ME3250 Fall 2011 Homework 9

Solution

Problem 1. A viscous fluid with dynamic viscosity μ is filled between two cylinders (a journal bearing) with radius of R1 and R2 respectively as shown in the figure. The two cylinders are rotating with angular velocity of Ω1 and Ω2 respectively. When the flow reaches steady state:

a) write down the momentum equation (three components) for a flow in cylindrical coordinate (r, θ, z). b) Make appropriate assumptions (ignore gravity) and simplify the equations to a simple form (1-D ordinary differential equation) that can be solved analytically. Refer to the assumptions used in solving the problems of viscous flows in Ch.6. c) Give sufficient number of boundary conditions such that the solution can be uniquely determined. d) Solve the simplified equation and determine the constants from integration by applying the appropriate boundary conditions.