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RESEARCH ARTICLE
Homogeneous-Heterogeneous Reactions inPeristaltic Flow with Convective ConditionsTasawar Hayat1,2, Anum Tanveer1, Humaira Yasmin1*, Ahmed Alsaedi2
1. Department of Mathematics, Quaid-I-Azam University, Islamabad, Pakistan, 2. NAAM Research Group,Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
*qau2011@gmail.com
Abstract
This article addresses the effects of homogeneous-heterogeneous reactions in
peristaltic transport of Carreau fluid in a channel with wall properties. Mathematical
modelling and analysis have been carried out in the presence of Hall current. The
channel walls satisfy the more realistic convective conditions. The governing partial
differential equations along with long wavelength and low Reynolds number
considerations are solved. The results of temperature and heat transfer coefficient
are analyzed for various parameters of interest.
Introduction
In the last few decades, the peristaltic motion of non-Newtonian fluids is a topic
of major contemporary interest both in engineering and biological applications.
To be more specific, such motion occurs in powder technology, fluidization,
chyme movement in the gastrointestinal tract, vasomotion of small blood vessels,
locomotion of worms, gliding motility of bacteria, passage of urine from kidney to
bladder, reproductive tracts, corrosive and sanitary fluids transport, roller, finger
and hose pumps and blood pump through heart lung machine. There is no doubt
that viscoelasticity has key role mostly in all the aforementioned applications.
Viscoelastic materials are non-Newtonian and possess both the viscous and elastic
properties. Most of the biological liquids such as blood at low shear rate, chyme,
food bolus etc. are viscoelastic in nature. Another aspect which has yet not been
properly addressed is the interaction of rheological characteristics of fluids in
peristalsis with convective effects. The significance of convective heat exchange
with peristalsis cannot be under estimated for instance in translocation of water in
tall trees, dynamic of lakes, solar ponds, lubrication and drying technologies,
diffusion of nutrients out of blood, oxygenation, hemodialysis and nuclear
reactors. The heat and mass transfer effects in such processes have prominent role.
OPEN ACCESS
Citation: Hayat T, Tanveer A, Yasmin H,Alsaedi A (2014) Homogeneous-HeterogeneousReactions in Peristaltic Flow with ConvectiveConditions. PLoS ONE 9(12): e113851. doi:10.1371/journal.pone.0113851
Editor: Sanjoy Bhattacharya, Bascom Palmer EyeInstitute, University of Miami School of Medicine,United States of America
Received: July 11, 2014
Accepted: November 2, 2014
Published: December 2, 2014
Copyright: � 2014 Hayat et al. This is an open-access article distributed under the terms of theCreative Commons Attribution License, whichpermits unrestricted use, distribution, and repro-duction in any medium, provided the original authorand source are credited.
Data Availability: The authors confirm that all dataunderlying the findings are fully available withoutrestriction. All relevant data are within the paper.
Funding: The authors have no support or fundingto report.
Competing Interests: The authors have declaredthat no competing interests exist.
PLOS ONE | DOI:10.1371/journal.pone.0113851 December 2, 2014 1 / 24
The magnetohydrodynamic (MHD) character of fluid especially in physiological
and industrial processes seems too much important. Such consideration is useful
for blood pumping and magnetic resonance imaging (MRI), cancer therapy,
hyperthermia etc. The controlled application of low intensity and frequency
pulsating fields modify the cell and tissue behavior. Magnetically susceptible of
chyme is satisfied from the heat generated by magnetic field or the ions contained
in the chyme. Also the magnetotherapy is an application of magnets to human
body which is used for the treatment of diseases. The magnets could heal
inflammations, ulceration, several diseases of bowel (intestine) and uterus. With
all such motivations in mind, the recent researchers are engaged in the
development of model of peristalsis of non-Newtonian liquids through different
aspects including heat/mass transfer, MHD etc. Few representative attempts in
this direction can be mentioned through the recent researchers [1–13] and several
studies therein.
To our knowledge, no study has been undertaken yet to discuss the effects of
homogeneous-heterogeneous reactions in peristaltic flows of non-Newtonian
fluids. Even such study is yet not presented for the viscous fluid. However such
consideration is quite important because many chemically reacting systems
involve both homogeneous and heterogeneous reactions, with examples occurring
in combustion, biochemical systems, catalysis, crops damaging through freezing,
cooling towers, fog dispersion, hydrometallurgical processes etc. Hence the main
objective of present investigation is to model and analyze the peristalsis of Carreau
fluid in a compliant wall channel with convective conditions and homogeneous-
heterogeneous reactions. Effects of Hall current and viscous dissipation are also
considered. The resulting mathematical systems are solved and examined in the
case of long wavelength and small Reynolds number. This article is structured as
follows. Section two consists of mathematical modelling and solution expressions
up to first order. The behaviors of sundry variables on the temperature, heat
transfer coefficient and concentration are discussed graphically in section three.
Main results of present study are also included in this section.
