Levitation of heavy particles against gravity in asymptotically downward flows

TL;DR

This paper demonstrates that heavy particles can be levitated against gravity in downward flows through vortex interactions, challenging the expectation that particles follow the flow direction.

Contribution

The study introduces a new theory showing heavy particles can be lifted by vortices in downward flows, supported by analytical and simulation results.

Findings

Heavy particles can be levitated against gravity in downward flows.

Vortex interactions enable particles to move contrary to flow direction.

The effect has implications for water droplet transport and sediment lifting.

Abstract

In the fluid transport of particles, it is generally expected that heavy particles carried by a laminar fluid flow moving downward will also move downward. We establish a theory to show, however, that particles can be dynamically levitated and lifted by interacting vortices in such flows, thereby moving against gravity and the asymptotic direction of the flow, even when they are orders of magnitude denser than the fluid. The particle levitation is rigorously demonstrated for potential flows and supported by simulations for viscous flows. We suggest that this counterintuitive effect has potential implications for the air-transport of water droplets and the lifting of sediments in water.

Equations

$$\psi (\mathbf{r},t)=\frac{{\Gamma }_A}{4\pi }\ln \frac{|\mathbf{r}-{\bar{\mathbf{r}}}_A{|}^2}{|\mathbf{r}-{\mathbf{r}}_A{|}^2}+\frac{{\Gamma }_B}{4\pi }\ln \frac{|\mathbf{r}-{\bar{\mathbf{r}}}_B{|}^2}{|\mathbf{r}-{\mathbf{r}}_B{|}^2}+w_{0}y,$$

$$\psi(\mathbf{r},t)\!=\!\psi_{0}(\mathbf{r})+\frac{4\gamma\varepsilon^{2}}{(1\!+\!\gamma)^{2}}\,\!\!\Big{[}\psi_{2c}(\mathbf{r})\cos\frac{2t}{\varepsilon^{2}}+\psi_{2s}(\mathbf{r})\sin\frac{2t}{\varepsilon^{2}}\Big{]}\!\!\!$$

$$\dot{\mathbf{r}}={\mathbf{v}}_0(\mathbf{r})+{\epsilon }^2{\mathbf{v}}_2(\mathbf{r},t/{\epsilon }^2;\gamma )+O({\epsilon }^3),$$

$$\text{vspace}-2mm{H}_{n+1}-H_n\simeq {\epsilon }^{2}M(t_n),$$

$$\vspace{-2mm}M(t_{n})=\frac{4\gamma(a+\widetilde{V}_{T}b)}{(1+\gamma)^{2}}\Big{[}\sin\frac{2t_{n}}{\varepsilon^{2}}-2{\rm St}\cos\frac{2t_{n}}{\varepsilon^{2}}\Big{]}-m\,{\rm St}$$

$${\rm St}_{c}(\gamma)=\frac{4\mid\gamma\mid}{m(1+\gamma)^{2}}\Big{[}a(\varepsilon)+\widetilde{V}_{T}b(\varepsilon)\Big{]},$$

$$\frac{{\mathrm{St}}_c(\gamma )}{{\mathrm{St}}_c(1)}=\frac{4\mid \gamma \mid }{(1+\gamma {)}^2},$$

$${\psi }_0(\mathbf{r})$$

$${\psi }_{2c}(\mathbf{r})$$

$${\psi }_{2s}(\mathbf{r})$$

$$\ddot{\mathbf{R}}=\frac{1}{\mbox{St}}\big{(}\mathbf{W}-\dot{\mathbf{R}}\big{)}+\frac{V_{T}}{\mbox{St}}\hat{\mathbf{g}}+\mathbf{R}-2\hat{\mathbf{X}}\times\dot{\mathbf{R}},$$

$$\widetilde{\text{mbox}St}={\epsilon }^2\text{mbox}St,\text{mbox}and{\tilde{V}}_T=V_T/\epsilon ,$$

$$\mathbf{W}=\mathbf{W}_{0}(\mathbf{R},\gamma)+\varepsilon^{2}\big{[}\mathbf{W}_{2c}(\mathbf{R},\gamma)\cos 2t+\mathbf{W}_{2s}(\mathbf{R},\gamma)\sin 2t\big{]},$$

