1)=Ωa12,g(b1)=0. Equation (3.3)
denotes that the inner cylinder is rotating with angular velocity
Ω and the outer one is at rest. The solution of (3.1)
can be written in the following form:
(3.4)y=C1M(−R2,1,x)+C2U(−R2,1,x), where M(a,b,x) and U(a,b,x) are the confluent hypergeometric functions [27, 28]. The solution has three coefficients; therefore, one needs three boundary
conditions. Thus, unless an additional condition is prescribed over
the conditions (2.10), one has a parametric solution. However, the
third coefficient can be determined by the properties of the
confluent hypergeometric functions. For R>0,x becomes negative; then for
x<0,U(a,b,x) is not acceptable and therefore C2 must be zero and (3.4) takes the form
(3.5)dgdr=C1rb12M(−R2,1,−r2/b122εR).Using the
identity in [28],
(3.6)M(−R2,1,−r2/b122εR)=e−r2/b12/2εRM(1+R2,1,r2/b122εR).The integration
gives
(3.7)g=C∫e−zM(1+R2,1,z)dz+D,where z=(r2/b12)/2εR.
Using the identity in [28] which
is
(3.8)∫e−zM(1+R2,1,z)dz=ze−zM(2+R2,2,z)+constant,and by the
boundary condition (3.3), one obtains
(3.9)gΩa12=ξ2e−ξ2/2εRM(2+R/2,2,ξ2/2εR)−e−1/2εRM(2+R/2,2,1/2εR)σ2e−σ2/2εRM(2+R/2,2,σ2/2εR)−e−1/2εRM(2+R/2,2,1/2εR), where u/Ωa1=(σ/ξ)(g/Ωa12) and σ=a1/b1.
When ε goes to zero, using the asymptotic expression
of M(a,b,x) in [28], g/Ωa12 becomes
(3.10)lim⁡ε→0gΩa12=lim⁡ε→0(r2/b12)(σ2/b12/2εR)R/2−(1/2εR)R/2σ2(σ2/2εR)R/2−(1/2εR)R/2=1−ξR+21−σR+2,
or
(3.11)uΩa1=σ1−σ2+R(1ξ−ξ1+R),which is the
expression of the velocity of a Newtonian fluid. The variation of
u/Ωa1 with respect to ζ=(ξ−σ)/(1−σ) for various values of R and ε is illustrated in Figure 1. The value of σ is taken as 0.2 and the values of ε are taken as 0 and 1.
Equation (3.9) is valid for all values of R and ε.
When R goes to infinity, using the expression given in
[28] which isFigure 1
The variation of the azimuthal velocity for various values of the
cross-Reynolds number and the elastic number.(ε=0); (ε=1), ζ=(ξ−σ)/(1−σ); σ≤ξ≤r/b1, σ=0.2.(3.12)lim⁡a→∞1Γ(b)M(a,b,xa)=x1/2−(1/2)bIb−1(2x),
equation (3.9) takes the following form:
(3.13)uΩa1=σξξI1(ε-1/2ξ)−I1(ε-1/2)σI1(ε-1/2σ)−I1(ε-1/2), where Γ(x) is the Gamma function and I1(x) is the modified Bessel function of the first
kind of order one. The variation of u/Ωa1 with respect to ζ for various values of ε is illustrated in Figure 1. Since the
asymptotic form of I1(x) is ex/2πx when ε goes to zero, (3.13) becomes
(3.14)uΩa1=σξ. Equation (3.14)
satisfies the boundary condition at ξ=σ,
but it does not satisfy the boundary condition at ξ=1.
It is clearly seen from Figure 1
that when R goes to +∞,
the flow shows a boundary layer character near the outer cylinder
for a Newtonian fluid, but it does not behave the same way for a
fluid of second grade. A similar situation occurs for the axial flow
between two circular cylinders with porous walls of a second-grade
fluid [15].ForR<0, x is positive; then for x>0, M(a,b,x) becomes an increasing function of x;
therefore, C1 must be zero, then (3.4) takes the form
(3.15)dgdr=C2rb12(N2,1,r2/b122εN),where N=−R(>0).
Using the identity in [28] which
is
(3.16)∫U(a,1,z)dz=11−aU(a−1,0,z)+constant,and by the
boundary condition, the integration gives
(3.17)gΩa12=U(N/2−1,0,r2/b12/2εR)−U(N/2−1,0,1/2εN)U(N/2−1,0,σ2/2εN)−U(N/2−1,0,1/2εN),where u/Ωa1=(σ/ξ)(g/Ωa12).
When ε goes to zero, using the asymptotic expression
of U(a,b,x) in [28], one obtains
(3.18)uΩa1=σ1−σ2−N(1ξ−ξ1−N)which is the
expression of the velocity for a Newtonian fluid. The variation of
u/Ωa1,
with respect to ζ=(ξ−σ)/(1−σ) for various values of −R and ε,
is illustrated in Figure 1. The
value of σ is taken as 0.2 and the values of ε are taken as 0 and 1.
Equation (3.17) is valid for all values of −R and ε.
When N goes to infinity, using the asymptotic
expression given in [28] which is
(3.19)lim⁡a→∞Γ(1+a−b)U(a,b,xa)=2x1/2−(1/2)bKb−1(2x). Equation (3.17) takes
the following form:
(3.20)uΩa1=σξξK1(ε-1/2ξ)−K1(ε-1/2)σK1(ε-1/2σ)−K1(ε-1/2), where K1(x) is the modified Bessel function of the second
kind of order one. The variation of u/Ωa1 with respect to ζ for various values of ε is illustrated in Figure 1. Since the
asymptotic form of K1(x) is π/2xe−x when ε goes to zero,  (3.20) becomes
(3.21)uΩa1=0. Equation (3.21)
satisfies the boundary condition at ξ=1,
but it does not satisfy the boundary condition at ξ=σ.
The reason is that a boundary layer occurs at ξ=σ for a Newtonian fluid, but it does not behave
the same way for a fluid of second grade.The case ofR=−2,
namely N=2,
cannot be obtained by (3.17). For N=2,
by the definition of U(a,b,x), U(1,1,x) can be found as
(3.22)U(1,1,x)=exEi(x),where Ei(x) is the exponential integral defined in the
following form
(3.23)Ei(x)=∫x∞e−ttdt.Inserting U(1,1,x) into the expression of dg/dr and by integration and by using the boundary
conditions, one finds
(3.24)uΩa1=σξeξ2/4εEi(ξ2/4ε)−e1/4εEi(1/4ε)+ln⁡(ξ2)eσ2/4εEi(σ2/4ε)−e1/4εEi(1/4ε)+ln⁡(σ2). The variation of Ωa1 with respect to ζ for various values of ε is illustrated in Figure 1. The value of σ is taken as 0.2 and the values of ε are taken as 0 and 1.
When ε goes to zero, the argument of the exponential
integral becomes large, then Ei(x) can be written as e−x/x and (3.24) becomes
(3.25)uΩa1=σξln⁡ξln⁡σ.


