Relaxation dynamics of a flexible rod in a fluid

The relaxation dynamics of a bent rod immersed in a fluid is studied experimentally for various rod materials and sizes and fluid viscosities. One extremity of the rod is clamped while its free end is displaced from its equilibrium position by a controlled distance. For large bending stiffness or low viscosity, the dynamics is underdamped and the rod oscillates around its equilibrium position with a well-defined frequency and a damped amplitude. In contrast, for low bending stiffness or large viscosity, the dynamics is overdamped and the rod relaxes to its equilibrium position without oscillating. We show the existence of two overdamped regimes where the relaxation dynamics is characterized by two different polynomial decays showing that the viscous force is not proportional to the rod velocity. The system is modeled using the dynamical beam equation supplemented by the viscous force experienced by a rigid cylinder moving at a constant speed in a fluid. In spite of this approximation, the model describes the dynamics in good approximation and provides a simple explanation for the existence of two overdamped regimes which originate from a change of the viscous force behavior as the Reynolds number varies. The model is then used to characterize the overdamped relaxation dynamics of the papillae of the bee tongues observed during nectar feeding. We show that the papillae relaxation is not complete when the sugar concentration exceeds 30% which impacts the amount of nectar collected per lap and yields an optimal concentration around 55% for the energy intake rate.

Figure 1

(a). Snapshots of a deflected stainless steel rod of length $L=54$ mm oscillating in water at various times as indicated. The horizontal dashed line indicates the equilibrium position of the rod whereas the red solid line represents the initial shape (6). (b), (c) Evolution of the rescaled position of the tip of rod $w_E\left(t\right)/d$ as a function of time. (b) Representative temporal variations of $w_E\left(t\right)/d$ as the rod materials and the fluid are changed, showing three types of evolution: overdamped, underdamped, and critically damped. $L=9.0\mathrm{cm}$ and $R=0.4\mathrm{mm}$ for the steel rod and $R=0.2\mathrm{mm}$ for PLA rod. The viscosity of the silicone I and II are, respectively, 1.1 and 5.5 Pa s. The crossing times $T$ are also indicated. (c) Representative evolution of $w_E\left(t\right)/d$ as a function of the rescaled time $t/T$ in the overdamped regime, showing two types of relaxation dynamics characterized either by a nearly exponential decay or by a polynomial decay: $L=4.5\mathrm{cm}$ and $R=0.15\mathrm{mm}$ for PP and $L=9.0\mathrm{cm}$ and $R=0.40\mathrm{mm}$ for steel. (d) Evolution of the crossing time $T$ as a function of the rod length $L$ for various fluids and rod materials and geometry: 5 GPa $\le E\le 165$ GPa, $1400\mathrm{kg}/{\mathrm{m}}^3\le {\rho }_s\le 8300\mathrm{kg}/{\mathrm{m}}^3$, $0.15\mathrm{mm}\le R\le 0.4\mathrm{mm}$, $1.1\mathrm{cm}\le L\le 9.0\mathrm{cm}$, ${10}^{-3}$ Pa s $\le \mu \le 10$ Pa s and $950\mathrm{kg}/{\mathrm{m}}^3\le {\rho }_l\le 1100\mathrm{kg}/{\mathrm{m}}^3$. Insets: representative temporal variations of $w_E\left(t\right)/d$.

Figure 2

(a) Evolution of the rescaled viscous force ${\overline{F}}_{\mu }=F_{\mu }/\left(4\pi \mu v_c\right)$, given by Eq. (2), as a function of the rescaled velocity $\overline{v}=v/v_c$ together with two power-law approximations: ${\overline{F}}_{\mu }=1.02{\overline{v}}^{1.34}$ for $\overline{v}\in [1,200]\times {10}^{-3}$ and ${\overline{F}}_{\mu }=0.288{\overline{v}}^{1.1}$ for $\overline{v}<{10}^{-3}$. The gray shaded area indicated the region where Eq. (2) is not valid. (b) Number of experimental data for given types of fluids as a function of $v/v_c$ where the velocity is estimated as $v=d/T$. The orange shaded area indicated data for which the experimental profiles $w_E\left(t\right)$ have 2 zeros (critically damped regime). Outside this region, the profiles have either 1 zero (overdamped regime) or more than 2 zeros (underdamped regime).

