Finite Element Algorithm for Solving a Fourth Order Nonlinear Lubrication Equation for Droplet Evaporation

A computer-readable medium encoded with instructions for a processor to perform a method for simulating the evolution of a height of an evaporating droplet comprising:

• generating a simulation space, wherein the simulation space is described in terms of three axes, a x-axis, a y-axis, and a z-axis, the three axes are orthogonal to each other;
• the simulation space is bounded by a xz-plane defined by the x-axis and the z-axis;
• the simulation space is also bounded by a yz-plane defined by the y-axis and the z-axis;
• generating a height function that is representative of the height of the droplet at a first point in time at a plurality of points in the simulation space based upon a lubrication equation that is a differential function describing variation of the height function over time;
• and
• determining the height function at a second point in time by finding an approximate solution that satisfies the lubrication equations and boundary conditions, the boundary conditions along the xz-plane includes equating to zero a function of a partial derivate along the y-axis of a Laplacian of the height function, the boundary conditions along the xz-plane further includes equating to zero the partial derivative of the height function along the y-axis;
• the boundary conditions along the yz-plane includes equating to zero a function of a partial derivate along the x-axis of a Laplacian of the height function, the boundary conditions along the yz-plane further includes equating to zero the partial derivate of the height function along the x-axis.