Soret and Dufour effects in peristaltic transport of physiological fluids with chemical reaction: A mathematical analysis

Computers & Fluids 89 (2014) 242–253

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Computers & Fluids j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p fl u i d

Soret and Dufour effects in peristaltic transport of physiological fluids with chemical reaction: A mathematical analysis T. Hayat a,b,⇑, Humaira Yasmin a, Maryem Al-Yami b a b

Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

a r t i c l e

i n f o

Article history: Received 15 July 2013 Received in revised form 17 September 2013 Accepted 28 October 2013 Available online 11 November 2013 Keywords: Peristalsis Casson model Convective conditions Soret and Dufour effects

a b s t r a c t The peristaltic transport of Casson fluid in a two-dimensional asymmetric channel with convective conditions is investigated. The Soret and Dufour effects are studied in the presence of chemical reaction. The relevant flow analysis is modeled for Casson fluid in a wave frame of reference. Computations of solutions are made for the velocity, temperature and concentration fields. Here two yield planes exist due to asymmetry in the channel. These planes are calculated by solving the transcendental equation in terms of the core width. Closed form expression of stream function is constructed. Plots are prepared for a parametric study reflecting the effects of Casson fluid parameter, chemical reaction parameter, Prandtl, Schmidt, Soret, Dufour and Biot numbers. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Peristalsis is popular amongst the recent physiologists in view of efficient mechanisms for fluid transport in many biological systems. It is the phenomenon in which a circumferential progressive wave of contraction or expansion (or both) propagates along the channel/tube. Peristalsis appears in many organisms and a variety of organs. In particular, peristaltic mechanism is involved in urine transport from the kidney to the bladder through the ureter, movement of chyme in the gastrointestinal tract, transport of spermatozoa in the ductus efferentes of the male reproductive tract, movement of ova in the fallopian tubes and in the vasomotion of small blood vessels. These flows also provide efficient means for sanitary fluid transport and are thus exploited in industrial pumping and medical devices. Based on the principle of peristalsis, the mechanical roller pumps, heart–lung machines, cell separators etc. have been fabricated. The main motivation for any mathematical analysis of physiological fluid flows is to ultimately have a better understanding of the particular flow being modeled. If there is similarity between the results obtained from the analysis and experimental and clinical data, then the mechanism of flow can at least be explained. An accurate mathematical study can help explain the major contributing factors to many flows in the human body because peristalsis is ⇑ Corresponding author at: Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan. Tel.: +92 51 90642172. E-mail address: [email protected] (T. Hayat). 0045-7930/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compfluid.2013.10.038

evident in many physiological flows. When comparing results between the mathematical model and the experimental data, it is desirable that the data obtained from experimental research be as close as possible to the actual physiological parameter being analyzed. The study of the mechanism of peristalsis in both mechanical and physiological situations has become the subject of scientific research since the initial attempt by Latham [1]. The earliest models by Shapiro et al. [2] and Fung and Yih [3] were idealized and represented the peristalsis by an infinite wave train in a two-dimensional channel, thus, they could pretend to only a qualitative relationship with the ureter. Since then several theoretical and experimental attempts have been made to understand peristaltic action in different situations for Newtonian fluids [4–6]. Most of the theoretical investigations have been carried out by assuming blood and other physiological fluids as a Newtonian fluid. Although this approach may provide a satisfactory understanding of peristaltic motion in the ureter, it fails to provide satisfactory model when this mechanism is involved in small blood vessels, lymphatic vessels etc. It has now been accepted that most of the physiological fluids behave like non-Newtonian fluids. Hence Raju and Devanathan [7] have attempted the analysis of peristaltic flow to understand the features of a non-Newtonian fluid in the case of small wave amplitude. The study of blood through arteries is of prime importance because it provides an insight into physiological situations. The blood can be viewed as a Newtonian or a non-Newtonian fluid, as the case may be [8–11]. Under diseased conditions, abnormal and unnatural growth develops in the lumen at various locations of the cardiovascular system.

