#### Peristaltic transport of an Oldroyd-B fluid in a planar channel

Hindawi Publishing Corporation
Mathematical Problems in Engineering
T. HAYAT 0
Y. WANG 0
K. HUTTER 0
S. ASGHAR 0
A. M. SIDDIQUI 0
0 S. Asghar: Department of Mathematics, Quaid-i-Azam University , Islamabad 45320, Pakistan E-mail address: s
The effects of an Oldroyd-B fluid on the peristaltic mechanism are examined under the long wavelength assumption. Analytical expressions for the stream function, the axial velocity, and the pressure rise per wavelength are obtained up to the second order in the dimensionless wave number. The effects of the various parameters of interest on the flow are shown and discussed.
1. Introduction
The word peristalsis derives from the Greek word περισταλτικos which means clasping
and compressing. It is used to describe a progressive wave of contraction along a channel
or tube whose cross-sectional area consequently varies. Peristalsis is regarded as having
considerable relevance in biomechanics and especially as a major mechanism for fluid
transport in many biological systems (as it is in the human). It appears in the ureter, in
the intestines, and in the oviducts, to name just a few instances.
Great strives have been undertaken, both experimentally and theoretically, to study
the propagation of waves in peristaltic motion [
3, 12, 14, 18, 26, 52, 53
]. Arbitrary shapes
of these waves [
27, 29, 31, 32, 33
] as well as sinusoidal waves [
1, 10, 12, 15, 23, 49
] have
been analyzed and measuring techniques [9, 24] were designed to test and verify early
hydrodynamic models [
15, 40, 41
].
The governing equations are nonlinear, so assumptions are made about the amplitude
ratio, the wavenumber, and the Reynolds number. The amplitude ratio is the ratio of the
amplitude of the wave to the half-width of the channel and is usually taken to be small.
The case of vanishingly small Reynolds number has also received considerable attention
[
3, 12, 41
]. To include nonlinear effects due to nonvanishing Reynolds number, solutions
are usually presented as expansions in terms of a small parameter. They are generally of
two types:
(1) expansion parameter is the amplitude of the wave that disturbs the wall; such an
expansion was pursued in [
15
] for the channel and in [
13, 55
] for the pipe, up to
second order. Only zero-mean flow was considered for the second-order terms,
a restriction that was removed in [
35
]. This approach, while valid for all Reynolds
numbers and wavelengths of the disturbing wall, is restricted to small amplitudes
and has been applied only to a sinusoidal wave;
(2) expansion in terms of the wavenumber and the Reynolds number for all wave
amplitudes. This has been done only for sinusoidal waves, and up to first order in
the square of the wavenumber (the natural parameter) and second order in the
Reynolds number [22, 28, 56]. The nonsinusoidal wave was studied in [27] for
small intestine and for zero Reynolds number only.
To obtain information about flows at moderate Reynolds number it has been
necessary to use numerical methods. Several investigators [
2, 10, 21, 49
] used numerical
methods for the solution of the Newtonian hydrodynamical equations. The results, in general,
agree with the analytical perturbation solutions in their range of validity with the
exception of the calculations of the pressure field by Takabatake and Ayukawa [49]. It is noted
that the higher-order terms of Reynolds and wavenumbers do not significantly extend the
range of validity of the results.
The application of the theory of particle-fluid mixture is also very useful in
understanding a number of diverse physical problems concerning peristalsis. An interesting
example is the particulate suspension theory of blood [
5, 20, 34, 36, 46, 47, 50
]. Peristaltic
transport of solid particles with fluid has first been attempted in [21]. Various geometric
and dynamic effects on the particle transport in a channel with flexible walls were
examined. The peristaltic motion for the case of two-phase flow was studied in [48] where a
perturbation solution for a small amplitude ratio is given.
Most studies on the peristaltic motion assume the physiological fluids to behave like
Newtonian fluids with constant viscosity. However, this approach fails to give an
adequate understanding of the peristaltic mechanism involved in small blood vessels,
lymphatic vessels, intestine, and ductus efferentes of the male reproductive tracts. In these
body organs, the viscosity of the fluid varies across the thickness of the duct [
11, 16, 19
].
Also, the assumption that the chyme in small intestine is a Newtonian material of
variable viscosity is not adequate in reality. Chyme is undoubtedly a non-Newtonian fluid.
Some authors (see, e.g., [51]) feel that the main factor responsible for moving the chyme
along the intestine is a gradient in the frequency of segmentation (a process of oscillating
contraction and relaxation of smooth muscles in the intestine wall) along the length of
intestine. Moreover, peristaltic waves die out after travelling a very short distance;
peristaltic waves which travel the entire length of small intestine do not occur in humans
except under abnormal conditions. Also, in transport of spermatozoa in the cervical canal,
there are some other important factors, responsible for the transport of semen in ductus
efferentes. One of the major factors is cilia, which keep semen moving towards the
epididymis [
7, 17, 25, 30, 54
]. The phenomenon of peristalsis has also been proposed as a
mechanism for the transport of spermatic fluid (semen) in vas deferens [
39
]. Movement
through vas deferens is accomplished by means of peristaltic action of contractile cells in
the duct wall [
39, 51
]. However, there is no doubt that peristalsis aids in moving semen
in ductus efferentes, the chyme in the intestine, and flow of semen in vas deferens.
The above review of physiological flows indicates that non-Newtonian viscoelastic
rheology is the correct way of properly describing the peristaltic flow through channels and
tubes. Only a few studies [
4, 8, 42, 43, 44, 45, 47
] have considered this aspect of the
problem. Although the second, and third-order models in [
43, 44, 45
] take into account
normal stress differences and shear-thinning/thickening effects, they lack other features
such as stress relaxation. The Oldroyd-B fluid, which includes elastic and memory
effects exhibited by dilute solutions, has been extensively used in many applications, and
also results of simulations fit experimental data quite well [
6, 37, 38
]. However, so far,
no attempt has been made to understand the peristaltic motion for an Oldroyd-B fluid.
We propose to study the effects of an Oldroyd-B fluid on the mechanism of peristaltic
transport in a planar channel. Of course the natural coordinate system is
axisymmetric; however, the planar case has been predominantly studied. Qualitatively the transport
phenomenon of the fluid is similar for both configurations [23]. Also, experimental data
are available for channel flows [26, 53]. Therefore, the present mathematical model
considers an Oldroyd-B fluid between parallel walls on which a sinusoidal travelling wave
is imposed. The assumption for the present analysis is that the length of the peristaltic
wave is large compared with the half-width of the channel. This assumption is similar
to those used in [22, 43] for the peristaltic motion of Newtonian and second-order
fluids, respectively. A regular perturbation technique is adopted to solve the present
problem and solutions are expanded in a power series of the small dimensionless
wavenumber. The Reynolds number and material time constants are left arbitrary. The analysis is
completely analytical but lengthy, and closed-form solutions up to second order of the
wavenumber are presented. The effects of the nonlinear terms of the governing equations
on the fluid transport are constructed. Comparison is made between the results for the
Newtonian and Oldroyd-B fluids. The explicit non-Newtonian terms are obtained and
their effect on peristaltic motion is examined. The results for Maxwell and Newtonian
fluids are obtained as special cases of the presented analysis.
2. Basic equations
Consider an incompressible fluid whose balance laws of mass and linear momentum are
given by
div V¯ = 0,
ρV˙ = div T¯ + ρf¯,
T = −p¯I + S¯,
¯
where ρ, V¯, T¯, and f¯ are mass density, velocity, Cauchy stress tensor, and specific body
force and the dot (·) denotes material time derivative. In the ensuing analysis, body forces
will be ignored and isothermal conditions will be implied. The above system of equations
will be closed by a constitutive equation for the stress tensor. The constitutive equation
for the Cauchy stress T¯ in an Oldroyd-B fluid is given by [
37
]
where the extra stress tensor S¯ is given by
(2.4)
in which −p¯I is the spherical part of the stress due to the constraint of incompressibility,
d/dt¯ denotes material time derivative, µ is the viscosity, and Λ1 and Λ2 are material time
constants referred to as relaxation and retardation time, respectively. It is assumed that
Λ1 ≥ Λ2 ≥ 0. The tensors L¯ and A¯ 1 are defined as follows:
L = grad V¯,
¯
¯
A1 = L¯ + L¯ T ,
where V¯ is the velocity vector. It should be noted that this model includes the
classical linear case for Λ1 = Λ2 = 0, and when Λ2 = 0, the model reduces to the Maxwell
model.
3. Formulation of the problem and flow equations
Consider a two-dimensional flow of an Oldroyd-B fluid in an infinite channel having
width 2a. Assume an infinite wave train travelling with velocity c along the walls. Choose
a rectangular coordinate system for the channel with X¯ along the central line in the
direction of wave propagation, and Y¯ transverse to it. Let the geometry of the wall surface
be defined as
h¯(X¯ , t¯) = a + b sin
2π (X¯ − ct¯) ,
λ
where b is the wave amplitude and λ the wavelength. Assume, moreover, that there is no
motion of the wall in the longitudinal direction (this assumption constrains the
deformation of the wall; it does not necessarily imply that the channel is rigid against longitudinal
motions, but is a convenient simplification that can be justified by a more complete
analysis. The assumption implies that for the no-slip condition U¯ = 0 at the wall).
