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mech7220-hw1-solns

# mech7220-hw1-solns - 1 Conservation Equations Exercises 1...

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1 Conservation Equations Exercises 1. Fill in the details in the derivation of Eq. (12) from Eq. (11). ∂t ( ρ u ) + ∇ · ( ρ uu ) = ρ u ∂t + u ± ∂ρ ∂t + ∇ · ( ρ u ) ² | {z } =0 + ρ ( u · ∇ ) u = ρ D u Dt 2. Derive Eq. 14 beginning with Eq. 13. Apply the divergence theorem to Eq. (13): ∂t ³ ρ ± e + u 2 2 ²´ + ∇ · ³ ρ u ± e + u 2 2 ²´ = ∇ · ( σ · u ) + ρ f · u - ∇ · q 00 + q 000 Apply the chain rule on the convective terms and use the continuity equation: ∂t ³ ρ ± e + u 2 2 ²´ + ∇ · ³ ρ u ± e + u 2 2 ²´ = ± e + u 2 2 ²± ∂ρ ∂t + ∇ · ( ρ u ) ² | {z } =0 + ρ D Dt ± e + u 2 2 ² The viscous dissipation term expands into ∇ · ( σ · u ) = ( ∇ · σ ) · u + σ : u Dot the momentum equation into u , to get ρ D Dt ( u 2 / 2) = ( ∇ · σ ) · u + ρ f · u which is a balance of mechanical energy (or, equivalently, a work balance). Subtract this from the energy equation to get ρ De Dt = σ : u - ∇ · q 00 + q 000 Now use e = i - P/ρ : ρ D Dt ( i - P/ρ

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mech7220-hw1-solns - 1 Conservation Equations Exercises 1...

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