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  • 1. Balestra, G.
    et al.
    Zhu, Lailai
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Gallaire, F.
    Viscous Taylor droplets in axisymmetric and planar tubes: from Bretherton’s theory to empirical models2018In: Microfluidics and Nanofluidics, ISSN 1613-4982, E-ISSN 1613-4990, Vol. 22, no 6, article id 67Article in journal (Refereed)
    Abstract [en]

    The aim of this study is to derive accurate models for quantities characterizing the dynamics of droplets of non-vanishing viscosity in capillaries. In particular, we propose models for the uniform-film thickness separating the droplet from the tube walls, for the droplet front and rear curvatures and pressure jumps, and for the droplet velocity in a range of capillary numbers, Ca, from 10 - 4 to 1 and inner-to-outer viscosity ratios, λ, from 0, i.e. a bubble, to high-viscosity droplets. Theoretical asymptotic results obtained in the limit of small capillary number are combined with accurate numerical simulations at larger Ca. With these models at hand, we can compute the pressure drop induced by the droplet. The film thickness at low capillary numbers (Ca< 10 - 3) agrees well with Bretherton’s scaling for bubbles as long as λ< 1. For larger viscosity ratios, the film thickness increases monotonically, before saturating for λ> 10 3 to a value 2 2 / 3 times larger than the film thickness of a bubble. At larger capillary numbers, the film thickness follows the rational function proposed by Aussillous and Quéré (Phys Fluids 12(10):2367–2371, 2000) for bubbles, with a fitting coefficient which is viscosity-ratio dependent. This coefficient modifies the value to which the film thickness saturates at large capillary numbers. The velocity of the droplet is found to be strongly dependent on the capillary number and viscosity ratio. We also show that the normal viscous stresses at the front and rear caps of the droplets cannot be neglected when calculating the pressure drop for Ca> 10 - 3.

  • 2.
    Gallino, Giacomo
    et al.
    Ecole Polytech Fed Lausanne, Lab Fluid Mech & Instabil, CH-1015 Lausanne, Switzerland..
    Zhu, Lailai
    KTH, Centres, SeRC - Swedish e-Science Research Centre. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Gallaire, Francois
    Ecole Polytech Fed Lausanne, Lab Fluid Mech & Instabil, CH-1015 Lausanne, Switzerland..
    The Hydrodynamics of a Micro-Rocket Propelled by a Deformable Bubble2019In: FLUIDS, ISSN 2311-5521, Vol. 4, no 1, article id 48Article in journal (Refereed)
    Abstract [en]

    We perform simulations to study the hydrodynamics of a conical-shaped swimming micro-robot that ejects catalytically produced bubbles from its inside. We underline the nontrivial dependency of the swimming velocity on the bubble deformability and on the geometry of the swimmer. We identify three distinct phases during the bubble evolution: immediately after nucleation the bubble is spherical and its inflation barely affects the swimming speed; then the bubble starts to deform due to the confinement gradient generating a force that propels the swimmer; while in the last phase, the bubble exits the cone, resulting in an increase in the swimmer velocity. Our results shed light on the fundamental hydrodynamics of the propulsion of catalytic conical swimmers and may help to improve the efficiency of these micro-machines.

  • 3. Hadikhani, P.
    et al.
    Hashemi, S. M. H.
    Balestra, G.
    Zhu, Lailai
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre. Princeton University, United States.
    Modestino, M. A.
    Gallaire, F.
    Psaltis, D.
    Inertial manipulation of bubbles in rectangular microfluidic channels2018In: Lab on a Chip, ISSN 1473-0197, E-ISSN 1473-0189, Vol. 18, no 7, p. 1035-1046Article in journal (Refereed)
    Abstract [en]