Mathematical Formulation
Consider the peristaltic transport of an incompressible Carreau fluid in two-
dimensional compliant wall channel. The channel walls satisfy the convective
conditions. The Cartesian coordinates x and y are considered along and transverse
to the direction of fluid flow respectively. The flow is generated by the peristaltic
wave of speed c travelling along the channel walls. The Hall effects are also
considered in the flow analysis. Further we consider the flow in the presence of a
simple homogeneous and heterogeneous reaction model. There are two chemical
species �A and �B with concentrations �a and �b respectively. The physical model of
the wall surface can be analyzed by the expression:
y~+g(x,t)~dza sin2pl
(x{ct), ð1Þ
Homogeneous-Heterogeneous Reactions with Peristalsis
PLOS ONE | DOI:10.1371/journal.pone.0113851 December 2, 2014 2 / 24
where a represents the wave amplitude, l the wavelength, d the half width of
symmetric channel, t the time, g displacement of upper wall and {g displacement
of lower wall.
Consider the uniform magnetic field in the form
B0~(0,0,B0): ð2Þ
Application of generalized Ohm’s law leads to the following expression
J~s V|B0{1
en^ (J|B0)
� �, ð3Þ
where J represents the current density, s the electrical conductivity, V the velocity
field, e the electric charge and n^
the number density of electrons. Also the effects
of electric field are considered absent i.e. E~0:If V~½u,u,0� is the velocity with components u and u in the x and y directions
respectively then from Eqs. (2) and (3) we have
J|B0~{sB2
0
1zm2u{muð Þ, uzmuð Þ,0½ �, ð4Þ
where m~sB0
en^ serves as the Hall parameter. The reaction model is considered in
the form [14–16]:
�Az2�B?3�B, rate~kcab2,
while on the catalyst surface we have the single, isothermal, first order chemical
reaction.
�A?�B, rate~ksa,
in which kc and ks are the rate constants. Both reaction processes are assumed
isothermal.
The corresponding flow equations are as follows:
LuLx
zLu
Ly~0, ð5Þ
rdudt
~{LpLx
zLRxx
Lxz
LRxy
Ly{
sB20
1zm2u{muð Þ, ð6Þ
rdu
dt~{
LpLy
zLRyx
Lxz
LRyy
Ly{
sB20
1zm2uzmuð Þ, ð7Þ
rcpdTdt
~kL2TLx2
zL2TLy2
� �zRxx
LuLx
zRxyLu
LxzRyx
LuLy
zRyyLu
Ly, ð8Þ
Homogeneous-Heterogeneous Reactions with Peristalsis
PLOS ONE | DOI:10.1371/journal.pone.0113851 December 2, 2014 3 / 24
dadt
~DAL2aLx2
zL2aLy2
� �{kcab2, ð9Þ
dbdt
~DBL2bLx2
zL2bLy2
� �zkcab2, ð10Þ
where Rij are the components of extra stress tensor for the Carreau fluid and extra
stress tensor R (see refs. [17] and [25]) here is given by
R~½g?z(g0{g?)(1z(C _c)2)n{1
2 )� _c: ð11Þ
Here g? is the infinite shear-rate viscosity, g0 the zero shear-rate viscosity, C the
time constant and n the dimensionless form of power law index (nv1). Also _c is
defined as follows:
_c~
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12
Xi
Xj
_cij _cji
s~
ffiffiffiffiffiffiffiffi12P
r, ð12Þ
where P denotes the second invariant strain tensor defined by
P~tr½+Vz(+V)t�2 and _c~+Vz(+V)t: Here we consider the case for which
g?~0 and C _cv1: Therefore the extra stress tensor takes the form
R~g0 1z(C _c)2� �n{12
� �_c: ð13Þ
It is worth mentioning that the above model reduces to viscous model for n~1 or
C~0: The component forms of extra stress tensor are
Rxx ~ 2m0½1zn{1
2(C _c)2� Lu
Lx, ð14Þ
Rxy ~ m0½1zn{1
2(C _c)2�( Lu
Lyz
Lu
Lx)~Ryx, ð15Þ
Ryy ~ 2m0½1zn{1
2(C _c)2� Lu
Ly: ð16Þ
In above equationsddt
is the material time derivative, r the density of fluid, m0
the fluid viscosity, n the kinematic viscosity, T the fluid temperature, C the
concentration of fluid, T0 and T1 the temperatures at the lower and upper walls
respectively, K the measure of the strength of homogeneous reaction, DA and DB
the diffusion coefficients for homogeneous and heterogeneous reactions, cp the
specific heat at constant pressure, k the thermal conductivity, a and b the
concentrations of homogeneous and heterogeneous reactions with a0 serves as
uniform concentration of reactant A and kc the rate constant.
Homogeneous-Heterogeneous Reactions with Peristalsis
PLOS ONE | DOI:10.1371/journal.pone.0113851 December 2, 2014 4 / 24
The exchange of heat at the walls is given by
kLTLy
~ {h1(T{T1), at y~g,
kLTLy
~ {h2(T0{T), at y~{g:
ð17Þ
Here h1 and h2 indicate the heat transfer coefficients at the upper and lower
walls respectively.
