**1. Introduction**

The study of peristaltic mechanism has gained considerable attention during the past few decades [1–10]. Peristaltic mechanism involves certain physiological phenomena, like swallowing food through the esophagus, vasomotion of small blood vessels, transport of urine from kidney to bladder, chyme motion in the gastrointestinal tract, and movement of spermatozoa in human reproductive tract.

Peristaltic pumping is a form of liquid transport that occurs when a progressive wave of area contraction or expansion propagates along the length of distensible duct. There are many engineering processes in which peristaltic pumps are used to handle a wider range of fluids, particularly in the chemical and pharmaceutical industries. This mechanism is also used in the transport of slurries, sanitary fluids, and noxious fluids in the nuclear industry [11–13]. Extensive analytical, numerical, and experimental studies have been undertaken involving such flows. Important studies to the present topic include the works done by [14–19]. In all previous studies, fluid viscosity is assumed to be constant. There are few attempts in which the variable viscosity in peristaltic phenomena has been used. Mention may be made of the works by [20–22].

There are various analytical techniques to solve the differential equations arising in physics and engineering. Thus, various perturbation and non perturbation techniques are in use. Recently, Adomian decomposition has acquired great credence in tackling the linear and non-linear problems, and sometimes gives the closed form solution in the form of general functions like trigonometric

functions, Bessel functions, and so on. The impressive bibliography of the work done by the Adomian decomposition method has been presented in papers by [23–30].

The intent of the paper is to present an integrated solution for different facets of the problem. These include application of endoscopy in a viscous fluid with the variable viscosity and closed form Adomian solutions, which are presented for unknown (general μ(*r*)) variable viscosity. In Section 2, mathematical formulation of the present problem is described. Section 3 deals with the solution of the problem using the Adomian decomposition method. Three typical examples were chosen and their closed form solutions were presented, and comparison is given with the existing literature. In Section 4, graphical results are presented to gauge the effects of certain physical parameters. Finally, streamlines for the flow problems are also drawn.