For unsteady two-dimensional flows,
(2.5)
(3.1)
V¯ = U¯ (X¯ , Y¯ , t¯), V¯ (X¯ , Y¯ , t¯), 0 ,
(3.2)
and we find that (2.1)–(2.5), in the absence of body forces, take the following form:
∂U¯ ∂V¯
∂X¯ + ∂Y¯ = 0,
ρ
ρ
∂t¯ ∂X¯ + V¯ ∂∂Y¯
∂ + U¯ ∂
U¯ = −
+ ∂S¯X¯X¯ + ∂∂S¯YX¯¯Y¯ ,
∂X¯
+ ∂S¯X¯Y¯ + ∂∂S¯YY¯¯Y¯ ,
∂X¯
S¯X¯X¯ + Λ1
∂∂t¯ + U¯ ∂∂X¯ + V¯ ∂∂Y¯ S¯X¯X¯ − 2 ∂∂UX¯¯ S¯X¯X¯ − 2 ∂∂UY¯¯ S¯X¯Y¯
= 2µ ∂U¯
∂X¯ + 2µΛ2
∂∂t¯ + U¯ ∂∂X¯ + V¯ ∂∂Y¯ ∂∂UX¯¯ − 2 ∂∂UX¯¯ 2
∂U¯ ∂U¯ ∂V¯
− ∂Y¯ ∂Y¯ + ∂X¯
,
S¯X¯Y¯ + Λ1
∂∂t¯ + U¯ ∂∂X¯ + V¯ ∂∂Y¯ S¯X¯Y¯ − ∂∂UY¯¯ S¯Y¯ Y¯ − ∂∂VX¯¯ S¯X¯X¯
∂U¯ ∂V¯
= µ ∂Y¯ + ∂X¯
+ µΛ2
∂t¯ ∂X¯ + V¯ ∂∂Y¯
∂ + U¯ ∂
− 2 ∂∂UX¯¯ ∂∂VX¯¯ + ∂∂UY¯¯ ∂∂VY¯¯
∂U¯ ∂V¯
∂Y¯ + ∂X¯
,
S¯Y¯ Y¯ + Λ1
∂∂t¯ + U¯ ∂∂X¯ + V¯ ∂∂Y¯ S¯Y¯ Y¯ − 2 ∂∂VX¯¯ S¯X¯Y¯ − 2 ∂∂VY¯¯ S¯Y¯ Y¯
= 2µ ∂∂VY¯¯ + 2µΛ2
∂∂t¯ + U¯ ∂∂X¯ + V¯ ∂∂Y¯ ∂∂VY¯¯ − 2 ∂∂VY¯¯ 2
∂V¯ ∂U¯ ∂V¯
− ∂X¯ ∂Y¯ + ∂X¯
where U¯ and V¯ are the longitudinal and transverse velocity components.
In the laboratory frame (X¯, Y¯ ), the flow in the channel is unsteady, but if we choose
moving coordinates (x¯, y¯) which travel in the positive X¯-direction with the same speed as
the wave, then the flow can be treated as steady. This coordinate system is known as the
wave frame. The coordinate frames are related through
and the velocity components in the laboratory and wave frames are related by
x¯ = X¯ − ct¯,
y¯ = Y¯ ,
u¯ = U¯ − c,
v¯ = V¯ ,
,
(3.3)
(3.4)
(3.5)
where u¯ and v¯ are dimensional velocity components in the directions of x¯ and y¯,
respectively. Employing these transformations in (3), we obtain
∂∂ux¯¯ + ∂∂vy¯¯ = 0,
ρ u¯ ∂∂x¯ + v¯ ∂∂y¯ u¯ = − ∂∂xp¯¯ + ∂∂S¯x¯x¯x¯ + ∂∂S¯yx¯¯y¯ ,
ρ u¯ ∂∂x¯ + v¯ ∂∂y¯ v¯ = − ∂∂py¯¯ + ∂∂S¯xx¯¯y¯ + ∂∂S¯y¯y¯y¯ ,
¯
Sx¯x¯ + Λ1
u¯ ∂∂x¯ + v¯ ∂∂y¯ S¯x¯x¯ − 2 ∂∂ux¯¯ S¯x¯x¯ − 2 ∂∂uy¯¯ S¯x¯y¯
= 2µ ∂U¯ + 2µΛ2
∂x¯
u¯ ∂∂x¯ + v¯ ∂∂y¯ ∂∂ux¯¯ − 2 ∂∂ux¯¯ 2
− ∂∂uy¯¯ ∂∂uy¯¯ + ∂∂xv¯¯
¯
Sx¯y¯ + Λ1
u¯ ∂∂x¯ + v¯ ∂∂y¯ S¯x¯y¯ − ∂∂uy¯¯ S¯y¯y¯ − ∂∂xv¯¯ S¯x¯x¯
= µ ∂∂uy¯¯ + ∂∂xv¯¯
+ µΛ2
¯ ∂ + v¯ ∂
u ∂x¯ ∂y¯
∂∂uy¯¯ + ∂∂xv¯¯ − 2 ∂∂ux¯¯ ∂∂xv¯¯ + ∂∂uy¯¯ ∂∂vy¯¯
,
S¯y¯y¯ + Λ1
¯ ∂ + v¯ ∂
u ∂x¯ ∂y¯
S¯y¯y¯ − 2 ∂v¯ ¯
∂x¯ Sx¯y¯ − 2 ∂∂vy¯¯ S¯y¯y¯
∂v¯
= 2µ ∂y¯
+ 2µΛ2
u¯ ∂∂x¯ + v¯ ∂∂y¯ ∂∂vy¯¯ − 2 ∂∂vy¯¯ 2
− ∂∂xv¯¯ ∂∂uy¯¯ + ∂∂xv¯¯
,
.
The formulation of the boundary conditions is postponed until Section 5.
4. Dimensionless formulation
To set the important parameters of the outlined problem in evidence, a scale analysis is
performed and the equations are nondimensionalized. Using the dimensionless variables
in (3), we arrive at
λx ,
x¯ = 2π
y¯ = ay,
u¯ = cu,
v¯ = cv,
S¯ = µac S,
λµc
p¯ = 2πa2 p,
¯
h = ah
δ ∂∂ux + ∂∂vy = 0,
e δu ∂∂x + v ∂∂y u = − ∂∂xp + δ ∂∂Sxxx + ∂∂Syxy ,
δ e δu ∂∂x + v ∂∂y v = − ∂∂py + δ2 ∂∂Sxxy + δ ∂∂Syyy ,
Sxx + λ1 δu ∂∂x + v ∂∂y Sxx − 2δ ∂∂ux Sxx − 2 ∂∂uy Sxy
= 2δ ∂∂ux + 2λ2 δ δu ∂∂x + v ∂∂y ∂∂ux − 2δ2 ∂∂ux 2 − ∂∂uy ∂∂uy + δ ∂∂xv
(4.3)
(4.4)
(4.5)
= ∂∂uy + δ ∂∂xv
+ λ2 δu ∂∂x + v ∂∂y
∂u + δ ∂v
∂y ∂x
− 2 δ2 ∂u ∂v + ∂u ∂v
∂x ∂x ∂y ∂y
Syy + λ1 δu ∂∂x + v ∂∂y
Syy − 2δ ∂v
∂x Sxy − 2 ∂∂vy Syy
= 2 ∂∂vy + 2λ2 δu ∂∂x + v ∂∂y ∂∂vy − 2 ∂y
− δ ∂∂xv ∂∂uy + δ ∂∂xv
,
where the dimensionless wavenumber δ, the Reynolds number e, and the Weissenberg
numbers λ1 and λ2 are defined, respectively, as
e = µc/aρ ,
These have easy physical interpretations: δ is a measure of how large the semidepth of
the peristaltic motion is, as compared to its wavelength. It is an aspect ratio and thus an
expression of shallowness. The Reynolds number e is formed with the wave speed, the
amplitude, and the kinematic viscosity of the Newtonian part of the constitutive
behavior; λ1 and λ2 measure the elastic contributions of the stress behavior.
