scholarly journals Latitudinal libration driven flows in triaxial ellipsoids

2015 ◽  
Vol 771 ◽  
pp. 193-228 ◽  
Author(s):  
S. Vantieghem ◽  
D. Cébron ◽  
J. Noir
Keyword(s):  
Numerical Simulations ◽  
Equations Of Motion ◽  
Liquid Core ◽  
Three Dimensional ◽  
Base Flow ◽  
Upper And Lower Bounds ◽  
Earth Planet ◽  
Ekman Layers ◽  
Core Dynamics ◽  
Inertial Instability

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 Chan et al. (Phys. Earth Planet. Inter., vol. 187, 2011, pp. 404–415) and Zhang et 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. Fluids A, 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.

scholarly journals Ion Confinement in a Toroidal Heliac

1998 ◽  
Vol 51 (1) ◽  
pp. 67
Author(s):  
J. L. V. Lewandowski
Keyword(s):  
Numerical Simulations ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Larmor Radius ◽  
Collision Term ◽  
Magnetic Topology ◽  
Finite Larmor Radius ◽  
Confinement Mode

A model to describe an unmagnetised plasma in three-dimensional magnetic topology is presented. Ion trajectories are integrated numerically and all finite-Larmor radius effects are retained exactly. A velocity-dependent collision term is included in the equations of motion. Numerical simulations relevant to the low-confinement mode of H1-NF are presented and discussed.

Proposed Solution Methodology for the Dynamically Coupled Nonlinear Geared Rotor Mechanics Equations

1985 ◽  
Vol 107 (1) ◽  
pp. 112-116 ◽  
Author(s):  
L. D. Mitchell ◽  
J. W. David
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Three Dimensional ◽  
Dynamic Coupling ◽  
State Response ◽  
Shaft System ◽  
Coupling Terms ◽  
Solution Methodology

The equations which describe the three-dimensional motion of an unbalanced rigid disk in a shaft system are nonlinear and contain dynamic-coupling terms. Traditionally, investigators have used an order analysis to justify ignoring the nonlinear terms in the equations of motion, producing a set of linear equations. This paper will show that, when gears are included in such a rotor system, the nonlinear dynamic-coupling terms are potentially as large as the linear terms. Because of this, one must attempt to solve the nonlinear rotor mechanics equations. A solution methodology is investigated to obtain approximate steady-state solutions to these equations. As an example of the use of the technique, a simpler set of equations is solved and the results compared to numerical simulations. These equations represent the forced, steady-state response of a spring-supported pendulum. These equations were chosen because they contain the type of nonlinear terms found in the dynamically-coupled nonlinear rotor equations. The numerical simulations indicate this method is reasonably accurate even when the nonlinearities are large.

On the energy instability of liquid crystals

1998 ◽  
Vol 9 (1) ◽  
pp. 23-36
Author(s):  
V. A. VLADIMIROV ◽  
K. I. ILIN
Keyword(s):  
Growth Rate ◽  
Initial Data ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Upper And Lower Bounds ◽  
Direct Lyapunov Method ◽  
Distortion Energy ◽  
The Stability ◽  
Energy Instability ◽  
General Equilibria

The direct Lyapunov method is used to investigate the stability of general equilibria of a nematic liquid crystal. First, we prove the converse Lagrange theorem stating that an equilibrium is unstable to small perturbations if the distortion energy has no minimum at this equilibrium (i.e. if the second variation of the distortion energy evaluated at the equilibrium is not positive definite). The proof is constructive rather than abstract: we explicitly construct a functional that grows exponentially with time by virtue of linearized equations of motion provided the condition of the theorem is satisfied. We obtain an explicit formula that gives the dependence of the perturbation growth rate upon the equilibrium considered and the initial data for the perturbation. Secondly, we obtain the upper and lower bounds for growing solutions of the linearized problem, and we identify the initial data corresponding to the most unstable mode (i.e. to the perturbation with maximal growth rate). All results are obtained in quite a general formulation: a nematic is inside a three-dimensional domain of an arbitrary shape and strong anchoring on the boundary is supposed; the standard equations of nematodynamics are employed as the governing equations.