$$\mathbf{R}(t)={\mathbf{R}}_{eq}+\mathbf{h}(t),$$

$$\displaystyle\ddot{\mathbf{h}}=\frac{1}{\mbox{St}}\Big{\{}({\text{D}}\mathbf{W}_{0}^{eq})\,\mathbf{h}+\varepsilon^{2}\Big{[}\mathbf{W}_{2c}(\mathbf{R}_{eq},\gamma)\cos 2t+\mathbf{W}_{2s}(\mathbf{R}_{eq},\gamma)\sin 2t$$

$$\displaystyle-\widetilde{V}_{T}(\hat{\mathbf{Y}}\sin t+\hat{\mathbf{Z}}\cos t)\Big{]}-\dot{\mathbf{h}}\Big{\}}+{\mathbf{h}}-2\hat{\mathbf{X}}\times\dot{\mathbf{h}}\,,$$

$${\mathbf{h}}_a(t)=\sum _{n=1,2}({\mathbf{p}}_n\cos nt+{\mathbf{q}}_n\sin nt),$$

$${\mathbf{L}}_1{\mathbf{p}}_1-{\mathbf{M}}_1{\mathbf{q}}_1$$

$${\mathbf{L}}_1{\mathbf{q}}_1+{\mathbf{M}}_1{\mathbf{p}}_1$$

$${\mathbf{L}}_2{\mathbf{p}}_2-{\mathbf{M}}_2{\mathbf{q}}_2$$

$${\mathbf{L}}_2{\mathbf{q}}_2+{\mathbf{M}}_2{\mathbf{p}}_2$$

$$\big{(}\mathbf{M}_{1}^{-1}\mathbf{L}_{1}+\mathbf{L}_{1}^{-1}\mathbf{M}_{1}\big{)}\mathbf{p}_{1}=\varepsilon^{2}\frac{\widetilde{V}_{T}}{\mbox{St}}\big{(}\mathbf{L}_{1}^{-1}\hat{\mathbf{Y}}+\mathbf{M}_{1}^{-1}\hat{\mathbf{Z}}\big{)}.$$

$$H_{n+1}-H_n={\int }_{{\tau }_{2n}}^{{\tau }_{2n+1}}\frac{d}{dt}{\psi }_p(\mathbf{r}(t))dt={\int }_{{\tau }_{2n}}^{{\tau }_{2n+1}}\nabla {\psi }_p(\mathbf{r}(t))\cdot \frac{d\mathbf{r}(t)}{dt}dt.$$

$$M(t_{n})=\int_{-\infty}^{\infty}\negmedspace\negmedspace\negmedspace\negmedspace{\mathbf{v}}_{0}({\mathbf{q}}(t))\times{\mathbf{v}}_{2}\Big{(}\mathbf{q}(t),\frac{t+t_{n}}{\varepsilon^{2}}\Big{)}\,dt.$$

$$M(t_n)=I_1+I_2+I_3+O({\tilde{V}}_T^2)+O({\tilde{V}}_T\text{mbox}St),$$

$$\displaystyle I_{1}=\int_{-\infty}^{\infty}\negmedspace\negmedspace\negmedspace\negmedspace\dot{\mathbf{q}}_{0}(t)\times{\mathbf{U}}_{2}\Big{(}\mathbf{q}_{0}(t),\frac{t+t_{n}}{\varepsilon^{2}}\Big{)}\,dt-\mbox{St}\int_{-\infty}^{\infty}\negmedspace\negmedspace\negmedspace\negmedspace\dot{\mathbf{q}}_{0}(t)\times\ddot{\mathbf{q}}_{0}(t)\,dt,$$

$$\displaystyle I_{2}=\widetilde{V}_{T}\int_{-\infty}^{\infty}\negmedspace\negmedspace\negmedspace\negmedspace\dot{\mathbf{q}}_{1}(t)\times\mathbf{U}_{2}\Big{(}\mathbf{q}_{0},\frac{t+t_{n}}{\varepsilon^{2}}\Big{)}\,dt,$$

$$\displaystyle I_{3}=\widetilde{V}_{T}\int_{-\infty}^{\infty}\negmedspace\negmedspace\negmedspace\negmedspace\dot{\mathbf{q}}_{0}(t)\times\Big{(}({\text{D}}\mathbf{U}_{2})\,\mathbf{q}_{1}(t)\Big{)}\,dt.$$

$$I_{1}=\frac{4\gamma}{(1+\gamma)^{2}}a(\varepsilon)\Big{(}\sin\frac{2t_{n}}{\varepsilon^{2}}-2\mbox{St}\cos\frac{2t_{n}}{\varepsilon^{2}}\Big{)}-m\,\mbox{St},$$

