Table 1 Expressions of the coefficients in Eqs (5–7) in the main text.
From: Electrification in granular gases leads to constrained fractal growth
Coefficient | Expression |
|---|---|
\( {\mathcal B} \) | \({k}_{e}\langle \delta {q}^{2}\rangle /(Td)\) |
\(l\) | \(\sqrt{4-{ {\mathcal B} }^{2}}\) |
\({l}_{1}\) | \(\sqrt{1-{ {\mathcal B} }^{2}}\) |
\({C}_{res}^{T}\) | \(80\,{2}^{1/5}\,{d}^{2}{C}_{\epsilon }m/(21\sqrt{\pi }{m}^{8/5})\) |
\({C}_{agg}^{T}\) | \({d}^{2}\mathrm{/(8}\sqrt{\pi m})\) |
\({C}_{res}^{q}\) | \(1.4080\,{d}^{2}{C}_{{\rm{\Delta }}q}^{2}\mathrm{(2}+\eta \mathrm{)(4/}m{)}^{\eta +\mathrm{1/2}}/\pi \) |
\({C}_{agg}^{q}\) | \(4{d}^{2}/\sqrt{\pi m}\) |
\({C}_{agg}^{n}\) | \({d}^{2}\mathrm{/(2}\sqrt{\pi m})\) |
\({a}_{1}^{T}\) | \( {\mathcal B} \mathrm{(8}-5{ {\mathcal B} }^{2})l+\pi \mathrm{(16}-10{ {\mathcal B} }^{2}+3{ {\mathcal B} }^{4})\) |
\({a}_{2}^{T}\) | \(-\,32+20{ {\mathcal B} }^{2}-6{ {\mathcal B} }^{4}\) |
\({a}_{3}^{T}\) | \(l\mathrm{(2}\pi {({ {\mathcal B} }^{2}-\mathrm{4)}}^{2}+ {\mathcal B} (\,-\,32+28{ {\mathcal B} }^{2}+{ {\mathcal B} }^{4}))\) |
\({a}_{4}^{T}\) | \(-\,\mathrm{4(32}-20{ {\mathcal B} }^{2}+9{ {\mathcal B} }^{4})\) |
\({a}_{5}^{T}\) | \(l {\mathcal B} (\,-\,32+28{ {\mathcal B} }^{2}+{ {\mathcal B} }^{4})\) |
\(+\,2\pi ({ {\mathcal B} }^{2}\mathrm{(20}-8l)+{ {\mathcal B} }^{4}(\,-\,9+l)+\mathrm{16(}\,-\,2+l))\) | |
\({a}_{6}^{T}\) | \(\mathrm{4(32}-20{ {\mathcal B} }^{2}+9{ {\mathcal B} }^{4})\) |
\({a}_{1}^{q}\) | \(2 {\mathcal B} l({ {\mathcal B} }^{2}-\mathrm{1)}-\pi \mathrm{(4}-2{ {\mathcal B} }^{2}+{ {\mathcal B} }^{4})\) |
\({a}_{2}^{q}\) | \(\mathrm{2(4}-2{ {\mathcal B} }^{2}+{ {\mathcal B} }^{4})\) |
\({a}_{3}^{q}\) | \(16l\sqrt{\pi }{l}_{1}{\mathrm{(4}-5{ {\mathcal B} }^{2}+{ {\mathcal B} }^{4})}^{2}\) |
\({a}_{4}^{q}\) | \(-\,\sqrt{\pi } {\mathcal B} l(\pi ({ {\mathcal B} }^{2}-\mathrm{4)(1}+2{ {\mathcal B} }^{2})\) |
\(+\,2 {\mathcal B} {l}_{1}(\,-\,8-54{ {\mathcal B} }^{2}+33{ {\mathcal B} }^{4}+2{ {\mathcal B} }^{6}))\) | |
\({a}_{5}^{q}\) | \(2\sqrt{\pi } {\mathcal B} {l}^{5}\mathrm{(1}+2{ {\mathcal B} }^{2})\) |
\({a}_{6}^{q}\) | \(16\sqrt{\pi } {\mathcal B} {l}_{1}^{5}\mathrm{(4}+5{ {\mathcal B} }^{2})\) |
\({a}_{7}^{q}\) | \(8l\sqrt{\pi }{l}_{1}{\mathrm{(4}-5{ {\mathcal B} }^{2}+{ {\mathcal B} }^{4})}^{2}\) |
\({a}_{8}^{q}\) | \(-\,\sqrt{\pi } {\mathcal B} {l}^{5}\mathrm{(1}+2{ {\mathcal B} }^{2})\) |
\({a}_{9}^{q}\) | \(\sqrt{\pi } {\mathcal B} {l}_{1} {\mathcal B} l\mathrm{(8}+54{ {\mathcal B} }^{2}-33{ {\mathcal B} }^{4}-2{ {\mathcal B} }^{6})\) |
\({a}_{10}^{q}\) | \(8\sqrt{\pi } {\mathcal B} {l}_{1}{({ {\mathcal B} }^{2}-\mathrm{1)}}^{2}\mathrm{(4}+5{ {\mathcal B} }^{2})\) |
\({a}_{1}^{n}\) | \(-\,{ {\mathcal B} }^{2}{l}^{3}\) |
\({a}_{2}^{n}\) | \({l}_{1}l {\mathcal B} (\,-\,4+7{ {\mathcal B} }^{2})+{l}_{1}\pi ({ {\mathcal B} }^{2}-1)(8-4l+{ {\mathcal B} }^{2}(\,-\,6+l))\) |
\({a}_{3}^{n}\) | \(4{l}_{1}\mathrm{(4}-7{ {\mathcal B} }^{2}+3{ {\mathcal B} }^{4})\) |
\({a}_{4}^{n}\) | \(-\,{ {\mathcal B} }^{2}{l}^{3}\) |
\({a}_{5}^{n}\) | \({l}_{1}l( {\mathcal B} (\,-\,4+7{ {\mathcal B} }^{2})+\pi \mathrm{(4}-5{ {\mathcal B} }^{2}+{ {\mathcal B} }^{4}))\) |
\({a}_{6}^{n}\) | \(-\,4{l}_{1}(4-7{ {\mathcal B} }^{2}+3{ {\mathcal B} }^{4})\) |
