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Mixed convective peristaltic transport of Carreau-Yasuda nanofluid in a tapered asymmetric channel
Mixed convective peristaltic transport of Carreau-Yasuda nanofluid in a tapered asymmetric channel T. Hayat, Rija Iqbal, A. Tanveer, A. Alsaedi PII: DOI: Reference:
S0167-7322(16)31215-6 doi: 10.1016/j.molliq.2016.08.003 MOLLIQ 6161
To appear in:
Journal of Molecular Liquids
Received date: Accepted date:
16 May 2016 1 August 2016
Please cite this article as: T. Hayat, Rija Iqbal, A. Tanveer, A. Alsaedi, Mixed convective peristaltic transport of Carreau-Yasuda nanofluid in a tapered asymmetric channel, Journal of Molecular Liquids (2016), doi: 10.1016/j.molliq.2016.08.003
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ACCEPTED MANUSCRIPT
Mixed convective peristaltic transport of Carreau-Yasuda nanoßuid in a tapered asymmetric channel d
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T. Hayatd>e , Rija Iqbald , Anum Tanveerd>1 and A. Alsaedie Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan
e
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Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of
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Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
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Abstract: Here modeling and analysis for mixed convective peristaltic ßow of CarreauYasuda nanoßuid have been carried out. The asymmetric channel walls are tapered. E ects
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of Brownian motion and thermophoresis are taken into consideration. The ßow equations for the relevant problem are Þrst constructed and then non-dimensionalized. The resulting non-linear coupled systems subject to lubrication approach are solved. Graphs reßecting
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the variations of pertinent parameters on the velocity, temperature, nanoparticle volume
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fraction are examined. Shear stress distribution and pumping and trapping phenomena are discussed. Results indicate that amplitude of shear stress decreases for larger Weissenberg
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number and power-law index whereas it enhances through viscosity ratio parameter. Keywords: Peristaltic ßow, Carreau-Yasuda nanoßuid, tapered asymmetric channel, mixed convection.
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Introduction
The study of nanoßuids has gained much popularity amongst the recent researchers. It is due to the fact that thermal conductivity of nanoßuids has a vital role in heat transfer phenomenon. Thermal conductivity of traditional heat transfer ßuids like oil, water and ethylene glycol mixture are insu!cient to meet present day requirements. Thus various techniques have been suggested for thermal conductivity enhancement of traditional ßuids. Amongst such techniques there is one through suspension of nano-particles in the traditional ßuids. The word "nanoßuid" was Þrst used by Choi [1] which refers to a liquid suspension containing nanometer-sized particles (having diameters less than 50 nm). Due to their enhanced thermal properties, the nanoßuids can be used as coolants in heat transfer equipment 1
Corresponding author. Tel.: + 92 51 90642172.
e-mail address: [email protected]
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ACCEPTED MANUSCRIPT such as heat exchangers, electronic cooling system and radiators. Nanoßuids have numerous biomedical applications as well. Cancer diagnosis, drug delivery systems, neuro electronic interfaces and photodynamic therapy are some examples in this direction. Nanoßuids can be
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used in advanced nuclear systems due to their signiÞcant feature of enhancing the thermal conductivity. Masuda et al. [2] Þrst worked on this application of nanoßuids. Buongiorno
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[3> 4] presented a systematic model for the study of convective transport in nanoßuids with
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combined e ects of Brownian motion and thermophoresis. He revealed that the only important baseßuid/nanoparticle slip mechanisms are the Brownian motion and thermophoresis.
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Study of peristalsis of nanoßuids has also gained signiÞcant interest of the researchers in recent times due to its wide ranging applications. Few attempts in this direction can be
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consulted by the studies [5 17].
Recently the peristaltic phenomenon has become a topic of great interest for the researchers in view of its widespread applications in physiological, medical and industrial
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processes. Its occurrence can be found in many common physiological processes such as
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in the urine passage from kidney to bladder, food digestion, chyme movement in gastrointestinal tract, blood circulation in small blood vessels and also in the human reproductive
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systems. In biomedical Þeld the dialysis and heart lung machines worked on the peristaltic principle. Roller and Þnger pumps and waste management pumps in nuclear industry are applications of peristalsis in industrial area. Several researchers have worked for peristaltic motion of viscous and non-Newtonian materials subject to di erent aspects and conditions. Much focus in the literature has been made on the peristaltic transport of viscous ßuids. Even several bioßuids have been considered as viscous ßuids by many workers. This in not adequate in the sense that microßuidic devices have been usually employed to analyze bioßuids, which are often solutions of long chain molecules imparting a non-Newtonian behavior characterized by variable viscosity, memory e ects, normal stress e ects, yield stress and hysteresis of ßuid properties. Also, the non-Newtonian materials are involved in many processes including food mixing, chyme movement in intestine, blood ßow, ßow of liquid metals, nuclear slurries and alloys. Few studies in this regard have been mentioned (see refs. [18 23]). Study of convective ßow in channel has also gained popularity due to its numerous industrial and technological applications including transport processes, solar collectors, electronic equipment cooled by fans, nuclear reactors cooled during emergency shutdown, transistors 2
ACCEPTED MANUSCRIPT and biomedical engineering. Keeping in mind the aforementioned applications, several researchers have made analysis for mixed convective peristaltic transport (see refs. [24 26] and several studies therein). Another important aspect which has not been yet explored
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properly in peristalsis is the non-uniform (tapered) ßow conÞguration. This conÞguration occurs in most practical applications such as in many physiological body organs, small blood
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vessels, intestines, lymphatic vessels and ductus a erents [27 29].
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It is remarkable that very little focus has been made on the study of peristaltic ßow of Carreau-Yasuda (CY) ßuid. The importance of this model is that it consists of Þve
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constants predicting the rheological characteristics of non-Newtonian ßuids through shear thinning/thickening parameters. That is why CY model is preferred over the power-law
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model. Few studies dealing with peristalsis of Carreau-Yasuda ßuid have been mentioned in the refs. [30 33].
To our knowledge, not much has been said yet about the peristalsis of Carreau-
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Yasuda ßuid. Such consideration further narrowed down when peristalsis of Carreau-Yasuda
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nanoßuid is examined [34]. Therefore main focus here is to make some advancement in this direction. SpeciÞcally we have intention to investigate the following salient features
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in the peristalsis of Carreau-Yasuda nanoßuid. Firstly to consider the mixed convective e ect. Secondly to analyze the tapered channel properties. Thirdly to examine Brownian motion and thermophoresis in tapered channel ßow. Fourth to predict shear thinning and shear thickening properties of ßuid in a tapered ßow conÞguration. The continuity, momentum, energy and concentration equations comprising mixed convection, Brownian motion and thermophoresis e ects have been constructed here. The relevant equations have been simpliÞed using low Reynolds number and long wavelength assumptions. The solutions for the stream function, velocity, temperature and concentration are calculated. The e ects of involved parameters on the velocity, temperature, concentration, pressure rise, pumping and streamlines have been addressed by the graphical illustrations. Further the shear stress has been graphically examined.
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Mathematical formulation
The ßow of an incompressible Carreau-Yasuda nanoßuid in a tapered asymmetric channel is considered. Brownian motion and thermophoresis e ects are present. The peristaltic waves 3
ACCEPTED MANUSCRIPT ¯ are travelling along the [-axis with speed f. Here \¯ -axis is taken perpendicular to the ¯ 2 represent the positions ¯ 1 and \¯ = k ¯ [axis. The wall conÞguration is such that \¯ = k of left and right walls of the channel. The amplitudes and phases of peristaltic waves are
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di erent. Geometry of the wall surfaces is deÞned below: ¸ 2 ¯ ¯k1 ([> ¯ ¯ w¯) = g p ¯ ¯ ([ fw) + ! > ¯ [ e1 sin ¸ 2 ¯ ¯k2 ([> ¯ ¯ w¯) = g + p ¯ ¯ [ + e2 sin ([ fw¯) =
(1) (2)
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Here w¯ denotes the dimensional time, ¯e1 and ¯e2 the waves amplitudes along the left and right walls, p(¿ ¯ 1) the non-uniform parameter of the tapered channel, the wavelength, g the half-width of channel and ! 5 [0> ] the phase di erence. Here the amplitudes ¯el (l = 1> 2)> g
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and ! satisfy the condition
(3)
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¯e2 + ¯e2 + 2¯e1¯e2 g cos(!) 6 (2g)2 = 1 2
Fig. 1. Schematic diagram of the tapered asymmetric channel.
The fundamental equations governing the present ßow of an incompressible Carreau-Yasuda nanoßuid are
¯ C Y¯ CX + = 0> ¯ C[ C \¯
(4)
¯ ¯¸ CX CX C S¯ C ¯ C ¯ C X¯ ¯ ¯ + X + Y = + ( V (V ¯ ¯ ) + (1 F0 ) i j(W W0 ) i ¯[ ¯) + [ ¯ ¯ ¯ ¯ C w¯ C[ C\ C[ C[ C \¯ [ \ +( s i )j(F F0 )> (5) 4
ACCEPTED MANUSCRIPT ¸ ¯ C Y¯ C S¯ C CY C Y¯ C ¯ ¯ + Y = + ¯ (V¯\¯ [¯ ) + ¯ (V¯\¯ \¯ )> i + X 0 ¯ ¯ ¯ Cw C[ C\ C\ C[ C\
(6)
¯ ¸ 2¯ ¸ µ ¯ ¯ ¶ CW C W¯ C W C 2 W¯ CF CW C F¯ C W¯ C W¯ 0 ¯ ¯ ( f )i +X ¯ +Y ¯ = ¯ 2 + C \¯ 2 + ( f )s GE C [ ¯ C[ ¯ + C \¯ C \¯ C w¯ C[ C\ C[ "µ ¶2 µ ¯ ¶2 # ¯ G C W CW W +( f0 )s + > (7) ¯ Wp C[ C \¯
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0
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¯ µ 2¯ ¶ 2¯ ¸ 2¯ 2¯ ¯ ¯¸ CF C F C F F G C W W C F C C W ¯ ¯ +X (8) ¯ + Y C \¯ = GE C [ ¯ 2 + C \¯ 2 + Wp C [ ¯ 2 + C \¯ 2 = C w¯ C[ Here X¯ and Y¯ are the velocity components in the axial and transverse directions respectively,
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S¯ the pressure, V¯[¯ [¯ > V¯[¯ \¯ > V¯\¯ [¯ > V¯\¯ \¯ the components of extra stress tensor, Wp the ßuid mean temperature, i the density of ßuid, s the density of nanoparticles, f0 the volumetric volume
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expansion coe!cient, the thermal conductivity of ßuid, W¯ and F¯ the ßuid temperature and nanoparticle concentration, GE the Brownian di usion coe!cient and GW the thermophoretic
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di usion coe!cient.
¯ = (ˆ S )A1 >
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Expression of extra stress tensor for Carreau-Yasuda ßuid is [33] :
where A1 represents the Þrst Rivlin-Erickson tensor and the apparent viscosity (ˆ ) is given by
q31 ¯ + (grad V) ¯ W > (ˆ ) = " + (0 " )[1 + (Kˆ )d ] d > A1 = grad V
in which
p 1 ¯ + ( grad V) ¯ W ]= 2wu(D2 )> D = [grad(V) 2 ¯ \¯ > w¯)> Y¯ ([> ¯ \¯ > w¯)> 0] is the velocity Þeld, " and 0 the inÞnite and zero ¯ = [X¯ ([> Here V ˆ =
shear-rate viscosities, d and K the non-dimensional and material parameters and q the nondimensional power law index for Carreau-Yasuda ßuid. The above mentioned ßuid model describes the pseudoplastic ßow with asymptotic viscosities at zero and inÞnite shear rates. The model also predicts the results of shear thinning, shear thickening and Newtonian ßuids for q ? 1> q A 1 and q = 1. Also for d = 2 the model reduces to Carreau ßuid model. Here we have taken the Yasuda parameter d = 1 to examine the rheological aspects of non-Newtonian ßuids.
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ACCEPTED MANUSCRIPT DeÞning the non-dimensional quantities through the deÞnitions below [35]: ¯1 ¯2 ¯ k k [ \¯ fw0 X¯ Y¯ g > | = > w = > x = > y = > = > k1 = > k2 = > g f f g g ¯e1 i fg W¯ W0 g2 S¯ g ¯ F¯ F0 X= > s= > vlm = V¯~¯m > l = > Re = > e1 = > W1 W0 f f F1 F0 g 0 ¯e2 fi p ¯ GE (F1 F0 ) GW (W1 W0 ) > Pr = > Qe = > Qw = > e2 = > p = g g Wp (1 F0 ) i jg2 (W1 W0 ) ( s i )jg2 (F1 F0 ) Jw = > Jq = > Vf = f f GE
(9)
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{=
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the continuity equation is identically satisÞed and Eqs. (5) (8) become ¸ C Cx Cx Cs C Cx + x +y = + (v{{ ) + (v{| ) + Jw X + Jq l> Re Cw C{ C| C{ C{ C| ¸ Cy Cx Cs C C Cy + x +y = + 2 (v{| ) + (v|| )> Re Cw C{ C| C| C{ C|
(10) (11)
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¸ ¸ ¸ 2 CX CX 1 Cl CX C 2X CX 2C X 2 Cl CX + x +y = + + Re + Qe Cw C{ C| Pr C{2 C{2 C{ C{ C| C| " µ ¶2 µ ¶2 # CX CX > (12) +Qw 2 + C{ C| ¸ ¸ 2 Cl C 2 l Qw 2 C 2 C 2 Cl Cl 2C l + x +y = + 2 + + Re Vf = (13) Cw C{ C| C{2 C| Qe C{2 C| 2
Here the non-dimensional parameters signify the following quantities: w the time, { and | the axial and transverse coordinates, x and y the axial and transverse components of velocity, p the non-uniform parameter, e1 and e2 the amplitudes of left and right walls, the ratio of e ective heat capacity of nanoparticle material to heat capacity of the ßuid, X the temperature, l the concentration, s the pressure, Re the Reynolds number, Pr the Prandtl number, Qe the Brownian motion parameter, Qw the thermophoresis parameter, Vf the Schmidt number and Jw and Jq the local temperature and local nanoparticle Grashof numbers.
