On a toroidal pendulum

We consider the problem of motion of a heavy particle on the surface of a torus with horizontal axis of rotation.     

Two-phase natural circulation in a toroidal loop

This is an experimental study of the overall behaviour of two-phase flow and heat transfer in a closed toroidal loop. Heat input and output are at opposite ends of a diameter. The magnitude of the heat input and the inclination of the loop are the two independent parameters whose influence is determined. Three different regions of operation with phase change are observed. Boiling at the evaporator section can be without bubbling, or with bubbling but with or without liquid carry over. The efficiency of the system as a heat transport device is determined and found to be better under liquid carry over conditions. A simple theoretical model of bubble formation and rise is also discussed.     

Experimental investigation of fluid convection in a closed toroidal channel

The results of investigating convective fluid motion in a closed toroidal channel are analyzed and compared with the results of calculations based on the Lorenz model. It is shown that this model adequately describes the fluid convection only in the case of a large aspect ratio of the mean radius of the torus to its cross-sectional dimension and small heating power. The measurements and visual observations record substantial deviations from the assumptions underlying the Lorenz model in the domain of problem parameters in which, in accordance with the calculations, bifurcations and instability of steady solutions of the Lorenz model should take place.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 20–28, July–August, 1995.     

Inflation of a bonded non-linear viscoelastic toroidal membrane

The inflation of a bonded viscoelastic toroidal membrane under finite deformations is considered. Three new variables, viz. the two principal stretch ratios and the angle between the normal vector of a deformed membrane and the axis of symmetry are introduced as dependent variables. The governing equations are reduced thereafter to a set of three first-order partial differential integral equations. The constitutive equation developed by Pipkin and Rogers for the non-linear response of a viscoelastic material is used. The creep phenomenon for an inflated viscoelastic toroidal membrane under a constant pressure is presented.     

Conditions of toroidal equilibrium of a moving plasma

Necessary conditions for stationary plasma flow in a magnetic field of toroidal geometry are derived in a magnetohydrodynamic approximation.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 31–35, January–February, 1976.     

Numerical investigation of toroidal shock wave focusing in a cylindrical chamber

In this paper, focusing of a toroidal shock wave propagating from an annular shock tube into a cylindrical chamber was investigated numerically with the dispersion controlled dissipation (DCD) scheme. The first case for an incident Mach number of 1.5 was conducted and compared with experiments for validation. Then, several cases were calculated for higher incident Mach numbers varying from 2.0 to 5.0, and complicated flow structures were observed. The numerical study was mainly focused on two aspects: focusing process and flow structures. The process, including diffraction, focusing, and reflection, is displayed to reveal the focusing mechanism, and the flow structures at different incident. Mach numbers are used to demonstrate shock reflection styles and focusing characteristics. PACS 47.40.Ki; 47.40.Nm; 52.35.Tc     

Limit points in the free inflation of a magnetoelastic toroidal membrane

One common phenomenon native to inflation of membranes is the elastic limit-point instability—a bifurcation point at which the membrane begins to deform enormously at the slightest increase of pressure. In the case of magnetoelastic materials, there is another possible phenomenon which we call magnetic limit-point instability, a state referring to the non-existence of an equilibrium state – either stable or unstable. In this work, we are concerned with such instabilities in an incompressible isotropic magnetoelastic toroidal membrane with an initial circular cross-section. A non-uniform magnetic field is generated using a circular current carrying loop placed inside the membrane in addition to inflation by a uniform hydrostatic pressure. An energy formulation based on magnetization is used to model the magneto-mechanical coupling along with a Mooney–Rivlin constitutive model for the elastic strain energy density. Computations show that the magnetic field strongly influences the location of elastic limit points and in some cases can cause them to vanish. Multiple equilibrium states are obtained as solutions of the governing equations and a criterion based on second variation is employed to determine their stability. Existence and dependence of magnetic limit point on the magnetic field is demonstrated. While the quantitative results obtained here are specific to the toroidal geometry, the deformation behaviour can be generalized to any magnetoelastic membrane.     

Unsteady natural convection in a toroidal vessel of arbitrary cross section

Unsteady natural convection in toroidal vessels, whose meridional cross section is a quadrangle with arbitrary curvilinear boundaries, is considered. The gravity vector is parallel to the axis of symmetry. The Navier-Stokes and energy equations are reduced to a universal form that describes the convective processes in both the plane and axisymmetric cases. A numerical solution of the problem is obtained on the basis of a multistep implicit finite-difference method. The unsteady natural convection in a toroidal vessel whose cross section is formed by segments of straight lines is calculated. The need to investigate natural convection processes in vessels of toroidal shape has arisen in connection with the expansion of design work on solar collectors and vessels used in cryogenic engineering. An experimentally based system of similarity criteria, which makes it possible to estimate the principal thermodynamic characteristics of the convection process in a toroidal reservoir partially filled with cryogenic fluid, was constructed in [1, 2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. A, pp. 16–22, July–August, 1990.     

Finite inflation of a hyperelastic toroidal membrane over a cylindrical rim

The present paper is devoted to the study of finite inflation of a hyperelastic toroidal membrane on a cylindrical rim under uniform internal pressure. Both compliant and rigid frictionless rims have been considered. The compliant cylindrical rim is modeled as a linear distributed stiffness. The initial cross-section of the torus is assumed to be circular, and the membrane material is assumed to be a homogeneous and isotropic Mooney–Rivlin solid. The problem is formulated as a two point boundary value problem and solved using a shooting method by employing the Nelder–Meads search technique. The optimization function is constructed on a two (three) dimensional search space for the compliant cylinder (rigid cylinder). The effect of the inflation pressure, material properties and elastic properties of the rim on the state of stretch and stress, and on the geometry of the inflated torus have been studied, and some interesting results have been obtained. The stability of the inflated configurations in terms of occurrence of the impending wrinkling state in the membrane has also been studied.     

Numerical simulation of magnetohydrodynamic flow in a toroidal duct of square cross-section

We present numerical simulation results of the quasi-static magnetohydrodynamic (MHD) flow in a toroidal duct of square cross-section with insulating Hartmann walls and conducting side walls. Both laminar and turbulent flows are considered. In the case of steady flows, we present a comprehensive analysis of the secondary flow. It consists of two counter-rotating vortex cells, with additional side wall vortices emerging at sufficiently high Hartmann number. Our results agree well with existing asymptotic analysis. In the turbulent regime, we make a comparison between hydrodynamic and MHD flows. We find that the curvature induces an asymmetry between the inner and outer side of the duct, with higher turbulence intensities occurring at the outer side wall. The magnetic field is seen to stabilize the flow so that only the outer side layer remains unstable. These features are illustrated both by a study of statistically averaged quantities and by a visualization of (instantaneous) coherent vortices.     

Axisymmetric equilibrium shapes of a fluid in a toroidal vessel under conditions of weightlessness

The axisymmetric problem of the evolution of the free surface of a fluid during the filling of a toroidal vessel under conditions of weightlessness is considered. Despite its interest [1, 2], this topical problem of the hydrodynamics of weightlessness remains unsolved for lack of an effective method of solution. This paper employs the iteration-difference approach proposed in [3–5] for calculating simply and doubly connected axisymmetric equilibrium figures. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 154–156, November–December, 1986.