The no-slip condition at the boundary wall is represented by the following
expressions
u~0, u~+gt at y~+g: ð18Þ
The compliant wall properties are described through the expression
{tL3
Lx3zm�1
L3
LxLt2zd0
L2
LtLx
� �g~
{rdudt
zLRxx
Lxz
LRxy
Ly{
sB20
1zm2u{muð Þ at y~+g,
ð19Þ
in which t is the elastic tension in the membrane, m�1 the mass per unit area and d’the coefficient of viscous damping. The mass conditions under the homogeneous
and heterogeneous reactions are given through the following expressions:
DALaLy
~ksa, at y~+g, ð20Þ
Figure 1. Plot of temperature h for wall parameters E1, E2, E3, with E~0:1, t~0:01, x~0:2, ª1~4, ª2~6,Br~1, m1~2, n~0:4, We~0:4 and m~0:04
doi:10.1371/journal.pone.0113851.g001
Homogeneous-Heterogeneous Reactions with Peristalsis
PLOS ONE | DOI:10.1371/journal.pone.0113851 December 2, 2014 5 / 24
DBLbLy
~{ksa, at y~+g, ð21Þ
where ks indicates the rate constant.
PerformingLLy
(6){LLx
(7) we get
rddt
LuLy
{Lu
Lx
� �~
L2Rxx
LyLx{
L2Ryx
Lx2z
L2Rxy
Ly2{
L2Ryy
LxLy{
sB20
1zm2
LuLy
{mLu
Ly
� �z
sB20
1zm2
Lu
Lxzm
LuLx
� �:
ð22Þ
Figure 2. Plot of temperature h for Brinkman number Br with E~0:1, t~0:01, x~0:2, ª1~4, ª2~6, E1~0:4,E2~0:2, E3~0:3, m1~2, n~0:4, We~0:4 and m~0:04
doi:10.1371/journal.pone.0113851.g002
Figure 3. Plot of temperature h for Biot number ª1 with E~0:1, t~0:01, x~0:2, Br~1, ª2~6, E1~0:4,E2~0:2, E3~0:3, m1~2, n~0:4, We~0:4 and m~0:04
doi:10.1371/journal.pone.0113851.g003
Homogeneous-Heterogeneous Reactions with Peristalsis
PLOS ONE | DOI:10.1371/journal.pone.0113851 December 2, 2014 6 / 24
Introducing the stream function y x,y,tð Þ and defining the following
dimensionless variables:
u~Ly
Ly, u~{
Ly
Lx,
y�~y
cd,x�~
xl
,y�~yd
, t�~ctl
, g�~g
d,h~
T{T0
T1{T0,b_c~ d
c_c,j�~
DB
DA,
g�~aa0
,h�~ba0
,R�xx~l
m0cRxx, R�xy~
dm0c
Rxy, R�yx~d
m0cRyx,R�yy~
dm0c
Ryy: ð23Þ
Eqs. (8), (9) and (22) yield
dReddt
L2y
Ly2zd2 L2y
Lx2
� �� �~d2 L2Rxx
LyLx{d2 L2Ryx
Lx2z
L2Rxy
Ly2zd
L2Ryy
LxLy
{m2
1
1zm2
L2y
Ly2zd2 L2y
Lx2z2md
L2y
LxLy
� �,
ð24Þ
d Pr Redh
dt~
d2 L2h
Lx2z
L2h
Ly2zBr d2Rxx
L2y
Ly2{d2Rxy
L2y
Lx2zRyx
L2y
Ly2{d3Ryy
L2y
LxLy
� �,
ð25Þ
Reddgdt
~1Sc
d2 L2gLx2
zL2gLy2
� �{Kgh2, ð26Þ
Figure 4. Plot of temperature h for Biot number ª2 with E~0:1, t~0:01, x~0:2, ª1~4, Br~1 E1~0:4,E2~0:2, E3~0:3, m1~2, n~0:4, We~0:4 and m~0:04:
doi:10.1371/journal.pone.0113851.g004
Homogeneous-Heterogeneous Reactions with Peristalsis
PLOS ONE | DOI:10.1371/journal.pone.0113851 December 2, 2014 7 / 24
Reddhdt
~j
Scd2 L2h
Lx2z
L2hLy2
� �zKgh2, ð27Þ
with the dimensionless conditions
g ~ 1zE sin 2p x{tð Þ, ð28Þ
Lh
Lyzc1(h{1) ~ 0 at y~g,
Lh
Ly{c2h ~ 0 at y~{g, ð29Þ
LgLy
{Mg~0 at y~+g, ð30Þ
jLhLy
zMh~0 at y~+g, ð31Þ
yy~0 at y~+g, ð32Þ
E1L3
Lx3zE2
L3
LxLt2zE3
L2
LxLt
� �g~{Red
ddt
(Ly
Ly)zd2 L
LxRxxz
LLy
Rxy
{m2
1
1zm2
Ly
Lyzmd
Ly
Lx
� �at y~+g: ð33Þ
Figure 5. Plot of temperature h for Hall parameter m with E~0:1, t~0:01, x~0:2, ª1~4, ª2~6, E1~0:4,E2~0:2, E3~0:3, m1~2, n~0:4, We~0:4 and Br~1:
doi:10.1371/journal.pone.0113851.g005
Homogeneous-Heterogeneous Reactions with Peristalsis
PLOS ONE | DOI:10.1371/journal.pone.0113851 December 2, 2014 8 / 24
Also Eqs. (14–16) become
Rxx~ 2(1zn{1
2We2b_c2
)yxy, ð34Þ
