3.1. Levitation condition

Different types of objects, including solid spherical beads, oblate spheroidal liquid droplets and bubbles, were suspended by a moving flow inside a cylindrical tube of inner radius $R$ , whose axis is oriented vertically (see figure 1 b,d). All spherical objects are taken to have radius $a$ , while oblate spheroidal objects are assumed to have smallest diameter (usually the height) equal to $2a$ and largest diameter (usually the width) equal to $2\epsilon a$ . In what follows, we will derive the levitation condition for general objects of aspect ratio $\epsilon \geqslant 1$ , and will set $\epsilon =1$ when dealing specifically with spherical objects.

The sum of gravity and buoyancy forces felt by the object is given by

(3.1) \begin{equation} F_g = \frac {4}{3}\pi \epsilon ^{2}a^3\Delta \rho g, \end{equation}

where $g$ is the acceleration due to gravity and $\Delta \rho$ is the density difference between the object material, $\rho _o$ , and the suspending fluid, $\rho$ . The object also experiences a drag force inside the moving fluid which opposes its motion. This force is dependent upon the shape and type of object and can be written as

(3.2) \begin{equation} F_d = 6\pi \eta \epsilon a\gamma \lambda v(r), \end{equation}

where $\eta$ is the viscosity of the suspending fluid, $v(r)$ is the speed of the fluid at a distance $r$ from the tube centre (see figure 1 b), $\gamma$ is a blocking factor and $\lambda =\lambda (a,R)$ is a form factor that accounts for the effects of the drag on the object caused by its shape and proximity to the walls.

The blocking factor $\gamma$ arises because the presence of the object in the tube blocks the flow. The restriction in flow caused by the blockage leads to a pressure build-up across the object in the axial direction of the tube. This build-up of pressure has the effect of increasing the drag on the object in a tube relative to that in an unbounded flow and gives rise to a doubling in the effective drag force (Clift et al. Reference Clift, Grace and Webster1978). We will hence set $\gamma =2$ in the drag force (3.2).

Values of the parameter $\lambda$ have been calculated for solid spherical particles and slightly distorted fluid spheres in a moving fluid inside a cylinder (Stokes’ approximation for the hydrodynamic equations for slow flow; Haberman & Sayre Reference Haberman and Sayre1958). This approach leads to power-law expansions for the parameter $\lambda$ for the levitated objects of the form

(3.3) \begin{equation} \lambda = \frac {2\big(1+\frac {2}{3}\sigma\big) - \frac {2}{3}\xi ^2 - 0.96074(1-\sigma)\xi ^5} {(1+\sigma) - 2.105\big(1+\frac {2}{3}\sigma\big)\xi + 2.0865\,\xi ^3 - 1.7068\big(1-\frac {2}{3}\sigma\big)\xi ^5 + 0.726(1-\sigma)\xi ^6}, \end{equation}

where $\xi ={a}/{R}$ and $\sigma ={\eta }/{\eta _o}$ is the viscosity ratio between the suspending fluid and the levitated object. In the case of solid particles, we set $\eta _o$ to be infinite and $\sigma =0$ . Values of $\sigma$ for bubbles and liquid droplets were determined using the data in table 1. Figure 4 shows the variation in $\lambda$ for solid spheres, liquid droplets and air bubbles in tubes of different internal diameters. As expected, the functional form of the drag force on the particle (cf. (3.2) and (3.3)) is consistent with motion in an unbounded flow when the objects are small, and increases monotonically as the object size approaches the tube diameter and hence wall effects become more significant.

Figure 4. Variation of parameter $\lambda$ as a function of the reduced half-width $(\epsilon \xi = {\epsilon a}/{R})$ of the levitating object, obtained from (3.3) for air bubbles in 5 mm (blue line), 8 mm (red line) and 10 mm (green line) inner diameter tubes. Curves are also plotted for solid PTFE spheres (black solid line) and liquid tetrabromoethane droplets (black dashed line) in 10 mm diameter tubes. For each curve, the value of the viscosity ratio $\sigma$ was determined from table 1. The values of the aspect ratio $\epsilon$ used for bubbles and droplets were obtained from the fits to the data in figure 2, while $\epsilon =1$ was used for the solid spheres.

Far from the object, the speed of the suspending fluid is assumed to be determined by Poiseuille flow, i.e.

