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Peristaltic flow of a realistic fluid in a compliant channel
Journal Pre-proof Peristaltic flow of a realistic fluid in a compliant channel Maryiam Javed, R. Naz
PII: DOI: Reference:
S0378-4371(19)32162-4 https://doi.org/10.1016/j.physa.2019.123895 PHYSA 123895
To appear in:
Physica A
Received date : 23 May 2019 Revised date : 6 November 2019 Please cite this article as: M. Javed and R. Naz, Peristaltic flow of a realistic fluid in a compliant channel, Physica A (2020), doi: https://doi.org/10.1016/j.physa.2019.123895. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
Journal Pre-proof *Highlights (for review)
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Highlights: Peristaltic transport of a Jeffrey fluid model has been discussed in an asymmetric channel with compliant walls. Small amplitude assumption is taken into account to solve the required boundary value problem. The possibility of flow reversal increases in the narrow part of the channel and increases in the wider part of the channel.
Journal Pre-proof *Manuscript Click here to view linked References
Peristaltic flow of a realistic fluid in a compliant channel
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Maryiam Javed and R. Naz Department of Applied Mathematics and Statistics,
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Institute of Space Technology 2750 Islamabad-44000, Pakistan
Abstract: Goal of the present analysis is to explore the elastic wall effect on the peri-
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staltic flow in an asymmetric channel. Formulation of the flow problem is based upon the constitutive equation of a Jeffrey fluid. Resulting problem is solved for a stream function and the peristaltic mean flow. The salient features of involved key parameters are also discussed. Keywords: Asymmetric channel, Jeffrey fluid, compliant walls. Nomenclature Main author Tel.: + 92-51-9075585
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E-mail address: [email protected]
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Symbol
Description
SI Units
Symbol
Description
e V
velocity;
m/sec
t
time;
m/sec
x
velocity components in x,
of
u, v
y-directions, respectively;
a1
N/m2
extra stress tensor;
amplitude of the wave along
a2
m
amplitude of the wave along
y
the channel walls;
spatial coordinate nor
the channel walls; ρ
fluid density;
m
δ
dimensionless wave num
m
c
wave speed;
m
k
spring stiffness coefficie
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the upper channel wall;
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S
spatial coordinate alo
the lower channel wall; η1
vertical displacement corresponding to the upper wall;
vertical displacement corresponding
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η2
to the lower wall; phase difference;
radians
p
pressure;
d
coefficient of viscous damping;
Kg/m2 sec
d1
upper half channel wid
¯ B
flexural rigidity of the plate;
Kg m2 /sec2
d2
lower half channel widt
T
elastic tension in the membrane;
Kg/sec2
m
plate mass per unit are
relaxation time
sec−1
λ2
retardation time
wavelength
m
Ψ
stream function.
λ
1
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λ1
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θ
Introduction
It is known that physical activity of peristaltic transport is nature’s way of moving the contents by repeated contractions of muscular fibers. It is an essential property of several
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biomedical and engineering systems. In physiology, it appears in transport of urine towards bladder from kidney, chyme motion in the gastrointestinal tract, motion of small blood ves-
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sels, ovum movement in the female fallopian tube, transport of male reproductive tract and in many glandular ducts. Engineers have used the peristaltic activity in pumping technol-
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ogy that has numerous industrial applications. Particularly, roller and finger pumps operate under this principle. Besides this the peristalsis has been used for sanitary fluid transport of corrosive fluids when the contact of the fluid is prohibited with the parts of machinery.
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After the pioneering work of Latham [1], the peristaltic mechanism has been widely studied via theoretical and experimental approaches. Mitra and Prasad [3] extended Fung and Yih [2] model and considered the two-dimensional analysis of peristaltic motion with flexible (elastic or viscoelastic) wall. Much attention has been given to the development of mathematical models for peristaltic transport in a two-dimensional infinitely long axisymmetric
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tubes or symmetric channel containing hydrodynamic/magnetohydrodynamic viscous and non-Newtonian fluids. Few recent attempts in this direction are made by the investigators
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in the studies [4 − 22].
It has been recently observed by the physiologists that in view of myometrial contractions, the intrauterine fluid flow represents a peristaltic mechanism. By De Varies et al. [23], these
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contractions occur in both asymmetric and symmetric directions. During the poliferative and the menstruation phase, the myometrial activity is symmetric but asymmetric contractions are seen when embryo goes into the uterus for implementation. With these facts in mind, some authors [24 − 30] discussed the peristaltic phenomena of viscous and non-Newtonian fluids in an asymmetric channel. Consideration of wall properties in peristalsis is of special value in study of blood flow in arteries and veins, urine flow in the urethras and air flow in the lungs. However no investigation is presented yet for the peristaltic transport of a
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non-Newtonian fluid in an asymmetric compliant wall channel. The main purpose of present study is to examine such analysis for a Jeffrey fluid. Hence, we introduce the model by
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analyzing the governing equations and related boundary conditions in Section 2. Section 3 derives the perturbation solution for small amplitude ratio. Graphical discussion is stated in
2
Problem development
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Section 4. Section 5 includes the final remarks.
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An incompressible Jeffrey fluid is considered in a channel having width d¯1 + d¯2 . The asymmetry is induced by considering peristaltic waves with different phases and amplitudes on the channel walls. Further, the channel walls are of elastic nature. We consider a Cartesian coordinate system in such a manner that x-axis is along the centre line and y-axis is taken
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normal to it.
Fig. 1: Coordinate Geometry of the physical model of the problem.
The equations governing the flow are e div V=0,
ρ
e dV = −∇e p + div S, dt
(1) (2)
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e denotes the velocity field, ρ denotes the density, d/dt defines the material derivawhere V
tive and pe is the pressure. For Jeffrey fluid, the extra stress tensor S is ∂ 1 + λ1 ∂t
∂ ¯ S=µ 1 + λ2 A1 , ∂t
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(3)
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in which t is the time, µ defines the dynamic viscosity, λ1 is the relaxation time, λ2 is the ¯ 1. retardation time and the first Rivlin-Erickson tensor is A Invoking Eqs. (1) and (3) into Eq. (2) we obtain
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∂ ∂ u¯ 1 ∂ ∂e p ∂ u¯ ∂ u¯ 1 + λ1 u¯ =− 1 + λ1 + v¯ + ∂t ∂x ∂y ∂t ρ ∂t ∂x ∂ +ν 1 + λ2 ∇2 u, ∂t ∂¯ v 1 ∂ ∂e p ∂¯ v ∂¯ v ∂ u¯ =− 1 + λ1 + v¯ + 1 + λ1 ∂t ∂x ∂y ∂t ρ ∂t ∂y ∂ +ν 1 + λ2 ∇2 v, ∂t
(5)
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(4)
where ν (= µ/ρ) is the viscosity.
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Defining the stream function as u¯ =
∂Ψ , ∂y
v¯ = −
∂Ψ ∂x
(6)
equation
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and using it in Eqs. (4) and (5) we obtained, after removing the pressure the following
∂ 1 + λ1 ∂t
∂ 2 ∂ 2 2 ∇2 ∇2 Ψ, ∇ Ψ + Ψ y ∇ Ψ x − Ψ x ∇ Ψ y = ν 1 + λ2 ∂t ∂t
(7)
in which ∇2 represents the Laplacian and subscripts shows partial differentiation. The condition of incompressibility (1) is satisfied automatically. The compliant wall is modeled such that it is restricted to move in the vertical direction. Let η1 and η 2 represents the vertical displacements related to the upper and lower walls.
