University:
Massachusetts Institute of Technology
Course:
2.06 | Fluid Dynamics
Academic year:

2022
Views:

496

Pages:

2
Author:

Montagutip9

2.06 Equation Sheet for Quiz 2 (Spring 2013) Conservation Relations for Open Systems (using ϑ for velocity) Mass conservation dM CV = ∑ m in − ∑ m out dt in out Linear momentum conservation     d( M ϑ )CV = ∑ m inϑ in − ∑ m outϑ out + ∑ Fext dt Bernoulli Equation (using ϑ for velocity) For steady, frictionless flows, P1 ϑ12 P ϑ2 + + gz1 = 2 + 2 + gz2 ρ 2 ρ 2 Differential Form  ∂ρ + ∇ ⋅( ρϑ ) = 0 ∂t Mass conservation: (for incompressible) Navier-Stokes equation: Total Derivative operator:  ∇ ⋅ϑ = 0   Dϑ  ρ = −∇P + ρ g + µ∇ 2ϑ Dt D Dt Vector Operators Cartesian Coordinates (x, y, z) ∇ψ = ∂ψ ˆ ∂ψ ∂ψ i+ ĵ + k̂ ∂x ∂y ∂z  ∂ Ax ∂ Ay ∂ Az ∇⋅ A = + + ∂x ∂ y ∂z ∇ 2ψ = ∂ 2ψ ∂ x2 + ∂ 2ψ ∂ y2 + ∂ 2ψ ∂ z2 #∂  & = % + ϑ .∇( $ ∂t ' Cartesian coordinates: Mass conservation (continuity equation): ∂ϑx ∂ϑy ∂ϑz + + =0 ∂x ∂y ∂z Navier-Stokes equations: ⎛ ∂ 2ϑx ∂ 2ϑx ∂ 2ϑx ⎞ ⎛ ∂ϑx ∂ϑx ∂ϑx ∂ϑx ⎞ ∂P + ϑx ρ⎜ + ϑy + ϑz ⎟ = − + ρ g x + ⎜µ 2 + 2 + 2 ⎟ ∂x ∂y ∂z ⎠ ∂x ∂y ∂z ⎠ ⎝ ∂t ⎝ ∂x ∂ϑ ∂ϑ ∂ϑ ⎛ ∂ϑ ρ ⎜ y + ϑx y + ϑy y + ϑz y ∂x ∂y ∂z ⎝ ∂t ⎞ ∂P + ρ gy + ⎟=− ∂y ⎠ ⎛ ∂ 2ϑy ∂ 2ϑy ∂ 2ϑy ⎞ µ + 2 + 2 ⎟ ⎜⎜ ∂x 2 ∂y ∂z ⎟⎠ ⎝ ⎛ ∂ 2ϑz ∂ 2ϑz ∂ 2ϑz ⎞ ⎛ ∂ϑz ∂ϑz ∂ϑz ∂ϑz ⎞ ∂P + ϑx ρ⎜ + ϑy + ϑz ⎟ = − + ρ g z + ⎜µ 2 + 2 + 2 ⎟ ∂z ∂y ∂z ⎠ ∂x ∂y ∂z ⎠ ⎝ ∂t ⎝ ∂x Newtonian viscous shear stresses : ⎛ ∂ϑ ∂ϑ ⎞ ∂ϑ ∂ϑ τ xy = τ yx = µ ⎜ x + y ⎟ τ xz = τ zx = µ ⎛⎜ x + z ⎟⎞ ∂x ⎠ ∂x ⎠ ⎝ ∂z ⎝ ∂y ⎛ ∂ϑy ∂ϑz ⎞ + ⎟ ∂y ⎠ ⎝ ∂z τ yz = τ zy = µ ⎜ Reduced Equations Above equations reduce to much simpler 2D form when the thickness of the fluid film is much smaller its length. When flow direction is x and thickness direction is y Mass Conservation ∂ϑ x =0 ∂x Navier-Stokes Equation # ∂2ϑ & ∂P − + ρ g x + µ %% 2x (( = 0 ∂x $ ∂y '

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