Abstract
The primary objective of present investigation is to introduce the novel aspect of thermophoresis in the mixed convective peristaltic transport of viscous nanofluid. Viscous dissipation and Joule heating are also taken into account. Problem is modeled using the lubrication approach. Resulting system of equations is solved numerically. Effects of sundry parameters on the velocity, temperature, concentration of nanoparticles and heat and mass transfer rates at the wall are studied through graphs. It is noted that the concentration of nanoparticles near the boundaries is enhanced for larger thermophoresis parameter. However reverse situation is observed for an increase in the value of Brownian motion parameter. Further, the mass transfer rate at the wall significantly decreases when Brownian motion parameter is assigned higher values.
Introduction
Mixed convection in vertical channels is of considerable importance for the enhancement of cooling systems in engineering. This includes modern heat exchangers, nuclear reactors, solar cells and many other electronic devices. Such flows are mainly affected by buoyancy. MHD mixed convective heat transfer analysis in vertical channels is of considerable importance due to its applications in self-cooled or separately cooled liquid metal blankets, cooling systems for electronic devices, solar energy collection and chemical processes. Use of nanoparticles as means to enhance the heat transfer in low thermal conductivity fluids has proven to be a novel technique [1]. Addition of nanoparticles has proven to facilitate the heat transfer phenomena by enhancing the thermal conductivity of low thermal conductivity fluids e.g. water, engine oil etc. Presently the nanofluids are largely used in medical, engineering and electrical processes. Such fluids find applications in hyperthermia, cryosurgery, modern drug delivery systems, heating and cooling systems, batteries of electronic devices, etc. Wide utility of nanofluids has paved the way for a new science named nanofluid mechanics. The science of nanofluids mechanics has gained considerable attention of motivated researchers from all over the world (see refs. [2]–[5]). Recently, Turkyilmazoglu [6] investigated the heat and mass transfer of MHD slip flow in nanofluids. In this problem the author computed the exact solutions of the involved equations. Natural convective boundary layer flow of nanofluid past a vertical surface is studied by Rashidi et al. [7]. Turkyilmazoglu [8] obtained the analytical solution for MHD mixed convection heat transfer flow of viscoelastic fluid over a stretching surface. Sheikholeslami et al. [9] investigated the nanofluid filled enclosure in presence of magnetic field with heat flux boundary. Turkyilmazoglu [10] analyzed the nanofluid flow and heat transfer due to rotating disk. Rashidi et al. [11] reported the second law analysis in flow of nanofluid induced by a rotating disk. Sheikholeslami and Ganji [12] studied the magnetohydrodynamic flow of viscous nanofluid in a permeable channel.
Peristaltic pumping is a mechanism of the fluid transport in a flexible tube by a progressive wave of contraction or expansion from a region of lower pressure to higher pressure. Peristalsis is one of the major mechanisms for fluid transport in physiology. It is an involuntary and key mechanism that moves food through the digestive tract, bile from the gallbladder into the duodenum, transport of blood through the artery with mild stenosis, urine from the kidneys through the ureters into the bladder and sperm through male reproductive track. Further, several engineering appliances including roller and finger pumps, hose pumps, dialysis and heart-lung machines are designed on the principle of peristalsis. Subject to such extensive applications, several investigators studied peristaltic flows under different flow configurations [13]–[17]. Recent developments in hyperthermia, cryosurgery and laser therapy as means to destroy the undesirable tissues in cancer therapy has triggered great interest in the analysis of peristaltic flows with heat transfer (see refs. [18]–[21]). Peristaltic transport of nanofluids is significant in modern drug delivery systems and cancer therapy. Review of the available literature indicates that studies focusing the peristaltic transport of nanofluids are scarce. Some of the available studies for peristaltic transport of nanofluids can be seen through refs. [22]–[24].