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Volume ßow rate
¯ is deÞned as The time-averaged ßow at a Þxed position [ Z W 1 ¯= T gw¯ T W 0
(14)
we Þnd ¯ = t + e1 f sin[ 2 ([ fw0 ) + !] + e2 f sin[ 2 ([ fw0 )]> T 6
(15)
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Z
K2
x¯(¯ {> |¯)g¯ |=
K1
is the dimensional volume ßow rate in the wave frame.
Using the deÞnitions of non-
¯ T t > = > fg fg
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we Þnd that Eq. (16) becomes [7> 28> 35] :
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I =
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dimensional mean ßows I in the laboratory frame and X in the wave frame
in which
(16)
k2
x g|=
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I =
Z
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I ({> w) = + d sin[2 (¯ { w) + !] + e sin[2 (¯ { w)]>
(17)
k1
Using the deÞnitions of stream function (x =
C# >y C|
= C# ), Eqs. (10) (13) subject to C{
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long wavelength and low Reynolds number approximation become:
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Cs C = (v{| ) + Jw X + Jq l> C{ C|
(18)
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Cs = 0> C| µ ¶ µ ¶2 Cl CX C2X CX + Qw Pr + Qe Pr = 0> C| 2 C| C| C| C 2 l Qw C 2 X + = 0= C| 2 Qe C| 2
" 0
is the viscosity ratio parameter and Z h =
(20) (21)
The non-dimensional form of the stress tensor is given by ¸ C2# C 2# v{| = 1 + (1 )(q 1)Z h 2 > C| 2 C| where =
(19)
Kf g
(22) the Weissenberg number
respectively. Eliminating pressure from Eqs. (19) and (20), we get the following systems C2 (v{| ) + Jw X + Jq l = 0> C| 2 µ ¶ µ ¶2 Cl CX C2X CX + Qe Pr + Qw Pr = 0> C| 2 C| C| C| C 2 l Qw C 2 X + = 0> C| 2 Qe C| 2 7
(23)
(24) (25)
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¸ C 2# C2# 1 + (1 )(q 1)Z h 2 > = C| 2 C|
(26)
subject to the boundary conditions
I C# > = 0> X = 1 and l = 1 at | = k2 = 1 + p{ + e2 sin[2 ({ w)]= 2 C|
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#=
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I C# #= > = 0> X = 0 and l = 0 at | = k1 = 1 p{ e1 sin[2 ({ w) + !]> (27) 2 C| (28)
Solution technique
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The dimensionless pressure rise per wavelength is given by Z 1Z 1µ ¶ Cs {s = g{gw. C{ |=0 0 0
Eqs. (25) and (26) can be solved for the exact solutions subject to the appropriate boundary
are
Frvk[Dk1 ] + Vlqk[Dk1 ] Frvk[Dk1 ] Frvk[Dk2 ] + Vlqk[Dk1 ] Vlqk[Dk2 ] Frvk[D|] + Vlqk[D|] > Frvk[Dk1 ] Frvk[Dk2 ] + Vlqk[Dk1 ] Vlqk[Dk2 ]
(29)
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X =
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conditions. Thus the exact solutions for the temperature and nanoparticle volume fraction
µ ¶ Frvk[Dk1 ] + Vlqk[Dk1 ] (Qe + Qw)(k1 + |) Qw l = (k1 + k2 )Qe Qe Frvk[Dk1 ] Frvk[Dk2 ] + Vlqk[Dk1 ] Vlqk[Dk2 ] µ ¶ Qw Frvk[D|] + Vlqk[D|] + = (30) Qe Frvk[Dk1 ] Frvk[Dk2 ] + Vlqk[Dk1 ] Vlqk[Dk2 ]
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Perturbation solution
Since the problems involved are nonlinear. It seems di!cult to obtain the exact solutions. Thus we are interested to derive the approximate analytical solutions by perturbation technique. Perturbation technique is better than other techniques as it can be used to solve any set of complicated problems containing small parameter. The expressions developed by perturbation method are not exact but these can lead to accurate results when the expansion parameter is small. Since Eq. (24) is non-linear in Weissenberg number and hence cannot be solved to obtain an exact solution. Therefore we apply the perturbation technique to Þnd
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ACCEPTED MANUSCRIPT approximate solutions for small Weissenberg number Z h. Thus we perturbed #> s> I and v|{ about Z h. Moreover the series solution are retained only up to Þrst order i.e. $ = $0 + Z h$ 1 + R(Z h2 )
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in which $ = #> s> I and v|{ respectively. Zeroth order systems and solutions
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3.1.1
(31)
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The corresponding systems at zero order are ¸ DFrvk[D|] + DVlqk[D|] C2 (v0{| ) Jw + Jq C| 2 Frvk[Dk1 ] Frvk[Dk2 ] + Vlqk[Dk1 ] Vlqk[Dk2 ] ¸ (Q e + Qw) Qw DFrvk[D|] + DVlqk[D|] + = 0 >(32) (k1 + k2 )Qe Qe Frvk[Dk1 ] Frvk[Dk2 ] + Vlqk[Dk1 ] Vlqk[Dk2 ]
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¸ Frvk[Dk1 ] + Vlqk[Dk1 ] Cs0 C Jq = (v0{| ) + Jw + C{ C| Frvk[Dk1 ] Frvk[Dk2 ] + Vlqk[Dk1 ] Vlqk[Dk2 ] Qe ¸ (Qe + Qw)(k1 |) Qw(Frvk[Dk1] Frvk[D|] + Vlqk[Dk1] Vlqk[D|]) (33) > (k1 k2 ) Frvk[Dk1 ] Frvk[Dk2 ] + Vlqk[Dk1 ] Vlqk[Dk2 ] v0{| =
C 2 #0 > C| 2
I0 C#0 > = 0 at | = k1 > 2 C| I0 C#0 #0 = > = 0 at | = k2 = 2 C|
#0 =
(34) (35) (36)
The solutions at zeroth order are
1 Jq (Qe + Qw)| 4 ( + 24Qe k1 k2 12(h3(1@2)D(k1 +k2 32|) (Jw Qe Jq Qw)Fvfk[ 12 D(k1 k2 )]) )> D3
#0 = F1 + F2 | + F3 | 2 + F4 | 3 +
(37)
µ ¶ gs0 ((hDk1 hD| )Jw ) Jq (hDk1 + hD| )Qw (Qe + Qw)(k1 |) 1 = 6F4 + + + + Dk Dk Dk Dk 1 2 1 2 g{ h h Qe h h k1 k2 2Qe µ ¶ 2Jq (Qe + Qw)| 1 + h3(1@2)D(k1 +k2 32|) (Jw Qe Jq Qw)Fvfk[ D(k1 k2 )] > (38) (k1 k2 ) 2
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First order systems and solutions
The corresponding systems at Þrst order are
(40) (41)
I1 C#1 > = 0 at | = k1 > 2 C|
(42)
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C 2 #1 C 2 #0 + (1 )(q 1) > C| 2 C| 2
#1 = #1 =
(39)
I1 C#1 > = 0 at | = k2 = 2 C|
The solutions at Þrst order are
(43)
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v1{| =
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Cs1 C = (v1{| )> C{ C|
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C2 (v1{| ) = 0> C| 2
h3D(k1 +k2 ) (1 + q) (30Dh2D| (k1 k2 )2 (Jw Qe Jq Qw)2 5 2 2 480D (k1 k2 ) Qe 5 2Dk1 4 +D h | (20(k1 k2 )Qe(18F42 (k1 k2 )Qe + F3 Jq (Qe + Q w)) + 36F4 Jq (k1 k2 )
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#1 = E1 + |(E2 + |(E3 + E4 |)) +
Qe(Qe + Qw)| + J2q (Qe + Qw)2 | 2 ) + D5 h2Dk2 | 4 (20(k1 k2 )Qe(18(F42 (k1 k2)Qe
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+F3 Jq (Qe + Qw)) + 36F4 Jq (k1 k2 )Qe(Qe + Qw)| + J2q (Qe + Qw)2 | 2 ) 2D5 hD(k1 +k2 ) (| 4 (20(k1 k2 )Qe(18F42 (k1 k2 )Qe + F3 Jq (Qe + Qw)) + 36F4 Jq (k1 k2 ) Qe(Qe + Qw)| + J2q (Qe + Qw)2 | 2 ) + 120hD(k1 +|) (k1 k2 )(Jw Qe Jq Qw)(4D(k1 k2 ) Qe(DF3 6F4 + 3DF4 |) + Jq 120hD(k2 +|) (k1 k2 ) + (Qe + Qw)(6 + D|(4 + D|))) +(Jw Qe + Jq Qw)(4D(k1 k2 )Qe(DF3 6F4 + 3DF4 |) + Jq (Qe + Qw) 1 (6 + D|(4 + D|))))(1 + )Fvfk[ D(k1 k2 )]2 > 2
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(44)
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gs1 1 1 = 6E4 + 2(q 1)(1 )(6F4 + Fvfk[ D(k1 k2 )](2Jq (Qe + Qw)| g{ 2(k1 k2 )Qe 2 Fvfk[ 12 D(k1 k2 )] 1 Vlqk[ D(k1 k2 )] + E5 (Frvk[D|] + Vlqk[D|])))(2F3 + 6F4 | + 2 2(k1 k2 )Qe 1 E5 (Frvk[D|] + Vlqk[D|]) (Jq (Qe + Qw)| 2 Vlqk[ D(k1 k2 )] + ) + (E6 (Frvk[2D|] 2 D +Vlqk[2D|])(D|(E7 + 18F4Jq (k1 k2 )Qe(Qe + Qw)| + J2q (Qe + Qw)2 | 2 ) 1 1 (Frvk[ D(k1 3k2 )] + Frvk[ D(3k1 + k2 )]) + (E8 E9 + E10 + E12 )| 2 2 1 +| 2 (E11 + E13 ) | 3 (E14 + E15 + E16 ) + E17 (Frvk[ D(k1 + k2 4|)] 2 1 1 +Vlqk[ D(k1 + k2 4|)]) + (E18 + |E19 + | 2 E20 )(Frvk[ D(k1 k2 2|)] 2 2 1 1 +Vlqk[ D(k1 k2 + 2|)]) + (E21 + |E22 + | 2 E23 )(Frvk[ D(k1 k2 + 2|)] 2 2 1 1 +Vlqk[ D(k1 k2 + 2|)]) + (|E24 + | 2 E25 + | 3 E26 )(Vlqk[ D(3k1 + k2 )] 2 2 1 1 1 +Vlqk[ D(k1 3k2 )] + Vlqk[ D(k1 + k2 )])))@(4D(k1 k2 )2 Qe2 (Frvk[ D 2 2 2 1 (k1 + k2 + 4|)] + Vlqk[ D(k1 + k2 + 4|)]))= (45) 2
v{|
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The dimensionless shear stress at the left wall of the channel is reduced as follows: ¸ C2# C2# = 1 + (1 )(q 1) 2 C| 2 C| 1 = (4D(k1 k2 )Qe((k1 k2 )(Jw Qe Jq Qw) + D(4F3 (k1 k2 ) 8D2 (k1 k2 )2 Qe2 Qe + k1 (12F4 (k1 k2 )Qe + Jq k1 (Qe + Qw)) + 4(E3 + 3E4 k1 )(k1 k2 )QeZ h)) +4(k1 k2 )(Jw Q e Jq Qw)((k1 k2 )(1 + q)(Jw Qe Jq Qw)Z h(1 + ) + D(k1 Qe (1 + 4(F3 3F4 k2 )(1 + q)Z h(1 + )) + E29 + 2(1 + q)Z h(1 + )E27 + D2 k21 E28 1 1 +(k1 k2 )2 (Jw Qe Jq Qw)2 Fvfk[ D(k1 k2 )]2 ))(1 + (1 q)Z h 2 2 8D (k1 k2 )2 Qe2 (1 )(4D(k1 k2 )Qe((k1 k2 )(Jw Qe Jq Qw) + D(4F3 (k1 k2 )Qe + k1 (12F4 (k1 k2 ) Qe + Jq k1 (Qe + Qw)) + 4(E3 + 3E4 k1 )(k1 k2 )QeZ h)) + 4(k1 k2 )(Jw Qe Jq Qw)((k1 k2 )(1 + q)(Jw Qe Jq Qw)Z h(1 + ) + D(k1 Qe(1 + 4(F3 3F4 k2 )(1 + q)Z h(1 + )) +E29 + 2(1 + q)Z h(1 + )(2(k1 k2 )2 (Jw Qe Jq Qw)2 + 2D(k1 k2 )(Jw Qe Jq Qw) (4F3 (k1 k2 )Qe + k1 (12F4 (k1 k2 )Qe + Jq k1 (Qe + Qw))) + D2 k21 (8F3 Jq (k1 k2 )Qe (Qe + Qw) + (12F4 (k1 k2 )Qe + Jq k1 (Qe + Qw))2 ) + ((k1 k2 )2 )(Jw Q e Jq Qw)2 1 Fvfk[ D(k1 k2 )]2 )))> 2 11
(46)
ACCEPTED MANUSCRIPT where the involved quantities have been described in Appendix.
Graphical results
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Figs. 2 38 have been sketched to analyze the physical signiÞcance of pertinent ßow pa-
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rameters on the peristaltic transport of Carreau-Yasuda nanoßuid in a tapered asymmetric
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channel. Particularly the plots of pressure rise, velocity, temperature, concentration and
4.1
Pumping characteristics
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shear stress have been drawn and discussed physically.
