Rapid communication Novel solution for acceleration motion of a vertically falling spherical particle by HPM–Padé approximant Mohsen Torabi, Hessameddin Yaghoobi ⇑ Department of Mechanical Engineering, Faculty of Engineering, Semnan University, Semnan, Iran article info Article history: Received 29 December 2010 Received in revised form 21 February 2011 Accepted 28 February 2011 Available online 12 March 2011 Keywords: Solid spherical particle Acceleration motion Solid–liquid system Sedimentation Homotopy perturbation method (HPM) Padé approximant abstract In this paper the acceleration motion of a vertically falling spherical particle in incompressible Newtonian media is investigated. The velocity is evaluated by using homotopy perturbation method (HPM) and Padé approximant which is an analytical solution technique. The current results are then compared with those derived from HPM and the established fourth order Runge–Kutta method in order to verify the accuracy of the proposed method. It is found that this method can achieve more suitable results in comparison to HPM. Crown Copyright Ó 2011 The Society of Powder Technology Japan. Published by Elsevier B.V. All rights reserved. 1. Introduction Non-linear phenomena play a crucial role in applied mathemat- ics and physics. We know that most of engineering problems are non-linear, and it is difficult to solve them analytically. Various powerful mathematical methods have been proposed for obtaining exact and approximate analytic solutions. Recently, He [1,2] proposed the homotopy perturbation method (HPM) and variational iteration method (VIM) for solving linear, non-linear, initial, and boundary value problems. It is worth men- tioning that the origin of variational iteration method can be traced back to Inokuti et al. [3], but the real potential of this technique was explored by He. Moreover, some researchers realized the physical significance of HPM and VIM, its compatibility with the physical problems and applied this promising technique to a wide class of linear and non-linear, ordinary, partial differential equa- tions [4–11]. The problem of describing the accelerated motion of a falling sphere in Newtonian fluids is relevant to many situations of prac- tical interest. Typical examples include unit operations, such as classification, centrifugal and gravity collection and separation, where it is often important to know the detailed trajectories of the accelerating particles for purposes of design or improved oper- ation. In other practical situations, for example raindrop terminal velocity measurements, or viscosity measurements in Newtonian fluids using the falling ball method, it is also necessary to know the time and distance required to reach terminal velocity for a gi- ven sphere-fluid combination prior to making the reliable determi- nation of the sphere settling velocity. Owing to the importance of the aforementioned applications, considerable attention has been devoted to the study of the accelerated motion of a sphere in a fluid, and an excellent account of the theoretical developments in this area has been given by Clift et al. [12] for Newtonian fluids. Re- cently, analytical methods [13–17] have been used to describe the transient motion of the falling sphere and non-sphere in Newto- nian fluids. Jalaal and Ganji [13] studied the unsteady motion of a spherical particle rolling down an inclined plane submerged in a Newtonian environment using a drag of the form given by Chhabra and Ferreira [18], for wide range of Reynolds numbers by HPM. Jalaal et al. [14] applied VIM on the acceleration motion of a non-spherical particle in an incompressible Newtonian environment for a wide range of Reynolds numbers using a drag coefficient as defined by Chien [19]. In [4,15] the unsteady motion of a spherical particle falling in a Newtonian fluid was analyzed using HPM. Jalaal et al. [16] analyzed the motion of a spherical particle in a plane couette flow. Jalaal et al. [17] applied homotopy analysis method (HAM) to obtain exact analytical solutions for unsteady motion of a spherical particle rolling down an inclined tube submerged in an incompressible Newtonian environment. To obtain precise velocity of a falling particle, an accurate rela- tionship between Reynolds numbers and drag coefficient is re- quired. In this work, we study the accelerated motion of a falling spherical particle with a general drag coefficient of form given by 0921-8831/$ - see front matter Crown Copyright Ó 2011 The Society of Powder Technology Japan. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.apt.2011.02.013 ⇑ Corresponding author. Tel.: +98 91 28055461; fax: +98 21 77180590. E-mail address: Yaghoobi.Hessam@gmail.com (H. Yaghoobi). Advanced Powder Technology 22 (2011) 674–677 Contents lists available at ScienceDirect Advanced Powder Technology journal homepage: www.elsevier.com/locate/apt