$$a(\varepsilon)=\int_{-\infty}^{\infty}\negmedspace\negmedspace\negmedspace\negmedspace\dot{\mathbf{q}}_{0}(t)\times\Big{[}{\mathbf{u}}_{2s}(\mathbf{q}_{0}(t))\cos\frac{2t}{\varepsilon^{2}}-{\mathbf{u}}_{2c}(\mathbf{q}_{0}(t))\sin\frac{2t}{\varepsilon^{2}}\Big{]}\,dt,$$

$$m={\int }_{-\infty }^{\infty }{\dot{\mathbf{q}}}_0(t)\times {\ddot{\mathbf{q}}}_0(t)dt.$$

$$a(\varepsilon)=\frac{e^{-\alpha_{e}/\varepsilon^{2}}}{\varepsilon^{2}}\big{(}\alpha_{0}+\alpha_{2}\varepsilon^{2}+\alpha_{4}\varepsilon^{4}\big{)},$$

$$I_{2}+I_{3}=\frac{4\gamma}{(1+\gamma)^{2}}\,\widetilde{V}_{T}\big{[}b_{2}(\varepsilon)+b_{3}(\varepsilon)\big{]}\Big{(}\sin\frac{2t_{n}}{\varepsilon^{2}}-2\mbox{St}\cos\frac{2t_{n}}{\varepsilon^{2}}\Big{)},$$

$$\displaystyle b_{2}(\varepsilon)=\int_{-\infty}^{\infty}\negmedspace\negmedspace\negmedspace\negmedspace\dot{\mathbf{q}}_{1}(t)\times\Big{[}{\mathbf{u}}_{2s}(\mathbf{q}_{0}(t))\cos\frac{2t}{\varepsilon^{2}}-{\mathbf{u}}_{2c}(\mathbf{q}_{0}(t))\sin\frac{2t}{\varepsilon^{2}}\Big{]}\,dt,$$

Full text

Levitation—the action of rising and hovering in apparent defiance of gravity—is a fascinating phenomenon with many practical implications. A classic demonstration is the Bernoulli ball levitation, in which a macroscopic particle heavier than air (such as a ping pong ball) can levitate in response to an inclined upward air stream that appears to only partially balance gravity. A key aspect of that form of levitation is the transversal stability due to the Coandǎ effect,[1] which relies on the tendency of the flow to curve around the surface of the ball and sustains stable levitation when the upward air stream is tilted. Here, we report a new form of fluid-dynamical levitation that can be observed even for a downward stream (i.e., in the direction of gravity) and that allows heavy particles to be levitated by a flow regardless of whether they are 10, 100, or 1,000 times denser than the fluid. This phenomenon is fundamentally different from the Coandǎ effect in that it concerns microscopic heavy particles and requires no disturbance of the flow by the particles.

Many natural and industrial flows transport small particles, like droplets, sediments, and microorganisms. [2] Inertial effects cause the trajectories of such particles to deviate from the streamlines of the flow, making the study of particle-laden flows challenging both theoretically and experimentally. [3, 4, 5, 6] Key to such studies are the dissipative nature of the advection dynamics and the consequent tendency of the particles to accumulate in specific zones of the flow domain, both in closed flows [7, 8, 9, 10, 11, 12, 13, 16, 14, 17, 14, 15] and in open flows, [22, 21, 19, 20, 18] even when the flows are incompressible. For example, particles less dense than the fluid tend to be attracted to the interior of vortices, [23, 24] which can lead to the formation of attractors for the particle dynamics independently of the global properties of flow. Particles denser than the fluid, on the other hand, tend to be repelled by vortices, which is a mechanism that can lead to the formation of attractors if the flow is closed; this effect, which is also related to the preferential trajectories phenomenon, [9] has been widely investigated over the past two decades. [3] However, much less is known about dense particles moving in flows that have unbounded streamlines and are therefore open. In open flows, an outstanding problem of particular interest concerns the transport of small particles much denser than the fluid, which we term heavy particles and which can represent for example water droplets in the air.

