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Carbon nanotube (CNT)-suspended nanofluid analysis due to metachronal beating of cilia with entropy generation

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Abstract

A mathematical model is presented for the hydromagnetic wavy (sinusoidal) flow of CNT-suspended nanofluid in a flexible ciliated tube. Entropy generation analysis has been also taken into account. This study is motivated by biomagnetic transport phenomena (of relevance to drug targeting and endoscopy) and also industrial applications such as electrically conducting rheological waste conveyance and transport of corrosive magnetized nanofluids where fluid contact with machinery parts is prohibited. Such systems can be controlled effectively with applied magnetic fields which generate a Lorentzian drag force in the flow. As geometry of the problem is ciliated tube, flow equations are modeled considering cylindrical coordinates. Governing partial differential equations are simplified and converted into differential equations using non-dimensionless variables with low Reynolds number (Re ≪ 0, i.e., inertial forces are small as compared to the viscous forces) and long wavelength approximations. Mathematica software is employed to evaluate the exact solutions of velocity profile, temperature profile, axial velocity profile, pressure gradient and entropy generation number. The influence of the pertinent control parameters on velocity profiles, temperature profile, pressure rise, pressure gradient and entropy generation number are illustrated graphically. It is observed that with an increase in the Hartmann number decreases the velocity of the fluid whereas an increment in the solid nanoparticle volume fraction of the fluid increases the velocity of the fluid.

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Abbreviations

\((\bar{R},\bar{Z})\) :

Coordinates in moving frame

ε :

Ratio w.r.t cilia length

\(\bar{P}\) :

Pressure

r :

Variable along the tube

\(\bar{S}\) :

Extra stress tensor

λ :

Wavelength

β :

Wave number

M :

Hartmann number

α :

Measure of the eccentricity

\(\mu_{\text{CNT}}\) :

Nanoparticles viscosity

\(k_{\text{CNT}}\) :

Thermal conductivity of the nanoparticles

Pr:

Prandtl number

t :

Time

ρ f :

Density of the fluid

\(\bar{\varPhi }\) :

Viscous dissipation term

Ns :

Entropy generation number

\(\theta_{0}\) :

Constant temperature

\((\bar{r},\bar{z})\) :

Coordinates in fixed frame

a :

Radius of the tube

\(\bar{u},\,\bar{w}\) :

Velocities along \(\bar{r}\) and \({\bar{z}}\) direction

\(\bar{U},\,\bar{W}\) :

Velocities along \({\bar{R}}\) and \({\bar{Z}}\) direction

\(\mu_{nf}\) :

Effective viscosity

c :

Wave speed

\(k_{nf}\) :

Effective thermal conductivity

\(B_{r}\) :

Brickman number

\(\mu_{f}\) :

Fluid viscosity

\(k_{f}\) :

Thermal conductivity of the fluid

φ :

Nanoparticles volume fraction

Ec :

Eckert number

\(\rho_{nf}\) :

Effective density

\(\rho_{\text{CNT}}\) :

Density of the nanoparticles

Be :

Bejan number

h :

Height of the tube

Re :

Reynold’s number

References

  1. Bejan A (1979) A study of entropy generation in fundamental convective heat transfer. J Heat Transfer 101:718–725

    Article  Google Scholar 

  2. Benedetti PL, Sciubba E (1993) Numerical calculation of the local rate of entropy generation in the flow around a heated finned tube. ASME AES 3

  3. Pakdemirli M, Yilbas BS (2006) Entropy generation in a pipe due to non-Newtonian fluid flow: constant viscosity case. Sadhana 31:21–29

    Article  Google Scholar 

  4. Bianco V, Manca O, Nardini S (2014) Entropy generation analysis of turbulent convection flow of Al2O3-water nanofluid in a circular tube subjected to constant wall heat flux. Energy Convers Manag 77:306–314

    Article  Google Scholar 

  5. Lam PAK, Prakash KA (2015) A numerical investigation of heat transfer and entropy generation during jet impingement cooling of protruding heat sources without and with porous medium. Energy Convers Manag 89:626–643

    Article  Google Scholar 

  6. Eiyad A-N (2006) Entropy generation due to heat and fluid flow in backward facing step flow with various expansion ratios. Int J Energy 3(4)

  7. Oztop HF, Sahin AZ, Dagtekin I (2004) Entropy generation through hexagonal cross-sectional duct for constant wall temperature in laminar flow. Int J Energy Res 28(8):725–737

    Article  Google Scholar 

  8. Oztop HF, Al-Salem K (2012) A review on entropy generation in natural and mixed convection heat transfer for energy systems. Renew Sustain Energy Rev 16(1):911–920