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Figs. 2 9 are sketched to analyze the impacts of Weissenberg number, power-law index, local nanoparticle and local temperature Grashof numbers, left wall wave amplitude> non-
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uniform parameter, thermophoresis parameter and Brownian motion parameter on average
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pressure rise {s= The plots have been constructed so that the upper right-hand quadrant (L) represents the peristaltic pumping region, where X A 0 (positive pumping) and {s A 0
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(adverse pressure gradient). Quadrant (LL) in which X A 0 (positive pumping) and {s ? 0 (favorable pressure gradient) is marked as the co-pumping region (or augmented ßow). Quadrant (LLL) where {s ? 0> X ? 0 is the region where there is no ßow. Quadrant (LY ) represents the region of backward pumping (or retrograde region) where {s A 0 and X ? 0. Fig. 2 illustrates the impact of Z h on {s. It is noticed that in the peristaltic pumping region, the pumping rate decreases with increasing Z h while opposite behavior is observed in the co-pumping region. It is observed through Fig. 3 that the pumping rate increases with higher values of q in the peristaltic pumping region which indicates that the pumping rate increases as the behavior of ßuid changes from shear thinning (q ? 1) to Newtonian (q = 1) and to shear thickening (q A 1)= Particularly for q = 1 in Eq. (22) > the results for viscous nanoßuid in a tapered channel [28] are recovered. The pumping rate enhances with escalating values of Jq and Jw in the peristaltic pumping and backward (retrograde) pumping regions (see Figs. 4 5). The results for Jq and Jw are in good agreement with those obtained by Kothandapani and Prakash [28]= Fig. 6 shows the plot for average rise in pressure for variation in e1 . It can be seen that the peristaltic pumping increases via e1 in the peristaltic and backward pumping regions. The plot to study the e ect of p on {s is sketched in Fig. 7. The plot shows elevation in pumping rate for larger p in the co-pumping 12
ACCEPTED MANUSCRIPT region {s ? 0 and free pumping region {s = 0= The plots of {s for variation in Qw and Qe are given in Figs. (8) and (9). Opposite behavior of peristaltic pumping is observed for
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variations in Qw and Qe.
Fig. 3. Average pressure rise for variation in q=
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Fig. 2. Average pressure rise for variation in Z h=
Fig. 4. Average pressure rise for variation in Jq =
Fig. 5. Average pressure rise for variation in Jw =
Fig. 6. Average pressure rise for variation in e1 =
Fig. 7. Average pressure rise for variation in p=
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Flow characteristics
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Fig. 9. Average pressure rise for variation in Qe=
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Fig. 8. Average pressure rise for variation in Qw=
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Figs. 10 18 are prepared to analyze the impacts of Weissenberg number> viscosity ratio parameter, mean ßow rate, local temperature and local nanoparticle Grashof number,
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thermophoresis parameter, Brownian motion parameter, power-law index and non-uniform
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parameter on the axial velocity x. Values of { and w are Þxed when { = 0=4 and w = 0=2. The
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behavior of velocity proÞle is seen parabolic through graphs. Fig. 10 shows the plot of axial velocity for variations in Z h. An increase in Z h corresponds to dual response towards x. Impact of on the velocity is sketched in Fig. 11= For larger the velocity proÞle decreases near the left wall while opposite behavior is seen towards the right wall. Fig. 12 shows that at the centre of channel the axial velocity of the ßuid is maximum and then it progressively reduces to zero at the walls for larger . It is observed from graphical results that increasing values of parameters have nonuniform response towards x= It can be clearly seen through Fig. 13 that the velocity proÞle decreases in the region | 5 [0=664> 0=284] while it shows opposite trend in the remaining part for an increase in Jw . Fig. 14 is sketched to see the e ect of Jq on x. The results for the impacts of Jq and Jw on x are in good agreement with those obtained in the ref. [28]. Velocity proÞle is observed increasing in the region | 5 [1=036> 0=267]= However decrease in velocity is observed in the rest frame. The velocity proÞle is found to decrease in the region | 5 [1=19> 0=168] while it enhances in the remaining region for larger values of Qw (see Fig. 15). Fig. 16 shows that velocity proÞle increases for large Qe in the region | 5 [1=284> 0=269] and opposite behavior is observed in the rest part of channel. The plot to study the e ect of q on x is displayed in Fig. 17. It can be clearly seen from the Fig. that the velocity proÞle decreases with an increase of q in the region 14
ACCEPTED MANUSCRIPT | 5 [0=567> 0=609] and it increases for larger q. we It can be seen through Fig. 18 that the velocity proÞle is non-symmetric i.e., the velocity proÞle decreases at the core part of the tapered channel and it increases near channel walls for larger p. Moreover the velocity
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for divergent channel (p A 0) is higher when compared with a uniform channel (p = 0). Particularly if we take q = 1 in Eq. (22), we recover the results for viscous ßuid similar to
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those obtained by Kothandapani and Prakash [28] for a tapered asymmetric channel.
Fig. 11. Axial velocity proÞle for =
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Fig. 10. Axial velocity proÞle for Z h=
Fig. 12. Axial velocity proÞle for =
Fig. 13. Axial velocity proÞle for Jw =
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Fig. 15. Axial velocity proÞle for Qw=
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Fig. 14. Axial velocity proÞle for Jq =
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Fig. 16. Axial velocity proÞle for Qe=
Fig. 17. Axial velocity proÞle for q=
Fig. 18. Axial velocity proÞle for p= 4.2.1
Trapping phenomenon
The trapping phenomenon for various values of local temperature and local nanoparticle Grashof numbers and Weissenberg number is captured in the Figs. 19 21. Fig. 19 illustrates the impact of Jw on streamline pattern. For larger values of Jw the trapped bolus 16
ACCEPTED MANUSCRIPT contracts in size. Fig. 20 is sketched to study the inßuence of Jq on the streamlines. We found that near the left and right parts of channel, the size of trapping bolus enhances with the ascending values of Jq . Fig. 21 displays the impact of Z h on the streamlines.
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No signiÞcant change in the size of trapped bolus is observed when the ßuid character is
(b)
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(a)
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changed from Newtonian ßuid (Z h = 0) to Carreau-Yasuda ßuid (Z h A 0)=
Fig. 19. Streamlines for d = 0=2> e = 0=4> p = 0=1> ! = @3> X = 1=5> = 0=3> Z h = 0=2>
q = 1> Qw = 0=5> Qe = 1> Pr = 0=6> Jq = 1> w = 0=4 (a) Jw = 0 and (b) Jw = 0=7.
(a)
(b)
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ACCEPTED MANUSCRIPT Fig. 20. Streamlines for d = 0=2> e = 0=3> p = 0=1> ! = @3> X = 1=5> = 0=3> Z h = 0=2>
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q = 1> Qw = 0=5> Qe = 1> Pr = 0=6> Jw = 0=7> w = 0=4 (a) Jq = 0 and (b) Jq = 0=4.
(a) 21.
Streamlines for d = 0=1> e = 0=3> p = 0=1> ! = @3> X = 1=2> = 0=3> q = 1>
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Qw = 0=3> Qe = 0=6> Pr = 2> Jw = 0=8> Jq = 0=5> w = 0=4 (a) Z h = 0 and (b) Z h = 0=2.
Temperature proÞle
To analyze the e ects of Prandtl number Pr, thermophoresis parameter Qw, Brownian motion parameter Qe> phase di erence !> right wall amplitude e2 and non-uniform parameter p on the temperature proÞle X, the Figs. 22 27 have been made for Þxed values of { = 0=4 and w = 0=2= Fig. 22 exhibits the impact of Pr on the temperature proÞle. An enhancement of X is noticed via increase in Pr = Temperature distribution for various values of Qw has been sketched in Fig. 23= An increase in X is observed for larger Qw= The plot for impact of Qe on X shows increasing behavior (see Fig. 24)= An increase in X is noticed when Qe enhances= It is due to the fact that with increasing thermophoretic and Brownian motion e ects, a rapid movement of nanoparticle from wall to the ßuid occurs which results in a momentous increase of the temperature. Impacts of ! on temperature is decreasing (see Fig. 25). Fig. 26 illustrates the impact of left wall amplitude e2 on the temperature. Magnitude of temperature decreases via e2 in the vicinity of the wall. The temperature proÞle shows a dual behavior for increasing p (see Fig. 27). Temperature proÞle is observed to increase in the region | 5 [1=579> 0=148] while reverse behavior is seen in the remaining portion. Our results are in good agreement with those obtained for viscous nanoßuid in a tapered 18
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asymmetric channel by Kothandapani and Prakash [28].
Fig. 23. Temperature proÞle for Qw=
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Fig. 22. Temperature proÞle for Pr =
Fig. 24. Temperature proÞle for Qe=
Fig. 25. Temperature proÞle for !=
Fig. 26. Temperature proÞle for e2 =
Fig. 27. Temperature proÞle for p=
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Nanoparticle volume fraction
The variations of nanoparticle volume fraction l for di erent values of thermophoresis parameter, Brownian motion parameter, left wall amplitude and non-uniform parameter have
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been made graphically through Figs. 28 31 when { = 0=4 and w = 0=2= Decrease in l is
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observed with an increase in Qw (see Fig. 28). Since an increase in Qw means a decrease in ßuid’s viscosity which causes a reduction in nanoparticle volume fraction by the less dense
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particles. Fig. 29 is sketched to explore the impact of Qe on l. It is observed that the nanoparticle volume fraction increases for larger values of Q e. The larger values of Qe corre-
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spond to an increase in density of nanoßuid particles which causes the nanoparticle volume fraction to enhance. Increasing values of e1 results in an enhancement of l (see Fig. 30).
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Plot of p on l is sketched in Fig. 31. The nanoparticle volume fraction shows a dual behavior for increasing p. It is seen that in the region | 5 [1=39> 0=105] the nanoparticle volume
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fraction enhances for larger p however opposite trend is seen in the region | 5 [0=106> 1=669].
Fig. 28. Nanoparticle volume fraction for Qw=
Fig. 29. Nanoparticle volume fraction for Qe=
Fig. 30. Nanoparticle volume fraction for e1 =
Fig. 31. Nanoparticle volume fraction for p=
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Amplitude of the shear stress distribution
Figs. 32 38 are constructed to see the impact of pertinent parameters on the shear stress distribution v{| at the left wall of channel. Fig. 32 is prepared to study the impact of
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non-uniform parameter p on v{| . Decrease in shear stress distribution with growing p is
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examined. As expected, the amplitude of shear stress distribution is seen larger in case of uniform channel than the tapered channel. Fig. 33 shows the e ect of Z h on v{| . It is
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concluded that for larger Z h the shear stress amplitude decreases. In addition shear stress is greater in case of Newtonian ßuids (Z h = 0) when compared with non-Newtonian ßuids.
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Impact of ! on v{| is explored in Fig. 34. The amplitude of shear stress shows an oscillatory behavior for increasing values of phase di erence !. Fig. 35 illustrates the inßuence of power-
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law index q on the shear stress distribution. Decline in shear stress is observed for larger values of q (see Fig. 35). The plot to investigate the impact of viscosity ratio parameter on
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shear stress distribution is secured in Fig. 36. Larger values of have an increasing e ect
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on v{| . Fig. 37 38 show the e ects of local temperature and local nanoparticle Grashof numbers on v{| . Amplitude of shear stress is a decreasing function of Jw and Jq . The results
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are found well matched with those obtained by Kothandapani and Prakash [7]=
Fig. 32. Shear stress distribution for varying p. Fig. 33. Shear stress distribution for varying Z h.
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Fig. 34. Shear stress distribution for varying !. Fig. 35. Shear stress distribution for varying q.
Fig. 36. Shear stress distribution for varying . Fig. 37. Shear stress distribution for varying Jw .
Fig. 38. Shear stress distribution for varying Jq .
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Conclusions
In this paper we examined the mixed convective peristaltic transport of Carreau-Yasuda nanoßuid in a tapered asymmetric channel. Major points of the presented analysis are listed 22
ACCEPTED MANUSCRIPT below • Pumping rate decreases with increasing Z h in the peristaltic pumping region while an
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• Pumping rate enhances via increase in the q, Jw and Jq .
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• The non-uniform behavior of velocity is captured towards an increase in and q. • An increase in Z h corresponds to dual response towards x.
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• The velocity for divergent channel (p A 0) is found higher when compared with uniform channel (p = 0). Further the velocity is lowest for convergent channel (p ? 0).
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• Temperature proÞle enhances with increasing Q w, Qe and Pr and it decreases for ! and e2 =
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• Nanoparticle volume fraction decreases with growing values of Pr and Qw whereas
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opposite behavior is observed for increasing Qe and e1 .