It is noticed that none of the above mentioned studies and others in the existing literature investigate the novel aspect of peristaltic transport with the thermophoresis boundary condition. This study aims to fill this void. Therefore this article investigates the peristaltic transport of viscous nanofluid under the influence of constant applied magnetic field. Incompressible fluid is taken in a channel with thermophoresis condition at the boundaries [4], [5]. Mixed convection, viscous dissipation and Joule heating are also taken into account. Mathematical modelling is carried out using the long wavelength and low Reynolds number approximations. Resulting coupled equations are numerically solved using Mathematica 8.0. Graphical analysis is carried out to examine the influence of various embedded parameters on the velocity, temperature, concentration of nanoparticles and heat and mass transfer rates at the wall.
Mathematical Analysis
Consider a viscous nanofluid in two-dimensional symmetric channel of width 
Walls of the channel are maintained at constant temperature 
A uniform magnetic field of strength 
is applied in a direction normal to the channel walls. Effects of induced magnetic field are discarded under the assumption of low magnetic Reynolds number. Cartesian coordinate system is defined in such a manner that the 
axis lies along the length of channel and the 
axis is normal to the 
axis. Sinusoidal waves with amplitude 
and wavelength 
propagate along the channel walls in the 
direction with constant speed 
Flow within the channel is induced due to the propagation of such waves. Geometry of the peristaltic walls is given below:
 |
(1) |
where 
is time and 
and 
are the walls lying in the positive and negative 
direction respectively. Velocity field for the two-dimensional flow in Cartesian coordinate system is given by 
Law of conservation of mass, linear momentum, energy and concentration are given below:
 |
(2) |
 |
(3) |
 |
(4) |
 |
(5) |
 |
(6) |
Here 
is the pressure, 
the density of nanofluid, 
the density of base fluid, 
the density of nanoparticles, 
the dynamic viscosity of nanofluid, 
the acceleration due to gravity, 
the thermal expansion coefficient, 
the concentration expansion coefficient, 
the fluid temperature, 
the concentration, 
the specific heat of nanofluid, 
the thermal conductivity of nanofluid, 
the mass diffusivity, 
the mean temperature and 
where 
designates the effective heat capacity of nanoparticles. We transform our problem into a frame of reference (wave frame) moving along the wave with same speed. The quantities from the fixed frame 
to wave frame 
are related by the following transformations:
 |
(7) |
in which 
and 
are the velocity components and pressure in wave frame 
Transformed set of equations takes the form
 |
(8) |
 |
(9) |
 |
(10) |
 |
(11) |
 |
(12) |
We define the following dimensionless quantities in order to non-dimensionalize the problem
 |
(13) |
The assumptions of long wavelength and small Reynolds number give 
and 
It should be pointed out that the theory of long wavelength and zero Reynolds number remains applicable for case of chyme transport in small intestine [25]. In this case c = 2 cm/min, 
cm and 
cm. Here half width of intestine is small in comparison to wavelength. i.e. 
