Study of heat transfer on physiological driven movement with CNT nanofluids and variable viscosity

Accepted Manuscript Title: Study of heat transfer on physiological driven movement with CNT nanofluids and variable viscosity Author: Noreen Sher Akbar, Naeem Kazmi, Dharmendra Tripathi, Nazir Ahmed Mir PII: DOI: Reference:

S0169-2607(16)30555-7 http://dx.doi.org/doi: 10.1016/j.cmpb.2016.08.001 COMM 4221

To appear in:

Computer Methods and Programs in Biomedicine

Received date: Revised date: Accepted date:

31-5-2016 16-7-2016 3-8-2016

Please cite this article as: Noreen Sher Akbar, Naeem Kazmi, Dharmendra Tripathi, Nazir Ahmed Mir, Study of heat transfer on physiological driven movement with CNT nanofluids and variable viscosity, Computer Methods and Programs in Biomedicine (2016), http://dx.doi.org/doi: 10.1016/j.cmpb.2016.08.001. 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.

Study of heat transfer on physiological driven movement with CNT nanofluids and variable viscosity Noreen Sher Akbar1, Naeem Kazmi 2, Dharmendra Tripathi3, and Nazir Ahmed Mir 2 1

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

2

Mathematics & Statistics Department Riphah International University I-14, Islamabad, Pakistan

3

Department of Mechanical Engineering, Manipal University Jaipur, Rajasthan-303007, India

Research Highlights: 

Ongoing interest in CNT nanofluids and materials in biotechnology, energy and environment, microelectronics, composite materials etc.



The current investigation is carried out to analyze the effect of variable viscosity and thermal conductivity of CNT nanofluids driven by cilia induced movement through a circular tube.



The problem is formulated in the form of non linear partial differential equations, which are simplified by using the dimensional analysis to avoid the complicacy of dimensional homogeneity.



Lubrication theory is employed to linearize the governing equations and which are also physically appropriate for cilia movement.



Analytical solutions for velocity, temperature and pressure gradient and stream function are obtained.



The analytical results are numerically simulated by using the Mathematica Software and plotted the graphs for velocity profile, temperature profile, pressure gradient and stream lines to better discussion and visualization.



This model is applicable in physiological transport phenomena to explore the nanotechnology in engineering the artificial cilia and ciliated tube/pipe

Abstract: Background and Objectives: Ongoing interest in CNT nanofluids and materials in biotechnology, energy and environment, microelectronics, composite materials etc., the current investigation is carried out to analyze the effects of variable viscosity and thermal conductivity

Page 1 of 30

of CNT nanofluids flow driven by cilia induced movement through a circular cylindrical tube. Metachronal wave is generated by beating of cilia and mathematically modelled as elliptical wave propagation by Blake (1971). Methods, Results and Conclusions: The problem is formulated in the form of non linear partial differential equations, which are simplified by using the dimensional analysis to avoid the complicacy of dimensional homogeneity. Lubrication theory is employed to linearize the governing equations and it is also physically appropriate for cilia movement. Analytical solutions for velocity, temperature and pressure gradient and stream function are obtained. The analytical results are numerically simulated by using the Mathematica Software and plotted the graphs for velocity profile, temperature profile, pressure gradient and stream lines to better discussion and visualization. This model is applicable in physiological transport phenomena to explore the nanotechnology in engineering the artificial cilia and ciliated tube/pipe.

Keywords: CNT nanofluids, heat and mass transfer, ciliated walls, variable viscosity; Nanotechnology; trapping.

Introduction Increasing the interest in field of nanofluids and nano materials to see the huge number of applications of nanofluids, researchers are doing theoretical as well experimental work on synthesizing the nanofluids and fabricating the nanoparticles. There are many types of nanoparticles in market however the single and multi-walled nanotubes nanoparticle are much attention of enhancing the thermal conductivity of nanofluids.

Homogeneous and stable

nanofluids are prepared by suspending well dispersible single and multi-walled carbon nanotubes (CNTs) into base Newtonian and non-Newtonian fluids. Ding et al. [1] studied the behaviour of heat transfer of CNT nanofluids flowing through horizontal tube and observed that significant enhancement of the convective heat transfer depends on Reynolds number, CNT concentration and the pH. Xie and Chen [2] experimentally studied the effect of thermal conductivity by suspension of CNT nanotubes in ethylene glycol base fluid. Experimental results concluded that CNT nanofluids enhance the thermal conductivity, and the enhancement ratios increase with the