## 4. The Torque Exerted by the Fluid on the Inner Cylinder

The torque
exerted by the flow on the inner cylinder is given in the following
form:(4.1)M=2πha12[μ(dudr−ur)+αα1rddr(dudr+ur)]r=a1, where h is the length of the inner cylinder.
The moment coefficient is
(4.2)(Cm)r=a1=σ[(dudξ−uξ)+εRξddξ(dudξ+uξ)]ξ=σ. Equation (4.2)
shows that the moment coefficient depends on the cross-Reynolds
number, the elastic number, and the ratio of the radii of the
cylinders. The variation of the moment coefficient with these
numbers is illustrated in Figure 2. There are appreciable differences between that of the Newtonian fluid and that of the second-grade fluid. The curves
in Figure 2 show the effect of the second-grade fluid. The moment
coefficient decreases with the cross-Reynolds number for a Newtonian
fluid and increases with the cross-Reynolds number for a
second-grade fluid, for all values of the ratio of the radii of the
cylinders.Figure 2
The variation of the moment coefficient with the cross-Reynolds number. The
ratio of the radii of the cylinders is taken as 0.2.(ε=0);
(ε=1),


## 5. Conclusions

An exact solution of the governing equation for the flow of a fluid of
second grade between two coaxial cylinders with porous walls is
given. The flow of a second-grade fluid for which the inner cylinder
is rotating and the outer one is at rest is considered. The solution
is given in terms of the confluent hypergeometric functions and is
valid for all values of the cross-Reynolds number and the elastic
parameter. The boundary conditions are the no-slip boundary
conditions and an additional boundary condition. The additional
boundary condition is that the solution tends to the Newtonian value
as the elastic number in the solution approaches zero. The solution
is discussed for positive, negative, infinite, and−2 values of the cross-Reynolds number. The torque exerted by the fluid on the inner
cylinder is calculated. It is shown that the moment coefficient
depends on the cross-Reynolds number, the elastic number, and the
ratio of the radii of the cylinders. For a given value of this
ratio, it decreases with the cross-Reynolds number for a Newtonian
fluid and increases with the cross-Reynolds number for a
second-grade fluid.



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*Source: 867906-2008-03-26.xml*