    Inertial microfluidics is an active field of research that deals with crossflow positioning of the suspended entities in microflows. Until now, the majority of the studies have focused on the behavior of rigid particles in order to provide guidelines for microfluidic applications such as sorting and filtering. Deformable entities such as bubbles and droplets are considered in fewer studies despite their importance in multiphase microflows. In this paper, we show that the trajectory of bubbles flowing in rectangular and square microchannels can be controlled by tuning the balance of forces acting on them. A T-junction geometry is employed to introduce bubbles into a microchannel and analyze their lateral equilibrium position in a range of Reynolds (1 < Re < 40) and capillary numbers (0.1 < Ca < 1). We find that the Reynolds number (Re), the capillary number (Ca), the diameter of the bubble (D), and the aspect ratio of the channel are the influential parameters in this phenomenon. For instance, at high Re, the flow pushes the bubble towards the wall while large Ca or D moves the bubble towards the center. Moreover, in the shallow channels, having aspect ratios higher than one, the bubble moves towards the narrower sidewalls. One important outcome of this study is that the equilibrium position of bubbles in rectangular channels is different from that of solid particles. The experimental observations are in good agreement with the performed numerical simulations and provide insights into the dynamics of bubbles in laminar flows which can be utilized in the design of flow based multiphase flow reactors.

  • 4.
    Horvath, Daniel G.
    et al.
    Santa Clara Univ, Dept Chem & Biochem, Santa Clara, CA 95053 USA..
    Braza, Samuel
    Santa Clara Univ, Dept Chem & Biochem, Santa Clara, CA 95053 USA..
    Moore, Trevor
    Santa Clara Univ, Dept Chem & Biochem, Santa Clara, CA 95053 USA..
    Pan, Ching W.
    Santa Clara Univ, Dept Chem & Biochem, Santa Clara, CA 95053 USA..
    Zhu, Lailai
    KTH, School of Engineering Sciences (SCI), Mechanics, Fluid Physics. Princeton Univ, Dept Mech & Aerosp Engn, Princeton, NJ 08544 USA..
    Pak, On Shun
    Santa Clara Univ, Dept Mech Engn, Santa Clara, CA 95053 USA..
    Abbyad, Paul
    Santa Clara Univ, Dept Chem & Biochem, Santa Clara, CA 95053 USA..
    Sorting by interfacial tension (SIFT): Label-free enzyme sorting using droplet microfluidics2019In: Analytica Chimica Acta, ISSN 0003-2670, E-ISSN 1873-4324, Vol. 1089, p. 108-114Article in journal (Refereed)
    Abstract [en]

    Droplet microfluidics has the ability to greatly increase the throughput of screening and sorting of enzymes by carrying reagents in picoliter droplets flowing in inert oils. It was found with the use of a specific surfactant, the interfacial tension of droplets can be very sensitive to droplet pH. This enables the sorting of droplets of different pH when confined droplets encounter a microfabricated trench. The device can be extended to sort enzymes, as a large number of enzymatic reactions lead to the production of an acidic or basic product and a concurrent change in solution pH. The progress of an enzymatic reaction is tracked from the position of a flowing train of droplets. We demonstrate the sorting of esterase isoenzymes based on their enzymatic activity. This label-free technology, that we dub droplet sorting by interfacial tension (SIFT), requires no active components and would have applications for enzyme sorting in high-throughput applications that include enzyme screening and directed evolution of enzymes.

  • 5.
    Liu, Ying
    et al.
    Princeton Univ, Dept Mech & Aerosp Engn, Princeton, NJ 08544 USA..
    Rallabandi, Bhargav
    Princeton Univ, Dept Mech & Aerosp Engn, Princeton, NJ 08544 USA.;Univ Calif Riverside, Dept Mech Engn, Riverside, CA 92521 USA..
    Zhu, Lailai
    KTH, Centres, SeRC - Swedish e-Science Research Centre. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. Princeton Univ, Dept Mech & Aerosp Engn, Princeton, NJ 08544 USA..
    Gupta, Ankur
    Princeton Univ, Dept Mech & Aerosp Engn, Princeton, NJ 08544 USA..
    Stone, Howard A.
    Princeton Univ, Dept Mech & Aerosp Engn, Princeton, NJ 08544 USA..
    Pattern formation in oil-in-water emulsions exposed to a salt gradient2019In: Physical Review Fluids, E-ISSN 2469-990X, Vol. 4, no 8, article id 084307Article in journal (Refereed)
    Abstract [en]

