Axisymmetric Ellipsoids Research Articles

Precession-driven flows in stress-free ellipsoids

Motivated by modelling rotating turbulence in planetary fluid layers, we investigate precession-driven flows in ellipsoids subject to stress-free boundary conditions (SF-BC). The SF-BC could indeed unlock numerical constraints associated with the no-slip boundary conditions (NS-BC), but are also relevant for some astrophysical applications. Although SF-BC have been employed in the pioneering work of Lorenzani & Tilgner (J. Fluid Mech., vol. 492, 2003, pp. 363–379), they have scarcely been used due to the discovery of some specific mathematical issues associated with angular momentum conservation. We revisit the problem using asymptotic analysis in the low-viscosity regime, which is validated with numerical simulations. First, we extend the reduced model of uniform-vorticity flows in ellipsoids to account for SF-BC. We show that the long-term evolution of angular momentum is affected by viscosity in triaxial geometries, but also in axisymmetric ellipsoids when the mean rotation axis of the fluid is not the symmetry axis. In a regime relevant to planets, we analytically obtain the primary forced flow in triaxial geometries, which exhibits a second inviscid resonance. Then, we investigate the bulk instabilities existing in precessing ellipsoids. We show that using SF-BC would be useful to explore the non-viscous instabilities (e.g. Kerswell, Geophys. Astrophys. Fluid Dyn., vol. 72, 1993, pp. 107–144), which are presumably relevant for planetary applications but are often hampered in experiments or simulations with NS-BC.

Anisotropy of random motions of gas in Messier 33

Context. The ellipsoid of random motions of the gaseous medium in galactic disks is often considered isotropic, as appropriate if the gas is highly collisional. However, the collisional or collisionless behavior of the gas is a subject of debate. If the gas is clumpy with a low collision rate, then the often observed asymmetries in the gas velocity dispersion could be hints of anisotropic motions in a gaseous collisionless medium. Aims. We study the properties of anisotropic and axisymmetric velocity ellipsoids from maps of the gas velocity dispersion in nearby galaxies. This data allow us to measure the azimuthal-to-radial axis ratio of gas velocity ellipsoids, which is a useful tool to study the structure of gaseous orbits in the disk. We also present the first estimates of perturbations in gas velocity dispersion maps by applying an alternative model that considers isotropic and asymmetric random motions. Methods. High-quality velocity dispersion maps of the atomic medium at various angular resolutions of the nearby spiral galaxy Messier 33, are used to test the anisotropic and isotropic velocity models. The velocity dispersions of hundreds of individual molecular clouds are also analyzed. Results. The HI velocity dispersion of M 33 is systematically larger along the minor axis, and lower along the major axis. Isotropy is only possible if asymmetric motions are considered. Fourier transforms of the H I velocity dispersions reveal a bisymmetric mode which is mostly stronger than other asymmetric motions and aligned with the minor axis of the galaxy. Within the anisotropic and axisymmetric velocity model, the stronger bisymmetry is explained by a radial component that is larger than the azimuthal component of the ellipsoid of random motions, thus by gaseous orbits that are dominantly radial. The azimuthal anisotropy parameter is not strongly dependent on the choice of the vertical dispersion. The velocity anisotropy parameter of the molecular clouds is observed highly scattered. Conclusions. Perturbations such as HI spiral-like arms could be at the origin of the gas velocity anisotropy in M 33. Further work is necessary to assess whether anisotropic velocity ellispsoids can also be invoked to explain the asymmetric gas random motions of other galaxies.

