Lecture-18-more_(11-18-10) - APPH 4200 Physics of Fluids...

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Unformatted text preview: APPH 4200 Physics of Fluids More Fluid Waves/Instabilities (Ch. 7) November 18, 2010 1.! ! Tides 2.! Jeans instability 1 Tides Thirty days of tide heights at Bridgeport Connecticut U.S.A. as calculated from the Harmonic Constituent data aligned with 0h Sunday 1'st September 1991. 2 10-component tide-predicting machine, conceived by Lord Kelvin Tide predicting machine, by Sir William Thomson in 1876. The first full sized machine for predicting tides, combined ten tidal components (one pulley for each component). It could trace the heights of the tides for one year in about four hours. 3 The M2 tidal constituent. Amplitude is indicated by color, and the white lines are cotidal differing by 1 hr. The curved arcs around the amphidromic points show the direction of the tides, each indicating a synchronized 6 hour period. R. Ray, TOPEX/Poseidon: Revealing Hidden Tidal Energy en:GSFC,en:NASA. Redistribute with credit to R. Ray, and NASA - Goddard Space Flight Center NASA - Jet Propulsion Laboratory Scientific Visualization Studio Semi-Diurnal Tide Map 4 Tides: Driven Shallow Water Waves 5 Surface Waves with a Lateral Force 6 (Same Eq using “integrated” velocity potential) 7 What is the lateral force? 8 Net Force with Period 2Ω 9 Driven Surface Waves 10 Star Formation 11 The Stability of a Spherical Nebula J. H. Jeans PHILOSOPHICAL TRANSACTIONS. 1. The Stability of a Spherical Nebula. By J. H. JEANS, B.A., Fellow of Trinity College, asnd Isaac Newton Student in the University of Cambridge. Communicated b)yProfessor G. H. DAvwN, F. R.?S. _ead June 20, 1901. Revised February 28, 1902. Received June 15,--R INTRODUUCTION. ? 1. object of the present paper can be best explained by referring to a sentence : which occurs in a paper by Professor G. H. DARWIN. This is as follows : " The principal question involved in the nebular hypothesis seems to be the stability of a rotating mass of gas; but, unfortunately, this has remained up to now an untouched field of mathematical research. We can only judge of probable results from the investigations which have been made concerning the stability of a rotating mass of liquid." In so far as the two cases are parallel, the argument by analogy will, of course, be valid enough, but the compressibility of a gas makes possible in the gaseous nebula a whole series of vibrations which have no counterpart in a liquid, and no infe:rence as to the stability of these motions call be drawn from an examination of the behaviour of a liquid. Thus, although there will be unstable vibrations in a rotating mass of gas similar to those which are known to exist in a rotating liquid, it does not at all follow that a rotating gas will become unstable, in the first place, through vibrations T-IE 12 Jeans Instability: Driven Sound Waves 13 Linearized Self-Gravitational Response 14 Summary • Linearization of dynamical equations provides a powerful methodology to understand disturbances • Harmonic analysis (ω) and modal analysis (k) provide understanding of linear PDEs 15 ...
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