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As the growth projects into lumen of the artery, blood flow will be restricted. Hence the detailed knowledge of the flow field may help in proper understanding and prevention of arterial diseases. In small blood vessels, blood exhibits shear-dependent viscosity and requires a finite yield stress before flow can commence, thereby making the non-Newtonian nature of blood an important factor in the modeling. Therefore Casson [12] has elaborated the flow equation for pigment oil-suspensions of the printing ink type. Srivastava and Srivastava [13] investigated the problem of peristaltic transport of blood assuming a single-layered Casson fluid and ignored the presence of a peripheral layer. A mathematical study of peristaltic transport of Casson fluid is given by Mernone and Mazumdar [14] and Mernone et al. [15] in two-dimensional axisymmetric channel using the generalized form of the constitutive equation for Casson fluid [16]. Later Nagarani and Sarojamma studied the peristaltic transport of Casson fluid and asymmetric channel [17] and in an annulus [18]. They discussed the effects of yield stress of the fluid and asymmetry of the channel. Sreenadh et al. [19] have investigated the Casson fluid flow in inclined tube of non-uniform cross-section with multiple stenoses. Recently, investigations of heat and mass transfer in peristalsis have been considered by some researchers due to its applications in the biomedical sciences. Srinivas and Kothandapani [20] investigated the effects of heat and mass transfer in MHD (magnetohydrodynamic) peristaltic flow through a porous space with compliant walls. Eldabe et al. [21] have discussed the heat and mass transfer in MHD peristaltic flow of a couple stress fluid through porous medium. Effects of heat and mass transfer on peristaltic flow of a nanofluid between eccentric cylinders are investigated by Nadeem et al. [22]. Hayat et al. [23] discussed the heat and mass transfer effects in peristaltic flow of Oldroyd-B fluid in a channel with compliant walls. Hydromagnetic flow and heat transfer of a conducting Casson fluid in a rectangular channel is observed by Attia and Sayed-Ahmed [24]. Hina et al. [25] studied the MHD non-linear peristaltic flow in a compliant walls channel with heat and mass transfer. It has been observed that an energy flux can be generated not only by the temperature gradient but also by the concentration gradient. The energy flux caused by a concentration gradient is termed as the diffusion-thermo (Dufour) effect. On the other hand, mass fluxes can also be created by temperature gradients and it is termed as thermo-diffusion (Soret) effect. Peristaltic motion with Soret and Dufour effects is discussed by Hayat et al. [26]. Srinivas et al. [27] have investigated the influence of heat and mass transfer in the peristaltic flow of a viscous fluid in a vertical asymmetric channel with wall slip. Bég and Tripathi [28] studied peristaltic pumping with double-diffusive convection in nanofluids. Peristaltic slip flow of viscoelastic fluid with heat and mass transfer in a tube is observed by Sobh [29]. In the present paper, we propose to study peristaltic transport of physiological fluids in a planar channel using the most generalized form of the constitutive equation of Casson fluid [18]. We have also considered the Soret and Dufour effects and chemical reaction. The Soret and Dufour effects are important for the high temperature and concentration gradients. The Soret effects for instance has been utilized for isotope separation and in mixture between gases with very high molecular weight (H2 ; He ) and of medium molecular weight (N 2 , air). The Dufour effects are found to be of order of considerable magnitude such that it cannot be ignored in certain conditions. The problem is modeled subject to long wavelength approximation. Closed form solutions are obtained for pressure gradient, velocity, temperature and concentration profiles. The inclusion of Biot numbers, Soret, Dufour and the non-Newtonian fluid characteristics lead to some interesting effects on the velocity, temperature, concentration and trapping phenomena. The presented analysis here differs from the work of Rani and Sarojamma [17] in the sense of heat and mass transfer effects with Soret and Dufour

features. In addition convective conditions are utilized here. Chemical reaction effects are also not considered in Ref. [17]. 2. Constitutive equation The constitutive equation corresponding to the flow of a Casson fluid is given by

 

s2 ¼ s2y þ l1 1

1



1 2 @U   @Y 

Ps y ; if s

ð1Þ

@U 6s y ; ¼ 0 if s @Y

ð2Þ

y is the yield stress. where l1 is the viscosity at high shear rate and s It is seen from relation (2) that the velocity gradient vanishes in the region where the shear stress is less than the yield stress. As a result 6s y . These relations between shear plug flow sets in whenever s  rate @U and shear stress are appropriate for positive values of s @Y and negative values of @U . For more general situation, the equivalent @Y form of these relations where the shear stress and shear rate can change the sign may be written as 1 ! s2y  @U sy l1 ¼  1 þ   2 1=2 s if s P sy ; jsj j @Y js

@U 6s y : ¼ 0 if s @Y

ð3Þ ð4Þ

3. Problem formulation Consider the peristaltic motion of a non-Newtonian fluid, modeled as a Casson fluid in the two dimensional asymmetric channel by taking ðX; YÞ as the Cartesian coordinates with X being measured in the direction of wave propagation and Y in the direction normal to the X-axis. The motion is induced due to the propagation of sinusoidal wave trains with a constant speed c along the channel 1 and Y ¼ h 2 be the upper and lower boundaries of walls. Let Y ¼ h the channel respectively (see Fig. 1). Then

  1 ðX; tÞ ¼ d1 þ a1 cos 2p ðX  ctÞ ; upper wall; h k   2 ðX; tÞ ¼ d2  a2 cos 2p ðX  ctÞ þ / ; lower wall; h k

ð5Þ ð6Þ

where a1 ; a2 are the waves amplitudes, k is the wavelength, d1 þ d2 is the width of the asymmetric channel and the phase difference / varies in the range 0 6 / 6 p. Here / ¼ 0 corresponds to symmetric channel with waves out of phase and / ¼ p the waves are in phase. Further a1 ; a2 ; d1 ; d2 and / satisfy the condition 2

a21 þ a22 þ 2a1 a2 cos / 6 ðd1 þ d2 Þ :

ð7Þ

From Eqs. (3) and (4) it is clear that the flow of Casson fluid in an asymmetric channel is three phase in nature in which the central core region corresponds to the plug flow region. If the plug flow re2 6  1 and the gion is represented by  k1 6 Y 6  k2 where h k1 ;  k2 6 h 2 6 Y 6   1 , then the two shear flow regions by h k1 and  k2 6 Y 6 h Casson’s fluid constitutive Eqs. (3) and (4) in these regions can be rewritten as

l1

@U þs  y  2s 1=2 j1=2 ¼ s js y @Y

2 6 Y 6 k1 ; if h

@U ¼ 0 if k1 6 Y 6 k2 ; @Y   @U 1 ; j1=2 if k2 6 Y 6 h l1 ¼  s þ sy  2s1=2 js y @Y