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• The amplitude of shear stress distribution v{| shows decreasing behavior with increasing p, Z h, q, Jw and Jq while reverse is noticed for larger . • Size of trapped bolus enhances with larger values of Jq while it reduces with Jw . • No signiÞcant change is observed in the size of trapped bolus for large values of Z h=
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Appendix Here we provide the involved values in the solutions. S u(Qe + Qw) > k1 k2 1 (12Dk1 (k1 k2 )2 k2 (Jw Qe Jq Qw) + 12(k1 + k2 ) = 3 24D (k1 k2 )3 Qe (k21 4k1 k2 + k22 )(Jw Qe Jq Qw) + D3 (12I (k1 + k2 )(k21 4k1 k2 + k22 )Qe
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F4 =
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+Jq k21 (k1 k2 )2 k22 (Q e + Qw)) + 12(k1 k2 )(k22 + k21 (1 + Dk2 ) 1 +k1 k2 (2 + Dk2 ))(Jw Qe Jq Qw)Frwk[ D(k1 k2 )])> 2 1 (6D(k1 k2 )2 (k1 + k2 )(Jw Qe Jq Qw) + 72k1 k2 12D3 (k1 k2 )3 Qe (Jw Qe + Jq Qw) + (D3 )k2(72I k1 Qe + Jq k1 (k1 k2 )2 (k1 + k2 )(Qe + Qw)) 1 +6D(k1 k2 )(k21 + 4k1 k2 + k22 )(Jw Qe Jq Qw)Frwk[ D(k1 k2 )])> 2 1 2 (12D(k1 k2 ) (Jw Q e Jq Qw) + 72(k1 + k2 )(Jw Qe 24D3 (k1 k2 )3 Qe +Jq Qw) + D3 (72I (k1 + k2 )Qe + Jq (k1 k2 )2 (k21 + 4k1 k2 + k22 ) 1 (Qe + Qw)) + 36D(k1 k2 )(k1 + k2 )(Jw Qe Jq Qw)Frwk[ D(k1 k2 )])> 2 1 (24(Jw Qe Jq Qw) + D3 (24I Qe Jq ((k1 k2 )2 12D3 (k1 k2 )3 Qe 1 (k1 + k2 )(Qe + Qw)) 12D(k1 k2 )(Jw Qe Jq Qw)Frwk[ D(k1 k2 )])> 2 1 ((1 + q)(1 + )(Fvfk[ D(k1 k2 )]2 )((Frvk[Dk1 ] + Vlqk[Dk1 ])2 ) 2 (Frvk[2Dk1 ] Vlqk[2Dk1 ])((D5 k41 (360F42 (k1 k2 )2 Qe2 + 20F3 Jq (k1 k2 ) 1 Qe(Qe + Qw) + 36F4 Jq k1 (k1 k2 )Qe(Qe + Qw) + J2q k21 (Qe + Qw)2 )Frvk[ D 4 1 (7k1 5k2 )] (2D(30J2w (k1 k2 )2 Qe2 + 60Jq Jw ((k1 k2 )2 )QeQw 30J2q 2 2 2 (k1 k2 ) Qw + D4 k41 (360F42 (k1 k2 )2 Qe2 + 20F3 Jq (k1 k2 )Qe(Q e + Qw) 1 +36F4 Jq k1 (k1 k2 )Qe(Qe + Qw) + J2q k21 (Qe + Qw)2 ))Frvk[ D(k1 k2 )] 2 5 4 2 2 2 D k1 (360F4 (k1 k2 ) Qe + 20F3 Jq (k1 k2 )Qe(Qe + Qw) + 36F4 Jq k1 3 (k1 k2 )Qe(Qe + Qw) + J2q k21 (Qe + Qw)2 )Frvk[ D(k1 k2 )] + 20D5 F3 Jq 2 5 2 5 2 6 2 k1 Qe Frvk[D(2k1 k2 )] + 360D F4 k1 Qe Frvk[D(2k1 k2 )](480D4 1 1 ((k1 k2 )2 Qe2 (Frvk[ D(k1 k2 )] + Vlqk[ D(k1 k2 )]))))@(6k2 (3k41 k22 2 2 3 3 2 4 5 6 +12k1 k2 18k1 k2 + 12k1 k2 3k2 ))> 28
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+20F3 Jq (4 + Dk1 )(k1 k2 )Qe(Qe + Qw) + J2q k21 (6 + Dk1 )((Qe + Qw)2 )) 1 Frvk[ D(7k1 5k2 )] + (30D(48F4 Jw J2w + 12(D3 F42 k31 (4 + Dk1 )) 4 (k1 k2 )2 Qe2 + J2q (480(k1 k2 )Qw(Qe + Qw) 6D4 k51 (Qe + Qw)2 J2q (480(k1 k2 )Qw(Qe + Qw) 6D4 k51 (Q e + Qw)2 + D5 k61 (Q e + Qw)2 + 30
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+Dk1 )Qe + (8 + 3Dk1 + Dk2 )Qw ) + D(360F4 (k1 k2 )Qw 5D3 k31 (4F3 1 1 +9F4 k1 )480D4 ((k1 k2 )2 Qe2 (Frvk[ D(k1 k2 )] + Vlqk[ D(k1 k2 )]) 2 2 Dk2 6 Dk2 ((Frvk[ ] + Vlqk[ ]) )))))@(3k41 k22 + 12k31 k32 18k21 k42 + 12k1 k52 2 2 3k62 )> 1 1 = (3k22 (((1 + q)(1 + )Fvfk[ D(k1 k2 )]2 (Frvk[Dk1 ] 2 2 6k1 k2 + 6k1 k2 2 2 4 3 +Vlqk[Dk1 ]) (Frvk[2Dk1 ] Vlqk[2Dk1 ])(D k1 (360F42 (4 + Dk1 )(k1 k2 )2 Qe2 + 36F4 Jq k1 (5 + Dk1 )(k1 k2 )Qe(Q e + Qw) + 20F3 Jq (4 + Dk1 )(k1 1 k2 )Qe(Qe + Qw) + J2q k21 (6 + Dk1 )(Qe + Qw)2 )Frvk[ D(7k1 5k2 )] 4 2 3 2 3 +(30D(48F4 Jw Jw + 12D F4 k1 (4 + Dk1 ))(k1 k2 )2 Qe2 + J2q (480 (k1 k2 )Qw(Qe + Qw) 6D4 k51 (Qe + Qw)2 + D5 k61 (Qe + Q w)2 + 30D(k1 k2 ) Qw(8k1 Qe + 7k1 Qw + k2 Qw)) + 4Jq (k1 k2 )Qe(15Jw (4(2 + Dk1 )Qe +(8 + 3Dk1 + Dk2 )Qw) + D(360F4 (k1 k2 )Qw 5D3 k31 (4F3 + 9F4 k1 ) 1 (Qe + Qw) + D4 k41 (5F3 + 9F4 k1 )(Q e + Qw))))Frvk[ D(k1 3k2 )] 4 1 1 2 480Jq Jw k1 Qe Frvk[ D(3k1 k2 )] + 1440DF4 Jw k21 Qe2 Frvk[ D(3 4 4 1 1 Dk2 k1 k2 )](Frvk[ D(k1 k2 )] + Vlqk[ D(k1 k2 )])(Frvk[ ] 2 2 2 Dk2 6 +Vlqk[ ]) )))))@((6k21 k2 + 6k1 k22 )(3k41 k22 + 12k31 k32 18k21 k2 4 2 +12k1 k52 3k62 ))>
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ACCEPTED MANUSCRIPT 1 1 (((1 + q)(1 + )(Fvfk[ D(k1 k2 )]2 )(Frvk[2Dk1 ] Vlqk[2Dk1 ]) 2 3k2 2 4 3 2 2 (D k2 (360(F4 )((k1 k2 ) )(4 + Dk2 )Q e2 + 36F4 Jq (k1 k2 )k2 (5 + Dk2 )
E4 =
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Qe(Qe + Qw) + 20F3 Jq (k1 k2 )(4 + Dk2 )Qe(Qe + Qw) + J2q k22 (6 + Dk2 ) 1 (Qe + Qw)2 )Frvk[ D(5k1 7k2 )] + 2(360DF4 (k1 k2 )2 (2Jw + D3 F4 k32 4 2 (4 + Dk2 ))Qe J2q (Qe + Qw)(240(k1 k2 )Qw 120D(k1 k2 )k2 Qw
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6D4 k52 (Qe + Qw) + D5 k62 (Qe + Qw)) + 4Jq (k1 k2 )Qe(180DF4 (k1 k2 )
E8 =
E10 =
E12 =
E14 =
E16 =
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E6 =
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E5 =
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Qw 30Jw (2 + Dk2 )(Qe + Q w) + 5D4 k32 (4F3 + 9F4 k2 )(Qe + Qw) D5 k42 (5F3 1 1 +9F4 k2 )(Qe + Qw)))Frvk[ D(k1 3k2 )] 480Jq Jw k1 Qe2 Frvk[ D(3k1 4 4 1 1 2 2 2 2 2 k2 )] + 1440DF4 Jw k1 Qe Frvk[ D(3k1 k2 )] 30D(Jw k1 Q e Frvk[ D 4 4 1 1 4 2 2 (3k1 k2 )](480D (k1 k2 ) Qe (Frvk[ D(k1 k2 )] + Vlqk[ D(k1 k2 )]) 2 2 Dk2 6 Dk2 4 2 3 3 ((Frvk[ ] + Vlqk[ ]) )))@(3k1 k2 (3k1 k2 + 12k1 k2 18k21 k42 + 12k1 k52 3k62 ))> 2 2 1 1 (k1 k2 )(Jw Qe Jq Qw)(Frvk[ D(k1 + k2 )] Vlqk[ D(k1 + k2 )])> 2 2 1 2 2 (1 + q)(1 + )Fvfk[ D(k1 + k2 )] > E7 = 72F4 (k1 k2 )2 Q e2 + 4F3 2 Jq (k1 k2 )Q e(Qe + Qw)> 1 8DF3 Jq (k2 k1 )Qe2 Frvk[ D(k1 + k2 )]> E9 = 144DF42 (k21 + k22 )Qe2 2 1 Frvk[ D(k1 + k2 )]> 2 1 288DF42 k1 k2 Qe2 Frvk[ D(k1 + k2 )]> E11 = 36DF4 Jq (k2 k1 )Qe2 2 1 Frvk[ D(k1 k2 )]> 2 1 8DF3 Jq (k2 k1 )QeQwFrvk[ D(k1 k2 )]> E13 = 36DF4 Jq (k2 k1 ) 2 1 QeQwFrvk[ D(k1 k2 )]> 2 1 2DJ2q (Qw2 + Qe2 )Frvk[ D(k1 k2 )]> E15 = 4DJ2q QeQw 2 1 Frvk[ D(k1 k2 )]> 2 1 2DJ2q Qw2 Frvk[ D(k1 k2 )]> E17 = 2J2w (k21 + k22 )Qe2 4k1 k2 (J2q Qw2 2 2 2 +Jw Qe ) 4Jq Jw (k21 + k22 )QeQw + 2J2q (k21 + k22 )Qw2 + 8Jq Jw k1 k2 QeQw>
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ACCEPTED MANUSCRIPT
E18 = 8DF3 (Jw Qe Jq Qw)k1 k2 Qe + 24F4 k1 k2 Qe(Jw Qe Jq Qw)
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+4DF3 (Jq Qw Jw Qe)(k21 + k22 )Qe + 12F4 (Jq Qw Jw Qe) (k21 + k22 )Q e>
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E19 = 2Jq Jw (k2 k1 )(Qe + Qw)Qe + 24DF4 (Jw Qe Jq Qw)k1 k2 Qe
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+12DF4 (Jq Qw Jw Qe)(k21 + k22 )Qe + 2J2q (k1 k2 )(Qw + Qe)Qw> E20 = DJq Jw (k2 Qe k1 Qw)Qe + DJq Jw k2 QeQw + DJ2q (k1 k2 )(Qe + Qw)Qw
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+4DF3 Jw k21 Qe2 >
E21 = 12F4 (Jw Qe Jq Qw)(k21 + k22 )Qe + 8DF3 (Jq Qw Jw Qe)k1 k2 Q e
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+24F4 (Jq Qw QeJw )k1 k2 Q e + 4DF3 (Jw Qe Jq Qw)(k21 + k22 )Qe> E22 = 2Jq Jw (k1 k2 )(Qe Qw)Qe + 12DF4 (k21 + k22 )(Jw Qe Jq Qw)Qe
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+24DF4 (Q wJq QeJw )k1 k2 Qe + 2J2q (k2 k1 )(Qe + Qw)Qw>
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E23 = DJq Jw (k1 k2 )(Qw + Qe)Q e + DJ2q (k2 k1 )(Qw + Qe)Qw>
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E24 = 72DF42 (k21 + k22 )Qe2 + 4DF3 Jq (k2 k1 )(Qe + Qw)Qe +144D(F42 )k1 k2 (Q e2 )> E25 = 18DF4 Jq (k2 k1 )(Qe + Qw)Qe> E26 = DJ2q (Qe2 Qw2 ) 2DJ2q QeQw 4DF3 Jq k1 Qe2 72DF42 k21 Qe2 > E27 = 2(k1 k2 )2 (Jw Qe Jq Q w)2 + 2D(k1 k2 )(Jw Qe Jq Qw)(4F3 (k1 k2 )Qe +k1 (12F4 (k1 k2 )Qe + Jq k1 (Qe + Qw)))> E28 = 8F3 Jq (k1 k2 )Qe(Qe + Qw) + (12F4 (k1 k2 )Qe + Jq k1 (Qe + Qw))2 > E29 = k21 (1 + q)(12F4 Qe + Jq (Qe + Qw))Z h(1 + ) + k2 Qe(1 + 4F3 Z h 1 (1 + q + q)))Frwk[ D(k1 k2 )]= 2
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ACCEPTED MANUSCRIPT Highlights
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Peristalsis of Carreau-Yasuda nanofluid is considered. Effects of mixed convection are present. Tapered asymmetry in the channel walls is outlined. Series solutions are developed. Amplitude of shear stress distribution and pumping characteristics are emphasized.
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· · · · ·
S0167-7322(16)31215-6 doi: 10.1016/j.molliq.2016.08.003 MOLLIQ 6161
To appear in:
Journal of Molecular Liquids
Received date: Accepted date:
16 May 2016 1 August 2016
Please cite this article as: T. Hayat, Rija Iqbal, A. Tanveer, A. Alsaedi, Mixed convective peristaltic transport of Carreau-Yasuda nanofluid in a tapered asymmetric channel, Journal of Molecular Liquids (2016), doi: 10.1016/j.molliq.2016.08.003
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. 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.
ACCEPTED MANUSCRIPT
Mixed convective peristaltic transport of Carreau-Yasuda nanoßuid in a tapered asymmetric channel d
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T. Hayatd>e , Rija Iqbald , Anum Tanveerd>1 and A. Alsaedie Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan
e
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Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of
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Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
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Abstract: Here modeling and analysis for mixed convective peristaltic ßow of CarreauYasuda nanoßuid have been carried out. The asymmetric channel walls are tapered. E ects
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of Brownian motion and thermophoresis are taken into consideration. The ßow equations for the relevant problem are Þrst constructed and then non-dimensionalized. The resulting non-linear coupled systems subject to lubrication approach are solved. Graphs reßecting
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the variations of pertinent parameters on the velocity, temperature, nanoparticle volume
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fraction are examined. Shear stress distribution and pumping and trapping phenomena are discussed. Results indicate that amplitude of shear stress decreases for larger Weissenberg
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number and power-law index whereas it enhances through viscosity ratio parameter. Keywords: Peristaltic ßow, Carreau-Yasuda nanoßuid, tapered asymmetric channel, mixed convection.
