AXIAL FLOW BETWEEN SLIDING, NON-CONCENTRIC CYLINDERS WITH APPLICATIONS TO THREAD INJECTION by ANDREW G. WALTON † (Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ) and JANE LABADIN and SIEW PING YIIONG (Department of Computational Science and Mathematics, Faculty of Computer Science & Information Technology, Universiti Malaysia Sarawak (UNIMAS), 94300 Kota Samarahan, Sarawak, Malaysia) [Received 23 November 2009. Revised 3 March 2010. Accepted 16 April 2010] Summary An investigation is conducted into the nature of the pressure-driven axial flow between cylinders with the inner cylinder moving in the axial direction. Such a flow is often referred to in the literature as a ‘thread-annular’ flow and is relevant to the procedure of thread injection: a surgical technique that allows the injection of porous medical implants, consisting of synthetic biocompatible materials, into the body in a minimally invasive way. A perturbation solution for the fluid flow is derived under the assumption that the inner cylinder is displaced slightly from a concentric position within the outer cylinder. From the basic solution, expressions for the force on the thread and the friction factor for the flow are derived. Our results are compared with the concentric case, experimental results and also the exact solution to the problem. It is found that reported discrepancies between experimental measurements and theory based on a concentric flow model can be explained by our inclusion of thread eccentricity. 1. Introduction The prediction of the behaviour of a fluid when it is injected into the body using a needle or syringe is a challenging problem for theoretical fluid dynamicists. Nowadays, a cosmetic plastic surgeon in- jects not only fluid into the body but also specially designed surgical threads that consist of synthetic biocompatible materials. Through modelling the procedure mathematically, a better understanding can be obtained of the way in which key parameters, such as the speed of injection, affect the fluid flow characteristics within the syringe. Ultimately, it is hoped that the entire procedure can be carried out more proficiently with a minimum of surgical trauma. A simplified version of the injection process can be modelled mathematically by considering the axial flow between concentric cylinders with the inner cylindrical core (representing the thread of material) moving at a constant velocity. The stability of this flow was first considered by † 〈a.walton@imperial.ac.uk〉 Q. Jl Mech. Appl. Math, Vol. 63. No. 3 c  The author 2010. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org Advance Access publication 19 May 2010. doi:10.1093/qjmam/hbq009 Downloaded from https://academic.oup.com/qjmam/article/63/3/315/1886826 by guest on 08 April 2022