Recursive algorithms for the computation of the potential harmonic coefficients of a constant density polyhedron

The three fundamental principles for the variation of static pressure p throughout a body of fluid at rest are (a) the pressure at a point is the same in all directions (Pascal’s law), (b) the pressure is the same at all points on the same horizontal level, and (c) the pressure increases with depth z according to the hydrostatic equation. dp/dz= ρ‎g For a fluid with constant density ρ‎, the increase in pressure over a depth increase h is ρ‎gh, a result which can be used to analyse the response of simple barometers and manometers to applied pressure changes and differences. In situations where very large changes in pressure occur an equation of state may be required to relate pressure and density together with an assumption about the fluid temperature. The hydrostatic equation is still valid but more difficult to integrate, as illustrated by consideration of the earth’s atmosphere.

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Laminar natural convection about an isothermally heated sphere at small Grashof number

Journal of Fluid Mechanics ◽

10.1017/s0022112068001825 ◽

1968 ◽

Vol 34 (1) ◽

pp. 163-176 ◽

Cited By ~ 36

Author(s):

Francis E. Fendell

Keyword(s):

Natural Convection ◽

Three Dimensional ◽

Perturbation Expansion ◽

Outer Sphere ◽

Convective Transport ◽

Convection Flow ◽

Constant Density ◽

Boussinesq Model ◽

Grashof Number ◽

Recirculating Flow

The flow induced by gravity about a very small heated isothermal sphere introduced into a fluid in hydrostatic equilibrium is studied. The natural-convection flow is taken to be steady and laminar. The conditions under which the Boussinesq model is a good approximation to the full conservation laws are described. For a concentric finite cold outer sphere with radius, in ratio to the heated sphere radius, roughly less than the Grashof number to the minus one-half power, a recirculating flow occurs; fluid rises near the inner sphere and falls near the outer sphere. For a small heated sphere in an unbounded medium an ordinary perturbation expansion essentially in the Grashof number leads to unbounded velocities far from the sphere; this singularity is the natural-convection analogue of the Whitehead paradox arising in three-dimensional low-Reynolds-number forced-convection flows. Inner-and-outer matched asymptotic expansions reveal the importance of convective transport away from the sphere, although diffusive transport is dominant near the sphere. Approximate solution is given to the nonlinear outer equations, first by seeking a similarity solution (in paraboloidal co-ordinates) for a point heat source valid far from the point source, and then by linearization in the manner of Oseen. The Oseen solution is matched to the inner diffusive solution. Both outer solutions describe a paraboloidal wake above the sphere within which the enthalpy decays slowly relative to the rapid decay outside the wake. The updraft above the sphere is reduced from unbounded growth with distance from the sphere to constant magnitude by restoration of the convective accelerations. Finally, the role of vertical stratification of the ambient density in eventually stagnating updrafts predicted on the basis of a constant-density atmosphere is discussed.

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A static generalization of the Einstein universe

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1958.0077 ◽

1958 ◽

Vol 245 (1241) ◽

pp. 202-212

Keyword(s):

General Problem ◽

Constant Density ◽

De Sitter ◽

Einstein Field Equations ◽

Field Equations ◽

Einstein Universe

Solutions of the Einstein field equations are found for the problem of a sphere of constant density surrounded by matter of different constant density. The solutions are discussed and particular attention paid to the topology of the surrounding matter. The Schwarzschild, de Sitter, and Einstein solutions emerge as particular cases of the general problem.

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APPLICATION OF AN ADAPTIVE STEP-SIZE ALGORITHM IN MODELS OF HYPERINFLATION

Macroeconomic Dynamics ◽

10.1017/s136510051000088x ◽

2012 ◽

Vol 16 (S3) ◽

pp. 355-375 ◽

Cited By ~ 14

Author(s):

Olena Kostyshyna

Keyword(s):

New York ◽

Mean Squared Error ◽

American Economic Review ◽

Sufficient Information ◽

Recursive Algorithms ◽

Time Varying ◽

Step Size ◽

Adaptive Step Size ◽

Squared Error ◽

Adaptive Step

An adaptive step-size algorithm [Kushner and Yin,Stochastic Approximation and Recursive Algorithms and Applications, 2nd ed., New York: Springer-Verlag (2003)] is used to model time-varying learning, and its performance is illustrated in the environment of Marcet and Nicolini [American Economic Review93 (2003), 1476–1498]. The resulting model gives qualitatively similar results to those of Marcet and Nicolini, and performs quantitatively somewhat better, based on the criterion of mean squared error. The model generates increasing gain during hyperinflations and decreasing gain after hyperinflations end, which matches findings in the data. An agent using this model behaves cautiously when faced with sudden changes in policy, and is able to recognize a regime change after acquiring sufficient information.

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Effect of radiative heat loss on steady hypersonic flow past a blunt body

Journal of Fluid Mechanics ◽

10.1017/s0022112067000989 ◽

1967 ◽

Vol 29 (3) ◽

pp. 485-494 ◽

Cited By ~ 2

Author(s):

M. I. G. Bloor

Keyword(s):

Numerical Solution ◽

Heat Loss ◽

Blunt Body ◽

Radiative Heat ◽

Constant Density ◽

Axially Symmetric ◽

Radiative Heat Loss ◽

Stagnation Region ◽

Expansion Procedure ◽

Axis Of Symmetry

Using the grey gas approximation, the effect of radiative heat loss on axially symmetric flows is studied. Using an expansion procedure about the axis of symmetry, a numerical solution for the stagnation region is found taking the shock to be spherical. The results of this calculation are compared with the results of Lighthill's non-radiative constant density solution.

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A multi-layer model for nonlinear internal wave propagation in shallow water

Journal of Fluid Mechanics ◽

10.1017/jfm.2012.24 ◽

2012 ◽

Vol 695 ◽

pp. 341-365 ◽

Cited By ~ 10

Author(s):

Philip L.-F. Liu ◽

Xiaoming Wang

Keyword(s):

Wave Propagation ◽

Internal Wave ◽

Shallow Water ◽

Internal Waves ◽

Bottom Topography ◽

Laboratory Data ◽

Constant Density ◽

Layer Model ◽

Fluid System ◽

Nonlinear Internal Wave

AbstractIn this paper, a multi-layer model is developed for the purpose of studying nonlinear internal wave propagation in shallow water. The methodology employed in constructing the multi-layer model is similar to that used in deriving Boussinesq-type equations for surface gravity waves. It can also be viewed as an extension of the two-layer model developed by Choi & Camassa. The multi-layer model approximates the continuous density stratification by an $N$-layer fluid system in which a constant density is assumed in each layer. This allows the model to investigate higher-mode internal waves. Furthermore, the model is capable of simulating large-amplitude internal waves up to the breaking point. However, the model is limited by the assumption that the total water depth is shallow in comparison with the wavelength of interest. Furthermore, the vertical vorticity must vanish, while the horizontal vorticity components are weak. Numerical examples for strongly nonlinear waves are compared with laboratory data and other numerical studies in a two-layer fluid system. Good agreement is observed. The generation and propagation of mode-1 and mode-2 internal waves and their interactions with bottom topography are also investigated.

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