Dimensionless scaling methods for capillary rise

In this article the different dimensionless scaling methods for capillary rise of liquids in a tube or a porous medium are discussed. A systematic approach is taken, and the possible options are derived by means of the Buckingham ππ theorem. It is found that three forces (inertial, viscous and hydrostatic forces) can be used to obtain three different scaling sets, each consisting of two dimensionless variables and one dimensionless basic parameter. From a general point of view the three scaling options are all equivalent and valid for describing the problem of capillary rise. Contrary to this we find that for certain cases (depending on the time scale and the dominant forces) one of the options can be favorable. Individually the different scalings have been discussed and used in literature previously, however, we intend to discuss the three different sets systematically in a single paper and try to evaluate when which scaling is most useful. Furthermore we investigate previous analytic solutions and determine their ranges of applicability when compared to numerical solutions of the differential equation of motion (momentum balance).

Graphical abstractDifferent methods for scaling capillary rise are examined using the Buckingham π theorem.Figure optionsDownload full-size imageDownload as PowerPoint slide