IJTC2007-????? DEFORMATION CHARACTERISTICS OF THE ULTRATHIN LIQUID FILM SURFACE CONSIDERING THE EFFECTS OF POLAR ENDGROUPS

ABSTRACT To examine deformations of ultra-thin but continuum liquid film, the long wave theory was employed. The long wave theory uses the time-evolution equation for the shape and deformation of the thin liquid film and includes the surface tensions and surface forces such as the van der Waals (vdW) force. By numerically solving the time-dependent long wave equation, deformations of the ultra-thin lubricant film considering the vdW pressure with initial/boundary configurations of the liquid surfaces were obtained

A universal method of solving differential equations in spreadsheet software

We propose a method for solving differential equations implicitly using iterative calculations in spreadsheet software. This method uses difference equations by discretizing the coordinates of the independent variables with cells of spreadsheet for analysis. Thus, the equations are embedded in the cells performing substitution of grid point. In addition, the embedded equations are solved by Newton’s method, and the iterative function of the spreadsheet software is used in the calculation. This method can be applied even when the differential equation is discretized by an implicit solution method. Therefore, it can be applied to ordinary and partial differential equations of different types. The advantages of discretization using the implicit method (high computational accuracy, large time increments) are also presented in this paper. The analysis procedure is almost the same regardless of the type of equation. In this paper, as an example, a simple differential equation was analyzed using Microsoft Excel, a typical spreadsheet software, to illustrate the method. The results are also shown by the analysis of the gas lubrication equation