Rigid Ellipsoids Research Articles

The morphology of cell spheroids in simple shear flow

Cell spheroids are a widely used model to investigate cell-cell and cell-matrix interactions in a 3D microenvironment in vitro. Most research on cell spheroids has been focused on their response to various stimuli under static conditions. Recently, the effect of flow on cell spheroids has been investigated in the context of tumor invasion in interstitial space. In particular, microfluidic perfusion of cell spheroids embedded in a collagen matrix has been shown to modulate cell-cell adhesion and to represent a possible mechanism promoting tumor invasion by interstitial flow. However, studies on the effects of well-defined flow fields on cell spheroids are lacking in the literature. Here, we apply simple shear flow to cell spheroids in a parallel plate apparatus while observing their morphology by optical microscopy. By using image analysis techniques, we show that cell spheroids rotate under flow as rigid prolate ellipsoids. As time goes on, cells from the outer layer detach from the sheared cell spheroids and are carried away by the flow. Hence, the size of cell spheroids declines with time at a rate increasing with the external shear stress, which can be used to estimate cell-cell adhesion. The technique proposed in this work allows one to correlate flow-induced effects with microscopy imaging of cell spheroids in a well-established shear flow field, thus providing a method to obtain quantitative results which are relevant in the general field of mechanobiology.

The Spin–Spin Problem in Celestial Mechanics

We study the dynamics of two homogeneous rigid ellipsoids subject to their mutual gravitational influence. We assume that the spin axis of each ellipsoid coincides with its shortest physical axis and is perpendicular to the orbital plane. Due to such assumptions, the problem is planar and depends on particular parameters of the ellipsoids, most notably, the equatorial oblateness and the flattening with respect to the shortest physical axes. We consider two models for such configuration: while in the full model, there is a coupling between the orbital and rotational motions, in the Keplerian model, the centers of mass of the bodies are constrained to move on coplanar Keplerian ellipses. The Keplerian case, in the approximation that includes the coupling between the spins of the two ellipsoids, is what we call spin–spin problem, that is a generalization of the classical spin–orbit problem. In this paper we continue the investigations of Misquero (Nonlinearity 34:2191–2219, 2021) on the spin–spin problem by comparing it with the spin–orbit problem and also with the full model. Beside detailing the models associated to the spin–orbit and spin–spin problems, we introduce the notions of standard and balanced resonances, which lead us to investigate the existence of periodic and quasi-periodic solutions. We also give a qualitative description of the phase space and provide results on the linear stability of solutions for the spin–orbit and spin–spin problems. We conclude by providing a comparison between the full and the Keplerian models with particular reference to the interaction between the rotational and orbital motions.

Chaos over order: mapping 3D rotation of triaxial asteroids and minor planets

ABSTRACT Celestial bodies approximated with rigid triaxial ellipsoids in a two-body system can rotate chaotically due to the time-varying gravitational torque from the central mass. At small orbital eccentricity values, rotation is short-term orderly and predictable within the commensurate spin–orbit resonances, while at eccentricity approaching unity, chaos completely takes over. Here, we present the full three-dimensional rotational equations of motion around all three principal axes for triaxial minor planets and two independent methods of numerical solution based on Euler rotations and quaternion algebra. The domains of chaotic rotation are numerically investigated over the entire range of eccentricity with a combination of trial integrations of Euler’s equations of motion and the GALI(k) (Generalized Alignment Index) method. We quantify the dependence of the order–chaos boundaries on shape by changing a prolateness parameter, and find that the main 1:1 spin–orbit resonance disappears for specific moderately prolate shapes already at eccentricities as low as 0.3. The island of short-term stability around the main 1:1 resonance shrinks with increasing eccentricity at a fixed low degree of prolateness and completely vanishes at approximately 0.8. This island is also encroached by chaos on longer time-scales, indicating longer Lyapunov exponents. Trajectories in the close vicinity of the 3:2 spin–orbit resonance become chaotic at smaller eccentricities, but separated enclaves of orderly rotation emerge at eccentricities as high as 0.8. Initial perturbations of rotational velocity in latitude away from the exact equilibrium result in a spectrum of free libration, nutation, and polar wander, which is not well matched by the linearized analysis omitting the inertial terms.

