Chapter 18: Hydrodynamic Methods

Question 1

Dissipative Particle Dynamics

Answer 1

• coarse-grained particle
• 3 pairwise, internal forces
  
  • conservative F_C
  • friction/dissipative FD
  • random/stochastic FS
• relationship between dissipative and stochastic forces fixed via Dissipation-Fluctuation Theorem (see below)
• Failures:​
  
  • ​multi-interface
  • complex geometry
  • bubbles in fluid

Question 2

Lattice Boltzmann Method

Answer 2

• model fluid population discretely on-lattice
  
  • statistical (i.e. mesoscopic) description
• F_i ≡ fluid distribution at lattice point i
• Lattice types
  
  • α ≡ dimensionality
  • β ≡ number of discrete velocities
    
    • D₂Q₉, D₃Q₁₉

Question 3

LBM Algorithm

Answer 3

1. streaming ≡ moving from site-to-site in time ∆t
2. collision ≡ interaction amongst fluid densities
   
   • τ ≡ relaxation time
   • F_eq ≡ lattice-based equilibrium distribution

Question 4

LBM Input Parameters

Answer 4

• c_s ≡ sound velocity in fleuid
• v ≡ kinematic viscosity
• ∆t,τ are tuned

Question 5

LBM Boundary Conditions

Answer 5

slip length S

Question 6

LBM Measurements

Answer 6

can measure macroscopic obsersavbles

• e.g. density ρ

Question 7

LBM + MD (Overview)

Answer 7

two parts

• MD part
• LB part

Question 8

LBM + MD (MD Part)

Answer 8

• four MD forces
  
  • F_C ≡ bead-bead conservative
  • F_k ≡ bead-bead rigid-body constraint
  • F_s ≡ stochastic fluctuations of fluid
  • F_D ≡ soulte-solvent interaction

Question 9

LBM + MD (LB Part)

Answer 9

• S_i ≡ particle-fluid reaction → coupling
  
  • _{<span><i>w</i></span>i} ≡ lattice-based weights

Question 10

LBM + MD (Coupling)

Answer 10

• spatial:
  
  • interpolate fluid velocity off-lattice to particle
• temporal:
  
  • ∆t_LB = M∆t_MD

Question 11

LBM + MD (Algorithm)

Answer 11

1. Interpolate fluid velocity off-lattice to particle
2. Perform MD for M timesteps
3. Extrapolate molecular foruces to fluid lattice
4. Perform LBM timestep