An investigation of an Emden-Fowler equation from thin film flow

A third-order ordinary differential equation (ODE) for thin film flow with both Neumann and Dirichlet boundary conditions is transformed into a second-order nonlinear ODE with Dirichlet boundary conditions. Numerical solutions of the nonlinear second-order ODE are investigated using finite difference schemes. A finite difference formulation to an Emden-Fowler representation of the second-order nonlinear ODE is shown to converge faster than a finite difference formulation of the standard form of the second-order nonlinear ODE. Both finite difference schemes satisfy the von Neumann stability criteria. When mapping the numerical solution of the second-order ODE back to the variables of the original third-order ODE we recover the position of the contact line. A nonlinear relationship between the position of the contact line and physical parameters is obtained.