Y. V. K. Ravi Kumar 1 Department of Mathematics, Stanley College of Engineering and Technology for Women, Hyderabad 500001, India e-mail: yvkravi@rediffmail.com S. V. H. N. Krishna Kumari.P e-mail: krishnagannamaraju@gmail.com M. V. Ramana Murthy e-mail: mv.rm50@gmail.com Department of Mathematics, Osmania University, Hyderabad 500007, India S. Sreenadh Department of Mathematics, Sri Venkateswara University, Tirupati 517502, India e-mail: drsreenadh@yahoo.co.in Unsteady Peristaltic Pumping in a Finite Length Tube With Permeable Wall Peristaltic transport due to a sinusoidal wave traveling on the boundary of a tube filled with an incompressible fluid is presented. Solution is obtained under infinite wavelength and zero Reynolds number in a finite length tube which extends the study of Li and Brasseur (1993, “Non-Steady Peristaltic Transport in Finite-Length Tubes,” J. Fluid Mech., 248, pp. 129–151). Boundary conditions are changed to include wall permeabil- ity. Analysis of pressure profile is described. DOI: 10.1115/1.4002518 Keywords: peristaltic pumping, finite length tube, permeable wall, Saffman slip condition 1 Introduction Peristaltic pumping is a form of fluid transport in a tube when a progressive wave of contraction or expansion propagates along its length. The mechanism of peristaltic pumping is interesting in the study of certain vital human physiological phenomena related to transportation of fluids through flexible tubes. This mechanism is involved in urine transport from kidney to bladder, movement of ovum in the fallopian tubes and in the vasomotion of small blood vessels. Engineers have developed pumps having industrial and physiological applications adopting the principle of peristalsis. Several mathematical models are obtained to study the effects of a train of periodic sinusoidal waves on the walls of an infinitely long two dimensional channel or axisymmetric tubes containing a Newtonian or non-Newtonian fluid. Some theoretical and experi- mental investigations have been made on the peristaltic motion of blood considering it as a non-Newtonian fluid. Nicoll and Webb 1 and Nicoll 2 reported that peristalsis plays an important role in blood circulation. The investigation of peristaltic pumping from a mechanical point of view was launched with an experiment by Latham 3 who also examined the problem analytically. The results of that experiment were gen- erally in good agreement with the theoretical investigations of Shapiro 4. Gupta and Seshadri 5 discussed the peristaltic flow through nonuniform channels and tubes with a particular reference to the flow of spermatic fluid in vas deferens considering the inertia terms to be small in comparison to the viscous terms. Peristaltic transport in a tapered tube has been studied by Mishra and Pandey 6. Vajravelu et al. 7 studied the peristaltic transport of a Herschel–Bulkley fluid in an inclined tube. Influence of lateral walls on peristaltic flow in a rectangular duct was discussed by Subba Reddy et al. 8. Tang and Fung 9 and Gopalan 10 modeled microscopic blood vessel as a channel with permeable walls. They called the blood space as the channel and the tissue space as the porous layer. Chaturani and Ranganatha 11 discussed a mathematical model for solute transfer in ultrafiltering glomerular capillaries, where capillaries are taken as permeable tubes. Furthermore, it is re- ported that the blood flow in small vessels occurs due to peristal- sis. First, Li and Brasseur 12 studied nonsteady peristaltic trans- port in a finite length tube. They observed that the fluctuations in pressure and shear stress arise due to a nonintegral number waves in the finite length tube. Retrograde motion of fluid particles dur- ing peristaltic transport reflux has an inherently different behav- ior with single peristaltic wave as compared with multiple train waves. Intrauterine fluid movements, induced by myometrial contrac- tions, are responsible for the embryo transport to a successful implantation site at the fundus. These contractions change direc- tion during the menstrual cycle, aid the transport of spermatozoa toward the fallopian tube, and direct the embryo transport toward the implantation site. The mechanism that controls their coordina- tion, however, could be due to peristalsis as are the dynamic char- acteristics of the intrauterine fluid wall interface. Eytan and Elad 13 formulated a mathematical model to explain this phenom- enon. In this model, peristaltic fluid flow in a two dimensional channel with wave trains having a phase difference moving inde- pendently on the upper and lower walls is investigated to simulate intrauterine fluid motion in a sagittal cross section of the uterus. Consequently, Eytan et al. 14 extended this work for a nonuni- form channel and discussed the application to embryo transport within uterine cavity. Most of the tubular organs in the living body such as gas- trointestinal tract, intrapleural membranes, and capillary blood vessels contain a coating of a thick mucus secretion at the inner surface of the walls. The inner walls of the boundary are coated with a fluid having different properties from that of the core fluid. It is observed that the primary function of the gastrointestinal tract is to absorb nutrients from the mix of food and liquid that move 1 Corresponding author. Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received June 26, 2009; final manuscript received September 4, 2010; published online October 6, 2010. Assoc. Editor: Paul Durbin. 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