Torque-free Conditions Research Articles

The motion of a self-propelling two-sphere swimmer in a weakly viscoelastic fluid

We study analytically the propulsion of a force- and torque-free swimmer composed of two counterrotating spheres of differing radii through a viscoelastic fluid described by the Giesekus constitutive model. Our analysis includes both swimmers composed of directly touching spheres and those composed of spheres separated by some finite distance. The propulsion speed of the swimmer is calculated by first expanding the equations of motion and the Giesekus constitutive model as a power series in the Weissenberg number, and then using the Lorentz reciprocal theorem to determine the first-order propulsion speed using the known flow fields for rotating and translating two-sphere geometries at zeroth order. We calculate the relative rotation speeds of the two spheres necessary to maintain the torque-free condition, including an approximate correction for fluid elasticity. The impact of the separation distance between the two spheres, the ratio of their radii, and the value of the Giesekus mobility parameter α on the propulsion speed are all examined; we find that the propulsion speed of the swimmer is maximized for two touching spheres with a radius ratio of approximately 0.7, with a Giesekus mobility parameter of α=0, corresponding to an Oldroyd-B fluid. We also quantify how the increased shear-thinning in the fluid, represented by increasing values of α, results in a significant decrease in the swimmer speed. Finally, through calculations of the fluid stresses around the two-sphere swimmer, we demonstrate the development of enhanced hoop stresses around the smaller sphere, which drive the expulsion of stretched fluid behind the smaller sphere and induce motion of the swimmer in the direction of the larger sphere.

Numerical study of cutterhead-soil interaction in dry sand

ABSTRACT This paper presents a numerical study on cutterhead-soil interaction and its influence on tunnel face stability for stationary and rotating cutterheads. A parametric study was carried out for different friction angles, cover depths, and face pressure. The results indicate that the required face pressure is influenced significantly by the cutterhead-opening-soil interaction. As face pressure is progressively reduced from a high value, stability of the soil across an opening is initially maintained by arch action. The transfer of load from the opening to the springers of the arch leads to an increase in cutterhead thrust. This reaches a peak value after which the arch collapses. A higher friction angle of the soil allows the arch to remain arched at lower face pressure. Under torque-free condition, a higher cutterhead-soil interface friction coefficient enhances stability slightly by mobilizing friction resistance on the soil at the springers of the arch.

Drag and lift forces on a rigid sphere immersed in a wall-bounded linear shear flow

Drag and lift forces experienced by a rigid spherical particle translating close to a flat wall in a linear shear flow are determined through fully resolved simulations. Particles are assumed to obey either a non-rotating or a torque-free condition and may either lead the fluid or lag behind it. The simulations cover a wide range of flow regimes, from low-but-finite Reynolds numbers to conditions close to the onset of wake unsteadiness. Results are used to build accurate semiempirical expressions for the two force components valid at arbitrary Reynolds number, relative shear rate, and separation distance from the wall.

Fast algorithms for large dense matrices with applications to biofluids

Numerical simulation of biofluids entails solving equations of fluid-structure interactions. At zero Reynolds number, solvers such as the Method of Regularized Stokeslets (MRS) give rise to large and dense matrices in practical applications where the number of structures immersed in the fluid is large. Building on previous work for an unbounded fluid domain, we first extend the Kernel-Independent Fast Multipole Method (KIFMM) for MRS to compute the matrix-vector products for the fluid flow induced by point forces above a stationary wall. In this case, the use of a regularized image system introduces additional terms to the solution which cause the matrix-vector multiplication to be quite challenging. In addition, we study the case where a linear system needs to be solved for the unknown forces that structures with known velocities exert on the fluid. Our main contribution is proposing several preconditioning techniques for the matrices associated with a few variants of MRS, including the case where a force-free, torque-free condition is imposed. They take advantage of the data-sparsity of FMM matrices as well as properties of Krylov subspaces. Our approach is memory efficient, capable of handling non-uniformly distributed structures and applicable to all FMM matrices. It enables efficient computation of the flow field surrounding a large group of dynamic micro-structures; in particular, we study the effects of fluid mixing caused by the periodic beating of a dense carpet of lung cilia.

