基于细胞和粒子在微流控器件中的输运特性对其进行识别，对于生物学和医学中的众多应用至关重要。在中性浮力粒子在微流控通道中输运时，由于惯性效应，它们会横向迁移至稳定位置。然而，粒子与流动特性对这些聚焦位置的影响在很大程度上仍属未知。我们利用耗散粒子动力学进行了大规模模拟，证明自由移动的粒子与被阻止旋转的粒子在聚焦模式上表现出显著差异。在圆管中，我们观察到旋转与不旋转的聚焦位置发生了剧烈变化。我们证明，与先前的认知不同，由旋转引起的横向升力是显著的，且与旋转幅度呈线性关系。我们提出了一个扩展现有理论的简单唯象解释，该解释与我们的数值发现吻合良好。在方形管道中，我们报告了旋转粒子的四个面心稳定位置，这与在 50 < Re < 200 的雷诺数范围内的实验研究相一致。然而，不旋转的粒子保持分散在一个同心的一维环面上，揭示出与自由粒子相比在定性上不同的行为。我们的发现为基于惯性的微流控器件中的微粒和细胞分选提出了新设计。
英文摘要：
The identification of cells and particles based on their transport properties in microfluidic devices is crucial for numerous applications in biology and medicine. Neutrally buoyant particles transported in microfluidic channels, migrate laterally towards stable locations due to inertial effects. However, the effect of the particle and flow properties on these focusing positions remain largely unknown. We conduct large scale simulations with dissipative particle dynamics, demonstrating that freely moving particles exhibit significant differences in their focusing patterns from particles that are prevented from rotation. In circular pipes, we observe drastic changes in rotating versus non-rotating focusing positions. We demonstrate that rotation-induced lateral lift force is significant, unlike previously believed, and is linearly dependent on the rotation magnitude. A simple phenomenological explanation extending existing theories is presented, that agrees well with our numerical findings. In square ducts, we report four face-centered stable positions for rotating particles, in accordance with experimental studies on a range of Reynolds numbers 50 < Re < 200. However, non-rotating particles stay scattered on a concentric one-dimensional annulus, revealing qualitatively different behavior with respect to the free ones. Our findings suggest new designs for micro-particle and cell sorting in inertia-based microfluidics devices.
In inertial microfluidics lift forces cause a particle to migrate across streamlines to specific positions in the cross section of a microchannel. We control the rotational motion of a particle and demonstrate that this allows to manipulate the lift-force profile and thereby the particle's equilibrium positions. We perform two-dimensional simulation studies using the method of multi-particle collision dynamics. Particles with unconstrained rotational motion occupy stable equilibrium positions in both halfs of the channel while the center is unstable. When an external torque is applied to the p…
We numerically study the inertial waves and zonal flows in spherical shells driven by longitudinal libration, an oscillatory variation of rotation rate. Internal shear layers are generated due to breakdown of the Ekman boundary layer at critical latitudes. Our numerical results validate the scaling laws of internal shear layers predicted by previous studies. Mean zonal flows are driven by the non-linear interaction in the boundary layers. Non-linear interaction of inertial waves in the interior fluids has no significant contribution to the zonal flow. Multiple geostrophic shear layers are gene…
Rotational dynamics often challenge physical intuition while enabling unique realizations, from the rotor of a gyroscope that maintains its orientation regardless of the outer gimbals, to a tennis racket that rotates around its handle when tossed face-up in the air. In the context of inertial mass sensing, which can measure mass with atomic precision, rotational dynamics are normally considered a complication hindering measurement interpretation. Here, we exploit the rotational dynamics of a microfluidic device to develop a new modality in inertial resonant sensing. Combining theory with exper…
We measure the rotational statistics of neutrally buoyant rods with lengths $2.8 < l/η<72.9 $ in turbulence. For particles with length in the inertial range, we derive a scaling relationship for the mean square rotation rate, $< \dot{p}_i \dot{p}_i > \propto l^{-4/3}$ and show that measurements approach this scaling. Deviations from the proposed scaling are explained as the effect of dissipation range scales. The correlation time of the Lagrangian autocorrelation of rod rotation rate scales as the turn over time of eddies of the size of the rod. Measuring rotational dynamics of single long rod…
We study swimming of small spherical particles who regulate fluid flow on their surface by applying tangential squirming strokes. We derive translational and rotational velocities for any given stroke which is not restricted by axial symmetry as assumed usually. The formulation includes inertia of both the fluid and the swimmer, motivated by inertia's relevance for large Volvox colonies. We show that inertial contribution to mean speed comes from dynamic coupling between translation and rotation, which occurs only for strokes that break axial symmetry. Remarkably, this effect enables overcomin…
