Khmelnik S.I. Mathematical Model of Dust Whirl

Transcription

1 Khmelnik S.I. Mathematial Model of Dust Whil Abstat The question of the soue of enegy in a dust whil is onsideed. Atmosphei onditions annot be the sole soue of enegy, as suh dust whils exist on Mas, whee the atmosphee is absent. Hee we show that the soue of enegy fo the dust whil is the enegy of the gavitational field. We pesent a mathematial model of the sand votex, whih uses a system of Maxwell-like gavitational equations. The model explains some of the popeties of the dust whil pesevation of ylindial vetial shape of the dust whil, motion of the dust whil as a whole. Contents 1. Intodution. Mathematial Model 3. The Enegy Flows 4. Vetial Stability 5. The Motion of the Dust Whil Appendix Refeenes 1. Intodution Thee exists a widely known dust dust whil, whih is an almost vetial loud of dust see Fig. 1. Suh a dust whil has a vetial axis of otation, height of a few tens of metes, diamete - a few metes, the time of existene - a few tens of seonds [1]. Thee ae simila phenomena - wate, ai, ash dust whils. The ause of thei existene is assumed to be vaious atmosphei phenomena (wind, heating of the atmosphee). Howeve, the vey existene of the dust whil its shape etention and movement, - ae diffiult to explain by the same easons. Futhemoe, suh dust whils ae also moving on Mas, whee thee is no atmosphee - see. Fig. [1]. Theefoe, in the explanation of the dust whils the main question is about the soue of enegy. 1

2 Fig. 1. Fig.. Thee ae muh moe poweful phenomena elated to dust whils - Sandy tsunami - see Fig. a and Fig.. The existing view that the auses of the movement of this olossus ae a beee and nonlinea medium seems unonvining. It seems that this "devie" has its own moto within, and the esistane of the medium is just a atalyst, a foe that pushes the gas pedal. Fig. а. Fig. в Below we pesent a mathematial model of the dust whil, whih uses a system of Maxwell-like gavitational equations. It is shown that the enegy soue fo the sand votex is the enegy of the gavitational field - see Appendix. In any ase, it is had to find any othe soue of enegy on the planet Mas.

3 The model is based on the following assumptions. Sandy dust whil is omposed of mateial patiles sand gains. The movement of these patiles is likened to mass uents. Mass uents in the gavitational field ae desibed by Maxwell-like gavitational equations [] (heeinafte - MLG-equations). The inteation between the moving masses is desibed by the Loent gavity-magneti (the GL-foe) simila to the Loent foes in eletodynamis ating between moving eletial hages. Cuents aising in the dust whil ae iulating (as shown) in the oss setion of the votex and along the vetial (up and down). The kineti enegy of suh iulation is spent on the losses fom ollisions of sand gains. It omes fom a gavitating body. Potential enegy of the dust whil is not hanged, and theefoe is not onsumed. I.e. in this ase thee is no onvesion of potential enegy into kineti enegy and vie vesa. Howeve, gavitating body expends its enegy on eating and maintaining a mass uent - see Appendix. Suppoting dust whil upight is explained as follows. Fom the analogy between the Maxwell equations and MLG it follows that thee may be a flow S of gavitational enegy. Suh flow an exist and hange ove time. Togethe with the flow thee is a gavitational momentum. If the body is in the flow of gavitational enegy (and this flow does not hange ove time), then on the body ats the foe S F=S\ (whee is the speed of light), dieted opposite to the flow dietion. It follows fom the law of onsevation of momentum. We emphasie one again that it is - a omplete analogy between gavitational and eletomagneti field. Fo the eletomagneti field, these elationships ae disussed in [3, 4]. In the body of the dust whil togethe with onstant mass uents exists (as shown below) a flow of gavitational enegy, onstant ove time. It is dieted downwad. In aodane with the above, an upwad foe ats on the body of the dust whil, thus holding it in an upight position. not. Mathematial Model MLG-equations fo gavity-magneti intensity H and density of mass uents J in stationay gavity-magneti field ae as follows: divh 0, (1) ot(h) J, () 3