Figure 3

Evolution of the experimental rescaled crossing time $T/\tau$ as a function of the control parameter $k$, defined by Eq. (10), for various types of rods and fluids. The data for the damping time $T^{★}/\tau$ are represented by star symbols. The black solid line represents the theoretical expression (24a) divided by 3. The orange shaded area highlights the critically damped regime as deduced theoretically whereas the horizontal dashed lines show the transition between the three regimes. The blue, green, and gray shaded areas and the dotted black curves show the variations of the theoretical scalings and of the numerical solution of Eqs. (1) and (2) as $\overline{d}$ is varied within its experimental range ($0.4\le \overline{d}\le 10.5$). Insets: positions of the tip of the rod as a function of time for some representative cases: overdamped, critically damped, and underdamped regimes.

Figure 4

(a), (b) Evolution of the rescaled position of the tip of the rod $w_E/d$ as a function of a rescaled time for the overdamped regimes I and II. Data obtained by solving numerically Eq. (12) with $\overline{d}=1$ and $\sigma =\frac{11}{10}$ (a) or $\sigma =\frac{4}{3}$ (b) are compared to the experimental data and the asymptotic theoretical curves (17). (c) Evolution of the crossing and damping times as a function of $k$ extracted from numerical solutions of Eq. (12) with $\overline{d}=1$ (symbols) together with analytical asymptotic behaviors: Eq. (18a) (dashed-dotted line), Eq. (18b) (solid line), Eq. (23) (dashed line), Eq. (24b) (short-dashed line). The orange shaded area highlights the critically damped regime. Inset: temporal evolution of $w_E/d$ at $k=k_c\simeq 8\times {10}^{-3}$. (d) Comparison between the temporal evolution of $w_E/d$ obtained by solving numerically Eq. (12) with $\sigma =\frac{4}{3}$ and $\overline{d}=1$ in the underdamped regime (solid curves) and the profiles (24a) obtained from a multiple-scale analysis (dashed curves). The dashed-dotted curve indicates the asymptotic theoretical curve (22) obtained in the undamped regime. The numerical and theoretical data have been shifted vertically for each value of $k$ for clarity.

Figure 5

Representative evolution of $w_E\left(t\right)/d$ as a function of time in the underdamped regime where the rod is immersed in water. The solid curves show the theoretical profiles (24a) with $\omega$ and $T^{★}$ adjusted for each rod. The values of $T^{★}$ obtained are the experimental values reported in Fig. 3 and $\omega$ is computed using Eq. (22) with $k$ and $\tau$ allowed to vary within their experimental errors. The theoretical profiles are limited to time intervals where $v/v_c$ is not too large so that the viscous force (2) is valid.

Figure 6

(a) Schematic of the papillae relaxation dynamics. (b) Comparison between the theoretical ingestion rate $Q$ [see (28)] and in vivo measurements for Bombus as a function of the sugar concentration together with the contribution $Q_P$ of the papillae and of the dragged film $Q_{\text{LLD}}$. The parameter values used are $T_L=(0.20\pm 0.02)$ s, $E=(1.1\pm 0.1)$ MPa, $R_G=(95\pm 5)\mu \mathrm{m}$, $R=(2.1\pm 0.1)\mu \mathrm{m}$, $L=(135\pm 5)\mu \mathrm{m}$, $d=Lsin(\pi /4)$, $L_I=2.0\pm 0.1\mathrm{mm}$, ${\rho }_l=(1175\pm 175)\mathrm{kg}/{\mathrm{m}}^3$. (c) Evolution of the energy intake rate $\dot{E}$ as a function of the sugar concentration for several bee species obtained from Eqs. (29) and (28) with the same parameter values. The vertical dotted line indicates the concentration at which $\dot{E}$ is maximum. The symbols represent data obtained by using (29) together with the in vivo measurements of $Q$ reported in (b). The shaded areas show the regions spanned by the theoretical curves when the parameters are varied within their error bars.

Figure 7

Comparison between ${\overline{w}}_0$ and $g_0/2$.

Figure 8

Evolution of the rescaled viscous force ${\overline{F}}_{\mu }=F_{\mu }/\left(4\pi \mu v_c\right)$, given by Eq. (2), as a function of the rescaled velocity $\overline{v}=v/v_c$, together with two power-law approximations in the overdamped II regime. The coefficient of determination $R^2$ is similar for both power laws.

Figure 9

(a) Viscosity of sugar solutions as a function of the sugar concentration at $20{}^{\circ }\mathrm{C}$ [78] together with the fit (C1). The fit at $30{}^{\circ }\mathrm{C}$ is shown for comparison. (b) Density of sugar solutions as a function of the viscosity [78].