Modelling initial motion of non-spherical sediment particles on inclined and seeped beds

Analytical and numerical solutions are developed to compute the threshold stress for the initiation of motion of sediment particles, from spherical to ellipsoidal, on inclined beds under boundary layer flows. The numerical model is based on the discrete element method and simulates the micro-mechanics of the sediment as an aggregate of rigid ellipsoids interacting by contact, while analytical solution establishes the basic dynamics for initiation of motion. In both models the coupling of grain motion with the flow dynamics is accomplished via drag, lift, buoyancy and seepage forces under turbulent velocity fluctuations by a one-way technique, since incipient motion does not modify substantially the flow.The comparison of both models and their extension to compute the threshold stress in terms of seepage, turbulent velocity fluctuations, bed inclination and particle orientations provide a better understanding of the onset of sediment particles. The parametric analyses cover a wide range of natural and usual conditions for incipient motion, showing that most of results fall within the variability range of the experiments. Additionally, these models are able to compute the initiation of motion for configurations not still tested in laboratory.

Acoustic and inertial modes in planetary-like rotating ellipsoids.

The bounded oscillations of rotating fluid-filled ellipsoids can provide physical insight into the flow dynamics of deformed planetary interiors. The inertial modes, sustained by the Coriolis force, are ubiquitous in rapidly rotating fluids and Vantieghem (2014, Proc. R. Soc. A, 470, 20140093. doi:10.1098/rspa.2014.0093) pioneered a method to compute them in incompressible fluid ellipsoids. Yet, taking density (and pressure) variations into account is required for accurate planetary applications, which has hitherto been largely overlooked in ellipsoidal models. To go beyond the incompressible theory, we present a Galerkin method in rigid coreless ellipsoids, based on a global polynomial description. We apply the method to investigate the normal modes of fully compressible, rotating and diffusionless fluids. We consider an idealized model, which fairly reproduces the density variations in the Earth's liquid core and Jupiter-like gaseous planets. We successfully benchmark the results against standard finite-element computations. Notably, we find that the quasi-geostrophic inertial modes can be significantly modified by compressibility, even in moderately compressible interiors. Finally, we discuss the use of the normal modes to build reduced dynamical models of planetary flows.

Compressible fluid modes in rigid ellipsoids: towards modal acoustic velocimetry

Motivated by planetary-driven applications and experiments in non-spherical geometries, we study compressible fluid modes in rotating rigid ellipsoids. Such modes are also required for modal acoustic velocimetry (MAV), a promising non-invasive method to track the velocity field components in laboratory experiments. To calculate them, we develop a general spectral method in rigid triaxial ellipsoids. The description is based on an expansion onto global polynomial vector elements, satisfying the non-penetration condition on the boundary. Then, we investigate the diffusionless compressible modes in rotating (and magnetised) rigid ellipsoids. The spectral description is successfully benchmarked against three-dimensional finite-element computations and analytical predictions. A spectral convergence is obtained. Our results have direct implications for MAV in experiments, for instance in the ZoRo experiment (gas-filled rigid spheroid). So far, deformation and rotational effects have been theoretically considered separately, as small perturbations of the solutions in non-rotating spheres. We carefully compare the perturbation approach, in this illustrative geometry, to the polynomial solutions. We show that second-order ellipticity effects are often present, even in weakly deformed ellipsoids. Moreover, high-order effects due to rotation and/or ellipticity should be observed for some acoustic modes in experimental conditions. Thus, perturbation theory should be used with care in MAV. Instead, the spectral polynomial method paves the way for future MAV applications in fluid experiments with rigid ellipsoids.

