Peristaltic Flow of a Couple-Stress Fluid with
Suspended Nanoparticles in an
Asymmetric Channel with Flexible Walls

V. P. Rathod, D. Sanjeevkumar "Peristaltic Flow of a Couple-Stress Fluid with
Suspended Nanoparticles in an
Asymmetric Channel with Flexible Walls", *International Journal of Mathematics Trends and Technology (IJMTT). *V32(2):112-126 April 2016. ISSN:2231-5373. www.ijmttjournal.org. Published by Seventh Sense Research Group.

**Abstract**

An analysis of peristaltic flow of a couple-stress fluid,
with immersed nanoparticles, in an asymmetric
channel having flexible walls is presented in this paper.
General boundary conditions on velocity, temperature
and concentration of nanoparticles are used for the
analysis. Thermophoresis and Brownian diffusion
effects govern the motion of nanoparticles. Long wave
length approximation and low Reynolds number
assumptions reduce the governing equations of motion
into a system of partial differential equations. Closed
form solutions for velocity, pressure gradient, pressure
rise over a wavelength, temperature distribution and
nanoparticle concentration distribution are presented.
Effects of slip, couple-stress, thermal and solutal
buoyancy, heat exchange, thermophoresis, Brownian
motion on velocity profiles, temperature distribution,
nanoparticle concentration distribution and pressure
rise are discussed. The results of the paper may lead
to possible technological applications in the field of
biomedicine.

**References**

[1] T. W. Latham, Fluid motion in a peristaltic pump [M.S.
thesis], MIT, Cambridge, UK, 1966.

[2] T.-F. Zien and S. Ostrach, “A long wave approximation
to peristaltic motion,” Journal of Biomechanics, vol. 3,
no. 1, pp. 63–75, 1970.

[3] J.-S. Lee and Y.-C. Fung, “Flow in nonuniform small
blood vessels,” Microvascular Research, vol. 3, no. 3,
pp. 272–287, 1971.

[4] P. Chaturani and V. P. Rathod, A theoretical model for
pulsatile blood flow with applications to cerebrovascular
diseases, Neurological Research 3 (1981) 289-303.

[5] L. M. Srivastava, V. P. Srivastava and S. N. Sinha,
“Peristaltic transport of a physiological fluid. Part I. Flow
in non-uniform geometry,” Biorheology, vol. 20, no. 2,
pp. 153–166, 1983.

[6] S. Takabatake, K. Ayukawa and A. Mori, “Peristaltic
pumping in circular cylindrical tubes: a numerical study
of fluid transport and its efficiency,” Journal of Fluid
Mechanics, vol. 193, pp. 267– 283, 1988.

[7] E. F. El Shehawey and K. S. Mekheimer, “Couplestresses
in peristaltic transport of fluids,” Journal of
Physics D, vol. 27, no. 6, pp. 1163–1170, 1994.

[8] R. A. Ramachandra and S. Usha, “Peristaltic transport of
two immiscible viscous fluids in a circular tube,” Journal
of Fluid Mechanics, vol. 298, pp. 271–285, 1995.

[9] K. S. Mekheimer, “Peristaltic transport of a couple stress
fluid in a uniform and non-uniform channels,”
Biorheology, vol. 39, no.6, pp. 755–765, 2002.

[10] K. Vajravelu, S. Sreenadh and V. R. Babu, “Peristaltic
transport of a Herschel-Bulkley fluid in an inclined tube,”
International Journal of Non-LinearMechanics, vol. 40,
no. 1, pp. 83–90, 2005.

[11] K. S. Mekheimer and Y. Abd elmaboud, “The influence
of heat transfer and magnetic field on peristaltic transport
of a Newtonian fluid in a vertical annulus: application of
an endoscope,” Physics Letters A, vol. 372, no. 10, pp.
1657–1665, 2008.

[12] V. P. Rathod and S. Tanveer, Pulsatile flow of couple
stress fluid through a porous medium with periodic body
acceleration and magnetic Field, Bull. Malays. Math. Sci.
Soc. 32 (2) (2009) 245–259.

[13] K. De Vries, E. A. Lyons, G. Ballard, C. S. Levi and D.
J. Lindsay, “Contractions of the inner third of
themyometrium,” American Journal of Obstetrics and
Gynecology, vol. 162, no. 3,pp. 679–682, 1990.

[14] A. Ramachandra Rao and M. Mishra, Nonlinear and
curvature effects on peristaltic flow of a viscous fluid in
an asymmetric channel, Acta Mechanica, 168 (2004), 35
- 59.

[15] E. F. Elshehawey, N. T. Eldabe, E. M. Elghazy and A.
Ebaid, “Peristaltic transport in an asymmetric channel
through a porous medium,” Applied Mathematics and
Computation, vol. 182, no. 1, pp. 140–150, 2006.

[16] M. V. Subba Reddy, A. Ramachandra Rao and S.
Sreenadh, “Peristaltic motion of a power-law fluid in an
asymmetric channel,” International Journal ofNon-
LinearMechanics, vol. 42, no. 10, pp. 1153–1161, 2007.

