Advertisement
Browse Subject Areas
?

Click through the PLOS taxonomy to find articles in your field.

For more information about PLOS Subject Areas, click here.

Mixed Convection Peristaltic Flow of Third Order Nanofluid with an Induced Magnetic Field

Mixed Convection Peristaltic Flow of Third Order Nanofluid with an Induced Magnetic Field

  • Saima Noreen
PLOS
x
  • Published: November 18, 2013
  • DOI: 10.1371/journal.pone.0078770

Correction

20 Dec 2013: Noreen S (2013) Correction: Mixed Convection Peristaltic Flow of Third Order Nanofluid with an Induced Magnetic Field. PLoS ONE 8(12): 10.1371/annotation/c18260c7-d0ff-4f79-82f7-55e9cf254f04. doi: 10.1371/annotation/c18260c7-d0ff-4f79-82f7-55e9cf254f04 View correction

Abstract

This research is concerned with the peristaltic flow of third order nanofluid in an asymmetric channel. The governing equations of third order nanofluid are modelled in wave frame of reference. Effect of induced magnetic field is considered. Long wavelength and low Reynolds number situation is tackled. Numerical solutions of the governing problem are computed and analyzed. The effects of Brownian motion and thermophoretic diffusion of nano particles are particularly emphasized. Physical quantities such as velocity, pressure rise, temperature, induced magnetic field and concentration distributions are discussed.

Introduction

Peristaltic motion is now an important research topic due to its immense applications in engineering and physiology. This type of rhythmic contraction is the basis of peristaltic pumps that move fluids through tubes without direct contact with pump components. This is a particular advantage in biological/medical applications where the pumped material need not to contact any surface except the interior of the tube. The word “peristalsis” comes from a Greek word “Peristaltikos”which means clasping and compressing. The peristaltic flow has specific involvement in the transport of urine from kidney to the bladder, chyme movement in gastrointestinal tract, movement of ovum in the female fallopian tubes, blood circulation in the small blood vessels, roller and finger pumps, sanitary fluid transport and many others. Latham [1] and Shapiro et al. [2] reported initial studies for the peristaltic flow of viscous fluid. Since then ample attempts have been made for peristalsis in symmetric flow geometry (seefewrecentstudies [3][8]). Recently, physiologists argued that the intra-uterine fluid flow (because of mymometrical contractions) represents peristaltic mechanism and the myometrical contractions may appear in both asymmetric and symmetric channels [9]. Hence some researchers [10][15] discussed the peristaltic transport in an asymmetric channel with regard to an application of intra-uterine fluid flow in a nonpregnant uterus.

Heat transfer in cooling processes is quite popular area of industrial research. Conventional methods for increasing cooling rates include the extended surfaces such as fins and enhancing flow rates. These conventional methods have their own limitations such as undesirable increase in the thermal management system's size and increasing pumping power respectively. The thermal conductivity characteristics of ordinary heat transfer fluids like oil, water and ethylene glycol mixture are not adequate to meet today's requirements. The thermal conductivity of these fluids have key role in heat transfer coefficient between the heat transfer medium and heat transfer surface. Hence many techniques have been proposed for improvement in thermal conductivity of ordinary fluids by suspending nano particles in liquids. The term “nano” introduced by Choi [16] describes a liquid suspension containing ultra-fine particles (diameter less than 50 nm). The nanoparticle can be made of metal, metal oxides, carbide, nitride and even immiscible nano scale liquid droplets. Although the literature on flow of viscous nanofluid has grown during the last few years (see [17][32] andmanyrefs.therein) but the information regarding peristaltic flow of nano fluids is yet scant. To our information, Akbar and Nadeem [33] studied the peristaltic flow of viscous nanofluid with an endoscope. Influence of partial slip in peristaltic flow of viscous fluid is explained by Akbar et al. [34].

The aim of present study is to venture further in the regime of peristalsis for fluids with nanoparticles. Therefore we examine here the mixed convective peristaltic transport of third order nanofluid in an asymmetric channel. Channel asymmetry is produced by peristaltic waves of different amplitude and phases. Mathematical modelling involves the consideration of induced magnetic field, Brownian motion and thermophorsis effects. Numerical solution of nonlinear problem is obtained using shooting method. Limiting case for viscous nanofluid in symmetric channel is also analyzed. Detailed analysis for the quantities of interest is seen.

