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crib_sheet_final - 1 Hydrostatics ∇ p = ρ g or dp dz =...

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Unformatted text preview: 1. Hydrostatics ∇ p = ρ g or dp dz =- ρg p 2- p 1 =- z 2 Z z 1 ρgdz Force on submerged plane surface, F = p cg A , where p cg is the pressure at the center of gravity. ξ cg = 1 A Z ξdA where ξ is the slant distance from the surface. The center of pressure is given by y cp =- I xx sin θ/h cg A, x cp =- I xy sin θ/h cg A I xy = R xydA, I xx = R y 2 dA The force on a curved surface equals the force acting on the projection of the surface onto a vertical plane. 2. Control Volume Analysis “Reynolds Transport Theorem” dN dt ¶ system = d dt ZZZ ηρdV + ZZ ηρ V · n dS Conservation of Mass: d dt ZZZ ρdV =- ZZ ρ V · n dS Conservation of Momentum: d dt ZZZ ρ V dV + ZZ ρ VV · n dS = F F pressure =- ZZ p n dS Conservation of Energy: ˙ Q- ˙ W shaft = d dt ZZZ ρedV + ZZ e + p ρ ¶ ρ V · n dS e = ˆ u + 1 2 V 2 + gz 3. Differential Equations of Motion Continuity: ∂ρ ∂t + ∇ · ( ρ V ) = 0 Momentum: ρ • ∂ V ∂t + ( V · ∇ ) V ‚ = ρ g- ∇ p + ∇ · τ ij ρ • ∂u i ∂t + u j ∂u i ∂x j ‚ = ρg i- ∂p ∂x i + ∂τ ij ∂x j 1 2 For incompressible flow, τ ij = μ ∂u i ∂x j + ∂u j ∂x i ¶ Then, ∂τ ij ∂x j = μ ∂ 2 u i ∂x j ∂x j = μ ∇ 2 V 4. Streamfunction Two-dimensional, incompressible flow ∇ · V = 0 u = ∂ψ ∂y , v =- ∂ψ...
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crib_sheet_final - 1 Hydrostatics ∇ p = ρ g or dp dz =...

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