Experimental test of unsteady Stokes’ drag force on a sphere

Abstract

Stokes calculated the force exerted by the surrounding fluid on a sphere and on a cylinder in oscillating motion. Although these results are valid only if the Reynolds number Re is very small, ${\text {Re}}\ll 1$, all the tests on macroscopic spheres have been made with Re larger than 20. Here, we describe an experiment which measures the drag force on an oscillating sphere with small values of the Reynolds number, down to ${\text {Re}}\approx 0.03$ for the smallest sphere studied here while the Stokes number St is large, between 150 and 1500. Our measurements are in very good agreement with Stokes’ result, and in particular, they exhibit the quadratic dependence of the force with the sphere radius when this radius is larger than the viscous penetration depth $\delta$.

Graphic abstract

Appendices

Appendix 1: Quasi-degeneracy of the normal modes of our pendulum

The pendulum has an almost perfect cylindrical symmetry around the wire axis. Such a pendulum has two normal modes with two oscillation frequencies which would be exactly equal if the cylindrical symmetry was perfect and which are slightly different because this symmetry is weakly broken. In our experiment, we observe a weak symmetry breaking: A possible explanation would be an elliptic cross section of the suspension wire, but we have found that the dominant contribution is due to the interaction of the magnet carried by the pendulum with the gradient of the laboratory magnetic field. We treat of this problem, in the case of a simple pendulum of mass m. Its kinetic energy term $E_K$ is isotropic, and the anisotropy comes from the potential energy term $E_P$

$$\begin{aligned} E_K= \, & {} \frac{m}{2} \left[ \left( \frac{du}{dt}\right) ^2 + \left( \frac{dv}{dt}\right) ^2\right] , \end{aligned}$$

(20)

$$\begin{aligned} E_P= \,& {} \frac{1}{2} \left[ k_u u^2 + k_v v^2\right] . \end{aligned}$$

(21)

Here, ${\mathbf {u}}$ and ${\mathbf {v}}$ are the axes which give a diagonal form to $E_P$ and u and v the coordinates of the pendulum center of mass in this coordinate system. The normal modes correspond to oscillations along the ${\mathbf {u}}$ and ${\mathbf {v}}$ axes with the angular frequencies

$$\begin{aligned} \omega _u= \sqrt{k_u/m} { \text{ and } } \omega _v= \sqrt{k_v/m}. \end{aligned}$$

(22)

We consider the displacement of the pendulum in the ${\mathbf {x}}$-direction. In the general case, the pendulum is in a superposition of the two normal modes

$$\begin{aligned} u = u_0 \cos \left( \omega _u t + \varphi _u\right) { \text{ and } } v = v_0 \cos \left( \omega _v t + \varphi _v\right). \end{aligned}$$

(23)

We note $\theta$ the angle between the ${\mathbf {u}}$ and ${\mathbf {x}}$ axes and $(\pi /2) -\theta$ the angle ${\mathbf {v}}$ and ${\mathbf {x}}$. The position of the pendulum along the ${\mathbf {x}}$-axis is

$$\begin{aligned} x=\, & {} x_u \cos \left( \omega _u t + \varphi _u\right) + x_v \cos \left( \omega _v t + \varphi _v\right) \end{aligned}$$

(24)

with $x_u=u_0 \cos \theta$ and $x_v=v_0 \sin \theta$. We introduce the phases $\psi _+$ and $\psi _-$ defined by

$$\begin{aligned} \psi _+=\, & {} \left[ \left( \omega _u + \omega _v \right) t +\varphi _u + \varphi _v\right] /2\nonumber \\ \psi _-=\, & {} \left[ \left( \omega _u -\omega _v \right) t +\varphi _u - \varphi _v\right] /2. \end{aligned}$$

(25)

$\psi _+$ describes the fast oscillation, while $\psi _-$ describes the envelope of the beats.

$$\begin{aligned} x= \left( x_u+ x_v \right) \cos \psi _- \cos \psi _+ + \left( -x_u + x_v \right) \sin \psi _- \sin \psi _+ \end{aligned}$$

(26)

