The Hausdorff Dimension and Capillary Imbibition release_obuhh2qwufbqbpggevatduwns4

by Didier Samayoa, Ernesto Pineda León, Lucero Damián Adame, Eduardo Reyes de Luna, Andriy Kryvko

Published in Fractal and Fractional by MDPI AG.

2022   Issue 332, p332

Abstract

The time scaling exponent for the analytical expression of capillary rise <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>ℓ</mo><mo>∼</mo><msup><mi>t</mi><mi>δ</mi></msup></mrow></semantics></math></inline-formula> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>d</mi><mi mathvariant="script">H</mi></msub></semantics></math></inline-formula>. Mapping is converted from the Euclidean metric into the geodesic metric for linear fractals <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula> governed by the geodesic dimension <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>d</mi><mi>g</mi></msub><mo>=</mo><msub><mi>d</mi><mi mathvariant="script">H</mi></msub><mo>/</mo><msub><mi>d</mi><mo>ℓ</mo></msub></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>d</mi><mo>ℓ</mo></msub></semantics></math></inline-formula> is the chemical dimension of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula>. The imbibition measured by the chemical distance <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>ℓ</mo><mi>g</mi></msub></semantics></math></inline-formula> is introduced. Approximate spatiotemporal maps of capillary rise activity are obtained. The standard differential equations proposed for the von Koch fractals are solved. Illustra [...]
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