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Mechanics & Industry 19, 401 (2018)© AFM, EDP Sciences 2018https://doi.org/10.1051/meca/2018022
Mechanics&IndustryAvailable online at:
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REGULAR ARTICLE
Flow analysis of particulate suspension on an asymmetricperistaltic motion in a curved configuration with heat andmass transferAhmed Zeeshan1,*, Nouman Ijaz1, and Muhammad Mubashir Bhatti2
1 Department of Mathematics and Statistics, FBAS, International Islamic University, Islamabad, Pakistan2 Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, PR China
* e-mail: a
Received: 31 October 2017 / Accepted: 9 April 2018
Abstract. This article addresses the influence of particulate-fluid suspension on asymmetric peristaltic motionthrough a curved configuration with mass and heat transfer. Amotivation for the current study is that such kindof theory is helpful to examine the two-phase peristaltic motion between small muscles during the propagation ofdifferent biological fluids. Moreover, it is also essential in multiple applications of pumping fluid-solid mixturesby peristalsis, i.e., Chyme in small intestine and suspension of blood in arteriole. Long wavelength, as well assmall Reynolds number, have been utilized to render the governing equations for particle and fluid phase. Exactsolutions are presented for velocity (uf,p), temperature (uf,p) and concentration distributions (’f,p). All theparameters such as Prandtl number (Pr), particle volume fraction (C), suspension parameter (M1), curvatureparameter (k), volumetric flow rate (Q), Schmidt number (Sc), phase difference (’), Eckert number (Ec), andSoret number (Sr) discussed graphically for peristaltic pumping (Dp), pressure gradient (dp/dx), velocity (uf,p),temperature (uf,p) and concentration distributions (’f,p). The streamlines are also plotted with the aid ofcontour.
Keywords: Peristaltic flow / particle-fluid / curved channel / mass and heat transfer / exact solutions
1 Introduction
The physiology of peristaltic flow (“sinusoidal motion”)grabs the concentration of many researchers because of itsmultiple applications in biomedical engineering andindustry. This process is substantial in the various processinvolved in a human body, i.e., propagation cilia wave, amotion of spermatozoa, vasomotion of small blood vessels,Chyme propagation, and urine propagation from a kidneyto bladder, etc. This phenomenon is wholly associated withfinger and roller pumps, the heart-lung machine, andcirculation of toxic and sanitary liquid. Due to theseincredible and vital applications of peristaltic flow, manyauthors explored this phenomenon using a viscous fluidmodel in different geometrical situations. Mishra and Rao[1] discussed the asymmetric peristaltic motion of Newto-nian fluid model propagating through a finite channel.They used the lubrication approach to model the governingequations and obtained the exact solutions. Nonlinearperistaltic motion of viscous fluid propagating through aporous medium was examined by Mekheimer [2]. The
three-dimensional flow with a viscous fluid model wasstudied by Reddy et al. [3]. Later, Elnaby and Haroun [4]presented a new model to analyze the wall behavior ofinclined planar channel. The finite planar channel filledwith a viscous fluid model and porous walls. Hayat et al. [5]analyzed the peristaltic flow of viscous fluid in a curvedcompliant channel. Ramanamurthy et al. [6] discussed theunsteady peristaltic motion of viscous fluid in the curvedchannel. They also used long wavelength and smallReynolds approximation and presented the exact solutions.Recently, Nadeem et al. [7] extended the work of Reddyet al. [3] and explored the behavior of viscous fluidpropagating in a three-dimensional duct with compliantwalls.