4 In the simulation of dust whil we shall use ylindial oodinates,,. Then the MLG equations will be: H 1 H H H 0, (3) 1 H H J, (4) H H J, (5) H H 1 H J, (6) The model is based on the following fats: 1. The intensity of the gavitational field is dieted along the axis of dust whil,. It eates a vetial flow of sand gains - a mass uent J. 3. Vetial mass uent J geneates annula gavity-magneti field H and adial gavity-magneti field H - see (6). 4. Gavity-magneti field H deflets by GL-foes sand gains of vetial flow in the adial dietion, eating a adial flow of sand gains - adial mass uent J. 5. Gavity-magneti field H deflets by GL-foes sand gains of adial flow pependiula to the adius, eating a vetial mass uent J. 6. Gavity-magneti field H deflets by GL-foes sand gains of vetial flow pependiula to the adius, eating a annula mass uent J. 7. Gavity-magneti field H deflets the GL-foes sand gains of annula flow is pependiula to the adius, eating a vetial mass uent J. 8. The mass uent J geneates a vetial gavity-magneti field H and annula gavity-magneti field H - see (4). 9. The mass uent J geneates a vetial gavity-magneti field H and adial gavity-magneti field H - see (5) 10. The mass uent J geneates a annula gavity-magneti field H and adial gavity-magneti field H - see (6). 4

5 Thus, the main mass uent J eates additional mass uents o J, J, J and gavity-magneti fields H, H, H. They must satisfy Maxwell equations (3-6). The uents must also satisfy the ontinuity ondition div( J ) 0, (8) o, in ylindial oodinates, J 1 J J 0. (9) Mass uents ae J nmv, (10) and thei kineti enegy - W nmv, (11) whee n - the numbe of sand gains in the flow, m - the mass of one sand gain, v - the speed of sand gains flow. Thus, equal mass uents may have diffeent kineti enegy. Fig. 4. The solution of system (3-6, 9) has been found in [5] and has the following fom: H. sin( ) h, (14) H. h os( ), (15) 5

6 whee H 1 sin( ), (16) J. os( ), (17) J. j sin( ), (18) 1 os sin J h. (19) j, h - some onstants, - an intege onstant. Fig. 4 shows the value of J on the setion plane fo 10, h 1, R 50, whee R is the dust whil adius. Hee it is impotant to note that the vetial uents iulate so that the sum of the uents on eah setion is equal to eo - see. (19). Thus, the dust patiles move along a losed path and gavity does not pefom wok on this tajetoy. Nevetheless, some wok is done to oveome the fitional foes between dust patiles when they ae moved by GLfoes. This wok is pefomed by the enegy of the gavitational field - see Appendix. We shall assume that the wok of fition foe between the sand gains P J, (0) whee - the esistivity of the mass uent, independent of its magnitude and dietion (simila to eletial esistane). Then, the entie wok an be defined in the same manne as in [5]. It is equal to 4 P R L j R 1/ 4 16 h 1 4, (1) whee R, L - adius and height of the dust whil aodingly. These fomulas ae simila to the fomulas fo a length of wie with a onstant uent. Assuming that fo mass uents (as well as fo eleti uents), the piniple of minimum themal losses is obseved, it is possible to find the atio of [5] j h R. () whee , (3) Then 4 P R Lh 1/ (4) That is thee powe that must ome fom the gavitational field fo the existene of dust whil. 6

7 3. The Enegy Flows By analogy with eletodynamis let us wite the onnetion between the mass uent and the gavity-eletial intensity in the fom of E J. (5) Also by analogy with eletodynamis let us detemine the density of gavitation enegy flows in the fom S E H. (6) Then we an find S J H. (7) The veto podut J H in ylindial oodinates looks as follows: S J H J H S J H S J H J H (8) S J H J H Enegy flows and fom stability wee used in simila mathematial models [6, 7]. By analogy, one ould ague that thee is no powe flow out of the body of the dust whil. Inside the body it is dieted along the adius fom peiphey to ente - S ; iumfeentially - S f ; vetial down - S. These enegy flows povide pesevation of the dust whil fom, fo the hange of its fom equies extenal enegy inflow [7], vetial stability, the dust whil motion. 4. Vetial Stability The body of the dust whil is pemeated by flows of gavitational enegy that ae eated by mass uents. A fomulai elationship between the uents and enegy flows is disussed in [5] fo diet uent. The same dependenies an be used in this ase. In patiula, in the body of a votex thee is a flow of enegy dieted vetially, with a density S j. (30) h 7