1
Introduction
The study of nanoßuids has gained much popularity amongst the recent researchers. It is due to the fact that thermal conductivity of nanoßuids has a vital role in heat transfer phenomenon. Thermal conductivity of traditional heat transfer ßuids like oil, water and ethylene glycol mixture are insu!cient to meet present day requirements. Thus various techniques have been suggested for thermal conductivity enhancement of traditional ßuids. Amongst such techniques there is one through suspension of nano-particles in the traditional ßuids. The word "nanoßuid" was Þrst used by Choi [1] which refers to a liquid suspension containing nanometer-sized particles (having diameters less than 50 nm). Due to their enhanced thermal properties, the nanoßuids can be used as coolants in heat transfer equipment 1
Corresponding author. Tel.: + 92 51 90642172.
e-mail address: [email protected]
1
ACCEPTED MANUSCRIPT such as heat exchangers, electronic cooling system and radiators. Nanoßuids have numerous biomedical applications as well. Cancer diagnosis, drug delivery systems, neuro electronic interfaces and photodynamic therapy are some examples in this direction. Nanoßuids can be
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used in advanced nuclear systems due to their signiÞcant feature of enhancing the thermal conductivity. Masuda et al. [2] Þrst worked on this application of nanoßuids. Buongiorno
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[3> 4] presented a systematic model for the study of convective transport in nanoßuids with
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combined e ects of Brownian motion and thermophoresis. He revealed that the only important baseßuid/nanoparticle slip mechanisms are the Brownian motion and thermophoresis.
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Study of peristalsis of nanoßuids has also gained signiÞcant interest of the researchers in recent times due to its wide ranging applications. Few attempts in this direction can be
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consulted by the studies [5 17].
Recently the peristaltic phenomenon has become a topic of great interest for the researchers in view of its widespread applications in physiological, medical and industrial
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processes. Its occurrence can be found in many common physiological processes such as
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in the urine passage from kidney to bladder, food digestion, chyme movement in gastrointestinal tract, blood circulation in small blood vessels and also in the human reproductive
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systems. In biomedical Þeld the dialysis and heart lung machines worked on the peristaltic principle. Roller and Þnger pumps and waste management pumps in nuclear industry are applications of peristalsis in industrial area. Several researchers have worked for peristaltic motion of viscous and non-Newtonian materials subject to di erent aspects and conditions. Much focus in the literature has been made on the peristaltic transport of viscous ßuids. Even several bioßuids have been considered as viscous ßuids by many workers. This in not adequate in the sense that microßuidic devices have been usually employed to analyze bioßuids, which are often solutions of long chain molecules imparting a non-Newtonian behavior characterized by variable viscosity, memory e ects, normal stress e ects, yield stress and hysteresis of ßuid properties. Also, the non-Newtonian materials are involved in many processes including food mixing, chyme movement in intestine, blood ßow, ßow of liquid metals, nuclear slurries and alloys. Few studies in this regard have been mentioned (see refs. [18 23]). Study of convective ßow in channel has also gained popularity due to its numerous industrial and technological applications including transport processes, solar collectors, electronic equipment cooled by fans, nuclear reactors cooled during emergency shutdown, transistors 2
ACCEPTED MANUSCRIPT and biomedical engineering. Keeping in mind the aforementioned applications, several researchers have made analysis for mixed convective peristaltic transport (see refs. [24 26] and several studies therein). Another important aspect which has not been yet explored
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properly in peristalsis is the non-uniform (tapered) ßow conÞguration. This conÞguration occurs in most practical applications such as in many physiological body organs, small blood
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vessels, intestines, lymphatic vessels and ductus a erents [27 29].
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It is remarkable that very little focus has been made on the study of peristaltic ßow of Carreau-Yasuda (CY) ßuid. The importance of this model is that it consists of Þve
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constants predicting the rheological characteristics of non-Newtonian ßuids through shear thinning/thickening parameters. That is why CY model is preferred over the power-law
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model. Few studies dealing with peristalsis of Carreau-Yasuda ßuid have been mentioned in the refs. [30 33].
To our knowledge, not much has been said yet about the peristalsis of Carreau-
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Yasuda ßuid. Such consideration further narrowed down when peristalsis of Carreau-Yasuda
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nanoßuid is examined [34]. Therefore main focus here is to make some advancement in this direction. SpeciÞcally we have intention to investigate the following salient features
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in the peristalsis of Carreau-Yasuda nanoßuid. Firstly to consider the mixed convective e ect. Secondly to analyze the tapered channel properties. Thirdly to examine Brownian motion and thermophoresis in tapered channel ßow. Fourth to predict shear thinning and shear thickening properties of ßuid in a tapered ßow conÞguration. The continuity, momentum, energy and concentration equations comprising mixed convection, Brownian motion and thermophoresis e ects have been constructed here. The relevant equations have been simpliÞed using low Reynolds number and long wavelength assumptions. The solutions for the stream function, velocity, temperature and concentration are calculated. The e ects of involved parameters on the velocity, temperature, concentration, pressure rise, pumping and streamlines have been addressed by the graphical illustrations. Further the shear stress has been graphically examined.
2
Mathematical formulation
The ßow of an incompressible Carreau-Yasuda nanoßuid in a tapered asymmetric channel is considered. Brownian motion and thermophoresis e ects are present. The peristaltic waves 3
ACCEPTED MANUSCRIPT ¯ are travelling along the [-axis with speed f. Here \¯ -axis is taken perpendicular to the ¯ 2 represent the positions ¯ 1 and \¯ = k ¯ [axis. The wall conÞguration is such that \¯ = k of left and right walls of the channel. The amplitudes and phases of peristaltic waves are
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di erent. Geometry of the wall surfaces is deÞned below: ¸ 2 ¯ ¯k1 ([> ¯ ¯ w¯) = g p ¯ ¯ ([ fw) + ! > ¯ [ e1 sin ¸ 2 ¯ ¯k2 ([> ¯ ¯ w¯) = g + p ¯ ¯ [ + e2 sin ([ fw¯) =
(1) (2)
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Here w¯ denotes the dimensional time, ¯e1 and ¯e2 the waves amplitudes along the left and right walls, p(¿ ¯ 1) the non-uniform parameter of the tapered channel, the wavelength, g the half-width of channel and ! 5 [0> ] the phase di erence. Here the amplitudes ¯el (l = 1> 2)> g
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and ! satisfy the condition
(3)
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¯e2 + ¯e2 + 2¯e1¯e2 g cos(!) 6 (2g)2 = 1 2
Fig. 1. Schematic diagram of the tapered asymmetric channel.
The fundamental equations governing the present ßow of an incompressible Carreau-Yasuda nanoßuid are
¯ C Y¯ CX + = 0> ¯ C[ C \¯
(4)
¯ ¯¸ CX CX C S¯ C ¯ C ¯ C X¯ ¯ ¯ + X + Y = + ( V (V ¯ ¯ ) + (1 F0 ) i j(W W0 ) i ¯[ ¯) + [ ¯ ¯ ¯ ¯ C w¯ C[ C\ C[ C[ C \¯ [ \ +( s i )j(F F0 )> (5) 4
ACCEPTED MANUSCRIPT ¸ ¯ C Y¯ C S¯ C CY C Y¯ C ¯ ¯ + Y = + ¯ (V¯\¯ [¯ ) + ¯ (V¯\¯ \¯ )> i + X 0 ¯ ¯ ¯ Cw C[ C\ C\ C[ C\
(6)
¯ ¸ 2¯ ¸ µ ¯ ¯ ¶ CW C W¯ C W C 2 W¯ CF CW C F¯ C W¯ C W¯ 0 ¯ ¯ ( f )i +X ¯ +Y ¯ = ¯ 2 + C \¯ 2 + ( f )s GE C [ ¯ C[ ¯ + C \¯ C \¯ C w¯ C[ C\ C[ "µ ¶2 µ ¯ ¶2 # ¯ G C W CW W +( f0 )s + > (7) ¯ Wp C[ C \¯
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¯ µ 2¯ ¶ 2¯ ¸ 2¯ 2¯ ¯ ¯¸ CF C F C F F G C W W C F C C W ¯ ¯ +X (8) ¯ + Y C \¯ = GE C [ ¯ 2 + C \¯ 2 + Wp C [ ¯ 2 + C \¯ 2 = C w¯ C[ Here X¯ and Y¯ are the velocity components in the axial and transverse directions respectively,
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S¯ the pressure, V¯[¯ [¯ > V¯[¯ \¯ > V¯\¯ [¯ > V¯\¯ \¯ the components of extra stress tensor, Wp the ßuid mean temperature, i the density of ßuid, s the density of nanoparticles, f0 the volumetric volume
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expansion coe!cient, the thermal conductivity of ßuid, W¯ and F¯ the ßuid temperature and nanoparticle concentration, GE the Brownian di usion coe!cient and GW the thermophoretic
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di usion coe!cient.
¯ = (ˆ S )A1 >
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Expression of extra stress tensor for Carreau-Yasuda ßuid is [33] :
where A1 represents the Þrst Rivlin-Erickson tensor and the apparent viscosity (ˆ ) is given by
q31 ¯ + (grad V) ¯ W > (ˆ ) = " + (0 " )[1 + (Kˆ )d ] d > A1 = grad V
in which
p 1 ¯ + ( grad V) ¯ W ]= 2wu(D2 )> D = [grad(V) 2 ¯ \¯ > w¯)> Y¯ ([> ¯ \¯ > w¯)> 0] is the velocity Þeld, " and 0 the inÞnite and zero ¯ = [X¯ ([> Here V ˆ =
shear-rate viscosities, d and K the non-dimensional and material parameters and q the nondimensional power law index for Carreau-Yasuda ßuid. The above mentioned ßuid model describes the pseudoplastic ßow with asymptotic viscosities at zero and inÞnite shear rates. The model also predicts the results of shear thinning, shear thickening and Newtonian ßuids for q ? 1> q A 1 and q = 1. Also for d = 2 the model reduces to Carreau ßuid model. Here we have taken the Yasuda parameter d = 1 to examine the rheological aspects of non-Newtonian ßuids.
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ACCEPTED MANUSCRIPT DeÞning the non-dimensional quantities through the deÞnitions below [35]: ¯1 ¯2 ¯ k k [ \¯ fw0 X¯ Y¯ g > | = > w = > x = > y = > = > k1 = > k2 = > g f f g g ¯e1 i fg W¯ W0 g2 S¯ g ¯ F¯ F0 X= > s= > vlm = V¯~¯m > l = > Re = > e1 = > W1 W0 f f F1 F0 g 0 ¯e2 fi p ¯ GE (F1 F0 ) GW (W1 W0 ) > Pr = > Qe = > Qw = > e2 = > p = g g Wp (1 F0 ) i jg2 (W1 W0 ) ( s i )jg2 (F1 F0 ) Jw = > Jq = > Vf = f f GE
(9)
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{=
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the continuity equation is identically satisÞed and Eqs. (5) (8) become ¸ C Cx Cx Cs C Cx + x +y = + (v{{ ) + (v{| ) + Jw X + Jq l> Re Cw C{ C| C{ C{ C| ¸ Cy Cx Cs C C Cy + x +y = + 2 (v{| ) + (v|| )> Re Cw C{ C| C| C{ C|
(10) (11)
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¸ ¸ ¸ 2 CX CX 1 Cl CX C 2X CX 2C X 2 Cl CX + x +y = + + Re + Qe Cw C{ C| Pr C{2 C{2 C{ C{ C| C| " µ ¶2 µ ¶2 # CX CX > (12) +Qw 2 + C{ C| ¸ ¸ 2 Cl C 2 l Qw 2 C 2 C 2 Cl Cl 2C l + x +y = + 2 + + Re Vf = (13) Cw C{ C| C{2 C| Qe C{2 C| 2
Here the non-dimensional parameters signify the following quantities: w the time, { and | the axial and transverse coordinates, x and y the axial and transverse components of velocity, p the non-uniform parameter, e1 and e2 the amplitudes of left and right walls, the ratio of e ective heat capacity of nanoparticle material to heat capacity of the ßuid, X the temperature, l the concentration, s the pressure, Re the Reynolds number, Pr the Prandtl number, Qe the Brownian motion parameter, Qw the thermophoresis parameter, Vf the Schmidt number and Jw and Jq the local temperature and local nanoparticle Grashof numbers.
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Volume ßow rate
¯ is deÞned as The time-averaged ßow at a Þxed position [ Z W 1 ¯= T gw¯ T W 0
(14)
we Þnd ¯ = t + e1 f sin[ 2 ([ fw0 ) + !] + e2 f sin[ 2 ([ fw0 )]> T 6
(15)
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Z
K2
x¯(¯ {> |¯)g¯ |=
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is the dimensional volume ßow rate in the wave frame.
