On the motion of a second grade fluid due to longitudinal and torsional oscillations of a cylinder: A numerical study Mehrdad Massoudi * , Tran X. Phuoc US Department of Energy, National Energy Technology Laboratory (NETL), P.O. Box 10940, Pittsburgh, PA 15236, USA article info Keywords: Continuum mechanics Second grade fluids Unsteady flows Drag reduction Torsional and longitudinal oscillations Cylinder Oil drilling abstract Unsteady problems involving the second grade fluids have received considerable attention in recent years. The present study is an attempt to look at the motion of an oscillating rod in a second grade fluid. Specifically, we solve numerically for the flow of a second grade fluid surrounding a solid cylindrical rod that is suddenly set into longitudinal and torsional motion. The equations are made dimensionless. The results are presented for the shear stresses at the wall, related to the drag force; these are physical quantities of interest, espe- cially in oil-drilling applications. Published by Elsevier Inc. 1. Introduction One of the most challenging engineering problems encountered in off-shore oil drillings, towing operations, or excitations of long rods (or cables), is obtaining an accurate expression and estimate for the (viscous) drag, or the damping force, due to the fluid, exerted on the rod or cable. In oil-drilling applications, Akyildiz [1], for example, lists the cooling and lubricating of the bit and the drilling string as one of the important design and operational parameters, since to a large extent this depends on the complex rheological structure of the surrounding fluid which is composed of water, mud, oil, rocks, sediments, etc. In fact, as Dareing and Livesay [9] point out the performance of the drill string under the dynamic conditions, i.e., longitudinal and torsional (angular) oscillations, is significantly influenced by the viscous (frictional) force of the fluid, which in turn can cause (thermal) stresses in the string which can affect the drilling rate and hole stability. The common practice has been to assume a linear relationship between the external damping force and the velocity of the rod, but as indicated by Caserella and Laura [8] this approach is not very accurate. Interest and research activities in drag-reduction techniques, whether in transportation systems (on ground, in or un- der water, or in air), oil-drilling industries, or materials handling have increased in the last few decades. Bushnell and Moore [6] describe that the various attempts to study these problems can generally be classified into (at least) three categories: (i) form-drag reduction, (ii) skin-friction drag reduction, and (iii) drag-due-to-lift reduction. Amongst the techniques commonly used in many chemical industries to reduce the skin friction drag is to use (surface) additives. It has also long been known that some additives can change the rheological properties of Newtonian fluids, such as water, to non-Newtonian fluids and that a significant reduction in drag of such non-Newtonian fluids flowing past solid objects has been observed (see [51,24,23]). As a result, in recent years, there have been many studies concerned with the calculation of the wall shear stresses and the drag for the flow of various non-Newtonian fluids past solid objects such as spheres and cylinders. The flow of a viscous fluid past a sphere was first studied by Stokes; for non-Newtonian fluids there have also been many such studies (see [19–21,18]). 0096-3003/$ - see front matter Published by Elsevier Inc. doi:10.1016/j.amc.2008.05.133 * Corresponding author. E-mail address: massoudi@netl.doe.gov (M. Massoudi). Applied Mathematics and Computation 203 (2008) 471–481 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc