Application of Successive Linearisation Method to Squeezing Flow with Bifurcation

This paper employs the computational approach known as successive linearization method (SLM) to tackle a fourth order nonlinear differential equation modelling the transient flow of an incompressible viscous fluid between two parallel plates produced by a simple wall motion. Numerical and graphical results obtained show excellent agreement with the earlier results reported in the literature. We obtain solution branches as well as a turning point in the flow field accurately. A comparison with numerical results generated using the inbuilt MATLAB boundary value solver, bvp4c, demonstrates that the SLM approach is a very efficient technique for tackling highly nonlinear differential equations of the type discussed in this paper. 1. Introduction Studies related to transient flows produced by a simple wall motion have been of interest for several years due to its practical importance in understanding several engineering and physiological flow problems. For instance, the entire conduits in human body are flexible and also collapsible. That is, when the external pressure exceeds the internal pressure, the cross-sectional area can be significantly reduced, if not fully diminished. The cross-section may eventually return to its original shape when the external pressure is reduced, and, consequently, normal internal fluid flow can be restored [1]. Other applications can be found in unsteady loading, which is met frequently in many hydrodynamical machines and apparatus [2]. In the light of these applications, squeezing flow in a channel has been studied by many authors; mention may be made of research studies [3–6]. This problem admits similarity variable [7, 8], thereby reducing the unsteady Navier-Stokes equations to a parameter dependent fourth order nonlinear ordinary differential equation for the similarity function. Generally speaking, nonlinear problems and their solutions provide an insight into inherently complex physical process in the system. The nonlinear nature of the model equations in most cases precludes its exact solution. Several approximation techniques have been developed to tackle this problem such as the homotopy analysis method [9–11], homotopy perturbation method [12, 13], spectral homotopy analysis method [14, 15], and variational iteration method [16]. In this paper, we employ the successive linearisation method [17–19] to tackle a fourth order nonlinear boundary value problem that governs the squeezing flow problem between parallel plates. In this work, we assess the applicability of the SLM approach in solving nonlinear problems with