The presence of particles in fluids is a ubiquitousphenomenon because it involves deeply in our daily life. Forinstance, the water we drink, the air we inhale, and fizzydrinks involve many particles in different forms. Moreover,blood an essential element promptly arises in our mind thatoccurs with white blood cells, plasma, and red blood cells.There are multiple numbers of examples that include thecollision between fluid and particles that can be encoun-tered in our daily life. According to the technologicalstandpoint, various activates in the industrial process
2 A. Zeeshan et al.: Mechanics & Industry 19, 401 (2018)
include particle-fluids systems, i.e., fluidization, powdertechnology, lunar ash flow, sedimentation, aerosol filtra-tion, and combustion, etc. Such kind of theory is alsobeneficial to analyze the hydrodynamics of biologicalsystems because it gives better information in bloodrheology, electrophoresis, dethroning of particles in therespiratory tract, separation of chromatographic, diffusionof proteins and macromolecules separation process.Multiphase flows are of multiple types, i.e., Gas-liquidflows (i.e., gas-droplet flows, separated flows and bubblyflows), Gas-solid flows (i.e., Gas-particle flows, Pneumaticmotion, and Fluidized beds) and solid-liquid flows (i.e.,Slurry flows, Sediment motion and Hydro transport).Various researchers have examined the peristaltic move-ment having solid particles in different geometricalsituations. Misra and Pandey [8] discussed the peristalticmovement through a cylindrical tube having smallspherical particles of similar size. They obtained thesolution using perturbation technique and observed theconverse flow appearance when a pressure gradient ishigher than the critical value and it occurs because of theoccupancy of particles. Saxena and Srivastava [9]addressed the particulate suspension peristaltic flowthrough a non-uniform axisymmetric tube. Mekheimeret al. [10] investigated the sinusoidal motion particlesthrough a planar channel. Recently, Mekheimer et al. [11]again considered the eccentric cylinders with threadannular and sinusoidal waves traveling containing smallparticles. Kamel et al. [12] investigated the peristalticmotion through a planar channel having small particles init. Few more studies are available in references [13–16].
Mass and heat transfer effects on peristaltic flow alsohave multiple applications in industry and biomedicalscience. Heat transfer includes multiple complex processesin tissues, i.e., heat conduction, heat convection because ofthe hemodynamics (“blood flow”) and radiation process.The mass transfer also has a valuable role in all theseprocesses. For instance, oxygenation and hemodialysisprocess has been examined by taking the peristaltic flowwith heat transfer. A complex relationship arises betweendriving potentials and fluxes in the presence of combinedmass and heat transfer effects. The energy flux persuadedby composition gradients and temperature gradientwhereas mass flux can be originated by a temperaturegradient (i.e., “soret effect”). The mass transfer has aremarkable role in the diffusion process, i.e., diffusion ofnutrients from the blood to neighboring tissues. Mass andheat transfer also arises in between fluid and solid particlesand flowing Newtonian/non-Newtonian media, i.e., multi-phase fluidized bed systems, bubble columns and boundarylayer flows. Furthermore, it can also encounter in themanufacturing of polymer alloys (via “liquid phase route”),devolatilization of films and food processing, etc. Mass andheat transfer process on Newtonian fluid models providesessential knowledge for single particles of highly romanti-cize geometry, i.e., cylinder, curved channels, flat plate, etc.but it becomes more complicated in multiparticle flows.Mass and heat transfer process arises in multiplegeometrical aspects with different boundary conditionsexamined extensively by various authors. Tripathi [17]analyzed the heat transfer effects on the peristaltic motion
of food bolus through an oesophagus and presented a newmathematical model. Further, they examined numericallythe behavior of two inherent mechanisms i.e., reflux andtrapping in the presence of heat transfer. Srinivas et al. [18]described the behavior of heat and mass transfer on theasymmetric peristaltic propulsion of Newtonian fluid in thepresence of slip impact. The three-dimensional non-uniform peristaltic motion of a viscous fluid with heatand mass transfer through a rectangular duct wasexamined by Ellahi et al. [19]. Bhatti et al. [20] studiedthe heat transfer on peristaltic blood flow containing smallparticles propagating in a non-uniform tube. They furtherassumed the variable viscosity of the blood and presentedthe exact results against particle and fluid phase. Recently,Bhatti et al. [21] studied the behavior of heat and masstransfer on the peristaltically induced motion of particle-fluid simultaneously through a finite channel with slipeffects. Again, Bhatti et al. [22] discussed the EMHD andnon-linear thermal radiation on peristaltic motion ofviscous dusty fluid through a porous medium.