The continuity equation (4.2), after defining the dimensionless stream function Ψ(x,
y) by the relations
u = ∂∂Ψy ,
v = −δ ∂∂Ψx ,
is identically satisfied and, from (4.3)–(4.7) we deduce
δ e
−δ3 e
∂Ψ ∂ ∂Ψ ∂ ∂Ψ
∂y ∂x − ∂x ∂y ∂y
∂Ψ ∂ ∂Ψ ∂ ∂Ψ
∂y ∂x − ∂x ∂y ∂x
= − ∂∂xp + δ ∂∂Sxxx + ∂∂Syxy ,
∂p + δ2 ∂Sxy + δ ∂Syy ,
= − ∂y ∂x ∂y
δ e
,
(4.6)
(4.7)
(4.9)
(4.10)
(4.11)
(4.12)
∂2Ψ ∂2Ψ 2 ∂2Ψ
− ∂y2 ∂y2 − δ ∂x2
Sxy + λ1 δ
= −2δ ∂∂x2∂Ψy + 2λ2 δ2 ∂∂Ψy ∂∂x − ∂∂Ψx ∂∂y
Sxx + λ1 δ
∂Ψ ∂ ∂Ψ ∂ ∂2Ψ ∂2Ψ
∂y ∂x − ∂x ∂y Sxx − 2δ ∂x∂y Sxx − 2 ∂y2 Sxy
= 2δ ∂∂x2∂Ψy + 2λ2 δ2 ∂∂Ψy ∂∂x − ∂∂Ψx ∂∂y ∂∂x2∂Ψy − 2δ2 ∂∂x2∂Ψy
2
(4.13)
(4.14)
(4.15)
(5.1)
(5.2)
(5.3)
where the compatibility equation (4.12) is obtained by eliminating p between (4.10) and
(4.11); it represents the vorticity transport equation. Notice that (4.10) and (4.11) are
formally decoupled from (4.12)–(4.15). So, the latter are used to determine Ψ and Sxx,
Sxy, Syy, while the former is then employed to determine the pressure field.
5. Rate of volume flow and boundary conditions
The dimensional rate of fluid flow in the laboratory frame is given by
where h¯, the position of the channel wall, is a function of X¯ and t¯. The rate of fluid flow
in the wave frame is given by
+ 2δ ∂∂x2∂Ψy ∂∂2yΨ2 + δ2 ∂2Ψ
∂x2
,
,
,
¯
h
Q = 0 U¯ (X¯, Y¯ , t¯)dY¯ ,
¯
h
q = 0 u¯(x¯, y¯)d y¯,
Q = q + ch¯.
where h¯ is now a function of x¯ alone. With the help of (3.4) and (3.5), one can show that
these two rates are related through
The time-averaged flow over a period T at a fixed position X¯ is given by
On using (5.3) in (5.4), we find that
If we define the dimensionless mean flows Θ, in the laboratory frame, and F, in the
wave frame, according to
one finds that (5.5) reduces to
Θ = F + 1,
Ψ(0) = 0,
Ψ(h) = F.
h(x) = 1 + Φ sin x,
b
Φ = a
(5.4)
(5.5)
(5.6)
(5.7)
(5.8)
(5.9)
(5.10)
(5.11)
(5.12)
(5.13)
(5.14)
where, according to the first equation of (4.9),
F =
h ∂Ψ
0 ∂y
d y = Ψ(h) − Ψ(0).
If we choose the zero value of the streamline along the central line (y = 0)
then the shape of the wave at the wall boundary is the streamline with value
The boundary conditions for the dimensionless stream function in the wave frame are
therefore
Ψ = 0 (by convention)
∂2Ψ
∂y2 = 0 (by symmetry)
∂Ψ
∂y = −1 (no-slip condition)
Ψ = F
on the central line y = 0,
at the wall y = h.
We also note that h represents the dimensionless form of the surface of the peristaltic wall
which will be chosen as a sinusoidal function, namely,
where
is the amplitude ratio or the occlusion and 0 < Φ < 1.
6. Perturbation solution
We note that (4.10)–(4.15) are higher-order nonlinear partial differential equations.
Therefore, it seems to be impossible to find the solution in closed form for arbitrary
values of all parameters. Even for Newtonian fluids [
22, 41
], all solutions obtained so far
are based on the assumption that one or several parameters are zero or small. Following
Jaffrin [22], we expand the flow quantities in a power series of the small parameter δ as
follows:
On substituting (6.1) into (4.10)–(4.15) and then collecting terms of equal powers of δ,
one obtains the following sets of perturbed equations.
(i) Zeroth-order equations
Ψ = Ψ0 + δΨ1 + δ2Ψ2 + · · · ,
p = p0 + δ p1 + δ2 p2 + · · · ,
S = S0 + δS1 + δ2S2 + · · · ,
F = F0 + δF1 + δ2F2 + · · · .
∂2S0xy
∂y2 = 0,
∂p0 + ∂S0xy
− ∂x ∂y
= 0,
∂p0
− ∂y = 0,
∂2Ψ0
S0xx − 2λ1 ∂y2 S0xy = −2λ2
∂2Ψ0 2,
∂y2
S0xy − λ1 ∂∂2yΨ20 S0yy = ∂∂2yΨ20 ,
S0yy = 0.