It is also declared by Lew et al. [26] that Reynolds number in small intestine was small. Further, the situation of intrauterine fluid flow due to myometrial contractions is a peristaltic type fluid motion in a cavity. The sagittal cross section of the uterus reveals a narrow channel enclosed by two fairly parallel walls [27]. The 1–3 mm width of this channel is very small compared with its 50 mm length [28], defining an opening angle from cervix to fundus of about 0.04 rad. Analysis of dynamics parameters of the uterus revealed frequency, wavelength, amplitude and velocity of the fluid-wall interface during a typical contractile wave were found to be 0.01–0.057 Hz, 10–30 mm, 0.05–0.2 mm and 0.5–1.9 mm/s. Applying the long wavelength and low Reynolds number approximations [25]–[30] the Eqs. (9–12) in view of Eq. (13) are reduced to
 |
(14) |
 |
(15) |
 |
(16) |
 |
(17) |
Here 
denotes the stream function, 
the wave number, 
the Hartman number, 
the Reynolds number, 
the thermal Grashoff number, 
the concentration Grashoff number, 
the Brinkman number, 
the Eckret number, 
the Prandtl number, 
dimensionless concentration, 
the dimensionless temperature, 
the Brownian motion parameter, 
the thermophoresis parameter and as a result of long wavelength and low Reynolds number approximations pressure becomes independent of transverse coordinate which can be seen through Eq. (15). the continuity equation is identically satisfied. Cross differentiation of Eqs. (14) and (15) gives
 |
(18) |
Defining 
and F as the dimensionless mean flows in the laboratory and wave frames by
 |
one can write
 |
where
 |
(19) |
Kuznetsov and Nield [4], [5] proposed that instead of considering constant concentration of nanoparticles at the boundaries, it is more realistic to take into account the thermophoresis at the boundaries. Hence we also assume that in presence of thermophoresis the normal fluxes of nanoparticles is zero at the boundaries. Mathematically, this condition is given as
 |
Thus the dimensionless boundary conditions in the present flow are
 |
(20) |
where 
is defined in Eq. (19) and 
The system of Eqs. (16–18) subject to boundary conditions Eq. (20) is solved numerically through NDSolve in Mathematica. Obtained numerical results are analyzed graphically in the next section.
Discussion
This portion of the article aims to analyze the numerical results through graphs. Plots for axial velocity, temperature and concentration are prepared and analyzed. Heat and mass transfer rates at the wall are studied through bar-charts.
Analysis of axial velocity
Figs. 1–5 are prepared to examine the influences of various parameters on velocity profile. All these Figs. show that velocity attains maximum value near the centre of channel. Also it can be seen from these Figs. that effects of different parameters on velocity are opposite near the channel walls compared with that of near the centre of channel. Fig. 1 depicts that velocity near the centre of channel decreases by increasing Hartman number. This signifies the reduction of maximum velocity subject to an increase in the strength of applied magnetic field. Thermal Grashoff number has an increasing effect of the velocity near the centre (see Fig. 2). Substantial increase in the maximum velocity is seen for an increase in thermal Grashoff number. Contrary to previous case, maximum velocity decreases when concentration Grashoff number is increased (see Fig. 3). Brownian motion and thermophoresis parameters have opposite effects on the velocity (see Figs. 4 and 5). Maximum velocity is found to increase when Brownian motion parameter is enhanced.
Figure 1. Effect of Hartman number on the velocity of nanofluid when 
, 
and 
.
Figure 5. Effect of thermophoresis parameter on the velocity of nanofluid when 
, 
and 
.
Figure 2. Effect of thermal Grashoff number on the velocity of nanofluid when 
, 
and 
.
Figure 3. Effect of concentration Grashoff number on the velocity of nanofluid when 
, 
and 
.
Figure 4. Effect of Brownian motion parameter on the velocity of nanofluid when 
, 
and 
.
Analysis of temperature
Temperature profile for variation in different parameters is studied through Figs. 6–10. Significant rise in the temperature of fluid is observed when Hartman number is increased (see Fig. 6). This fact is physically justified due to Joule heating. Fig. 7 shows that an increase in thermal Grashoff number results in an increase in temperature of fluid. Such increase in temperature however is small when compared with that of previous case. Fig. 8 indicates that increase in concentration Grashoff number slightly decreases the fluid temperature. Increase in Brownian motion parameter enhances the temperature (see Fig. 9). Such increase becomes insignificant for Nb>1. Fig. 10 shows a decrease in temperature when thermophoresis parameter is increased.
Figure 6. Effect of Hartman number on the temperature profile when 
, 
and 
.
Figure 10. Effect of thermophoresis parameter on the temperature profile when 
, 
and 
.
Figure 7. Effect of thermal Grashoff number on the temperature profile when 
, 
and 
.
Figure 8. Effect of concentration Grashoff number on the temperature profile when 
, 
and 
.
Figure 9. Effect of Brownian motion parameter on the temperature profile when 
, 
and 
.