To the best of our knowledge, no effort has been devotedto estimate the mass and heat transfer influence on theasymmetric peristaltic motion of viscous fluid having smallparticles through a curved channel. In the previous studies,it is found that different authors have discussed particle-fluid suspension, but asymmetric/symmetric peristalticmotion in the presence of small particles in curvedconfiguration was not discussed by anyone. To fill thisgap, the present study has been presented. Consequently, aremarkable effort has hitherto been dedicated to develop abetter understanding of the two-phase asymmetric peri-staltic motion of viscous fluid in the presence of mass andheat transfer. The inspiration of current study is that thetheory of multiphase flow is beneficial to analyze thesinusoidal muscular expansion and contraction in propa-gating of various biological fluids. The present flow ismodeled with a long wavelength and ignoring the inertialforces. Closed form solutions are obtained for particle andfluid phase.
2 Problem modelling
Let us consider an asymmetric peristaltic motion of aviscous fluid with incompressible, constant density andirrotational properties having small particles propagatingwith constant celerity. The geometry of walls (see Figure 1)is described as
Upper wall : h1ðx; tÞ ¼ a1 cos 2pl�1 x � ct½ � þ d1; ð1Þ
Lower wall : h2ðx; tÞ ¼ b1 cos 2pl�1 x þQ� ct½ � � d2: ð2ÞIn above equation, the phase difference Q having range
0�Q�p whereas Q=p is associated with a wave in aphase and Q=0 related to symmetric channel havingwaves out of phase. The other constants satisfy thefollowing condition.
a21 þ b21 þ 2a1b1cos � ðd1 þ d2Þ2: ð3Þ
Fig. 1. Flow structure.
A. Zeeshan et al.: Mechanics & Industry 19, 401 (2018) 3
2.1 Fluid phase
∂ ðrþR�Þvf� �
∂rþR� ∂uf
∂x¼ 0; ð4Þ
ð1� CÞrf∂vf∂t
þ vf∂vf∂r
þ ufR�
rþR�∂vf∂x
þ u2f
rþR�
!
¼ �ð1� CÞ ∂P∂r
þ 2msð1� CÞðrþR�Þ2
∂∂r
∂vf∂r
ðR� þ rÞ� �
þ msðR� �R�CÞðR� þ rÞ
∂∂x
∂uf
∂rþ R�
R� þ r
∂vf∂x
� uf
R� þ r
� �
� ð1� CÞ 2ms
rþR�
� �R�
R� þ r
∂uf
∂xþ vfR� þ r
� �
þ CS0ðvp � vfÞ; ð5Þ
ð1� CÞrf∂uf
∂tþ vf
∂uf
∂rþ R�
R� þ ruf
∂uf
∂xþ ufvfrþR�
� �
¼ �R�ð1� CÞrþR�
∂P∂x
þ msð1� CÞðrþR�Þ2
∂∂r
∂uf
∂rðR� þ rÞ2
� �þ ð1� CÞ ms
ðrþR�Þ2∂∂r
R�
R� þ r
∂vf∂x
� uf
rþR�
� �ðR� þ rÞ2
� �
þð1� CÞ 2msR�
R� þ r
� �∂∂x
∂uf
∂xR�
ðR� þ rÞ þvf
ðR� þ rÞ� �� �
þCS0ðup � ufÞ; ð6Þ
ð1� CÞrfCp∂Tf
∂tþ vf
∂Tf
∂rþ ufR
�
rþR�∂Tf
∂x
� �¼ Kð1� CÞ
∂2Tf
∂r2þ 1
R� þ r
∂Tf
∂rþ R�
R� þ r
� �2 ∂2Tf
∂x2
!
þ rfCpCðTp � TfÞ$T
þ CS0ðuf � upÞ2$r
; ð7Þ
ð1� CÞ ∂Ff
∂tþ vf
∂Ff
∂rþ ufR
�
rþR�∂Ff
∂x
� �¼ Dmð1� CÞ
∂2Ff
∂r2þ 1
R� þ r
∂Ff
∂rþ ∂2Ff
∂x2
R�
R� þ r
� �2!
