An Analysis of the Flow of a Newtonian Fluid between Two Moving Parallel Plates

We consider flow of an incompressible Newtonian fluid produced by two parallel plates, moving towards and away from each other with constant velocity. Inverse solutions of the equations of motion are obtained by assuming certain forms of the stream function. Analytical expressions for the stream function, fluid velocity components, and fluid pressure are derived. 1. Introduction Owing to the nonlinear nature of the Navier-Stokes equations, their exact solutions are far and few in number. Importance of the exact solutions lies in the fact that they serve as standards for validating the corresponding solutions obtained by numerical methods and other approximate techniques. The inverse or the indirect method, see for example, Neményi [1], is often used to compute these exact solutions. Finding exact solution using the inverse method consists of making an assumption on the general form of the stream function , involving certain unknown functions, without considering the shape of boundaries of the solution domain occupied by the fluid. We then substitute this assumed form of in the compatibility equation for the stream function to find the unknown functions involved in . This provides the stream function , and subsequently, the fluid velocity components. Once the fluid velocity components are available, then the second step is to compute the fluid pressure field using the component form of the Navier-Stokes equations. This kind of methods with applications in various fields of continuum mechanics are given in an article by Neményi [1]. Moreover, a number of reviews on the exact solutions for Navier-Stokes equations have been published, for example, Berker [2], Dryden et al. [3], Whitham [4], Schlichting [5], Wang [6], and Wang [7]. In this paper, we apply the technique described above to analyze the flow of a viscous incompressible fluid induced by the motion of two parallel plates. These plates are moving towards each other and in the opposite direction with a constant velocity , when size of the plates is much larger than distance between them. A large class of the processes such as the motion of liquid through a hydraulic pump and that of the underground water may mathematically be considered from this point of view, see Aristov and Gitman [8] and Siddiqui et al. [9, 10]. We apply the inverse method to solve this problem for Berker type flow, Riabouchinsky type flow and the potential flow with perturbation. The results obtained are compared with the known viscous solutions by setting the relative velocity of the disks equal to zero. 2. Basic Equations If