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Furthermore, η 1 and η 2 are taken in the form as 2π η 1 = a1 cos (x − ct) , λ
2π η 2 = a2 cos θ + (x − ct) . λ
(8)
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In above equation a1 and a2 show the amplitude of the waves, c is the speed of wave, λ is
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the wavelength, θ shows the phase difference that varies in range 0 ≤ θ ≤ π. Further, θ = 0 corresponds to symmetric channel case with waves out of phase and θ = π in phase. Also a1 , 2 a2 , θ, d¯1 , and d¯2 satisfies 2a1 a2 cos θ + a21 + a22 ≤ d¯1 + d¯2 .
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The compliant wall equation can be written as [16]: η1 4 2 2 ∂ ∂ ∂ ∂ ¯ B − T + m + d + K = pe − pe0 , ∂x4 ∂x2 ∂t2 ∂t η2
(9)
¯ is the flexural rigidity of plate, m is the mass of plate per unit area, K shows the where B spring stiffness, d is the coefficient of damping, T shows the tension and pe0 is the outside
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surface pressure of the wall. Taking the walls of the channel inextensible and pe0 = 0. The
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displacement along horizontal axis is taken zero. The relevant conditions are d¯1 + η 1 . Ψy = 0 at y = −d¯2 − η2
(10)
Continuity of the stresses requires that at the interfaces of the walls and fluid pe should be
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the same as it is acting on the fluid at the boundaries. Using Eq. (4) we get 4 2 2 η 1 ∂ ¯ ∂ ∂ ∂ ∂ ∂ B 4 −T 2 +m 2 +d +K 1 + λ1 ∂t ∂x ∂x ∂x ∂t ∂t η2 ∂ ∂ 2 = ρ ν 1 + λ2 ∇ Ψ y − 1 + λ1 (Ψyt + Ψy Ψyx − Ψx Ψyy ) . ∂t ∂t Introducing
(11)
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x ct u¯ v¯ pe y , xˆ = , tˆ = , b u¯ = , b v¯ = , b pe = 2 , d1 d1 d1 c c ρc ¯ B η ˆ Ψ m ˆ dd1 b¯ ηˆ = , Ψ = ,m ˆ = ,d= ,B= , d1 cd1 ρd1 ρν ρd1 ν 2 T d1 ˆ Kd31 ˆ c ˆ 2 = c λ2 , Tˆ = , K = , λ1 = λ1 , λ 2 2 ρν ρν d1 d1
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yˆ =
and removing the hats we obtain ∂ 1 + λ1 ∂t
∂ ∂ 2 1 2 2 1 + λ2 ∇2 ∇2 Ψ, ∇ Ψ + Ψy ∇ Ψx − Ψx ∇ Ψy = ∂t R ∂t
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(12)
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η1 = ǫ cos α (−t + x) , η 2 = aǫcos [α (−t + x) + θ] , −η 2 − h Ψy = 0 at y = , η1 + 1
K η1
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¯ 4 ∂ B ∂2 d ∂ T ∂2 ∂ ∂ + m + − + 1 + λ1 ∂t ∂x R2 ∂x4 ∂t2 R ∂t R2 ∂x2 R2 η2 Ψ +Ψ Ψ 1 + η1 yt y yx ∂ ∂ 1 2 at y = 1 + λ2 ∇ Ψ y − 1 + λ1 = R ∂t ∂t −h − η2 −Ψx Ψyy
(13) (14)
(15)
,
where h = d2 /d1 , ǫ = a1 /d1 , a = a2 /a1 , α = 2πd1 /λ represents the wave number and
3
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R = cd1 /ν represents the Reynolds number.
Problem solution
We obtain the solution for the stream function as a power series in terms of the small parameter ǫ, by expanding Ψ and
∂ pe ∂x
in the form (see Fung and Yih [2])
Ψ = Ψ0 + ǫΨ1 + ǫ2 Ψ2 + ..., 2 1 0 ∂e p ∂e p ∂e p ∂e p 2 +ǫ +ǫ + ..., = ∂x ∂x ∂x ∂x
(16) (17)
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where the first term on the R. H. S in Eq. (17) represents the imposed pressure gradient and the remaining terms represents the peristaltic motion.
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The ǫ0 set of differential equations subject to steady parallel flow and assumption of transverse symmetry corresponds to
(18)
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(h − 1) 2 y 3 Ψ0 (y) = K0 hy − + C, y − 2 3 0 R de p K0 = − , 2 dx where C is a constant.
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In order to solve second and third sets of differential equations in Ψ1 and Ψ2 we express
eiα(x−t) e−iα(x−t) ∗ Ψ1 (x, y, t) = Φ1 (y) , + Φ1 (y) 2 2 e−2iα(x−t) e2iα(x−t) Φ20 (y) ∗ . + Φ22 (y) + Φ22 (y) Ψ2 (x, y, t) = 2 2 2
(19) (20)
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In above equations the asterisk represents the complex conjugate. Invoking above mentioned expressions into the differential equations and the relevant boundary conditions, we have three
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sets of coupled differential equations in linear form with their related boundary conditions. (∂p/∂x)0 = 0 in case of free pumping i.e. K0 = 0 and hence we have
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d2 − α2 + iαR dy 2 −h ′ Φ1 = 0, 1
Φ′′′ 1
=
−h 1
1 − λ1 iα 1 − λ2 iα
−α
2
Φ′1
1 − λ1 iα 1 − λ2 iα
d2 2 − α Φ1 (y) = 0, dy 2
(21)
(22) −h
1 aeiθ Rδ, 1
+ iαR
1 − λ1 iα 1 − λ2 iα
Φ′1
−h 1 (23)
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iα 2 2 4 2 α R m + iαRd − α B − α T − K , R2 iαR ′ ∗ ′′ Φ′′′′ [Φ1 (y) Φ∗′′ 20 (y) = − 1 (y) − Φ1 (y) Φ1 (y)] , 2 −h 1 aeiθ −h 1 ae−iθ −h ′ ∗′′ ′′ ∓ Φ20 Φ Φ ∓ = 0, 2 1 1 2 1 1 1 1 1 −h −h −h −h −h iαR ∗′′ ∗ ′′ ′′′ Φ Φ − Φ Φ Φ20 =− 1 1 1 1 2 1 1 1 1 1 iθ −h −h ae 1 ∗′′′′ ∗′′ ± − α (α + iR) Φ1 Φ1 2 1 1 1 −iθ −h −h 1 ae ′′ Φ′′′′ , ± (±1) − α (α − iR) Φ 1 1 2 1 1 1
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δ=−
1
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2 d 1 − 2λ1 iα d2 2 2 − 4α + 2iαR − 4α Φ22 (y) dy 2 1 − 2λ2 iα dy 2 iαR 1 − 2λ1 iα [Φ′1 (y) Φ′′1 (y) − Φ1 (y) Φ′′′ = 1 (y)] , 2 1 − 2λ2 iα
Φ′22 (±1) = ∓
(24)
(25)
(26)
(27)
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Φ′′1 , (28) 2 aeiθ −h 1 1 1 1 − 2λ iα 1 − 2λ iα 1 1 ′ ′2 ′′′ + iαR Φ22 Φ1 = 4α 2α − iR 2Φ22 1 − 2λ2 iα 1 − 2λ2 iα −h −h −h 1 1 1 1 1 − 2λ1 iα Φ′′′′ −iαR Φ1 Φ′′1 ∓ 1 1 − 2λ2 iα −h −h aeiθ −h 1 1 − 2λ1 iα ′′ Φ1 . (29) −α (α − iR) 1 − 2λ2 iα −h
Here prime shows the derivative w.r.t. y.