Hydrodynamic rotlet dipole driven by spinning chiral liquid crystal droplets

Chirality is an essential evolutionary-conserved physical aspect of swimming microorganisms. However, the role of chirality on the hydrodynamics of such microswimmers is still being elucidated. Hydrodynamic theories have so far predicted that, under a torque-free condition satisfied in the system of microswimmers, a rotlet dipole generating a twisting flow is the leading-order singularity of the chiral flow field. Nevertheless, such a chiral flow field has never been experimentally detected. Here we explore a hydrodynamic field generated in a system of a chiral microswimmer, where a droplet of a cholesteric liquid crystal (CLC) exhibits helical and spinning motions in surfactant solutions due to a chiral nonequilibrium cross coupling between the rotation and the Marangoni flow. Combining measurement of the flow field around the spinning CLC droplets and a computational flow modeling, we revealed that the CLC droplets generate a flow field of a rotlet dipole. Remarkably, we found that the chiral component of the flow field decays with distance r as r^{-3}, which is consistent with the theoretical prediction for the flow field produced by a point singularity of a rotlet dipole. Our findings will promote the understanding of roles of chirality on the hydrodynamics in active matter as well as liquid crystals.

An innovative test bed for verification of attitude control system

Experimental validation of the attitude determination and control subsystem performance is an essential requirement for new satellites. It can provide a very low cost and low risk environment for evaluating the proposed design before launch. One of the greatest difficulties in developing an Attitude Determination and Control Subsystem (ADCS) test bed is to perform the ground-based experiments in different environments to provide the orbital conditions experienced by a real spacecraft. Simulating the space environment is not an easy task and also testing of the attitude determination and control is very complicated. The essential requirements of functional tests in ADCS hardware are 1) rotational frictionless free motions, 2) simulating orbital environment for sensors and actuators, 3) software implementation of algorithms, and 4) finally integrating of these parts. Three degree of freedom (DOF) air bearing based simulators are the most common method for simulation of a frictionless media. Air bearing offers a nearly torque-free condition, perhaps as close as possible to that of space and hence it is a preferred technology for ground-based research in space attitude dynamics and control, although it certainly cannot provide the full experience of microgravity. There are several simulators around the world that are developed for many different education and research purposes. One of these simulators is developed for testing the bifocal relay mirror spacecraft system.

Possible interpretation of the precession of comet 67P/Churyumov-Gerasimenko

Data derived from the reconstruction of the nucleus shape of comet 67P/Churyumov-Gerasimenko (67P) from images of the OSIRIS camera onboard ROSETTA show evidence that the nucleus rotates in complex mode. First, the orientation of the spin axis is not fixed in an inertial reference frame, which suggests a precessing motion around the angular momentum vector with a periodicity of approximately 257 h ± 12 h.Second, periodograms of the right ascension and declination (RA/Dec) coordinates of the body-frame Z axis show a very significant (higher than 99.99%) periodicity at 276 h ± 12 h, different from the rotational period of 12.40 h as previously determined from light-curve analysis. Aims. The main goal is to interpret the data and associated periodicities of the spin axis orientation in space. Methods. We analyzed the spin axis orientation in space and associated periodicities and compared them with solutions of Euler equations under the assumption that the body rotates in torque-free conditions. Statistical tests comparing the observationally derived spin axis orientation with the outcome from simulations were applied to determine the most likely inertia moments, excitation level, and periods. Results. Under the assumption that the body is solid-rigid and rotates in torque-free conditions, the most likely interpretation is that 67P is spinning around the principal axis with the highest inertia moment with a period of about 13 h. At the same time, the comet precesses around the angular momentum vector with a period of about 6.35 h. While the rotating period of such a body would be about 12.4 h, RA/Dec coordinates of the spin axis would have a periodicity of about 270 h as a result of the combination of the two aforementioned motions.

Rotation of slender swimmers in isotropic-drag media.

The drag anisotropy of slender filaments is a critical physical property allowing swimming in low-Reynolds number flows, and without it linear translation is impossible. Here we show that, in contrast, net rotation can occur under isotropic drag. We first demonstrate this result formally by considering the consequences of the force- and torque-free conditions on swimming bodies and we then illustrate it with two examples (a simple swimmers made of three rods and a model bacterium with two helical flagellar filaments). Our results highlight the different role of hydrodynamic forces in generating translational versus rotational propulsion.

Speed of a swimming sheet in Newtonian and viscoelastic fluids

We measure the swimming speed of a cylindrical version of Taylor's swimming sheet in viscoelastic fluids, and find that depending on the rheology, the speed can either increase or decrease relative to the speed in a Newtonian viscous fluid. The swimming stroke of the sheet is a prescribed propagating wave that travels along the sheet in the azimuthal direction. The measurements are performed with the sheet immersed in a fluid inside a cylindrical tank under torque-free conditions. Swimming speeds in the Newtonian case are found to be consistent with calculations using the Stokes equation. A faster swimming speed is found in a viscoelastic fluid that has a viscosity independent of shear rate. By contrast, a slower swimming speed is found with more complex shear-thinning viscoelastic fluids which have multiple relaxation time scales as well. These results are compared with calculations with Oldroyd-B fluids which find a decreasing swimming speed with Deborah number given by the product of the fluid elastic relaxation time scale and the driving frequency.