Using the deÞnitions of non-
¯ T t > = > fg fg
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we Þnd that Eq. (16) becomes [7> 28> 35] :
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I =
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dimensional mean ßows I in the laboratory frame and X in the wave frame
in which
(16)
k2
x g|=
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I =
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I ({> w) = + d sin[2 (¯ { w) + !] + e sin[2 (¯ { w)]>
(17)
k1
Using the deÞnitions of stream function (x =
C# >y C|
= C# ), Eqs. (10) (13) subject to C{
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long wavelength and low Reynolds number approximation become:
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Cs C = (v{| ) + Jw X + Jq l> C{ C|
(18)
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Cs = 0> C| µ ¶ µ ¶2 Cl CX C2X CX + Qw Pr + Qe Pr = 0> C| 2 C| C| C| C 2 l Qw C 2 X + = 0= C| 2 Qe C| 2
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is the viscosity ratio parameter and Z h =
(20) (21)
The non-dimensional form of the stress tensor is given by ¸ C2# C 2# v{| = 1 + (1 )(q 1)Z h 2 > C| 2 C| where =
(19)
Kf g
(22) the Weissenberg number
respectively. Eliminating pressure from Eqs. (19) and (20), we get the following systems C2 (v{| ) + Jw X + Jq l = 0> C| 2 µ ¶ µ ¶2 Cl CX C2X CX + Qe Pr + Qw Pr = 0> C| 2 C| C| C| C 2 l Qw C 2 X + = 0> C| 2 Qe C| 2 7
(23)
(24) (25)
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¸ C 2# C2# 1 + (1 )(q 1)Z h 2 > = C| 2 C|
(26)
subject to the boundary conditions
I C# > = 0> X = 1 and l = 1 at | = k2 = 1 + p{ + e2 sin[2 ({ w)]= 2 C|
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#=
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I C# #= > = 0> X = 0 and l = 0 at | = k1 = 1 p{ e1 sin[2 ({ w) + !]> (27) 2 C| (28)
Solution technique
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The dimensionless pressure rise per wavelength is given by Z 1Z 1µ ¶ Cs {s = g{gw. C{ |=0 0 0
Eqs. (25) and (26) can be solved for the exact solutions subject to the appropriate boundary
are
Frvk[Dk1 ] + Vlqk[Dk1 ] Frvk[Dk1 ] Frvk[Dk2 ] + Vlqk[Dk1 ] Vlqk[Dk2 ] Frvk[D|] + Vlqk[D|] > Frvk[Dk1 ] Frvk[Dk2 ] + Vlqk[Dk1 ] Vlqk[Dk2 ]
(29)
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X =
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conditions. Thus the exact solutions for the temperature and nanoparticle volume fraction
µ ¶ Frvk[Dk1 ] + Vlqk[Dk1 ] (Qe + Qw)(k1 + |) Qw l = (k1 + k2 )Qe Qe Frvk[Dk1 ] Frvk[Dk2 ] + Vlqk[Dk1 ] Vlqk[Dk2 ] µ ¶ Qw Frvk[D|] + Vlqk[D|] + = (30) Qe Frvk[Dk1 ] Frvk[Dk2 ] + Vlqk[Dk1 ] Vlqk[Dk2 ]
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Perturbation solution
Since the problems involved are nonlinear. It seems di!cult to obtain the exact solutions. Thus we are interested to derive the approximate analytical solutions by perturbation technique. Perturbation technique is better than other techniques as it can be used to solve any set of complicated problems containing small parameter. The expressions developed by perturbation method are not exact but these can lead to accurate results when the expansion parameter is small. Since Eq. (24) is non-linear in Weissenberg number and hence cannot be solved to obtain an exact solution. Therefore we apply the perturbation technique to Þnd
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ACCEPTED MANUSCRIPT approximate solutions for small Weissenberg number Z h. Thus we perturbed #> s> I and v|{ about Z h. Moreover the series solution are retained only up to Þrst order i.e. $ = $0 + Z h$ 1 + R(Z h2 )
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in which $ = #> s> I and v|{ respectively. Zeroth order systems and solutions
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3.1.1
(31)
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The corresponding systems at zero order are ¸ DFrvk[D|] + DVlqk[D|] C2 (v0{| ) Jw + Jq C| 2 Frvk[Dk1 ] Frvk[Dk2 ] + Vlqk[Dk1 ] Vlqk[Dk2 ] ¸ (Q e + Qw) Qw DFrvk[D|] + DVlqk[D|] + = 0 >(32) (k1 + k2 )Qe Qe Frvk[Dk1 ] Frvk[Dk2 ] + Vlqk[Dk1 ] Vlqk[Dk2 ]
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¸ Frvk[Dk1 ] + Vlqk[Dk1 ] Cs0 C Jq = (v0{| ) + Jw + C{ C| Frvk[Dk1 ] Frvk[Dk2 ] + Vlqk[Dk1 ] Vlqk[Dk2 ] Qe ¸ (Qe + Qw)(k1 |) Qw(Frvk[Dk1] Frvk[D|] + Vlqk[Dk1] Vlqk[D|]) (33) > (k1 k2 ) Frvk[Dk1 ] Frvk[Dk2 ] + Vlqk[Dk1 ] Vlqk[Dk2 ] v0{| =
C 2 #0 > C| 2
I0 C#0 > = 0 at | = k1 > 2 C| I0 C#0 #0 = > = 0 at | = k2 = 2 C|
#0 =
(34) (35) (36)
The solutions at zeroth order are
1 Jq (Qe + Qw)| 4 ( + 24Qe k1 k2 12(h3(1@2)D(k1 +k2 32|) (Jw Qe Jq Qw)Fvfk[ 12 D(k1 k2 )]) )> D3
#0 = F1 + F2 | + F3 | 2 + F4 | 3 +
(37)
µ ¶ gs0 ((hDk1 hD| )Jw ) Jq (hDk1 + hD| )Qw (Qe + Qw)(k1 |) 1 = 6F4 + + + + Dk Dk Dk Dk 1 2 1 2 g{ h h Qe h h k1 k2 2Qe µ ¶ 2Jq (Qe + Qw)| 1 + h3(1@2)D(k1 +k2 32|) (Jw Qe Jq Qw)Fvfk[ D(k1 k2 )] > (38) (k1 k2 ) 2
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First order systems and solutions
The corresponding systems at Þrst order are
(40) (41)
I1 C#1 > = 0 at | = k1 > 2 C|
(42)
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C 2 #1 C 2 #0 + (1 )(q 1) > C| 2 C| 2
#1 = #1 =
(39)
I1 C#1 > = 0 at | = k2 = 2 C|
The solutions at Þrst order are
(43)
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v1{| =
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Cs1 C = (v1{| )> C{ C|
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C2 (v1{| ) = 0> C| 2
h3D(k1 +k2 ) (1 + q) (30Dh2D| (k1 k2 )2 (Jw Qe Jq Qw)2 5 2 2 480D (k1 k2 ) Qe 5 2Dk1 4 +D h | (20(k1 k2 )Qe(18F42 (k1 k2 )Qe + F3 Jq (Qe + Q w)) + 36F4 Jq (k1 k2 )
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#1 = E1 + |(E2 + |(E3 + E4 |)) +
Qe(Qe + Qw)| + J2q (Qe + Qw)2 | 2 ) + D5 h2Dk2 | 4 (20(k1 k2 )Qe(18(F42 (k1 k2)Qe
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+F3 Jq (Qe + Qw)) + 36F4 Jq (k1 k2 )Qe(Qe + Qw)| + J2q (Qe + Qw)2 | 2 ) 2D5 hD(k1 +k2 ) (| 4 (20(k1 k2 )Qe(18F42 (k1 k2 )Qe + F3 Jq (Qe + Qw)) + 36F4 Jq (k1 k2 ) Qe(Qe + Qw)| + J2q (Qe + Qw)2 | 2 ) + 120hD(k1 +|) (k1 k2 )(Jw Qe Jq Qw)(4D(k1 k2 ) Qe(DF3 6F4 + 3DF4 |) + Jq 120hD(k2 +|) (k1 k2 ) + (Qe + Qw)(6 + D|(4 + D|))) +(Jw Qe + Jq Qw)(4D(k1 k2 )Qe(DF3 6F4 + 3DF4 |) + Jq (Qe + Qw) 1 (6 + D|(4 + D|))))(1 + )Fvfk[ D(k1 k2 )]2 > 2
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(44)
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gs1 1 1 = 6E4 + 2(q 1)(1 )(6F4 + Fvfk[ D(k1 k2 )](2Jq (Qe + Qw)| g{ 2(k1 k2 )Qe 2 Fvfk[ 12 D(k1 k2 )] 1 Vlqk[ D(k1 k2 )] + E5 (Frvk[D|] + Vlqk[D|])))(2F3 + 6F4 | + 2 2(k1 k2 )Qe 1 E5 (Frvk[D|] + Vlqk[D|]) (Jq (Qe + Qw)| 2 Vlqk[ D(k1 k2 )] + ) + (E6 (Frvk[2D|] 2 D +Vlqk[2D|])(D|(E7 + 18F4Jq (k1 k2 )Qe(Qe + Qw)| + J2q (Qe + Qw)2 | 2 ) 1 1 (Frvk[ D(k1 3k2 )] + Frvk[ D(3k1 + k2 )]) + (E8 E9 + E10 + E12 )| 2 2 1 +| 2 (E11 + E13 ) | 3 (E14 + E15 + E16 ) + E17 (Frvk[ D(k1 + k2 4|)] 2 1 1 +Vlqk[ D(k1 + k2 4|)]) + (E18 + |E19 + | 2 E20 )(Frvk[ D(k1 k2 2|)] 2 2 1 1 +Vlqk[ D(k1 k2 + 2|)]) + (E21 + |E22 + | 2 E23 )(Frvk[ D(k1 k2 + 2|)] 2 2 1 1 +Vlqk[ D(k1 k2 + 2|)]) + (|E24 + | 2 E25 + | 3 E26 )(Vlqk[ D(3k1 + k2 )] 2 2 1 1 1 +Vlqk[ D(k1 3k2 )] + Vlqk[ D(k1 + k2 )])))@(4D(k1 k2 )2 Qe2 (Frvk[ D 2 2 2 1 (k1 + k2 + 4|)] + Vlqk[ D(k1 + k2 + 4|)]))= (45) 2
v{|
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The dimensionless shear stress at the left wall of the channel is reduced as follows: ¸ C2# C2# = 1 + (1 )(q 1) 2 C| 2 C| 1 = (4D(k1 k2 )Qe((k1 k2 )(Jw Qe Jq Qw) + D(4F3 (k1 k2 ) 8D2 (k1 k2 )2 Qe2 Qe + k1 (12F4 (k1 k2 )Qe + Jq k1 (Qe + Qw)) + 4(E3 + 3E4 k1 )(k1 k2 )QeZ h)) +4(k1 k2 )(Jw Q e Jq Qw)((k1 k2 )(1 + q)(Jw Qe Jq Qw)Z h(1 + ) + D(k1 Qe (1 + 4(F3 3F4 k2 )(1 + q)Z h(1 + )) + E29 + 2(1 + q)Z h(1 + )E27 + D2 k21 E28 1 1 +(k1 k2 )2 (Jw Qe Jq Qw)2 Fvfk[ D(k1 k2 )]2 ))(1 + (1 q)Z h 2 2 8D (k1 k2 )2 Qe2 (1 )(4D(k1 k2 )Qe((k1 k2 )(Jw Qe Jq Qw) + D(4F3 (k1 k2 )Qe + k1 (12F4 (k1 k2 ) Qe + Jq k1 (Qe + Qw)) + 4(E3 + 3E4 k1 )(k1 k2 )QeZ h)) + 4(k1 k2 )(Jw Qe Jq Qw)((k1 k2 )(1 + q)(Jw Qe Jq Qw)Z h(1 + ) + D(k1 Qe(1 + 4(F3 3F4 k2 )(1 + q)Z h(1 + )) +E29 + 2(1 + q)Z h(1 + )(2(k1 k2 )2 (Jw Qe Jq Qw)2 + 2D(k1 k2 )(Jw Qe Jq Qw) (4F3 (k1 k2 )Qe + k1 (12F4 (k1 k2 )Qe + Jq k1 (Qe + Qw))) + D2 k21 (8F3 Jq (k1 k2 )Qe (Qe + Qw) + (12F4 (k1 k2 )Qe + Jq k1 (Qe + Qw))2 ) + ((k1 k2 )2 )(Jw Q e Jq Qw)2 1 Fvfk[ D(k1 k2 )]2 )))> 2 11
(46)
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Graphical results
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Figs. 2 38 have been sketched to analyze the physical signiÞcance of pertinent ßow pa-
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rameters on the peristaltic transport of Carreau-Yasuda nanoßuid in a tapered asymmetric
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channel. Particularly the plots of pressure rise, velocity, temperature, concentration and
4.1
Pumping characteristics
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shear stress have been drawn and discussed physically.
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Figs. 2 9 are sketched to analyze the impacts of Weissenberg number, power-law index, local nanoparticle and local temperature Grashof numbers, left wall wave amplitude> non-
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uniform parameter, thermophoresis parameter and Brownian motion parameter on average
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pressure rise {s= The plots have been constructed so that the upper right-hand quadrant (L) represents the peristaltic pumping region, where X A 0 (positive pumping) and {s A 0
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(adverse pressure gradient). Quadrant (LL) in which X A 0 (positive pumping) and {s ? 0 (favorable pressure gradient) is marked as the co-pumping region (or augmented ßow). Quadrant (LLL) where {s ? 0> X ? 0 is the region where there is no ßow. Quadrant (LY ) represents the region of backward pumping (or retrograde region) where {s A 0 and X ? 0. Fig. 2 illustrates the impact of Z h on {s. It is noticed that in the peristaltic pumping region, the pumping rate decreases with increasing Z h while opposite behavior is observed in the co-pumping region. It is observed through Fig. 3 that the pumping rate increases with higher values of q in the peristaltic pumping region which indicates that the pumping rate increases as the behavior of ßuid changes from shear thinning (q ? 1) to Newtonian (q = 1) and to shear thickening (q A 1)= Particularly for q = 1 in Eq. (22) > the results for viscous nanoßuid in a tapered channel [28] are recovered. The pumping rate enhances with escalating values of Jq and Jw in the peristaltic pumping and backward (retrograde) pumping regions (see Figs. 4 5). The results for Jq and Jw are in good agreement with those obtained by Kothandapani and Prakash [28]= Fig. 6 shows the plot for average rise in pressure for variation in e1 . It can be seen that the peristaltic pumping increases via e1 in the peristaltic and backward pumping regions. The plot to study the e ect of p on {s is sketched in Fig. 7. The plot shows elevation in pumping rate for larger p in the co-pumping 12
ACCEPTED MANUSCRIPT region {s ? 0 and free pumping region {s = 0= The plots of {s for variation in Qw and Qe are given in Figs. (8) and (9). Opposite behavior of peristaltic pumping is observed for
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variations in Qw and Qe.