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The solution of Eqs. (21)-(23) is (30)
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e1 sinh yα + L e2 cosh yα + L e3 sinh yβ + L e4 cosh yβ, Φ1 (y) = L "
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# e2 α sinh α + L e3 β cosh β + L e4 β sinh β L e1 = − L , α cosh α ! e3 + N5 N9 L3 e σRδ N e2 = − L , e1 N e5 N e1 N8 − N e3 N6 − aeiθ N e1 N5 σRδ N e3 = , L e e e N5 N6 N9 − N1 N10 e2 N e1 + L e3 N2 L e4 = L , e3 N 1 − λ1 iα 2 2 β = α − iαRσ, σ = 1 − λ2 iα
N7
N8
sinh α cosh αh + sinh αh cosh α = −iα Rσ , cosh α " # β 2 − α2 + iαRσ cosh α cosh βh − iαRσ cosh αh cosh β = β , cosh α # " β 2 − α2 + iαRσ cosh α sinh βh + iαRσ cosh αh sinh β , = −β cosh α 2
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N6
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sinh α cosh hα + cosh α sinh hα e , N1 = α cosh α cosh β cosh hα − cosh α cosh hβ N2 = β , cosh α e3 = β cosh α sinh hβ + sinh β cosh hα , N cosh α e5 = β β 2 − α2 sinh β, N4 = β β 2 − α2 cosh β, N
The solution of Eqs. (24)-(26) can be written as Φ′20 (y) = Fe (y) + e g (y) + c1 y 2 + c2 y + c3 .
(31)
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The peristaltic mean flow is given by ǫ2 ′ Φ (y) 2 20 ǫ2 e = F (y) + e g (y) + c1 y 2 + c2 y + c3 . 2
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u¯ (y) =
Fe (y) = −
2 2 e3 L e∗ + L e∗ L e4 L α − β 1 2 iαR
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In expressions (31) and (32) we have
(32)
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cosh (α + β) y 4 (α + β)2 ∗ 2 ∗2 2 ∗e 2 ∗ ∗ e e e e e e e α −β L1 L3 − L2 L4 β −α L2 L4 + L1 L3 − cosh y (α − β) + cosh y (α + β ∗ ) 2 2 ∗ (−β + α) (β + α) ∗2 2 e1 L e∗ − L e2 L e∗ e3 L e∗ + L e4 L e∗ β ∗2 − α2 L L β − β 3 4 3 4 ∗ − cosh y (α − β ) + cosh y (β + β ∗ ) 2 2 ∗ ∗ (α − β ) (β + β) e3 L e∗ − L e4 L e∗ β ∗2 − β 2 L 3 4 ∗ cosh y (β − β ) , − (β − β ∗ )2
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g (y) = − e
∗ 2 2 e1 L e4 + L e3 L e∗2 L α − β iαR
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sinh (α + β) y 4 (α + β)2 ∗ 2 ∗2 ∗ 2 2 ∗ ∗ e e e e e e e e L1 L4 − L3 L2 α −β β −α L2 L3 + L1 L4 − sinh (α − β) y + sinh (α + β ∗ ) y 2 2 ∗ (β − α) (β + α) ∗2 2 e1 L e∗ − L e2 L e∗ e3 L e∗ + L e∗ L e4 β ∗2 − α2 L L β − β 4 3 4 3 + sinh (α − β ∗ ) y + sinh (β + β ∗ ) y ∗ 2 ∗ 2 (α − β ) (β + β ) e3 L e∗4 − L e∗3 L e4 β ∗2 − β 2 L ∗ sinh (β − β ) y , + (β − β ∗ )2 c1 = c2 = c3 =
1 (D3 − s) , 2 2 −Fe (1) + D1 − 2 −Fe (−h) + D2 + 2e g (−h) − 2e g (1) − (−h2 + 1) (−s + D3 )
2 (h + 1) e e 2 −F (1) + D1 h + 2 D2 − F (−h) − 2e g (−h) − 2he g (1) − h (h + 1) (−s + D3 ) 2 (h + 1)
, ,
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α2 (L1 + L∗1 ) sinh α + α2 (L2 + L∗2 ) cosh α + β 2 L3 sinh β 2 +β ∗2 L∗3 sinh β ∗ + β 2 L4 cosh β + β ∗2 L∗4 cosh β ∗ ,
D1 = −
1 L1 e−iθ + L∗1 eiθ α2 sinh αh − α2 L2 e−iθ + L∗2 eiθ cosh hα + β 2 L3 e−iθ sinh hβ 2 +L∗3 eiθ β ∗2 sinh β ∗ h − L4 e−iθ β 2 cosh βh − L∗4 eiθ β ∗2 cosh β ∗ h ,
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D2 = −
iαR 2 α − β 2 (L∗1 L3 sinh α sinh β + L∗1 L4 sinh α cosh β) + α2 − β 2 (L∗2 L3 cosh α sinh β 4 L∗2 L4 cosh α cosh β) + β ∗2 − α2 (L1 L∗3 sinh α sinh β ∗ + L1 L∗4 sinh α cosh β ∗ )
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D3 = −
+ β ∗2 − α2 (L2 L∗3 cosh α sinh β ∗ + L2 L∗4 cosh α cosh β ∗ ) + β ∗2 − β 2 (L3 L∗3 sinh β sinh β ∗
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+L3 L∗4 sinh β cosh β ∗ ) + β ∗2 − β 2 (L∗3 L4 cosh β sinh β ∗ + L4 L∗4 cosh β cosh β ∗ )
1 4 α (L1 + L∗1 ) − α3 (α − iR) L1 − α3 (α + iR) L∗1 sinh α + α4 (L2 + L∗2 ) − α3 (α − iR) L2 2 −α3 (α + iR) L∗2 cosh α + β 4 − αβ 2 (α − iR) L3 sinh β + β ∗4 − αβ ∗2 (α + iR) L∗3 sinh β ∗
−
+ β 4 − αβ 2 (α − iR) L4 cosh β + β ∗4 − αβ ∗2 (α + iR) L∗4 cosh β ∗ ,
iαR 2 α − β 2 (L∗1 L3 + L∗2 L4 ) cosh (α + β) − α2 − β 2 (L∗1 L3 − L∗2 L4 ) cosh (α − β) 4 + β ∗2 − α2 (L1 L∗3 + L2 L∗4 ) cosh (α + β ∗ ) − β ∗2 − α2 (L1 L∗3 − L2 L∗4 ) cosh (α − β ∗ )
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s = −
+ β ∗2 − β 2 (L3 L∗3 + L4 L∗4 ) cosh (β + β ∗ ) − β ∗2 − β 2 (L3 L∗3 − L4 L∗4 ) cosh (β − β ∗ ) + α2 − β 2 (L∗1 L4 + L∗2 L3 ) sinh (α + β) − α2 − β 2 (L∗1 L4 − L∗2 L3 ) sinh (α − β)
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+ β ∗2 − α2 (L1 L∗4 + L2 L∗3 ) sinh (α + β ∗ ) + β ∗2 − α2 (L1 L∗4 − L2 L∗3 ) sinh (α − β ∗ )
+ β ∗2 − β 2 (L3 L∗4 + L∗3 L4 ) sinh (β + β ∗ ) + β ∗2 − β 2 (L3 L∗4 − L∗3 L4 ) sinh (β − β ∗ )
4
Graphical analysis
This section discusses the quantitative analysis of the sundry parameters. For this purpose the mean velocity and mean axial velocity distribution are calculated for K0 = 0. Computer codes were developed for the numerical evaluations of the analytical results and some important
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results are displayed graphically in Figs. 2 − 3. The constant D1 , which initially arose from the non-slip condition of the axial-velocity on the wall, is due to the value of Φ′20
u¯ (1) =
ǫ2 ′ Φ 2 20
(1) =
ǫ2 D1 . 2
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at the boundary and is related to the mean velocity at the boundaries of the channel by In Fig. 2a the effects of a on the mean velocity at the boundaries
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D1 with wave number α are observed. It is noted that D1 increases by increasing a. Fig. 2b indicates the influence of D1 with d for different values of the channel width h. One can see that there is an increase in D1 with an increase in h. We have found that D1 decreases with
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an increase in d, goes zero for some value of d and then remains negative until d reaches 1 which indicates that the damping causes the mean flow reversal. Fig. 2c depicts the effect of D1 with the elastance of the wall for different values of phase difference θ. It is found that D1 increases for 0 ≤ θ ≤ π/2 and decreases for π/2 < θ ≤ π. Fig. 2d shows the variation of D1 with R for different values of the wall tension T . Fig. 2d depicts an increase in the