þDm
TmðkT � kTCÞ
∂2Tf
∂r2þ 1
rþR�∂Tf
∂rþ R�
rþR�
� �2 ∂2Tf
∂x2
!
þ rfCpCS0ðFp � FfÞ$c
; ð8Þ
2.2 Particulate phase
∂ ðR�vp þ rvpÞ� �
∂rþ ∂up
∂xR� ¼ 0; ð9Þ
rp C∂vp∂t
þ C∂vp∂r
vp þ CupR
�
R� þ r
∂vp∂x
þ Cu2p
rþR�
!
¼ �C∂P∂r
� S0Cðvp � vfÞ; ð10Þ
rp C∂up
∂tþ vpC
∂up
∂rþ R�CR� þ r
up∂up
∂xþ upCvpR� þ r
� �
¼ � R�CR� þ r
∂P∂x
þ S0Cðuf � upÞ; ð11Þ
rfCpC∂Tp
∂tþ vp
∂Tp
∂rþ upR
�
rþR�∂Tp
∂xþ upvprþR�
� �
¼ rfCpCðTp � TfÞ$T
; ð12Þ
4 A. Zeeshan et al.: Mechanics & Industry 19, 401 (2018)
rpC∂Fp
∂tþ vp
∂Fp
∂rþ R�
rþR� up∂Fp
∂xþ upvprþR�
� �
¼ rpCpCðFf � FpÞ$c
: ð13Þwhere
S0 ¼ 9m0
2a2lðCÞ
lðCÞ ¼ 3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�3C2 þ 8C
pþ 4þ 3C
h i2� 3Cð Þ�2
ms ¼m0
1� xC0
x ¼ 0:07eC
1107
CTe1:69Cþ 2:49
� �
9>>>>>>>>>>=>>>>>>>>>>;; ð14Þ
To move from fixed frame to wave frame, let us definethe following variables as
x� ¼ x� ct;r� ¼ r;z� ¼ z; v� ¼ v;u�
¼ u� c;pðx�; z�Þ ¼ Pðx; z; tÞ: ð15Þ
Henceforward, the dimensionless quantities areSee equation 16 below pageUsing equation (15) and equation (16) in equations (3)–
(13), and assuming creeping flow with a long wavelength,the resulting equations for fluid phase can be written as
dp
dx¼ kþ r
k
� �1
kþ r
∂∂r
∂uf
∂rðkþ rÞ2
� ��
þ ∂∂r
� uf
rþ kðkþ rÞ2
� �Þ þM1Cðup � ufÞ; ð17Þ
∂2uf∂r2
þ 1
ðkþ rÞ∂uf∂r
¼ �PrM1ðup � ufÞ� PrEcM1Cðup � ufÞ2; ð18Þ
∂2ff
∂r2þ 1
ðkþ rÞ∂ff
∂rþNA
∂2uf∂r2
þ 1
ðrþ kÞ∂uf∂r
� �¼ �CM1ðfp � ffÞ; ð19Þ
uf ¼ u0 þ u1r þ u2r2 þ u3r
3 þ u4log r½r
up ¼ u0 þ u1r þ u2r2 þ u3r
3 þ u4log r þ k½ � þr þ k
x ¼ x�
�; r ¼ r�
d; u ¼ u�
c; v ¼ v�
c; p� ¼ d2p
�c�; � ¼ d
�; k ¼ R
d
h2� ¼ h2
d1; d� ¼ d2
d1; a� ¼ a1
d1; b� ¼ b1
d1; Re ¼ �f dc�
�; �f
Ec ¼ C 2
C T 1T 0ð Þ ; M 1 ¼ S 0a2
1Cð Þ�s
; Sc ¼ �s
�Dm; Sr ¼ �Dm
�sT
and for particulate phase
dp
dx¼ ð1� CÞM1ðuf � upÞ; ð20Þ
up ¼ uf ; ð21Þ
fp ¼ ff ; ð22Þ
Corresponding non-dimensional boundary conditionsare
ufðh1Þ ¼ �1ufðh2Þ ¼ �1
g; ð23Þ
where
h1ðxÞ ¼ 1þ acosðxÞ;h2ðxÞ ¼ �b� dcosðxþQÞ g; ð24Þ
The exact solutions for fluid and particle phase areobtained as
See equation (25) and (26) below page.
