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Genuine shock waves and adiabatic hypothesis

 

 

 

                          Genuine shock waves and adiabatic hypothesis.

          The author's account for the scientific working people about of his executed works

 

                                                         Content:                       

                                           1. The adiabatic hypothesis.

                                      2. The genuine shock waves

                                            3. The practical utilization of the theory

                                                         

                                                             !--break--

>

 

                                     Part 1. The adiabatic hypothesis.

                                                              

      Any natural phenomenon can be successfully used in the technics only then if the notions of scientists and engineers about this phenomenon - its  theory - are adequate to it. In contemporary hydrodynamics for any shock waves (in any material, with any amplitudes, et seq.) is used a theory based on the system of Rankine - Hugoniot (R. - H.) equations, the energy conservation law in which is introduced by the Hugoniot equation

 

                            e(p, V) - e0(p0, V0) = (1/2)(p + p0)(V0 - V) ,                 (1)

 

 where e is the specific inner energy of the enough small particle of environment, p - the pressure, V - the specific volume; the badge "0" corresponds to the initial state of particle (p0, V0)  //for the shock wave it is the state on X = Xs (t) + 0//, the absence of the badge - to the final state (p, V) //for shock wave it is the state on X = Xs  (t), the coordinate of shock front//. Equation (1) is disappeared on the base of adiabatic hypothesis: it always (a priori)  is supposed that the particle does not changes by the heat with its surroundings while it cross the shock front  //see any monograph or text - book on the question//. Such the adiabatic hypothesis can be come true at the elastic deformations only (see lower); for the genuine shock waves - with the non elastic deformations of the matter - the use of adiabatic hypothesis, and well then of the equation (1), leads to the false results.

      For any natural processes the law on energy conservation have a form

                                

                                de = dQ + dw ,                                                            (2)

 

where dw - the modification of e by mechanical work, dQ - its modification by all others, non - mechanical influences on the particle (so called "heat flow"). In adiabatic processes dQ = 0. Then dw appears as the full differential and w - as the function of state: its alteration is conditioned only by quantities of (p0, V0) and (p, V) and does not depend by the way of passage among them. Such the behaviour of w leads to the linear dependence

                                     p = p0+B(V0-V),   B=Const.                                                 (3)

//see lower in this blog: "Three problems about of the shock waves for the curious students (hydrodynamics). The second problem"//. The adiabatic deformations are the elastic ones by definition: if any object - for example, the particle - returns to its initial state (on the any way of transition), its inner energy also returns to its initial quantity e0(p0, V0) independently from the quantity e(p(t), V(t)) in the course of the transition  //because of here dw = de is the full differential!//. In a plane straight adiabatic wave of deformations with any continuous or even rupture profile the linear dependence (2) is the only physically justified dependence among p and V. The Hugoniot equation (1) is the integral from (2) with dQ = 0.     

Because of (3), the graph of the function p == f(V) , which leads from the resolving of (1), - so called the "Hugoniot adiabate" or the "shock adiabate" -  is the  straight line . The experimental researches confirm it: until the deformations are elastic, the p is the linear function from V independently from the profile of wave, - for instance, it is so at the rectangular profile. Naturally, it is so only until the Hugoniot equation itself conserve the physical sense, that is until dQ = 0.

 

The all said above had been strict detailed in the article [1]: 

[1]. Л.Г.Филиппенко. Следствия из уравнения Гюгонио. Сб. «Гидромеханика», вып. 36, Киев, «Наукова думка», 1977г.   (L.G.Philippenko. "The consequences from the Hugoniot equation". The English text of this article

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is placed on the author's
site "Shock Heat Transference"at URL :   http://www.leonid-philippenko.narod.ru/index.html

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).