8 In the Intodution it was shown that a flow with a given density pemeating a body eates a pessue foe ating on the body with a density (pessue) S F, (31) In a dietion opposite to the flow. Let us find the full foe of the pessue exeted in eah setion of the body of the dust whil of adius R, R R j hr Fo S d j h d. (3) As the flow of enegy (30) is dieted downwads, the foe of opposite (3) is dieted upwads and suppots the dust whil in an upight position. The gavity ounteats to the above foe and balanes it. 5. The Motion of the Dust whil The tajetoy of the dust whil is pooly peditable. We an say that the dust whil makes haoti movement. In ode to show that the motion of the dust whil is aomplished by the intenal enegy (and not by the foe of the wind) let us again tun to the onsideation of the intenal flow of eletomagneti enegy. In [5] it is shown that in the body of the dust whil thee is a flow of enegy dieted adially with density 1 3 S h j. (33) As fo the vetial enegy flow, a foe with the density S F. (34) also oesponds to this flow. Let us find the total foe ating in the dust whil's body along the adius: R 1 Fo S d. (35) 0 Fo a symmetial distibution of the adial flow total foe (35) is eo. If the axial symmety of the votex is boken, then thee appeas an unompensated foe. Let 1 - be a oeffiient haateiing the symmety beaking. Then unompensated foe an be found fom the fomula 8

9 o 1 R / R F o S d S d. (36) 0 R / F o o, in view of (33), F o R 1 S R / d R 1 h R / R R h j 3 5. (37) j 4 d. (38) This foe esults of the motion of the dust whil as a whole. It an be shown that the eason fo this distotion is ai esistane and sand gains inetia (but that is anothe topi). Appendix Consevative foes (by definition) do not pefom wok on a losed tajetoy. The foe of gavity is a onsevative foe (whih is poved mathematially). Hene the onlusion is eahed that 1) thee does not exist a moto using only onsevative foes (speifially, the foe of gavity) to pefom wok. Next an unpoven onlusion is made that ) thee does not exist a moto using the enegy of onsevative foes soue (inluding the gavity foes), fo pefoming the wok. Coulomb foes ae also onsevative. Fom this by analogy one an make the onlusion 1). Howeve, the onlusion ) is easily efuted: thee exists, fo example, a DC moto with self-exitation. Its enegy soue is a onstant voltage soue, i.e., a soue of Coulomb foes. Theefoe, in the geneal ase, the assetion ) is inoet, and the tue statement is as follows: 3) Thee an exist a moto using the enegy of onsevative foes soue fo pefoming wok. Nevetheless, the existene of the moto that uses enegy of the eletial onsevative foes soue (SECF) does not mean that thee is a moto that uses the enegy soue of the gavitational onsevative foes (SGCF). 9

10 Eletial foes eate the hages motion along a losed tajetoy eleti uent whih foms a magneti field. Due to this the enegy of SECF tuns into magneti enegy. It ous even if the enegy is not expended fo the motion of the hages on the losed path. Thus, the enegy of SECF exeeds the enegy of the mehanial motion of the hages. This is the eason fo the existene of a moto using the enegy SECF. Gavity foes also an eate a mass motion on a losed tajetoy mass uent. Let us assume that mass uent also foms a gavity magneti field (it is shown in []). Then by analogy with the pevious we may assume that 4) thee an exist a moto using the enegy of the soue of gavity onsevative foes fo pefoming wok. This does not ontadit the law of onsevation of enegy: it is the enegy of SGCF that is onveted into wok, and SGCF powe soue loses some of its enegy (it annot be said that the enegy of SGCF may be used only fo the movement of the masses). 10 Refeenes 1. Dust_devil, Khmelnik S.I. Expeimental Claifiation of Maxwell-Simila Gavitation Equations, "Papes of Independent Authos", publ. «DNA», pinted in USA, ISSN , Lulu In., ID , Isael-Russia, 014, iss. 5, ISBN , in Russian, R.P. Feynman, R.B. Leighton, M. Sands. The Feynman Letues on Physis, volume, Khmelnik S.I. Loent Foe, Ampee Foe and Momentum Consevation Law Quantitative. Analysis and Coollaies. "Papes of Independent Authos", publ. «DNA», ISSN , pinted in USA, Lulu In , Isael-Russia, 014, iss. 30, ISBN , in Russian, Khmelnik S.I. Stutue of Constant Cuent, 6. Khmelnik S.I. Математическая модель электрического торнадо, in Russian,

11 7. Khmelnik S.I. Mathematial Model of Ball Lightning 8. Khmelnik S.I. The Flow Stutue of the Eletomagneti Enegy in the Wie with Constant Cuent, in Rusian, 11