(Solved): Problem 3: The leakage flow of hydraulic fluid through the gap between a piston-cylinder arrangemen ...

Problem 3: The leakage flow of hydraulic fluid through the gap between a piston-cylinder arrangement may be modelled as laminar flow of fluid between infinite parallel plates, as shown in Figure.1. The problem is a practical example of the applications of FEA approach and this model offers reasonable results for relatively small gaps. The differential equation governing the flow is: $?\frac{d^{2}u}{d{y}^2}=\frac{dP}{dx}$ Figure.1: Laminar flow of fluid between infinite parallel plates where $?$ is the dynamic viscosity of the hydraulic fluid, $\mathrm{u}$ is the fluid velocity, and $\mathrm{dP}/\mathrm{dx}$ is the pressure drop and is constant. Derive the equation for the exact fluid velocities. Assume an approximate fluid velocity solution of the form: $u(y)=A\left(y^4?H^{3}y\right)$ Note that the domain is divided into 4 elements with equal lengths, use the following methods to evaluate ${\mathrm{u}}_2,{\mathrm{u}}_3,{\mathrm{u}}_4:\mathrm{a}$ ) variational method (b) the subdomain method (c) Least square method (d) One Dimensional FEM based on Galerkin method (element matrices-Galerkin method). Compare the approximate results to the exact solution at the nodes if $?=0.02N.s/{\mathbf{m}}^2,\mathbf{dp}/\mathbf{dx}=?1100^8\mathbf{pa}/\mathbf{m}$ and $H=1\times 10^{?5}\mathrm{m}$.