View source for Gas dynamics, equations of

Mathematical expressions of the fundamental laws of conservation of the mass, the momentum and the energy of a gas, which describe the state of the gas. A gas is a population of a large number of particles (molecules, atoms, ions) in continuous chaotic motion. Computations of the interactions and of the motions of the individual particles are exceedingly difficult, and for this reason a statistical, or continuous, approach is adopted in describing the state of the gas. When this is done, the overall state of the particles of the gas is characterized by a distribution function of the particles defined in a seven-dimensional phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g0433301.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g0433302.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g0433303.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g0433304.png" />) or in the four-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g0433305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g0433306.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g0433307.png" />). In the former case one considers a scalar distribution function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g0433308.png" /></td> </tr></table>

in which the magnitudes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g0433309.png" /> are continuously-varying arguments, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333011.png" /> vary in bounded or unbounded intervals, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333012.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333013.png" /> itself satisfies Boltzmann's integro-differential equation (cf. [[Boltzmann equation|Boltzmann equation]]; [[Kinetic equation|Kinetic equation]]) or else satisfies other equations, depending on the physical premises (cf. [[Bogolyubov chain of equations|Bogolyubov chain of equations]]; [[Vlasov kinetic equation|Vlasov kinetic equation]]). In the latter case the distribution function describing the state of the gas is a vector function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333014.png" /></td> </tr></table>

depending on four arguments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333015.png" /> which all vary continuously and independently in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333016.png" />. In such a case a  "particle"  is to be understood, strictly speaking, as a material element of the gas which occupies an infinitesimal volume and has a certain velocity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333017.png" />, which is a function of the arguments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333018.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333019.png" /> is the density of the gas, i.e. the mass of the gas per unit volume, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333020.png" /> is the overall energy of the mass of the gas and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333021.png" /> is the internal energy of a unit mass of the gas.

On the assumption of local thermodynamic equilibrium, the Boltzmann equation yield the laws of conservation of gas dynamics in their integral form. In an inertial orthonormal coordinate system

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333023.png" /> is the volume of the space bounded by the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333025.png" />. The relations (1) are valid for an arbitrary volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333026.png" /> with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333027.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333028.png" />-dimensional phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333029.png" />. The magnitudes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333032.png" /> have the following form in the three-dimensional case:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333034.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333035.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333036.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333037.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333038.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333039.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333040.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333041.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333042.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333043.png" /> is the pressure of the gas, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333044.png" /> is the temperature of the gas, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333045.png" /> is the Kronecker symbol, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333046.png" /> is the viscosity coefficient of compression, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333047.png" /> is the viscosity coefficient of motion, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333048.png" /> is the coefficient of thermal conduction. The notation of the formulas is that employed in tensor analysis.

A system of differential equations in divergence form is employed for smooth flows:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333049.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

After the equations of state have been added, this set of equations becomes closed. At thermodynamic equilibrium the equations of state assume the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333050.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333051.png" /></td> </tr></table>

If the system is not at equilibrium, these magnitudes may depend on the gradients of the flow function.

The representation (2) has a certain physical meaning:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333052.png" /></td> </tr></table>

correspond, respectively, to convective fluxes of matter, momentum, energy; to the spherical non-dissipative part of the stress tensor, i.e. to the pressure; and to the dissipative part of the stresses (viscosity, thermal diffusion). It is used in the splitting method to obtain effective integration schemes of the problems in gas dynamics.

A gas flow can be described in various coordinate systems. In addition to the coordinate system rigidly attached to the physical space — the Galilean (or Euler) coordinate system — various mobile coordinate systems, which need not be Cartesian or Galilean, are also employed. The Lagrangian coordinate system, connected with the gas particles, is very extensively used. In this system each material element has a fixed coordinate. In Euler's method, at each moment of time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333053.png" /> the parameters of the state of the gas are defined as functions of the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333054.png" /> (Euler coordinates) of points of some mobile coordinate system, while the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333055.png" /> denotes the velocity of the gas particle which is located at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333056.png" /> at the moment of time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333057.png" />. In the Lagrange method, the velocity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333058.png" /> and the thermodynamic magnitudes are specified for each particle as a function of the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333059.png" />. If the gas particle is fixed with the aid of parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333060.png" />, then the flow parameters of the gas are obtained as a function of the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333061.png" /> and of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333062.png" /> (Lagrange coordinates). The connection between the Euler and the Lagrange coordinates has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333063.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333064.png" /> are the Euler coordinates of the particle which is located at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333065.png" /> at the moment of time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333066.png" />. If there is one spatial variable, and if the gas-dynamic variables are continuously-differentiable functions, then the equations of a viscous heat-conducting gas have the following form:

in Euler coordinates:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333067.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333068.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333069.png" /></td> </tr></table>

in Lagrange coordinates

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333070.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333071.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333072.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333074.png" />.

In the case of an arbitrary mobile coordinate system it is often expedient to simultaneously transform the velocity components according to the tensor law. If

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333075.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

is a transformation of spatial coordinates under which the time coordinate remains unchanged, then the mapping (5) may be connected with the flow of the gas itself, and it will then define a field of local coordinate systems depending on this flow. More general transformations, including a change of the time coordinate, are also possible.