 

                             

                                     Part 2. The genuine shock waves

      With a desire, the elastic (adiabatic) wave with a rectangular profile it is possible, of course, to name "the shock wave". For the practice, however, the genuine shock waves, in which the deformations of the matter on shock front are non elastic ones, are much more important and interesting.  From the energy conservation law (2) it leads, that such the waves can be accomplished only if dQ not equals to nought. The integration of (2) from Xs to (Xs + 0) leads then to the shock curve equation

 

   e(p,V) - e0(p0, V0) = sQ(p, V, p0, V0) + (1/2)(p + p0)(V0 - V)                    (4)

 

where sQ is the measure of the Shock Heat Transference (SHT)  (or the "Shock Heat Exchange") - that is  the specific quantity of heat by which the infinitesimal particle is in time to exchange with its surroundings when it cross shock rupture. The SHT is the physical phenomenon which earlier had not been described in a literature: discovery of this phenomenon was at first fixated in the article [1].

The graph of the function p = f(V) , which is got from resolving (4), - that is the shock curve - is nonlinear one. Just the such graphs are received in experiments on shock waves with a not too small amplitudes; only general using (1), instead of the right equation for the non elastic deformations (4), so far by all users, leads to calling this experimental graphs as the shock (or Hugoniot) adiabat.

The presence of SHT in the genuine - with the non elastic deformations - shock waves leads to the necessity of the radical revision of the thesises of its contemporary theory. In particular:

      a) In the any adiabatic wave the mass velocity at initial (in front) point of its profile equals to the sound velocity in the matter in front of wave, and no mechanical signal from the wave can outstrip the wave itself; the heat signals are absent in view of dQ = 0. Therefore the matter in front of wave remains not excited, so that p0 = p(X>>Xs), and just the same for V0, e0, et seq.; therefore this all do not depend from the amplitude of the wave. On this fact in contemporary (adiabatic!) theory (which none the less is applied to the non elastic shock waves!) are based such operations as:

-- the graphic determination of the quantity of shock wave velocity;

-- dividing of modification of the full energy of particle at shock front on the kinetic and inner parts.

This operations are false if apply to the genuine - non elastic - shock waves. Here the presence of  SHT leads to appearance the heat flow in front of wave, why the p0 already not equals to p(X>>Xs), but depend from it and from the amplitude of wave: p0 = f(p, p(X>>Xs)); just the same it is right for V, e, et seq. The point (p0, V0) now is not placed on the shock curve and above-mentioned operations are impossible.

      b) For a contemporary technics the researches of the energy equation of state e(p, V) and also the satisfying to it the dependence p from V, in the different materials at the over high pressures, are actual. This pressures are created in the strong shock waves. For resolve the putting problems the contemporary theory offers the simple way: experimentally is measured  the dependence between of the mass velocities in front and behind of shock front, v0 and v,  for which as the good approximation is found the linear bond

                                      

                                         v0 = a + bv ;   "a" and "b" are Const.                     (5)

 

The energy equation of state e = e(p, V) then obtain by resolve the R. - H. system in common with (5). But on the adiabatic shock wave (it describes by R. - H. system!) v0 = v(X>>Xs ) and does not depend from the amplitude of wave, and hence from v = vs //see above//; it means that (5) is not compatible with the Hugoniot equation (1).

 But that simple way is false also if apply to the non elastic waves: as it one can see from (4), because of  presence of the term sQ, the term w(p, V, p0 , V0 ) in such waves equals to the sum (e - sQ) which present not any function of states; and for to obtain the dependence of p from V, it is necessary to resolve the equation (4) in common with (5), for what it is necessary to know the function sQ = f(p, V, p0 , V0 ).

      c) The possibility of existence of the shock waves of certain kind is determined by the second principle of thermodynamics. The most significant what can to offer the contemporary theory in that question - it is the known Zemplen theorem; from it, it is infered, in particular, that the stretching shock waves in majority materials are impossible in nature //see any text-book or monograph on that theme//. This inference is false //see lower in this blog: "Three problems about of the shock waves for the curious students (hydrodynamics). The third problem"//. The reason is simple: the Zemplen theorem is infered on the base of Hugoniot equation and, as such, is applicable only to reversible processes, how are the adiabatic - elastic - shock waves; but in the reversible processes the straight and reverse ways are equally possible. The genuine - non elastic - shock waves are nonreversible processes, and here the second principle of thermodynamics have the form:

                 ds = de s + di s;      de s = dQ/T ,    di s > 0  or di s = 0                    (6)