Drag coefficient and averaged Nusselt number of a scalene prolate ellipsoid

• Momentum and heat transfer characteristics of a scalene prolate ellipsoid were studied. • Effects of particle shape, orientation and Reynolds number were discussed. • Existing correlations for the drag coefficient and averaged Nu were improved. A cold fluid flowing past a hot stationary scalene prolate ellipsoid is numerically studied via a three-dimensional immersed boundary-lattice Boltzmann model (IB-LBM). First, axisymmetric ellipsoids are considered for verification purposes, and good agreement is found between obtained drag coefficient ( C d ) and averaged Nusselt number ( Nu ) and previously reported results. Subsequently, 81 case studies are conducted by changing shape (2 ≤ Ar 1, Ar 2 ≤ 6) and incident angle (0° ≤ θ ≤ 90°) of the ellipsoid as well as the Reynolds number (25 ≤ Re ≤ 200). In this study, we particularly consider scalene prolate ellipsoids with unequal Ar 1 and Ar 2, which have not been studied in detail before. Effects of different factors on momentum and heat transfer characteristics between solid and fluid phases are understood based on the analysis of numerical results. Available formulas reported in the literature are also briefly reviewed. Finally, new correlations for both C d and Nu are developed using the most important factors with average deviations of 3.875% for C d and 1.598% for Nu , respectively. These proposed correlations can be used to facilitate the phase coupling in the multi-phase flow modelling.

Anisotropic Particles in Turbulence

Anisotropic particles are common in many industrial and natural turbulent flows. When these particles are small and neutrally buoyant, they follow Lagrangian trajectories while exhibiting rich orientational dynamics from the coupling of their rotation to the velocity gradients of the turbulence field. This system has proven to be a fascinating application of the fundamental properties of velocity gradients in turbulence. When particles are not neutrally buoyant, they experience preferential concentration and very different preferential alignment than neutrally buoyant tracer particles. A vast proportion of the parameter range of anisotropic particles in turbulence is still unexplored, with most existing research focusing on the simple foundational cases of axisymmetric ellipsoids at low concentrations in homogeneous isotropic turbulence and in turbulent channel flow. Numerical simulations and experiments have recently developed a fairly comprehensive picture of alignment and rotation in these cases, and they provide an essential foundation for addressing more complex problems of practical importance. Macroscopic effects of nonspherical particle dynamics include preferential concentration in coherent structures and drag reduction by fiber suspensions. We review the models used to describe nonspherical particle motion, along with numerical and experimental methods for measuring particle dynamics.

Characterization of Self-Excited, Nearly Axisymmetric, Spinning Rigid-Body Motion as an Oblate Epicycloid

A unique formulation of the solution to a spinning, nearly axisymmetric rigid-body is presented. Direct integration of the linearized equations of motion gives accurate results for nearly axisymmetric inertia ellipsoids while avoiding the complexity of more general formulations. The simplicity of the formulation lends itself to a better understanding of the system behavior. Specifically, the motion of the spin axis for this nearly axisymmetric case is described by an oblate epicycloid, providing an extension of the classic epicycloid solution for axisymmetric objects.

Measurements of the solid-body rotation of anisotropic particles in 3D turbulence

We introduce a new method to measure Lagrangian vorticity and the rotational dynamics of anisotropic particles in a turbulent fluid flow. We use 3D printing technology to fabricate crosses (two perpendicular rods) and jacks (three mutually perpendicular rods). Time-resolved measurements of their orientation and solid-body rotation rate are obtained from four video images of their motion in a turbulent flow between oscillating grids with = 91. The advected particles have a largest dimension of 6 times the Kolmogorov length, making them a good approximation to anisotropic tracer particles. Crosses rotate like disks and jacks rotate like spheres, so these measurements, combined with previous measurements of tracer rods, allow experimental study of axisymmetric ellipsoids across the full range of aspect ratios. The measured mean square tumbling rate, , confirms previous direct numerical simulations that indicate that disks tumble much more rapidly than rods. Measurements of the alignment of a unit vector defining the orientation of crosses with the direction of their solid-body rotation rate vector provide the first direct observation of the alignment of anisotropic particles by the velocity gradients in a turbulent flow.

Dynamical description of vesicle growth and shape change

We systematize and extend the description of vesicle growth and shape change using linear nonequilibrium thermodynamics. By restricting the study to shape changes from spheres to axisymmetric ellipsoids, we are able to give a consistent formulation which includes the lateral tension of the vesicle membrane. This allows us to generalize and correct a previous calculation. Our present calculations suggest that, for small growing vesicles, a prolate ellipsoidal shape should be favored over oblate ellipsoids, whereas for large growing vesicles oblates should be favored over prolates. The validity of this prediction is examined in the light of the various assumptions made in its derivation.

