Khmelnik S.I. Unusual fountain and gravitomagnetism

Transcription

1 Khmelnik.I. Unusual ountain and gavitomagnetism Contents. Intoduction. Main mathematical model 3. Computational algoithm 4. Enegy lows in the unusual ountain 5. et shape calculation 6. Conclusion eeences Abstact This pape pesents an unusual submaine ountain []. Its mathematical model is ceated and it is demonstated that its shape may be attibuted to the existence o gavitomagnetic oces simila to Loent oces, signiicant in magnitude, and gavitomagnetic enegy low simila to electomagnetic enegy low.. Intoduction An unusual ountain [] is installed in England, which constitutes a votex within a tanspaent cylinde the Chaybdis votex ountain see ig.. Thee is also an aticle [] about anothe atiicial votex, less impessive, but stuctually moe tanspaent. The ig. shows this votex in a glass with pesentation o its stuctue. We may also indicate a natual phenomenon esembling the unusual ountain [3] see ig. 3. To the autho's knowledge such phenomena have no stict mathematical desciption. Peviously, the autho suggested a mathematical model o a wate low unning into a unnel and out o a pipe [4]. At that, gavitomagnetism euations wee used the ones simila to Maxwell s euations o electodynamics the Maxwellian-like gavity euations heeinate, MLG euations. Inteaction between the moving wate masses was descibed by gavitomagnetic Loent oces heeinate, GL oces simila to Loent oces in electodynamics [5]. Futhe ationale is simila to that o [4]. oweve,

2 thee is a undamental dieence between the jet lowing out o the pipe unde pessue and the jet ascending in the unusual ountain. In the ist case, the wate jet speads out and jet density changes. In the second case, the jet density euals to that o the suounding wate, as the latte is an incompessible liuid. Theeoe, explanation o the jet shape in the ountain should die om the same in [4]. Main attention is devoted to this subject below. Fig.. Fig..

3 Fig. 3.. Main mathematical model MLG euations o gavitomagnetic intensities and mass cuent densities in stationay gavitomagnetic ield ae witten as: div, ot, Fo votex modelling we will utilie cylindical coodinates,,. o MLG euations will be as ollows:, 3, 4, 5. 6 Besides, the cuents should meet the continuity condition div, 7 o, in cylindical coodinates,. 8 These euations essentially descibe inteaction o cuents, intensities and GL oces. The latte ae deined as F G, 9 L o 3

4 whee G is a gavitation constant, and is a gavitomagnetic pemeability o the medium [5], o. In cylindical coodinates, this coss poduct appeas as ollows: o o o. o ollows: whee 4 3. Computational algoithm In [4], it is demonstated that the euations.3-.6,.8 appea as. 8 exp. exp. 3 exp 3. exp, 4. 7 exp, 5 exp, 6 8 X 8, 7 X, 8, X, / exp g h,,, g some invaiables., Vaious unctions ae possible, wheeon depend 8,, unctions. Let us conside the simplest case, when the unction is constant, so h 8, 8,

5 ee, X is a eaviside unction appoximation, g value chaacteies appoximation leap width, is a jet adius, the coodinate value, wheein the unction changes its value om to. Function is to be detemined. Fig. 3 shows unctions -6 at h.,.5,.8, g.6,... j j - F j. Z FIG FIG. 4. 5

6 Let us conside now a vecto ield o cuents on cicle in the ountain hoiontal plane see ig. 4 with the same paamete values. Analyed points located on "dotted" adii ae denoted hee with ciclets. Continuous intevals indicate cuent vectos. 4. Enegy lows in the unusual ountain A stuctue o constant cuent electomagnetic enegy low in a cylindical wie with constant cuent was descibed in [6]. It was demonstated that the density o electomagnetic enegy low is. whee is a speciic electical esistivity. imilaly, let us detemine the density o gavitomagnetic enegy low in wate jet, whee is a speciic esistivity to mass cuent. Theeoe,, 3 o whee o is detemined in accodance with.. 6 Fig. 4. Fig. 4 shows vetical section o the ountain in the plane, and jet edge. Gavitomagnetic enegy low 4

7 7 ciculates within the jet. Fig. 4 shows pojections, o this steam and the sum o these pojections. Pojection is tangential to the jet cicumeence, and is not shown in ig. 4. It may be also that the aggegate pojection is pependicula to the edge in its vicinity and eual to eo at the vey jet edge. I this condition is ulilled in all edge points, the gavitomagnetic enegy low always emains inside the jet. Aguing as in [7], it should be noticed that density integal o this steam by volume V o the jet is popotional to the electomagnetic ield momentum Р in this volume, since in the I system, as is known, E c c dv dp. 5 By vitue o the momentum consevation law, jet etains its continuity, since the integal o the gavitomagnetic enegy low density is changing with the jet shape. 5. et shape calculation The condition omulated above enables jet shape calculation. Let us ind based on.,.-.6: exp exp exp exp 3 7 o exp exp exp exp 8 7 o o exp 3 7 o, 3 exp o. 4 Poviding o.7-.,.3 we obtain: h X o exp, 5 3 exp h X o. 6 Now, let us ind the angle shown in ig. 4: h h tg o o, 7

8 o tg h h, 8 whee is a jet adius. Let us designate the jet geneato unction as Q. I the angle is a slope angle with this unction o the tangent line, then d tg Q. 9 d Theeoe, Q h h, d On integating we obtain: h h Q ln Q o. Fig. 5, in the uppe window, the unction Q is shown, wheeas the unction log is shown by dots o eeence. In the lowe window, exp unction is shown, incopoated in the omulas And yet, it is assumed that h,.8, Q 3 o. -5 Q=Z x 5 expn* Z FIG. 5. 8

9 Fig. 6 shows pojections o the enegy density vecto, o, o and ull enegy vecto o, detemined using.. These values ae shown on the plane o jet vetical section, at h.,.5,.5, g.6. The igue shows jet edges. It is seen that the enegy low deceases om jet cente up- and sidewads. An enegy souce is located below to ceate the ountain and this gavitomagnetic enegy diection is natual. This enegy is spent to viscous ictional dop in mass wate lows. o F Z Z -.5 Z Z Z - Fig Conclusion I ig. and ig. 5 ae compaed, we may notice similaity o eal and simulated shapes o the unusual ountain. Thus, it is ai to say that gavitomagnetism euations ae conimed expeimentally. At that, existence o gavitomagnetic oces, signiicant in magnitude, and gavitomagnetic enegy low is conimed. 9

10 eeences. Bondaov M.N., avitchev V.I. Atiicial maelstom and its application, Khmelnik.I. About Flow o Wate Into the Funnel and om Pipe, in ussian 5. Khmelnik.I. Moe on Expeimental Claiication o Maxwellsimila Gavitation Euations, "The Papes o independent Authos", publ. «DNA», IN 5-677, Isael-ussia, 4, issue 8, IBN , pinted in UA, Lulu Inc., ID in ussian, see also 6. Khmelnik.I. The Flow tuctue o the Electomagnetic Enegy in the Wie with Constant Cuent, 7. Khmelnik.I. To the Question o Intanuclea Foces. "The Papes o independent Authos", publ. «DNA», IN 5-677, Isael-ussia, 4, issue 8, IBN , pinted in UA, Lulu Inc., ID in ussian see also