(3.4a,b) \begin{gather} v(r) = v_m\left (1-\left (\frac {r}{R}\right)^{2}\right)\!, \qquad \text{and} \qquad Q =\int _0^{2\pi }\hspace {-0.2cm}\int _0^R v(r)\,r\,\mathrm{d} r\mathrm{d}\theta =\frac {\pi R^2}{2}v_m, \end{gather}

where $v_m$ is the maximum speed achieved at the centre of the tube, and $Q$ is the volumetric flow rate. We note that, while the background flow speed is valid for $0\leqslant r\leqslant R$ , the object’s centre of mass only experiences fluid speeds for $0\leqslant r\leqslant R-\epsilon a$ . If the object’s terminal speed in the tube matches the speed of the suspending fluid, then the two forces in (3.1) and (3.2) balance and the object will appear to levitate in the frame of reference of an external observer. Setting $F_g=F_d$ and combining with (3.4a , b ), provides an expression for the flow rate that is required to levitate the object, given by

(3.5) \begin{equation} Q_{{lev}} = \frac {\pi \epsilon a^2R^2\Delta \rho g}{9\eta \lambda \gamma \left ( 1-\left (\dfrac {r}{R}\right)^{2} \right)}. \end{equation}

Figure 5. Flow rate for levitation of (a) air bubbles, (b) liquid tetrabromoethane droplets and (c) solid PTFE particles as a function of the normalised half-width, ${\epsilon a}/{R}$ , of the object being levitated. Panel (a) shows data for air bubbles levitated inside tubes with internal diameters of 5 mm (blue diamonds), 8 mm (red squares) and 10 mm (green circles). Equation (3.5) was used to generate the solid lines using material parameters given in table 1.

Figure 5 shows a comparison between the measured and calculated levitation flow rates for objects of different sizes. Results are presented for solid PTFE beads, liquid tetrabromoethane droplets and air bubbles. In all cases where the objects levitated off centre in the tube, the experimentally determined levitation flow rates were corrected by a factor of $1- ({r}/{R})^2$ , with $r$ set to the experimentally observed off-centre position. Correcting experimental flow rate data in this way enabled a simplified form of (3.5) to be used by setting $r=0$ for all solid lines presented in figure 5. The values of $\lambda$ used to generate the theoretical lines are calculated using (3.3), with viscosity ratios taken from the parameters of table 1. Density values were also taken from table 1 and values of $\epsilon$ were obtained from the fits to the data in figure 2.

The experimentally determined values of $\epsilon$ that were obtained for the bubbles and drops can be compared directly with the predictions of a simple model proposed by Manica et al. (Reference Manica, Klaseboera and Chan2016b ). These authors derived an expression for $\epsilon$ by balancing the inertial pressure due to the flow and the internal Laplace pressure of a bubble. We apply their approach to write the aspect ratio of the bubbles using the notation developed here, to obtain

(3.6) \begin{equation} \epsilon = \left ( 1 - \frac {9a\rho Q_{lev}^2}{16\pi ^2R^4\gamma _{lo}} \right)^{-1}, \end{equation}

where $\gamma _{lo}$ is the interfacial tension between the suspending liquid and the deformable object (bubble or drop). If we consider bubbles with $a=0.4R$ (corresponding to the rough position of the maximum in the flow rate curves) and use the results of figure 5, we obtain levitation flow rates of approximately $1.5$ , $7$ and $12\ \mathrm{ml\boldsymbol{\, }s^{-1}}$ for bubbles in 5, 8 and 10 mm diameter tubes, respectively. Assuming a value of $\gamma _{lo}=72\times 10^{-3}\ \mathrm{J\boldsymbol{\, }m^{-2}}$ (surface tension of water/air interfaces at room temperature), the corresponding values of $\epsilon$ that we calculate using (3.6) are 1.05, 1.32 and 1.57. These values are broadly in agreement with the experimental values obtained for air bubbles from figure 2. Application of the same approach to the tetrabromoethane droplets yields a value of $\epsilon =1.34$ using $\gamma _{lo}=72\times 10^{-3}\ \mathrm{J\boldsymbol{\, }m^{-2}}$ or $\epsilon =1.57$ using $\gamma _{lo}=50\times 10^{-3}\ \mathrm{J\boldsymbol{\, }m^{-2}}$ (a typical value for alkane/water and haloalkane/water interfaces; Israelachvili Reference Israelachvili1991). Some disagreement between the values of $\epsilon$ predicted by (3.6) and the data for tetrabromoethane is to be expected when using appropriate values for $\gamma _{lo}$ . This equation was developed by Manica et al. (Reference Manica, Klaseboera and Chan2016b ) for a compressible gas bubble and as such neglects the influence of viscosity and incompressibility that are expected to be important in a liquid droplet.

