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documentclass[12pt]{article} usepackage{amsmath} usepackage{amssymb} usepackage[cp866]{inputenc} usepackage[english]{babel} usepackage{hhline} usepackage{graphicx} usepackage{fancyhdr} extwidth 16cm extheight 24cm opmargin -7mm hoffset -1in oddsidemargin 30mm evensidemargin 20mm fancyhead{} fancyhead[RO,LE]{ hepage} fancyfoot{} pagestyle{fancy} enewcommand{ itle}[1]{setcounter{equation}{0}igskip egin{center} Large sf #1 end{center}} enewcommand{author}[1]{{centering em #1par}igskip} enewcommand{ ablename}[1]{} ewcommand{inst}[1]{parparbox[t]{15cm}{ extsl{#1}} igskip} ewcommand{add}[2]{fancyhead[LO,RE]{#1}addcontentsline{toc}{subsection}{sc #1 em #2}} egin{document} itle{THE SHOCK WAVE STRUCTURE IN THE HEAT-CONDUCTING INVISCID MEDIUM} author{E.,V. Ermolova, A.,N. Kraiko} inst{Central Institute of Aviation Motors of P.I. Baranov, Moscow} add{Ermolova E.,V., Kraiko A.,N.}{THE SHOCK WAVE STRUCTURE IN THE INVISCID MEDIUM WITHOUT HEAT-CONDUCTING} index{Ermolova E.,V.} index{Kraiko A.,N.} The complete theory of the shock wave (SW) structure was formulated for heat-conducting inviscid medium. Raley supposed such situation for common gas however it is more natural to neglect viscosity under taking into account heat-conducting for thermonuclear temperatures that are about tens and hundred millions degrees. For such temperatures the radiation transfer of energy ('radiant heat-conducting') is incomparably more important then all impulse transfer ways. Under thermonuclear temperatures the medium consists of completely ionized atoms, electrons and radiation, at that the last one can contribute significantly not only to energy transfer but also to pressure $p$, specific internal energy $e$, enthalpy $h$ and entropy $s$ of medium. In case 'degree symbol' marks dimensional quantities then $ ext{T}^{°}$ -- absolute temperature,$ ho^{°}$, $R^{°} = c_{p}^{°} - c_{v}^{°}$, $c_{p}^{°}$ and $c_{v}^{°}$ -- density, gas constant and specific heat capacities ideal gas -- electrically neutral plasma that consists of ions and electrons, $c^{°}$ -- velocity of light and $sigma^{°}$ -- Stefan-Boltsman constant, so for the equilibrium mixture of gas and radiation the state equations are cite{Kraiko:book1} egin{equation}label{Kraiko:main} p^{°}=R^{°} ho^{°} ext{T}^{°}+frac{4sigma^{°}}{3c^{°}} ext{T}^{°4},quad e^{°}=c_{v}^{°} ext{T}^{°}+frac{4sigma^{°}}{ ho^{°} c^{°}} ext{T}^{°4},quad h^{°}=e^{°}+frac{p^{°}}{ ho^{°}}=c_p^{°} ext{T}^{°}+frac{16sigma^{°}}{3 ho^{°}c^{°}} ext{T}^{°4}. end{equation} According to them when the temperatures are high the second items that reflect the radiation contribution can become the same order with the first items and even become greater then the last ones. From incoming in the state equations ( ef{Kraiko:main}), constants and gas density in front of the SW $ ho_{0}^{°}$, it is possible to make up combinations whose dimensions are the same to that of all parameters incoming in ( ef{Kraiko:main}) and in integral conversation laws that describe structure of the SW. Such combinations with dimensions of temperature, velocity