Wheat drying kinetics. Diffusivities for sphere and ellipsoid by finite elements

A two-dimensional finite element model was used to analyze isothermal thin layer drying of wheat representing the grains as axisymmetric ellipsoids. The effective diffusion coefficient was estimated by minimizing the sum of squares of the residuals between numerically predicted and experimental moistures. Estimations done for drying air temperatures of 35°C, 50°C, 60°C and 70°C and initial grain moistures of 0.2694, 0.2396, 0.2133 and 0.1891 d.b., were correlated by means of Arrhenius-type functions. These correlations were compared to those previously obtained with the same data by assuming grains as spheres. The results, all in the order of 10 −11 m 2/ s , indicated that the diffusion coefficients found for ellipsoidal geometry were some 9–13% lower than the values fitted for spheres. The ratio of diffusivities for ellipsoids to those for spheres was in the order of the wheat sphericity. However, further theoretical and experimental studies are required to verify this observation.

Average trapped surfaces on an arbitrary initial data set.

We study the behavior of the expansion [Theta] of an outgoing null geodesic congruence on an initial surface [Sigma]. If the intrinsic geometry (and/or topology) allows a global or local slicing of [Sigma] by two-dimensional closed surfaces, then the rate of change of [Theta] along the normal and tangential direction of each slice is derived. We find that the rate of change of [Theta] along the normal direction is governed by the initial data and also by the Gaussian, mean, and extrinsic curvature of each slice. Based on this differential equation, a formula is established that defines the total expansion along any two-dimensional slice. As it turns out, the total expansion depends upon the amount of rest mass enclosed, the amount of matter current flowing across the surface, as well as upon other geometrical characteristics of the slice. Using this formula a necessary and sufficient condition is established that guarantees whether or not a particular two-slice [ital S] is an average trapped surface. In the special case of time-symmetric initial data the criterion takes a simple and tractable form. Roughly, an average trapped surface forms when the interior rest mass [ital M] is greater than the value of an integralmore » involving only geometrical characteristics of the slice. In the special case of the spherical slices or particular slicing by axisymmetric ellipsoids, we are able to find the upper and lower bounds of the above-mentioned integral by quantities related to the size of [ital S]. Thus we are also able to establish a condition upon the amount of rest mass [ital M] needed to make [ital S] an average trapped surface.« less

Equilibrium solutions of the restricted problem of 2 + 2 axisymmetric rigid bodies

The restricted problem of 2 + 2 homogenous axisymmetric ellipsoids such that their equatorial planes coincide with the orbital plane of the centers of mass is considered. The equilibrium solutions of this problem are shown to exist. Six of these solutions are located about the collinear points of the restricted problem of three axisymmetric ellipsoids. A special case of this problem is studied and sixteen solutions are found in the neighborhood of the triangular Lagrangian points.

Integration theory for the restricted problem of three axisymmetric bodies

In this paper the circular planar restricted problem of three axisymmetric ellipsoids S i (i = 1, 2, 3), such that their equatorial planes coincide with the orbital plane of the three centres of masses, be considered. The equations of motion of infinitesimal body S 3 be obtained in the polar coordinates. Using iteration approach we have given an approximation for another integral, which independent of the Jacobian integral, in the case of P-type orbits (near circular orbits surrounding both primaries).

The equilibrium solutions of the restricted problem of three rigid bodies with variable mass

In this paper we consider the circular planar restricted problem of three rigid bodiesSi(i=1, 2, 3), two of them are axisymmetric ellipsoids and a third bodyS3 is a spherical satellite with decreasing mass, under the gravitational forces. The effect of small perturbations in the Coriolis force and the centrifugal forces on the location of equilibrium points has been studied. It is found only in the case when the primaries have equal differences between their respective principal moments of inertial the pointsL4 andL5 form nearly equilateral tringles with the primaries. The equilibrium pointsL1,L2,L3 remain collinear an ies on the line joining the primaries.