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mean velocity at the boundaries with increasing wall tension. Fig. 2e shows the effect of the relaxation time λ1 on D1 for different values of the α. One observes that D1 decreases with
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an increasing λ1 . The effect of λ2 on D1 is opposite to the effect of λ1 on D1 as is clear from Fig. 2f. Fig. 3a explains the effects of a on u¯. The possibility of flow reversal increases in the narrow part of the channel whereas an increase is observed in the wider part of the channel.
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Fig. 3b provides the effect of the channel width h on the mean velocity. From this figure it is observed that maximum reverse in the flow occurs in the channel’s lower part when compared with the channel’s upper part. Fig. 3c elucidates the effect of the phase difference θ on u¯. Obviously, the flow reversal decreases in the channel’s narrow part while it increases in the channel’s wider part. A similar behavior is observed in Fig. 3d which displays the effect of T on u¯, while the reverse situation is seen in Fig. 3g displaying the effect of the wall damping on u¯(y). Figs. 3e and 3f indicate the opposite behavior of the relaxation and the
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retardation time on u¯(y) respectively. From Fig. 3e it is seen that flow reversal decreases near the channel’s lower wall. However it increases in the remaining parts of channel. Note
Final remarks
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5
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that the situation is reversed in Fig. 3f.
In the present analysis, the effects of elastic walls are examined on the peristaltic flow of a realistic fluid in an asymmetric channel. Governing equations are first modeled and then
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solved analytically for the case of free pumping. The effects of various parameters on D1 and mean velocity distribution u¯(y) are analyzed. The main findings are listed below. • The variations of a, h, T and λ1 on D1 are quite opposite to λ2 . • The flow reversal occurs close to the boundaries.
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• The flow reversal close to the lower wall is much greater than that close to the lower
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wall.
• Results for viscous fluid can be deduced when λ1 = λ2 = 0. • The solution formally coincides with that of Fung and Yih [2] by choosing θ = λ1 =
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λ2 = 0 and a = h = 1; however, the results differ in many respects • The results of hydrodynamic viscous fluid [16] can be computed as the limiting cases of present study by choosing θ = λ1 = λ2 = 0 and a = h = 1. • The last solution (32) (when wall tension T and wall elastance K → 0 (thin plate); wall rigidity B and wall elastance K → 0 (membrane)) and by choosing θ = λ1 = λ2 = 0 and a = h = 1 agree with the work of Mitra and Prasad [3].
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Acknowledgment: The authors are grateful to the financial support of Higher Edu-
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cation Commission (HEC) of Pakistan (5854/Federal/NRPU/R&D/HEC/2016).
References
of Technology, Cambridge, AMA (1966).
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[1] T. W. Latham, Fluid motion in a peristaltic pump, MS thesis, Massachusetts Institute
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[2] Y. C. Fung and C. S. Yih, Peristaltic transport. J. Appl. Mech. 35 (1968) 669 − 75. [3] T. K. Mitra and S. N. Prasad, On the influence of wall properties and Poiseuille flow in peristalsis, J. Biomech. 6 (1973) 681 − 93.
[4] S. E. Ghasemi, M. Hatami, J. Hatami, S. A. R. Sahebi and D. D. Ganji, An efficient approach to study the pulsatile blood flow in femoral and coronary arteries by Differential
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Quadrature Method, Physica A: Statistical Mechanics and its Applications 443 (2016)
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406 − 414.
[5] S. E. Ghasemi, M. Vatani, M. Hatami and D. D. Ganji, Analytical and numerical investigation of nanoparticle effect on peristaltic fluid flow in drug delivery systems, Journal
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of Molecular Liquids 215 (2016) 88 − 97. [6] A. S. Dogonchi and D. D. Ganji, Impact of Cattaneo–Christov heat flux on MHD nanofluid flow and heat transfer between parallel plates considering thermal radiation effect, Journal of the Taiwan Institute of Chemical Engineers 80 (2017) 52 − 63. [7] M. Gholinia, Kh. Hosseinzadeh, H. Mehrzadi, D. D. Ganji and A. A. Ranjbar, Investigation of MHD Eyring–Powell fluid flow over a rotating disk under effect of homogeneous– heterogeneous reactions, Case Studies in Thermal Engineering 13 (2019) 100356.
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Research 118 (2018) 162 − 172.
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[12] S. Noreen, Quratulain and D. Tripathi, Heat transfer analysis on electroosmotic flow via peristaltic pumping in non-Darcy porous medium, Thermal Science and Engineering
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Chemical Engineers 58 (2016) 28 − 38. [14] M. Rafiq, H. Yasmin, T. Hayat and F. Alsaadi, Effect of Hall and ion-slip on the peristaltic transport of nanofluid: A biomedical application, Chinese Journal of Physics In press (2019) .
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Condensed Matter 529 (2018) 33 − 40. [16] M. A. Abd Elnaby and M. H. Haroun, A new model for study the effect of wall properties
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olar fluid flow and heat transfer in a channel with permeable walls, Journal of Molecular Liquids 204 (2015) 198 − 204.
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[19] J. Rahimi, D. D. Ganji, M. Khaki and Kh. Hosseinzadeh, Solution of the boundary layer flow of an Eyring-Powell non-Newtonian fluid over a linear stretching sheet by collocation method, Alexandria Engineering Journal 56 (2017) 621 − 627.