The results of figure 5 indicate that the shapes of the experimental data are broadly in agreement with those of the model predicted by (3.5). While the level of agreement shown is generally good, the model proposed here clearly fails to describe the behaviour of the levitating system for certain values of the normalised half-width of the levitated object. This is true for small $ ({\epsilon a}/{R} \lt 0.3)$ bubbles, and both large $ ( {\epsilon a}/{R} \gt 0.7)$ bubbles and liquid droplets, where the experimentally determined flow rates are higher than those predicted by the model. In the case of the spheres, the shape of the flow rate curve predicted by (3.5) is similar that observed in experiments. However, a simple scaling of the solid line in the bottom panel of figure 5 by a factor of $\sim\! 1.1$ is required to obtain good agreement between the model and the experimental data. This factor could be accounted for in terms of uncertainties in the parameters in table 1.

Disagreement between the data and the simple model is not entirely unexpected. For example, the model neglects the effects of compressibility and the time-dependent shape deformations of bubbles and droplets in the flow, which will significantly influence the values of $\lambda$ used for larger objects. The assumption that the far-field speed profile obtained for Poiseuille flow in a tube is valid close to the levitating objects is also a crude one. This is potentially further complicated by the use of a relatively short length of the straight section of the glass tube used to levitate the objects. The use of a short circulation loop is likely to make it more difficult to establish laminar flow in the presence of a levitating object. Ideally a longer vertical section of the tube would have been used to perform these experiments and allow the Poiseuille flow profile to be better established. However, disturbances caused by fluctuations in the flow due to the pump were found to have a significant effect on the stability of the levitation condition. A low noise pump with a low flow rate was therefore chosen to minimise background disturbances in the flow. The choice of pump limited the pressure range available and therefore the height of the fluid column that could be used to obtain a wide enough range of levitation flow rates in the experiments.

In addition to the above factors, the range of validity of the Reynolds numbers associated with (3.3) are a potential source of disagreement. We estimate the Reynolds numbers attained in these experiments to be ${Re} \sim 600$ , while (3.3) was developed for systems where ${Re}=\mathcal{O}(1)$ (Clift et al. Reference Clift, Grace and Webster1978).

3.2. Oscillatory motion of objects

The oscillatory motion of bubbles, drops and solid beads can be explained by the process of vortex shedding. This mechanism has been shown to be responsible for the zigzag and helical motion of bubbles rising in quiescent liquids (Lunde & Perkins Reference Lunde and Perkins1998; Liu et al. Reference Liu, Zhang, Wei, Mu, Yang and Farooq2023) and small diameter tubes (Ortiz-Villafuerte, Schmidl & Hassan Reference Ortiz-Villafuerte, Schmidl and Hassan2001). High-speed movies of solid particles levitated in the tubes studied here confirm that vortex shedding also occurs when mica powder particles are added to the water (see Supplementary Information Movie 1.mp4).

In what follows, a simple model is derived using a similar approach to that described by Baker (Reference Baker2016). We begin with the definition of the Strouhal number, $St$ , a dimensionless parameter that describes oscillating flow mechanisms. It can be used to set the frequency of vortex shedding, $f$ , behind an object of width $w$ in a flow of speed $v_o$ , and it is defined by

(3.7) \begin{equation} St=\frac {fw}{v_o}. \end{equation}

The Strouhal number has been reported in the literature to be approximately equal to $0.2$ for most Reynolds numbers (Baker Reference Baker2016). In the case of an incompressible fluid, the same flow rate should be maintained everywhere and hence the product of fluid speed and cross-sectional area should be constant, both when considered around the edges of the object or far away.

When an object of cross-sectional area $A$ moves in a tube of radius $R$ , this condition becomes

(3.8a,b) \begin{gather} v_o(\pi R^2-A) = v_m\pi R^2 \qquad \Leftrightarrow \qquad v_o = \frac {v_m\pi R^2}{(\pi R^2-A)}, \end{gather}

where we have assumed that the characteristic fluid speed far away from the object is set by $v_m$ from (3.4a ). Inserting (3.8b ) into (3.7) and using (3.4b ) to eliminate the maximum fluid speed, we obtain an equation for the vortex shedding frequency of an oblate spheroid of the form

(3.9) \begin{equation} f=\frac {Q_{{lev}}}{5\pi \epsilon a\big(R^2 - \epsilon ^2a^2\big)}, \end{equation}

where we have set $w=2\epsilon a$ and $A=\pi \epsilon ^2a^2$ for the width and cross-sectional area, respectively. Here, we have assumed that the frequency of vibration is set by the speed of the fluid around the edges of the object as this is where vortex shedding occurs. This is different to the speed used to determine the levitation condition, where form drag ((3.2) and (3.3)) is important and the speed of fluid incident on the surface of the object presented to the flow is more appropriate.