uf;p ¼ u0 þ u1log kþ r½ � þ u2rþ u3r2; ð27Þ
ff;p ¼ f0 þ f1log kþ r½ � þ f2rþ f3r2: ð28Þ
The constants with subscripts are defined in theappendix section.
The rate of volume flow is given by
Q ¼Z h2
h1
uf 1� Cð ÞdrþZ h2
h1
upCdr: ð29Þ
The pressure gradient can be evaluated with the help ofabove expression and we have
þ k� þ u5rlog r þ k½ � þ u6r2log r þ k½ �
þ k; ð25Þ
u5rlog r þ k½ � þ u6r2log r þ k½ � 1
ð1CÞM 1
dp
dx; ð26Þ
�; h1
� ¼ h1
d1; Pr ¼ �C
K; NA ¼ SrSc;
;p ¼ T f ;pT 0
T 1T 0; �f ;p ¼
F f ;pF 0
F 1F 0;
kT T 1T 0ð Þm F 1F 0ð Þ : : ð16Þ
9>>>>>=>>>>>;
Fig 2. Velocity curves for multiple values of C.
A. Zeeshan et al.: Mechanics & Industry 19, 401 (2018) 5
See equation (30) below page.The pressure rise Dp in dimensionless form is defined as
Dp ¼Z10
dp
dxdx: ð31Þ
3 Illustrations and physical discussion
The aim of this section is to see the variation of differentparameters on peristaltic pumping (Dp), streamlines,velocity (uf,p), pressure gradient (dp/dx), temperature(uf,p) and concentration distributions (’f,p). Numericalcomputation is used by means built-in “shooting algorithm”in “Mathematica (10.3v)” software. Particularly, weexplored the variation of all the physical quantities i.e.,particle volume fraction “C”, suspension parameter “M1”,curvature parameter “k”, volumetric flow rate “Q”, phasedifference “Q”, Prandtl number “Pr”, Eckert number “Ec”,Schmidt number “Sc” and Soret number “Sr”, respectively.It is worth to point out here that the current analysis is alsoapplicable for single phase flow by considering “C=0”.
3.1 Velocity behaviour and pumping characteristics
Figure 2a and b depicts the variation of particle volumefraction “C” on velocity profiles. It reveals from both figuresthat due to increment in the value of “C”, the variation invelocity is very small, however, the velocity of the fluid actsoppositely near the curvy walls. In the presence of solidparticles, the drag force rises and there is retardation in theflow. Moreover, for the other side of the channel (r> 0.1),the wall is asymmetric, and the drag force is less effective.The current results are also incorporated with previousstudies Bhatti and Zeeshan [23], Kamel et al. [12]. More
dp
dx¼
36 1þ Cð ÞM 1 �1�2ð Þ �2 þQ�1ð Þ �1 þðh
�1�2ð Þ2 2k þ �1 þ �2ð Þ 36þM 1 6 3þ Cð ÞK1��
þ36 1þ Cð ÞM 1 �1�2ð Þ Q þ �2�1ð Þ �1 þ �2 þ 2kð
h12M 1 �1 þ kð Þ2 2C �2�1ð Þ �2 þ kð Þ2 þ �2 þ kð
h
details are presented in Drew [23]. Figure 3a and b indicatesthat large values of curvature parameter “k” tend to resistthe flow markedly whereas on the opposite of the wall(r> 0.1), it fails to produce a significant resistance and as aresult, the velocity of the fluid rises. Figure 4a and b andFigure 5 present pressure gradient profiles which areplotted with the help of equation (29). It is observed fromFigure 4a and b that volumetric flow rate “Q” revealscompletely adverse attitude as compared to particlevolume fraction “C” on pressure gradient. It is observedthat higher flow rate causes less resistance in the flow andas a result, more pressure is required to control the flow.However, for particle volume fraction “C”, pressuregradient significantly declines due to the higher rate ofdrag force in the flow. In Figure 5 it is found that when thephase difference “Q” increases then pressure gradient alsoincreases. Figure 6a and b is drawn to visualize thepumping features. This is an important mechanism in ahuman body that