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[20] S. S. Ghadikolaei, Kh. Hosseinzadeh and D. D. Ganji, Analysis of unsteady MHD EyringPowell squeezing flow in stretching channel with considering thermal radiation and Joule heating effect using AGM, Case Studies in Thermal Engineering 10 (2017) 579 − 594. [21] S. S. Ghadikolaei, Kh. Hosseinzadeh, D. D. Ganji and B. Jafari, Nonlinear thermal radiation effect on magneto Casson nanofluid flow with Joule heating effect over an
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inner third of the myometrium, Am. J. Obstetrics Gynecol 162 (1990) 679 − 682. [24] M. Kothandapani and J. Prakash, Effect of radiation and magnetic field on peristaltic transport of nanofluids through a porous space in a tapered asymmetric channel, Journal of Magnetism and Magnetic Materials 378 (2015) 152 − 163. [25] M. Kothandapani, J. Prakash and V. Pushparaj, Nonlinear peristaltic motion of a
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Johnson–Segalman fluid in a tapered asymmetric channel, Alexandria Engineering Jour-
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nal 55 (2016) 1607 − 1618.
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University - Engineering Sciences 30 (2018) 86 − 95. [27] T. Hayat, S. Farooq, M. Mustafa and B. Ahmad, Peristaltic transport of Bingham plastic fluid considering magnetic field, Soret and Dufour effects, Results in Physics 7 (2017) 2000 − 2011. [28] G. C. Shit and N. K. Ranjit, Role of slip velocity on peristaltic transport of couple stress fluid through an asymmetric non-uniform channel: Application to digestive system, Journal of Molecular Liquids 221 (2016) 305 − 315.
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Fig. 2: The variation of D1 for different values of amplitude ratio a (panel (a)), channel width h (panel (b)), phase difference θ (panel (c)), wall tension T (panel (d)), relaxation time
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λ1 (panel (e)) and retardation time λ2 (panel (f )).
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Fig. 3: The variation of mean-velocity distribution and reversal flow for different values of amplitude ratio a (panel (a)), channel width h (panel (b)), difference θ (panel (c)), wall
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tension T (panel (d)), relaxation time λ1 (panel (e)), retardation time λ2 (panel (f )) and wall damping d (panel (g)).
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Figure 2c
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Figure 2e
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Figure 3e
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Figure 3g
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Declaration of interests ☐ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
PII: DOI: Reference:
S0378-4371(19)32162-4 https://doi.org/10.1016/j.physa.2019.123895 PHYSA 123895
To appear in:
Physica A
Received date : 23 May 2019 Revised date : 6 November 2019 Please cite this article as: M. Javed and R. Naz, Peristaltic flow of a realistic fluid in a compliant channel, Physica A (2020), doi: https://doi.org/10.1016/j.physa.2019.123895. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
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Highlights: Peristaltic transport of a Jeffrey fluid model has been discussed in an asymmetric channel with compliant walls. Small amplitude assumption is taken into account to solve the required boundary value problem. The possibility of flow reversal increases in the narrow part of the channel and increases in the wider part of the channel.
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Peristaltic flow of a realistic fluid in a compliant channel
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Maryiam Javed and R. Naz Department of Applied Mathematics and Statistics,
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Institute of Space Technology 2750 Islamabad-44000, Pakistan
Abstract: Goal of the present analysis is to explore the elastic wall effect on the peri-
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staltic flow in an asymmetric channel. Formulation of the flow problem is based upon the constitutive equation of a Jeffrey fluid. Resulting problem is solved for a stream function and the peristaltic mean flow. The salient features of involved key parameters are also discussed. Keywords: Asymmetric channel, Jeffrey fluid, compliant walls. Nomenclature Main author Tel.: + 92-51-9075585
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E-mail address: [email protected]
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Symbol
Description
SI Units
Symbol
Description
e V
velocity;
m/sec
t
time;
m/sec
x
velocity components in x,
of
u, v
y-directions, respectively;
a1
N/m2
extra stress tensor;
amplitude of the wave along
a2
m
amplitude of the wave along
y
the channel walls;
spatial coordinate nor
the channel walls; ρ
fluid density;
m
δ
dimensionless wave num
m
c
wave speed;
m
k
spring stiffness coefficie
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the upper channel wall;
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S
spatial coordinate alo
the lower channel wall; η1
vertical displacement corresponding to the upper wall;
vertical displacement corresponding
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η2
to the lower wall; phase difference;
radians
p
pressure;
d
coefficient of viscous damping;
Kg/m2 sec
d1
upper half channel wid
¯ B
flexural rigidity of the plate;
Kg m2 /sec2
d2
lower half channel widt
T
elastic tension in the membrane;
Kg/sec2
m
plate mass per unit are
relaxation time
sec−1
λ2
retardation time
wavelength
m
Ψ
stream function.
λ
1
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λ1
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θ
Introduction
It is known that physical activity of peristaltic transport is nature’s way of moving the contents by repeated contractions of muscular fibers. It is an essential property of several
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biomedical and engineering systems. In physiology, it appears in transport of urine towards bladder from kidney, chyme motion in the gastrointestinal tract, motion of small blood ves-
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sels, ovum movement in the female fallopian tube, transport of male reproductive tract and in many glandular ducts. Engineers have used the peristaltic activity in pumping technol-
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ogy that has numerous industrial applications. Particularly, roller and finger pumps operate under this principle. Besides this the peristalsis has been used for sanitary fluid transport of corrosive fluids when the contact of the fluid is prohibited with the parts of machinery.
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After the pioneering work of Latham [1], the peristaltic mechanism has been widely studied via theoretical and experimental approaches. Mitra and Prasad [3] extended Fung and Yih [2] model and considered the two-dimensional analysis of peristaltic motion with flexible (elastic or viscoelastic) wall. Much attention has been given to the development of mathematical models for peristaltic transport in a two-dimensional infinitely long axisymmetric
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tubes or symmetric channel containing hydrodynamic/magnetohydrodynamic viscous and non-Newtonian fluids. Few recent attempts in this direction are made by the investigators
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in the studies [4 − 22].
It has been recently observed by the physiologists that in view of myometrial contractions, the intrauterine fluid flow represents a peristaltic mechanism. By De Varies et al. [23], these
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contractions occur in both asymmetric and symmetric directions. During the poliferative and the menstruation phase, the myometrial activity is symmetric but asymmetric contractions are seen when embryo goes into the uterus for implementation. With these facts in mind, some authors [24 − 30] discussed the peristaltic phenomena of viscous and non-Newtonian fluids in an asymmetric channel. Consideration of wall properties in peristalsis is of special value in study of blood flow in arteries and veins, urine flow in the urethras and air flow in the lungs. However no investigation is presented yet for the peristaltic transport of a
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non-Newtonian fluid in an asymmetric compliant wall channel. The main purpose of present study is to examine such analysis for a Jeffrey fluid. Hence, we introduce the model by
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analyzing the governing equations and related boundary conditions in Section 2. Section 3 derives the perturbation solution for small amplitude ratio. Graphical discussion is stated in
2
Problem development
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Section 4. Section 5 includes the final remarks.
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An incompressible Jeffrey fluid is considered in a channel having width d¯1 + d¯2 . The asymmetry is induced by considering peristaltic waves with different phases and amplitudes on the channel walls. Further, the channel walls are of elastic nature. We consider a Cartesian coordinate system in such a manner that x-axis is along the centre line and y-axis is taken
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normal to it.