Page 2 of 30

nanotube loading and the temperature. In same pattern, Nasiri et al. [3] studied the effect of thermal conductivity of nanofluid and discussed the stability of nanofluids. Philip and Shima [4] analyzed the thermal properties of nanofluids and magnetic nanofluids. Babaei et al. [5] investigated the enhancement of thermal conductivity of paraffins through adding the CNT graphene. In this direction, a review on CNT nanfluids has been reported by Murshed and Castro [6]; a critical study on thermal conductivity models on CNT nanofluids is presented by Lamas et al [7]; a peristaltic flow model for nano fluids are discussed by Tripathi and Beg [8]; experimental studies to observe the thermal conductivity of CNT nanofluids is performed by Estellé et al [9]; a comparison of theoretical and experimental results of thermal conductivity of CNT nanofluids is discussed by Abbasi et al. [10]; stability analysis and thermal conductivity of water based CNT nanofluids are presented by Farbod et al [11]. In continuation of developments and researches in the field of nanofluids, many works [12-23] have been presented by several authors and still many works on nanofluids for some improvements and new findings are going on. Now a days, it is much attention to use technology and mathematical models for real life problems like environmental problems, agriculture problems, commercial problems, financial problems and biological problems etc. In recent years, it is highly demand of interdisciplinary works in research community. Cilia are rigid on the back stroke and pushing the organism forward however it bends due to forward stroke. Beating of cilia generate the wave known as Metachronal wave. Cilia are found in the majority of mammalian cells throughout development and in adult life of an organism, and associated with a variety, often overlapping, of clinical abnormalities affecting the function of numerous tissues and organs including epithelium of the respiratory tract, oviduct, testes, brain, kidney, eye, inner ear, and olfactory epithelium. Considering the experimental analysis of cilia induced flow, the first theoretical and analytical investigation of cilia induced flow was presented by Barton and Raynor [24]. Thereafter, Blake [25] proposed an interesting approach i.e. spherical envelope of cilia propulsion and he [26] studied the tubular flow due to cilia beating and again with coauthorship of Sleigh [27], he studied mechanism of cilia movement. He [28] further investigated the movement of mucus in lung. In continuation of above studies on cilia induce motion and artificial cilia movement, Khaderi et al. [29] discussed the Magnetically-actuated artificial cilia for microfluidics applications; Dauptain [30] analyzed the hydrodynamics study of cilia movement; Toonder [31]

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applied the artificial cilia for microfluidic mixing; Chung et al. [32] fabricated the Biomimetic silicone cilia for microfluidic applications; Hussong et al. [33] experimentally observed the behaviour of artificial cilia; Hussong et al. [34] further investigated the Cilia-driven particle and fluid transport over mucus-free mice tracheae. More recently some interesting mathematical models [35-42] on cilia induced movement of Newtonian fluids and non-Newtonian fluids like viscoelastic fluids and casson fluids with heat transfer and without heat transfer, and Nanofluids through channel, tube and curved channels have been presented by several authors. Motivated from the huge applications of nanofluids and cilia movement in biological science and biomedical engineering especially in fabrication of microfluidics devices, artificial cilia, and physiological transport pumps, we analyze a mathematical model to study the thermal conductivity of CNT nanofluids with variable viscosity transported by metachronal wave propagation due to beating of cilia. Since the cilia induced flow is very slow and inertia force is negligible in comparison to viscous and body force due to that governing equations are simplified using the long wavelength and low Reynolds number approximations. The numerical results are obtained by Mathematica Software and plotted the graphs for physical discussion. The present results are having good agreement with previous published work.

Mathematical formulation of the problem We consider a carbon nanotubes (CNT nanofluids) transported by metachronal wave propulsion generated due to collective beating of the cilia through the circular axisymmetric cylindrical tube. The inner surface of the circular tube is ciliated with metachronal waves propagating with constant wave velocity c and tube radius a and constant temperature at surface wall are taken into consideration. The geometry of the problem is depicted in Fig.1.

The envelopes of the cilia tips (elliptical waves) are mathematically considered [22-25] as:  2 R  H  f Z , t  a  a  co s   



 Z  c t   ,

(1a)

 2 Z  g Z , Z 0 , t  a  a  sin   

 Z  c t   ,

(1b)











where a denotes the mean radius of the tube,  is the non-dimensional measure with respect

Page 4 of 30

to the cilia length, 

and c are the wavelength and wave speed of the metachronal wave

respectively. Z 0 is the reference position of the nanoparticle and 

1

is the measure of the

eccentricity of the elliptical motion. If no slip condition is applied, then the velocities of the transporting fluid are just those caused by the cilia tips, which can be given [22-25] as: W 

Z   g  g Z g g    W,   t Z t Z  t t Z 0

(2a)

V 

R  f  f Z f f    W.   t  t Z  t t Z Z0

(2b)

Using Eq.(1) in Eq.(2), we get W 

(3a)

  ac sin    Z  c t   .  a cos Z  c t      

(3b)

2





2

1



2



1

2



2

V 

  ac cos    Z  c t   ,   a cos    Z  c t  

2





2



2



In the fixed coordinates system  R , Z , flow through circular cylindrical tube is unsteady. It becomes steady when we transform in a wave frame  r , z  moving with wave velocity c in the Z

direction. The transformations between the wave (moving) and laboratory (fixed) frames are

given as:







r  R , z  Z  c t , v  V , w  W  c, p z, r  p Z , R , t

.