    Flow instabilities can occur in a fluid system with two components that have significantly different diffusivities and that have opposite effects on the fluid density, as is the scenario in traditional double-diffusive convection. Here, we experimentally show that an oil-in-water emulsion exposed to salt concentration gradients generates a flowerlike pattern driven by vertical and azimuthal instabilities. We also report numerical and analytical studies to elaborate on the mechanism, the instability criteria, and the most unstable modes that determine the details of the observed patterns. We find that the instability is driven by buoyancy and stems from the differential transport between the dissolved salt and the suspended oil droplets, which have opposing effects on the density of the medium. Consequently, we identify a criterion for the development of the instability that involves the relative densities and concentrations of the salt and oil droplets. We also argue that the typical wave number of the pattern formed scales with the Peclet number of the salt, which here is equivalent to the Rayleigh number since the flow is driven by buoyancy. We find good agreement of these predictions with both experiments and numerical simulations.

  • 6.
    Pietrzyk, Kyle
    et al.
    Santa Clara Univ, Dept Mech Engn, Santa Clara, CA 95053 USA..
    Nganguia, Herve
    Indiana Univ Penn, Dept Math & Comp Sci, Indiana, PA 15705 USA..
    Datt, Charu
    Univ British Columbia, Dept Mech Engn, Vancouver, BC V6T 1Z4, Canada..
    Zhu, Lailai
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Elfring, Gwynn J.
    Univ British Columbia, Dept Mech Engn, Vancouver, BC V6T 1Z4, Canada..
    Pak, On Shun
    Santa Clara Univ, Dept Mech Engn, Santa Clara, CA 95053 USA..
    Flow around a squirmer in a shear-thinning fluid2019In: Journal of Non-Newtonian Fluid Mechanics, ISSN 0377-0257, E-ISSN 1873-2631, Vol. 268, p. 101-110Article in journal (Refereed)
    Abstract [en]

    Many biological fluids display shear-thinning rheology, where the viscosity decreases with an increasing shear rate. To better understand how this non-Newtonian rheology affects the motion of biological and artificial micro swimmers, recent efforts have begun to seek answers to fundamental questions about active bodies in shear-thinning fluids. Previous analyses based on a squirmer model have revealed non-trivial variations of propulsion characteristics in a shear-thinning fluid via the reciprocal theorem. However, the reciprocal theorem approach does not provide knowledge about the flow surrounding the squirmer. In this work, we fill in this missing information by calculating the non-Newtonian correction to the flow analytically in the asymptotic limit of small Carreau number. In particular, we investigate the local effect due to viscosity reduction and the non-local effect due to induced changes in the flow; we then quantify their relative importance to locomotion in a shear-thinning fluid. Our results demonstrate cases where the non-local effect can be more significant than the local effect. These findings suggest that caution should be exercised when developing physical intuition from the local viscosity distribution alone around a swimmer in a shear-thinning fluid.

  • 7. Reigh, Shang Yik
    et al.
    Zhu, Lailai
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Gallaire, Francois
    Lauga, Eric
    Swimming with a cage: low-Reynolds-number locomotion inside a droplet2017In: Soft Matter, ISSN 1744-683X, E-ISSN 1744-6848, Vol. 13, no 17, p. 3161-3173Article in journal (Refereed)
    Abstract [en]

    Inspired by recent experiments using synthetic microswimmers to manipulate droplets, we investigate the low-Reynolds-number locomotion of a model swimmer (a spherical squirmer) encapsulated inside a droplet of a comparable size in another viscous fluid. Meditated solely by hydrodynamic interactions, the encaged swimmer is seen to be able to propel the droplet, and in some situations both remain in a stable co-swimming state. The problem is tackled using both an exact analytical theory and a numerical implementation based on a boundary element method, with a particular focus on the kinematics of the co-moving swimmer and the droplet in a concentric configuration, and we obtain excellent quantitative agreement between the two. The droplet always moves slower than a swimmer which uses purely tangential surface actuation but when it uses a particular combination of tangential and normal actuations, the squirmer and droplet are able to attain the same velocity and stay concentric for all times. We next employ numerical simulations to examine the stability of their concentric co-movement, and highlight several stability scenarios depending on the particular gait adopted by the swimmer. Furthermore, we show that the droplet reverses the nature of the far-field flow induced by the swimmer: a droplet cage turns a pusher swimmer into a puller, and vice versa. Our work sheds light on the potential development of droplets as self-contained carriers of both chemical content and self-propelled devices for controllable and precise drug deliveries.