Concentration of nanoparticles
Impact of different parameters on the concentration of nanoparticles is examined through Fig. 11–16. These graphs indicate that the concentration of nanoparticles near the channel walls is largely influenced by the change in embedded parameters. This is mainly due to consideration of the thermophoresis condition for concentration. A common observation from these Figs. is that the impact of a parameter on concentration of nanoparticles is proportional to the impact of that parameter on fluid temperature. Fig. 11 depicts that the concentration of nanoparticles considerably increases with an increase in Hartman number. This is mainly the consequence of Joule heating and the thermophoresis boundary condition. Concentration of the nanoparticles enhances with an increase in the value of thermal Grashoff number (see Fig. 12). Contrary to this case the concentration of nanoparticles decreases with an increase in the value of concentration Grashoff number (see Fig. 13). The Brownian motion and thermophoresis parameters have opposite effects on the concentration of nanoparticles (see Figs. 14 & 15). Considerable increase in the concentration of nanoparticles is reported when the value of thermophoresis parameter increases. Such increase becomes large as we move from the center of channel towards the channel walls.
Figure 11. Concentration profile for variation in Hartman number when 
, 
and 
.
Figure 16. Effect of Hartman number on the heat transfer rate the wall 
when 
, 
and 
.
Figure 12. Concentration profile for variation in thermal Grashoff number when 
, 
and 
.
Figure 13. Concentration profile for variation in concentration Grashoff number when 
, 
and 
.
Figure 14. Concentration profile for variation in Brownian motion parameter when 
, 
and 
.
Figure 15. Concentration profile for variation in thermophoresis parameter when 
, 
and 
.
Heat and mass transfer rates
Effects of Hartman number, Brownian motion and thermophoresis parameters on the heat and mass transfer rates at the wall are studied through bar-charts given in Figs. 16–21. Considerable increase in the heat transfer rate at the wall is noted when the strength of applied magnetic field is increased (see Fig. 16). The Brownian motion and thermophoresis parameters have very little effect on heat transfer rate at the wall (see Figs. 17–18). This is primarily due to the fact that we have assumed the channels walls to possess constant temperature. Heat transfer rate at the wall slightly increases with an increase in Brownian motion parameter. However this observation is not true for 
Fig. 19 depicts that the mass transfer rate at the wall significantly increases with an increase in the value of Hartman number. This is primarily due to the effect of applied magnetic field. The Brownian motion and thermophoresis parameters have opposite effects on the temperature of fluid. Considerable decrease in mass transfer rate at the wall is seen for an increase in Brownian motion parameter. This is the consequence of considered thermophoresis condition for concentration.
Figure 21. Effect of thermophoresis parameter on the mass transfer rate the wall 
when 
, 
and 
.
Figure 17. Effect of Brownian motion parameter on the heat transfer rate the wall 
when 
, 
and 
.
Figure 18. Effect of thermophoresis parameter on the heat transfer rate the wall 
when 
, 
and 
.
Figure 19. Effect of Hartman number on the mass transfer rate the wall 
when 
, 
and 
.
Figure 20. Effect of Brownian motion parameter on the mass transfer rate the wall 
when 
, 
and 
.
Comparison with available results
In order to check the validity of the solution methodology and the obtained results, we provide the comparison of the special case of present study with the results of Ali et al. [13]. Comparison between the numerical values of pressure rise per wavelength obtained through exact solutions by Ali et al. [13] is provided with that of the special case of present study (see Table 1). It is seen that the two results are in good agreement which confirms the validity of present results.
Table 1. Comparison of the critical values of flow rate below which 
is positive and above which it is negative computed by Ali et al. [13] and limiting case of present study.
|
Ali et al. [13] | Present study |
| 0.0 | 0.3683 | 0.368529 |
| 0.03 | 0.3533 | 0.353565 |
| 0.06 | 0.3411 | 0.341398 |
Conclusions
MHD mixed convective peristaltic transport of viscous nanofluid in a channel with thermophoresis at the boundaries is examined. Key findings of this study are summarized below.