[17] N. Ali and T. Hayat, Peristaltic flow of a micropolar fluid
in an asymmetric channel. Comput Math Appl, 55
(2008), 589-608.

[18] A. Ebaid, “Effects of magnetic field and wall slip
conditions on the peristaltic transport of a Newtonian
fluid in an asymmetric channel,” Physics LettersA, vol.
372, no. 24, pp.4493–4499, 2008.

[19] A. M. Sobh, “Slip flow in peristaltic transport of a
Carreau fluid in an asymmetric channel,” Canadian
Journal of Physics, vol. 87, no. 8, pp. 957–965, 2009.

[20] G. C. Shit, M. Roy and E. Y. K. Ng, “Effect of induced
magnetic field on peristaltic flow of a micropolar fluid in
an asymmetric channel,” International Journal for
Numerical Methods in Biomedical Engineering, vol. 26,
no. 11, pp. 1380–1403, 2010.

[21] K. S. Mekheimer, S. Z. A. Husseny and Y. Abd
Elmaboud, “Effects of heat transfer and space porosity on
peristaltic flow in a vertical asymmetric channel,”
Numerical Methods for Partial Differential Equations,
vol. 26, no. 4, pp. 747–770, 2010.

[22] S. Srinivas and R. Muthuraj, “Effects of chemical
reaction and space porosity on MHD mixed convective
flow in a vertical asymmetric channel with peristalsis,”
Mathematical and Computer Modelling, vol. 54, no. 5-6,
pp. 1213–1227, 2011.

[23] S. Srinivas, R. Gayathri and M. Kothandapani, “Mixed
convective heat and mass transfer in an asymmetric
channel with peristalsis,” Communications in Nonlinear
Science and Numerical Simulation, vol. 16, no. 4, pp.
1845–1862, 2011.

[24] K. Das, “Influence of slip and heat transfer on MHD
peristaltic flow of a Jeffrey fluid in an inclined
asymmetric porous channel,” Indian Journal of
Mathematics, vol. 54, pp. 19–45, 2012.

[25] Y. Abd Elmaboud, S. Kh. Mekheimer, and A. I.
Abdellateef, “Thermal properties of couple-stress fluid
flow in an asymmetric channel with peristalsis,” Journal
of Heat Transfer, vol. 135, no. 4, 8 pages, 2013.

[26] S. U. S. Choi, “Enhancing thermal conductivity of fluids
with nanoparticles,” in The Proceedings of the ASME
International Mechanical Engineering Congress and
Exposition, ASME, San Francisco, Calif, USA.
Computational and Mathematical Methods in Medicine

[27] S. U. S. Choi, Z. G. Zhang, W. Yu, F. E. Lockwood, and
E. A. Grulke, “Anomalous thermal conductivity
enhancement in nanotube suspensions,” Applied Physics
Letters, vol. 79, no. 14, pp. 2252–2254, 2001.

[28] N. S. Akbar and S. Nadeem, “Endoscopic effects on
peristaltic flow of a nanofluid,” Communications
inTheoretical Physics, vol. 56, no. 4, pp. 761–768, 2011.

[29] N. S. Akbar and S. Nadeem, “Peristaltic flow of a Phan-
Thien-Tanner nanofluid in a diverging tube,” Heat
Transfer, vol. 41, no.1, pp. 10–22, 2012.

[30] N. S. Akbar, S. Nadeem, T. Hayat, and A. A. Hendi,
“Peristaltic flow of a nanofluid in a non-uniform tube,”
Heat and Mass Transfer/Waerme-und Stoffuebertragung,
vol. 48, no. 2, pp. 451– 459, 2012.

[31] N. S. Akbar, S. Nadeem, T. Hayat, and A. A. Hendi,
“Peristaltic flow of a nanofluid with slip
effects,”Meccanica, vol. 47,pp. 1283–1294, 2012.

[32] M. Mustafa, S.Hina, T.Hayat, and A. Alsaedi, “Influence
of wall properties on the peristaltic flow of a nanofluid:
analytic and numerical solutions,” International Journal
of Heat and Mass Transfer, vol. 55, pp. 4871–4877,
2012.

[33] O. A. Beg and D. Tripathi, “Mathematica simulation of
peristaltic pumping with double-diffusive convection in
nanofluids: a bio-nano-engineeringmodel,” Journal of
Nanoengineering and Nanosystems, 2012.

[34] V. K. Stokes, Couple Stresses in Fluids, Phys Fluids, vol.
9, pp. 1709-1715, 1966.

[35] S. C. Cowin, The theory of polar fluids. Advances in
Applied Mechanics, Editor: C.S. Yih, Academic Press,
New York, pp. 279-347, 1974.

[36] L. M. Srivastava, Peristaltic transport of couple stress
fluid, Rheologica Acta, vol. 25, pp. 638-641, 1986.

[37] N. Ali, T. Hayat and M. Sajid, Peristaltic flow of a
couple stress fluid in an asymmetric channel,
Biorheology, vol. 44(2), pp. 125-38, 2007.