Fig. 3. Average pressure rise for variation in q=
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Fig. 2. Average pressure rise for variation in Z h=
Fig. 4. Average pressure rise for variation in Jq =
Fig. 5. Average pressure rise for variation in Jw =
Fig. 6. Average pressure rise for variation in e1 =
Fig. 7. Average pressure rise for variation in p=
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Flow characteristics
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Fig. 9. Average pressure rise for variation in Qe=
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Fig. 8. Average pressure rise for variation in Qw=
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Figs. 10 18 are prepared to analyze the impacts of Weissenberg number> viscosity ratio parameter, mean ßow rate, local temperature and local nanoparticle Grashof number,
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thermophoresis parameter, Brownian motion parameter, power-law index and non-uniform
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parameter on the axial velocity x. Values of { and w are Þxed when { = 0=4 and w = 0=2. The
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behavior of velocity proÞle is seen parabolic through graphs. Fig. 10 shows the plot of axial velocity for variations in Z h. An increase in Z h corresponds to dual response towards x. Impact of on the velocity is sketched in Fig. 11= For larger the velocity proÞle decreases near the left wall while opposite behavior is seen towards the right wall. Fig. 12 shows that at the centre of channel the axial velocity of the ßuid is maximum and then it progressively reduces to zero at the walls for larger . It is observed from graphical results that increasing values of parameters have nonuniform response towards x= It can be clearly seen through Fig. 13 that the velocity proÞle decreases in the region | 5 [0=664> 0=284] while it shows opposite trend in the remaining part for an increase in Jw . Fig. 14 is sketched to see the e ect of Jq on x. The results for the impacts of Jq and Jw on x are in good agreement with those obtained in the ref. [28]. Velocity proÞle is observed increasing in the region | 5 [1=036> 0=267]= However decrease in velocity is observed in the rest frame. The velocity proÞle is found to decrease in the region | 5 [1=19> 0=168] while it enhances in the remaining region for larger values of Qw (see Fig. 15). Fig. 16 shows that velocity proÞle increases for large Qe in the region | 5 [1=284> 0=269] and opposite behavior is observed in the rest part of channel. The plot to study the e ect of q on x is displayed in Fig. 17. It can be clearly seen from the Fig. that the velocity proÞle decreases with an increase of q in the region 14
ACCEPTED MANUSCRIPT | 5 [0=567> 0=609] and it increases for larger q. we It can be seen through Fig. 18 that the velocity proÞle is non-symmetric i.e., the velocity proÞle decreases at the core part of the tapered channel and it increases near channel walls for larger p. Moreover the velocity
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for divergent channel (p A 0) is higher when compared with a uniform channel (p = 0). Particularly if we take q = 1 in Eq. (22), we recover the results for viscous ßuid similar to
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those obtained by Kothandapani and Prakash [28] for a tapered asymmetric channel.
Fig. 11. Axial velocity proÞle for =
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Fig. 10. Axial velocity proÞle for Z h=
Fig. 12. Axial velocity proÞle for =
Fig. 13. Axial velocity proÞle for Jw =
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Fig. 15. Axial velocity proÞle for Qw=
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Fig. 14. Axial velocity proÞle for Jq =
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Fig. 16. Axial velocity proÞle for Qe=
Fig. 17. Axial velocity proÞle for q=
Fig. 18. Axial velocity proÞle for p= 4.2.1
Trapping phenomenon
The trapping phenomenon for various values of local temperature and local nanoparticle Grashof numbers and Weissenberg number is captured in the Figs. 19 21. Fig. 19 illustrates the impact of Jw on streamline pattern. For larger values of Jw the trapped bolus 16
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No signiÞcant change in the size of trapped bolus is observed when the ßuid character is
(b)
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changed from Newtonian ßuid (Z h = 0) to Carreau-Yasuda ßuid (Z h A 0)=
Fig. 19. Streamlines for d = 0=2> e = 0=4> p = 0=1> ! = @3> X = 1=5> = 0=3> Z h = 0=2>
q = 1> Qw = 0=5> Qe = 1> Pr = 0=6> Jq = 1> w = 0=4 (a) Jw = 0 and (b) Jw = 0=7.
(a)
(b)
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q = 1> Qw = 0=5> Qe = 1> Pr = 0=6> Jw = 0=7> w = 0=4 (a) Jq = 0 and (b) Jq = 0=4.
(a) 21.
Streamlines for d = 0=1> e = 0=3> p = 0=1> ! = @3> X = 1=2> = 0=3> q = 1>
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Qw = 0=3> Qe = 0=6> Pr = 2> Jw = 0=8> Jq = 0=5> w = 0=4 (a) Z h = 0 and (b) Z h = 0=2.
Temperature proÞle
To analyze the e ects of Prandtl number Pr, thermophoresis parameter Qw, Brownian motion parameter Qe> phase di erence !> right wall amplitude e2 and non-uniform parameter p on the temperature proÞle X, the Figs. 22 27 have been made for Þxed values of { = 0=4 and w = 0=2= Fig. 22 exhibits the impact of Pr on the temperature proÞle. An enhancement of X is noticed via increase in Pr = Temperature distribution for various values of Qw has been sketched in Fig. 23= An increase in X is observed for larger Qw= The plot for impact of Qe on X shows increasing behavior (see Fig. 24)= An increase in X is noticed when Qe enhances= It is due to the fact that with increasing thermophoretic and Brownian motion e ects, a rapid movement of nanoparticle from wall to the ßuid occurs which results in a momentous increase of the temperature. Impacts of ! on temperature is decreasing (see Fig. 25). Fig. 26 illustrates the impact of left wall amplitude e2 on the temperature. Magnitude of temperature decreases via e2 in the vicinity of the wall. The temperature proÞle shows a dual behavior for increasing p (see Fig. 27). Temperature proÞle is observed to increase in the region | 5 [1=579> 0=148] while reverse behavior is seen in the remaining portion. Our results are in good agreement with those obtained for viscous nanoßuid in a tapered 18
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asymmetric channel by Kothandapani and Prakash [28].
Fig. 23. Temperature proÞle for Qw=
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Fig. 22. Temperature proÞle for Pr =
Fig. 24. Temperature proÞle for Qe=
Fig. 25. Temperature proÞle for !=
Fig. 26. Temperature proÞle for e2 =
Fig. 27. Temperature proÞle for p=
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Nanoparticle volume fraction
The variations of nanoparticle volume fraction l for di erent values of thermophoresis parameter, Brownian motion parameter, left wall amplitude and non-uniform parameter have
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been made graphically through Figs. 28 31 when { = 0=4 and w = 0=2= Decrease in l is
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observed with an increase in Qw (see Fig. 28). Since an increase in Qw means a decrease in ßuid’s viscosity which causes a reduction in nanoparticle volume fraction by the less dense
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particles. Fig. 29 is sketched to explore the impact of Qe on l. It is observed that the nanoparticle volume fraction increases for larger values of Q e. The larger values of Qe corre-
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spond to an increase in density of nanoßuid particles which causes the nanoparticle volume fraction to enhance. Increasing values of e1 results in an enhancement of l (see Fig. 30).
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Plot of p on l is sketched in Fig. 31. The nanoparticle volume fraction shows a dual behavior for increasing p. It is seen that in the region | 5 [1=39> 0=105] the nanoparticle volume
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fraction enhances for larger p however opposite trend is seen in the region | 5 [0=106> 1=669].
Fig. 28. Nanoparticle volume fraction for Qw=
Fig. 29. Nanoparticle volume fraction for Qe=
Fig. 30. Nanoparticle volume fraction for e1 =
Fig. 31. Nanoparticle volume fraction for p=
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Amplitude of the shear stress distribution
Figs. 32 38 are constructed to see the impact of pertinent parameters on the shear stress distribution v{| at the left wall of channel. Fig. 32 is prepared to study the impact of
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non-uniform parameter p on v{| . Decrease in shear stress distribution with growing p is
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examined. As expected, the amplitude of shear stress distribution is seen larger in case of uniform channel than the tapered channel. Fig. 33 shows the e ect of Z h on v{| . It is
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concluded that for larger Z h the shear stress amplitude decreases. In addition shear stress is greater in case of Newtonian ßuids (Z h = 0) when compared with non-Newtonian ßuids.
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Impact of ! on v{| is explored in Fig. 34. The amplitude of shear stress shows an oscillatory behavior for increasing values of phase di erence !. Fig. 35 illustrates the inßuence of power-
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law index q on the shear stress distribution. Decline in shear stress is observed for larger values of q (see Fig. 35). The plot to investigate the impact of viscosity ratio parameter on
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shear stress distribution is secured in Fig. 36. Larger values of have an increasing e ect
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on v{| . Fig. 37 38 show the e ects of local temperature and local nanoparticle Grashof numbers on v{| . Amplitude of shear stress is a decreasing function of Jw and Jq . The results
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Fig. 32. Shear stress distribution for varying p. Fig. 33. Shear stress distribution for varying Z h.
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Fig. 34. Shear stress distribution for varying !. Fig. 35. Shear stress distribution for varying q.
Fig. 36. Shear stress distribution for varying . Fig. 37. Shear stress distribution for varying Jw .
Fig. 38. Shear stress distribution for varying Jq .
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Conclusions
In this paper we examined the mixed convective peristaltic transport of Carreau-Yasuda nanoßuid in a tapered asymmetric channel. Major points of the presented analysis are listed 22
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increase in {s is observed in the co-pumping region.
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• Pumping rate enhances via increase in the q, Jw and Jq .
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• The non-uniform behavior of velocity is captured towards an increase in and q. • An increase in Z h corresponds to dual response towards x.
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• The velocity for divergent channel (p A 0) is found higher when compared with uniform channel (p = 0). Further the velocity is lowest for convergent channel (p ? 0).
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• Temperature proÞle enhances with increasing Q w, Qe and Pr and it decreases for ! and e2 =
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• The amplitude of shear stress distribution v{| shows decreasing behavior with increasing p, Z h, q, Jw and Jq while reverse is noticed for larger . • Size of trapped bolus enhances with larger values of Jq while it reduces with Jw . • No signiÞcant change is observed in the size of trapped bolus for large values of Z h=
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ßuid through porous medium in the presence of inclined magnetic Þeld in an inclined
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asymmetric channel, J. Mol. Liq., 219 (2016) 256—271. [20] T. Hayat, M. ShaÞque, A. Tanveer and A. Alsaedi, Hall and ion-slip e ects on peristaltic ßow of Je rey nanoßuid with Joule heating, J. Magn. Magn. Mater., 407 (2016) 51—59. [21] A. Sinha, G. C. Shit and N. K. Ranjit, Peristaltic transport of MHD ßow and heat transfer in an asymmetric channel: E ects of variable viscosity, velocity-slip and temperature jump, Alex. Eng. J., 54 (2015) 691—704. [22] T. Hayat, F. M. Abbasi, B. Ahmad and A. Alsaedi, Peristaltic transport of CarreauYasuda ßuid in a curved channel with slip e ects, PLoS ONE., 9 (2014) doi: 10.1371/journal.pone.0095070. [23] A. Riaz, R. Ellahi, and S. Nadeem, Peristaltic transport of Carreau ßuid in a compliant rectangular duct, Alex. Eng. J., 53 (2014) 475-484 (2014). [24] S. Srinivas, R. Gayathri and M. Kothandapani, Mixed convective heat and mass transfer in an asymmetric channel with peristalsis, Comm. Nonlinear. Sci. Numer. Simul., 16 (2011) 1845-1862.
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ACCEPTED MANUSCRIPT [25] S. Srinivas and R. Muthuraj, E ects of chemical reaction and space porosity on MHD mixed convective ßow in a vertical asymmetric channel with peristalsis, Math. Comput.
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Mod., 54 (2011) 1845-1862. [26] T. Hayat, F. M. Abbasi, M. Al-Yami and S. Monaquel, Slip and Joule heating e ects
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in mixed convection peristaltic transport of nanoßuid with Soret and Dufour e ects, J.
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Mole. Liq., 194 (2014) 93—99.
[27] M. Kothandapani, J. Prakash and V. Pushparaj, E ects of heat transfer, magnetic Þeld
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and space porosity on peristaltic ßow of a Newtonian ßuid in a tapered channel, Appl. Mech. Mater., 814 (2015) 679-684.
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[28] M. Kothandapani and J. Prakash, E ect of radiation and magnetic Þeld on peristaltic transport of nanoßuids through a porous space in a tapered asymmetric channel, J.
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Magn. Magn. Mater., 378 (2015) 152-163.
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[29] K. Vajravelu, S. Sreenadh and R. Saravana, Combined inßuence of velocity slip, temperature and concentration jump conditions on MHD peristaltic transport of a Carreau
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ßuid in a non-uniform channel, Appl. Math. Comput., 225 (2013) 656—676. [30] T. Hayat, F. M. Abbasi, A. Alsaedi and F. Alsaedi, Hall and Ohmic heating e ects on the peristaltic transport of Carreau-Yasuda ßuid in an asymmetric channel, ZNA., 69 (2014) 43-51.
[31] T. Hayat, A. Tanveer and A. Alsaedi, Mixed convective peristaltic ßow of Carreau— Yasuda ßuid with thermal deposition and chemical reaction, Int. J. Heat Mass. Transf., 96 (2016) 474—481. [32] F. M. Abbasi, T.Hayat and A.Alsaedi, Numerical analysis for MHD peristaltic transport of Carreau—Yasuda ßuid in a curved channel with Hall e ects, J. Magn. Magn. Mater., 382 (2015) 104—110. [33] T. Hayat, F. M. Abbasi, B. Ahmad and A. Alsaedi, Peristaltic transport of CarreauYasuda ßuid in a curved channel with slip e ects, PLoS ONE., 9 (2014) doi: 10.1371/journal.pone.0095070. [34] F. M. Abbasi, T. Hayat, B. Ahmad and G. Q. Chen, Peristaltic motion of a nonNewtonian nanoßuid in an asymmetric channel, ZNA., 69a (2014) 451—461. 26
ACCEPTED MANUSCRIPT [35] M. Kothandapani and J. Prakash, E ects of thermal radiation parameter and magnetic Þeld on the peristaltic motion of Williamson nanoßuids in a tapered asymmetric channel,
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Int. J. Heat Mass. Transf., 81 (2015) 234-245.