(s - the specific entropy); here de s -  the change of entropy owing to non mechanical interaction with an outward bodies, di s -  owing to inner processes in material.  As it is clear from (6), this law does not difference between a stretching and pressing shock waves: only the di s have the importance. For instance, in a plain straight shock wave along axis X the integration of di s/dt gives

                     T(delta)i s = Y(dv/dX)(deltaV)                                                 (7)

where Y is the viscosity coefficient ( all parameters at X = Xs ). In a pressing shock waves with the increasing amplitudes (dv/dX)<0 (the back layers of matter runs over the shock front). In a stretching shock waves (dv/dX)>0, if the amplitudes increase (this layers runs away from the front the faster the further from the front it are found). Decreasing of the amplitude is accompanied with (dv/dX)>0 at pressing and (dv/dX)<0 at stretching shock wave on its rupture. At last, if the shock wave is stationary then also stationary will be motion behind its front , and (dv/dX) = 0. Thus the viscous share of (delta)i s depends not only from the sign of (deltaV) but also from the direction of change of the shock wave amplitude. When the amplitudes of shock waves (pressing or stretching - with indifference) increase, the contribution from viscosity in (delta)i s is positive one, so that such shock waves all are thermodynamics permissible. At the decreasing of the amplitude the right term of (7) <0 , also with indifference from the sign of (deltaV) : the existence of not adiabatic shock waves ( both pressing and stretching) at the decreasing of its amplitudes is impossible.

      The all essential characteristic peculiarities of non elastic shock waves had been strict detailed in the monograph [2]:

[2] Л.Г.Филиппенко. Сильные ударные волны в сплошных телах. - Киев, УМК  ВО, 1992г. После опубликования монография существенно дополнена и переработана автором. (L.G.Philippenko. Strong Shock Waves in the Continuous Bodies. - Kiev, UMK VO, 1992. After of the publication the monograph had been essentially completed and corrected by author).

In particular, in the monograph [2] the conception  active stretching shock waves” had been introduced in the mechanics of the continuous environment and had been analysed there. This idea  have the serious practical consequences (see lower in this blog:  The sudden throws of rock, coal and gas in the mines “).

The said above in n0 (a) - (c) had been in detail analysed in [2] and, in a more brief account, in the article [3]: "The Shock Heat Transference", which had been sent to "International Journal on Shock Waves, Detonations and Explosions" at July 2007.

[3] L.G.Philippenko."The Shock Heat Transference". Now this article is placed

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 on the author's
site "Shock Heat Transference"at URL :   http://www.leonid-philippenko.narod.ru/index.html

 

 

 

                                 Part 3. The practical utilization of the theory

      As it is known, "It is not object more practical than a right theory". Some recommendations on this theme for the genuine (non elastic) shock waves was gave in monograph [2]. Here we note only on the next from that:

      a) Initiating of thermonuclear reaction by shock waves.

      b) The attempts of the transformation of individual chemical substances (as for example the graphite into the diamond).

      c) The problem of the struggle with the sudden throws of rock, coal and gas in the mines.

This questions were briefly elucidated in this blog before //see the earlier blog posts lower//. Its account contains also in the article "Three technical problems for genuine shock waves" which had been sent to European Journal of Mechanics  B / Fluids at July 2009 ; the text of this articles is contained in the hosting FilesAnywhere, above - mentioned  in references [1] and [3].

The brief survey of the principal peculiarities of genuine shock waves contains in the article "The genuine (non adiabatic) shock waves" which had been sent to Nature Physics at July 2009, which text also is contained in the FilesAnywhere, indicated in [1] and [3]; there also are contained the articles "The heat exchange with a shock deformed material" and "On the hypothesis about of the continuous structure"

  The English text of author's articles on shock waves see also on site "Shock Heat Transference" at URL:   

www.leonid-philippenko.narod.ru/index.html

                   

 

 

 

 

 

 

 

 

 

 

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