Physical Model

Extra stress tensor for third order fluid model is given by(1)(2)(3)in which and respectively stand for the identity tensor, the pressure, the fluid dynamic viscosity, the extra stress tensor and the first and second Rivlin Ericksin tensors in which the material parameters ; and ; must satisfyIn the absence of displacement current, the Maxwell's equations are(4)(5)

Mathematical Formulation

Consider third order nanofluid in an asymmetric channel of width . Let be the speed by which sinusoidal wavetrains propagate along the channel walls. The and -axes in the rectangular coordinates system are taken parallel and transverse to the direction of wave propagation, respectively. A constant magnetic field of strength acts in the transverse direction resulting in an induced magnetic field The total magnetic field is Further the lower wall is maintained at temperature and nano particles concentration while the temperature and nanoparticles concentration at the upper wall are and respectively. The wall surfaces satisfy(6)where are the wave amplitudes and the phase difference varies in the range The case is subjected to the symmetric channel with waves out of phase and the waves are in phase for Further is the wavelength, the time and and satisfy . Denoting the velocity components and along the and directions in the fixed frame, one can write as(7)The fundamental equations governing the flow of an incompressible fluid are(8)(9)(10)(11)(12)in which denotes the density of fluid, the thermophoretic diffusion coefficient, the temperature, the concentration, the thermal conductivity, the acceleration due to gravity, is the pressure, is Brownian diffusion coefficient, the ratio of the specific heat capacity of the nanoparticle material and heat capacity of the fluid, the thermal diffusivity, is the volumetric volume expansion coefficient, and is the density of the particle, are components of extra stress tensor and is the magnetic diffusivity.

To facilitate the analysis, we introduce the following transformations between fixed and wave frames(13)in which () are the velocity components in the wave frame.

Equations (1)(12) in terms of above transformations give(14)(16)(17)(18)(19)Defining as mass Grashof number, Prandtl number, local temperature Grashof number, Brownian motion parameter, thermophoresis parameter, magnetic Reynold number and Hartman number(20)and then employing long wavelength and low Reynolds number approximation, the dimensionless forms of above equations in terms of stream function and magnetic force function can be expressed as(21)(22)(23)(24)(25)where

The dimensionless boundary conditions are given by(26)with . The dimensionless time mean flow rate in the wave frame is related to the dimensionless time mean flow rate in the laboratory frame by the following expressions(27)

Results and Discussion

Our main interest in this section is to examine the velocity (, temperature (), concentration (), pressure rise per wavelength (), induced magnetic field for the influence of local Grashof number (), Deborah number (), mass Grashof number (), Prandtl number (), Brownian motion parameter (), Hartman number (), magnetic Reynolds number () and thermophoresis parameter (

4.1. Pumping characteristics

This subsection illustrates the behavior of emerging parameters , , and on pressure rise per wavelength . The dimensionless pressure rise per wavelength versus time-averaged flux has been plotted in the Figs. 16. Here the upper right-hand quadrant denotes the region of peristalsis pumping, where (positive pumping) and (adverse pressure gradient). Quadrant , where (favorable pressure gradient) and (positive pumping), is designated as augmented flow (copumping region). Quadrant , such that (adverse pressure gradient) and is called retrograde or backward pumping. The flow is opposite to the direction of the peristaltic motion and there is no flow in the last (Quadrant . There is an inverse linear relation between and . It is noticed from Figs. 12 and 45 that increases with , and in all the pumping regions. Fig. 3 shows that pumping rate increases by increasing in pumping region. There are specific values of for which there is no difference between viscous and third order nanofluids. On the other hand, in the copumping region the pumping rate decreases with the increase in Deborah number. Fig. 6 shows that decreases with in copumping region.

thumbnail
Figure 1. Influence of on

doi:10.1371/journal.pone.0078770.g001

thumbnail
Figure 2. Influence of on

doi:10.1371/journal.pone.0078770.g002

thumbnail
Figure 3. Influence of on

doi:10.1371/journal.pone.0078770.g003

thumbnail
Figure 4. Influence of on

doi:10.1371/journal.pone.0078770.g004

thumbnail
Figure 5. Influence of on

doi:10.1371/journal.pone.0078770.g005

thumbnail
Figure 6. Influence of on

doi:10.1371/journal.pone.0078770.g006

4.2 Flow characteristics

The variations of , , and on the velocity have been plotted in this subsection. Fig. 7 shows that there is an increase in velocity at the centre of the channel when increases. We see a little influence of Deborah number on velocity near the walls of channel. However, magnitude of the velocity of third order nanofluid is more than viscous nanofluid. Figs. 8 and 9 depict the influence of local and mass Grashof number. Clearly the velocity increases near the lower wall. Increase in supports the motion in the channel which is shown in Fig. 10. Fig. 11 shows the influence of on velocity distribution. Interestingly an increasing thermophoresis leads to an increase in the fluid velocity at the lower wall of channel. There is a considerable variation near the walls and for and (Figs. 1213).

