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April 6, 2017 | Aerospace | By admin | 0 Comments

By Greitzer E.M., Tan C.S., Graf M.B.

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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.

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