Direct simulation of rigid particles in a viscoelastic fluid

In this paper, we present a numerical method which performs the direct simulation of 2D viscoelastic suspensions. Objectives. Interactions between viscoelastic fluids of Oldroyd type, including inertial effects, and a large number of rigid disks or ellipsoids are addressed. Method. The numerical method is built upon a fictitious domain approach which consists in defining the fluid-structure problem over the whole domain (fluid+solid), see e.g. [1]. In the Newtonian case [2, 3], the resulting variational formulation, although classical, uses non-standard functional spaces which include the rigidity constraints. From the numerical point of view, these constraints can be addressed by penalty methods, leading to the possible use of standard finite elements solvers with fixed structured meshes. We show how this approach can be adapted to viscoelastic fluids of Oldroyd type. Results. Two types of configurations are investigated in dilute and dense regimes: sedimentation of circular particles and shear flows with ellipsoidal particles.

Analytical and discrete solutions for the incipient motion of ellipsoidal sediment particles

ABSTRACTThis work introduces analytical and numerical approaches to compute the incipient motion of ellipsoidal sediment particles. Initiation of motion of spherical particles is dominated by rolling mode. However, solutions for initiation of motion for non-spherical grains have to incorporate rolling, sliding, and mixed modes. The proposed approaches include a wide variety of shapes and inclinations that represent realistic configurations of sediment bed layers. The numerical procedure is based on the discrete element method, simulating the micro-mechanics of the sediment as an aggregate of rigid ellipsoids interacting by contact. The numerical solution covers a range of incipient movements that cannot be covered by the analytical approach. Hence, some trapped modes observed in analytical calculations are complemented by the numerical computation of threshold stresses. The main results are organized as novel extended Shields diagrams for non-spherical grains, where non-dimensional critical shear stress is represented in terms of friction Reynolds number.

Impact of Weak Particle Inertia on a Dilute Suspension of Ellipsoids

ABSTRACTIn this paper, we investigate asymptotical models for the motion of small rigid ellipsoids suspended in a Newtonian fluid. The microscale models account for particle inertia and form the basis for dilute suspension modeling. They were derived from the description of first principles for a coupled particle-fluid problem by asymptotical analysis in the size ratio in Vibe and Marheineke (2015), where the effect of inertia was incorporated via different scalings of the particle momentum balances. We classify three different inertial regimes and analyze numerically their impact on the particle motion. We observe reasonable physical translational and rotational behavior.

The influence of inertia on the rheology of a periodic suspension of neutrally buoyant rigid ellipsoids

We investigate the rheological properties of a suspension of neutrally buoyant rigid ellipsoids by fluid–structure interaction simulations of a particle in a periodic domain under simple shear using the curvilinear immersed-boundary (CURVIB) method along with a quaternion–angular velocity technique to calculate the dynamics of the particle’s motion. We calculate all the different terms of particle stress for the first time for non-spherical particles, i.e. in addition to the stresslet, we calculate the acceleration and Reynolds stress, which are typically ignored in previous similar works. Furthermore, we derive analytical expressions for all these terms to verify the numerical results and deduce the effect of inertia by comparing our numerical results with the analytical solution. The effect of particle Reynolds number ($\mathit{Re}$), volume fraction (${\it\phi}$), and the shape of particles has been studied on all mechanisms of stress generation, the intrinsic viscosity, and normal stress differences of the suspension for the range$0.008\leqslant {\it\phi}\leqslant 0.112$and$0.01\leqslant \mathit{Re}\leqslant 10.0$. We found that inertia increases the shear and the second normal difference of the stresslet (dominant term of the particle stress), and decreases the first normal difference that is generated due to the strain field. The contribution of acceleration stress to the total stress is found to be important in the second normal stress difference, with a cycle-average comparable to the stresslet component. We also discovered that the contribution of Reynolds stress in the first normal stress difference becomes important even when inertia is as low as$\mathit{Re}\sim O(0.1)$, and its value can be even greater than the stresslet when inertia increases, i.e. Reynolds stresses cannot be ignored for non-spherical particles. For concentrations in the range from dilute to semi-dilute, the effect of inertia on the intrinsic viscosity of a suspension is found to be comparable to the volume fraction. Furthermore, our calculations show that for a dilute concentration and the low-inertia regime ($\mathit{Re}<1.0$), the intrinsic viscosity of a suspension consisting of ellipsoids with an aspect ratio of five can be 20 % higher than its Stokesian analytical value.