ð8Þ ð9Þ ð10Þ

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Fig. 1. Geometry of the problem.

k1 and Y ¼  k2 are the two yield plane locations to be in which Y ¼  determined as part of solution to the problem under consideration. If the channel length is finite and equal to an integral number of wavelength and if the pressure difference between the ends of the channel is constant, the flow is steady in wave frame of reference (Shapiro et al. [2]). We assume that these conditions are met and we study the problem in the wave frame. The Galilean transformað Þ are tions between fixed frame OðX; Y; tÞ and moving frame o x; y given by

 ¼ Y; u  ðx; y Þ ¼ UðX; Y; tÞ  c; v ðx; y Þ ¼ VðX; Y; tÞ; y Þ ¼ TðX; Y; tÞ; Cðx; y Þ ¼ CðX; Y; tÞ; p ðx; y Þ ¼ PðX; Y; tÞ; Tðx; y x ¼ X  ct;

ð11Þ  ; v Þ and ðU; VÞ are the velocity components in the wave in which ðu  and P stand for pressure in the and the fixed frames respectively, p wave and fixed frames. Also T and C are the temperature and concentration of the fluid in dimensional form. Defining dimensionless variables, wave number ðdÞ, temperature ðhÞ, concentration ðrÞ, Prandtl number ðPrÞ, Dufour number ðDuÞ, Schmidt number ðScÞ, Soret number ðSrÞ and chemical reaction parameter ðcÞ by 2 x    y u v d p d1 sd1 x¼ ; y¼ ; u¼ ; v ¼ ; p¼ 1 ; d¼ ; s¼ ; k d1 c ckl1 cd k l1 c 1 2 h h ct T  Ta C  C0 sy d1 ; r¼ ; sy ¼ ; h1 ¼ ; h2 ¼ ; t ¼ ; h ¼ k d1 d1 Ta C1  C0 l1 c d2 a1 b1 k1 k2 lcp m ; Sc ¼ ; d ¼ ; a ¼ ; b ¼ ; k1 ¼ ; k2 ¼ ; Pr ¼ d1 d1 d1 d1 d1 k D

DkT ðC 1  C 0 Þ DkT Du ¼ ; Sr ¼ ; c¼ lc p C s T a mðC 1  C 0 Þ

2 k1 d1

m

; c ¼

2 k1 d 1 C 0

mðC 1  C 0 Þ

@u ¼ s þ sy  2s1=2 j sj1=2 y @y

; ð12Þ

the relevant dimensionless flow equations under long wavelength along with low Reynolds number assumptions can be expressed as

@p @ s  ¼ 0; @x @y @p ¼ 0; @y 

1 @2h @2r þ Du 2 ¼ 0; Pr @y2 @y 2

if h2 6 y 6 k1 ;

@u ¼ 0 if k1 6 y 6 k2 ; @y   @u if k2 6 y 6 h1 ; ¼  s þ sy  2s1=2 j sj1=2 y @y

ð17Þ ð18Þ ð19Þ

whereas the non-dimensional form of the peristaltic walls are given by

y ¼ h1 ¼ 1 þ a cosð2pxÞ;

ð20Þ

ð13Þ

y ¼ h2 ¼ d  b cosð2px þ /Þ:

ð21Þ

ð14Þ

In fixed frame of reference, the instantaneous volume flow rate is given by

ð15Þ

2

1 @ r @ h þ Sr 2  cr  c ¼ 0: Sc @y2 @y

Eq. (14) indicates that p – pðyÞ. In the limit Re ! 0, the inertialess flow corresponds to Poiseuille-like longitudinal velocity profile. The pressure gradient depends upon x and t only in laboratory frame. It does not depends on y. Such features can be expected because there is no streamline curvature to produce transverse pressure gradient when d ¼ 0. The assumptions of long wavelength and small Reynolds number gives d ¼ 0 and Re ¼ 0. It should be pointed out that the theory of long wavelength and zero Reynolds number remains applicable for case of chyme transport in male small intestine [30]. In this case c ¼ 2 cm/min, a ¼ 1:25 cm and k ¼ 8:01 cm. Here half width of intestine is small in comparison to wavelength. i.e. a=k ¼ 0:156. It is also declared by Lew et al. [31] that Reynolds number in small intestine was small. Further, the situation of intrauterine fluid flow due to myomaterial contractions is a peristaltic type fluid motion in a cavity. The sagittal crosssection of the uterus reveals a narrow channel enclosed by two fairly parallel walls [32]. The 1–3 mm width of this channel is very small compared with its 50 mm length [33], defining an opening angle from cervix to fundus of about 0:04 rad. Analysis of dynamics parameters of the uterus revealed frequency, wavelength, amplitude and velocity of the fluid-wall interface during a typical contractile wave were found to be 0.01–0.057 Hz, 10–30 mm, 0.05– 0.2 mm and 0.5–1.9 mm/s respectively. The Casson’s fluid constitutive equation in dimensionless form is expressed as