is very supportive to move variousbiological fluids (i.e., cilia motion, blood pumping andtransport of urine) in a human body. Moreover, inbiomedical engineering numerous devices are manufac-tured on the phenomena of peristaltic pumping. Thesefigures are divided into four regions i.e., peristalticpumping {Dp> 0, Q> 0}, retrograde pumping {Dp> 0,Q< 0}, co-pumping {Dp< 0, Q> 0} and free pumping{Dp< 0, Q< 0}. In Figure 6a it is observed that themagnitude of pumping rate is very high in peristalticpumping region as well as in retrograde pumping andincreases for higher values of particle volume fraction “C”.However, its attitude becomes reverse at Q=0.6 andreveals opposite influence in the co-pumping region. It isclear from Figure 6b that suspension parameter “M1”significantly enhances the pressure rise in retrograde andperistaltic pumping regions while its behaviour becomesconverse at Q=0.5 and depicts opposite behaviour in theco-pumping region.
2k þ �2Þ þ 2 �1 þ kð Þ �2 þ kð Þlog kþ�1kþ�2
h ii2 þ 3 3þ Cð Þk �1 þ �2ð Þ þ 4C �21 þ �1�2 þ �22
��Þ þ 2 �1 þ kð Þ �2 þ kð Þlog kþ�1
kþ�2
h iiÞ23 1þ Cð Þklog kþ�1
kþ�2
h iilog kþ�1
kþ�2
h i : ð30Þ
Fig 3. Velocity curves for multiple values of k.
Fig 4. Pressure gradient for multiple values of Q and C.
Fig 5. Velocity curves for multiple values of Q.
6 A. Zeeshan et al.: Mechanics & Industry 19, 401 (2018)
3.2 Temperature distributions
Figure 7a and b and Figure 8 represent the temperaturedistributions and consist of the present results obtained inequation (27). From Figure 7a and b, it is concluded thatparticle volume fraction “C” and curvature parameter “k”markedly enhance the temperature profile. Figure 8 showsthe variation of Prandtl number “Pr” and Eckert number“Ec” on temperature distribution. In this figure, we can seethat both the parameters (i.e., Eckert number “Ec” andPrandtl number “Pr”) enhance the temperature. The rationbetween momentum diffusivity to thermal diffusivity isknown as Prandtl number. Inspection of Figure 8 alsoreveals that when Prandtl number is high, then thermaldiffusivity is more prominent on momentum diffusivity. Itis worth mentioning here that such type of results is
substantial for those fluids which have a large Prandtlnumber and less significant for low Prandtl number fluidssuch as ionized gasses etc.
3.3 Concentration distributions
Figures 9–11 are plotted for concentration distributions tosee the physical effects of involved parameters. It can beviewed from Figure 9a that concentration distributionsignificantly diminishes due to amore significant influence ofparticle volume fraction. However, for large values ofcurvature parameter, concentration distribution markedlyrises (see Fig. 9b). From Figure 10 we can see that highervalues of Eckert number and Prandtl number produce amarked reduction in a concentration distribution. Figure 11is plotted for “NA=(ScSr)” (product of Schmidt number “Sc”
and a Soret number “Sr”. Schmidt number “Sc ¼ ms
rDm
�” is
the ratio between momentum and mass diffusivity and it isbeneficial to determine the convection process of mass andmomentum diffusion. This figure reveals that an incrementin “NA” tends to resist the concentration distribution. Ithappens because mass diffusivity becomes more dominatedover momentum diffusivity and due to higher Soret number,the particles move to cold region from hot region and as aresult, the concentration distribution diminishes.
