Permeability conditions for the physiological viscous nanoﬂuid:
endoscopic analysis for uniform and non-uniform tubes
M. U. Akbar
Received: 30 October 2016 / Accepted: 27 April 2017 / Published online: 24 June 2017
Ó The Brazilian Society of Mechanical Sciences and Engineering 2017
Abstract The analysis of endoscopic effect in uniform and
non-uniform tubes for the peristaltic ﬂow of a viscous
nanoﬂuid with permeable condition is discussed. We have
used copper as a nanoparticle and blood as its base ﬂuid.
The present problem is modeled and exact solutions for
non-dimensional differential equations are found under low
Reynolds number and long wavelength approximation. The
possessions of all physical parameters on peristaltic ﬂow
and heat transfer characteristics are witnessed from
graphical depictions. It is found that temperature proﬁle
decreases when we increase nanoparticle concentration
into our base ﬂuid for both uniform and non-uniform tubes.
It is also found that velocity proﬁle increases near the
endoscopic tube for increasing values of Darcy number.
Keywords Peristaltic ﬂow Á Viscous nanoﬂuid Á Mixed
convection Á Permeability conditions Á Endoscopic tube Á
Uniform and non-uniform tubes Á Exact solution
Nanoﬂuid is a suspension of solid nanoparticles of
diameter less than 100 nm in basic ﬂuids such as oil,
water, and ethylene glycol. Nanoparticles can be made
from various materials, such as metallic Cu, Hg, Ag, Au,
Fe, Ti, etc. and non-metallic CuO, TiO
etc. In recent days, nanoparticle research has become a
powerful tool for the scientists and engineers due to a
wide variety of potential uses in biomedical, electronic,
and optical ﬁelds. As the innovative work done by Choi
, various aspects of nanoﬂuid have been discussed.
Ellahi et al.  discussed suspended nanoparticle shape
effects in HFE-7100 over wedge in the presence of
entropy generation and mixed convection. Jiang et al. 
explored that dietary Cu accretion recovers the condition
of hypertrophic cardiomyopathy. Nadeem and Sadaf 
discussed theoretical analysis of Cu–blood nanoﬂuid for
metachronal wave of cilia motion in a curved channel.
For more aspect, see [5–11].
Peristalsis is one of the most essential techniques for
mixing as well as transporting ﬂuids, which is produced
by a progressive wave of area contraction or expansion
moving along the walls of the channel or tube. This
phenomenon is established in urinary system, reproduc-
tive tracts, gastrointestinal, and various other glandular
ducts in a living organism. Recent mechanical appli-
ances have been organized on the concept of peristaltic
pumping for transporting ﬂuids without internal moving
parts. The mechanism involved in the peristaltic trans-
port has concerned the attentionofmanyresearchers
since the experimental exploration of Latham .
Afterwards, a number of numerical, analytical, and
experimental studies related to peristaltic ﬂow of dif-
ferent ﬂuids have been described under different
Technical Editor: Jader Barbosa Jr.
& Hina Sadaf
DBS&H, CEME, National University of Sciences and
Technology, Islamabad, Pakistan
Department of Computer Engineering, Bahria University,
Islamabad 44000, Pakistan
Department of Mathematics, Quaid-i-Azam University,
45320, Islamabad 44000, Pakistan
J Braz. Soc. Mech. Sci. Eng. (2017) 39:3413–3423