The amplitude of the fast oscillation is a function of the time t. It is given by the modulus of the Fourier component of frequency $\omega _+ = \left( \omega _u + \omega _v \right) /2$:

$$\begin{aligned} {\tilde{x}}_{\omega _+ }(t)= \,& {} \sqrt{x_u^2+ x_v^2 + 2x_u x_v \cos \left( \omega _- t +\varphi _-\right) }\exp \left( -t/\tau \right) , \end{aligned}$$

(27)

where $\omega _-$ and $\varphi _-$ are the frequency and the initial phase of the beat pattern. We have taken into account damping by adding the factor $\exp \left( -t/\tau \right)$. This is correct if the damping forces are equal for the two normal modes: This assumption is excellent as the pendulum has an almost perfect cylindrical symmetry.

We now apply these results to the treatment of our experimental data. The measured amplitude $x_0(t)$ is an average of ${\tilde{x}}_{\omega _+ }(t)$, because of the use of the sliding fast Fourier transform. When the pendulum is under vacuum, the damping time constant $\tau$ is very long and we observe beats which are fitted by the following equation:

$$\begin{aligned} x_0(t)= \sqrt{A + B \cos \left( \frac{2\pi t}{T_0} +\phi \right) } \exp \left( -t/\tau \right) . \end{aligned}$$

(28)

This equation is equivalent to Eq. (27) but better suited for a fit. In this case, even if we record the oscillation for a long time, the final amplitude is not very small and there is no need to include a term representing the mean effect of the vibrations of the experiment. Figure 4 presents an example of the recorded data and its fit by Eq. (28).

Fig. 4

Damping of the oscillation of the sphere of radius $a=15$ mm under vacuum. The measured amplitude $x_0(t)$ is plotted as a function of the time t (s) (full red line) and its fit using Eq. (28) (dashed blue line). The damping time constant $\tau$ and the beat period $T_0$ given by the fit are $\tau = 7174$ s and $T_0 = 4061$. The amplitudes A and B  are almost equal in this experiment: This condition, which maximizes the beat contrast, occurs if $\theta \approx \pm \pi /4$

In a preliminary setup, we measured the oscillation amplitude in two perpendicular directions with two shadow detectors measuring the amplitude ${\tilde{x}}_{\omega _+ }(t)$ and ${\tilde{y}}_{\omega _+ }(t)$. By taking the square root of the sum of the squares of these two amplitudes, we eliminated the effect of the beats and the damping was exponential. The results of this preliminary experiment appear in Table 2 in the column $\tau _\mathrm{{exp1}}$.

In the present setup, we measure only the amplitude ${\tilde{x}}_{\omega _+ }(t)$. When the pendulum is at atmospheric pressure, the damping time constant $\tau$ is considerably shorter than under vacuum. We do not observe beats, and we use Eq. (13) to fit the measured amplitude. This technique was successful for all the spheres except for the smallest one with a radius $a=15$ mm. For this sphere, the measured damping time constant varied noticeably when we reduced the fraction $\alpha$ of the initial amplitude. We think that this was due to undetected beats: The beat period $T_0 \approx 4000$ s (see Fig. 4) is sufficiently large with respect to the damping time constant $\tau \approx 430$ s so that we do not observe beats for this sphere, but the hidden beat pattern makes that the damping is not exponential. (This effect appears to be negligible for larger spheres because the beat period $T_0$ increases when the pendulum mass increases, and at the same time, the damping time constant $\tau$ decreases as shown in Table 2.) For the sphere with $a=15$ mm, we have applied a dipolar magnetic field produced by a magnet in a geometry such that the excited oscillation is a normal mode and then, the beat amplitude is negligible. This technique restored an exponential decay of the amplitude, and we thus measured the value of $\tau _\mathrm{{exp2}}$ for this sphere.