and pressure are $$ ext{T}_{ ext{c}}^{°}={(R^{°} ho_{0}^{°} c^{°})}^{1/3}/{(4 sigma^{°})}^{1/3},quad u_{ ext{c}}^{°}=R^{°2/3}{( ho_{0}^{°}c^{°})}^{1/6}/{(4 sigma^{°})}^{1/6},quad p_{ ext{c}}^{°}={{ R^{°4} ho_{0}^{°4}c^{°}/{(4sigma^{°})} } }^{1/3}. $$ egin{table}[!h] centering caption{The values of $ ext{T}_{ ext{c}}^{°}$ , $u_{ ext{c}}^{°}$ and $p_{ ext{c}}^{°}$ for some gases} $$ egin{array}{|c|c|c|c|c|c|} hline ext{Gas} & ho_{0}^{°} ( extit{g}/ extit{cm}^{3})&0.01&0.1&1&10 hline & ext{T}_{ ext{c}}^{°}(° ext{K})&6.02*10^6&1.30*10^7&2.80*10^7&6.02*10^7hhline{~|- -|- -|- -|- -|- -|} ext{H}_{1}^{1}&u_{ ext{c}}^{°}( extit{m}/ extit{s})&3.15*10^5&4.62*10^5&6.79*10^5&9.96*10^5hhline{~|- -|- -|- -|- -|- -|} &p_{ ext{c}}^{°}( extit{Pa})&9.93*10^{11}&2.14*10^{13}&4.61*10^{14}&9.93*10^{15} hline & ext{T}_{ ext{c}}^{°}(° ext{K})&6.04*10^6&1.30*10^7&2.80*10^7&6.04*10^7hhline{~|- -|- -|- -|- -|- -|} ext{D}_{2}^{1}&u_{ ext{c}}^{°}( extit{m}/ extit{s})&2.00*10^5&2.93*10^5&4.30*10^5&6.31*10^5hhline{~|- -|- -|- -|- -|- -|} &p_{ ext{c}}^{°}( extit{Pa})&3.98*10^{11}&8.58*10^{12}&1.85*10^{14}&3.98*10^{15}hline & ext{T}_{ ext{c}}^{°}(° ext{K})&4.19*10^6&9.02*10^6&1.94*10^7&4.19*10^7hhline{~|- -|- -|- -|- -|- -|} ext{Tr}_{3}^{1}&u_{ ext{c}}^{°}( extit{m}/ extit{s})&1.52*10^5&2.24*10^5&3.28*10^5&4.82*10^5hhline{~|- -|- -|- -|- -|- -|} &p_{ ext{c}}^{°}( extit{Pa})&2.32*10^{11}&5.00*10^{12}&1.08*10^{14}&2.32*10^{15} hline & ext{T}_{ ext{c}}^{°}(° ext{K})&4.35*10^6&9.38*10^6&2.02*10^7&4.35*10^7hhline{~|- -|- -|- -|- -|- -|} ext{He}_{4}^{2}&u_{ ext{c}}^{°}( extit{m}/ extit{s})&1.65*10^5&2.42*10^5&3.55*10^5&5.22*10^5hhline{~|- -|- -|- -|- -|- -|} & p_{ ext{c}}^{°}( extit{Pa})&2.71*10^{11}&5.84*10^{12}&1.26*10^{14}&2.71*10^{15} hline & ext{T}_{ ext{c}}^{°}(° ext{K})&3.45*10^6&7.44*10^6&1.60*10^7&3.45*10^7hhline{~|- -|- -|- -|- -|- -|} ext{He}_{8}^{2} ext{ --}&u_{ ext{c}}^{°}( extit{m}/ extit{s})&1.04*10^5&1.52*10^5&2.23*10^5&3.29*10^5hhline{~|- -|- -|- -|- -|- -|} ext{'He-3'}&p_{ ext{c}}^{°}( extit{Pa})&1.08*10^{11}&2.32*10^{12}&4.99*10^{13}&1.08*10^{15} hline end{array} $$ label{mytab} end{table} The values of $ ext{T}_{ ext{c}}^{°}$, $u_{ ext{c}}^{°}$ and $p_{ ext{c}}^{°}$ for three hydrogen isotopes and two helium isotopes with different densities $ ho_{ ext{0}}^{°}$ are put in the table. If we take $ ext{T}_{ ext{c}}^{°}$, $p_{ ext{c}}^{°}$ and $ ho_{ ext{c}}^{°}= ho_{ ext{0}}^{°}$ as scales then we obtain the state equations in dimensionless form egin{equation}label{Kraiko:2} p= ho ext{T}+frac{ ext{T}^4}{3},quad h=frac{1+varepsilon}{2varepsilon} ext{T}+frac{4 ext{T}^4}{3 ho},quad varepsilon=frac{gamma-1}{gamma+1},quad gamma=frac{c_{p}}{c_{v}}. end{equation} According to ( ef{Kraiko:2}) in such thermodynamic model different media differ only by ratio of specific heats $gamma$ or by constant $gamma$ that is related to $varepsilon$. Values$0levarepsilonle0.5$ corresponds to $1 le gamma le 3$. If $a$ is a sound velocity then for the state equations ( ef{Kraiko:2}) egin{equation}label{Kraiko:3} a^{2} equiv {left( frac{partial p} {partial ho} ight)}_{s}=frac{9(1+varepsilon) ho^{2} +24varepsilon(5 