ð16Þ

Q¼

Z

 ðX;tÞ h 1

UðX; Y; tÞdY;

 ðX;tÞ h 2

whereas in wave frame of reference, it is given by

ð22Þ

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q¼

Z

h1 ðxÞ

ðx; y Þdy : u

ð23Þ

Substituting the expression for s from Eq. (39) in the constitutive Eqs. (17)–(19) and integrating with the help of distribution in different regions as

ð24Þ

uðyÞ ¼ u ðyÞ ¼ 1  dp 1 4 pffiffiffi 2 3=2  b½ðK  yÞ3=2  ðK  h2 Þ  ðK þ bÞðy  h2 Þ  ðy2  h2 Þ þ dx 2 3

h2 ðxÞ

From Eqs. (11), (22) and (23) we can write

 ðxÞ  ch  ðxÞ: Q ¼ q þ ch 1 2 The time-mean flow over a period T is defined as

Z

1 T

Q¼

for h2 6 y 6 k1 ;

T

Qdt ¼ q þ cd1 þ cd2 :

0

uðyÞ ¼ up ¼ constant for k1 6 y 6 k2 ;

Defining H and F as the dimensionless time-mean flows in the fixed and wave frames respectively by

H¼

Q ; cd1

F¼

q ; cd1

ð26Þ

H ¼ F þ 1 þ d; Z

ð44Þ

uðyÞ ¼ uþ ðyÞ ¼ 1  dp 1 2 4 pffiffiffi 3=2  ðh1  y2 Þ þ ðb  KÞðh1  yÞ  b½ðh1  KÞ  ðy  KÞ3=2  dx 2 3 for k2 6 y 6 h1 ;

we have

F¼

ð43Þ

ð25Þ

ð27Þ

h1 ðxÞ

uðyÞdy:

ð28Þ

h2 ðxÞ

Keeping in view the physical constraints of the problem, the boundary conditions in the fixed frame may be expressed mathematically as follows:

1 Þ ¼ 0 ¼ UðY ¼ h 2 Þ; UðY ¼ h  ðY ¼ k1 Þ ¼ s y ¼ s ðY ¼ k2 Þ; s

ð29Þ

UðY ¼ k1 Þ ¼ U p ¼ UðY ¼ k2 Þ;

ð31Þ

@T ¼ g1 ðT  T a Þ; @Y @T ¼ g2 ðT a  TÞ; k @Y

k

ð30Þ

1 ; C ¼ C 1 at Y ¼ h

ð32Þ

2 : C ¼ C 0 at Y ¼ h

ð33Þ

ð45Þ

where u and uþ represent the velocity in shear flow regions h2 6 y 6 k1 and k2 6 y 6 h1 respectively and up represents the velocity in the plug flow region k1 6 y 6 k2 which can be determined from Eq. (36). Here b ! 0 corresponds to the Newtonian fluid and a ¼ b; d ¼ 1 and / ¼ 0 in a symmetric channel case, the expression for axial velocity agrees with the expression obtained by Shapiro et al. [2]. The continuity condition (36) for velocity distribution at the interfaces y ¼ k1 and y ¼ k2 gives the relation

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2  3h1 þ 3h2  8 bK h1  K þ 8h2 K  h2  8 bK K  h2 pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  6h2 ðb þ KÞ þ h1 ð6b þ 8 b h1  K þ 6KÞ þ 8 bK K  k1 pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  8 bk1 K  k1 þ 8 bK k2  K  8 bk2 k2  K

 ð46Þ þ 3 ð2ðb þ KÞ  k1 Þk1 þ 2ðb  KÞk2 þ k22 ¼ 0:

uðy ¼ h1 Þ ¼ 1 ¼ uðy ¼ h2 Þ;

ð34Þ

The relations (41) and (46) constitute a system of equations in two unknowns k1 and k2 . We can determine k1 and k2 from these equa  2 tions with the help of the relation K ¼ k1 þk . 2 We can calculate the transverse velocity v from the continuity equation through

 sðy ¼ k1 Þ ¼ sy ¼ sðy ¼ k2 Þ; uðy ¼ k1 Þ ¼ up ¼ uðy ¼ k2 Þ; @h þ Bi1 h ¼ 0; r ¼ 1 at y ¼ h1 ; @y @h  Bi2 h ¼ 0; r ¼ 0 at y ¼ h2 ; @y

ð35Þ ð36Þ

@v @u ¼ : @x @y

ð37Þ

Defining the stream function as

Writing the above conditions through Eqs. (11) and (12), the dimensionless boundary conditions are

ð38Þ

where Bi1 ¼ g1 d1 =k and Bi2 ¼ g2 d1 =k are the Biot numbers at the upper and lower walls of channel respectively. 4. Solution methodology The solution of Eq. (13) through (35) is given by

dp s ¼  ðy  KÞ; dx

ð40Þ

Using Eqs. (35) and (39), we obtain

k2  k1 ¼ b; 2  

ð41Þ

dp dx



is the half width of the plug flow region.