thumbnail
Figure 7. Influence of on

doi:10.1371/journal.pone.0078770.g007

thumbnail
Figure 8. Influence of on

doi:10.1371/journal.pone.0078770.g008

thumbnail
Figure 9. Influence of on

doi:10.1371/journal.pone.0078770.g009

thumbnail
Figure 10. Influence of on

doi:10.1371/journal.pone.0078770.g010

thumbnail
Figure 11. Influence of on

doi:10.1371/journal.pone.0078770.g011

thumbnail
Figure 12. Influence of on

doi:10.1371/journal.pone.0078770.g012

thumbnail
Figure 13. Influence of on

doi:10.1371/journal.pone.0078770.g013

4.3 Heat transfer characteristics

Effect of heat transfer on peristalsis is shown in the Figs. 1416. Figs. 14 and 15 depict the effects of Brownian motion parameter () and thermophoresis parameter ( on the temperature profile. One can observe that the temperature profile is an increasing function of and between the walls and . In Fig. 16, we observed the effects of on the temperature profile by fixing the other parameters. This Fig. indicates that the temperature increases with the increase of .

thumbnail
Figure 14. Influence of on

doi:10.1371/journal.pone.0078770.g014

thumbnail
Figure 15. Influence of on

doi:10.1371/journal.pone.0078770.g015

thumbnail
Figure 16. Influence of on

doi:10.1371/journal.pone.0078770.g016

4.4 Mass transfer characteristics

Influence of mass transfer on peristalsis is shown in the Figs. 1719. Figs. 17 and 18 depict that the concentration distribution increases at the upper and lower walls of channel when and are increased. Fig. 19 shows the effect of on the concentration when the other parameters are fixed. It shows increasing behavior of on concentration distribution near the walls and

thumbnail
Figure 17. Influence of on

doi:10.1371/journal.pone.0078770.g017

thumbnail
Figure 18. Influence of on

doi:10.1371/journal.pone.0078770.g018

thumbnail
Figure 19. Influence of on

doi:10.1371/journal.pone.0078770.g019

4.5 Induced magnetic field characteristics

The variations of and on the induced magnetic field have been plotted in the Figs. 2022. Fig. 20 shows that there is an increase in when increases. We see that magnitude of the induced magnetic field in third order nanofluid is more than viscous nanofluid. Figs. 21 and 22 depict the influence of and . Clearly the increases near the lower half of channel.

thumbnail
Figure 20. Influence of on

doi:10.1371/journal.pone.0078770.g020

thumbnail
Figure 21. Influence of on

doi:10.1371/journal.pone.0078770.g021

thumbnail
Figure 22. Influence of on

doi:10.1371/journal.pone.0078770.g022

4.6 Trapping

Trapping phenomenon is shown in the Figs. 23 and 24 for different values of and respectively. Trapping is an interesting aspect of peristaltic motion. It is the formation of a bolus of fluid by the closed streamlines. Fig. 23 is made for increasing values of We note that trapping exists for , in the upper part of channel. It is observed that number of closed streamlines circulating the bolus reduce in number as we increase the values of local Grashof number. Meanwhile size of trapped bolus increases. Streamlines are plotted in Fig. 24 to see the effects of thermophoresis parameter ( Clearly, the size of trapped bolus increases when increases from to An upper shift and flatness of bolus along with reduced closed streamlines is observed.

thumbnail
Figure 23. Streamlines for (a): (b): and (c):

doi:10.1371/journal.pone.0078770.g023

thumbnail
Figure 24. Streamlines for (a): (b): and (c):

doi:10.1371/journal.pone.0078770.g024

Conclusions

A detailed analysis is presented for peristaltic transport of third order nanofluid in an asymmetric channel with an induced magnetic field and mixed convection. The main findings of the presented study are listed below.

