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Slow kinetics of water escape from randomly folded foils

Alexander S. Balankin, S. Matías Gutierres, D. Samayoa Ochoa, J. Patiño Ortiz, B. Espinoza Elizarraraz, and C. L. Martínez-González

Grupo “Mecánica Fractal,” Instituto Politécnico Nacional, 07738 México D.F., México

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Phys. Rev. E 83, 036310 – Published 17 March, 2011

DOI: https://doi.org/10.1103/PhysRevE.83.036310

Abstract

We study the kinetics of water escape from balls folded from square aluminum foils of different thickness and edge size. We found that the water discharge rate obeys the scaling relation $Q \propto V_{P} (M - M_{r})^{α}$ with the universal scaling exponents $α = 3 \pm 0.1$ , where $V_{P}$ is the volume of pore space, $M (t)$ is the actual mass of water in the ball, and $M_{r}$ is the mass of residual water. The last is found to be a power-law function of $V_{P}$ . The relation of these findings to the fractal geometry of randomly folded matter is discussed.

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References (28)

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The porosity of tested balls was varied in the range $0.802 < p (λ) < 0.987$ .
In this way, we were able to fill with water up to $97 %$ of pore space of a folded ball.
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It should be pointed out that the value of exponent $α$ obtained by the least-square fitting of experimental data with Eq. (1) varied from experiment to experiment in the range from 2.7 to 3.1 with mean 2.96 and standard deviation $σ_{α} = 0.073$
Specifically, we found that $M_{r} - M_{r} (BF) = ɛ ρ V_{P}$ , where the parameter $ɛ$ are varied from experiment to experiment in the range $0.003 ⩽ ɛ ⩽ 0.004$ .
Notice that the same amount of water escapes from a hollow reservoir with holes more than ten times faster than from a folded sheet with the same cross-sectional area available for water escape.
Strictly speaking, the best fit of all experimental data is $\bar{K} = 0.13 ρ V_{P}^{1.2}$ [see the top insert in Fig. 5(a)], whereas the data for balls with different contraction ratios are best fitted by Eq. (3) [see the bottom insert in Fig. 5(a)].
J. Bear, The Dynamics of Fluids in Porous Media (Dover, Mineola, NY, 1988).
This is possible, for example, if the height of water in a folded sheet does not decrease so far $M > M_{r}^{BF}$ , or if both $Δ P$ and $Δ l$ decrease as $M$ decreases, in such a way that $Δ P / Δ l = const .$ Alternatively, we can assume that $Δ P / Δ l ≪ ρ g$ , but this seems less probable. In any case, this point needs to be clarified in further studies.
R. H. Brooks and A. T. Corey, in Hydrology Papers, No. 3 (Colorado State University, Fort Collins, CO, 1964), p. 22.
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Of course, this statement needs to be verified by direct experiments.
Experimentally it was found that the mass fractal dimension of folded matter increases as $λ$ increases [see Y.-C. Lin et al., Phys. Rev. E 80, 066114, (2009)]. As far as the pore space is continuous, its physical fractal dimension is in the range $2 ⩽ D_{P} ⩽ 3$ , whereas $D_{P} < 2$ , if the porosity $p (λ)$ is less than its critical value $p_{C}$ at the percolating threshold, when the pore space becomes discontinuous; obviously the limit $D_{P} (p = 0) \equiv 0$ .
For many porous materials it was found that $D_{P} = D_{S}$ [see P. Pfeifer and D. Avnir, J. Chem. Phys. 79, 3558, (1983) and references therein]. In this way, we can speculate that randomly folded thin matter is also characterized by $D_{P} = D_{S}$ . If so, our finding $n = α = 3$ together with (7), (8) suggest that $ω = 1$ , while $ζ = 1 - χ$ .

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Slow kinetics of water escape from randomly folded foils

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