(ii) First-order equations
e
e
S1xx + λ1
= 2 ∂∂x2Ψ∂y0 − 4λ2 ∂∂2yΨ20 ∂∂2yΨ21 ,
∂∂Ψy0 ∂∂x − ∂∂Ψx0 ∂∂y ∂∂2yΨ20 + 2 ∂∂x2Ψ∂y0 ∂∂2yΨ20 ,
− ∂∂py2 + ∂S∂0xxy + ∂S∂1yyy = 0,
S2xy + λ1
with boundary conditions
∂4Ψ0
∂y4 = 0,
∂∂px0 = ∂∂3yΨ30 ,
∂p0
∂y = 0,
S0xy = ∂∂2yΨ20 ,
S0yy = 0
S0xx = 2 λ1 − λ2
∂∂2yΨ20 2,
Ψ0 = 0,
with boundary conditions
(6.22)
(6.23)
(6.24)
(6.25)
(6.27)
(6.28)
(iii) Second-order system
e
− 2 ∂∂2yΨ20 ∂∂2yΨ22 + ∂∂2xΨ20 ∂∂2yΨ20 −
,
(6.29)
(6.30)
(6.31)
S2xy = λ1
with boundary conditions
We will now solve each system, subject to the boundary conditions, and thereby
generate the series solution.
Zeroth-order solution. Mere inspection of the governing equations (6.15)–(6.21) shows
that with the exception of (6.18) this zeroth-order problem is described by the properties
of a linear viscous fluid flowing such that the long wavelength approximation is satisfied.
Only S0xx depends on the difference of the Weissenberg numbers and therefore exhibits
non-Newtonian properties. Effects of this contribution are not visible except possibly on
the walls if tractions are measurable. The reader may check that the solution of (6.15)–
(6.21) in terms of stream function Ψ0, the axial velocity u0, and the longitudinal pressure
gradient d p0/dx is given by
3
Ψ0 = − 2 F0 + h
L3
3 − L − y,
u0 = − 23h F0 + h L2 − 1 − 1,
where L = y/h, and from (6.17) it is clear that the transverse pressure gradient is zero.
It is also of some interest to calculate the pressure rise over a wavelength ( Pλ) in the
longitudinal direction on the axis (y = 0). At this order, we have
First-order solution. Inspection of (6.22)–(6.28) would lead us to assume that
nonNewtonian effects will now enter the first-order solution; however, on substituting
the zeroth-order solution (6.36) into (6.22) and (6.23), we find that
where
where
(6.39)
(6.40)
(6.41)
(6.42)
(6.43)
(6.44)
(6.45)
∂4Ψ1
∂y4 =
−
,
b1 = − e hx 3F02 + 5F0h + 2h2 ,
40
e hx 3F02 + 3F0h + h2 ,
8
b2 =
a01 =
where b1 and b2 are defined in (6.45), while b3 is given by
The axial velocity and the longitudinal pressure gradient at this order take the form
These formulas demonstrate that u1 shows no non-Newtonian response but d p1/dx does.
The pressure rise over a wavelength for this order turns out to be
(6.46)
(6.47)
(6.48)
(6.49)
Pλ1 = 235e 02π 27hF302 + 3hF20 − h3 hxdx
=0
We note that this expression is independent of the non-Newtonian effects; this comes
somewhat as a surprise.
Second-order solution. We have solved the second-order system. Computations are very
massive and mere inspection of the equations no longer suffices to deduce qualitative
behavior. We must restrict ourselves to presenting only the key steps and results,
respectively. Inserting the zeroth-order and first-order solutions in (6.29), we obtain
∂4Ψ2
∂y4 =
,
∂p2
∂y = −
3yhx 3hF40 + 2h ,
(6.51)
(6.52)
(6.53)
where
b1 = c11L7 + c12L5 + c13L3 + c14L,
b2 = y4 dh1112x −
b3 = e14L6 + e15L4 + e16L2,
b4 = y4 dh1132x −
h3
b6 =
2b0h1g1124 y5 + 2b01hg815 y3 + 2b0h1g416 y ,
b10 = h12 + g12 y3
g11 y5
h10 + gh138y ,
,
∂p2∗ y7 y
∂x = − b1x 7h7 − h
hx b1 +hb2 + b3 − b1x + b2x + b3x
= −
e28h0x 2 15hF02 − 3h − 3F21hhx +
1
c11 = h4 40a11b01 − 24a01b1 ,
1
c14 = h4 6b02a13 − 2a02b3 − 6a01 b1 + b2 + b3 − 2b01a14 ,
h3
−
Inspection of the inhomogeneous part of (6.51) shows that it is a polynomial of y. Thus,
there is hope to integrate (6.51) analytically. Indeed, after lengthy calculations, the
solution of (6.48) subject to the boundary conditions (6.35) turns out to be given by
Ψ2 =
+ 2 λ1 − λ2 m15h3 + m16h5 + m17h3 ,
3
C2 = 2h F2 + eh3m18 + 2λ1 λ1 − λ2 m19h2
The axial velocity at this order is given by
u2 =
e c71210yh170 + 3c3126yh85 + 1c2130yh63 + c2144yh4
It obviously depends on non-Newtonian (elastic) material behavior as it depends on the
difference of the Weissenberg numbers.