ho+4 ext{T}^{3}/3) ext{T}^3}{9 ho[(1-varepsilon) ho+ 8varepsilon ext{T}^{3}]} ext{T}. end{equation} Let the SW be rested, $D$ -- the approach flow velocity and other parameters also as $ ho_0 = 1$, are marked by '0'. So with account of ( ef{Kraiko:3}) egin{equation}label{Kraiko:4} 0 le a_{0}^{2} le D^2 le infty, quad a_0^{2}=frac{1+varepsilon+8(5+4sigma)varepsilon sigma}{1-varepsilon+24varepsilonsigma} ext{T}_0, quad sigma=frac{ ext{T}_0^{3}}{3}. end{equation} The equalities $ ext{T}_0 = 0$, $sigma = 0$ and $a_0 = 0$, that hold true concurrently are corresponding to the 'cold background', when with account of temperature values $ ext{T}_{ ext{c}}^{°}$ from the table under the thermonuclear temperatures behind the SW the approach flow temperature and pressure are so slight that they can be neglected. In such approach for any finite velocity $D$ the SW proves to be strong, as Mach number $Mequiv D/a_0=infty$. In the context of this model the SW structure analysis means to find out its features for the following values of parameters: $0levarepsilonle0.5$, $0lesigmaleinfty$ and $a_{0}le Dleinfty$. Executed analysis consisted of three stages. At first such cases were considered when temperatures are high enough for radiant heat-conducting effects prevalence but behind the SW $ ext{T}^{3}ll3 ho$ as yet. In such cases one can neglect the second items in ( ef{Kraiko:2}) and therefore admit items that are proportional to $ ext{T}^3$ and $sigma$ in the expression for sound velocity from ( ef{Kraiko:3}) and ( ef{Kraiko:4}). In such approach that had been considered already by Raley for each $varepsilon$ the SW structure is defined by the value of ratio $t_0 = ext{T}_0/D^2$. With the growing of velocity $D$ from $a_0$ to $infty$ this ratio is falling from $(1-varepsilon)/(1+varepsilon)$ to null. When $(1-2varepsilon)/(1+2varepsilon)D_{*}(varepsilon)$ the SW structure is continuous with finite harbinger and asymptotic tendency to constant parameters behind the SW. If $D=D_{*}(varepsilon)$, then there is no IS and continuous structure is the finite harbinger in front of the SW. It were essentially these conclusions that have been made by V.A. Belokogn in a very condensed form cite{Kraiko:Article1}. In general case of warm or even hot approach flow ($sigma>0$), the complete state equations ( ef{Kraiko:2}) and expressions for sound velocity are used. In this approach the droningly and unrestrictedly growing function when $varepsilon o 0$ $sigma_{**}(varepsilon)$ was found. When $sigmagesigma_{**}(varepsilon)$ there is no IS for any $a_0(sigma,varepsilon)le D leinfty$, thus the SW structure is continuous and asymptotic from both sides of the SW. For $sigmaD_{*1}(sigma, varepsilon)>a_0(sigma, varepsilon)$, that define the SW structure were calculated. When $a_0(sigma, varepsilon)D_{*2}(sigma, varepsilon)$, then the SW structure is continuous and asymptotic from both sides of the SW as it is in case when $sigmagesigma_{**}(varepsilon)$. For $D_{*1}(sigma, varepsilon) Note. Abstracts are published in author's edition

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