ð48Þ F 2

 2F

and using the conditions w ¼ at y ¼ h1 and w ¼ at y ¼ h2 , we obtain the stream function in the three regions as below.

wðyÞ ¼ w ðyÞ ¼ y   2    dp y 1 y3 2  ðK þ bÞ  h2 y   h2 y dx 2 3 2   4 pffiffiffi 2 3=2 5=2  b ðK  yÞ þ ðK  h2 Þ y þ C 1 for h2 6 y 6 k1 ; 3 5

ð49Þ

Table 1 Variation of yield plane location k1 with phase difference of the wave when the amplitudes of the waves are same (a ¼ b ¼ 0:5; d ¼ 1) for different values of yield stress. /

where

b ¼ sy

dw ¼ udy  v dx;

ð39Þ

where

  k1 þ k2 : K¼ 2

ð47Þ

ð42Þ

0

p/6 p/3 p/2 2p/3 5p/6 p

sy 0.0

0.05

0.1

0.15

2.0

2.5

0.000 0.033 0.125 0.250 0.376 0.466 0.500

0.05 0.016 0.075 0.200 0.325 0.416 0.450

0.100 0.066 0.025 0.150 0.275 0.366 0.4

0.150 0.116 0.025 0.100 0.225 0.316 0.350

2.00 0.166 0.075 0.050 0.175 0.266 0.300

0.250 0.216 0.125 0.000 0.125 0.216 0.250

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T. Hayat et al. / Computers & Fluids 89 (2014) 242–253

Fig. 2. Variation in dp=dx for b when a ¼ 0:7; b ¼ 1:2; d ¼ 2; / ¼ p=2; H ¼ 1:2.

Fig. 6. Variation in Dpk versus H for / when a ¼ 0:7; b ¼ 1:2; d ¼ 2; b ¼ 0:1.

Fig. 3. Variation in dp=dx for / when a ¼ 0:7; b ¼ 1:2; d ¼ 2; b ¼ 0:1; H ¼ 1:2.

Fig. 7. Variation in Dpk versus H for a when b ¼ 1:2; d ¼ 2; / ¼ p=2; b ¼ 0:1.

Fig. 4. Variation in dp=dx for H when a ¼ 0:7; b ¼ 1:2; d ¼ 2; / ¼ p=2; b ¼ 0:2.

Fig. 8. Variation in Dpk versus H for b when a ¼ 0:5; d ¼ 2; / ¼ p=2; b ¼ 0:1.

wðyÞ ¼ wp ¼ up y þ C p

for k1 6 y 6 k2 ;

ð50Þ

     dp 1 2 y3 y2 þ ðb  KÞ h1 y  wðyÞ ¼ wþ ðyÞ ¼ y  h1 y  dx 2 3 2   4 pffiffiffi 2 3=2 5=2 þ C 2 for k2 6 y 6 h1 ; b ðh1  KÞ y  ðy  KÞ  3 5 ð51Þ where

( ) 2 3 2 F dp Kh2 h2 bh2 4 pffiffiffi 3=2 C1 ¼ þ h2 þ þ   bðK  h2 Þ ð2K þ 3h2 Þ ; 2 dx 15 2 3 2 Fig. 5. Variation in Dpk versus H for b when a ¼ 0:7; b ¼ 1:2; d ¼ 2; / ¼ p=2.

ð52Þ

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Fig. 11. Temperature profile hðyÞ for various values a ¼ 0:5; b ¼ 0:7; d ¼ 1:2; / ¼ p=2; H ¼ 1:2; x ¼ 0:5; Bi1 ¼ 8 2:0; Sr ¼ Du ¼ c ¼ 0:2.

of Sc when , Bi2 ¼ 10; Pr ¼

2 k1 ¼ k2 ¼ h1 þh and in this case the stream function agrees with 2 the expression given by Mishra and Rao [4], when b ¼ 0 and a ¼ b; d ¼ 1; / ¼ 0 (corresponding to the Newtonian fluid flow in a symmetric channel case) the stream function matches with that of Shapiro et al. [2]. The pressure gradient is obtained by using Eq. (28) as follows:

dp ðF þ h1  h2 Þ ¼ ; dx gðxÞ

ð55Þ

where

gðxÞ ¼ Fig. 9. (a) Longitudinal velocity uðyÞ for b when a ¼ 0:5; b ¼ 0:5, d ¼ 1; / ¼ 0; x ¼ 0:25 and H ¼ 1. (b) Longitudinal velocity uðyÞ for b when a ¼ 0:5; b ¼ 1:2, d ¼ 1:2; / ¼ 0; x ¼ 0:25 and H ¼ 1.

(

2 h2

k21

3=2

 ðK  h2 Þ

!

