Fluid physics of telescoping cardboard boxes

The economics, environmental impact, and mechanical properties of paper-based storage containers have been widely studied. However, knowledge of the physical processes relevant to the end-user experience is unavailable. This paper outlines the main effects associated with the closing and opening of telescoping boxes, which are used, for instance, to store and transport board games, footwear, mobile phones, and tablet computers. The sliding motion of the lid is controlled by the flow in a thin film of air in the gap separating the lid and the base of the box. Based on a broad comparison between theory and experiments on real and synthetic boxes, we find that the process is primarily controlled by the shape of the gap between the base and the lid. We derive a master equation for the lid motion and identify the origin of three distinct experimental regimes. Finally, an optimal design for a rapidly closing box is identified.

Figure 1

Telescoping boxes. (a) Image sequence of a typical closing process and (b) conceptual sketch: air escapes the box of height $a$ at the rate $Q$ via the gap of width $h$ between the lid and base. The time-dependent overlap between the lid and base is denoted $L\left(t\right)$. (c) Closeup of the gap, highlighting three distinct cases: constant, tapering, and widening corresponding to the gap form factor $\alpha =\mathrm{\Delta }h/h_0=0, \alpha >0$ and $\alpha <0$, respectively, in Eq. (5). (d)–(f) Experimental data from 13 commercially available boxes (32 experiments): relative lid-base overlap $L/a$ plotted as a function of time $t$ scaled by the closing time $t_c$ and initial time $t_0$. Solid lines represent limiting theory cases in Eq. (13), Eq. (12), and Eq. (14), respectively. [An initial overlap $L/a={\ell }_0=0.25$ was assumed in panels (e) and (f).] See details of the data classification scheme in the text.

Figure 2

Normalized lid-base overlap $F$ [Eq. (10)] plotted as a function of normalized time $T$ [Eq. (11)]. Data from Figs. 1–1 are in reasonable accord with theory (solid black line) using the fitted shape factor $\langle \alpha \rangle$; see Table 1.

Figure 3

Controlled experiments. (a) Image sequence of a 7.8 mm diameter 3D printed rod penetrating an 8 mm inner diameter plexiglas cylinder. (b) Conceptual sketch: the weight of the rod causes liquid to escape the cylinder of height $a$ at the rate $Q$ via the gap of width $h$. The time-dependent overlap between the rod and cylinder is denoted $L\left(t\right)$. (c) Closeup of the gap, highlighting three distinct cases: constant, tapering, and widening corresponding to $\alpha =\mathrm{\Delta }h/h_0=0, \alpha >0$, and $\alpha <0$, respectively, in Eq. (5). The image sequence in panel (a) represents the $\alpha =0$ case. (d) Experimental data: relative lid-base overlap $L/a$ plotted as a function of time $t$ scaled by the closing time $t_c$ and initial time $t_0$ for rods. The gap shape parameter $\alpha$ varies from $\alpha =-13.6$ to 0.9.

Figure 4

(a) Normalized overlap lid-base overlap $F$ [Eq. (10)] plotted as a function of normalized time $T$ [Eq. (11)] for the controlled experiments. Data from Fig. 3 are in good accord with theory (solid black line). Error bars represent two standard deviations (5–10 repetitions). (b) Closing time $t_c$ as function of $\beta =\mathrm{\Delta }h/\overline{h}$ normalized by the closing time for the reference geometry (Table 2) and mean gap height $\overline{h}$, in reasonable agreement with Eq. (15) (thick solid line). The dashed vertical line indicates the optimal values $\beta =2/3$ where $t_c$ is reduced by $\sim 15\%$ when compared to the straight gap $\beta =0$.

Figure 5

Lid speed increases with mass. Closing time $t_c$ plotted as a function of the lid mass $m$ for the board game Photosynthesis (dots; see Table 1). The solid line illustrates the approximately inverse relation (i.e., $t_c\sim 1/m$) between the two parameters. The initial mass of the lid $\tilde{m}=(199\pm 2)\mathrm{g}$ was modified by adding weights to the top of the box. The mean closing time of the virgin container was ${\tilde{t}}_c=2.5\mathrm{s}$. Error bars indicate the standard deviation.

Figure 6

Geometric parameters in the controlled experiments.