    Article  Google Scholar 

  9. Dagtekin I, Oztop HF, Sahin AZ (2005) An analysis of entropy generation through circular duct with different shaped longitudinal fins of laminar flow. Int J Heat Mass Transf 48(1):171–181

    Article  MATH  Google Scholar 

  10. Sheikholeslami M, Ganji DD (2015) Entropy generation of nanofluid in presence of magnetic field using Lattice Boltzmann method. Phys A 417:273–286

    Article  Google Scholar 

  11. Latham TW (1966) Fluid motion in a peristaltic pump. Masc. Thesis, Massachusetts Institute of Technology, Cambridge, MA

  12. Fung YC, Yih CS (1968) Peristaltic transport. J Appl Mech 35(4):669–675

    Article  MATH  Google Scholar 

  13. Radhakrishnamacharya G (1982) Long wave length approximation to peristaltic motion of power law fluid. Rheol Acta 21:30–35

    Article  MATH  Google Scholar 

  14. El Shehawey EF, Mekheimer KS, Kaldas SF, Afif NAS (1999) Peristaltic transport through a porous medium. J Biomath 14(1):1–13

    MathSciNet  Google Scholar 

  15. Mishra M, Rao AR (2003) Peristaltic transport of a Newtonian fluid in an Asymmetric channel. Z Angew Math Phys 54:532–550

    Article  MathSciNet  MATH  Google Scholar 

  16. Mekheimer K, Abd Elmaboud Y (2008) Peristaltic flow through a porous medium in an annulus: application of an endoscope. Appl Math Inf Sci 2(1):103–121

    MathSciNet  MATH  Google Scholar 

  17. Choi SUS, Eastman JA (1995) Enhancing thermal conductivity of fluids with nanoparticles. ASME FED 231:99–103

    Google Scholar 

  18. Buongiorno J (2005) Convective transport in nanofluids. J Heat Transf 128(3):240–250

    Article  Google Scholar 

  19. Xue Q (2005) Model for thermal conductivity of carbon nanotube-based composites. Modern Phys Lett B 368:302–307

    Google Scholar 

  20. Hong H, Wright B, Wensel J, Jin S, Ye XR, Roy W (2007) Enhanced thermal conductivity by the magnetic field in heat transfer nanofluids containing carbon nanotube. Synth Met 157:437–440

    Article  Google Scholar 

  21. Meyer J, McKrell T, Grote K (2013) The influence of multi-walled carbon nanotubes on single-phase heat transfer and pressure drop characteristics in the transitional flow regime of smooth tubes. Int J Heat Mass Transf 58:597–609

    Article  Google Scholar 

  22. Xinfeng S, Balaji S, Quynh P, Feng L, Katherine W, Billups E, Wilson W, Mikos LJ, Antonios G (2007) Fabrication of porous ultra-short single-walled carbon nanotube nanocomposite scaffolds for bone tissue engineering. Biomaterials 28:4078–4090

    Article  Google Scholar 

  23. Sleigh MA (1962) The biology of cilia and flagella. MacMillian, New York

    Book  Google Scholar 

  24. Akbar NS, Butt AW (2014) Heat transfer analysis of viscoelastic fluid flow due to metachronal wave of cilia. Int J Biomath 7(06):1450066

    Article  MathSciNet  MATH  Google Scholar 

  25. Nadeem S, Sadaf H (2015) Metachronal wave of cilia transport in a curved channel. Zeitschrift für Naturforschung A 70:33

    Google Scholar 

  26. Akbar NS, Butt AW (2016) Entropy Generation analysis for metachronal beating of ciliated Cu-water nanofluid with magnetic field. Int J Energy 19(1):41–54

    Google Scholar 

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Authors and Affiliations

  1. DBS & H, CEME, National University of Sciences and Technology, Islamabad, Pakistan

    Noreen Sher Akbar

  2. DBS & H, MCE, National University of Sciences and Technology, Islamabad, Pakistan

    Adil Wahid Butt

Authors
  1. Noreen Sher Akbar
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  2. Adil Wahid Butt
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Correspondence to Noreen Sher Akbar.

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Technical Editor: Marcio S. Carvalho.

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Akbar, N.S., Butt, A.W. Carbon nanotube (CNT)-suspended nanofluid analysis due to metachronal beating of cilia with entropy generation. J Braz. Soc. Mech. Sci. Eng. 39, 2001–2012 (2017). https://doi.org/10.1007/s40430-016-0681-9

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  • DOI: https://doi.org/10.1007/s40430-016-0681-9

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