Rxy~ Ryx~2(1zn{1
2We2b_c2
)(yyy{d2yxx), ð35Þ
Ryy ~ {2d(1zn{1
2We2b_c2
)yxy: ð36Þ
In above equations asterisks have been omitted for simplicity. Here d is the
dimensionless wave number, the Reynolds number Re, the Prandtl number Pr, the
Figure 6. Plot of temperature h for Hartman number m1 with E~0:1, t~0:01, x~0:2, ª1~4, ª2~6, E1~0:4,E2~0:2, E3~0:3, Br~1, n~0:4, We~0:4 and m~0:04:
doi:10.1371/journal.pone.0113851.g006
Figure 7. Plot of temperature h for Weissenberg number We with E~0:1, t~0:01, x~0:2, ª1~4, ª2~6,E1~0:4, E2~0:2, E3~0:3, m1~2, n~0:4, Br~1 and m~0:04:
doi:10.1371/journal.pone.0113851.g007
Homogeneous-Heterogeneous Reactions with Peristalsis
PLOS ONE | DOI:10.1371/journal.pone.0113851 December 2, 2014 9 / 24
amplitude ratio E, the chemical reaction parameter c, the Hartman number m1,
the non-dimensional elasticity parameters E1, E2, E3, the Schmidt number Sc, the
Eckert number E, the Brinkman number Br, the heat transfer Biot numbers c1, c2,
the Weissenberg number We, the ratio of diffusion coefficient j, the strength
measuring parameters K and M (for homogeneous and heterogeneous reaction
respectively) and _c (non-dimensional form of _c) are given through the following
variables:
d ~dl
,Re~cdn
, Pr ~mcp
k,E~
ad
,We~Ccd
,E~c2
(T1{T0)cp,
Sc ~m0
rDA,E1~{
td3
l3m0c,E2~
m�1cd3
l3m0
,E3~d3d
0
ml2 ,Br~EPr,
b_c ~
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4d2 L2y
LxLy
� �2
zL2y
Ly2{d2 L2y
Lx2
� �s,c1~
h1dk
,c2~h2d
k,
m21 ~
sB20d2
m0,j~
DB
DA,K~
kca20d2
n,M~
ksdDA
:
ð37Þ
We now employ the approximations of long wavelength and low Reynolds
number [17–23] and equality of diffusion coefficients DA and DB i.e. j~1: The
assumption j~1 leads to the following relation:
g(g)zh(g)~1, ð38Þ
and we obtain the following set of equations
Figure 8. Plot of temperature h for power law index n with E~0:1, t~0:01, x~0:2, ª1~4, ª2~6, E1~0:4,E2~0:2, E3~0:3, m1~2, We~0:4, Br~1 and m~0:04:
doi:10.1371/journal.pone.0113851.g008
Homogeneous-Heterogeneous Reactions with Peristalsis
PLOS ONE | DOI:10.1371/journal.pone.0113851 December 2, 2014 10 / 24
L4y
Ly4z
32
(n{1)We2 L4y
Ly4
L2y
Ly2
� �2
z3(n{1)We2 L3y
Ly3
� �2L2y
Ly2{
m21
1zm2
L2y
Ly2~0,
ð39Þ
L2h
Ly2zBr
L2y
Ly2
� �2
1zn{1
2We2 L2y
Ly2
� �2 !
~0, ð40Þ
1Sc
L2gLy2
{Kg(1{g)2~0, ð41Þ
Figure 9. Plot of heat transfer coefficient Z for wall parameters E1, E2, E3, with E~0:1, t~0:01, ª1~4,ª2~6, m1~2, We~0:4, n~0:4, Br~1 and m~0:04:
doi:10.1371/journal.pone.0113851.g009
Figure 10. Plot of heat transfer coefficient Z for Brinkman number Br with E~0:1, t~0:01, ª1~4, ª2~6,E1~0:4, E2~0:2, E3~0:3, m1~2, n~0:4, We~0:4 and m~0:04:
doi:10.1371/journal.pone.0113851.g010
Homogeneous-Heterogeneous Reactions with Peristalsis
PLOS ONE | DOI:10.1371/journal.pone.0113851 December 2, 2014 11 / 24
Ly
Ly~ 0 at y~+g, ð42Þ
Lh
Lyzc1 h{1ð Þ ~ 0, at y~g,
Lh
Ly{c2h ~ 0, at y~{g,
ð43Þ
LgLy
{Mg ~ 0, at y~+g, ð44Þ
E1L3
Lx3zE2
L3
LxLt2zE3
L2
LxLt
� �g ~
L3y
Ly3z
32
(n{1)We2 L2y
Ly2
� �2L3y
Ly3
{m2
1
1zm2
Ly
Lyat y ~ +g:
ð45Þ
2.1 Method of solution
It is seen from Eqs. (39) and (41) that these Eqs. are non-linear and involve
Weissenberg number We and homogeneous reaction parameter K respectively.