Maintenance of a columnar vortex by inertial waves in rotational turbulence
Landau Institute for Theoretical Physics, Russian Academy of Sciences（朗道理论物理研究所，俄罗斯科学院）；National Research University Higher School of Economics（国立高等经济大学）；Osipyan Institute of Solid State Physics, Russian Academy of Sciences（奥西普扬固体物理研究所，俄罗斯科学院）
We propose a numerical method to compute the inertial modes of a container with near-spherical geometry based on the fully spectral discretisation of the angular and radial directions using spherical harmonics and Gegenbauer polynomial expansion respectively. This allows to solve simultaneously the Poincare equation and the no penetration condition as an algebraic polynomial eigenvalue problem. The inertial modes of an exact oblate spheroid are recovered to machine precision using an appropriate set of spheroidal coordinates. We show how other boundaries that deviate slightly from a sphere can…
We present a study of inertial modes in a differentially rotating spherical shell (spherical Couette flow) experiment with a radius ratio of $η= 1/3$. Inertial modes are Coriolis-restored linear wave modes which often arise in rapidly rotating fluids. Recent experimental work has shown that inertial modes exist in a spherical Couette flow for $Ω_{i}<Ω_{o}$, where $Ω_i$ and $Ω_o$ is the inner and outer sphere rotation rate. A finite number of particular inertial modes has previously been found. By scanning the Rossby number from $-2.5 < Ro = (Ω_{i}-Ω_{o})/Ω_{o} < 0$ at two fixed $Ω_{o}$, we rep…
We study numerically the inertial migration of a single rigid sphere and an oblate spheroid in straight square and rectangular ducts. A highly accurate interface-resolved numerical algorithm is employed to analyse the entire migration dynamics of the oblate particle and compare it with that of the sphere. Similarly to the inertial focusing of spheres, the oblate particle reaches one of the four face-centred equilibrium positions, however they are vertically aligned with the axis of symmetry in the spanwise direction. In addition, the lateral trajectories of spheres and oblates collapse into an…
We study the hydrodynamics of spherical spinners suspended in a Newtonian fluid at inertial regime. We observe a spontaneous condensation of the spinners into particle rich regions, at low but finite particle Reynolds numbers and volume fractions. The particle clusters have a coherent internal dynamics. The spinners form colloidal vortices surrounded by the fluid depleted of the particles. The formation of vortices is observed both in periodic simulation box and when the spinners are confined between two flat walls. The stabilisation of the observed states relies only on hydrodynamic interacti…
We study the rotational dynamics of {\it inertial} disks and rods in three-dimensional, homogeneous isotropic turbulence. In particular, we show how the alignment and the decorrelation time-scales of such spheroids depend, critically, on both the level of inertia and the aspect ratio of these particles. These results illustrate the effect of inertia---which leads to a preferential sampling of the local flow geometry---on the statistics of both disks and rods in a turbulent flow. Our results are important for a variety of natural and industrial settings where the turbulent transport of asymmetr…
The Earth is a rapidly rotating body. The centrifugal pull makes its shape resemble a flattened ellipsoid and Coriolis forces support waves in its fluid core, known as inertial waves. These waves can lead to global oscillations, or modes, of the fluid. Periodic variations of the Earth's rotation axis (nutations) can lead to an exchange of angular momentum between the mantle and the fluid core and excite these inertial modes. In addition to viscous torques that exist regardless of the shape of the boundaries, the small flattening of the core-mantle boundary (CMB) allows inertial modes to exert …
Spherical Couette flow (flow between concentric rotating spheres) is one of flows under consideration for the laboratory magnetic dynamos. Recent experiments have shown that such flows may excite Coriolis restored inertial modes. The present work aims to better understand the properties of the observed modes and the nature of their excitation. Using numerical solutions describing forced inertial modes of a uniformly rotating fluid inside a spherical shell, we first identify the observed oscillations of the Couette flow with non-axisymmetric, retrograde, equatorially anti-symmetric inertial mod…