Latitudinal libration driven flows in triaxial ellipsoids

Motivated by understanding the liquid core dynamics of tidally deformed planets and moons, we present a study of incompressible flow driven by latitudinal libration within rigid triaxial ellipsoids. We first derive a laminar solution for the inviscid equations of motion under the assumption of uniform vorticity flow. This solution exhibits a resonance if the libration frequency matches the frequency of the spin-over inertial mode. Furthermore, we extend our model by introducing a reduced model of the effect of viscous Ekman layers in the limit of low Ekman number (Noir & Cébron,J. Fluid Mech., vol. 737, 2013, pp. 412–439). This theoretical approach is consistent with the results of Chanet al. (Phys. Earth Planet. Inter., vol. 187, 2011, pp. 404–415) and Zhanget al. (J. Fluid Mech., vol. 692, 2012, pp. 420–445) for spheroidal geometries. Our results are validated against systematic three-dimensional numerical simulations. In the second part of the paper, we present the first linear stability analysis of this uniform vorticity flow. To this end, we adopt different methods (Lifschitz & Hameiri,Phys. FluidsA, vol. 3, 1991, p. 2644; Gledzer & Ponomarev,Acad. Sci., USSR, Izv., Atmos. Ocean. Phys., vol. 13, 1977, pp. 565–569) that allow us to deduce upper and lower bounds for the growth rate of an instability. Our analysis shows that the uniform vorticity base flow is prone to inertial instabilities caused by a parametric resonance mechanism. This is confirmed by a set of direct numerical simulations. Applying our results to planetary settings, we find that neither a spin-over resonance nor an inertial instability can exist within the liquid core of the Moon, Io and Mercury.

Modified collective rearrangement sphere-assembly algorithm for random packings of nonspherical particles: Towards engineering applications

A modified collective rearrangement algorithm, based on the sphere-assembly representation of the particle shape, is presented for simulating random packings of nonspherical particles and arbitrary shapes. Compared to other collective rearrangement methods, the modification introduced by this algorithm consists in i) avoiding the particle-to-container shrinking procedure by allowing particles to sequentially settle down and ii) in a stability constraint applied to all the particles in the packing. The coupling of these criteria ensures that all the particles are stable and contacting each other, allowing for an unambiguous detection of contacts, which is important in the evaluation of the effective properties desired in many engineering applications, such as percolation thresholds and effective conductivity. The effect of the internal parameters of the algorithm is investigated, showing that random close packings can be obtained. The algorithm is applied to simulate packings of rigid ellipsoids and cylinders with different aspect ratios, which are compared with simulation results provided by other packing algorithms, showing the consistency of our method. Simulations of inter-penetrating particles, mixtures of particles with different shapes and packings of agglomerates are shown, which confirm the applicability of the method to a broad range of packing problems of practical interest and, in particular, for fuel cell applications.