Therefore the problem at hand cannot be solved exactly, but can be linearized
about "small" parameter to the mathematical description of the exactly solvable
problem. The technique is referred as perturbation. Perturbation method
represent a very powerful tool in modern mathematical physics and, in particular,
in fluid dynamics and leads to a series solution of resulting system of equations
having small paramter. Therefore we have applied this method to form the series
Figure 11. Plot of heat transfer coefficient Z for Biot number ª1 with E~0:1, t~0:01, Br~1, ª2~6, E1~0:4,E2~0:2, E3~0:3, m1~2, n~0:4, We~0:4 and m~0:04:
doi:10.1371/journal.pone.0113851.g011
Homogeneous-Heterogeneous Reactions with Peristalsis
PLOS ONE | DOI:10.1371/journal.pone.0113851 December 2, 2014 12 / 24
solutions for stream function y, temperature h and concentration g corre-
sponding to the involved non-linear quantities (We and K). For this we write the
flow quantities in the forms:
y~y0zWe2y1zO We4� �
,
h~h0zWe2h1zO We4� �
,
g~g0zKg1zO K2� �
,
Z~Z0zWe2Z1zO(We4):
2.2 Zeroth order system and its solution
The zeroth order system is given by
L4y0
Ly4{
m21
1zm2
L2y0
Ly2~ 0, ð46Þ
L2h0
Ly2zBr
L2y0
Ly2
� �2
~ 0, ð47Þ
1Sc
L2g0
Ly2~0, ð48Þ
Ly0
Ly~0, at y~+g, ð49Þ
Figure 12. Plot of heat transfer coefficient Z for Biot number ª2 with E~0:1, t~0:01, ª1~4, Br~1, E1~0:4,E2~0:2, E3~0:3, m1~2, n~0:4, We~0:4 and m~0:04:
doi:10.1371/journal.pone.0113851.g012
Homogeneous-Heterogeneous Reactions with Peristalsis
PLOS ONE | DOI:10.1371/journal.pone.0113851 December 2, 2014 13 / 24
Lh0
Lyzc1(h0{1)~0 at y~g,
Lh0
Ly{c2h0~0 at y~{g, ð50Þ
Lg0
Ly{Mg0~0, at y~+g, ð51Þ
Figure 13. Plot of heat transfer coefficient Z for Hall parameter m with E~0:1, t~0:01, ª1~4, ª2~6,E1~0:4, E2~0:2, E3~0:3, m1~2, n~0:4, We~0:4 and Br~1:
doi:10.1371/journal.pone.0113851.g013
Figure 14. Plot of heat transfer coefficient Z for Hartman number m1 with E~0:1, t~0:01, ª1~4, ª2~6,E1~0:4, E2~0:2, E3~0:3, Br~1, n~0:4, We~0:4 and m~0:04:
doi:10.1371/journal.pone.0113851.g014
Homogeneous-Heterogeneous Reactions with Peristalsis
PLOS ONE | DOI:10.1371/journal.pone.0113851 December 2, 2014 14 / 24
E1L3
Lx3zE2
L3
LxLt2zE3
L2
LxLt
� �g~
L3y0
Ly3{
m21
1zm2
Ly0
Lyat y~+g: ð52Þ
The solutions of Eqs. (46–48) subject to the boundary conditions (49–52) are
y0~A2(ffiffiffiffiHp
y{sech(ffiffiffiffiHp
g) sinh (ffiffiffiffiHp
y)), ð53Þ
h0~{L4z4L5
Hy2{
(L4c2{L2)
1zc2gy{
2L5 cosh (2ffiffiffiffiHp
y)
H2, ð54Þ
g0~B1zB2y, ð55Þ
Z0~gxh0y(g),
~gx½{4L5({2
ffiffiffiffiHp
gz sinh (2ffiffiffiffiHp
g))
H3=2{
L4c2
1zc2g
{2L5({2Hg(2zc2g)zc2 cosh (2
ffiffiffiffiHp
g)z2ffiffiffiffiHp
sinh (2ffiffiffiffiHp
g))
H2(1zc2g)�:
ð56Þ
2.3 First order system and its solution
At this order we have
L4y1
Ly4z
32
(n{1)L4y0
Ly4
L2y0
Ly2
� �2
z3(n{1)L3y0
Ly3
� �2L2y0
Ly2{
m21
1zm2
L2y1
Ly2~0, ð57Þ
Figure 15. Plot of heat transfer coefficient Z for Weissenberg number We with E~0:1, t~0:01, ª1~4,ª2~6, E1~0:4, E2~0:2, E3~0:3, m1~2, n~0:4, Br~1 and m~0:04:
doi:10.1371/journal.pone.0113851.g015
Homogeneous-Heterogeneous Reactions with Peristalsis
PLOS ONE | DOI:10.1371/journal.pone.0113851 December 2, 2014 15 / 24
L2h1
Ly2z2Br
L2y0
Ly2
L2y1
Ly2z
Br(n{1)
2L2y0
Ly2
� �4
~0, ð58Þ
1Sc
L2g1
Ly2{g0(1{g0)2~0, ð59Þ
Ly1
Ly~0, at y~+g, ð60Þ
Lh1
Lyzc1h1~0 at y~g,
Lh1
Ly{c2h1~0 at y~{g, ð61Þ
Lg1
Ly{Mg1~0, at y~+g, ð62Þ
L3y1
Ly3z
32
(n{1)L3y0
Ly3
L2y0
Ly2
� �2