Thermal and concentration Grashoff numbers have opposite effects on the axial velocity and temperature of nanofluid.
Axial velocity decreases with an increase in the value of Brownian motion parameter whereas opposite is reported for thermophoresis parameter.
Temperature of the nanofluid decreases with increase in the strength of applied magnetic field and Brownian motion of nanoparticles.
Increase in the strength of applied magnetic field results in an increase in nanoparticles concentration near the channel walls.
The Brownian motion and thermophoresis parameters have large but opposite effects on the concentration of nanoparticles.
Significant decrease in the mass transfer rate at the wall is noted when Brownian motion parameter is assigned higher values.
Data Availability
The authors confirm that all data underlying the findings are fully available without restriction. All relevant data are within the paper.
Funding Statement
This paper is funded by the Deanship of Scientific Research King Abdulaziz University, Jeddah, Saudi Arabia under grant no. 37-16-35-HiCi. The funder has no role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript.
References
- 1. Choi SUS (1995) Enhancing thermal conductivity of fluid with nanofluid, in: D. A. Siginer, Wang HP (Eds.), Developments and applications of non-Newtonian flows. ASME J Heat Transfer 66: 99–105. [Google Scholar]
- 2. Khanafer K, Vafai K, Lightstone M (2003) Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. Int J Heat Mass Transfer 46: 3639–3653. [Google Scholar]
- 3. Buongiorno J (2006) Convective transport in nanofluids. ASME J Heat Transfer 128: 240–250. [Google Scholar]
- 4. Kuznetsov AV, Nield DA (2013) The Cheng-Minkowycz problem for natural convective boundary layer flow in a porous medium saturated by a nanofluid: A revised model. Int J Heat Mass Transfer 65: 682–685. [Google Scholar]
- 5. Kuznetsov AV, Nield DA (2014) Natural convective boundary-layer flow of a nanofluid past a vertical plate: A revised model. Int J Thermal Sci 77: 126–129. [Google Scholar]
- 6. Turkyilmazoglu M (2012) Exact analytical solutions for heat and mass transfer of MHD slip flow in nanofluids. Chemical Eng Science 84: 182–187. [Google Scholar]
- 7. Rashidi MM, Bég OA, Asadi M, Rastegari MT (2012) DTM- Padé modeling of natural convective boundary layer flow of a nanofluid Past a vertical surface. Int J Thermal Environmental Engin 4: 13–24. [Google Scholar]
- 8. Turkyilmazoglu M (2013) The analytical solution of mixed convection heat transfer and fluid flow of a MHD viscoelastic fluid over a permeable stretching surface. Int J Mech Sci 77: 263–268. [Google Scholar]
- 9. Sheikholeslami M, Gorji-Bandpy M, Ganji DD, Soleimani S (2014) Heat flux boundary condition for nanofluid filled enclosure in presence of magnetic field. J Mol Liq 193: 174–184. [Google Scholar]
- 10. Turkyilmazoglu M (2014) Nanofluid flow and heat transfer due to a rotating disk. Comput. Fluids 94: 139–146. [Google Scholar]
- 11. Rashidi MM, Abelman S, Mehr NF (2013) Entropy generation in steady MHD flow due to a rotating porous disk in a nanofluid. Int J Heat Mass Transfer 62: 515–525. [Google Scholar]
- 12. Sheikholeslami M, Ganji DD (2014) Magnetohydrodynamic flow in a permeable channel filled with nanofluid. Scientia Iranica B 21: 203–212. [Google Scholar]
- 13. Ali N, Hussain Q, Hayat T, Asghar S (2008) Slip effects on the peristaltic transport of MHD fluid with variable viscosity. Phys Lett A 372: 1477–1489. [Google Scholar]
