S. Nadeem 1 Department of Mathematics, Quaid-i-Azam University, 45320, Islamabad, 44000 Pakistan e-mail: snqau@hotmail.com Safia Akram Department of Humanities and Basic Sciences, Military College of Signals, National University of Sciences and Technology, Rawalpindi 46000, Pakistan T. Hayat Department of Mathematics, Quaid-i-Azam University, 45320, Islamabad, 44000 Pakistan; Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia Awatif A. Hendi Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia Peristaltic Flow of a Carreau Fluid in a Rectangular Duct In the present investigation we have studied the peristaltic flow of a Carreau fluid in a rectangular duct. The flow is investigated in the wave frame of reference moving with the velocity c away from the fixed frame. The peristaltic wave propagating on the horizontal side walls of a rectangular duct is studied under long wave length and low Reynolds num- ber approximation. The analytical solutions of velocity and pressure gradient have been found under lubrication approach with the help of Homotopy perturbation method. Graphical results are displayed to see the behavior of various emerging parameters of Carreau fluid. The comparison of the present work is also made with the existing literature. [DOI: 10.1115/1.4005727] Keywords: peristaltic flow, rectangular duct, Carreau fluid, homotopy perturbation solution Introduction In recent years, the peristaltic flows of Newtonian and non- Newtonian fluids with different flow geometries have attracted the attention of a number of researchers because of its useful applica- tions in medical and engineering sciences. Such applications are urine transport from the kidney to bladder, swallowing food through the esophagus, food mixing, chyme movement in the gas- trointestinal tract, and circulation of blood in small blood vessels. It is also used in sanitary fluid transport, blood pumps in heart lung machine and transport of corrosive fluids where the contact of the fluid with the machinery parts is prohibited. With different flow geometries, the peristaltic flows have been reported analyti- cally, numerically and experimentally [1–6]. To reduce some dif- ficultly level of these peristaltic flow problems, Shapiro et al. [7] studied the peristaltic flow of Newtonian fluid with long wave- length and low Reynolds number approximation. They discussed the pressure, mechanical efficiency, reflux limit and trapping limit in both two-dimensional and axisymmetric cases by assuming in- finite length of vessels. Later on, a detailed discussion of the peri- staltic flows under various assumptions has been given in the studies present in Refs. [8–11]. A vast amount of literature is available on two-dimensional peristaltic flow problems. However, very limited attention has been given to the peristaltic flow in a rectangular channel. The study of lateral walls on peristaltic flow of Newtonian fluid in a rectangular channel is examined by Reddy et al. [12]. According to their opinion, the sagital cross section of the uterus may be better approximated by a tube of rectangular cross section than a two-dimensional channel. Tsangris and Vla- chakis [13] have presented the study of pulsating flow of Newto- nian fluid in a rectangular duct in order to obtain the blood flow in fiber membranes used for artificial kidney. Recently, Nadeem and Akram [14] have highlighted the study of peristaltic flow of a non-Newtonian fluid in a rectangular channel under the assump- tions of long wave length and low Reynolds number approxima- tions. The purpose of this paper is to study the peristaltic flow of Carreau fluid in a rectangular duct. The solution of highly nonlin- ear partial differential equations have been found with the help of Homotopy perturbation method under the assumptions of long- wave length and low Reynolds number approximations. The expressions for velocity and pressure gradient have been com- puted and discussed through graphs and tables. Mathematical Formulation Let us consider the peristaltic flow of an incompressible Carreau fluid in a duct of rectangular cross section having the channel width 2d and height 2a. We are considering the Cartesian coordinates sys- tem in such a way that the X  axis is taken along the axial direction, the Y  axis is taken along the lateral direction and the Z  axis is along the vertical direction of a rectangular duct. Schematic diagram of peristaltic flow with waves propagating on horizontal walls in a rectangular duct is shown in Fig. 1 . The peristaltic waves on the walls are represented as Z ¼ HX; t ð Þ¼ 6a6b cos 2p k X  ct ð Þ   (1) 1 Corresponding author. Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received December 15, 2010; final manuscript received May 3, 2011; published online March 27, 2012. Assoc. Editor: Neelesh A. Patankar. Journal of Fluids Engineering APRIL 2012, Vol. 134 / 041201-1 Copyright V C 2012 by ASME Downloaded 16 Apr 2012 to 116.246.2.244. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm