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02fluid_eqns07part1

# 02fluid_eqns07part1 - FLUID MODELS Part 1 Basic Equations...

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FLUID MODELS: Part 1 Basic Equations Consider a control volume of volume V fixed in space. Fluid is free to cross the surface of the volume. A surface element is denoted as dA and a unit normal vector outward from the surface is denoted by ˆ n . Continuity Equation: Let nm ρ = be the mass density, where n is the number density and m is the particle mass. The mass in the control volume is V dV ρ . The time rate of change of the mass inside the volume is given by - = losses sources V L S dV ρ t where S represents sources of material and L represents losses of material. Since fluid can flow across the surface this represents both a source and/or a loss of material. This can be represented by the amount of mass carried across the surface by the fluid. This flow effect can be represented by a flux of mass density across the surface - A dA n ˆ u ρ where the negative sign is because the unit normal is outward. Using this we can write - - = losses sources A V L S dA n ˆ u ρ dV ρ t With no sources or losses : - = A V dA n ˆ u ρ dV ρ t This is the integral form of the continuity equation . Using Gauss's Theorem a ˆ ndA A = ∇⋅ a dV V we can rewrite dV u ρ dA n ˆ u ρ V A = and rewrite the continuity equation with sources or losses 1 u n ˆ T , p , ρ dA

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2 FLUID MODELS: Part 1 Basic Equations - = +
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