Appendix 2: Calculation of the energy lost $\varDelta E_\mathrm{{h}}$

We calculate the energy lost $\varDelta E_\mathrm{{h}}$ because of the hydrodynamic forces acting on the sphere and on the wire. The drag force on the sphere given by Eq. (4) can be rewritten $F'_x\left( t\right) =- \beta _\mathrm{{S}} v_x\left( t\right)$. The velocity of the sphere center being $v_x\left( t\right) = v_\mathrm{{m}}\left( t\right) \cos \left( \omega t\right)$, where $v_\mathrm{{m}}\left( t\right)$ varies slowly with t, the work $\varDelta E_\mathrm{{S}}(t)$ done by this drag force during one period of oscillation $T= 2\pi /\omega$ is given by a straightforward integration:

$$\begin{aligned} \varDelta E_\mathrm{{S}}(t) \approx \beta _\mathrm{{S}} \frac{\pi v_\mathrm{{m}}^2\left( t\right) }{\omega }. \end{aligned}$$

(29)

The drag force per unit length of the wire given by Eq. (6) is used to calculate the force on each element dz of the wire as if it was part of an infinite cylinder moving with a uniform oscillating velocity

$$\begin{aligned} \frac{{\text {d}}F'_x }{\mathrm{{d}}z}= & {} -\beta _\mathrm{{W}} v_x\left( z,t\right) {\text{ with } } \beta _\mathrm{{W}} = 2\pi \eta \left( \frac{R}{\delta }\right) ^2 k'. \end{aligned}$$

(30)

If the wire is approximated by a straight line, the velocity $v\left( z,t\right)$ is proportional to the distance z and related to the velocity of the sphere center by

$$\begin{aligned} v_x\left( z,t\right) = \frac{z}{l_\mathrm{{S}}}v_\mathrm{{m}}\left( t\right) \cos \left( \omega t\right). \end{aligned}$$

(31)

The work $\varDelta E_\mathrm{{W}}(t)$ done by the drag force on the wire during one period of oscillation is then given by integration over the length of the wire and over one period of oscillation

$$\begin{aligned} \varDelta E_\mathrm{{W}}(t) \approx \beta _\mathrm{{W}}\frac{l_\mathrm{{W}}^3}{3 l_\mathrm{{S}}^2} \frac{\pi v_\mathrm{{m}}^2\left( t\right) }{\omega }. \end{aligned}$$

(32)

We thus get

$$\begin{aligned} \varDelta E_\mathrm{{h}}(t)=\, & {} \varDelta E_\mathrm{{S}}(t) + \varDelta E_\mathrm{{W}}(t) = \beta \frac{\pi v_\mathrm{{m}}^2\left( t\right) }{\omega } \end{aligned}$$

(33)

$$\begin{aligned} {\text{ with } } \beta=\, & {} \beta _{S1} + \beta _{S2} + \beta _\mathrm{{W}} \end{aligned}$$

(34)

$$\begin{aligned} \beta _{S1}= \,& {} 6 \pi \eta a { \text{ and } } \beta _{S2} = 6 \pi \eta a^2/\delta \end{aligned}$$

(35)

$$\begin{aligned} \beta _\mathrm{{W}}=\, & {} 2\pi \eta \left( \frac{R}{\delta }\right) ^2 k' \frac{l_\mathrm{{W}}^3}{3l_\mathrm{{S}}^2}. \end{aligned}$$

(36)

In order to compare this theoretical value $\beta _\mathrm{{th.}}$ to its experimental values $\beta _\mathrm{{exp}}$, we will take into account the correction due to the fluid confinement given by Eq. (18) and this correction is applied only to the sphere terms $\beta _{S1}$ and $\beta _{S2}$.

Appendix 3: Refined theory of the pendulum

We have developed the theory of a flexible-beam pendulum (Dolfo and Vigué 2015), i.e., a rigid pendulum body suspended by a suspension spring, as used in traditional clocks. This theory predicts the existence of two resonances corresponding to different motions of the pendulum: We are interested here in the lowest frequency resonance which is associated with the usual pendular motion. The theoretical values of these two frequencies have been compared to experimental results (Dolfo et al. 2016), and the agreement is good, especially for the lowest frequency resonance, which is the one we are interested here. We refer the reader to our paper (Dolfo and Vigué 2015) for the equations which are used here to calculate the frequency and total energy of our pendulum. To apply this theory, we must know the masses of the pendulum components and their dimensions as well as the elastic properties of the suspension wire.

