Hydrostatic equilibrium

About: Hydrostatic equilibrium is a research topic. Over the lifetime, 2451 publications have been published within this topic receiving 62172 citations.

The Toroidal Iron Atmosphere of a Protoneutron Star: Numerical Solution

Abstract: A numerical method presented by Imshennik et al. (2002) is used to solve the two-dimensional axisymmetric hydrodynamic problem on the formation of a toroidal atmosphere during the collapse of an iron stellar core and outer stellar layers. An evolutionary model from Boyes et al. (1999) with a total mass of 25M⊙ is used as the initial data for the distribution of thermodynamic quantities in the outer shells of a high-mass star. Our computational region includes the outer part of the iron core (without its central part with a mass of 1M⊙ that forms the embryo of a protoneutron star at the preceding stage of the collapse) and the silicon and carbon-oxygen shells with a total mass of (1.8–2.5)M⊙. We analyze in detail the results of three calculations in which the difference mesh and the location of the inner boundary of the computational region are varied. In the initial data, we roughly specify an angular velocity distribution that is actually justified by the final result—the formation of a hydrostatic equilibrium toroidal atmosphere with reasonable total mass, Mtot=(0.117–0.122)M⊙, and total angular momentum, Jtot=(0.445–0.472)×1050 erg s, for the two main calculations. We compare the numerical solution with our previous analytical solution in the form of toroidal atmospheres (Imshennik and Manukovskii 2000). This comparison indicates that they are identical if we take into account the more general and complex equation of state with a nonzero temperature and self-gravitation effects in the atmosphere. Our numerical calculations, first, prove the stability of toroidal atmospheres on characteristic hydrodynamic time scales and, second, show the possibility of sporadic fragmentation of these atmospheres even after a hydrodynamic equilibrium is established. The calculations were carried out under the assumption of equatorial symmetry of the problem and up to relatively long time scales (∼10 s).

Computer Simulation of Hydrostatic Co-Extrusion of Bimetallic Compounds.

Abstract: 2.2 .1 Analytical Approaches ............................................................... 50 2 .2 .1 .1 Energy Method .............................................................. 50 2 .2 .1 .2 Slab A n a l y s i s .............................................................. 53 2 .2 .1 .3 Lower Bound Analysis ............................................. 54 2 .2 .1 .4 Upper Bound Analysis ............................................. 54 2 .2 .2 Numerical Approaches .................................................................... 55 CHAPTER 3 FINITE ELEMENT ANALYSIS OF METAL FORMING PROCESSES . . 57 3.1 I n t r o d u c t i o n ................................................................................................ 57 3.2 T h e o r y ............................................................................................................ 59 3.2 .1 Basic M e c h a n i c s .......................................................................... . 60 3 .2 .1 .1 Displacements .............................................................. 60 3 .2 .1 .2 S t ra in Measures ............................................................ 62 3 .2 .1 .3 S t ra in Rates ................................................................... 67 3 .2 .1 .4 S t ress Measures and S t ra in Rates ...................... 68 3 .2 .2 Governing Equations fo r E la s to -p la s t i c D e f o r m a t i o n .................................................................................... 70 3 .2 .3 Const i tu t ive Equations fo r E la s to -P la s t i c D e f o r m a t i o n .................................................................................... 72 3.3 The F in i te Element Program ..................................................................... 74 3.4 Simulation Parameters ............................................................................... 78 CHAPTER 4 HYDROSTATIC EXTRUSION EXPERIMENTS ...................................... 81 4.1 I n t r o d u c t i o n ................................................................................................ 81 4.2 Hydrostatic Extrusion Experiments ...................................................... 81 4.2 .1 Experimental F a c i l i t i e s .......................................................... 81 4 .2 .2 Experimental Procedure .............................................................. 83 4.3 Residual S t ress Measurements ............................................................... 86 4 .3 .1 Residual S t resses ....................................................................... 86 4 .3 .2 Sachs' Boring-out Technique .................................................. 87 4 .3 .2 .1 Radial S t resses ........................................................... 87 4 .3 .2 .2 Tangential S t resses ................................................... 89 4 .3 .2 .3 Longitudinal S tresses ............................................... 91 4 .3 .3 Electro-chemical Machining ...................................................... 92 4.4 Experimental S t ress Measurements ...................................................... 96 4.4 .1 Experimental F a c i l i t i e s and Materia ls ............................. 96 4 .4 .2 Sample Preparat ion ....................................................................... 99 4 .4 .3 Cal ib ra t ion o f Electro-chemical Machining .................... 100 4 .4 .4 Experimental Procedure ............................................................... 101

Size and shape of a celestial body, definition of a planet

Abstract: Experience shows that celestial bodies have a nearly round shape only from a certain size. At IAU resolution B5, item b, that shape serves as an indicator for a distinct mechanism of its forming, caused by a minimum of mass. Rigid body forces should have been overcome by self gravity and a hydrostatic equilibrium shape should have been achieved. A new approach to correlate the size and shape of a solid in hydrostatic equilibrium by balancing self gravity and rigid body forces leads to a real, not to an arbitrary lower limit of size. No arbitrary criterion is required as it is often the case. Above this limit the shape of a solid body is restricted by a maximum of its surface area, and this maximum vanishes only at infinite size. Therefore the shape like that of a related fluid in hydrostatic equilibrium can only be reached in the solid state, if its surface area is larger than this maximum. This applies only to the four giant planets and Haumea. All the known nearly round celestial bodies in the solar System cant have achieved their present shape while being solids. The dwarf Haumea is a questionable exception. The present shapes have been formed before solidification and are frozen. The gas giants are not solids at all and the terrestrial planets have at least partially melted in their thermal history for millions of years. The smaller nearly round objects such as asteroids and satellites were at least partially melted combined with internal differentiation and resurfacing of a mechanically unstable crust in their very early thermal history. No rigid body forces had to be overcome. Item b of the current planet definition is somewhat undetermined. All planets and dwarfs in the solar system match this requirement only at very generous interpretation. It should be completely deleted. The number of dwarfs could be restricted by an arbitrary minimum of mass.