  • 8.
    Shukla, Isha
    et al.
    Ecole Polytech Fed Lausanne, Lab Fluid Mech & Instabil, CH-1015 Lausanne, Switzerland..
    Kofman, Nicolas
    Ecole Polytech Fed Lausanne, Lab Fluid Mech & Instabil, CH-1015 Lausanne, Switzerland..
    Balestra, Gioele
    Ecole Polytech Fed Lausanne, Lab Fluid Mech & Instabil, CH-1015 Lausanne, Switzerland..
    Zhu, Lailai
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre. Ecole Polytech Fed Lausanne, Lab Fluid Mech & Instabil, CH-1015 Lausanne, Switzerland.;Princeton Univ, Dept Mech & Aerosp Engn, Princeton, NJ 08544 USA.
    Gallaire, Francois
    Ecole Polytech Fed Lausanne, Lab Fluid Mech & Instabil, CH-1015 Lausanne, Switzerland..
    Film thickness distribution in gravity-driven pancake-shaped droplets rising in a Hele-Shaw cell2019In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 874, p. 1021-1040, article id PII S0022112019004531Article in journal (Refereed)
    Abstract [en]

    We study here experimentally, numerically and using a lubrication approach, the shape, velocity and lubrication film thickness distribution of a droplet rising in a vertical Hele-Shaw cell. The droplet is surrounded by a stationary immiscible fluid and moves purely due to buoyancy. A low density difference between the two media helps to operate in a regime with capillary number $Ca$ lying between $0.03$ and $0.35$ , where $Ca=\unicode[STIX]{x1D707}_{o}U_{d}/\unicode[STIX]{x1D6FE}$ is built with the surrounding oil viscosity $\unicode[STIX]{x1D707}_{o}$ , the droplet velocity $U_{d}$ and surface tension $\unicode[STIX]{x1D6FE}$ . The experimental data show that in this regime the droplet velocity is not influenced by the thickness of the thin lubricating film and the dynamic meniscus. For iso-viscous cases, experimental and three-dimensional numerical results of the film thickness distribution agree well with each other. The mean film thickness is well captured by the Aussillous & Quere (Phys. Fluids, vol. 12 (10), 2000, pp. 2367-2371) model with fitting parameters. The droplet also exhibits the 'catamaran' shape that has been identified experimentally for a pressure-driven counterpart (Huerre et al., Phys. Rev. Lett., vol. 115 (6), 2015, 064501). This pattern has been rationalized using a two-dimensional lubrication equation. In particular, we show that this peculiar film thickness distribution is intrinsically related to the anisotropy of the fluxes induced by the droplet's motion.

  • 9.
    Ungarish, Marius
    et al.
    Technion Israel Inst Technol, Dept Comp Sci, IL-32000 Haifa, Israel..
    Zhu, Lailai
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Stone, Howard A.
    Princeton Univ, Dept Mech & Aerosp Engn, Princeton, NJ 08544 USA..
    Inertial gravity current produced by the drainage of a cylindrical reservoir from an outer or inner edge2019In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 874, p. 185-209Article in journal (Refereed)
    Abstract [en]

