University:
Massachusetts Institute of Technology
Course:
12.520 | Geodynamics
Academic year:

2023
Views:

92

Pages:

4
Author:

Levi Dalton

Lecture Notes 21 Fluids Fluids – no memory of shape – flow under applied tractions, body forces, stop flowing (don’t reverse flow) when “driving forces” removed. Newton’s concept of viscosity – subject fluid to shearing X2 . u1 X1 Figure 21.1 σ 12 = µ ∂u1 ∂x2 where µ is shear viscosity. Substance µ (Pa ⋅ sec) at 20 o C Poise (10 Pa ⋅ sec) air 2 ⋅10 −5 2 ⋅10 −4 water 10 −3 10 −2 glycerine 1 10 ice ( 0 o C ) 1013 1014 glass 1017 1018 “Earth” 1019 −10 21 10 20 −10 22 Physical cause – gasses – (vertical) motion of particles with different horizontal velocities. • Fluids – elastic resistance to distortion of atomic “cages” as atoms and molecules “slide by”. Figure 21.2 • Solids – diffusion of defects in the lattice (vacancies or interstitials); motion of dislocations in lattice structure. E E E E Figure 21.3 For a general stress-strain, for a Newtonian fluid σ ij = − p 'δij + Dijklε kl where Dijkl is viscosity tensor and ε kl is strain rate tensor. For an isotropic fluid σ ij = − p'δij + λε kkδij + 2µε ij For volumetric strain rate σ kk = −3p '+ (3λ + 2µ )ε kk σ kk 2 = − p '+ (λ + µ )ε kk 3 3 Mean normal stress: 2 where λ + µ is bulk viscosity. 3 2 For many applications, λ + µ = 0 (Stokes fluid) 3 2 3 σ ij = − p 'δij + 2µεij − µε kkδij For many applications, ε kk 0 σ ij = − p 'δij + 2µε ij Often η is used for viscosity σ ij = − p 'δij + 2ηεij Sometimes η → 0 σ ij = − p 'δij (“perfect fluid”)