- 14. Mekheimer KhS, Abd Elmaboud Y, Abdellateef AI (2013) Particulate suspension flow induced by sinusoidal peristaltic waves through eccentric cylinders: thread annular. Int J Biomath 06: 1350026. [Google Scholar]
- 15. Mekheimer KhS, Komy SR, Abdelsalamd SI (2013) Simultaneous effects of magnetic field and space porosity on compressible Maxwell fluid transport induced by a surface acoustic wave in a microchannel. Chin Phys B 22: 124702. [Google Scholar]
- 16.Abbasi FM, Alsaedi A, Hayat T (2014) Peristaltic transport of Eyring-Powell fluid in a curved channel. J. Aerosp Eng doi:10.1061/(ASCE)AS.1943–5525.0000354
- 17. Hayat T, Abbasi FM, Ahmad B, Alsaedi A (2014) Peristaltic transport of Carreau-Yasuda fluid in a curved channel with slip effects. Plos One 9: e95070. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 18. Srinivas S, Kothandapani M (2008) Peristaltic transport in an asymmetric channel with heat transfer-A note. Int Commun Heat Mass Transfer 35: 514–522. [Google Scholar]
- 19. Tripathi D (2013) Study of transient peristaltic heat flow through a finite porous channel. Math Comput Modell 57: 1270–1283. [Google Scholar]
- 20. Hayat T, Abbasi FM, Alsaedi A, Alsaadi F (2014) Hall and Ohmic heating effects on the peristaltic transport of Carreau-Yasuda fluid in an asymmetric channel. Z Naturforsch 69a: 43–51. [Google Scholar]
- 21. Hayat T, Abbasi FM, Alsaedi A (2014) Soret and Dufour effects on peristaltic flow in an asymmetric channel. Arab J Sci Eng 39: 4341–4349. [Google Scholar]
- 22. Abbasi FM, Hayat T, Ahmad B, Chen GQ (2014) Slip effects on mixed convective peristaltic transport of copper-water nanofluid in an inclined channel. Plos One 9: e105440. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 23. Hayat T, Abbasi FM, Al-Yami M, Monaquel S (2014) Slip and Joule heating effects in mixed convection peristaltic transport of nanofluid with Soret and Dufour effects. J Mol Liq 194: 93–99. [Google Scholar]
- 24. Abbasi FM, Hayat T, Ahmad B, Chen GQ (2014) Peristaltic motion of non-Newtonian nanofluid in an asymmetric channel. Z Naturforsch 69a: 451–461. [Google Scholar]
- 25. Srivastava LM, Srivastava VP (1988) Peristaltic transport of a power law fluid: Applications to the ductus efferentes of the reproductive tract. Rheol Acta 27: 428–433. [Google Scholar]
- 26. Lew SH, Fung YC, Lowenstein CB (1971) Peristaltic carrying and mixing of chyme. J Biomech 4: 297–315. [DOI] [PubMed] [Google Scholar]
- 27. Eytan O, Elad D (1999) Analysis of intra-uterine fluid motion induced by uterine contractions. Bull Math Biol 61: 221–238. [DOI] [PubMed] [Google Scholar]
- 28. Strohmer H, Obruca A, Rander KM, Feichtinger W (1994) Relationship of the individual uterine size and the endometrial thickness in stimulated cycles. Fertil Steril 61: 972–975. [DOI] [PubMed] [Google Scholar]
- 29.Akbar NS, Khan ZH (2014) Heat transfer analysis on the peristaltic instinct of biviscosity fluid with the impact of thermal and velocity slips. Intl Commun Heat Mass Transfer doi:10.1016/j.icheatmasstransfer.2014.08.036
- 30. Mustafa M, Abbasbandy S, Hina S, Hayat T (2014) Numerical investigation on mixed convective peristaltic flow of fourth grade fluid with Dufour and Soret effects. J Taiwan Institute of Chemical Engineers 45: 308–316. [Google Scholar]
Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
The authors confirm that all data underlying the findings are fully available without restriction. All relevant data are within the paper.