Such a dependence does now also arise in the pressure rise per wavelength along the
central line of the channel (y = 0). It can be obtained by integrating the second-order
pressure gradient d p2/dx from (6.52):
After lengthy calculations, the pressure rise per wavelength along the central line of the
channel (y = 0) at this order is given by
Pλ2 =
Φ dx + 6λ1 λ1 − λ2
Φ dx + 6 λ1 − λ2
Φ∗dx,
(6.59)
2π
0
2π
0
where
+
+
+
+ 2a025bh018b01x + 6a015bh028b01x + 4a023xhb071b02
16a02b01b02hx + 2b01xa02b02 ,
3h8 3h7
h
Φ∗ = 280 e11x −
6e1h1hx + 1h20 e12x −
4e1h2hx + 3h6 e13x −
2e13hx
h
17a12b01 + 9a02b1
7h5 7h5 −
24a01b2
35h5 −
3a11b02
h5
Summary. We summarize our results of the perturbation series through order 2.
Recalling (6.36)–(6.39), (6.46)–(6.50), and (6.52)–(6.59), we have shown that
L3
3 − L − y + δ b1
L7
7 − L + b2
L5
5 − L + b3
L3
3 − L
(6.61)
+ δ2u2,
(6.62)
(6.63)
(6.64)
(6.65)
− 23h F0 + h L2 − 1 − 1 +
+
,
Pλ = −3 F0I3 + I2 − 3δF1I3 + δ2
Pλ2 .
Ψ =
7. Discussion of some results
The above equations have been programmed to construct explicit results. In this section,
some results of the perturbation solutions (6.61) are graphically displayed.
In Figure 7.1, the pressure drops per wavelength along the flow direction (or the
pressure rises against the flow direction) ∆ Pλ are illustrated in terms of the wave amplitude Φ
of the wall disturbance with various values of the Weissenberg numbers λ1 (Figure 7.1(a)),
λ2 (Figure 7.1(b)), the dimensionless wavenumber δ (Figure 7.1(c)), and the total flux F
(Figure 7.1(d)), respectively. Firstly, it is obvious that for Φ = 0 the pressure drop is
independent of the Weissenberg numbers λ1 and λ2; in such a case, the flow behaves just as a
Newtonian fluid. Secondly, with increasing occlusion Φ the pressure drop increases, that
is, an increasing pressure gradient is needed to push the same flux to pass the channel.
Thirdly, increasing the Weissenberg number λ1 causes a decrease of the pressure drop to
maintain the same flux (Figure 7.1(a)), whereas with an increase of the Weissenberg
number λ2 the pressure drop ∆ Pλ increases (Figure 7.1(b)). Fourthly, the larger the
wavenumber δ is, the less pressure drop is required (Figure 7.1(c)). Furthermore, as expected, the
larger the flow flux is, the larger the pressure drop needed to press it to pass the channel
will be (Figure 7.1(d)).
The transverse distributions of the longitudinal velocity at the most narrow position
of the channel, which is at x = 3π/2 with a channel half-width of h = 0.7 for Φ = 0.3, are
shown in Figure 7.2 for different values of the Weissenberg numbers λ1 (Figure 7.2(a)),
λ2 (Figure 7.2(b)), dimensionless wavenumber δ (Figure 7.2(c)), and total flux F (Figure
7.2(d)). Owing to the limit of the no-slip boundary condition in (5.12), the velocity at the
wall has the same value u(y = h) = −1 in the wave frame for all values of the parameters.