Orbital drift of capsules and red blood cells in shear flow

Many numerical studies have considered the dynamics of capsules and red blood cells in shear flow under the condition that the axis of revolution of such bodies remained aligned in the shear plane. In contrast, several experimental studies have shown that the axis of revolution of red blood cells could drift away from the shear plane in a certain range of controlling parameters. In this article, we present three-dimensional numerical simulations on the orientation dynamics of capsules in simple shear flow with different initial undeformed shapes, namely, prolate, oblate, and biconcave disk. It is observed that unlike rigid ellipsoids in Stokes flow, capsules reorient their axis of revolution either towards the vorticity axis while undergoing a precessing motion or towards the shear plane while undergoing a kayaking-type motion. The specific dynamics are observed to depend on initial shape, capillary number, and the ratio of the internal to external fluid viscosity. Near the physiological values of the viscosity ratio, the biconcave shape performs a rolling motion like a wheel. If the viscosity ratio is reduced below the physiological range, a transition to the kayaking dynamics is observed with increasing capillary number. The critical shear stress at which the rolling-to-kayaking transition occurs is found to be dependent on the viscosity ratio.

A general approach for modeling the motion of rigid and deformable ellipsoids in ductile flows

A general approach for modeling the motion of rigid or deformable objects in viscous flows is presented. It is shown that the rotation of a 3D object in a viscous fluid, regardless of the mechanical property and shape of the object, is defined by a common and simple differential equation, d Q / d t = − Θ ˜ Q , where Q is a matrix defined by the orientation of the object and Θ ˜ is the angular velocity tensor of the object. The difference between individual cases lies only in the formulation for the angular velocity. Thus the above equation, together with Jeffery's theory for the angular velocity of rigid ellipsoids, describes the motion of rigid ellipsoids in viscous flows. The same equation, together with Eshelby's theory for the angular velocity of deformable ellipsoids, describes the motion of deformable ellipsoids in viscous flows. Both problems are solved here numerically by a general approach that is much simpler conceptually and more economic computationally, compared to previous approaches that consider the problems separately and require numerical solutions to coupled differential equations about Euler angles or spherical (polar coordinate) angles. A Runge–Kutta approximation is constructed for solving the above general differential equation. Singular cases of Eshelby's equations when the object is spheroidal or spherical are handled in this paper in a much simpler way than in previous work. The computational procedure can be readily implemented in any modern mathematics application that handles matrix operations. Four MathCad Worksheets are provided for modeling the motion of a single rigid or deformable ellipsoid immersed in viscous fluids, as well as the evolution of a system of noninteracting rigid or deformable ellipsoids embedded in viscous flows.

Dynamic rheology of a dilute suspension of elastic capsules: effect of capsule tank-treading, swinging and tumbling

Three-dimensional numerical simulations are used to study the effect of unsteady swinging and tumbling motion on the rheology of a dilute suspension of oblate-shaped elastic capsules. Unlike a suspension of initially spherical capsules undergoing the steady tank-treading motion for which the rheology is constant in time, the suspension of non-spherical capsules is time-dependent due to the unsteady capsule motion. In a simple shear flow, the non-spherical capsules undergo a transition from the tank-treading/swinging to the tumbling motion with a reduction in the shear rate or an increase in the ratio of the internal to external fluid viscosities. We find that the time-averaged rheology obtained for the non-spherical capsules undergoing the unsteady motion is qualitatively similar to that obtained for the spherical capsules undergoing the steady tank-treading motion, and that the tank-treading-to-tumbling transition has only a marginal effect. The time-averaged rheology exhibits a shear viscosity minimum when the capsules are in a swinging motion at high shear rates but not at low shear rates. This is a remarkable departure from the behaviour of a vesicle suspension which exhibits a shear viscosity minimum at the point of transition. We find that the shear viscosity in a capsule suspension can decrease as well as increase with increasing viscosity ratio during both tank-treading and tumbling motions, while that of a vesicle suspension always decreases in tank-treading motion and increases in tumbling motion. We then seek to connect the time-dependent rheology with the time-dependent membrane tension, capsule orientation, deformation and tank-treading velocity. At low shear rates, the numerical results exhibit a similar trend to that predicted by analytical theory for rigid ellipsoids undergoing tumbling motion. The trend differs during swinging motion due to the periodic deformation and time-dependent variation of the membrane stress. The elastic component of the shear stress is minimum when the capsules are maximally compressed, and is maximum when the capsules are maximally elongated. In contrast, the viscous component is related to the periodic variation of the tank-treading velocity synchronized with the swinging motion, and the rate of capsule elongation or compression. The swinging or tumbling velocity makes no contribution to the time-dependent rheology.