{m2
1
1zm2
Ly1
Ly~0 at y~+g: ð63Þ
Solving Eqs. (57–59) and then applying the corresponding boundary conditions
we get the solutions in the forms given below:
Figure 16. Plot of heat transfer coefficient Z for power law index n with E~0:1, t~0:01, ª1~4, ª2~6,E1~0:4, E2~0:2, E3~0:3, m1~2, Br~1, We~0:4 and m~0:04:
doi:10.1371/journal.pone.0113851.g016
Homogeneous-Heterogeneous Reactions with Peristalsis
PLOS ONE | DOI:10.1371/journal.pone.0113851 December 2, 2014 16 / 24
y1~A3A4 exp ({ffiffiffiffiHp
(3yz2g))½{3 exp (2ffiffiffiffiHp
y)z3 exp (4ffiffiffiffiHp
y)
z exp (2ffiffiffiffiHp
g)z exp (4ffiffiffiffiHp
g){ exp (2ffiffiffiffiHp
(3yzg))
{ exp (2ffiffiffiffiHp
(3yz2g)){3 exp (2ffiffiffiffiHp
(yz3g))
z3 exp (2ffiffiffiffiHp
(2yz3g))z12(ffiffiffiffiHp
(y{g){1) exp (4ffiffiffiffiHp
(yzg))
z12(ffiffiffiffiHp
(y{g)z1) exp (2ffiffiffiffiHp
(yzg))
z12(ffiffiffiffiHp
(yzg){1) exp (2ffiffiffiffiHp
(2yzg))
z12(ffiffiffiffiHp
(yzg)z1) exp (2ffiffiffiffiHp
(yz2g))�,
ð64Þ
Figure 17. Plot of concentration g for homogeneous reaction parameter K with E~0:2, t~0:1, x~0:1,Sc~1:5 and M~2.
doi:10.1371/journal.pone.0113851.g017
Figure 18. Plot of concentration g for heterogeneous reaction parameter M with E~0:2, t~0:1, x~0:1,Sc~1:5 and K~0:5.
doi:10.1371/journal.pone.0113851.g018
Homogeneous-Heterogeneous Reactions with Peristalsis
PLOS ONE | DOI:10.1371/journal.pone.0113851 December 2, 2014 17 / 24
h1~L6 exp ({4ffiffiffiffiHp
(yzg))½(9Br{4)L1fexp (4ffiffiffiffiHp
g)z exp (6ffiffiffiffiHp
g)
z exp (4ffiffiffiffiHp
(2yzg))z exp (2ffiffiffiffiHp
(4yz3g))g{12BrL1fexp (2ffiffiffiffiHp
(yzg))
z exp (2ffiffiffiffiHp
(3yzg))z exp (2ffiffiffiffiHp
(yz4g))z exp (2ffiffiffiffiHp
(3yz4g))g
z4L1 exp (2ffiffiffiffiHp
(yz2g))(16z3Br({3z4ffiffiffiffiHp
(y{g))
{4L1 exp (6ffiffiffiffiHp
(yzg))({16z3Br(3z4ffiffiffiffiHp
(y{g))
z exp (4ffiffiffiffiHp
y)f(4z3Br)L1{4(3Br{4)ffiffiffiffiHp
(y(c1{c2){g(c1zc2){2)g
z exp (2ffiffiffiffiHp
(2yz5g))f(4z3Br)L1z4(3Br{4)ffiffiffiffiHp
(y(c1{c2){g(c1zc2){2))g
z4L1 exp (2ffiffiffiffiHp
(yz3g))(16z3Br(4ffiffiffiffiHp
(yzg){3))
{4L1 exp (2ffiffiffiffiHp
(3yz2g))({16z3Br(4ffiffiffiffiHp
(yzg)z3))
z exp (4ffiffiffiffiHp
(yz2g))f3(9Br{20)L1{28(3Br{4)ffiffiffiffiHp
(y(c1{c2){g(c1zc2){2)
z48BrH(L1y2z2y(c1{c2)g{4g{3g2(c1zc2){2c1c2g3)g
z exp (2ffiffiffiffiHp
(2yzg))f3(9Br{20)L1z28(3Br{4)ffiffiffiffiHp
(y(c1{c2){g(c1zc2){2)
z48BrH(L1y2z2y(c1{c2)g{4g{3g2(c1zc2){2c1c2g3)g
z16 exp (2ffiffiffiffiHp
(2yz3g))f(3Br{4)L1z2ffiffiffiffiHp
((4{3Br)(y(c1zc2){2)
z2(3Br{2)zg(c1zc2)z6Brc1c2g2)
z12BrH3=2g(2yg(c2{c1){L1y2zg(4z3g(c1zc2)z2c1c2g2))
z12H(({1zBr)L1y2z(Br{2)(c1{c2)ygz4gz3g2(c1zc2)
z2c1c2g3{2gBr(1zc1g)(1zc2g))g{16 exp (4ffiffiffiffiHp
(yzg))f(4{3Br)L1
{2ffiffiffiffiHp
((3Br{4)(y(c1{c2){2){2g(3Br{2)(c1zc2){6Brc1c2g2)
{12H((Br{1)L1y2zg(c1{c2)(Br{2)yz4gz3g2(c1zc2)z2c1c2g3
{2gBr(1zc1g)(1zc2g))z12BrH3=2g(2yg(c2{c1){L1y2
z4gz3g2c2zc1g2(3z2gc2)g�,
ð65Þ
g1~H1zC1yzC2y2zC3y3zC4y4zC5y5, ð66Þ
and the heat transfer coefficient is
Homogeneous-Heterogeneous Reactions with Peristalsis
PLOS ONE | DOI:10.1371/journal.pone.0113851 December 2, 2014 18 / 24
Z1~gxh1y(g),
~gxL7½{4(1z exp (2ffiffiffiffiHp
g)(8 exp (2ffiffiffiffiHp
g){8 exp (6ffiffiffiffiHp
g)
z exp (8ffiffiffiffiHp
g)z24ffiffiffiffiHp
g exp (4ffiffiffiffiHp
g){1)
z3Brfexp (10ffiffiffiffiHp
g){1z( exp (8ffiffiffiffiHp
g)z exp (2ffiffiffiffiHp
g))(8ffiffiffiffiHp
g{7)
{8 exp (6ffiffiffiffiHp
