Nearly Incompressible Laminar Flow – Equations and Solution Technique Author: John M. Cimbala, Penn State University Latest revision: 31 October 2007 Assumptions and Approximations •The fluid is Newtonianwith constant properties (μ, ν, k, α, κ, CP). •The flow is laminarrather than transitional or turbulent. •The fluid is nearly incompressible– either an incompressibleliquidor anideal gas at very low Mach numbers. Differential Equations of Motion for Nearly Incompressible Flow (general review)•Conservation of mass: 0iiux∂=∂. •Momentum equation: 2iiijiijijDuuuupρρuρgμDttxxx x⎛⎞∂∂∂∂=+=−++⎜⎟∂∂∂⎝⎠j. •Conservation of energy (first law): Forincompressible liquid:2pDTTρCkDtx xφ∂=+, where 2ij ijee=. Forideal gas at very low Ma: 2pDTTρDtx x∂=∂ ∂, or 2DTTκDtx x∂=∂ ∂(pkκρC≡= thermal diffusivity).
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