(4)

The governing equations for the flow of an incompressible nanofluid can be written as: 1  r u  r

r



u



r

 nf  u

w



w



w



r

 nf  u

 0,

z

(5)

u  p   u  2  u u     u w     2  nf      nf          , (6)   nf        r  r r  r  r  r r  z  z    r

w

w  p 1    u w      nf    r   z  z r r  r  z

  w   2  nf           z    z 

 nf





g T  T0 ,

(7) u

T r

w

T z



k nf

  cp

nf

2   2T Q0 1 T  T    .  2 2     r r  r  z  cp   nf

(8)

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where r

and z

are the axial and radial coordinates in wave frame. u

velocity components in the r the fluid. Further, (   ) n f (  c p ) nf

and z

and w

are the

directions respectively, T is the local temperature of

is the effective density,

is the heat capacitance,  n f

 nf

is the effective dynamic viscosity,

is the effective thermal diffusivity, and

k nf

is the

effective thermal conductivity of the nanofluid, which are defined as (see Ref. [43] ):

   nf

 1     

 nf    

 f 

1   

k nf

 nf 

c  p

c  p

k nf

nf

 2.5

f

   

k

(9a)

, CNT

(9b)

,

(9c)

, nf

 1      c p      c p  f

 (1   )   kf   (1   )  

2 kCNT kCNT  k f 2 k f kCNT  k f

 log  log

kCNT  k f 2k f kCNT  k f 2k f

kCNT

(9d)

,

  ,  

(9e)

where  is the solid volume fraction of the Carbon nanotube. We introduce the following nondimensional variables: r 

r

, z

a

z



, w

w

, u 

c

u ac

2

, p

a p c  f

,  

T

 T0  T0

, t

ct



,

(10)

Making use of these non-dimensional variables in Eqs. (5-8), and using the assumptions of low Reynolds number and long wavelength approximation, the governing equations (5-8) are reduced as: 1   ru  r p r

p z

r



w z

 0,

(11)

 0,



1    nf      w  r r r   0  r

(12)

   nf  ,    Gr    f 

(13)

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 (1   )  1      0 r   (1   )  r r  r  

Where

 nf   0



e

log

2  k CNT k CNT  k f

   0,  

k CNT  k f 2k f k CNT  k f 2k f

(14)

 

1   

,

2 .5

e

 

 1   ,

a

in which  is viscosity parameter.  

number,  

 log 

2k f k CNT  k f





1,

 nf g  a T0 2

is wave number, G r 

,

c f

is Grashof

2

Q0a

is the heat absorption parameter.

k f T0

The non-dimensional boundary conditions on the wall surface and center of tube are considered as: w

 0,

r w

 r

 0 at r  0,

(15a)

 2  1 1  1 cos  2  z  1  2  1 1  1 cos  2  z 

 1,   0, at r  h  z   1   cos  2  z  .

(15b)

Analytical Solutions Solving Eqs. (12-14) together with boundary conditions (15) , the axial velocity is obtained as: w  r, z   1  S 

P z

 (  1f      

(1   )

2 .5

1 1 2 2 4 4  2 .5   3  r  h    4  r  h    1  G r (1   ) 8 4 

 CNT )   1

CNT



  7 r  h   6 r  h   5 r  h ,  6 4 2  

f

6

1

6

4

1

4

2

(16)

2

where, 1 

 kf

2 

,

4 k nf

 4  1, 7  S 

 1 4

1 4

5  8 

,

4

 2  1  1 cos  2  z  1  2  1  1 cos  2  z 

h

2

kf

 3    2  1,

,

k nf  2 3

    6   1 3  2 4 2  4

,

2 2

h 4 24



3 16

,

9 

Gr 4



4

h 7 4



2

h 6 3



5 2

 , 

,

,

and the temperature field is obtained as:

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 1  (1   )   r, z   4  (1   )  

2 k f kCNT  k f 2 kCNT kCNT  k f

 log  log

kCNT  k f 2k f kCNT  k f 2k f

  h   

2

r

2

.

(17)

The volumetric flow rate is defined as: h z 

F 



(18)

rw dr ,

0

which implies that p z

F  h A9 (1   ) 4



5/2

1    f

h A8 (1   ) 4

5/2

 (

kC N T



k

CNT

)



 1  S 

h

2

.