Fig. 1: Coordinate Geometry of the physical model of the problem.
The equations governing the flow are e div V=0,
ρ
e dV = −∇e p + div S, dt
(1) (2)
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e denotes the velocity field, ρ denotes the density, d/dt defines the material derivawhere V
tive and pe is the pressure. For Jeffrey fluid, the extra stress tensor S is ∂ 1 + λ1 ∂t
∂ ¯ S=µ 1 + λ2 A1 , ∂t
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(3)
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in which t is the time, µ defines the dynamic viscosity, λ1 is the relaxation time, λ2 is the ¯ 1. retardation time and the first Rivlin-Erickson tensor is A Invoking Eqs. (1) and (3) into Eq. (2) we obtain
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∂ ∂ u¯ 1 ∂ ∂e p ∂ u¯ ∂ u¯ 1 + λ1 u¯ =− 1 + λ1 + v¯ + ∂t ∂x ∂y ∂t ρ ∂t ∂x ∂ +ν 1 + λ2 ∇2 u, ∂t ∂¯ v 1 ∂ ∂e p ∂¯ v ∂¯ v ∂ u¯ =− 1 + λ1 + v¯ + 1 + λ1 ∂t ∂x ∂y ∂t ρ ∂t ∂y ∂ +ν 1 + λ2 ∇2 v, ∂t
(5)
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(4)
where ν (= µ/ρ) is the viscosity.
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Defining the stream function as u¯ =
∂Ψ , ∂y
v¯ = −
∂Ψ ∂x
(6)
equation
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and using it in Eqs. (4) and (5) we obtained, after removing the pressure the following
∂ 1 + λ1 ∂t
∂ 2 ∂ 2 2 ∇2 ∇2 Ψ, ∇ Ψ + Ψ y ∇ Ψ x − Ψ x ∇ Ψ y = ν 1 + λ2 ∂t ∂t
(7)
in which ∇2 represents the Laplacian and subscripts shows partial differentiation. The condition of incompressibility (1) is satisfied automatically. The compliant wall is modeled such that it is restricted to move in the vertical direction. Let η1 and η 2 represents the vertical displacements related to the upper and lower walls.
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Furthermore, η 1 and η 2 are taken in the form as 2π η 1 = a1 cos (x − ct) , λ
2π η 2 = a2 cos θ + (x − ct) . λ
(8)
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In above equation a1 and a2 show the amplitude of the waves, c is the speed of wave, λ is
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the wavelength, θ shows the phase difference that varies in range 0 ≤ θ ≤ π. Further, θ = 0 corresponds to symmetric channel case with waves out of phase and θ = π in phase. Also a1 , 2 a2 , θ, d¯1 , and d¯2 satisfies 2a1 a2 cos θ + a21 + a22 ≤ d¯1 + d¯2 .
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The compliant wall equation can be written as [16]: η1 4 2 2 ∂ ∂ ∂ ∂ ¯ B − T + m + d + K = pe − pe0 , ∂x4 ∂x2 ∂t2 ∂t η2
(9)
¯ is the flexural rigidity of plate, m is the mass of plate per unit area, K shows the where B spring stiffness, d is the coefficient of damping, T shows the tension and pe0 is the outside
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surface pressure of the wall. Taking the walls of the channel inextensible and pe0 = 0. The
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displacement along horizontal axis is taken zero. The relevant conditions are d¯1 + η 1 . Ψy = 0 at y = −d¯2 − η2
(10)
Continuity of the stresses requires that at the interfaces of the walls and fluid pe should be
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the same as it is acting on the fluid at the boundaries. Using Eq. (4) we get 4 2 2 η 1 ∂ ¯ ∂ ∂ ∂ ∂ ∂ B 4 −T 2 +m 2 +d +K 1 + λ1 ∂t ∂x ∂x ∂x ∂t ∂t η2 ∂ ∂ 2 = ρ ν 1 + λ2 ∇ Ψ y − 1 + λ1 (Ψyt + Ψy Ψyx − Ψx Ψyy ) . ∂t ∂t Introducing
(11)
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x ct u¯ v¯ pe y , xˆ = , tˆ = , b u¯ = , b v¯ = , b pe = 2 , d1 d1 d1 c c ρc ¯ B η ˆ Ψ m ˆ dd1 b¯ ηˆ = , Ψ = ,m ˆ = ,d= ,B= , d1 cd1 ρd1 ρν ρd1 ν 2 T d1 ˆ Kd31 ˆ c ˆ 2 = c λ2 , Tˆ = , K = , λ1 = λ1 , λ 2 2 ρν ρν d1 d1
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yˆ =
and removing the hats we obtain ∂ 1 + λ1 ∂t
∂ ∂ 2 1 2 2 1 + λ2 ∇2 ∇2 Ψ, ∇ Ψ + Ψy ∇ Ψx − Ψx ∇ Ψy = ∂t R ∂t
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(12)
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η1 = ǫ cos α (−t + x) , η 2 = aǫcos [α (−t + x) + θ] , −η 2 − h Ψy = 0 at y = , η1 + 1
K η1
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¯ 4 ∂ B ∂2 d ∂ T ∂2 ∂ ∂ + m + − + 1 + λ1 ∂t ∂x R2 ∂x4 ∂t2 R ∂t R2 ∂x2 R2 η2 Ψ +Ψ Ψ 1 + η1 yt y yx ∂ ∂ 1 2 at y = 1 + λ2 ∇ Ψ y − 1 + λ1 = R ∂t ∂t −h − η2 −Ψx Ψyy
(13) (14)
(15)
,
where h = d2 /d1 , ǫ = a1 /d1 , a = a2 /a1 , α = 2πd1 /λ represents the wave number and
3
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R = cd1 /ν represents the Reynolds number.
Problem solution
We obtain the solution for the stream function as a power series in terms of the small parameter ǫ, by expanding Ψ and
∂ pe ∂x
in the form (see Fung and Yih [2])
Ψ = Ψ0 + ǫΨ1 + ǫ2 Ψ2 + ..., 2 1 0 ∂e p ∂e p ∂e p ∂e p 2 +ǫ +ǫ + ..., = ∂x ∂x ∂x ∂x
(16) (17)
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where the first term on the R. H. S in Eq. (17) represents the imposed pressure gradient and the remaining terms represents the peristaltic motion.
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The ǫ0 set of differential equations subject to steady parallel flow and assumption of transverse symmetry corresponds to
(18)
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(h − 1) 2 y 3 Ψ0 (y) = K0 hy − + C, y − 2 3 0 R de p K0 = − , 2 dx where C is a constant.