Figure 6. Vibrational frequencies of (a) air bubbles, (b) liquid tetrabromoethane droplets and (c) solid PTFE particles as a function of the normalised half-width, ${\epsilon a}/{R}$ , of the levitated objects. Panel (a) shows data for air bubbles levitated inside tubes with internal diameters of 5 mm (blue diamonds), 8 mm (red squares) and 10 mm (green circles). Equation (3.9) was used to generate the dashed lines using material parameters given in table 1.

Figure 6 shows a comparison between the results of (3.9) and the measured vibrational frequencies of air bubbles, tetrabromoethane drops and solid PTFE particles as a function of the size of the levitating object in tubes of different radii. The values of the levitation flow rate obtained using (3.5) (see also figure 5) were inserted into (3.9) and used to generate the dashed lines in figure 6. The frequencies predicted by the simple model are clearly comparable to the measured vibrational frequencies, and the functional form predicted using vortex shedding arguments captures the essential physics of the oscillatory motion.

If we assume that the behaviour of small objects can be taken as indicative of the behaviour of an object in an unbounded flow, we see from figure 6 that the experimentally determined vibrational frequency (a proxy for the rate of vortex shedding) for bubbles is suppressed for intermediate object widths, but then increases as the width of the bubbles approaches the diameter of the tube. In the case of the liquid drops and solid particles, the agreement between data and model is improved. While there is a tendency for the vortex shedding rate (vibrational frequency) to plateau (with a small dip for the spheres around ${\epsilon a}/{R} \sim 0.7$ ), we again see an enhancement in the rate of vortex shedding when the flow around the objects becomes constricted.

Disagreement is observed between theory and experiments for the vibrational frequencies of small bubbles and large spheres. This is likely caused by similar failings of the model that was used to predict the flow rates shown in figure 5. Given that (3.9) predicts $f\propto Q_{{lev}}$ , any failure to reproduce the experimentally determined levitation flow rates with the simple model will propagate and generate disagreement between the predicted and measured vibrational frequencies. An additional limitation of the vortex shedding model is that it neglects the parabolic flow profile inside the tube, using only the maximum speed, $v_m$ , as the characteristic speed. This is likely to introduce some disagreement between the model and data in figure 6 for small objects that are offset from the centre of the tube and for large deformable objects such as bubbles, where the fluid speed changes considerably across their front surfaces.

Figure 7. Examples of shape oscillations in air bubbles and liquid tetrabromoethane droplets inside 10 mm diameter tubes. Data in the first two columns correspond to air bubbles with values of ${\epsilon a}/{R}=0.64$ (column a) or ${\epsilon a}/{R}=0.80$ (column b), while data in the last two columns are for tetrabromoethane droplets with ${\epsilon a}/{R}=0.42$ (column c) or ${\epsilon a}/{R}=0.69$ (column d). The rows correspond to images captured at 23 ms time intervals. The scale bar in each image corresponds to 3 mm.

The issue of shape oscillations in large deformable objects such as air bubbles and tetrabromoethane droplets is a particularly important consideration. Figure 7 shows examples of some of the shape changes observed in 10 mm diameter tubes. These images clearly show that the shapes of smaller bubbles and liquid droplets maintain a shape that is close to that of an oblate spheroid during the oscillatory motion (columns a and c). However, larger objects with ${\epsilon a}/{R} \sim 0.7$ and greater undergo large oscillations in shape and exhibit large deviations from a spheroidal shape (columns b and d). In cases such as these, we note that the values of ${\epsilon a}/{R}$ quoted represent time averages. Interestingly, these large deviations from the spheroidal shape correspond to regions in figures 5 and 6 where the experimental levitation flow rates and vibrational frequencies show deviations from the predictions of the simple model. A more detailed analysis of shape changes for large droplets and bubbles could provide better agreement between the model and the experimental data. This will be the subject of future work.

In future we also hope to expand the work presented here to include levitation of non-spheroidal objects e.g. cylinders and other shapes with different aspect ratios. In particular, we will study their vortex shedding and vibrational properties. We will also develop experiments to levitate and manipulate multiple objects by application of pulsed flows and using shaped tubes.