In this article, we demonstrate the possibility of levitation and upward transport of heavy particles by a flow moving asymptotically downward, even in the presence of gravity. Because at first these conditions seem to facilitate downward advection, one might expect that all particles would necessarily fall, which is in sharp contrast with the effect we report. The starting point of our analysis is the observation that such a flow can support pairs of mutually interacting vortices traveling in a direction that opposes the flow. We thus focus on asymptotically simple flows that move downward and have a pair of vortices moving upward. Using this class of flows we show that attracting points (dimension-zero attractors) formed near the center of vorticity can capture heavy particles released at any distance above the vortices. For this to occur, particle inertia must allow for particles to approach the vortices and for the existence of attractors to retain them in that region; we show that these two conditions are satisfied for a wide range of Stokes number in the class of flows we consider. This is demonstrated analytically using asymptotic analysis and Melnikov functions, and is illustrated numerically using simulations in both inviscid and viscous laminar flows.

Our analysis is inspired by previous experimental realizations of flows with pairs of interacting vortices [25, 26, 27, 28, 29] and theoretical work on particle advection in such flows. [19, 16, 17, 18, 15, 33, 32, 31, 30] Several studies have shown, both numerically [19] and analytically, [16, 17, 18] the formation of attractors for the dynamics of heavy particles in the vicinity of identical co-rotating vortices. Different work has shown that such attractors persist for non-identical vortices in closed potential flows in the absence of gravity, [15] but no results exist for open flows, viscous regimes, or the effects of gravity. Here, in order to demonstrate the proposed levitation of heavy particles, we first generalize the special results in open flows, previously established for nongeneric vortex pairs, to the case of (i) generic pairs of both co-rotating and counter-rotating vortices of arbitrary vortex strength ratio, (ii) for vortices moving against a background flow that is co-directional with gravity, and (iii) for both potential and viscous flows. Under these general conditions we then show the existence of attracting points that capture heavy particles from both the closed and open flow regions and carry them against gravity and the background flow.

The model flow is depicted in Fig. 1. It consists of two vortices, $A$ and $B$, with strengths $\Gamma_{A}$ and $\Gamma_{B}$, respectively, plus mirror vortices $A^{\prime}$ and $B^{\prime}$ with opposite strengths and symmetric positions with respect to the vertical axis $Oz$. This axis can be regarded as a “wall” in the framework of potential flow theory used in our calculations. The position $\mathbf{r}_{I}$ of the center of vorticity $I$ of the pair $(A,B)$ is $\mathbf{r}_{I}=({\Gamma_{A}\mathbf{r}_{A}+\Gamma_{B}\mathbf{r}_{B}})/({\Gamma_{A}+\Gamma_{B}}),$ where $\mathbf{r}_{A}$ is the position of $A$ and $\mathbf{r}_{B}$ is the position of $B$. In the absence of mirror vortices, point $I$ remains fixed and $AB$ rotates steadily around $I$ with angular velocity $\Omega_{0}=({\Gamma_{A}+\Gamma_{B}})/({8\pi d_{0}^{2}}),$ where $2d_{0}=|AB|$ is the distance between the vortices, which remains constant in this case. When the two pairs $(A,B)$ and $(A^{\prime},B^{\prime})$ are close enough to interact, elementary vortex dynamics show that point $I$ moves vertically, and its distance to the symmetry axis $Oz$ remains equal to its initial value, $L_{0}$. For $d_{0}\ll L_{0}$, the (vertical) velocity of $I$ approaches $w_{0}=({\Gamma_{A}+\Gamma_{B}})/{4\pi L_{0}}$ in the frame of the fluid at infinity. The streamfunction of the exact 2D potential flow induced by the four vortices, in the frame translating at constant speed $w_{0}\,\hat{\mathbf{z}}$, where $\hat{\mathbf{z}}$ is the upward unit vector, is

[TABLE]

where $\mathbf{r}=(y,z)$, and $\mathbf{\bar{r}}_{A}$ and $\mathbf{\bar{r}}_{B}$ are the positions of the mirror vortices $A^{\prime}$ and $B^{\prime}$, respectively. The instantaneous velocity of fluid elements in this frame trace closed streamlines near the vortices (closed flow region) and open streamlines further away from them (open flow region). As indicated in Fig. 1, these two regions meet along the separatrix streamline $\Sigma$ joining the stagnation points located on the $z$-axis, $S_{1}$ and $S_{2}$. The structure formed by $\Sigma\cup S_{2}S_{1}$ is a heteroclinic cycle for the dynamics of fluid elements; heteroclinic (and homoclinic) cycles are generally expected to play a role in the transport of both non-inertial and inertial particles. [34, 3]