2

(19)

Using the transformation between wave frame and laboratory frame for flow rate and also taking 1

the time average as Q   F dt , the relation between time averaged volume flow rate and volume 0

flow rate is computed as: F Q

2 1   1   . 2 2 

(20)

The pressure rise across the one wavelength in axial direction is defined as: 1

P 

 p 

   z  dz .

(21)

0

Results and Discussions: In this section, the effects of physical parameters which defined to study cilia induced flow CNT nanofluids through circular cylindrical tube, are discussed through the graphical illustrations. Exact solutions obtained for the velocity, temperature, pressure gradient, pressure rise and stream function are graphically illustrated through the Figs.2-9. Figs.2(a-c) depict the velocity profile for different values of heat absorption parameter (  ), Grashof number ( G r ) and viscosity parameter (  ). It is found that the velocity profile is parabolic which confirms the validity of present model because the velocity profile for pressure driven flow is always parabolic. It is observed that velocity in case of pure water is more for all cases as compared to velocity for the same cases of single-walled carbon nano tubes.

Page 8 of 30

Furthermore, velocity is greater towards the center of the tube and slightly lower at the edges near the walls of the tube. We also note that with an increment in the heat absorption parameter, Grashof number, and viscosity parameter, the velocity of the governing fluids enhances in center of tube and diminishes near to tube walls . However the velocity of pure water changes more rapidly as compared to single-walled carbon nano tubes.

Temperature profiles are shown through the Figs.3(a-c). Similar pattern like the velocity of the fluid, the temperature rise is more in pure water as compared to the single-walled carbon nano tubes. This clearly shows that the addition of single-walled carbon nano tubes to the base fluids enhances the thermal conductivity of the base fluid which further proof the valididty of present model. It is also revealed that increasing the amplitude ratio (  ), the temperature profile increases however temperature profile decreases with the going towards the axial direction. It is further inferred that the temperature profile reduces with increasing the magnitude of nanoparticle volume fraction (  ). Finally it is pointed out that for pure water temperature changes more rapidly as compared to single-walled carbon nano tubes.

The variations of pressure gradient along the tube length are shown illustrated through the Figs. (4a-c). The effects of heat absorption parameter (  ), viscosity parameter (  ) and volume fraction (  ) on pressure gradient are discussed. Because when we rise heat absorption parameter, viscosity parameter and nanoparticle volume fraction there will be more resistance. So it is noticed that the variation of pressure gradient along the tube length is similar to metachronal wave propagation which is trivial because pressure is created due to beating of cilia. It is also seen that pressure gradient is more for pure water as compared to single-walled carbon nano tubes. Fig.4a depicts that pressure gradient enhances with increasing the magnitude of heat absorption parameter for pure water as well as SWNT. Fig.4b reveals that pressure gradient goes up for more viscous fluids. Fig.4c shows that pressure gradient diminishes with increasing the magnitude of volume fraction. The impacts of heat absorption parameter (  ), viscosity parameter (  ) and volume fraction (  ) on pressure rise across one wavelength against the time averaged volumetric flow rate are represented through the Figs.5(a-c). And the difference in pressure rise for pure water and for

Page 9 of 30

single-walled carbon nano tubes are also depicted for all three parameters. It is observed that the relation between pressure rise and time averaged volumetric flow rate is linear which are similar in many existing works (Refs. [12-23]). It is also noticed that volumetric flow rate is maximum at zero pressure and vice versa. Three regions are classified on the basis of the values of pressure rise, pumping region (  p  0 ) i.e. Q  0, free pumping region (  p  0 ) and augmented pumping region (  p  0 ) i.e. Q  0 . Fig.5a shows that the effect of heat absorption parameter (  ) on pressure rise for pure water as well as SWNT. It is noticed that pressure rise is less for higher values of heat absorption parameter in pumping region because rise in heat absorption parameter pressure rise decreases and it is also depicted that pressure is more for SWNT as compared to pure water for all values of heat absorption parameter in pumping region. Fig.5b illustrates that the effect of viscosity parameter on pressure rise for both SWNT and pure water. It is inferred that the effect of viscosity parameter on pressure rise is similar to that of heat absorption parameter. Fig.5c represents the influence of volume fraction on pressure rise and it is observed that the pressure elevates with increasing the value of volume fraction in pumping region. It is also revealed that the effects of all parameters on pressure rise in pumping region are opposite in augmented pumping region. It is further noticed that the values of pressure for all values of various parameters coincide.

