Abstract
This study presents a high-order numerical method based on a uniform mesh in order to obtain a highly accurate solution to the fourth-order nonlinear degenerate diffusion equation that arises in the context of the surface tension-driven flow of thin liquid films. This study aims to demonstrate that thin liquid film flow under the influence of surface tension is a common occurrence, as well as to examine the underlying physics of contact line motion. Secondly, to emphasize the numerous advantages of compact finite difference schemes, in which the derivative is approximated by solving a tri-diagonal or penta-diagonal set of equations at each time step. This research delves into two fields: fluid mechanics and numerical methods. A thin liquid film flowing down an inclined solid substrate under the influence of surface tension is considered. The challenges posed by the region close to the advancing contact line and the effect of surface tension are covered in a brief overview of thin liquid film flow. We give a brief overview of numerical methods. The porous media equation and the fourth-order nonlinear degenerate diffusion equation are introduced, linearized, and numerically treated using a sixth-order compact finite difference scheme. The quasilinearization method is used to linearize both equations. We semi-discretize using the Crank-Nicolson method in the temporal direction, while in the spatial direction, we discretize using the sixth-order compact finite difference scheme. A stability analysis is carried out to prove that the proposed method is unconditionally stable. The credibility of the proposed method is proved using the verification method, which is a process whereby the solution of the proposed method is compared to a well-established solution. Several numerical examples are carried out for the porous media equation, and a good agreement between our solutions and the exact solutions is observed. Numerical results, visualized in terms of tables and graphs, are presented to validate the proposed method. A comparative study is performed between the proposed method and other very good numerical methods in order to show that the proposed method provides accurate and efficient numerical solutions for thin film flow problems. A considerable computational advantage for the proposed method over other numerical methods is observed.
M.Sc. (Applied Mathematics)