By Greitzer E.M., Tan C.S., Graf M.B.
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Additional info for Internal flow
1), which becomes V ∂ ∂ρ (ρu i ) dV = 0. 2) The volume V is arbitrary. 2) to hold, therefore, the integrand must be zero everywhere, so ∂ ∂ρ (ρu i ) = 0 + ∂t ∂ xi ∂ρ + ∇ · (ρu) = 0, in vector notation . 3) is the differential form of the mass conservation, or continuity, equation. It can also be expressed in terms of the substantial derivative of the density as ∂u i 1 Dρ + =0 ρ Dt ∂ xi 1 Dρ + ∇ · u = 0, ρ Dt in vector notation . 4) The continuity equation for an incompressible ﬂuid can be written as an explicit statement that the density of a ﬂuid particle remains constant: Dρ = 0.
3), the notation D( )/Dt has been used to indicate a derivative deﬁned following the ﬂuid particle. This notation is conventional, and the quantity D( )/Dt, which occurs throughout the description of ﬂuid motion, is known variously as the substantial derivative, the material derivative, or the convective derivative. Noting that in Cartesian coordinates the ﬁrst three terms of the derivative are formally equivalent to u · ∇c, the substantial derivative can be written more compactly as ∂c ∂c Dc ∂c = + (u · ∇) c = + ui .
13) for a system to which there is no heat transfer is dS ≥ 0 (for a system with – d Q = 0). 13) can also be written as a rate equation in terms of the heat transfer rate and temperature of the ﬂuid particles which comprise the system. With s the speciﬁc entropy or entropy per unit mass, DS D = Dt Dt sdm ≥ 1– dQ . 15), the summation is taken over all locations at which heat enters or leaves the system. 15) will be developed in terms of ﬂuid motions and temperature ﬁelds later in this chapter. The ﬂuids considered in this book are those described as simple compressible substances.