Trapping phenomenon is inherent properties of oscillating flow (may be cilia induced flow or peristaltic flow) which is defined as it is process of circulation of center stream lines at good combination of the values of time averaged volumetric flow rate and amplitude of metachronal wave. Figs. 6-9 (a & b) are plotted for stream lines in wave frame at fixed values of amplitude and time averaged flow rate and other parameters. Figs.6 (a & b) shows the effect of Grashof number (Gr) on trapping, Grashof number is the ratio of the buoyancy to viscous force acting on a fluid so rise in Grashof number buoyancy forces will dominant to viscous forces and it is found that rise in buoyancy forces size of the trapped bolus starts to decrease in size but no of bolus increases when we increase the magnitude of Grashof number. Figs.7 (a &b) shows that the effect of heat absorption parameter on trapping and it is pointed out that with an increase in heat absorption parameter; fluid will thin trapped bolus starts to increase in size. Figs.8 (a&b) illustrates that the impact of viscosity parameter and it is noticed that rise in viscosity parameter fluid will be more resistive due to that size and number of the trapped bolus starts to increase

Page 10 of 30

with increasing the magnitude of viscosity parameter. Figs.9 (a &b) are drawn for different values of time averaged volumetric flow and it is inferred that when flow rate will increase fluid flow with more speed due to which trapped bolus increases in size but number of bolus decreases with increasing flow rate.

Conclusion We analyze the cilia induced motion of CNT nanofluids with base fluid as water through circular cylindrical tube. In above section, the effect of relevant physical parameters on velocity, temperature, pressure gradient, pressure rise and trapping phenomenon are discussed. The key points of the performed analysis are as follows: 1. Velocity in case of pure water is more compared to velocity for the single-wall carbon nano tubes at all the values of other parameters. 2. Velocity is greater towards the center of the tube and slightly lower at the edges near the surface wall with increasing the magnitude of Grashof number, absorption parameter and Viscosity parameter. 3. The changes in the velocity of pure water are more in comparison to single-walled carbon nano tubes under the influence of physical parameters. 4. Temperature is more for pure water as compared to the single-walled carbon nano tubes.

Page 11 of 30

5. Temperature diminishes with increasing the magnitude of Nanoparticle volume fraction however opposite trends are noticed for amplitude ratio.

6. Pressure gradient is higher for pure water as compared to single-walled carbon nano tubes. 7. Pressure gradient increases with an increase in heat absorption parameter, and viscosity parameter however the effect of Nanoparticle volume fraction on pressure gradient is opposite to other two parameters. 8. Pressure rise reduces with increasing the magnitude of

heat absorption parameter,

viscosity parameter in the peristaltic pumping region however the effect of Nanoparticle volume fraction is opposite to others. 9. Opposite trends are noticed for all parameters in augmented pumping region in comparison to pumping region. 10. With an increase in heat absorption parameter, trapped bolus reduces in size but increases the numbers, and increasing the viscosity parameter, size of the trapped bolus expands however increasing the time averaged flow rate, trapped bolus enlarges in size.

References 1. Y. Ding, H. Alias, D. Wen, & R.A. Williams, Heat transfer of aqueous suspensions of carbon nanotubes (CNT nanofluids). International Journal of Heat and Mass Transfer, 49(1) (2006)240-250. 2. H. Xie, & L. Chen, Adjustable thermal conductivity in carbon nanotube nanofluids. Physics Letters A, 373(21) (2009)1861-1864. 3. A. Nasiri, M. Shariaty-Niasar, M. Rashidi, A. Amrollahi, & R. Khodafarin, R. Effect of dispersion method on thermal conductivity and stability of nanofluid. Experimental thermal and fluid science, 35(4)(2011)717-723. 4. J. Philip, & P. D. Shima, Thermal properties of nanofluids. Advances in colloid and interface science, 183(2012)30-45. 5. H. Babaei, P. Keblinski, & J. M. Khodadadi, Thermal conductivity enhancement of paraffins

by

increasing

the

alignment

of

molecules

through

adding

CNT/graphene. International Journal of heat and mass transfer, 58(1) (2013)209-216.

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6. S. Murshed, & C. N.

Castro, Superior thermal features of carbon nanotubes-based

nanofluids–A review. Renewable and Sustainable Energy Reviews, 37(2014)155-167. 7. B. Lamas, B.Abreu, A. Fonseca, N. Martins, & M. Oliveira, Critical analysis of the thermal conductivity models for CNT based nanofluids. International Journal of Thermal Sciences, 78(2014)65-76. 8. D. Tripathi, & O. A. Bég, A study on peristaltic flow of nanofluids: Application in drug delivery systems. International Journal of Heat and Mass Transfer, 70(2014) 61-70. 9. P. Estellé, S. nanofluids:

Halelfadl, & T. Maré. Thermal conductivity of CNT water based

Experimental

trends

and

models

overview. Journal

of

Thermal

Engineering, 1(2) (2015)381-390. 10. S. Abbasi, S. M. Zebarjad, S. H. N. Baghban, , A. Youssefi, Comparison between experimental and theoretical thermal conductivity of nanofluids containing MWCNTs decorated with TiO2 nanoparticles. Experimental Heat Transfer, (2015) (just-accepted). 11. M. Farbod, A. Ahangarpour, & S. G. Etemad, Stability and thermal conductivity of water-based carbon nanotube nanofluids. Particuology, 22(2015)59-65. 12. S. Mukherjee, P.C. Mishra, S.K.S. Parashar, & P. Chaudhuri, Role of temperature on thermal conductivity of nanofluids: a brief literature review. Heat and Mass Transfer, 4(2016)1-11. 13. R. Walvekar, M. K. Siddiqui, S. Ong, & A.F. Ismail, Application of CNT nanofluids in a turbulent flow heat exchanger. Journal of Experimental Nanoscience, 11(1) (2016)1-17. 14. A.O.C. Gómez, A.R.K. Hoffmann, & E.P. Bandarra Filho. Experimental evaluation of CNT nanofluids in single-phase flow. International Journal of Heat and Mass Transfer, 86(2015)277-287. 15. B. H. Thang, P. H. Khoi, & P. N. Minh. A modified model for thermal conductivity of carbon nanotube-nanofluids. Physics of Fluids (1994-present), 27(3) (2015) 032002. 16. L. Fan, J.Q.Li, Y. Zhang, L., Yu, Z. T., & Cen, K. F. The effect of concentration on transient pool boiling heat transfer of graphene-based aqueous nanofluids. International Journal of Thermal Sciences, 91(2015)83-95. 17. S. Kakaç, & A. Pramuanjaroenkij. Single-phase and two-phase treatments of convective heat transfer enhancement with nanofluids–A state-of-the-art review. International Journal of Thermal Sciences, 100(2016)75-97.

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18. O. A. Bég, & D. Tripathi, Mathematica simulation of peristaltic pumping with doublediffusive convection in nanofluids: a bio-nano-engineering model. Proceedings of the Institution of Mechanical Engineers, Part N: Journal of Nanoengineering and Nanosystems, 1740349912437087(2012). 19. N. S. Akbar, M. Raza, & R. Ellahi. Copper oxide nanoparticles analysis with water as base fluid for peristaltic flow in permeable tube with heat transfer. Computer Methods and Programs in Biomedicine, 130(2016)22-30. 20. N. S. Akbar, Z. Khan, S. Nadeem, & W. Khan, Double-diffusive natural convective boundary-layer flow of a nanofluid over a stretching sheet with magnetic field. International Journal of Numerical Methods for Heat & Fluid Flow, 26(1) (2016)108-121. 21. N. S. Akbar, Dharmendra Tripathi and O. Anwar Beg, Modeling Nanoparticle Geometry Effects on Peristaltic Pumping of Medical Magnetohydrodynamic Nanofluids with Heat Transfer, Journal of Mechanics in Medicine and Biology, Vol. 16, No. 6 (2016) 1650088 (20 pages) 22. N. S. Akbar, & A. W. Butt, Entropy generation analysis for metachronal beating of ciliated Cu-water nanofluid with magnetic field. International Journal of Exergy, 19(1) (2016) 41-54. 23. N. S. Akbar, & Z. H. Khan. Influence of magnetic field for metachoronical beating of cilia for nanofluid with Newtonian heating. Journal of Magnetism and Magnetic Materials, 381(2015) 235-242. 24. C. Barton, & S. Raynor, Analytical investigation of cilia induced mucous flow. The Bulletin of mathematical biophysics, 29(3) (1967)419-428. 25. J. R. Blake, A spherical envelope approach to ciliary propulsion, J. Fluid Mechanics, 47 (1971). 26. J. R. Blake, Flow in tubules due to ciliary activity, Bull Math Biol. 35 (1973) 513-523. 27. J. R. Blake, & M. A. Sleigh, Mechanics of ciliary locomotion. Biological Reviews, 49(1) (1974)85-125. 28. J. R. Blake. On the movement of mucus in the lung. Journal of biomechanics, 8(3) (1975)179-190. 29. S.N. Khaderi, C.B. Craus, J. Hussong, N. Schorr,d J. Belardi, J. Westerweel, O. Prucker,d

