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Hydrostatic equilibrium
About: Hydrostatic equilibrium is a research topic. Over the lifetime, 2451 publications have been published within this topic receiving 62172 citations.
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TL;DR: In this article, the authors proved that when the depth goes to 0, the solution of the stationary Navier-Stokes equations with adherence at the bottom and traction by wind at the surface, once conveniently normalized, goes to a three-dimensional limit which is the solution for an incompressible model with vertical diffusion.
Abstract: The motion of a fluid in a lake with small depth compared to width is investigated. We prove that when the depth goes to 0, the solution of the stationary Navier--Stokes equations with adherence at the bottom and traction by wind at the surface, once conveniently normalized, goes to a three-dimensional limit which is the solution of an incompressible model with vertical diffusion. The limit velocity is given in terms of the vertical coordinate and of the limit pressure. This pressure, which depends only on the horizontal coordinates, is driven by a two-dimensional equation on the surface degenerating on the shore, which is solved in a weighted space. Thus, a three-dimensional approximation is obtained by a simple two-dimensional computation.
16 citations
TL;DR: In this article, the internal structure of Mercury is assumed to be similar to that of the earth and the internal temperature distribution in the interior is calculated from the structural model of Mercury.
Abstract: The internal structure of Mercury is assumed to be similar to that of the earth. A review of the astronomical data gives the mean radius as 2446 km and the mean density as 5.31 g/cm3. Mercury must thus have both a core and a mantle. An iterative procedure gives a core radius of 2112 km. The pressure, density, and gravity are calculated for a small mantle and core. Numerical integration of the equation of hydrostatic equilibrium yields 7.37 g/cm3 as the density and 0.327×1012 dynes/cm2 as the pressure at the center of the planet. From the structural model of Mercury the temperature distribution in the interior is calculated. Two extreme values of the temperature are calculated because of the uncertainty in the heat generation and thermal conductivity terms. The highest temperature (1380°K) occurs on the sunward side of Mercury at the core-mantle boundary. This indicates that neither the core nor the mantle is molten. Certain geophysical and astronomical effects may arise because of the peculiar temperature distribution and the synchronous rotation of Mercury about the sun.
16 citations
TL;DR: In this article, the geometry and gravity potential of the reference equipotential surface of Mars (Areoid) were computed based on a constrained optimization problem, which best fits to the shape of Mars in least squares sense.
Abstract: Using the shape model of Mars GTM090AA in terms of spherical harmonics complete to degree and order 90 and gravitational field model of Mars GGM2BC80 in terms of spherical harmonics complete to degree and order 80, both from Mars Global Surveyor (MGS) mission, the geometry (shape) and gravity potential value of reference equipotential surface of Mars (Areoid) are computed based on a constrained optimization problem. In this paper, the Areoid is defined as a reference equipotential surface, which best fits to the shape of Mars in least squares sense. The estimated gravity potential value of the Areoid from this study, i.e. W0 = (12,654,875 ± 69) (m2/s2), is used as one of the four fundamental gravity parameters of Mars namely, {W0, GM, ω, J20}, i.e. {Areoid’s gravity potential, gravitational constant of Mars, angular velocity of Mars, second zonal spherical harmonic of gravitational field expansion of Mars}, to compute a bi-axial reference ellipsoid of Somigliana-Pizzetti type as the hydrostatic approximate figure of Mars. The estimated values of semi-major and semi-minor axis of the computed reference ellipsoid of Mars are (3,395,428 ± 19) (m), and (3,377,678 ± 19) (m), respectively. Finally the computed Areoid is presented with respect to the computed reference ellipsoid.
16 citations
TL;DR: In this paper , the authors investigated the failure process and evolution mechanisms in circular tunnels under a complex stress path of deep in-situ stress + excavation unloading + stress adjustment using a simulation test based on loading first and then drilling (LFTD) mode.
16 citations