Fine particle percolation dynamics in porous media

The influences of restitution coefficient, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:msub> <a:mi>e</a:mi> <a:mi>n</a:mi> </a:msub> </a:math> , interparticle friction, <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>μ</b:mi> </b:math> , and size ratio, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>R</c:mi> </c:math> , on gravity-driven percolation of fine particles through static beds of larger particles in the free-sifting regime ( <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mi>R</d:mi> <d:mo>≳</d:mo> <d:mn>6.5</d:mn> </d:mrow> </d:math> ) remain largely unexplored. Here, we use discrete element method simulations to study the fine particle percolation velocity, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:msub> <e:mi>v</e:mi> <e:mi>p</e:mi> </e:msub> </e:math> , and velocity fluctuations, <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:msub> <f:mi>v</f:mi> <f:mi>rms</f:mi> </f:msub> </f:math> , for <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:mn>7</g:mn> <g:mo>≤</g:mo> <g:mi>R</g:mi> <g:mo>≤</g:mo> <g:mn>50</g:mn> </g:mrow> </g:math> and a range of <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:msub> <h:mi>e</h:mi> <h:mi>n</h:mi> </h:msub> </h:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mi>μ</i:mi> </i:math> . Varying <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:msub> <j:mi>e</j:mi> <j:mi>n</j:mi> </j:msub> </j:math> modulates the degree of particle excitation and thereby alters the nature of particle trajectories: at low <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:msub> <k:mi>e</k:mi> <k:mi>n</k:mi> </k:msub> </k:math> , percolation is dominated by gravity, whereas increasing <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:msub> <l:mi>e</l:mi> <l:mi>n</l:mi> </l:msub> </l:math> amplifies velocity fluctuations thereby reducing the mean percolation velocity. Increasing <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>μ</m:mi> </m:math> decreases <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"> <n:msub> <n:mi>v</n:mi> <n:mi>rms</n:mi> </n:msub> </n:math> but its influence on <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"> <o:msub> <o:mi>v</o:mi> <o:mi>p</o:mi> </o:msub> </o:math> varies with <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"> <p:msub> <p:mi>v</p:mi> <p:mi>rms</p:mi> </p:msub> </p:math> , decreasing <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"> <q:msub> <q:mi>v</q:mi> <q:mi>p</q:mi> </q:msub> </q:math> for low <r:math xmlns:r="http://www.w3.org/1998/Math/MathML"> <r:msub> <r:mi>v</r:mi> <r:mi>rms</r:mi> </r:msub> </r:math> and increasing <s:math xmlns:s="http://www.w3.org/1998/Math/MathML"> <s:msub> <s:mi>v</s:mi> <s:mi>p</s:mi> </s:msub> </s:math> for high <t:math xmlns:t="http://www.w3.org/1998/Math/MathML"> <t:msub> <t:mi>v</t:mi> <t:mi>rms</t:mi> </t:msub> </t:math> . Although the influence of size ratio is weaker, larger values of <u:math xmlns:u="http://www.w3.org/1998/Math/MathML"> <u:mi>R</u:mi> </u:math> increase both <v:math xmlns:v="http://www.w3.org/1998/Math/MathML"> <v:msub> <v:mi>v</v:mi> <v:mi>p</v:mi> </v:msub> </v:math> and <w:math xmlns:w="http://www.w3.org/1998/Math/MathML"> <w:msub> <w:mi>v</w:mi> <w:mi>rms</w:mi> </w:msub> </w:math> . We also assess the influence of different excitation mechanisms, specifically using static, randomly excited, and sheared beds, finding that an inverse correlation between <x:math xmlns:x="http://www.w3.org/1998/Math/MathML"> <x:msub> <x:mi>v</x:mi> <x:mi>p</x:mi> </x:msub> </x:math> and <y:math xmlns:y="http://www.w3.org/1998/Math/MathML"> <y:msub> <y:mi>v</y:mi> <y:mi>rms</y:mi> </y:msub> </y:math> persists across all cases and is well-described by the Drude model, where increased scattering reduces mobility, when <z:math xmlns:z="http://www.w3.org/1998/Math/MathML"> <z:msub> <z:mi>v</z:mi> <z:mi>rms</z:mi> </z:msub> </z:math> is large. However, for weakly excited particles with low <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML"> <ab:msub> <ab:mi>v</ab:mi> <ab:mi>rms</ab:mi> </ab:msub> </ab:math> , the Drude analogy breaks down. In this regime, we introduce a staircase-inspired model that accounts for the gravitationally dominated percolation behavior. These findings provide fundamental insight into the mechanisms governing percolation dynamics in porous media and granular systems.