Damping of a pendulum: An experimental test of the Stokesian unsteady friction force on a cylinder

In 1851, Stokes calculated the force exerted by a viscous fluid on an infinite cylinder in oscillating motion. Although the calculation was restricted to vanishingly small Reynolds numbers $\text{Re}$, most of the experimental tests performed up to now have been made with $\text{Re}>1$. Here we present a series of experiments involving a cylindrical pendulum oscillating in air at different pressures and with $\text{Re}<1$. We deduce the unsteady friction force from the measurement of the damping time of the oscillations. We compare our experimental results for the unsteady force (a) to Stokes' predictions for an infinite cylinder and (b) to predictions for finite-length cylinders, using an expression of the unsteady force derived from the works of Lawrence and Weinbaum and Loewenberg. The agreement is quite satisfactory and proves that macroscopic objects like a pendulum can be used as probes of the Stokesian flow regime.

Figure 1

Schematic drawing of our setup, with the $R\approx 19$ mm cylinder. The spring thickness, the oscillation amplitude, and the size of the laser beam of the shadow detector have been grossly exaggerated in order to be visible. The double arrow gives the length scale.

Figure 2

Schematic drawing of the pendulum (not to scale). The spring and the cylindrical body are represented in two positions. The rest position, $\theta =0$, is used to calculate the moment of inertia for rotation around $C$ and the position with $\theta \ne 0$ exhibits the shape of the spring and the center of rotation $C$ of the pendulum body. $O$ is the upper end of the spring where it is clamped in the support and $D$ its lower end, where it is clamped in the pendulum body, while $G$ is the center of mass of the pendulum body. The length $L_1$ is equal to $CD$. The $x$ axis, the $z$ axis, and the oscillation angle $\theta$ are represented. The angle $\theta$ has been grossly exaggerated as, in our experiments, it never exceeds 1 mrad.

Figure 3

Semilogarithmic plot of the amplitude $a\left(t\right)$ in millimeters as a function of $t$ in seconds: measurements (solid red line) and their best fit (solid blue line). As shown by the good agreement of the fit with the measured data, the amplitude is well an exponential function of time up to $t\approx 6.5\times {10}^3$ s, i.e., for about $5{\tau }^{\text{expt}}$ where the measured value of the damping time constant is ${\tau }^{\text{expt}}\approx 1334$ s. Cylinder radius $R\approx 19$ mm; pressure $P\approx 665$ mbar.

Figure 4

Plot of $K^{\prime }$ as a function of the ratio $R/\delta$. The measured values $K^{\prime \text{expt}}$ are represented by red squares. The dashed blue curve is Stokes' value given by Eq. (5) while the solid black curve represents the finite-cylinder value given by Eq. (12) (these three values are listed in Tables VII and VIII of the Supplemental Material [14]). Upper panel: data of the $R\approx 5$ mm, $\varphi =52$ pendulum; lower panel: data of the $R\approx 19$ mm, $\varphi =13.3$ pendulum.

Figure 5

Position of our experiments as well as previous ones in the $\text{Re}\text{−}\text{St}$ plane (the upper scale gives $R/\delta$): (i) the violet solid line represents the results of Berg et al. [13] as discussed in the Supplemental Material [14]; (ii) the red bullets represent the results of Williams and Hussey [8]; and (iii) the green $+$ signs represent the results of Martin [12] and the blue X represents the result of Stuart and Woodgate [10]. The green bullets, connected by green lines, represent the initial and final values of the Reynolds number during our experiments. Finally, the dashed gray line $\text{Re}=\text{St}$ suggests that most of the previous experiments agreeing with the Stokes predictions are far from satisfying condition (31).