Arab Journal of Nuclear Science and Applications, 94 (4), (129-141) 2016 921 Unsteady Axial Flows of Upper-Convected Maxwell Fluid in Pipe M. Saleh Yousef 1,2 , M. H. M. Soleiman 2 , and A. M. O. Elshini 2,* (1) Department of Physics, Taibah University, Almadinah Almunawwarah, Saudi Arabia (2) Department of Physics, Cairo University, Giza 12613, Egypt Received: 5/11/2015 Accepted: 4/1/2016 ABSTRACT 1 The present research involves an analytical examination of the unsteady axial flow of an incompressible viscoelastic upper-convected Maxwell fluid through a straight tube of circular cross-section in the absence of external forces. The flow is considered initially at rest and the flow pattern is investigated through the velocity profiles for four cases of pressure -gradient field: (i) Constant pressure -gradie nt, (ii) Exponentially rising pressure - gradient with time, (iii) Exponentially falling pressure-gradient with time, and (iv) Periodic pressure-gradient. Fourier-Bessel series solution is assumed in a general form for the velocity field. The integral-transforms and the inverse derivative technique are used to obtain the exact solution of the considered initial-boundary value problem. The limiting flow of the Newtonian fluids is well described in this approach by removing the relaxation-time dependence. The velocity profiles have been determined. Keywords: Upper-Convected Maxwell Fluid / Viscoelastic / Circularcross-section /Fourier-BesselSeries / Periodic Pressure-Gradient / Traverse-Time. 1- INTRODUCTION The study of the pipe flows of fluids is of immense importance and serves a wide variety of practical applications, e.g. food, chemical, petroleum, mechanical, material processing, and nuclear industries. The simplest rheological studies are carried out on the N ewtonian fluids as Newton’s law for the viscous fluids exhibits a direct proportionality between stress and strain rate in laminar flow for which the constitutive law is,  =  (1) The stress T versus the strain rate D graph presents a straight line that passes the origin and the viscosityis independent of the strain rate although it may be affected by other physical parameters, e.g. temperature and pressure, for a studied fluid system (1) . The unsteady pipe flows of Newtonian fluids have been studied by several authors, e.g. (2–6) . In practice, the applications of Newtonian fluid are limited as very few fluids obey Newton’s law of viscosity. Moreover, there are several fluids of importance, both in technology and in nature, that deviate greatly from Newtonian fluid in behavior which and are called non-linear viscoelastic fluids. Many authors studied those fluids (7–11) . Pascal and Pascal (7) studied the power-law fluid and formulated it using non-linear shear flows of non-Newtonian fluids. Dabe et al. (8) studied Rivlin-Ericksen fluid in tube of a little deformation in cross-section with mass and heat transfer. Avinash et al. (9) studied Bingham fluid flow through a tapered tube with a permeable wall. Tong et al. (10) studied unsteady helical flows of a generalized Oldroyd-B fluid. Kamran et al. (11) obtained exact solutions for the unsteady rotational flow of a generalized second grade fluid through a circular cylinder. The upper-convected Maxwell (UCM) fluid is a non-linear viscoelastic fluid model, which exhibits a simple combination of the Newton’s law for viscous fluids and the derivative of Hook’s law for elastic solids (12) . 1 Corresponding author e-mail:aymanelshini@gmail.com