# Indeed we can now formulate the dirichlet problem as

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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, ∂ρ + ∇ · (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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