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Compressible Viscous Equations Also known as the compressible Navier-Stokes equations: Mass: ∂ ∂ρ (ρv j ) = 0 + ∂t ∂x j Momentum: ∂ ( ρν i ) ∂ ∂p ∂ ρν iν j ) = − + + ( (τ ij ) , i = 1, 2,3 ∂t ∂x j ∂xi ∂x j Energy: ∂ 1 ∂ (ρ e + ρ v2 ) + ∂t ∂x j 2 ∂ν i τ ij = µ ∂x j ∂Τ , q = k ∂x j + 1 2 ∂ ∂ ∂ + ( ρ e ρ v ) v p ν τ ν ( q j ) = − + + ( ) ( ) j j ij i 2 ∂x j ∂x j ∂x j ∂ν j ∂ν k + δ ij λ ∂xi ∂xk e = e ( p, Τ ) ← state relationship (ideal gas ) Incompressible Viscous Equations In this case, we assume ρ = const. Mass: ∂ν j ∂x j Momentum: ρ =0 ∂ν i ∂ (ν iν j ) = − ∂p + ∂ +ρ ∂t ∂x j ∂xi ∂x j ∂ν i ∂ν j + µ ∂x j ∂xi Usually, µ = µ (Τ ) . Often, when temperature variations are small, µ =const. is assumed. ∂ν j ∂ ∂ν i ∂ν j ∂ν i ⇒ + = + µ µ ∂x j ∂x j ∂x j ∂xi ∂x j ∂x j ∂xi ∂ ∂ν j =0 ∂xi ∂x j ∂ν i =µ ∂x j ∂x j Usual form of momentum for incompressible flow: ρ ∂vi ∂vi ∂p ∂ +µ (v i v j ) = − +ρ ∂xi ∂x j ∂x j ∂x j ∂t In this case, the energy equation is not needed to find vi & p . Incompressible Inviscid Equations In this case we assume that the effects of viscous stresses are small compared to acceleration and pressure forces: Mass: ∂ν j =0 ∂x j Momentum: ρ ∂ν i ∂ ∂p ν iν j ) = − +ρ ( ∂t ∂x j ∂xi These are known as the incompressible Euler equations. Incompressible Potential Flow K In potential flow, we assume the flow is irrotational (i.e. ∇ × V = 0 ). This allows the velocity to be written as the gradient of a scalar potential: νi = Mass: ∂φ , ∂xi ∂φ j ∂x j ∂x j φ = Potential =0 ∂ 2φ ∂ 2φ ∂ 2φ = 0 + + Also written out as: ∂z 2 ∂y 2 ∂x 2 or ∇2φ = 0, ∇2 ≡ Laplacian This is a single equation for a single unknown φ . It is the same for steady and unsteady flows. What happens to momentum? 16.100 2