1.1 Information on the pendulum and calculation of its frequency

Figure 1 presents a schematic drawing of the pendulum. The free length OD of the suspension wire is noted $l_\mathrm{{W}}$. The distance OS, when the pendulum is at rest, is noted $l_\mathrm{{S}} = l_\mathrm{{W}} +l_N + a$ where $l_N= 10$ mm is the nut length (see fig. 1 of our letter). The length DG is noted h, and the gyration ratio of the pendulum body is noted $\rho$. We have weighted the various parts of the pendulum and measured the various dimensions to calculate the total mass M of the pendulum, the distance $DG=h$ and the gyration ratio $\rho$ of the pendulum body. All these values are given in Table 3.

The pendulum suspension wire is a piano string of diameter $2R= 0.49\pm 0.01$ mm, length $l_\mathrm{{W}} = 404\pm 0.5$ mm and mass $m_\mathrm{{W}}= 0.60$ g. Moreover, the theory of Dolfo and Vigué (2015) which takes into account the bending rigidity of this wire requires the product ${\text {EI}}_\mathrm{{s}}$ for the suspension wire, where E is the Young modulus of the wire material and $I_\mathrm{{s}}$ is the second moment of the area of its cross section, $I_\mathrm{{s}}= \pi R^4/4$. We have measured this quantity as in our previous work (Dolfo et al. 2016) by measuring the deflection of a portion of this wire initially horizontal by a series of small test masses. We thus get ${\text {EI}}_\mathrm{{s}}= (6.17\pm 0.07)\times 10^{-4}$ N $\hbox {m}^2$, in good agreement with the calculated value ${\text {EI}}_\mathrm{{s}}= (5.94\pm 0.48)\times 10^{-4}$ N $\hbox {m}^2$, using the value $E\approx 210$ GPa of Young’s modulus for high-carbon content steel (value from the website http://www.matweb.com/)

In our paper (Dolfo and Vigué (2015)), the mass of the flexible beam was neglected with respect of the pendulum body and this was a good approximation in our experimental test (Dolfo et al. 2016). It is no more the case in the present experiment, because the wire mass (0.60 g) is not negligible with respect to the other masses, especially for the smallest spheres. As a consequence, in the calculation of $\rho$ and M, we have included the contribution of the wire as if it was straight. From these parameters, we deduce the parameter $\kappa = \sqrt{Mg/\left( {\text {EI}}_\mathrm{{s}}\right) }$ and the dimensionless parameter $\chi =\kappa l_\mathrm{{W}}$. Using the equations (23) and (24) of our paper (Dolfo and Vigué (2015)), we calculate the frequency $\omega /(2\pi )$ of its lowest resonance and the length $\lambda$. The agreement of the calculated frequency with its measured value is very good.

Table 3 For each sphere, we give the mean value of its radius a, with an uncertainty interval $2\varDelta a$ covering the variations of the diameter measured along 7 well-spaced directions, the total pendulum mass M (the present work is a part of the PhD thesis of one of us (GD). We found that the electronic mass balance used to measure the masses of the pendulum components gave wrong values, $\approx 80$% of the correct value. This error has been corrected and we have redone all the experiments), the lengths $l_\mathrm{{S}}$ and h and the gyration ratio $\rho$

1.2 Calculation of the drag force coefficient $\beta$

Using our theory of this type of pendulum (Dolfo and Vigué 2015), we calculate the total energy $E_\mathrm{{tot}}$ and the energy lost $\varDelta E_\mathrm{{h}}$ by the drag forces during one period of oscillation as a function of the drag force coefficient $\beta$. In our calculation, we take into account the exact shape of the wire. The results of this calculation are given in Table 4. In this table, we also give the theoretical value $\beta _\mathrm{{th.}}$ of this coefficient, calculated using Eq. (33) and taking into account the correction for the fluid confinement given by Eq. (18), assuming that the vessel enclosing the pendulum is equivalent to a sphere of radius $b=140$ mm.

Table 4 For each pendulum, this table gives the rounded value of the sphere radius a, the value of the viscous penetration depth $\delta$, the value of the Stokes number St, the experimental value $\beta _\mathrm{{exp}}$ deduced from the value of the damping time constant $\tau _\mathrm{{h}}$ given in Table 2 and the theoretical value $\beta _\mathrm{{th}}$ (see text)