To proceed, we define $\gamma=\Gamma_{A}/\Gamma_{B}$ as the vortex strength ratio, where $-1<\gamma\leq 1$. It can be checked that up to order $\varepsilon^{2}$, when $\varepsilon\equiv d_{0}/L_{0}<(\gamma+1)/2$, $A$ and $B$ rotate around $I$ approximately as a rigid body with angular velocity $\Omega_{0}$. This allows us to write $\mathbf{r}_{A,B}(t)$ and ${\mathbf{\bar{r}}_{A,B}}(t)$ in the form of $2\pi/\Omega_{0}$-periodic functions plus $O(\varepsilon^{2})$ corrections. Throughout the rest of the article we operate with the equations in non-dimensional form, using as units $L_{0}$ for lengths and $w_{0}/2$ for velocities since these choices capture the appropriate orders of magnitude for the flow near the heteroclinic cycle $\Sigma$. (No new notation is introduced for non-dimensional variables.) Expanding Eq. (1) in powers of $\varepsilon$, in non-dimensional form the streamfunction reads

[TABLE]

plus $O(\varepsilon^{3})$ terms, where the components $\psi_{0},\,\psi_{2c},\,\psi_{2s}$ are identical to those for the case $\gamma=1$ [35] given that the $\gamma$-dependence is accounted for by the prefactor (see supplementary material). The remainder in Eq. (2) can be shown to be $O(\varepsilon^{4})$ when $\gamma=1$. The velocity field—defined as $\mathbf{u}=\mathbf{\nabla}\times(\psi\,\hat{\mathbf{x}})$, where $\hat{\mathbf{x}}$ is the right handed unit vector orthogonal to the $yz$ plane—is therefore of the form $\mathbf{u}(\mathbf{r},t)=\mathbf{u}_{0}(\mathbf{r})+\varepsilon^{2}\mathbf{u}_{2}(\mathbf{r},t/\varepsilon^{2};\gamma)+O(\varepsilon^{3})$, where $\mathbf{u}_{2}$ is time-periodic with period $T=\pi\varepsilon^{2}$ and corresponds to the leading perturbation induced by the rotation of the vortices.

Having established the fluid flow equations, we now write the particle equation of motion in this flow. For a heavy particle with small particle Reynolds number, the non-dimensional equation is $\ddot{\mathbf{r}}={\widetilde{\rm St}}^{-1}\left[{\mathbf{u}}(\mathbf{r},t)+\widetilde{\mathbf{V}}_{T}-\dot{\mathbf{r}}\right]$, where $\mathbf{r}$ is the particle position, $\widetilde{\mathbf{V}}_{T}=-\widetilde{V}_{T}\hat{\mathbf{z}}=-2(g\,\tau_{p}/w_{0})\hat{\mathbf{z}}$ is the free-fall terminal velocity, and $\widetilde{\rm St}$ is the Stokes number (i.e., the response time of the particle, $\tau_{p}$, divided by the time-scale of the flow, $2L_{0}/w_{0}$). [36] We also introduce another Stokes number, ${\rm St}=\Omega_{0}\,\tau_{p}$, to describe the dynamics of particles directly influenced by the rotation of the vortices around each other. The formation of attractors near the vortices requires that ${\rm St}$ be no larger than order one since drag has to balance centrifugal force in this case. We therefore assume ${\rm St}=O(1)$, so that $\widetilde{\rm St}\equiv\varepsilon^{2}{\rm St}\ll 1$ and the equation for $\mathbf{r}$ can be reduced to

[TABLE]

where $\mathbf{v}_{0}(\mathbf{r})={\mathbf{u}}_{0}(\mathbf{r})+\widetilde{\mathbf{V}}_{T}$ and \mathbf{v}_{2}(\mathbf{r},\tau;\gamma)\simeq\mathbf{u}_{2}({\mathbf{r}},\tau;\gamma)-{\rm St}\big{[}({\mathbf{u}}_{0}+\widetilde{\mathbf{V}}_{T}).\nabla{\mathbf{u}}_{0}+{\partial{\mathbf{u}}_{2}(\mathbf{r},\tau;\gamma)}/{\partial\tau}\big{]}, for $\tau=t/\varepsilon^{2}$ (see also Ref. [37]). We show that the particle dynamics described by this equation have at least one (for $-1<\gamma\leq 1$) and possibly two attracting points (for $0<\gamma\leq 1$), provided that the Stokes number is not too large (see supplementary material).