A Fractional Step Immersed Boundary Method for Stokes Flow with an Inextensible Interface Enclosing a Solid Particle

In this paper, we develop a fractional step method based on the immersed boundary (IB) formulation for Stokes flow with an inextensible (incompressible) interface enclosing a solid particle. In addition to solving for the fluid variables such as the velocity and pressure, the present problem involves finding an extra unknown elastic tension such that the surface divergence of the velocity is zero along the interface, and an extra unknown particle surface force such that the velocity satisfies the no-slip boundary condition along the particle surface. While the interface moves with local fluid velocity, the enclosed particle hereby undergoes a rigid body motion, and the system is closed by the force-free and torque-free conditions along the particle surface. The equations are then discretized by standard centered difference schemes on a staggered grid, and the interactions between the interface and particle with the fluid are discretized using a discrete delta function as in the IB method. The resultant linear system of equations is symmetric and can be solved by fractional steps so that only fast Poisson solvers are involved. The present method can be extended to Navier--Stokes flow with moderate Reynolds number by treating the nonlinear advection terms explicitly for the time integration. The convergent tests for a Stokes solver with or without an inextensible interface are performed and confirm the desired accuracy. The tank-treading to tumbling motion for an inextensible interface enclosing a solid particle with different filling fractions under a simple shear flow has been studied extensively, and the results here are in good agreement with those obtained in literature. (A corrected version is attached.)

A two-sphere model for bacteria swimming near solid surfaces

We present a simple model for bacteria like Escherichia coli swimming near solid surfaces. It consists of two spheres of different radii connected by a dragless rod. The effect of the flagella is taken into account by imposing a force on the tail sphere and opposite torques exerted by the rod over the spheres. The hydrodynamic forces and torques on the spheres are computed by considering separately the interaction of a single sphere with the surface and with the flow produced by the other sphere. Numerically, we solve the linear system which contains the geometrical constraints and the force-free and torque-free conditions. The dynamics of this swimmer near a solid boundary is very rich, showing three different behaviors depending on the initial conditions: (1) swimming in circles in contact with the wall, (2) swimming in circles at a finite distance from the wall, and (3) swimming away from it. Furthermore, the order of magnitude of the radius of curvature for the circular motion is in the range 8-50μm, close to values observed experimentally.

Effect of inertia on the hydrodynamic interaction between two liquid capsules in simple shear flow

Three-dimensional numerical simulations using front-tracking method are performed to study the hydrodynamic interaction between two liquid capsules suspended in simple shear flow in presence of inertia. Capsules are modeled as liquid drops surrounded by neo-Hookean elastic membranes. In the limit of zero inertia, it has been known from past research that the hydrodynamic interaction between two deformable particles (drops/capsules) suspended in shear flow results in an irreversible shift in the trajectories of the particles as one particle rolls over the other. In this article, we show that the presence of inertia can significantly alter the capsule trajectories. When inertia is small but finite, the capsules do undergo an irreversible displacement, but the lateral separation between them first decreases before they roll over each other, unlike in Re ≪ 1. For moderate to high inertia, the capsules reverse their directions of motion before coming close to each other. The reversal of motion occurs progressively earlier in time (that is, the capsules come less closer to each other) with increasing inertia. The long-time behavior of the capsule–capsule interaction at finite inertia showed that the capsules engage in spiraling motions. Based on our simulations, four different regimes of capsule–capsule interaction at finite inertia are identified: (i) a self-diffusive type interaction, (ii) an outwardly spiraling motion, (iii) a fixed-orbit spiraling motion, and (iv) an inwardly spiraling motion in which the capsules settle with zero relative velocity. The reversal of motion, and the spiraling trajectories at finite inertia have no analogy in the limit of zero inertia. Such motions are explained by analyzing the flow field around a deformed capsule which shows reverse flow regions and off-surface stagnation points, similar to those previously reported in case of rigid spheres and cylinders under torque-free condition.

Chaotic rotations of a liquid-filled solid

The disturbed Hamiltonian equations of a solid filled with a rotating ellipsoidal mass of a liquid and subjected to small-applied moments are revisited using Deprit's variables. We investigate the chaotic dynamics of the orbiting liquid-filled solid and of the liquid-filled solid sliding and rolling on a perfectly smooth plane, in either energy-conservative or energy-dissipative conditions, when appropriately perturbed. Criteria for the judgment of potential chaotic rotations of the perturbed system are formulated by means of Melnikov–Holmes–Marsden (MHM) integrals. Strategies for the solution of heteroclinic orbits of the symmetrical liquid-filled solid under torque-free conditions are outlined theoretically. Physical parameters that will probably trigger the onset of chaotic motions can be determined accordingly. Results from MHM algorithms are crosschecked with Poincare sections together with Lyapunov characteristic exponents.