An iterative method for the boundary element solution of the exterior acoustic problem

In this paper we discuss the Burton and Miller's boundary integral formulation of the exterior Neumann problem for the Helmholtz equation, which is uniquely solvable for all frequencies as well as a similar formulation of Brakhage and Werner, Lies and also Panich, for the exterior Dirichlet problem. We present an iterative procedure for the efficient solution of the full linear systems arising from the discretisation of such boundary integral equations. Numerical results will also be presented for the Neumann problem for a class of axisymmetric ellipsoids.

Boundary-layer separation from a smooth slip surface

Numerical results are presented for the streaming motion of an incompressible Newtonian fluid past axisymmetric ellipsoids of fixed shape, subject to the condition of zero tangential stress on the body surface. It is demonstrated that a substantial, closed-streamline wake appears behind the ellipsoid at moderate Reynolds numbers, O(20–50), when the ellipsoid is sufficiently nonspherical, and that this wake initially increases in size with increase of Reynolds number but then eventually decreases in size and disappears when the Reynolds number is sufficiently large, O(200–250) in our cases. The existence of streaming flow past a smooth, slip surface without separation is consistent with predictions from laminar boundary-layer theory. It is concluded that the recirculating wake that appears behind moderately deformed bubbles is strictly a finite Reynolds number phenomenon that cannot be predicted (or understood) via a boundary-layer analysis.

On the motion of three rigid bodies: Central configurations

In the present paper, the motion of three rigid bodies is considered. With a set of new variables, and the 10 first integrals of the motion, the problem is reduced to a system of order 25 and one quadrature. The plane motions are characterized, and finally, an equation for the existence of central configurations (in particular, Lagrangian and Eulerian solutions) has been found. Besides, the case of three axisymmetric ellipsoids is studied.

On the equilibrium solutions in the circular planar restricted three rigid bodies problem

In this paper, the restricted problem of three rigid bodies under central forces is considered, and the collinear and triangular equilibrium solutions are obtained. Finally, an application to the case of axisymmetric ellipsoids is made.

Rotation of small non-axisymmetric particles in a simple shear flow

The equations for the rotation of non-axisymmetric ellipsoids in a simple shear flow at low Reynolds numbers are derived in terms of Euler angles. Numerical solutions of this third-order system of equations show a doubly periodic structure to the rotation, with a change in the general nature of the solutions when a certain planar rotation of the particle becomes unstable. Some analytic progress can be made for nearly spherical ellipsoids and for nearly axisymmetric ellipsoids. The near spheres show the same qualitative behaviour as the general ellipsoids. Quite small deviations from axial symmetry are found to produce large changes in the rotation.

Three-dimensional dynamics of protostellar evolution

view Abstract Citations (41) References (12) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Three-dimensional dynamics of protostellar evolution. Cook, T. L. ; Harlow, F. H. Abstract A three-dimensional finite-difference numerical technique is outlined for self-gravitating rotating gaseous systems. The full nonlinear time-dependent equations describing a rotating self-gravitating fluid are solved in rotating cylindrical coordinates, and the time-dependent adiabatic collapse of gravitationally bound rotating protostellar clouds is analyzed for specified uniform and nonuniform initial conditions. It is found that uniform clouds can form axisymmetric rotating toroidal configurations, that nonuniform clouds can collapse to axisymmetric ellipsoids if the thermal pressure is high, and that the collapsing cloud is unstable to perturbations for low thermal pressures. The ensuing fragmentation of unstable protostellar clouds is examined by studying the response of self-gravitating equilibrium toroids to nonaxisymmetric perturbations. The results show that the detailed evolution of a fragmenting toroid depends on a nondimensional function of the initial entropy, as well as the total mass in the toroid, the rotational angular velocity, and the number of perturbation wavelengths around the circumference of the toroid. Publication: The Astrophysical Journal Pub Date: November 1978 DOI: 10.1086/156568 Bibcode: 1978ApJ...225.1005C Keywords: Finite Difference Theory; Protostars; Stellar Evolution; Stellar Models; Chandrasekhar Equation; Dynamic Models; Gravitational Collapse; Rotating Matter; Toroids; Astrophysics; Collapse:Protostellar Clouds; Hydrodynamics:Star Formation full text sources ADS |