In Eq. (3), the leading term, ${\mathbf{v}}_{0}$, represents the conservative dynamics of non-inertial particles in a steady flow induced by the equivalent to a single vortex with strength $\Gamma_{A}+\Gamma_{B}$ (together with its mirror) plus a uniform flow $-\widetilde{V}_{T}\hat{\mathbf{z}}$. The particle streamfunction for this term, $\psi_{p}(y,z)=\psi_{0}(y,z)+y\widetilde{V}_{T}$, is time-independent (a similar streamfunction has been used to describe plankton dynamics [38]). The first perturbative term, $\varepsilon^{2}{\mathbf{v}}_{2}$, contains the contribution $\mathbf{u}_{2}$ of the unsteadiness of the flow due to the fact that for $\varepsilon>0$ there are two vortices rather than one, and also the effect of inertia in the ${\rm St}$ terms. It is thus convenient to regard this system as a time-independent Hamiltonian $\psi_{p}$ perturbed by dissipative and fast periodic terms. [39] To leading order in $\varepsilon$, the trajectories of the particles coincide with the curves $\psi_{p}(y,z)=$ cte. These curves correspond to open streamlines separated from closed streamlines by a heteroclinic cycle, which we denote $\Sigma_{p}$ and which is the particle analog of $\Sigma$ in Fig. 1 except that $\Sigma_{p}$ does not include the time-dependent perturbation terms.

An important necessary condition for particles from the open flow to be captured by attracting points in the vicinity of the vortices is that they cross $\Sigma_{p}$ (the separatrix of the unperturbed dynamics) under the effect of the motion of the vortices. The occurrence of separatrix crossing can be predicted employing a construction based on separatrix maps. [40, 41] We consider a solution $\mathbf{r}(t)$ of the perturbed system (up to order $\varepsilon^{2}$) and define $t_{n}\,(n=1,2,...)$ as the times at which the particle crosses the axis $Oy$ downward. We also use $\tau_{2n}$ and $\tau_{2n+1}$ to denote the times the particle passes closest to the saddle points $S_{1}$ and $S_{2}$, respectively, and $H_{n}=\psi_{p}[\mathbf{r}(\tau_{2n})]$ and $H_{n+1}=\psi_{p}[\mathbf{r}(\tau_{2n+1})]$ to denote the corresponding values of the unperturbed Hamiltonian. Oscillations of $H_{n+1}-H_{n}$ around zero as $n$ varies will indicate that the separatrix $\Sigma_{p}$ is crossed. Assuming that $\mathbf{r}(t)\simeq\mathbf{q}(t-t_{n})$, for $\mathbf{q}(t)$ denoting the solution of the unperturbed system along $\Sigma_{p}$, we obtain

[TABLE]

where $M(t_{n})$ is the Melnikov function associated with $\Sigma_{p}$:

[TABLE]

(see supplementary material for details). Here, the amplitudes $a=a(\varepsilon)$ and $b=b(\varepsilon)$ are functions of $\varepsilon$ only, and $m$ is a constant accounting for the centrifugal effect along $\Sigma_{p}$ due to the particle’s inertia.

The Melnikov function in Eq. (5) is either strictly negative or oscillates between positive and negative values. [42] When the function has a constant negative sign, we have $H_{n+1}<H_{n}$ for all $n$, which indicates that particles are centrifuged away from the vortices. When the function has simple zeros, particles can enter and exit the closed flow. Thus, the central prediction of our theory is that a heavy particle in the open flow may be captured by an attracting point near the vortices provided that the Stokes number ${\rm St}$ is below the critical value

[TABLE]

which is the condition for $M(t_{n})$ to change sign (and in fact have infinitely many isolated zeros). We therefore predict that the levitation of heavy particles released in the open flow region above the vortices is possible for ${\rm St}<{\rm St}_{c}(\gamma)$. These results imply that heavy particles with densities across many orders of magnitude can be levitated by the same flow. [43]

Remarkably, Eq. (6) shows that ${\rm St}_{c}$ is an increasing function of $\widetilde{V}_{T}$ and hence that gravity facilitates levitation. This means that a particle that would be too inertial to penetrate inside $\Sigma_{p}$ in the absence of gravity can be captured by the attracting points when gravity is present. This equation also shows that ${\rm St}_{c}$ depends on the vortex strength ratio $\gamma$ and in particular that the normalized critical Stokes number is

[TABLE]

irrespective of $\widetilde{V}_{T}$ and $\varepsilon$. Thus, not only levitation is possible for both co- and counter-rotating vortices of arbitrary $\gamma\neq-1$, but also the phenomenon is more pronounced for counter-rotating vortices with sufficiently small $\gamma$ than for identical co-rotating vortices ($\gamma=1$).

