The Hausdorff Dimension and Capillary Imbibition
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Didier Samayoa, Ernesto Pineda León, Lucero Damián Adame, Eduardo Reyes de Luna, Andriy Kryvko
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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