Problem 1. Flow of a Falling Film (30 points). Consider the geometry depicted below. A film of thickness, d density, and viscosity, is coating a stationary and solid vertical plane. Initially, at some t=0 the film is at rest. For t>0, gravity acts on the film, causing it to accelerate, eventually reaching some steady velocity distribution. The fluid outside the film is air, so the boundary condition at x=0 is just the zero shear stress condition. You may take the flow to be unidirectional in the z-direction. Your objective is to solve for the transient and steady-state velocity profiles. a.) Develop the appropriate transient unidirectional differential equations, initial condition, and boundary conditions necessary to solve for the velocity profile and render them dimensionless. What is the characteristic time scale for this problem, and what is its physical interpretation? b.) Solve for the velocity distribution at steady-state. c.) DO THIS PART: Develop an expression for the velocity distribution valid at all times using the separation of variables technique, obtaining both eigenfunctions and eigenvalues. Plot this transient profile for different dimensionless time points in Maple (or your software of choice)