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In order to solve second and third sets of differential equations in Ψ1 and Ψ2 we express
eiα(x−t) e−iα(x−t) ∗ Ψ1 (x, y, t) = Φ1 (y) , + Φ1 (y) 2 2 e−2iα(x−t) e2iα(x−t) Φ20 (y) ∗ . + Φ22 (y) + Φ22 (y) Ψ2 (x, y, t) = 2 2 2
(19) (20)
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In above equations the asterisk represents the complex conjugate. Invoking above mentioned expressions into the differential equations and the relevant boundary conditions, we have three
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sets of coupled differential equations in linear form with their related boundary conditions. (∂p/∂x)0 = 0 in case of free pumping i.e. K0 = 0 and hence we have
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d2 − α2 + iαR dy 2 −h ′ Φ1 = 0, 1
Φ′′′ 1
=
−h 1
1 − λ1 iα 1 − λ2 iα
−α
2
Φ′1
1 − λ1 iα 1 − λ2 iα
d2 2 − α Φ1 (y) = 0, dy 2
(21)
(22) −h
1 aeiθ Rδ, 1
+ iαR
1 − λ1 iα 1 − λ2 iα
Φ′1
−h 1 (23)
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iα 2 2 4 2 α R m + iαRd − α B − α T − K , R2 iαR ′ ∗ ′′ Φ′′′′ [Φ1 (y) Φ∗′′ 20 (y) = − 1 (y) − Φ1 (y) Φ1 (y)] , 2 −h 1 aeiθ −h 1 ae−iθ −h ′ ∗′′ ′′ ∓ Φ20 Φ Φ ∓ = 0, 2 1 1 2 1 1 1 1 1 −h −h −h −h −h iαR ∗′′ ∗ ′′ ′′′ Φ Φ − Φ Φ Φ20 =− 1 1 1 1 2 1 1 1 1 1 iθ −h −h ae 1 ∗′′′′ ∗′′ ± − α (α + iR) Φ1 Φ1 2 1 1 1 −iθ −h −h 1 ae ′′ Φ′′′′ , ± (±1) − α (α − iR) Φ 1 1 2 1 1 1
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δ=−
1
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1 1
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2 d 1 − 2λ1 iα d2 2 2 − 4α + 2iαR − 4α Φ22 (y) dy 2 1 − 2λ2 iα dy 2 iαR 1 − 2λ1 iα [Φ′1 (y) Φ′′1 (y) − Φ1 (y) Φ′′′ = 1 (y)] , 2 1 − 2λ2 iα
Φ′22 (±1) = ∓
(24)
(25)
(26)
(27)
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Φ′′1 , (28) 2 aeiθ −h 1 1 1 1 − 2λ iα 1 − 2λ iα 1 1 ′ ′2 ′′′ + iαR Φ22 Φ1 = 4α 2α − iR 2Φ22 1 − 2λ2 iα 1 − 2λ2 iα −h −h −h 1 1 1 1 1 − 2λ1 iα Φ′′′′ −iαR Φ1 Φ′′1 ∓ 1 1 − 2λ2 iα −h −h aeiθ −h 1 1 − 2λ1 iα ′′ Φ1 . (29) −α (α − iR) 1 − 2λ2 iα −h
Here prime shows the derivative w.r.t. y.
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The solution of Eqs. (21)-(23) is (30)
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e1 sinh yα + L e2 cosh yα + L e3 sinh yβ + L e4 cosh yβ, Φ1 (y) = L "
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# e2 α sinh α + L e3 β cosh β + L e4 β sinh β L e1 = − L , α cosh α ! e3 + N5 N9 L3 e σRδ N e2 = − L , e1 N e5 N e1 N8 − N e3 N6 − aeiθ N e1 N5 σRδ N e3 = , L e e e N5 N6 N9 − N1 N10 e2 N e1 + L e3 N2 L e4 = L , e3 N 1 − λ1 iα 2 2 β = α − iαRσ, σ = 1 − λ2 iα
N7
N8
sinh α cosh αh + sinh αh cosh α = −iα Rσ , cosh α " # β 2 − α2 + iαRσ cosh α cosh βh − iαRσ cosh αh cosh β = β , cosh α # " β 2 − α2 + iαRσ cosh α sinh βh + iαRσ cosh αh sinh β , = −β cosh α 2
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N6
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sinh α cosh hα + cosh α sinh hα e , N1 = α cosh α cosh β cosh hα − cosh α cosh hβ N2 = β , cosh α e3 = β cosh α sinh hβ + sinh β cosh hα , N cosh α e5 = β β 2 − α2 sinh β, N4 = β β 2 − α2 cosh β, N
The solution of Eqs. (24)-(26) can be written as Φ′20 (y) = Fe (y) + e g (y) + c1 y 2 + c2 y + c3 .
(31)
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The peristaltic mean flow is given by ǫ2 ′ Φ (y) 2 20 ǫ2 e = F (y) + e g (y) + c1 y 2 + c2 y + c3 . 2
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u¯ (y) =
Fe (y) = −
2 2 e3 L e∗ + L e∗ L e4 L α − β 1 2 iαR
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In expressions (31) and (32) we have
(32)
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cosh (α + β) y 4 (α + β)2 ∗ 2 ∗2 2 ∗e 2 ∗ ∗ e e e e e e e α −β L1 L3 − L2 L4 β −α L2 L4 + L1 L3 − cosh y (α − β) + cosh y (α + β ∗ ) 2 2 ∗ (−β + α) (β + α) ∗2 2 e1 L e∗ − L e2 L e∗ e3 L e∗ + L e4 L e∗ β ∗2 − α2 L L β − β 3 4 3 4 ∗ − cosh y (α − β ) + cosh y (β + β ∗ ) 2 2 ∗ ∗ (α − β ) (β + β) e3 L e∗ − L e4 L e∗ β ∗2 − β 2 L 3 4 ∗ cosh y (β − β ) , − (β − β ∗ )2
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g (y) = − e
∗ 2 2 e1 L e4 + L e3 L e∗2 L α − β iαR
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sinh (α + β) y 4 (α + β)2 ∗ 2 ∗2 ∗ 2 2 ∗ ∗ e e e e e e e e L1 L4 − L3 L2 α −β β −α L2 L3 + L1 L4 − sinh (α − β) y + sinh (α + β ∗ ) y 2 2 ∗ (β − α) (β + α) ∗2 2 e1 L e∗ − L e2 L e∗ e3 L e∗ + L e∗ L e4 β ∗2 − α2 L L β − β 4 3 4 3 + sinh (α − β ∗ ) y + sinh (β + β ∗ ) y ∗ 2 ∗ 2 (α − β ) (β + β ) e3 L e∗4 − L e∗3 L e4 β ∗2 − β 2 L ∗ sinh (β − β ) y , + (β − β ∗ )2 c1 = c2 = c3 =
1 (D3 − s) , 2 2 −Fe (1) + D1 − 2 −Fe (−h) + D2 + 2e g (−h) − 2e g (1) − (−h2 + 1) (−s + D3 )
2 (h + 1) e e 2 −F (1) + D1 h + 2 D2 − F (−h) − 2e g (−h) − 2he g (1) − h (h + 1) (−s + D3 ) 2 (h + 1)
, ,
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α2 (L1 + L∗1 ) sinh α + α2 (L2 + L∗2 ) cosh α + β 2 L3 sinh β 2 +β ∗2 L∗3 sinh β ∗ + β 2 L4 cosh β + β ∗2 L∗4 cosh β ∗ ,
D1 = −
1 L1 e−iθ + L∗1 eiθ α2 sinh αh − α2 L2 e−iθ + L∗2 eiθ cosh hα + β 2 L3 e−iθ sinh hβ 2 +L∗3 eiθ β ∗2 sinh β ∗ h − L4 e−iθ β 2 cosh βh − L∗4 eiθ β ∗2 cosh β ∗ h ,
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D2 = −
iαR 2 α − β 2 (L∗1 L3 sinh α sinh β + L∗1 L4 sinh α cosh β) + α2 − β 2 (L∗2 L3 cosh α sinh β 4 L∗2 L4 cosh α cosh β) + β ∗2 − α2 (L1 L∗3 sinh α sinh β ∗ + L1 L∗4 sinh α cosh β ∗ )
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D3 = −
+ β ∗2 − α2 (L2 L∗3 cosh α sinh β ∗ + L2 L∗4 cosh α cosh β ∗ ) + β ∗2 − β 2 (L3 L∗3 sinh β sinh β ∗
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+L3 L∗4 sinh β cosh β ∗ ) + β ∗2 − β 2 (L∗3 L4 cosh β sinh β ∗ + L4 L∗4 cosh β cosh β ∗ )
1 4 α (L1 + L∗1 ) − α3 (α − iR) L1 − α3 (α + iR) L∗1 sinh α + α4 (L2 + L∗2 ) − α3 (α − iR) L2 2 −α3 (α + iR) L∗2 cosh α + β 4 − αβ 2 (α − iR) L3 sinh β + β ∗4 − αβ ∗2 (α + iR) L∗3 sinh β ∗
−
+ β 4 − αβ 2 (α − iR) L4 cosh β + β ∗4 − αβ ∗2 (α + iR) L∗4 cosh β ∗ ,
iαR 2 α − β 2 (L∗1 L3 + L∗2 L4 ) cosh (α + β) − α2 − β 2 (L∗1 L3 − L∗2 L4 ) cosh (α − β) 4 + β ∗2 − α2 (L1 L∗3 + L2 L∗4 ) cosh (α + β ∗ ) − β ∗2 − α2 (L1 L∗3 − L2 L∗4 ) cosh (α − β ∗ )
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s = −
+ β ∗2 − β 2 (L3 L∗3 + L4 L∗4 ) cosh (β + β ∗ ) − β ∗2 − β 2 (L3 L∗3 − L4 L∗4 ) cosh (β − β ∗ ) + α2 − β 2 (L∗1 L4 + L∗2 L3 ) sinh (α + β) − α2 − β 2 (L∗1 L4 − L∗2 L3 ) sinh (α − β)
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+ β ∗2 − α2 (L1 L∗4 + L2 L∗3 ) sinh (α + β ∗ ) + β ∗2 − α2 (L1 L∗4 − L2 L∗3 ) sinh (α − β ∗ )
+ β ∗2 − β 2 (L3 L∗4 + L∗3 L4 ) sinh (β + β ∗ ) + β ∗2 − β 2 (L3 L∗4 − L∗3 L4 ) sinh (β − β ∗ )
4
Graphical analysis