F dp  1 3 Cp ¼ þ h2  ðK þ bÞ þ ðk31  h2 Þ 2 dx 3 2 4 pffiffiffi 3=2 b½ðK  k1 Þ3=2 ð2K þ 3h2 Þ  ðK  h2 Þ ð2K þ 3h2 Þ ;  15

1 3 K 2 b 2 3 2 ðh  h2 þ k31  k32 Þ þ ðh2  h1 þ k22  k21 Þ þ ðh1 3 1 2 2 4 pffiffiffi 2 b½ðK  k1 Þ3=2 ð2K þ 3k1 Þ þ h2  k21  k22 Þ  15  ðk2  KÞ

3=2

ð2K þ 3h2 Þ þ ðh1  KÞ

3=2

ð3h1 þ 2KÞ

ð2K þ 3k2 Þ:

ð56Þ

The pressure rise per wavelength ðDPk Þ is

ð53Þ

DP k ¼

Z

k 0

( ) 3 2 F dp h1 h1 4 pffiffiffi 3=2 C 2 ¼ þ h1 þ þ ðb  KÞ  bðh1  KÞ ð3h1 þ 2KÞ : 2 dx 3 2 15 ð54Þ The expression for w in symmetric channel case agrees with that obtained by Mernone et al. [15]. As b ! 0 (Newtonian case) we have

dp dx: dx

ð57Þ

The solutions to Eqs. (15) and (16) corresponding to boundary conditions (37) and (38) are given by

hðyÞ ¼

1 ½DuPr csc hA3 fcðA1 ð1 þ Bi1 ðy  h2 ÞÞ ðA2 A4 cÞ

þA1 ð1 þ Bi2 ðy  h2 ÞÞ cosh A3 þ A2 ðBi1 ð1 þ Bi2 ðy  h2 ÞÞ sinh A3   ðh2  yÞA1 þ ðA1 ð2 þ Bi1 ðy  h1 Þ  Bi2 ðy  h2 ÞÞ þA4 sinh A2 þA1 ð2  Bi1 ðy  h1 Þ þ Bi2 ðy  h2 ÞÞ cosh A3     ðh1  yÞA1 ðh2  yÞA1 þA2 A4 sinh A3  sinhð Þ þ sinh c ; A2 A2 ð58Þ    ðh2  yÞA1 csc hA3 c sinh c A2     ðh1  yÞA1 ðh2  yÞA1 Þ þ sinh c ; þ sinh A3  sinhð A2 A2

rðyÞ ¼ 

1



where

Fig. 10. Temperature profile hðyÞ for various values a ¼ 0:5; b ¼ 0:7; d ¼ 1:2; / ¼ p=2; H ¼ 1:2; x ¼ 0:5; Bi1 ¼ 8, Bi2 ¼ 10; Sc ¼ Sr ¼ Du ¼ c ¼ 0:2.

of

Pr

when

A1 ¼

pffiffiffiffiffiffiffiffi cSc;

A3 ¼

ðh1  h2 ÞA1 ; A2

A2 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  DuPrScSr; A4 ¼ Bi1 þ Bi2 þ Bi1 Bi2 ðh1  h2 Þ:

ð59Þ

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Fig. 12. Temperature profile hðyÞ for various values a ¼ 0:5; b ¼ 0:7; d ¼ 1:2; / ¼ p=2; H ¼ 1:2; x ¼ 0:5; Bi1 ¼ 8; 2:0; Sc ¼ Sr ¼ Du ¼ c ¼ 0:2.

of c when Bi2 ¼ 10; Pr ¼

Fig. 15. Temperature profile hðyÞ for various values of Sr when a ¼ 0:5; b ¼ 0:7; d ¼ 1:2; / ¼ p=2; H ¼ 1:2; x ¼ 0:5; Bi1 ¼ 8, Bi2 ¼ 10; Pr ¼ 2:0; Sc ¼ Du ¼ c ¼ 0:2.

two yield planes determine the width of the core region. From Eq. (42) it is seen that the width 2b of the plug flow region depends on the shear stress sy and the pressure gradient and is independent of the asymmetry of the channel walls. Table 1 shows the location of the first yield plane k1 with phase angle / of the wave imposed on the lower wall. The location of the second yield plane k2 can be calculated from Table 1 and using the relation given by Eq. (41). It is observed from the table that as / increases for a fixed value of sy the location of yield plane (k1 ) shifts towards upper boundary and as sy increases, the width of the plug region increases. We can see that these results are in an excellent agreement with the results obtained by Rani and Sarojamma [17]. 5.2. Pumping characteristics Fig. 13. Temperature profile hðyÞ for various values of Bi1 when a ¼ 0:5; b ¼ 0:7; d ¼ 1:2; / ¼ p=2; H ¼ 1:2; x ¼ 0:5; Bi2 ¼ 10; Pr ¼ 2:0; Sc ¼ Sr ¼ Du ¼ c ¼ 0:2.

Fig. 14. Temperature profile hðyÞ for various values of Bi2 when a ¼ 0:5; b ¼ 0:7; d ¼ 1:2; / ¼ p=2; H ¼ 1:2; x ¼ 0:5; Bi1 ¼ 10; Pr ¼ 2:0; Sc ¼ Sr ¼ Du ¼ c ¼ 0:2.