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Appendix Here we provide the involved values in the solutions. S u(Qe + Qw) > k1 k2 1 (12Dk1 (k1 k2 )2 k2 (Jw Qe Jq Qw) + 12(k1 + k2 ) = 3 24D (k1 k2 )3 Qe (k21 4k1 k2 + k22 )(Jw Qe Jq Qw) + D3 (12I (k1 + k2 )(k21 4k1 k2 + k22 )Qe
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+Jq k21 (k1 k2 )2 k22 (Q e + Qw)) + 12(k1 k2 )(k22 + k21 (1 + Dk2 ) 1 +k1 k2 (2 + Dk2 ))(Jw Qe Jq Qw)Frwk[ D(k1 k2 )])> 2 1 (6D(k1 k2 )2 (k1 + k2 )(Jw Qe Jq Qw) + 72k1 k2 12D3 (k1 k2 )3 Qe (Jw Qe + Jq Qw) + (D3 )k2(72I k1 Qe + Jq k1 (k1 k2 )2 (k1 + k2 )(Qe + Qw)) 1 +6D(k1 k2 )(k21 + 4k1 k2 + k22 )(Jw Qe Jq Qw)Frwk[ D(k1 k2 )])> 2 1 2 (12D(k1 k2 ) (Jw Q e Jq Qw) + 72(k1 + k2 )(Jw Qe 24D3 (k1 k2 )3 Qe +Jq Qw) + D3 (72I (k1 + k2 )Qe + Jq (k1 k2 )2 (k21 + 4k1 k2 + k22 ) 1 (Qe + Qw)) + 36D(k1 k2 )(k1 + k2 )(Jw Qe Jq Qw)Frwk[ D(k1 k2 )])> 2 1 (24(Jw Qe Jq Qw) + D3 (24I Qe Jq ((k1 k2 )2 12D3 (k1 k2 )3 Qe 1 (k1 + k2 )(Qe + Qw)) 12D(k1 k2 )(Jw Qe Jq Qw)Frwk[ D(k1 k2 )])> 2 1 ((1 + q)(1 + )(Fvfk[ D(k1 k2 )]2 )((Frvk[Dk1 ] + Vlqk[Dk1 ])2 ) 2 (Frvk[2Dk1 ] Vlqk[2Dk1 ])((D5 k41 (360F42 (k1 k2 )2 Qe2 + 20F3 Jq (k1 k2 ) 1 Qe(Qe + Qw) + 36F4 Jq k1 (k1 k2 )Qe(Qe + Qw) + J2q k21 (Qe + Qw)2 )Frvk[ D 4 1 (7k1 5k2 )] (2D(30J2w (k1 k2 )2 Qe2 + 60Jq Jw ((k1 k2 )2 )QeQw 30J2q 2 2 2 (k1 k2 ) Qw + D4 k41 (360F42 (k1 k2 )2 Qe2 + 20F3 Jq (k1 k2 )Qe(Q e + Qw) 1 +36F4 Jq k1 (k1 k2 )Qe(Qe + Qw) + J2q k21 (Qe + Qw)2 ))Frvk[ D(k1 k2 )] 2 5 4 2 2 2 D k1 (360F4 (k1 k2 ) Qe + 20F3 Jq (k1 k2 )Qe(Qe + Qw) + 36F4 Jq k1 3 (k1 k2 )Qe(Qe + Qw) + J2q k21 (Qe + Qw)2 )Frvk[ D(k1 k2 )] + 20D5 F3 Jq 2 5 2 5 2 6 2 k1 Qe Frvk[D(2k1 k2 )] + 360D F4 k1 Qe Frvk[D(2k1 k2 )](480D4 1 1 ((k1 k2 )2 Qe2 (Frvk[ D(k1 k2 )] + Vlqk[ D(k1 k2 )]))))@(6k2 (3k41 k22 2 2 3 3 2 4 5 6 +12k1 k2 18k1 k2 + 12k1 k2 3k2 ))> 28
ACCEPTED MANUSCRIPT 1 E2 = (3k22 (k21 + k22 ) + 2k2 (k31 + k32 ))(3k22 (((1 + q)(1 + )(Fvfk[ D 2 2 2 4 3 (k1 k2 )] )(Frvk[Dk1 ] + Vlqk[Dk1 ]) (Frvk[2Dk1 ] Vlqk[2Dk1 ])(D k1 (360
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F42 (4 + Dk1 )(k1 k2 )2 Qe2 + 36F4 Jq k1 (5 + Dk1 )(k1 k2 )Qe(Qe + Qw)
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+20F3 Jq (4 + Dk1 )(k1 k2 )Qe(Qe + Qw) + J2q k21 (6 + Dk1 )((Qe + Qw)2 )) 1 Frvk[ D(7k1 5k2 )] + (30D(48F4 Jw J2w + 12(D3 F42 k31 (4 + Dk1 )) 4 (k1 k2 )2 Qe2 + J2q (480(k1 k2 )Qw(Qe + Qw) 6D4 k51 (Qe + Qw)2 J2q (480(k1 k2 )Qw(Qe + Qw) 6D4 k51 (Q e + Qw)2 + D5 k61 (Q e + Qw)2 + 30
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D(k1 k2 )Qw(8k1 Qe + 7k1 Qw + k2 Qw)) + 4Jq (k1 k2 )Qe(15Jw (4(2
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+Dk1 )Qe + (8 + 3Dk1 + Dk2 )Qw ) + D(360F4 (k1 k2 )Qw 5D3 k31 (4F3 1 1 +9F4 k1 )480D4 ((k1 k2 )2 Qe2 (Frvk[ D(k1 k2 )] + Vlqk[ D(k1 k2 )]) 2 2 Dk2 6 Dk2 ((Frvk[ ] + Vlqk[ ]) )))))@(3k41 k22 + 12k31 k32 18k21 k42 + 12k1 k52 2 2 3k62 )> 1 1 = (3k22 (((1 + q)(1 + )Fvfk[ D(k1 k2 )]2 (Frvk[Dk1 ] 2 2 6k1 k2 + 6k1 k2 2 2 4 3 +Vlqk[Dk1 ]) (Frvk[2Dk1 ] Vlqk[2Dk1 ])(D k1 (360F42 (4 + Dk1 )(k1 k2 )2 Qe2 + 36F4 Jq k1 (5 + Dk1 )(k1 k2 )Qe(Q e + Qw) + 20F3 Jq (4 + Dk1 )(k1 1 k2 )Qe(Qe + Qw) + J2q k21 (6 + Dk1 )(Qe + Qw)2 )Frvk[ D(7k1 5k2 )] 4 2 3 2 3 +(30D(48F4 Jw Jw + 12D F4 k1 (4 + Dk1 ))(k1 k2 )2 Qe2 + J2q (480 (k1 k2 )Qw(Qe + Qw) 6D4 k51 (Qe + Qw)2 + D5 k61 (Qe + Q w)2 + 30D(k1 k2 ) Qw(8k1 Qe + 7k1 Qw + k2 Qw)) + 4Jq (k1 k2 )Qe(15Jw (4(2 + Dk1 )Qe +(8 + 3Dk1 + Dk2 )Qw) + D(360F4 (k1 k2 )Qw 5D3 k31 (4F3 + 9F4 k1 ) 1 (Qe + Qw) + D4 k41 (5F3 + 9F4 k1 )(Q e + Qw))))Frvk[ D(k1 3k2 )] 4 1 1 2 480Jq Jw k1 Qe Frvk[ D(3k1 k2 )] + 1440DF4 Jw k21 Qe2 Frvk[ D(3 4 4 1 1 Dk2 k1 k2 )](Frvk[ D(k1 k2 )] + Vlqk[ D(k1 k2 )])(Frvk[ ] 2 2 2 Dk2 6 +Vlqk[ ]) )))))@((6k21 k2 + 6k1 k22 )(3k41 k22 + 12k31 k32 18k21 k2 4 2 +12k1 k52 3k62 ))>
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ACCEPTED MANUSCRIPT 1 1 (((1 + q)(1 + )(Fvfk[ D(k1 k2 )]2 )(Frvk[2Dk1 ] Vlqk[2Dk1 ]) 2 3k2 2 4 3 2 2 (D k2 (360(F4 )((k1 k2 ) )(4 + Dk2 )Q e2 + 36F4 Jq (k1 k2 )k2 (5 + Dk2 )
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Qe(Qe + Qw) + 20F3 Jq (k1 k2 )(4 + Dk2 )Qe(Qe + Qw) + J2q k22 (6 + Dk2 ) 1 (Qe + Qw)2 )Frvk[ D(5k1 7k2 )] + 2(360DF4 (k1 k2 )2 (2Jw + D3 F4 k32 4 2 (4 + Dk2 ))Qe J2q (Qe + Qw)(240(k1 k2 )Qw 120D(k1 k2 )k2 Qw
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6D4 k52 (Qe + Qw) + D5 k62 (Qe + Qw)) + 4Jq (k1 k2 )Qe(180DF4 (k1 k2 )
E8 =
E10 =
E12 =
E14 =
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Qw 30Jw (2 + Dk2 )(Qe + Q w) + 5D4 k32 (4F3 + 9F4 k2 )(Qe + Qw) D5 k42 (5F3 1 1 +9F4 k2 )(Qe + Qw)))Frvk[ D(k1 3k2 )] 480Jq Jw k1 Qe2 Frvk[ D(3k1 4 4 1 1 2 2 2 2 2 k2 )] + 1440DF4 Jw k1 Qe Frvk[ D(3k1 k2 )] 30D(Jw k1 Q e Frvk[ D 4 4 1 1 4 2 2 (3k1 k2 )](480D (k1 k2 ) Qe (Frvk[ D(k1 k2 )] + Vlqk[ D(k1 k2 )]) 2 2 Dk2 6 Dk2 4 2 3 3 ((Frvk[ ] + Vlqk[ ]) )))@(3k1 k2 (3k1 k2 + 12k1 k2 18k21 k42 + 12k1 k52 3k62 ))> 2 2 1 1 (k1 k2 )(Jw Qe Jq Qw)(Frvk[ D(k1 + k2 )] Vlqk[ D(k1 + k2 )])> 2 2 1 2 2 (1 + q)(1 + )Fvfk[ D(k1 + k2 )] > E7 = 72F4 (k1 k2 )2 Q e2 + 4F3 2 Jq (k1 k2 )Q e(Qe + Qw)> 1 8DF3 Jq (k2 k1 )Qe2 Frvk[ D(k1 + k2 )]> E9 = 144DF42 (k21 + k22 )Qe2 2 1 Frvk[ D(k1 + k2 )]> 2 1 288DF42 k1 k2 Qe2 Frvk[ D(k1 + k2 )]> E11 = 36DF4 Jq (k2 k1 )Qe2 2 1 Frvk[ D(k1 k2 )]> 2 1 8DF3 Jq (k2 k1 )QeQwFrvk[ D(k1 k2 )]> E13 = 36DF4 Jq (k2 k1 ) 2 1 QeQwFrvk[ D(k1 k2 )]> 2 1 2DJ2q (Qw2 + Qe2 )Frvk[ D(k1 k2 )]> E15 = 4DJ2q QeQw 2 1 Frvk[ D(k1 k2 )]> 2 1 2DJ2q Qw2 Frvk[ D(k1 k2 )]> E17 = 2J2w (k21 + k22 )Qe2 4k1 k2 (J2q Qw2 2 2 2 +Jw Qe ) 4Jq Jw (k21 + k22 )QeQw + 2J2q (k21 + k22 )Qw2 + 8Jq Jw k1 k2 QeQw>
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E18 = 8DF3 (Jw Qe Jq Qw)k1 k2 Qe + 24F4 k1 k2 Qe(Jw Qe Jq Qw)
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+4DF3 (Jq Qw Jw Qe)(k21 + k22 )Qe + 12F4 (Jq Qw Jw Qe) (k21 + k22 )Q e>
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E19 = 2Jq Jw (k2 k1 )(Qe + Qw)Qe + 24DF4 (Jw Qe Jq Qw)k1 k2 Qe
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+12DF4 (Jq Qw Jw Qe)(k21 + k22 )Qe + 2J2q (k1 k2 )(Qw + Qe)Qw> E20 = DJq Jw (k2 Qe k1 Qw)Qe + DJq Jw k2 QeQw + DJ2q (k1 k2 )(Qe + Qw)Qw
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+4DF3 Jw k21 Qe2 >
E21 = 12F4 (Jw Qe Jq Qw)(k21 + k22 )Qe + 8DF3 (Jq Qw Jw Qe)k1 k2 Q e
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+24F4 (Jq Qw QeJw )k1 k2 Q e + 4DF3 (Jw Qe Jq Qw)(k21 + k22 )Qe> E22 = 2Jq Jw (k1 k2 )(Qe Qw)Qe + 12DF4 (k21 + k22 )(Jw Qe Jq Qw)Qe
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+24DF4 (Q wJq QeJw )k1 k2 Qe + 2J2q (k2 k1 )(Qe + Qw)Qw>
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E23 = DJq Jw (k1 k2 )(Qw + Qe)Q e + DJ2q (k2 k1 )(Qw + Qe)Qw>
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E24 = 72DF42 (k21 + k22 )Qe2 + 4DF3 Jq (k2 k1 )(Qe + Qw)Qe +144D(F42 )k1 k2 (Q e2 )> E25 = 18DF4 Jq (k2 k1 )(Qe + Qw)Qe> E26 = DJ2q (Qe2 Qw2 ) 2DJ2q QeQw 4DF3 Jq k1 Qe2 72DF42 k21 Qe2 > E27 = 2(k1 k2 )2 (Jw Qe Jq Q w)2 + 2D(k1 k2 )(Jw Qe Jq Qw)(4F3 (k1 k2 )Qe +k1 (12F4 (k1 k2 )Qe + Jq k1 (Qe + Qw)))> E28 = 8F3 Jq (k1 k2 )Qe(Qe + Qw) + (12F4 (k1 k2 )Qe + Jq k1 (Qe + Qw))2 > E29 = k21 (1 + q)(12F4 Qe + Jq (Qe + Qw))Z h(1 + ) + k2 Qe(1 + 4F3 Z h 1 (1 + q + q)))Frwk[ D(k1 k2 )]= 2
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Peristalsis of Carreau-Yasuda nanofluid is considered. Effects of mixed convection are present. Tapered asymmetry in the channel walls is outlined. Series solutions are developed. Amplitude of shear stress distribution and pumping characteristics are emphasized.
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