# Interdisciplinary Applied Mathematics

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(2.5a)

De

p— = -pV-v — V-q + Ф,

(2.5b)

where Ф = т • Vv is the dissipation function and D/Dt = d/dt + v • V is the material derivative.

In addition    to    the    governing    conservation    laws,    an equation    of    state    is

required. For ideal gases, it has the simple form

p = pRT,    (2.6)

where R is the ideal gas constant defined as the difference of the constant specific heats; that is, R = Cp — Cv, where Cv = ^p and Cp = jCv with Y the adiabatic index. For ideal gases, the energy equation can be rewritten in terms of the temperature, since e = p/(p(Y — 1)) = CvT, and so equation (2.5b) becomes

pCv ^ = -pV-v + V- [kVT] + Ф.    (2.7)

The system of equations (2.4a; 2.5a), (2.6), and (2.7) is called compressible Navier-Stokes equations, contains six unknown variables (p, v,p,T) with six scalar equations. Mathematically, it is an incomplete parabolic system, since there are no second-order derivative terms in the continuity equation.

A hyperbolic system arises in the case of inviscid flow, that is, p = 0 (assuming that    we    also    neglect    heat    losses    by    thermal diffusion,    that    is,

к = 0). In that case we obtain the Euler equations, which in the absence of external forces or heat sources have the form

dp

dt

+ V-(pv)

0,

d(pv)

dt

+ V-(pvv)

—Vp,

dE

~dt

+ V-[(E + p)v]

0.

(2.8a)

(2.8b)

(2.8c)

This system admits discontinuous solutions, and it can also describe the transition from a supersonic flow (where |v| > cs) to subsonic flow (where |v| < cs), where cs = (pRT)1/2 is the speed of sound. Typically, the transition is obtained through a shock wave, which represents a discontinuity in flow variables. In such a region the integral form of the equations should be used by analogy with equations (2.1a)-(2.1c).

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