3.4 Trapping mechanism
This subsection deals with the trapping phenomenawhich are observed through the streamlines. Streamlinesrepresent the “family of curves” that are promptly tangent
Fig 7. Temperature profile for multiple values of C and k.
Fig 8. Temperature profile for multiple values of Pr and Ec.
Fig. 6. Pressure rise vs. volume flow rate for various values of C and M1.
A. Zeeshan et al.: Mechanics & Industry 19, 401 (2018) 7
to velocity vectors of a flow. Moreover, it is composed ofbolus circulating inside. The composition of bolus circulat-ing the fluid is embedded by numerous streamlines that areknown as “Trapping”. These inner circulating bolusespropagate along the peristaltic wave. This mechanism isbeneficial in blood flow situations, i.e., composition ofthrombus and propagation of food bolus in a gastrointesti-nal tract. Figures 12–14 depict the streamlines for multiplevalues of associated parameters in momentum equations. Itcan be seen from Figures 12 and 13 that by increasingparticle volume fraction “C” and volumetric flow rate “Q”the number of streamlines reduces significantly. However,we have observed that the number of streamlines increasesvery slowly with the increment in curvature parameter “k”.
4 Concluding remarks
The viscous particle-fluid (“two-phase”) model has beenused to analyse the simultaneous impact of mass and heattransfer on peristaltically induced motion through anasymmetric curve channel. The exact solution expressionsof velocity, concentration, pressure gradient and tempera-ture are obtained whereas numerical integration has beencarried to explore the pumping features. Graphicalillustrations are presented against multiple values ofinvolved several parameters. The critical findings aredescribed below
– there is a critical value of r around which the velocitydistribution acts opposite;–
pressure gradient enhances due to the increment involume flow rate and phase difference;–
pressure gradient tends to diminish significantly for largevalues of C;–
peristaltic pumping and retrograde pumping regiondecreases for higher values of C and opposite forsuspension parameter M1;–
Prandtl number, particle volume fraction, and Eckertnumber have an increasing impact on temperatureprofile;–
curvature parameter fails to provide a significantresistance temperature distribution, while its attitudeis converse for concentration profile;–
concentration profile acts in similar form against C NA(product of SrSc).
Fig 11. Concentration profile for multiple values of NA.Fig 10. Concentration profile for multiple values of Pr and Ec.
Fig. 12. Graph of stream lines for multiple values of C=0.1, 0.2, 0.3.
Fig 9. Temperature profile for multiple values of C and k.
8 A. Zeeshan et al.: Mechanics & Industry 19, 401 (2018)
The present study has neglected rheological workingfluids. It will be examined in the next studies with differentbiological non-Newtonian fluids. Moreover, the presentstudy addresses various engrossing results and character-istics that warrant further investigation on two-phase flowmodel. The subject reveals a platform for the mathematicalformulation, and hopefully it will help further to examinethe two-phase non-Newtonian fluids experimentally withmass and heat transfer.
Nomenclature
u, v
Velocity components Tm Mean temperatureM1
Suspension parameter Dm Mass diffusivity kT Thermal diffusivity Ec Eckert number P Pressure R* Radius of curvature Pr Prandtl number C Volume fraction density ~t Time Re Reynolds number Sc Schmidt number Cp Effective heat capacity Sr Soret number K Thermal conductivity F Concentration
Fig. 14. Graph of stream lines for multiple values of k=0.5, 0.6, 0.7.
Fig. 13. Graph of stream lines for multiple values of Q=0.2, 0.4, 0.6.