An important role in the theory of the equations of gas dynamics is played by the analysis of the small parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333079.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333080.png" /> is the compressibility coefficient) forming part of (3). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333081.png" />, then (3) represents the equations of the dynamics of an ideal gas; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333082.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333083.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333084.png" />, equation (3) represents the [[Navier–Stokes equations|Navier–Stokes equations]] of an incompressible liquid. These equations are not of Cauchy–Kovalevskaya type. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333087.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333088.png" />, one obtains a system of parabolic equations of Cauchy–Kovalevskaya type which is not strongly parabolic. In turbulence theory and in the theory of non-Newtonian liquids the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333090.png" /> may depend on the gradients of gas-dynamic magnitudes.

The defining relations (2) and, in particular, the state equations (4), characterize the type of the system of equations (3) of gas dynamics and a number of its qualitative features. Thus, for an ideal compressible gas (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333091.png" />) the system of equations (3) is hyperbolic if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333092.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

where the entropy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333093.png" /> is defined by the relation (the second law of thermodynamics)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333094.png" /></td> </tr></table>

Condition (6) is local, depends on the solution, and may be violated in certain cases. Thus, in the case of the van der Waals equation of state, condition (6) may be violated, the equations become elliptic and the solution is unstable.

The conservation laws (1) make it possible to formulate a generalized solution of the equations of gas dynamics which is no longer necessarily continuous and does not satisfy the differential equations of gas dynamics (3). A complete theory of generalized equations of gas dynamics has not been developed, but the simplest generalized solutions have been thoroughly studied; these include, for example, shock waves, centred rarified waves, contact flows, etc. There exists a hypothesis according to which a generalized solution of the equations of an ideal compressible gas is the limit of the corresponding solution for the viscous gas as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333095.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333096.png" />. This statement has been rigorously proved for one-dimensional shock waves and certain special cases for equations of the type

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333097.png" /></td> </tr></table>

Of special interest are the stationary equations of gas dynamics, which are basically connected with the problems of stationary flow around bodies in an infinite space or stationary flow through channels. In such cases the solution of the system of equations (3) is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333098.png" />, and the system takes the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g04333099.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

For the stationary system of equations (7) a certain boundary value problem, which may be very complicated, is posed, and equation (7) itself may be both of elliptic and of mixed type. For instance, in the problem of the flow of an ideal compressible gas and on the assumption that this flow is potential, the following equation is obtained in the two-dimensional case:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g043330100.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g043330101.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g043330102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g043330103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g043330104.png" /> is the velocity potential, and the square of the velocity of sound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g043330105.png" /> may be found from the Bernoulli integral:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g043330106.png" /></td> </tr></table>

For equation (8) one may pose the problem of flow around a given contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g043330107.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g043330108.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g043330109.png" /> is the normal component of the velocity vector along the normal to the contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g043330110.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g043330111.png" />, equation (8) is hyperbolic; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043330/g043330112.png" />, it is elliptic. Transition from elliptic to hyperbolic type is possible (transonic flow). It can be shown that the boundary value problem is ill-posed (cf. [[Ill-posed problems|Ill-posed problems]]) in the case of transonic flow, since the smallest change in the contour may render the boundary value problem unsolvable in the class of continuous functions.

Of major interest are problems of gas-dynamic instability and turbulence, which are usually described in the framework of the theory of non-self-adjoint equations (Orr–Sommerfeld equation, Reynolds equation). In the context of practical applications the equations of gas dynamics which describe the motion of more complex media have become very important (multi-phase media, non-Newtonian liquids, magnetical hydrodynamics). Most equations of mathematical physics are the result of a linearization of equations of gas dynamics. For the numerical solution of problems in gas dynamics cf. [[Gas dynamics, numerical methods of|Gas dynamics, numerical methods of]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Serrin,   "Mathematical principles of classical fluid mechanics"  S. Flügge (ed.) , ''Handbuch der Physik'' , '''8/1''' , Springer  (1959)  pp. 125–263  {{MR|0108115}} {{MR|0108116}} {{MR|0108117}} {{MR|0108118}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.I. Sedov,   "A course in continuum mechanics" , '''1–4''' , Wolters-Noordhoff  (1971–1972)  (Translated from Russian)  {{MR|}} {{ZBL|0308.73006}} {{ZBL|0284.73001}} {{ZBL|0241.76001}} {{ZBL|0308.73005}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.D. Landau,   E.M. Lifshits,   "Mechanics" , Pergamon  (1965)  (Translated from Russian)  {{MR|0183138}} {{ZBL|0178.57901}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.E. Kochin,   I.A. Kibel',   N.V. Roze,   "Theoretical hydrodynamics" , Interscience  (1964)  (Translated from Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  G.G. Branover,   A.M. Tsinober,   "Magnetic hydrodynamics of incompressible media" , Moscow  (1970)  (In Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B.L. Rozhdestvenskii,   N.N. Yanenko,   "Systems of quasilinear equations and their applications to gas dynamics" , Amer. Math. Soc.  (1983)  (Translated from Russian)  {{MR|0694243}} {{ZBL|}} </TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Courant,   K.O. Friedrichs,   "Supersonic flow and shock waves" , Interscience  (1948)  {{MR|0029615}} {{ZBL|0041.11302}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.W. Liepmann,   A. Roshko,   "Elements of gas dynamics" , Wiley  (1957)  {{MR|0118167}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W.R. Sears (ed.) , ''General theory of high speed aerodynamics'' , Princeton Univ. Press  (1954)  {{MR|0064569}} {{ZBL|0057.18302}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  C. Cercignani,   "The Boltzmann equation and its applications" , Springer  (1988)  {{MR|1313028}} {{ZBL|0646.76001}} </TD></TR></table>