    We consider the time-dependent flow of a fluid of density rho(1) in a vertical cylindrical container embedded in a fluid of density rho(2) (<rho(1)) whose side boundary is suddenly removed and the fluid drains freely from the edge. We show that in the inertial-buoyancy regime (large initial Reynolds number) the flow is modelled by the shallow-water equations and bears similarities to a gravity current released from a lock (the dam-break problem) driven by the reduced gravity g' = (1 - rho(2)/rho(1))g. This formulation is amenable to an efficient finite-difference solution. Moreover, we demonstrate that similarity solutions exist, and show that the flow created by the dam break approaches the predicted self-similar behaviour when the volume ratio nu(t)/nu(0) approximate to 1/2 where t is time elapsed from the dam break. We considered two cases of drainage: (i) outward from the outer boundary in a full-radius reservoir; and (ii) inward from the inner radius in an annular-shaped reservoir. For the first case the similarity solution is expressed analytically, while the second case is more complicated and requires a numerical solution. In both cases nu(t)/nu(0) decays like t(-2), but the details are different. The similarity solutions admit an adjustable virtual-origin constant, which we determine by matching with the finite-difference solution. The analysis is valid for both Boussinesq and non-Boussinesq systems, and a wide range of geometric parameters (inner and outer radii, and height). The importance of the neglected viscous terms increases with time, and eventually the inertial-buoyancy model becomes invalid. An estimate for this occurrence is also provided. The predictions of the model are compared to results of direct numerical simulations; there is good agreement for the position of the interface and for the averaged radial velocity, and excellent agreement for nu(t)/nu(0). A box model is used for estimating the effect of a partial (over a sector) dam break. This study is an extension of the work for a rectangular reservoir of Momen et al. (J. Fluid Mech., vol. 827, 2017, pp. 640-663). We demonstrate that there are some similarities, but also significant differences, between the rectangular and the cylindrical reservoirs concerning the velocity, shape of the interface and rate of drainage, which are of interest in applications. The overall conclusion is that this simple model captures very well the flow field under consideration.

  • 10. Xue, Nan
    et al.
    Khodaparast, Sepideh
    Zhu, Lailai
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Nunes, Janine K.
    Kim, Hyoungsoo
    Stone, Howard A.
    Laboratory layered latte2017In: Nature Communications, ISSN 2041-1723, E-ISSN 2041-1723, Vol. 8, article id 1960Article in journal (Refereed)
    Abstract [en]

    Inducing thermal gradients in fluid systems with initial, well-defined density gradients results in the formation of distinct layered patterns, such as those observed in the ocean due to double-diffusive convection. In contrast, layered composite fluids are sometimes observed in confined systems of rather chaotic initial states, for example, lattes formed by pouring espresso into a glass of warm milk. Here, we report controlled experiments injecting a fluid into a miscible phase and show that, above a critical injection velocity, layering emerges over a time scale of minutes. We identify critical conditions to produce the layering, and relate the results quantitatively to double-diffusive convection. Based on this understanding, we show how to employ this single-step process to produce layered structures in soft materials, where the local elastic properties vary step-wise along the length of the material.

  • 11.
    Yu, Yingxian Estella
    et al.
    Princeton Univ, Dept Mech & Aerosp Engn, Princeton, NJ 08544 USA..
    Zhu, Lailai
    KTH, School of Engineering Sciences (SCI), Mechanics. Princeton Univ, Dept Mech & Aerosp Engn, Princeton, NJ 08544 USA.;KTH Mech, Linne Flow Ctr, SE-10044 Stockholm, Sweden.;KTH Mech, Swedish E Sci Res Ctr SeRC, SE-10044 Stockholm, Sweden..
    Shim, Suin
    Princeton Univ, Dept Mech & Aerosp Engn, Princeton, NJ 08544 USA..
    Eggers, Jens
    Univ Bristol, Sch Math, Bristol BS8 1TW, Avon, England..
    Stone, Howard A.
    Princeton Univ, Dept Mech & Aerosp Engn, Princeton, NJ 08544 USA..
    Time-dependent motion of a confined bubble in a tube: transition between two steady states2018In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 857, article id R4Article in journal (Refereed)
    Abstract [en]

    When a confined bubble translates steadily in a cylindrical capillary tube, without the consideration of gravity effects, a uniform thin film of liquid separates the bubble surface and the tube wall. In this work, we investigate how this steady state is established by considering the transitional motion of the bubble as it adjusts its film thickness profile between two steady states, characterized by two different bubble speeds. During the transition, two uniform film regions coexist, separated by a step-like transitional region. The transitional motion also requires modification of the film solution near the rear of the bubble, which depends on the ratio of the two capillary numbers. These theoretical results are verified by experiments and numerical simulations.