This letter presents a theory on the coalescence of two spherical liquid droplets that are initially stationary. The evolution of the radius of a liquid neck formed upon coalescence was formulated as an initial value problem and then solved to yield an exact solution without free parameters, with its two asymptotic approximations reproducing the well-known scaling relations in the viscous and inertial regimes. The viscous-to-inertial crossover observed by Paulsen et al. [Phys. Rev. Lett. 106, 114501 (2011)] is also recovered by the theory, rendering the collapse of data of different viscositie…
Flows between concentric, counter rotating spherical boundaries are under investigation in the gap with size equal to inner sphere radius. Outer sphere rotational rate is fixed, while inner sphere rotational rate has time periodic variations. The amplitudes and frequencies of these variations are small relative to both spheres averaged rotational rates. With amplitude increase transition from initial periodical flow to chaos is occurred. To determine state of the flow time series of azimuthal velocity, taken with laser Doppler anemometry, were used. We demonstrate appearance of flow states in …
We investigate the properties of small-amplitude inertial waves propagating in a differentially rotating incompressible fluid contained in a spherical shell. For cylindrical and shellular rotation profiles and in the inviscid limit, inertial waves obey a second-order partial differential equation of mixed type. Two kinds of inertial modes therefore exist, depending on whether the hyperbolic domain where characteristics propagate covers the whole shell or not. The occurrence of these two kinds of inertial modes is examined, and we show that the range of frequencies at which inertial waves may p…
Inertial cavitation in soft matter is an important phenomenon featured in a wide array of biological and engineering processes. Recent advances in experimental, theoretical, and numerical techniques have provided access into a world full of nonlinear physics, yet most of our quantitative understanding to date has been centered on a spherically symmetric description of the cavitation process. However, cavitation bubble growth and collapse rarely occur in a perfectly symmetrical fashion, particularly in soft materials. Predicting the onset of dynamically arising, non-spherical instabilities has …
A novel mechanism for the transport of microscale particles in viscous fluids is demonstrated. The mechanism exploits the trapping of such particles by rotational streaming cells established in the vicinity of an oscillating cylinder, recently analyzed in previous work. The present work explores a strategy of transporting particles between the trapping points established by multiple cylinders undergoing oscillations in sequential intervals. It is demonstrated that, by controlling the sequence of oscillation intervals, an inertial particle is effectively and predictably transported between the …
Microswimmers, and among them aspirant microrobots, generally have to cope with flows where viscous forces are dominant, characterized by a low Reynolds number ($Re$). This implies constraints on the possible sequences of body motion, which have to be nonreciprocal. Furthermore, the presence of a strong drag limits the range of resulting velocities. Here, we propose a swimming mechanism, which uses the buckling instability triggered by pressure waves to propel a spherical, hollow shell. With a macroscopic experimental model, we show that a net displacement is produced at all $Re$ regimes. An o…
Inertial waves propagate in homogeneous rotating fluids, and constitute a challenging and simplified case study for the broader class of inertio-gravity waves, present in all geophysical and astrophysical media, and responsible for energetically costly processes as diapycnal and angular momentum mixing. However, a complete analytical description and understanding of internal waves in arbitrarily shaped enclosed domains, such as the ocean, or a planet liquid core, is still missing. In this work, the inviscid, linear inertial wave field is investigated by means of three dimensional ray tracing i…
The chapter begins with a description of the annulus model of rotationg convection. The basic equations for the spherical problem are then introduced and the onset of columnar convection in spherical shells is discussed. The onset of inertial mode convection is described which prevails at very low Prandtl numbers. The properties of finite amplitude convection are outlined for a wide range of Prandtl numbers. Equatorially attached convection is considered which evolves from inertial mode convection.