g)(1{2ffiffiffiffiHp
gz4Hg2)z8 exp (4ffiffiffiffiHp
g)(1z2ffiffiffiffiHp
gz4Hg2)g�,
ð67Þ
in which
H~m2
1
1zm2,L~2(E1zE2)p cos 2p(t{x)zE3 sin 2p(t{x),A2~
4p2ELH3=2
,
A3~{(n{1)p6E3
2 1z exp (2ffiffiffiffiHp
g)H5=2� � ,A4~L3sech3(
ffiffiffiffiHp
g),L1~c1zc2z2gc1c2,
L2~2L5(2Hg(2zc2g){c2 cosh (2
ffiffiffiffiHp
g){2ffiffiffiffiHp
sinh (2ffiffiffiffiHp
g))
H2,
L3~{c1{2L5c1({2Hg2z cosh (2
ffiffiffiffiHp
g)
H2
{4BrL2p4E2sech(
ffiffiffiffiHp
g)2({2ffiffiffiffiHp
gz sinh (2ffiffiffiffiHp
g))
H32
,
Figure 19. Plot of concentration g for Schmidt number Sc with E~0:2, t~0:1, x~0:1 K~0:5 and M~2.
doi:10.1371/journal.pone.0113851.g019
Homogeneous-Heterogeneous Reactions with Peristalsis
PLOS ONE | DOI:10.1371/journal.pone.0113851 December 2, 2014 19 / 24
L4~L2(1zc1g)zL3(1zc2g)
L1, L5~BrL2p4E2sech(
ffiffiffiffiHp
g)2,
L6~sech(
ffiffiffiffiHp
g)4L4(n{1)p8E4
8(1z exp 2ffiffiffiffiHp
g� �
)H3L1,
L7~exp ({4
ffiffiffiffiHp
g)sech(ffiffiffiffiHp
g)4L4(n{1)(1zc2g)c1p8E4
(1z exp (2ffiffiffiffiHp
g))L1H5=2,
H1~Sc
480M8g3|(60{60M(3z4M)gz270M2g2z480M3g2z240M4g2{150M3g3
{360M4g3{240M5g3{M4(33z40M(2zM))g4z5M5(9z4M(7z6M)g5),
B1~1{Mg
2M2g,B2~
12Mg
,C1~Sc
480M8g3|(60M{60M2(3z4M)gz210M3g2
z480M4g2z240M5g2{60M4g3{240M5g3(1zM){M5(33z40M(2zM))g4),
C2~Sc
480M8g3|(30M2{30M3(3z4M)gz90M4g2z240M5g2
z120M6g2{30M5g3{120M6g3{120M7g3),
C3~Sc
480M8g3|(30M3{20M4(3z4M)gz30M5g2z40M6g2(2zM)),
C4~Sc
480M8g3|(15M4{5M2(3z4M)g,C5~
Sc160M3g3
:
Results and Discussion
This section is prepared to explore the effects of influential parameters on the
temperature, heat transfer coefficient and concentration.
3.1 Temperature profile
Figs. (1–8) are formulated to examine the impact of various involved parameters
on temperature distribution h: Fig. 1 indicates the increasing behavior of
temperature profile with wall parameters E1 and E2 while E3 corresponds to
reduction in temperature profile. It is in view of the fact that elastic properties of
the wall depicted by E1 and E2 cause less resistance to flow of fluid velocity as well
as energy. On the other hand the damping characteristic of the wall identified by
E3 reduces the velocity and temperature of the fluid (see Fig. 1). The temperature
Homogeneous-Heterogeneous Reactions with Peristalsis
PLOS ONE | DOI:10.1371/journal.pone.0113851 December 2, 2014 20 / 24
profile is an increasing function of Brinkman number Br (see Fig. 2). This is
because of the increase in internal resistance of fluid particles which increases the
fluid temperature. The Biot numbers c1 and c2 on the lower and upper walls have
similar effect on the temperature profile i:e: increase in c1 and c2 decreases the
temperature profile near upper and lower channel walls respectively (see Figs. 3
and 4). It is seen that increasing c1 and c2 reduces the thermal conductivity which
causes reduction of temperature profile. The Hall parameter m increases the
temperature. This is due to the fact that electrical conductivity increases with
increasing values of m (see Fig. 5). It is observed from Fig. 6 that Hartman
number m1 lessens the temperature distribution. Also the results drawn in Figs. 7
and 8 show opposite effects of Weissenberg number We and the power law index
n i:e:, increasing values of We reduces the temperature whereas an increase in nenhances the temperature of fluid. The obtained results are in good agreement
with the articles presented in [17–19].