Three-dimensional instability of viscoelastic elliptic vortices

1997 ◽  
Vol 353 ◽  
pp. 357-381 ◽  
Author(s):  
H. HAJ-HARIRI ◽  
G. M. HOMSY
Keyword(s):  
Plane Waves ◽  
Three Dimensional ◽  
Deborah Number ◽  
Base Flow ◽  
Necessary Condition ◽  
Inertial Instability ◽  
Viscoelastic Instability ◽  
Dimensional Instability ◽  
Model Oscillator ◽  
Vortex Axis

An analysis of the three-dimensional instability of two-dimensional viscoelastic elliptical flows is presented, extending the inviscid analysis of Bayly (1986) to include both viscous and elastic effects. The problem is governed by three parameters: E, a geometric parameter related to the ellipticity; Re, a wavenumber-based Reynolds number; and De, the Deborah number based on the period of the base flow. New modes and mechanisms of instability are discovered. The flow is generally susceptible to instabilities in the form of propagating plane waves with a rotating wavevector, the tip of which traces an ellipse of the same eccentricity as the flow, but with the major and minor axes interchanged. Whereas a necessary condition for purely inertial instability is that the wavevector has a non-vanishing component along the vortex axis, the viscoelastic modes of instability are most prominent when their wavevectors do vanish along this axis. Our analytical and numerical results delineate the region of parameter space of (E, ReDe) for which the new instability exists. A simple model oscillator equation of the Mathieu type is developed and shown to embody the essential qualitative and quantitative features of the secular viscoelastic instability. The cause of the instability is a buckling of the ‘compressed’ polymers as they are perturbed transversely during a particular phase of the passage of the rotating plane wave.

Validation of a Belt Model for Prediction of Hub Forces from a Rolling Tire4

2009 ◽  
Vol 37 (2) ◽  
pp. 62-102 ◽  
Author(s):  
C. Lecomte ◽  
W. R. Graham ◽  
D. J. O’Boy
Keyword(s):  
Internal Pressure ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Prediction Method ◽  
Analytical Models ◽  
Beam Model ◽  
Impedance Boundary ◽  
Bending Waves ◽  
Tire Rotation ◽  
Three Dimensional Models

Abstract An integrated model is under development which will be able to predict the interior noise due to the vibrations of a rolling tire structurally transmitted to the hub of a vehicle. Here, the tire belt model used as part of this prediction method is first briefly presented and discussed, and it is then compared to other models available in the literature. This component will be linked to the tread blocks through normal and tangential forces and to the sidewalls through impedance boundary conditions. The tire belt is modeled as an orthotropic cylindrical ring of negligible thickness with rotational effects, internal pressure, and prestresses included. The associated equations of motion are derived by a variational approach and are investigated for both unforced and forced motions. The model supports extensional and bending waves, which are believed to be the important features to correctly predict the hub forces in the midfrequency (50–500 Hz) range of interest. The predicted waves and forced responses of a benchmark structure are compared to the predictions of several alternative analytical models: two three dimensional models that can support multiple isotropic layers, one of these models include curvature and the other one is flat; a one-dimensional beam model which does not consider axial variations; and several shell models. Finally, the effects of internal pressure, prestress, curvature, and tire rotation on free waves are discussed.

scholarly journals Development and Validation of Quasi-Eulerian Mean Three-Dimensional Equations of Motion Using the Generalized Lagrangian Mean Method

2021 ◽  
Vol 9 (1) ◽  
pp. 76
Author(s):  
Duoc Nguyen ◽  
Niels Jacobsen ◽  
Dano Roelvink
Keyword(s):  
Surface Waves ◽  
Deep Water ◽  
Surf Zone ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Breaking Waves ◽  
Successful Implementation ◽  
Random Waves ◽  
Mean Motion ◽  
Generalized Lagrangian

This study aims at developing a new set of equations of mean motion in the presence of surface waves, which is practically applicable from deep water to the coastal zone, estuaries, and outflow areas. The generalized Lagrangian mean (GLM) method is employed to derive a set of quasi-Eulerian mean three-dimensional equations of motion, where effects of the waves are included through source terms. The obtained equations are expressed to the second-order of wave amplitude. Whereas the classical Eulerian-mean equations of motion are only applicable below the wave trough, the new equations are valid until the mean water surface even in the presence of finite-amplitude surface waves. A two-dimensional numerical model (2DV model) is developed to validate the new set of equations of motion. The 2DV model passes the test of steady monochromatic waves propagating over a slope without dissipation (adiabatic condition). This is a primary test for equations of mean motion with a known analytical solution. In addition to this, experimental data for the interaction between random waves and a mean current in both non-breaking and breaking waves are employed to validate the 2DV model. As shown by this successful implementation and validation, the implementation of these equations in any 3D model code is straightforward and may be expected to provide consistent results from deep water to the surf zone, under both weak and strong ambient currents.

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