The Hausdorff Dimension and Capillary Imbibition

Abstract

The time scaling exponent for the analytical expression of capillary rise ℓ∼tδ for several theoretical fractal curves is derived. It is established that the actual distance of fluid travel in self-avoiding fractals at the first stage of imbibition is in the Washburn regime, whereas at the second stage it is associated with the Hausdorff dimension dH. Mapping is converted from the Euclidean metric into the geodesic metric for linear fractals F governed by the geodesic dimension dg=dH/dℓ, where dℓ is the chemical dimension of F. The imbibition measured by the chemical distance ℓg is introduced. Approximate spatiotemporal maps of capillary rise activity are obtained. The standard differential equations proposed for the von Koch fractals are solved. Illustrative examples to discuss some physical implications are presented.

Keywords

• capillary rise
• spontaneous imbibition
• Hausdorff dimension
• geodesic distance
• Euclidean distance
• von Koch curve