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Unformatted text preview: x, y, z )-space.1 Euler’s equations are
expressed in terms of the quantities,
v(x, y, z, t) = (velocity of the gas at (x, y, z ) at time t),
1 For a complete derivation of these equations, see Chapter 9 of Alexander Fetter and John
Walecka, Theoretical mechanics of particles and continua , McGraw-Hill, New York, 1980, or
Chapter 1 of Alexandre Chorin and Jerrold Marsden, A mathematical introduction to ﬂuid
mechanics , third edition, Springer, 1993. 130 ρ(x, y, z, t) = (density at (x, y, z ) at time t),
p(x, y, z, t) = (pressure at (x, y, z ) at time t).
The ﬁrst of the Euler equations is the equation of continuity,
+ ∇ · (ρv) = 0.
∂t (5.16) To derive this equation, we represent the ﬂuid ﬂow by the vector ﬁeld
F = ρv,
so that the surface integral
F · NdA
S represents the rate at which the ﬂuid is ﬂowing across S in the direction of N.
We assume that no ﬂuid is being created or destroyed. Then the rate of change
of the mass of ﬂuid within D is given by two expressions, D ∂...
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