[38] Kh. S. Mekheimer and Y. Abd elmaboud, Peristaltic flow
of a couple stress fluid in an annulus: Application of an
endoscope, Physica A: Statistical Mechanics and its
Applications, vol. 387(11), pp. 2403–2415, 2008.

[39] V. P. Rathod and S. K. Asha, Peristaltic transport of a
couple stress fluid in a uniform and non uniform annulus,
International journal of mathematical
modeling ,simulation and application, vol. 2, pp. 414-426,
2009.

[40] V. P. Rathod and S. K. Asha, Effect of couple stress fluid
and an endoscope in peristaltic motion, Ultra Science,
vol. 21, pp. 83–90, 2009.

[41] Sohail Nadeem and Safia Akram, Peristaltic flow of a
couple stress fluid under the effect of induced magnetic
field in an asymmetric channel, Archive of Applied
Mechanics, vol. 81 (1), pp. 97-109, 2011.

[42] Y. Abd elmabound, Kh. S. Mekheimer and A. I.
Abdellateef, Thermal properties of couple-stress fluid
flow in an asymmetric channel with peristalsis, J. Heat
Transfer, vol. 135(4), art. id. 044502, 2013.

[43] V. P. Rathod and N. G. Sridhar, Peristaltic transport of
couple stress fluid in uniform and non-uniform annulus
through porous medium, International Journal of
Mathematical Archive, vol. 3(4), pp. 1561-1574, 2012.

[44] V. P. Rathod and S. K. Asha, Effect of couple stress fluid
on peristaltic motion in a uniform and non-uniform
annulus, International Journal of Computer &
Organization Trends, vol. 3(11), pp. 482-490, 2013.

[45] V. P. Rathod, N. G. Sridhar and M. Mahadev, Peristaltic
pumping of couple stress fluid through non-erodible
porous lining tube wall with thickness of porous material,
Advances in Applied Science Research, vol. 3 (4), pp.
2326-2336, 2012.

[46] T. Raghunath Rao and D. R. V. Prasada Rao, Peristaltic
transport of a couple stress fluid permeated with
suspended particles, International Journal of Advances in
Applied Mathematics and Mechanics, vol. 1(2), pp. 86-
102, 2013.

[47] A. Alsaedi, N. Ali, D. Tripathi and T. Hayat , Peristaltic
flow of couple stress fluid through uniform porous
medium, Applied Mathematics and Mechanics, vol.
35(4), pp. 469-480, 2014.

[48] G. C. Shit and M. Roy, Hydromagnetic effect on inclined
peristaltic flow of a couple stress fluid, Alexandria
Engineering Journal, vol. 53, pp. 949–958, 2014.

[49] M. Vidhya, E. P. Siva and A. Govindarajan, MHD
peristaltic flow of a couple stress fluid permeated with
suspended particles through a porous medium under long
wavelength approximation, ARPN Journal of
Engineering and Applied Sciences, vol. 10(7), pp. 3072-
3077, 2015.

[50] S. Hina, M. Mustafa and T. Hayat, On the exact solution
for peristaltic flow of couple-stress fluid with wall
properties, Bulgarian Chemical Communications, vol.
47(1), pp. 30 – 37, 2015.

[51] M. Majumder, N. Chopra, R. Andrews, and B. J. Hinds,
“Nanoscale hydrodynamics: enhanced flow in carbon
nanotubes,” Nature, vol. 438, no. 7064, p. 44, 2005.

[52] N.S. Akbar and S. Nadeem, “Thermal and velocity slip
effects on the peristaltic flow of a six constant Jeffrey’s
fluid model,” International Journal of Heat and Mass
Transfer, vol. 55, no. 15-16, pp. 3964–3970, 2012.

[53] Dharmendra Tripathi, Peristaltic hemodynamic flow of
couple-stress fluids through a porous medium with
slip effect, Transport in porous media, vol. 92(3), pp.
559-572, 2012.

[54] V.P. Rathod and Pallavi Kulkarni, The effect of slip
condition and heat transfer on MHD peristaltic transport
through a porous medium with compliant wall, Int. J.
Applied Mathematical Sciences, vol. 5, pp. 47-63, 2011.

[55] V. P. Rathod and Laxmi Devindrappa, Slip effect on
peristaltic transport of a conducting fluid through a
porous medium in an asymmetric vertical channel by
Adomian decomposition method, International Journal
of Mathematical Archive, vol. 4, pp. 133-141, 2013.

[56] A. Ebaid and E.H. Aly, Exact analytical solution of the
peristaltic nanofluids flow in an asymmetric channel with
flexible walls and slip condition: Application to the
cancer treatment, Computational and Mathematical
Methods in Medicine, vol. 2013, art. id. 825376, 2013.

[57] V. P. Rathod and D. Sanjeevkumar, Closed form solution
of peristaltic flow of a magnetohydrodynamic nanofluid
in an asymmetric channel having flexible walls,
International Journal of Mathematical Archive, vol. 6,
no.4, pp. 1-14, 2015.

**Keywords**

peristaltic motion, asymmetric channel,
couple-stress fluid, nano particles, thermophoresis,
Brownian motion, slip parameter.