  • 12.
    Zhu, Lailai
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre. Ecole Polytech Fed Lausanne, Lab Fluid Mech & Instabil, CH-1015 Lausanne, Switzerland.;Princeton Univ, Dept Mech & Aerosp Engn, Princeton, NJ 08540 USA.
    Gallaire, Francois
    Ecole Polytech Fed Lausanne, Lab Fluid Mech & Instabil, CH-1015 Lausanne, Switzerland..
    Bifurcation Dynamics of a Particle-Encapsulating Droplet in Shear Flow2017In: Physical Review Letters, ISSN 0031-9007, E-ISSN 1079-7114, Vol. 119, no 6, article id 064502Article in journal (Refereed)
    Abstract [en]

    To understand the behavior of composite fluid particles such as nucleated cells and double emulsions in flow, we study a finite-size particle encapsulated in a deforming droplet under shear flow as a model system. In addition to its concentric particle-droplet configuration, we numerically explore other eccentric and time-periodic equilibrium solutions, which emerge spontaneously via supercritical pitchfork and Hopf bifurcations. We present the loci of these solutions around the codimension-two point. We adopt a dynamic system approach to model and characterize the coupled behavior of the two bifurcations. By exploring the flow fields and hydrodynamic forces in detail, we identify the role of hydrodynamic particle-droplet interaction which gives rise to these bifurcations.

  • 13.
    Zhu, Lailai
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre. Princeton Univ, Dept Mech & Aerosp Engn, Princeton, NJ 08544 USA.
    Stone, Howard A.
    Princeton Univ, Dept Mech & Aerosp Engn, Princeton, NJ 08544 USA..
    Propulsion driven by self-oscillation via an electrohydrodynamic instability2019In: Physical Review Fluids, ISSN 2469-990X, Vol. 4, no 6, article id 061701Article in journal (Refereed)
    Abstract [en]

    Oscillations of flagella and cilia play an important role in biology, which motivates the idea of functional mimicry as part of bioinspired applications. Nevertheless, it still remains challenging to drive their artificial counterparts to oscillate via a steady, homogeneous stimulus. Combining theory and simulations, we demonstrate a strategy to achieve this goal by using an elastoelectrohydrodynamic instability (based on the Quincke rotation instability). In particular, we show that applying a uniform dc electric field can produce self-oscillatory motion of a microrobot composed of a dielectric particle and an elastic filament. Upon tuning the electric field and filament elasticity, the microrobot exhibits three distinct behaviors: a stationary state, undulatory swimming, and steady spinning, where the swimming behavior stems from an instability emerging through a Hopf bifurcation. Our results imply the feasibility of engineering self-oscillations by leveraging the elastoviscous response to control the type of bifurcation and the form of instability. We anticipate that our strategy will be useful in a broad range of applications imitating self-oscillatory natural phenomena and biological processes.

  • 14.
    Zhu, Lailai
    et al.
    KTH, School of Engineering Sciences (SCI), Mechanics.
    Stone, Howard A.
    Princeton Univ, Dept Mech & Aerosp Engn, Princeton, NJ 08544 USA..
    Rotation of a low-Reynolds-number watermill: theory and simulations2018In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 849, p. 57-75Article in journal (Refereed)
    Abstract [en]

    Recent experiments have demonstrated that small-scale rotary devices installed in a microfluidic channel can be driven passively by the underlying flow alone without resorting to conventionally applied magnetic or electric fields. In this work, we conduct a theoretical and numerical study on such a flow-driven 'watermill' at low Reynolds number, focusing on its hydrodynamic features. We model the watermill by a collection of equally spaced rigid rods. Based on the classical resistive force (RF) theory and direct numerical simulations, we compute the watermill's instantaneous rotational velocity as a function of its rod number N, position and orientation. When N >= 4, the RF theory predicts that the watermill's rotational velocity is independent of N and its orientation, implying the full rotational symmetry (of infinite order), even though the geometrical configuration exhibits a lower-fold rotational symmetry; the numerical solutions including hydrodynamic interactions show a weak dependence on N and the orientation. In addition, we adopt a dynamical system approach to identify the equilibrium positions of the watermill and analyse their stability. We further compare the theoretically and numerically derived rotational velocities, which agree with each other in general, while considerable discrepancy arises in certain configurations owing to the hydrodynamic interactions neglected by the RP theory. We confirm this conclusion by employing the RP-based asymptotic framework incorporating hydrodynamic interactions for a simpler watermill consisting of two or three rods and we show that accounting for hydrodynamic interactions can significantly enhance the accuracy of the theoretical predictions.

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