5. Results and discussion 5.1. Yield plane locations The effect of yield stress is that the fluid exhibits a solid-like behavior (or a plug core) in a region where the magnitude of the shear stress is less than the value of the yield stress. The location of a point where the magnitude of actual shear stress is equal to the yield value, called a yield point and the locus of all such points is the yield plane. In view of the asymmetry in the two boundary walls the location of the two yield planes are calculated. These

In this subsection our aim is to analyze the pressure gradient dp=dx and pressure rise per wavelength Dpk for different embedding parameters in the present problem. Fig. 2 is plotted to serve the effects of Casson fluid parameter b on pressure gradient dp=dx. It is anticipated that the pressure gradient increases at the center of the channel while it decreases near the walls with an increase in Casson fluid parameter b. It is interesting to note that the assistance or resistance from pressure gradient for a Casson fluid (b – 0) is higher when compared to a Newtonian fluid (b ¼ 0). Fig. 3 depicts the variation in pressure gradient dp=dx for different values of phase difference /. We observe that by increasing the values of the phase difference / the pressure gradient dp=dx decreases. The effects of various values of flow rate H on pressure gradient dp=dx are shown in Fig. 4. It is observed that pressure gradient increases with an increase in flow rate H. Fig. 5 shows the variation in pressure rise per wavelength Dpk against flow rate H for various values of Casson fluid parameter b. For such purpose, the numerical integration has been carried out in Eq. (57) using ‘‘MATHEMATICA’’. It is seen that by increasing the value of b the pressure rise increases in the peristaltic pumping region and it decreases in the copumping region when b is increased. This means peristalsis works against greater pressure rise for a Casson fluid (b – 0) when compared to a Newtonian fluid (b ¼ 0). The opposite behavior is observed for different values of phase difference / in Fig. 6. That is pressure rise per wavelength Dpk decreases in peristaltic pumping region and it increases in copumping region when phase difference / is increased. Figs. 7 and 8 are plotted to see the effects of upper and lower wave amplitudes a and b on pressure rise per wavelength Dpk respectively. These Figs. show that pressure rise increases in the peristaltic pumping region with an increase in upper wave amplitude a as well as lower wave amplitude b. However it has opposite behavior in the copumping region.

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Fig. 16. Temperature profile hðyÞ for various values of Du when a ¼ 0:5; b ¼ 0:7; d ¼ 1:2; / ¼ p=2; H ¼ 1:2; x ¼ 0:5; Bi1 ¼ 8, Bi2 ¼ 10; Pr ¼ 2:0; Sc ¼ Sr ¼ c ¼ 0:2.

Fig. 17. Concentration profile rðyÞ for various values of Pr when a ¼ 0:5; b ¼ 0:7; d ¼ 1:2; / ¼ p=2; H ¼ 1:2; x ¼ 0:5; Sc ¼ 2:0; Sr ¼ Du ¼ c ¼ 0:2.

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Fig. 19. Concentration profile rðyÞ for various values of Sr when a ¼ 0:5; b ¼ 0:7; d ¼ 1:2; / ¼ p=2; H ¼ 1:2; x ¼ 0:5; Pr ¼ Sc ¼ 2:0; c ¼ Du ¼ 0:2.

Fig. 20. Concentration profile rðyÞ for various values of Du when a ¼ 0:5; b ¼ 0:7; d ¼ 1:2; / ¼ p=2; H ¼ 1:2; x ¼ 0:5; Pr ¼ Sc ¼ 2:0; c ¼ Sr ¼ 0:2.

Fig. 21. Concentration profile rðyÞ for various values of Sc when a ¼ 0:5; b ¼ 0:7; d ¼ 1:2; / ¼ p=2; H ¼ 1:2; x ¼ 0:5; Pr ¼ 2:0, c ¼ Sr ¼ Du ¼ 0:2. Fig. 18. Concentration profile rðyÞ for various values of c when a ¼ 0:5; b ¼ 0:7; d ¼ 1:2; / ¼ p=2; H ¼ 1:2; x ¼ 0:5; Pr ¼ Sc ¼ 2:0; Sr ¼ Du ¼ 0:2.

5.4. Temperature profile 5.3. Velocity distribution The axial velocity in a symmetric channel (a ¼ b ¼ 0:5; d ¼ 1; / ¼ 0) is shown in Fig. 9(a) and (b) shows the axial velocity in an asymmetric channel (a ¼ 0:5; b ¼ 1:2; d ¼ 1:2; / ¼ p=4). The velocity in a symmetric channel is seen to be symmetric while the profiles are skewed towards lower boundary in an asymmetric channel. As the yield stress increases we notice that there is a reduction in the magnitude of the velocity and plug flow dominates over the cross-section.

This subsection deals with the effects of various emerging parameters on temperature field hðyÞ. It is noteworthy that the temperature profile hðyÞ increases when there is an increase in the Prandtl Pr and Schmidt numbers Sc (see Figs. 10 and 11). Effects of chemical reaction parameter c on temperature profile hðyÞ are shown in Fig. 12. It is observed that by increasing the chemical reaction parameter c the temperature profile increases. Here we have considered c > 0 which represents the destructive chemical reaction process. In Figs. 13 and 14, the effects of the Biot numbers Bi1 and Bi2 on temperature profile hðyÞ are observed. We see that by

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Fig. 22. Streamlines for different values of b (panels ðaÞ; ðcÞ and ðeÞ for / ¼ 0 and panels ðbÞ; ðdÞ and ðf Þ for / ¼ p=4).