Effective viscosity of semidilute suspensions of rigid ellipsoids

Hard and refractory particles with various shapes have been added to ceramic materials to enhance creep resistance. Also, many of the particles in food emulsions, magnetic particle suspensions, and other industrial systems may have nonspherical shapes. Hence, it is important to understand the effects of particle shape on the viscosity of suspensions. While the shape of many particles can be approximated as prolate or oblate spheroids, the purpose of the present study is to model the effective viscosity of semidilute suspensions of rigid ellipsoids. Closed-form expressions are obtained; however, their formulations are formidable. For the special cases of rodlike or disklike particles, simple closed-form expressions can be obtained. The present solutions are compared with existing solutions.

Computational fluid dynamics of aggregating red blood cells in postcapillary venules

Aggregate formation of red blood cells (RBCs) in a postcapillary venular bifurcation is investigated with three-dimensional computer simulations using the Chimera grid method. Interaction energy between the RBCs is modelled by a depletion interaction theory; RBCs are modelled as rigid oblate ellipsoids. The cell–cell interactions of RBCs are strongly dependent on vessel geometry and shear rates. The experimental data on vessel geometry, pseudoshear rates, and Dextran concentration obtained in our previous in vivo RBC aggregation study in postcapillary venules of the rat spinotrapezius muscle were used to simulate RBC aggregation. The computational results were compared to the experimental results from the in vivo study. The results show that cells have a larger tendency to form an aggregate under reduced flows. Aggregate formation also depends on the angle and location of the cells before they enter the bifurcation region. Comparisons with experimental data are discussed.

ParticleSync: A program for simulating multiparticle interactions

A Windows application is presented that emulates a model previously developed. The model simulates the rotational disturbances of rigid ellipsoids subject to simple shear as a result of interactions between neighbours. A summary of the model along with instructions on how use the software are presented as well as results from simulations which reveal the particles' preferred orientation at an angle to the shear direction.

Rheology of semidilute suspensions of rigid polystyrene ellipsoids at high Peclet numbers

The rheology of semidilute suspensions of rigid polystyrene ellipsoids at rotational Peclet numbers greater than 103 was studied for two different aspect ratios. The ellipsoid suspensions exhibit shear thinning behavior for both aspect ratios, similar to previous experiments on suspensions of rigid fibers. Possible mechanisms for the rate dependent rheology are discussed and evaluated through a comparison with the rheology of suspensions of spheres with the same material properties as the ellipsoids. For the largest Peclet number, the relative viscosity scales linearly with dimensionless number density and is independent of the aspect ratio.

Shear Thinning and Tumbling Dynamics of Single Polymers in the Flow-Gradient Plane

The conformational changes imparted on single, flexible DNA polymers by a steady, simple shear flow were directly visualized in the flow-gradient plane. Two fluorescently stained DNA double-strand sizes of 22 μm and 80 μm in contour length were employed, and Wi values of up to 584 were probed (Wi = shear rate × longest polymer relaxation time). By exploitation of the linear proportionality between polymer density and its recorded image, the accessible radius of gyration tensor elements (Gij) were measured. Of those, the ensemble-averaged 〈G22〉 and 〈G12〉 were related to the bulk shear viscosity and first normal stress coefficient, respectively, via the Giesekus stress tensor. We found their respective behaviors to follow power-law decays of Wi -0.52 and Wi -1.28 at large Wi. Polymer dynamics were also investigated. Like rigid ellipsoids of revolution, polymers displayed a constant partition between positive and negative orientations irrespective of shear rate at Wi ≫ 1. Unlike them, however, polymers prefe...