The largest difference occurs near the central line y = 0. With increasing Weissenberg
68
64
60
56
Pλ52
48
44
40
36
70
60
50
λ 40
P∆30
20
10
0
(b)
δ = 0
δ = 0.05
δ = 0.09
(c)
(d)
number λ1 (Figure 7.2(a)) and increasing wavenumber δ (Figure 7.2(c)), the velocity
increases, especially near the center of the channel, while an increase of the Weissenberg
number λ2 produces a decrease of the velocity (Figure 7.2(b)). Besides, as a matter of
course, a large flux possesses a large velocity, as shown in Figure 7.2(d).
In Figure 7.3, the distributions of the velocity along the central line of the channel (y =
0) within a wavelength x ∈ (0, 2π) are displayed for various values of λ1 and δ. Obviously,
u
u
−1.0
−2.0
−3.0
−4.0
−5.0
−6.0
−1.0
−2.0
−3.0
−4.0
−5.0
−6.0
0
0
δ = 0
δ = 0.05
δ = 0.09
in the narrow part of the channel, x ∈ [π, 2π], especially near the most narrow position
x = 3π/2, the velocity depends more conspicuously on these parameters than in the wide
part of the channel, x ∈ [0, π]. As shown in Figure 7.3(a), in the narrow part, the behavior
of an Oldroyd-B fluid is obviously different from that of a Newtonian fluid (λ1 = λ2 = 0),
while in the wide part, their difference may be negligible if the Weissenberg number λ1 is
not extremely large. Similarly, the strong dependence of the velocity on the wavenumber
(d)
u −5.0
−3.0
λ1 = 0.9
0
1
2
δ = 0
δ = 0.1
δ = 0.2
(b)
3
4
x
5
6
3
4
x
5
6
of the wall disturbance occurs also mainly near the narrow part of the channel, as is seen
in Figure 7.3(b).
8. Summary and conclusions
Peristaltic motion has been studied for two-dimensional geometry in the limit of long
wavelength and low frequency. Asymptotic expansions in terms of a dimensionless
wavenumber δ have been constructed, and solutions to (δ2) have been obtained in
closed form.
The limiting solution (i.e., the zeroth-order solution) has been found to be identical to
that for an infinite wavelength or Newtonian behavior. At this order, it is found that the
Weissenberg numbers only contribute to S0xx. Higher-order solutions have been studied
to reveal the effects of non-Newtonian behavior on peristaltic waves with long but
finite wavelengths. The results indicate that, for the first-order solution, the Weissenberg
numbers do not include any contribution to the stream function, the axial velocity, and
pressure rise per wavelength; however, these do give rise to contributions to the
longitudinal pressure gradient. The solution for terms of (δ2) depends strongly upon the
Weissenberg numbers.
It is hoped that the present analysis may be used with confidence to describe
physiological flows in humans with proper geometric modifications.
The following results are found.
(i) The perturbation analysis is valid for large values of wavelength.
(ii) The solution determined here holds for all values of the Reynolds number and the
Weissenberg numbers.
(iii) The terms of orders (δ2), (δ2 e2), (δ2 e), [λ1(λ1 − λ2) e δ2], and [(λ1 −
λ2) eδ2] represent, respectively, the curvature, inertia, and the non-Newtonian character
of the fluid.
(iv) With increasing occlusion Φ, the pressure drop per wavelength along the flow
direction increases, that is, an increase of the pressure gradient is needed to push the
same flux to pass the channel.
(v) With a decrease of the Weissenberg number λ1 or an increase of the Weissenberg
number λ2, the pressure drop per wavelength required to maintain the same flux
increases.
(vi) In the narrow part of the channel, the behavior of an Oldroyd-B fluid is much
more different from that of a Newtonian fluid than in the wide part of the channel.
(vii) The results for a Maxwell fluid can be obtained as a special case of the presented
analysis by taking λ2 = 0. To the best of our knowledge, the peristaltic motion of Maxwell
fluid has not been discussed so far.
Acknowledgment References
T. Hayat wishes to acknowledge the support of the Alexander von Humboldt Foundation
(Bonn, Germany) for this research.
[16]
[17]
[18]
[19]
[45]
T. Hayat: Department of Mechanics, Darmstadt University of Technology, Hochschulstraße 1,
64289 Darmstadt, Germany
Current address: Department of Mathematics, Quaid-i-Azam University, Islamabad 45320,
Pakistan
E-mail address: t
Y. Wang: Department of Mechanics, Darmstadt University of Technology, Hochschulstraße 1,
64289 Darmstadt, Germany
E-mail address:
K. Hutter: Department of Mechanics, Darmstadt University of Technology, Hochschulstraße 1,
64289 Darmstadt, Germany
E-mail address:
A. M. Siddiqui: Department of Mathematics, Pennsylvania State University, York Campus, York,
PA 17403, USA
E-mail address:
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