Motion of elongated particles in shear is studied. In applications where particles are much heavier than the carrying fluid, e.g. aerosols, the influence of particle inertia dominates the particle dynamics. Assuming that the particle only experiences a local linear velocity profile, its rotational and translational motion are independent. However, we show that quadratic terms of the local velocity profile combined with particle inertia cause a lateral drift of prolate spheroidal particles. We find that this drift is maximal when particle inertial forces are of the same order of magnitude as vi…
Inertial focusing dynamics of spherical particles in curved microfluidic ducts with a trapezoidal cross-section
School of Mathematics and Statistics, Victoria University of Wellington（维多利亚大学惠灵顿校区数学与统计学院）；School of Mathematical Sciences, University of Adelaide（阿德莱德大学数学科学学院）
We analyse the angular dynamics of a neutrally buoyant nearly spherical particle immersed in a steady general linear flow. The hydrodynamic torque acting on the particle is obtained by means of a reciprocal theorem, regular perturbation theory exploiting the small eccentricity of the nearly spherical particle, and assuming that inertial effects are small, but finite.
We investigate the asymptotic properties of axisymmetric inertial modes propagating in a spherical shell when viscosity tends to zero. We identify three kinds of eigenmodes whose eigenvalues follow very different laws as the Ekman number $E$ becomes very small. First are modes associated with attractors of characteristics that are made of thin shear layers closely following the periodic orbit traced by the characteristic attractor. Second are modes made of shear layers that connect the critical latitude singularities of the two hemispheres of the inner boundary of the spherical shell. Third ar…
Several planetary bodies in our solar system undergo a forced libration owing to gravitational interactions with their orbital companions, leading to complex fluid motions in their metallic liquid cores or subsurface oceans. In this study, we numerically investigate flows in longitudinally librating spherical shells. We focus on the Ekman number dependencies of several shear layers when the libration frequency is less than twice of the rotation frequency and the libration amplitude is small. Time-dependent flows mainly consist of inertial waves excited at the critical latitudes due to the Ekma…
Wave kinetic theory for rapidly rotating flows is developed in this paper using a rigorous application of multiple scales perturbation theory. The governing equations are an asymptotically reduced set of equations that are derived from the incompressible Navier-Stokes equations. These equations are applicable for rapidly rotating flow regimes and are best suited to describe anisotropic dynamics of rotating flows. The independent variables of these equations inherently reside in a helical wave basis that is the most suitable basis for inertial waves. A coupled system of equations for the two gl…
A popular model for a generic fat-cored vortex ring or eddy is Hill's spherical vortex (Phil. Trans. Roy. Soc. A vol. 185, 1894, p. 213). Here we find an exact solution for such a spherical vortex steadily propagating along the axis of a rotating ideal fluid. The spherical vortex swirls in such a way as to exactly cancel out the background rotation of the system. The flow external to the spherical vortex exhibits fully nonlinear inertial wave motion. We show that above a critical rotation rate, closed streamlines form in this outer fluid region and hence fluid is carried along with the spheric…
The inertial collapse of two interacting and non-translating spherical bubbles of equal size is considered. The exact analytic solution to the nonlinear ordinary differential equation that governs the bubble radii during collapse is first obtained via a slowly converging power series. An asymptotic approximant is then constructed that accelerates convergence of the series and imposes the asymptotic collapse behavior when the radii are small. The solution generalizes the classical 1917 Rayleigh problem of single bubble collapse, as this configuration is recovered when the distance between the b…
We investigate the asymptotic properties of inertial modes confined in a spherical shell when viscosity tends to zero. We first consider the mapping made by the characteristics of the hyperbolic equation (Poincaré's equation) satisfied by inviscid solutions. Characteristics are straight lines in a meridional section of the shell, and the mapping shows that, generically, these lines converge towards a periodic orbit which acts like an attractor. We then examine the relation between this characteristic path and eigensolutions of the inviscid problem and show that in a purely two-dimensional prob…
The statistics of rotational motion of small, inertialess triaxial ellipsoids are computed along Lagrangian trajectories extracted from direct numerical simulations of homogeneous isotropic turbulence. The particle angular velocity and its components along the three principal axes of the particle are considered, expanding on the results presented by \citet{ChevillardMeneveau13}. The variance of the particle angular velocity, referred to as the particle enstrophy, is found to increase for particles with elongated shapes. This trend is explained by considering the contributions of vorticity and …