  • Pumping rate increases with , and while it decreases with in all pumping regions.
  • Velocity distribution is increasing functions of Deborah number at the centre of channel. Absolute value of axial velocity and pressure rise in third order nanofluid is larger than viscous nanofluid.
  • Influence of and on mass distribution is opposite to temperature distribution.
  • Temperature distribution is an increasing function of Brownian motion parameter () and thermophoresis parameter (
  • Induced magnetic field increases with and it decreases with

Author Contributions

Conceived and designed the experiments: SN. Performed the experiments: SN. Analyzed the data: SN. Contributed reagents/materials/analysis tools: SN. Wrote the paper: SN. Design of problem: SN. Mathematical formulation: SN.

References

  1. 1. Latham TW (1966) Fluid motion in a peristaltic pump. MIT Cambridge MA, 1966.
  2. 2. Shapiro AH, Jaffrin MY, Weinberg SL (1969) Peristaltic pumping with long wavelengths at low Reynolds number J. Fluid Mech 37: 799–825. doi: 10.1017/s0022112069000899
  3. 3. Tripathi D, Pandey SK, Das S (2010) Peristaltic flow of viscoelastic fluid with fractional Maxwell model through a channel. Appl Math Comput 215: 3645–3654. doi: 10.1016/j.amc.2009.11.002
  4. 4. Tripathi D, Pandey SK, Das S (2011) Peristaltic transport of a generalized Burgers' fluid: Application to the movement of chyme in small intestine. Acta Astronautica 69: 30–39. doi: 10.1016/j.actaastro.2010.12.010
  5. 5. Abd Elmaboud Y, Mekheimer KhS (2011) Non-linear peristaltic transport of a second-order fluid through a porous medium. Applied Mathematical Modelling 35: 2695–2710. doi: 10.1016/j.apm.2010.11.031
  6. 6. Hayat T, Mehmood OU (2011) Slip effects on MHD flow of third order fluid in a planar channel. Comm nonlinear Sci Num Simulation 16: 1363–1377. doi: 10.1016/j.cnsns.2010.06.034
  7. 7. Hayat T, Noreen S, Alsaedi A (2012) Slip and induced magnetic field effects on the peristaltic transport of a Johnson-Segalman fluid. App Math Mech 33: 1035–1048. doi: 10.1007/s10483-012-1603-6
  8. 8. Mekheimer KhS, Abd Elmaboud Y (2008) Peristaltic flow of a couple stress fluid in an annulus: application of an endoscope. Phys Lett A 387: 2403–2415. doi: 10.1016/j.physa.2007.12.017
  9. 9. De Vries K, Lyons EA, Bavard J, Levi CS, Lindsay DJ (1990) Contractions of inner third of the myometrium. Am J Obslet Gynecol 162: 679–682. doi: 10.1016/0002-9378(90)90983-e
  10. 10. Das K (2012) Effects of Slip and Heat Transfer on MHD Peristaltic Flow in An Inclined Asymmetric Channel. Iranian J Math Sci Informatics 7: 35–52.
  11. 11. Kothandapani M, Srinivas S (2008) Peristaltic transport of a Jeffrey fluid under the effect of magnetic field in an asymmetric channel,. Int J Non-LinearMech 43: 915–924. doi: 10.1016/j.ijnonlinmec.2008.06.009
  12. 12. Akbar NS, Nadeem S, Hayat T, Obaidat S (2012) Peristaltic flow of a Williamson fluid in an inclined asymmetric channel with partial slip and heat transfer. Int J of Heat and Mass Transfer 55: 1855.1862. doi: 10.1016/j.ijheatmasstransfer.2011.11.038
  13. 13. Noreen S, Hayat T, Alsaedi A, Qasim M (2013) Mixed convection, heat and mass transfer in peristaltic flow with chemical reaction and inclined magnetic field,. Int J Phy 87: 889–896. doi: 10.1007/s12648-013-0316-2
  14. 14. Srinivas S, Gayathri R, Kothandapani M (2011) Mixed convection heat and mass transfer in peristaltic flow an asymmetric channel with peristalsis. Commun Nonlin Sci Num Simul 16: 1845–1862. doi: 10.1016/j.cnsns.2010.08.004
  15. 15. Srinivas S, Kothandapani M (2008) Peristaltic transport in an asymmetric channel with heat transfer- A note,. Int Comm Heat Mass Transfer 35: 514–522. doi: 10.1016/j.icheatmasstransfer.2007.08.011