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J. Ruhe, J. M. J. den Toondere and P.R. Onck, Magnetically-actuated artificial cilia for microfluidic propulsion, Lab Chip, 11(2011)2002 30. J. Favier Dauptain, and A. Battaro, Hydrodynamics of ciliary propulsion, J. Fluids Struct. 24 (2008) 1156–1165. 31. Toonder Den, Jaap, Femke Bos, Dick Broer, Laura Filippini, Murray Gillies, Judith de Goede, Titie Mol et al. "Artificial cilia for active micro-fluidic mixing." Lab on a Chip 8, no. 4 (2008): 533-541. 32. K. O Chung, J. H., Devasia, S., & Riley, J. J. Bio-mimetic silicone cilia for microfluidic manipulation. Lab on a Chip, 9(11) (2009)1561-1566. 33. J. Hussong, N. Schorr, J. Belardi, O. Prucker, , J. Rühe, & J. Westerweel, Experimental investigation of the flow induced by artificial cilia. Lab on a Chip, 11(12) (2011). 20172022. 34. J. Hussong, R. Lindken, P. Faulhammer, K. Noreikat, K.V. Sharp, W. Kummer, & J. Westerweel.

Cilia-driven

particle

and

fluid

transport

over

mucus-free

mice

tracheae. Journal of biomechanics, 46(3) (2013) 593-598. 35. L.C. Espinha, D.A. Hoey, P.R. Fernandes, H.C. Rodrigues, and C.R.Jacobs, Oscillatory fluid flow influences primary cilia and microtubule mechanics. Cytoskeleton, 71(7) (2014)435-445. 36. N.S. Akbar, & A. W. Butt, Heat transfer analysis of viscoelastic fluid flow due to metachronal

wave

of

cilia. International

Journal

of

Biomathematics, 7(06)

(2014)1450066. 37. S. Nadeem, & H. Sadaf, Metachronal wave of cilia transport in a curved channel. Zeitschrift für Naturforschung A, 70(2015)33. 38. M. Corrigan, K. Lee, M. Labour, C. Jacobs, and D. Hoey. Fluid flow-induced bending of the primary cilium triggers a distinct intraciliary calcium flux in mesenchymal stem cells. Cilia, 4(Suppl 1), (2015) P22. 39. C.Y. Chen, L. Y. Cheng, C. C. Hsu, & K. Mani, Microscale flow propulsion through bioinspired

and

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actuated

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cilia.

Biomicrofluidics, 9(3)

(2015)034105. 40. S. Nadeem, and Hina Sadaf. "Theoretical analysis of Cu-blood nanofluid for metachronal wave of cilia motion in a curved channel." NanoBioscience, IEEE Transactions on 14.4

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(2015) 447-454. 41. H. Sadaf, & S. Nadeem. Influences of slip and Cu-blood nanofluid in a physiological study of cilia. Computer Methods and Programs in Biomedicine, 131(2016)169-180. 42. Noreen Sher Akbar, M. Raza, R. Ellahi, Copper oxide nanoparticles analysis with water as base fluid for peristaltic flow in permeable tube with heat transfer, Computer Methods and Programs in Biomedicine. 1 3 0 ( 2 0 1 6 ) 22–30. 43. Noreen Sher Akbar,Z.H. Khan,S. Nadeem,

Influence of magnetic field and slip on

Jeffrey fluid in a ciliated symmetric channel with metachronal wave pattern , Journal of Applied Fluid Mechanics, 9(2) (2016)565-572.

Page 16 of 30

44. Figure 1: Schematic view of cilia induced transport of CNT nanofluids with variable viscosity through the circular cylindrical tube subjected to constant temperature at the left and right surfaces. The above schematic depicts a metachronal wave propagating due to beating of cilia from the bottom reservoir to the top reservoir with a wave velocity c.

Page 17 of 30

0.975 0.980 0.985

Gr Gr

2.0, Gr 4.0, Gr

3.0, 5.0

0.990 0.995

0.005, 0.1 , z

1.000 0.5

2.0 , 0.5 , Q

0.6 , 0.1

0.0

0.5

 a

0.980 0.985 0.990

0.5, 0.9,

0.7, 1.1

0.995 0.01, 2.0, Gr 2.0, 0.1, z 0.5, Q 0.1

1.000 0.5

0.0

0.5

 b

Page 18 of 30

0.965 0.970 0.975

0.5, 2.5,

1.5, 3.5

0.980 0.985 0.990 0.995

0.01, 0.1, z

1.000 0.5

1.7, Gr 2.0, 0.5, Q 0.1 0.0 r

0.5

 c

Fig.2. Velocity profile Grashof number

Gr

w r, z 

for different values of the (a)

(b) Heat absorption parameter

(c) Viscosity parameter





and

.