We investigate the properties of forced inertial modes of a rotating fluid inside a spherical shell. Our forcing is tidal like, but its main property is that it is on the large scales. Our solutions first confirm some analytical results obtained on a two-dimensional model by Ogilvie (2005). We also note that as the frequency of the forcing varies, the dissipation varies drastically if the Ekman number E is low (as is usually the case). We then investigate the three-dimensional case and compare the results to the foregoing model. These solutions show, like their 2D counterpart, a spiky dissipat…
In this paper, we present an analytical calculation of the rotational mobility functions of a particle rotating on the centerline of an elastic cylindrical tube whose membrane exhibits resistance towards shearing and bending. We find that the correction to the particle rotational mobility about the cylinder axis depends solely on membrane shearing properties while both shearing and bending manifest themselves for the rotational mobility about an axis perpendicular to the cylinder axis. In the quasi-steady limit of vanishing frequency, the particle rotational mobility nearby a no-slip rigid cyl…
In a shear flow particles migrate to their equilibrium positions in the microchannel. Here we demonstrate theoretically that if particles are inertial, this equilibrium can become unstable due to the Saffman lift force. We derive an expression for the critical Stokes number that determines the onset of instable equilibrium. We also present results of lattice Boltzmann simulations for spherical particles and prolate spheroids to validate the analysis. Our work provides a simple explanation of several unusual phenomena observed in earlier experiments and computer simulations, but never interpret…
Rotational mixing, the key process in stellar evolution, transports angular momentum and chemical elements in stellar radiative zones. In the past two decades, an emphasis has been placed on the turbulent transport induced by the vertical shear instability. However, instabilities arising from horizontal shear and the strength of the anisotropic turbulent transport that they may trigger remain relatively unexplored. This paper investigates the combined effects of stable stratification, rotation, and thermal diffusion on the horizontal shear instabilities in the context of stellar radiative zone…
Through laboratory measurements, we compare the rotation of spherical and ellipsoidal particles in homogeneous, isotropic turbulence. We find that the particles' angular velocity statistics are well described by an Ornstein-Uhlenbeck (OU) process. This theoretical model predicts that the Lagrangian autocovariance of particles' angular velocity will decay exponentially. We measure the autocovariance using stereoscopic particle image velocimetry (SPIV) applied to particles whose size is within the inertial subrange of the ambient turbulence. The SPIV resolves the motion of points interior to the…
We study the translational and rotational dynamics of neutrally-buoyant finite-size spheroids in hydrodynamic turbulence by means of fully resolved numerical simulations. We examine axisymmetric shapes, from oblate to prolate, and the particle volume dependences. We show that the accelerations and rotations experienced by non-spherical inertial-scale particles result from volume filtered fluid forces and torques, similar to spherical particles. However, the particle orientations carry signatures of preferential alignments with the surrounding flow structures, which is reflected in distinct axi…
The existence of triadic resonances in the magnetized spherical Couette system (MSC) is related to the development of modulated rotating waves, which are quasiperiodic flows understood in terms of bifurcation theory in systems with symmetry. In contrast to previous studies in spherical geometry the resonant modes are not inertial waves but related with the radial jet instability which is strongly equatorially antisymmetric. We propose a general framework in which triadic resonances are generated through successive Hopf bifurcations from the base state. The study relies on an accurate frequency…
The dynamics of a spherical chemically-powered synthetic colloidal motor that operates by a self-diffusiophoretic mechanism and has a catalytic domain of arbitrary shape is studied using both continuum theory and particle-based simulations. The motor executes active rotational motion when self-generated concentration gradients and interactions between the chemical species and colloidal motor surface break spherical symmetry. Local variations of chemical reaction rates on the motor catalytic surface with catalytic domain sizes and shapes provide such broken symmetry conditions. A continuum theo…
Recent experimental and numerical observations have shown the significance of the Basset--Boussinesq memory term on the dynamics of small spherical rigid particles (or inertial particles) suspended in an ambient fluid flow. These observations suggest an algebraic decay to an asymptotic state, as opposed to the exponential convergence in the absence of the memory term. Here, we prove that the observed algebraic decay is a universal property of the Maxey--Riley equation. Specifically, the particle velocity decays algebraically in time to a limit that is $\mathcal O(ε)$-close to the fluid velocit…