3.2 Heat transfer coefficient
Figs. 9–16 demonstrate the influence of embedded parameters on the heat transfer
coefficient Z: The graphs signify the oscillatory behavior of Z because of the
propagation of peristaltic waves. Fig. 9 reveals that magnitude of heat transfer
coefficient increases for compliant wall parameters E1, E2 and E3. Since E1, E2 and
E3 describes the elastic nature of wall that offer less resistance to heat transfer.
Increasing values of Brinkman number Br show similar behavior on heat transfer
as of wall parameters. However the results obtained are much more distinguished
in case of Br (see Fig. 10). The Biot number c1 causes reduction in magnitude of
heat transfer coefficient on the upper wall. Here thermal conductivity decreases
with an increase in c1 which lessens the impact of heat transfer coefficient near
positive side (xw{0:1) as depicted in Fig. 11. Reverse effect of Biot number c2
has been observed in the region from Fig. 12 as heat transfer being directly related
to Biot number dominates with an increase in c2 which in turn increases the heat
transfer distribution. Fig. 13 shows decrease in heat transfer coefficient Z with
Hall parameter m. Also in absence of Hall parameter (m~0) the results are much
more distinguished. The Hartman number m1 is an increasing function of heat
transfer coefficient Z as fluid viscosity decreases with an increase in m1. The less
viscous fluid particles will move through gain of higher kinetic energy that causes
rise in transfer of heat (see Fig. 14). The effects of Weissenberg number We are
displayed in Fig. 15. The obtained results show increase in transfer of heat when
We increases as speed of wave increases with an increase in We that supports the
transfer of heat. The increasing values of power law index show decline in heat
transfer distribution (see Fig. 16).
3.3 Homogeneous-Heterogeneous reactions effects
Effects of homogeneous and heterogeneous reaction parameters M and K and
Schmidt number Sc are displayed in the Figs. 17–19. The results drawn in Fig. 17
Homogeneous-Heterogeneous Reactions with Peristalsis
PLOS ONE | DOI:10.1371/journal.pone.0113851 December 2, 2014 21 / 24
illustrates the dual behavior of homogeneous reaction parameter K on the
concentration profile. It is observed that concentration increases in the region
xw0:5 as in this region increase in K enhances the fluid density hence
concentration rises while in the region xv0:5 the concentration decreases because
viscosity reduces. On the other hand the heterogeneous reaction parameter Mshows the opposite behavior when compared with K i:e: it increases along positive
side of the coordinate axes xv0ð Þ (since diffusion reduces with an increase in Mand less diffused particles will rise the concentration) and decreases along negative
side of coordinate axes (xw0) (as increase in rate of reaction dominates the
decrease in diffusion in this region). It is evident from Figs. 17 and 18 that the
concentration distribution of reactants increases from {g to g in both cases and
after a certain value of g it starts decaying. This critical value of g depends on the
strength of homogeneous reaction and it is prominent for increasing K . The
effects of Schmidt number Sc are depicted in Fig. 19. The exhibited results are
quite similar to Fig. 17. The drawn results follow by the fact that viscosity of fluid
increases with an increase in Schmidt number that provides resistance to flow of
fluid. The slow moving fluid particles have small molecular vibrations which
lessen the concentration of fluid. As Schmidt number defines the ratio of viscous
diffusion rate to molecular diffusion rate. Hence increasing values of Sc enhances
the viscous diffusion rate for fixed molecular diffusion rate which in turn helps to
increase the concentration of fluid (see Fig. 19). The similar findings are reported
by Shaw et al. [24].
3.4 Concluding remarks
The present analysis explores the effects of homogeneous and heterogeneous
reactions in the peristalsis of Carreau fluid. Such analysis even for viscous fluid is
yet not available. The major results of this study are listed below.
N Similar behavior is observed for compliant wall parameters on temperature
profile and heat transfer coefficient.
N Temperature is increasing function of Brinkman number and Hall parameter.
N The Biot numbers and Hartman number decrease the temperature of fluid.
N Opposite effects of Weissenberg number We and power law index n are
observed on the temperature profile and heat transfer coefficient.
N Concentration of the reactants is more signified in case of homogeneous
reaction parameter K than heterogeneous reaction parameter M.
Author Contributions
Conceived and designed the experiments: TH AT HY AA. Performed the
experiments: TH AT HY AA. Analyzed the data: TH AT HY AA. Contributed
reagents/materials/analysis tools: TH AT HY AA. Wrote the paper: TH AT HY
AA.
Homogeneous-Heterogeneous Reactions with Peristalsis
PLOS ONE | DOI:10.1371/journal.pone.0113851 December 2, 2014 22 / 24
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