Mechanics of Films The Lubrication Approximation

The dynamical properties of objects in motion are described by Newton s law, which states that the sum of all forces applied to an object is equal to its mass times its acceleration. The first difficulty one runs into when dealing with liquids is that different elements of volume making up the liquid can move independently of one another. This forces us it to work with small elements of volume, rather than with a rigid body moving as a whole. Under these circumstances, the fundamental equation reads 

The above equation, together with a condition related to the incompressibility of the liquid (Vi7 = 0), constitutes what is known as the Navier-Stokes equation. It is normally extremely difficult to solve for three reasons  

It is a vector equation, and in three-dimensional space, three separate equations must be solved in order to determine the velocity vector, 

FIGURE 5.1. Thin liquid film flowing down an incline. 

It is a non-linear equation because of the presence of an inertial term. For a flow characterized with an average velocity V over a characteristic length L, the inertial term is pV fL, which is quadratic in V, 

---