A. Zeeshan et al.: Mechanics & Industry 19, 401 (2018) 9
T
Temperature S’ Drag coefficient Q Volume flow rate d1+ d2 Channel width a1, b1 Waves amplitudes k Curvature parameter c Wave celerityGreek symbols
uf,p
Temperature $T Thermal equilibrium time ms Viscosity of the fluid Q Phase difference ff;p Concentration l Wavelength r Fluid density $v Relaxation time of the particle f Stream functionSubscripts
p
Particulate phase f Fluid phaseReferences
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Cite this article as: A. Zeeshan, N. Ijaz, M.M. Bhatti, Flow analysis of particulate suspension on an asymmetric peristaltic motionin a curved configuration with heat and mass transfer, Mechanics & Industry 19, 401 (2018)
A. Zeeshan et al.: Mechanics & Industry 19, 401 (2018) 11
Appendix
u0 ¼2k h1 � h2ð Þ 6þ C �6þ dp=dx 3k h1 þ h2ð Þ þ 2 h21 þ h1h2 þ h22
� � � �6 �1þ Cð Þ h1 � h2ð Þ 2kþ h1 þ h2ð Þ
þ dp
dx
3 �1þ Cð Þk kþ h1ð Þ2h2log h1 þ k½ � h2 þ 2kð Þ � 2kh1 þ h21
h2 þ kð Þ2log h2 þ k½ �h i
6 �1þ cð Þ h1� h2ð Þ 2K1þ h1þ h2ð Þ ;
u1 ¼�h2 þ h1ð Þ 3þ C �3þ dp=dx �3k2 þ h21 þ h1h2 þ h22
� �� �3 �1þ cð Þ h1 � h2ð Þ 2kþ h1 þ h2ð Þ
þ3 1� Cð Þk2dp=dx kþ h1ð Þ2log h1 þ k½ � � h2 þ kð Þ2log h2 þ k½ �
h i3 �1þ cð Þ �h2 þ h1ð Þ 2kþ h1 þ h2ð Þ ;
u2 ¼ dp
dx
�2C h1 � h2ð Þ 2kþ h1 þ h2ð Þ þ 3 1� Cð Þ k kþ h1ð Þ2log h1 þ k½ � � h2 þ kð Þ2klog h2 þ k½ �h i
6 �1þ Cð Þ h1 � h2ð Þ 2kþ h1 þ h2ð Þ ;
u3 ¼ C
3 1� Cð Þ ; u4 ¼ dp
dx
k3 �1þ Cð Þ h1 � h2ð Þ h1 þ h2ð Þ þ 2k �1þ Cð Þh1 þ h2 � Ch2ð Þð Þ2 �1þ Cð Þ h1 � h2ð Þ 2kþ h1 þ h2ð Þ ;
u5 ¼ k2dp
dx; u6 ¼ k
2
dp
dx;
u0 ¼4þ CEcM1 dp=dxð Þ2Prh2 2kþ h2ð Þ �
log kþ h1½ � � CEcM1 dp=dxð Þ2 2Prh1kþ Prh1h1ð Þlog kþ h2½ �4 log h1 þ k½ � � log h2 þ k½ �ð Þ ;
u1 ¼ �4þ CEcM1 dp=dxð Þ2Pr h1 � h2ð Þ 2kþ h1 þ h2ð Þ4 log kþ h1½ � � log kþ h2½ �ð Þ ; u2 ¼ � 1
2CEcM1k
dp
dx
� �2
Pr; u3 ¼ � 1
2CEcM1
dp
dx
� �2
Pr;
f0 ¼� �4þ CEcM1NA dp=dxð Þ2Prh2 2kþ h2ð Þ �
log kþ h1½ � � CEcM1 dp=dxð Þ2NAPrh1 2kþ h1ð Þlog kþ h2½ �4 log kþ h1½ � � log kþ h2½ �ð Þ ;
f1 ¼�4þ CEcM1NA dp=dxð Þ2Pr h1 � h2ð Þ 2kþ h1 þ h2ð Þ
4 log kþ h1½ � � log kþ h2½ �ð Þ ; f2 ¼ �NA
2CEcM1k
dp
dx
� �2
Pr; f3 ¼ �NA
2CEcM1
dp
dx
� �2
Pr;