This section discusses the quantitative analysis of the sundry parameters. For this purpose the mean velocity and mean axial velocity distribution are calculated for K0 = 0. Computer codes were developed for the numerical evaluations of the analytical results and some important
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results are displayed graphically in Figs. 2 − 3. The constant D1 , which initially arose from the non-slip condition of the axial-velocity on the wall, is due to the value of Φ′20
u¯ (1) =
ǫ2 ′ Φ 2 20
(1) =
ǫ2 D1 . 2
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at the boundary and is related to the mean velocity at the boundaries of the channel by In Fig. 2a the effects of a on the mean velocity at the boundaries
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D1 with wave number α are observed. It is noted that D1 increases by increasing a. Fig. 2b indicates the influence of D1 with d for different values of the channel width h. One can see that there is an increase in D1 with an increase in h. We have found that D1 decreases with
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an increase in d, goes zero for some value of d and then remains negative until d reaches 1 which indicates that the damping causes the mean flow reversal. Fig. 2c depicts the effect of D1 with the elastance of the wall for different values of phase difference θ. It is found that D1 increases for 0 ≤ θ ≤ π/2 and decreases for π/2 < θ ≤ π. Fig. 2d shows the variation of D1 with R for different values of the wall tension T . Fig. 2d depicts an increase in the
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mean velocity at the boundaries with increasing wall tension. Fig. 2e shows the effect of the relaxation time λ1 on D1 for different values of the α. One observes that D1 decreases with
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an increasing λ1 . The effect of λ2 on D1 is opposite to the effect of λ1 on D1 as is clear from Fig. 2f. Fig. 3a explains the effects of a on u¯. The possibility of flow reversal increases in the narrow part of the channel whereas an increase is observed in the wider part of the channel.
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Fig. 3b provides the effect of the channel width h on the mean velocity. From this figure it is observed that maximum reverse in the flow occurs in the channel’s lower part when compared with the channel’s upper part. Fig. 3c elucidates the effect of the phase difference θ on u¯. Obviously, the flow reversal decreases in the channel’s narrow part while it increases in the channel’s wider part. A similar behavior is observed in Fig. 3d which displays the effect of T on u¯, while the reverse situation is seen in Fig. 3g displaying the effect of the wall damping on u¯(y). Figs. 3e and 3f indicate the opposite behavior of the relaxation and the
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retardation time on u¯(y) respectively. From Fig. 3e it is seen that flow reversal decreases near the channel’s lower wall. However it increases in the remaining parts of channel. Note
Final remarks
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that the situation is reversed in Fig. 3f.
In the present analysis, the effects of elastic walls are examined on the peristaltic flow of a realistic fluid in an asymmetric channel. Governing equations are first modeled and then
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solved analytically for the case of free pumping. The effects of various parameters on D1 and mean velocity distribution u¯(y) are analyzed. The main findings are listed below. • The variations of a, h, T and λ1 on D1 are quite opposite to λ2 . • The flow reversal occurs close to the boundaries.
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• The flow reversal close to the lower wall is much greater than that close to the lower
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wall.
• Results for viscous fluid can be deduced when λ1 = λ2 = 0. • The solution formally coincides with that of Fung and Yih [2] by choosing θ = λ1 =
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λ2 = 0 and a = h = 1; however, the results differ in many respects • The results of hydrodynamic viscous fluid [16] can be computed as the limiting cases of present study by choosing θ = λ1 = λ2 = 0 and a = h = 1. • The last solution (32) (when wall tension T and wall elastance K → 0 (thin plate); wall rigidity B and wall elastance K → 0 (membrane)) and by choosing θ = λ1 = λ2 = 0 and a = h = 1 agree with the work of Mitra and Prasad [3].
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Acknowledgment: The authors are grateful to the financial support of Higher Edu-
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cation Commission (HEC) of Pakistan (5854/Federal/NRPU/R&D/HEC/2016).
References
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Fig. 2: The variation of D1 for different values of amplitude ratio a (panel (a)), channel width h (panel (b)), phase difference θ (panel (c)), wall tension T (panel (d)), relaxation time
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λ1 (panel (e)) and retardation time λ2 (panel (f )).
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b
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Fig. 3: The variation of mean-velocity distribution and reversal flow for different values of amplitude ratio a (panel (a)), channel width h (panel (b)), difference θ (panel (c)), wall
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tension T (panel (d)), relaxation time λ1 (panel (e)), retardation time λ2 (panel (f )) and wall damping d (panel (g)).
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Figure 1
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Figure 2a
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Figure 2b
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Figure 2c
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Figure 2d
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Figure 2e
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Figure 2f
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Figure 3a
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Figure 3b
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Figure 3c
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Figure 3d
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Figure 3e
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Figure 3f
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Figure 3g
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Declaration of interests ☐ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.