Figure 2 shows the analytical prediction in Eq. (7) along with a numerical verification for particles released in the open flow above the vortices (red region in Fig. 1), where bisection in ${\rm St}$ was used to determine the critical value at which particles start crossing $\Sigma_{p}$. The agreement with the numerics is good, particularly for co-rotating vortices. Importantly, the numerical ${\rm St}_{c}$ is always higher than the theoretical prediction, widening the range of the effect. [44]

To further illustrate the theoretical results above we have done a series of computations by choosing the parameters according to the diagram of Fig. 2. Figure 3 shows the evolution of clouds of particles for two different Stokes numbers in the case of co-rotating vortices: ${\rm St}<{\rm St}_{c}$ [Fig. 3(a, b)] and ${\rm St}>{\rm St}_{c}$ [Fig. 3(c, d)], corresponding to the points $P_{1}$ and $P_{2}$ in Fig. 2, respectively. The particles are released in the open-streamline region above the vortices (red) and in the closed-streamline region near the vortices (blue), as sketched in Fig. 1. The clouds are shown after $16$ turnover times [Fig. 3(a, c)] and after $95$ turnover times [Fig. 3(b, d)], which was chosen purposely large to facilitate visualization of the long-term behavior. For $P_{1}$, a fraction of the blue particles as well as a fraction of the red particles accumulate near the two attracting points after long times. (For visualization, particles near the attracting points were slightly dispersed). In contrast, for $P_{2}$ only blue particles are captured by the attracting points; red particles remain outside $\Sigma_{p}$ and are transported downstream. These observations are in complete agreement with our predictions.

Similar agreement is observed for counter-rotating vortices, as shown in Fig. 4 for ${\rm St}<{\rm St}_{c}$ ($Q_{1}$ in Fig. 2) and ${\rm St}>{\rm St}_{c}$ ($Q_{2}$ in Fig. 2), but with two important differences. First, in the case of $Q_{1}$ there is only one attracting point around which a portion of blue and red particles accumulate. This is expected for the range of Stokes number considered, but studies of isolated counter-rotating pairs suggest that other attractors may exist for different parameters. [15] Second, for $Q_{2}$, there is no attractor, so that not only do red particles not cross $\Sigma_{p}$ inward, but also all blue particles are centrifuged outward across $\Sigma_{p}$. As a result, in this case no particle is captured by an attractor independently of the initial condition.

To verify the significance of our predictions for more realistic flows, we have simulated viscous flow solutions of the Navier-Stokes equations in the setup of Fig. 1. As shown in Fig. 5, for ${\rm St}<{\rm St}_{c}$, particles with initial positions in both the closed flow [5(a, c)] and open flow [5(a, d)] are captured and levitated by attracting points in the vicinity of the vortices. The main difference from the case of idealized potential flows considered above is that levitation is not permanent in viscous flows since the vortices eventually coalesce. For the conditions considered in Fig. 5, which corresponds to approximately 15 turnover times before vortex merging, particles from the open flow are accelerated downward by vortices alone but, when capture occurs, up to $20\%$ of the particles are levitated by the vortices until they merge [dashed versus continuous lines in Fig. 5(b)]. A partial view of the basins of attraction [Fig.  5(e)] provides further insight into the initial conditions of the particles that can be levitated by this mechanism. Simulations were performed using OpenFOAM. [46] For more details and simulation movies, we refer to the supplementary material.

Our demonstration that heavy particles can be levitated shows that they can be transported in any direction relative to the asymptotic flow and gravity. Exploring this effect in more complex problems (possibly involving non-laminar flows), such as the air-transport of water droplets and aerosols, [47, 49, 48] the resuspension of sediments by coherent vortical structures, [50, 51] and industrial applications for particle sorting, [22, 21] are among the questions of great interest for future research.

Supplementary Material

The supplementary material includes the components of the streamfunction in Eq. (2), analysis of the attracting points, additional details on the Melnikov function calculation and viscous flow simulations, Supplementary Figures for viscous flows, and Supplementary Movies showing animated versions of the dynamics in Fig. 5.

Acknowledgements.

This research was supported by NSF Grant PHY-1001198.