Page 19 of 30

0.030 0.025 0.020 0.015 0.1,

0.2,

0.3,

0.4

0.010 0.005

0.02,

1.0

0.1, z 0.8

0.5

0.0

0.5

1.0

 b

0.10

0.05 z z z z

0.00 0.04 ,

1.0

0.5

0.1, 0.2, 0.3, 0.4

0.1 ,

0.0

0.4

0.5

1.0

 c

Page 20 of 30

0.0

0.025

0.01 0.02

0.020

0.03 0.04

0.015 0.010

0.0 , 0.01 , 0.02, 0.03, 0.04

0.005 0.01 ,

1.0

0.5

0.1 , z 0.8

0.0 r

0.5

1.0

 d

Fig.3. Temperature profile

 r , z 

against the radial axis

for different values of (a) amplitude ration, coordinate,

z



, (b) axial

, and (d) Nanoparticle volume fraction,

.

Page 21 of 30

20 1.0 , 4.0 ,

0

2.5, 5.5

dP dz

20 40 60 0.11, 80

2.0

2.1, Gr 2.0,

1.5

1.0

0.5 z

0.1, Q 0.0

2.0 0.5

1.0

 a

20 0.8, 2.8,

0

1.8, 3.8

dP dz

20 40 60 0.1, 80

2.0

1.5

2.8, Gr 2.0, 1.0

0.5 z

0.1, Q 0.0

2.0 0.5

1.0

 b

Page 22 of 30

40

0

20

0,

0

0.06

0.06 , 0.12, 0.18 , 0.24

0.12 0.18

dP dz

20

0.24

40 60 80 100

0.8 , 2.0

1.5

2.8 , Gr 1.0

2.0 , 0.5 z

0.1 , Q 0.0

2.0

0.5

1.0

 c

Fig.4

.

Pressure gradient against the axial distance for

different values of (a) Heat absorption parameter, viscosity parameter,  ,



, (b)

(c) Nanoparticle volume fraction,

.

Page 23 of 30

1.0, 2.0, 3.0, 4.0

40

p

20

0

20

0.2, Gr 2.0 , 2.0

1.5

2.1,

1.0

0.1

0.5 Q

0.0

0.5

1.0

 a

0.1, 0.8, 1.5, 2.2

40

p

20

0

20

0.2, Gr 2.0

1.5

2.0,

3.3,

1.0

0.5 Q

0.1 0.0

0.5

1.0

 b

Page 24 of 30

0.0, 0.1, 0.2, 0.3

60 40

0.0 0.1 0.2 0.3

p

20 0 20 40 60

0.1, Gr 2

2.0,

3.3,

1

0 Q

0.1 1

2

 c

Fig.5. Pressure rise against the time averaged flow rate for different values Heat absorption parameter, viscosity parameter,  ,



, (b)

(c) Nanoparticle volume fraction,

.

Page 25 of 30

1.5

1.5

1.0

1.0

0.5

0.5

0.0

0.0

0.5

0.5 4

2

0

2

4

4

2

0

2

4

Fig.6. Stream lines in wave frame for different values of Grashof number

G r  1, 2 .

Page 26 of 30

1.5

1.5

1.0

1.0

0.5

0.5

0.0

0.0

0.5

0.5 4

2

0

2

4

4

2

0

2

4

Fig.7. Stream lines in wave frame for different values of heat absorption parameter

  9,10.

Page 27 of 30

1.5

1.5

1.0

1.0

0.5

0.5

0.0

0.0

0.5

0.5 4

2

0

2

4

4

2

0

2

4

Fig.8. Stream lines in wave frame for different values of viscosity parameter

  1, 3.

Page 28 of 30

1.5

1.5

1.0

1.0

0.5

0.5

0.0

0.0

0.5

0.5 4

2

0

2

4

4

2

0

2

4

Fig.9. Stream lines in wave frame for different values of time averaged flow rate

Q  1, 2 .

Page 29 of 30

Table.1: Thermo physical properties of pure water and SWCNT. P hysical P roperties

P ure w ater

SW CNT

c p ( J / kgK )

4179

425

 ( kg / m )

997.1

2600

K (W / m K )

0.613

6600

3

5

  10 (1 / K )

21

1.5

Table 2. Comparison of present work with the existing literature for pure fluid. Present work (Pure fluid i.e φ=0, Gr=0)

Ref.[36], λ1=0

Ref. [37] k = 0

-1.0

-1.0000

-1.0000

-1.0000

-0.8

0.0815

0.0814

0.0816

-0.6

0.9173

0.9172

0.9174

-0.4

1.5115

1.5114

1.5116

-0.2

1.8668

1.8667

1.8666

0.0

1.9851

1.9850

1.9852

0.2

1.8668

1.8667

1.8666

0.4

1.5115

1.5114

1.5116

0.6

0.9173

0.9172

0.9174

0.8

0.0815

0.0814

0.0816

1.0

-1.0000

-1.0000

-1.0000

r

Page 30 of 30