Effect of free rotation on the motion of a solid sphere in linear shear flow at moderate Re

The effect of free rotation on the drag and lift forces on a solid sphere in unbounded linear shear flow is investigated. The sphere Reynolds number, Re=|ur|d/ν, is in the range 0.5–200, where ur is the slip velocity. Direct numerical simulations of three-dimensional flow past an isolated sphere are performed using spectral methods. The sphere is allowed to rotate and translate freely in the shear flow in response to the hydrodynamic forces and torque acting on it. The effect of free rotation is studied in a systematic way by considering three sets of simulations. In the first set of simulations, we study how fast a pure rotational or translational motion of the sphere approaches steady state. The “history” effect of rotational and translational motions are compared. Results at high Re are found to be significantly different from the analytical prediction based on low Re theory. In steady simulations, the sphere is allowed to rotate in a torque-free condition. The torque-free rotation rate and the drag and lift forces under such a condition are reported. Comparisons are drawn with the forces on a nonrotating sphere. The effect of rotation is observed to be high in the range 5≲Re≲100. The total lift force is shown to be the sum of the lift force on a nonrotating sphere in shear flow and the lift force on a sphere that is forced to spin at the torque-free rotation rate in a uniform stream (Magnus effect). Finally, we consider the effect of combined free rotation and translation. It is observed that under such combined motion, the sphere achieves translational equilibrium with the local fluid much earlier than it can achieve the zero torque state. The sphere rotates at a rate much lower than the torque-free rotation rate. Free rotation is shown to have a negligible effect on the unsteady drag and lift forces.

Asymmetric flows of spherical particles in a cylindrical tube

To study the rheological behavior of blood cells in various flow patterns through narrow vessels, we analyzed numerically the motion of blood cells arranged in one row or two rows in tube flow, at low Reynolds numbers. The particles are assumed to be identical rigid spheres placed periodically along the vessel axis at off-axis positions with equal spacings. The flow field of the suspending fluid in a circular cylindrical tube is analyzed by a finite element method applied to the Stokes equations, and the motion of each particle is simultaneously determined by a force-free and torque-free condition. In both cases of single- and two-file arrangements of the particles, their longitudinal and angular velocities are largely affected by the radial position and the axial spacing between neighboring particles. The apparent viscosity of the asymmetric flows is higher than that of the symmetric flow where particles are located on the tube centerline, and this is more pronounced when particles are placed farther from the tube centerline and when the axial distance between neighboring particles is reduced.

Stability of large damped flexible spacecraft with stored angular momentum

Vibrational stability of large flexible structurally damped spacecraft carrying internal angular momentum and undergoing large rigid body rotations is analysed modeling the systems as elastic continua. Initially, analytical solutions to the motion of rigid gyrostats under torque-free conditions are developed. The solutions to the gyrostats modeled as axisymmetric and triaxial spacecraft carrying three and two constant speed momentum wheels, respectively, with spin axes aligned with body principal axes are shown to be complicated. These represent extensions of solutions for simpler cases existing in the literature. Using these solutions and modal analysis, the vibrational equations are reduced to linear ordinary differential equations. Equations with periodically varying coefficients are analysed applying Floquet theory. Study of a few typical beam- and plate-like spacecraft configurations indicate that the introduction of a single reaction wheel into an axisymmetric satellite does not alter the stability criterion. However, introduction of constant speed rotors deteriorates vibrational stability. Effects of structural damping and vehicle inertia ratio are also studied.

Stability of large flexible damped spacecraft modeled as elastic continua

Vibrational stability of a large flexible, structurally damped spacecraft subject to large rigid body rotations is analysed modelling the system as an elastic continuum. Using solution of rigid body attitude motion under torque free conditions and modal analysis, the vibrational equations are reduced to ordinary differential equations with time-varying coefficients. Stability analysis is carried out using Floquet theory and Sonin-Polya theorem. The cases of spinning and non-spinning spacecraft idealized as a flexible beam plate undergoing simple structural vibration are analysed in detail. The critical damping required for stabilization is shown to be a function of the spacecraft's inertia ratio and the level of disturbance.

The decay of torque free toroidal magnetic fields confined beneath the solar convection zone

Calculation similar to those of Mestel and Moss (1983) are performed to investigate the decay of a toroidal field through configurations satisfying the torque free condition, imposed by the presence of a poloidal field of dipolar form confined beneath the solar convection zone. It is found that initially stable field configurations diffuse into unstable configurations on time-scales of order a few x 108 yr. The results are similar to those of Tayler (1982) for a simpler field model without any dynamical constraints.