  16. 16. Choi SUS (1995) The Proceedings of the ASME. Int Mech Eng Congress and Exposition, San Francisco, USA, ASME, FED 231/MD 66: 99–105.
  17. 17. Nield DA, Kuznetsov AV (2009) The Cheng-Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid. Int J Heat and Mass Transf 52: 5792.5795. doi: 10.1016/j.ijheatmasstransfer.2009.07.024
  18. 18. Nield DA, Kuznetsov AV (2011) The Cheng-Minkowycz problem for the double-diffusive natural convective in a porous medium saturated by a nanofluid,. Int J Heat Mass Transfer 54: 374–378. doi: 10.1016/j.ijheatmasstransfer.2010.09.034
  19. 19. Motsumi TG, Makinde OD (2012) Effects of thermal radiation and viscous dissipation on boundary layer flow of nano fluids over a permeable moving flat plate,. Phys Scr 86: 045003–0450010. doi: 10.1088/0031-8949/86/04/045003
  20. 20. Mehmoodi M (2011) Numerical simulation of free convection of nano fluid in a square cavity within inside heater,. Int J Thermal Sci 50: 2161–2175. doi: 10.1016/j.ijthermalsci.2011.05.008
  21. 21. Makinde OD, Aziz A (2011) Boundary layer flow of a nano fluid past a stretching sheet with a convective boundary condition,. Int J Thermal Sci 50: 1326–1332. doi: 10.1016/j.ijthermalsci.2011.02.019
  22. 22. Arife AS, Vanani SK, Soleymani F (2013) The Laplace Homotopy Analysis Method for solving a general fractional diffusion eqution arising in nano-hydrodynamics,. J Comput Theor NanoSci 10: 33–36. doi: 10.1166/jctn.2013.2653
  23. 23. Asproulis N, Drikakis D (2010) Surface roughness effectin micro and nanaofluidic devices,. J Comput Theor NanoSci 7: 1825–1830. doi: 10.1166/jctn.2010.1547
  24. 24. Gavili A, Ebrahimi S, Sabbaghzadeh J (2011) The enhancement of heat in a two- dimensional enclosure utilized with nanofluids containing cylindrical nanoparticles,. J Comput Theor NanoSci 8: 2362–2375. doi: 10.1166/jctn.2011.1969
  25. 25. Jancar J, Jancarova E, Zidek J (2010) Combining reputation dynamics and percolation inmodeling viscoelastic response of collagen based nanoparticles,. J Comput Theor NanoSci 7: 12557–1264.
  26. 26. Nusrati R, Hadigol M, Raisee M, Nourbakhsh A (2012) Numerical investigationon laminar flow due to suddenexpansion usind nanofluids,. J Comput theo NanoSci 9: 2217–2217.
  27. 27. Ozsoy O, Harigaya K (2011) Theoratical calculation of shrinking and stretching in bond structure of monolayer graphite flake via hole doping treatment,. J Comput Theor NanoSci 8: 31–37. doi: 10.1166/jctn.2011.1654
  28. 28. Ono T, Fujimoto Y, Tsukamoto S (2012) First-Principles calculation method for obtaining scattering waves to investigate transport properties of nanostructures,. Quantum Matter 1: 4–19. doi: 10.1166/qm.2012.1002
  29. 29. Bose PK, Paitya N, Bhattachary S, De D, Saha S, et al. (2012) Influence of light waves on the effective electron mass in quantum wells, wires, inversion layers and superlattices,. Quantum Matter 1: 89–126. doi: 10.1166/qm.2012.1009
  30. 30. Tüzün B, Erkoç S (2012) Structural and electronic properties of unusual carbon Nanorods,. Quantum Matter 1: 136–148. doi: 10.1166/qm.2012.1012
  31. 31. Narayanan M, Peter AJ (2012) Pressure and temperature induced non-linear optical properties in a narrow band gap quantum dot,. Quantum Matter 1: 53–58. doi: 10.1166/qm.2012.1005
  32. 32. Paitya N, Bhattacharya S, De D, Ghatak KP (2012) Influence of quantizing magnetic field on the Fowler-Nordheim field emission from non-parabolic materials,. Quantum Matters 1: 63–85. doi: 10.1166/qm.2012.1007
  33. 33. Akbar NS, Nadeem S (2011) Endoscopic effects on peristaltic flow of a nanofluid,. Commun Theor Phys 56: 761–768. doi: 10.1088/0253-6102/56/4/28
  34. 34. Akbar NS, Nadeem S, Hayat T, Hendi AA (2012) Peristaltic flow of a nanofluid with slip effects,. Meccanica 47: 1283–1294. doi: 10.1007/s11012-011-9512-3