increasing the values of Bi1 the temperature decreases near the upper wall while it has no significant effect near the lower wall of channel. Also temperature decreases near the lower wall of the channel when there is an increase in Bi2 while it has no effect near the upper wall of channel. Here we have considered the nonuniform temperature fields within the fluid (i.e., Biot numbers are much greater than 1) because problems involving small Biot numbers are thermally simple due to the uniform temperature fields within the material. Effects of Soret and Dufour on temperature profile hðyÞ are depicted in Figs. 15 and 16. We illustrate that by

increasing the value of Soret number Sr the temperature profile gradually decreases (see Fig. 15) while it increases with an increase in Dufour number Du (see Fig. 16). Physically we can observe that the diffusion-thermo or Dufour effect describes the heat flux created when a chemical system is under a concentration gradient. These effects depend upon thermal diffusion which is generally very small but can be sometimes significant when the participating species are of widely differing molecular weights. Mass diffusion occurs if the species are initially distributed unevenly i.e., when a concentration gradient exists. A temperature gradient can also

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Fig. 23. Streamlines for different values of H (panels ðaÞ; ðcÞ and ðeÞ for / ¼ 0 and panels ðbÞ; ðdÞ and ðf Þ for / ¼ p=4).

work as a driving force for mass diffusion called thermo-diffusion or Soret effects. Therefore the higher the temperature gradient, the larger the Soret effects. 5.5. Concentration profile This subsection deals with the variation in the concentration profile rðyÞ for different values of embedding parameters in the problem. Here it is noted that by increasing the values of Prandtl number Pr, chemical reaction parameter c, Soret Sr and Dufour Du numbers and Schimdt number Sc concentration profile rðyÞ de-

creases (see Figs. 17–21). Schmidt number Sc is a dimensionless number defined as the ratio of momentum diffusivity (viscosity) and mass diffusivity and is used to characterize fluid flows in which there are simultaneous momentum and mass diffusion convection processes. The effect of chemical reaction parameter c is very important in concentration field rðyÞ. Chemical reaction increases the rate of interfacial mass transfer. Reaction reduces the local concentration, thus increases its concentration gradient and its flux when we have constructive chemical reaction. In our problem we have considered c > 0 which corresponds to the destructive chemical reaction.

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5.6. Streamlines and trapping The shape of the streamlines is generally the same as that of the walls in the wave frame of reference. However some of the streamlines split and enclose a bolus under certain conditions which moves as a whole with the wave. This is known as trapping. Figs. 22 and 23 are sketched to see the effects of Casson fluid parameter b and volume flow rate H on trapping. The panels on left side show the trapping phenomenon for the case of symmetric channel (a ¼ b ¼ 0:5; d ¼ 1; / ¼ 0) and the panels on right side show the trapping for the case of asymmetric channel (a ¼ 0:5; b ¼ 0:7, d ¼ 1:2; / ¼ p=4). From Fig. 22 it is revealed that trapped bolus exists and decreases in size when there is an increase in the Casson fluid parameter b. In this figure b ¼ 0:0 (panels ðaÞ and ðbÞ), b ¼ 0:2 (panels ðcÞ and ðdÞ) and b ¼ 0:3 (panels ðeÞ and ðf Þ). Fig. 23 shows that the trapped bolus increases by increasing the value of volume flow rate H. For this figure H ¼ 1:2 (panels ðaÞ and ðbÞ), H ¼ 1:5 (panels ðcÞ and ðdÞ) and H ¼ 1:7 (panels ðeÞ and ðf Þ). A comparative study of these panels reveals that the trapped bolus is symmetric with respect to the channel for / ¼ 0 (left panel). On the other hand for / ¼ p=4 the trapped bolus shifts towards left (lower wall) because of the asymmetry of channel (right panel).

6. Conclusions This article addresses the Soret and Dufour effects in peristaltic transport of Casson fluid in an asymmetric channel with convective conditions. The conducted study leads to the following observations:  Due to asymmetry in the channel, the yield stress sy of the fluid tends to form two yield planes in the plug core region. The yield planes are found to be located symmetrically when channel is symmetric and they are skewed towards the boundary wall with higher amplitude (or phase difference) in an asymmetric channel.  The magnitude of the pressure gradient dp=dx increases by increasing the values of Casson fluid parameter (b). Also assistance or resistance from pressure gradient for a Casson fluid (b – 0) is higher than that of a Newtonian fluid (b ¼ 0).  Pressure rise per wavelength Dpk increases in peristaltic pumping region and it decreases in copumping region when there is an increase in the Casson fluid parameter (b).  By increasing the values of Casson fluid parameter b the magnitude of velocity increases at the center of the channel and it decreases near the channel walls. The velocity in a symmetric channel (/ ¼ 0) is seen to be symmetric while the profiles are skewed towards lower boundary in an asymmetric channel (/ ¼ p=4).  There is a reduction in the magnitude of the velocity uðyÞ and plug flow dominates over the cross-section when b increases.  By increasing the values of Prandtl (Pr), Schimdt (Sc) and Soret (Sr) numbers, temperature profile hðyÞ increases. On the other hand temperature profile hðyÞ decreases when Biot numbers (Bi1 ; Bi2 ), chemical reaction parameter (c) and Dufour number (Du) are increased.  Concentration profile rðyÞ is a decreasing function of Prandtl (Pr), Soret (Sr), Dufour (Du), Schimdt (Sc) and chemical reaction (c) parameters.  Trapping decreases with an increase in Casson fluid parameter b while it increases by increasing the volume flow rate H. The present material provides a basis for future interesting studies on the topic. Few such works include the relevant studies

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