Download Fluid mechanics formula sheet and more Cheat Sheet Fluid Mechanics in PDF only on Docsity! 31446 Mechanics of fluids: Formula sheet to be used at written examinations ◮The ǫ− δ identity reads ǫinmǫmjk = ǫminǫmjk = ǫnmiǫmjk = δijδnk − δikδnj ◮Strain rate tensor, vorticity tensor ∂vi ∂xj = 1 2 ∂vi ∂xj + ∂vi ∂xj 2∂vi/∂xj + ∂vj ∂xi − ∂vj ∂xi =0 = 1 2 ( ∂vi ∂xj + ∂vj ∂xi ) + 1 2 ( ∂vi ∂xj − ∂vj ∂xi ) = Sij +Ωij ◮The vorticity vector is computed as ω = ∇× v ωi = ǫijk ∂vk ∂xj ◮The material derivative ρ dΨ dt = ρ ∂Ψ ∂t + ρvj ∂Ψ ∂xj where Ψ = vi, u, T, k, v′iv ′ j . . . ◮The balance equation for mass dρ dt + ρ ∂vi ∂xi = 0 ◮The balance equation for linear momentum ρ dvi dt = ∂σji ∂xj + ρfi ◮The balance equation for internal energy ρ du dt = σji ∂vi ∂xj − ∂qi ∂xi ◮The equation for kinetic energy reads ρ dk dt = ∂viσji ∂xj − σji ∂vi ∂xj + ρvifi ◮The constitutive law for Newtonian viscous fluids σij = −pδij + 2µSij − 2 3 µSkkδij , σij = −pδij + τij qi = −k ∂T ∂xi ◮Viscosity 1 µ: dynamic viscosity ν: kinematic viscosity (ν = µ/ρ) ◮The continuity equation and the Navier-Stokes equation for incompressible flow with constant viscosity read (conservative form, p denotes the hydrostatic pressure, i.e. p = 0 if vi = 0) ∂vi ∂xi = 0 ρ ∂vi ∂t + ρ ∂vivj ∂xj = − ∂p ∂xi + µ ∂2vi ∂xj∂xj ◮The Navier-Stokes equation for incompressible flow with constant viscosity read (non-conservative form) ρ ∂vi ∂t + ρvj ∂vi ∂xj = − ∂p ∂xi + µ ∂2vi ∂xj∂xj The viscous stress tensor then reads τij = 2µSij = µ ( ∂vi ∂xj + ∂vj ∂xi ) ◮The equation for internal energy reads ρ du dt = −p ∂vi ∂xi + 2µSijSij − 2 3 µSkkSii Φ + ∂ ∂xi ( k ∂T ∂xi ) ◮Streamfunction, Ψ; potential, Φ v1 = ∂Ψ ∂x2 , v2 = − ∂Ψ ∂x1 vk = ∂Φ ∂xk ◮The Rayleigh problem η = x2 2 √ νt , f = v1 V0 d2f dη2 + 2η df dη = 0 ◮Blasius solution ξ = ( V1,∞ νx1 )1/2 x2, Ψ = (νV1,∞x1) 1/2 g 1 2 gg′′ + g′′′ = 0 ◮The Navier-Stokes (different form of the convective term) ∂vi ∂t + ∂k ∂xi − εijkvjωk = −1 ρ ∂P ∂xi + ν ∂2vi ∂xj∂xj + fi 2
