du Langevin Langevin + = u n ( t ) Equation Equation dt - PowerPoint PPT Presentation

! Introduction to Aerosols ! Introduction to Aerosols ! Drag Forces ! Drag Forces ! Cunningham Corrections ! Cunningham Corrections ! Lift Forces ! Lift Forces ! Brownian Motion ! Brownian Motion ! Particle Deposition Mechanisms ! ! Gravitational Sedimentation ! ! Aerosol Coagulation ! ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi du Langevin Langevin + β = u n ( t ) Equation Equation dt β = πµ = τ 3 d / C m 1 / c N(t) = White Noise N(t) = White Noise β 2 kT = S nn Spectral Intensity Spectral Intensity π m A Particle under Random Molecular Impact A Particle under Random Molecular Impact ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi 1

ω = ω ω 2 S ( ) | H ( ) | S ( ) τ = + τ ) Autocorrelation Autocorrelation R ( ) u ( t u ( t ) uu nn Function Function 1 System System ω = H ( ) ω + β Function Function +∞ i 1 ∫ τ = ωτ ω ω i R ( ) e S ( ) d uu uu 2 β π Response Power Response Power 2 kT / m − ∞ Fourier Fourier ω = S ( ) Spectrum Spectrum uu ω + β 2 2 Transform Transform +∞ 1 ∫ ω = − ωτ τ τ i S ( ) e R ( ) d π uu uu kT Autocorrelation Autocorrelation τ = − β τ | | R ( ) e − ∞ uu m ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi Mass t Mass 1 d ∫ = − τ τ τ 2 = Variance Variance x ( t ) 2 ( t ) R ( ) d 2 D x ( t ) Diffusivity Diffusivity uu 2 dt 0 ∞ ∫ = τ τ t D R ( ) d ∫ uu = x ( t ) u ( t 1 dt ) Position Position 0 1 Diffusivity Diffusivity 0 kTC kT = = c t t D ∫∫ Variance Variance = τ − τ τ τ 2 β πµ x ( t ) R ( ) d d m 3 d uu 1 2 1 2 0 0 ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi 2

Fokker- -Planck Equation Planck Equation F ( x ) Fokker Langevin Langevin + β − = & & & x x n ( t ) Equation Equation m ∂ ∂ β ∂ 2 f kT f − β = ( uf ) ∂ V ( x ) ∂ ∂ ∂ = − 2 t u m F ( x ) u ∂ x Fokker- Fokker -Planck Equation Planck Equation mu 2 1 − = Probability Probability ∂ ∂ ∂ β ∂ f e 2 kT & 2 f ( x f ) 1 kT f = − + β − + π & [( x F ( x )) f ] Density Density 2 kT / m ∂ ∂ ∂ ∂ & & 2 t x x m m x ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi = n ( t ) 0 x & 2 White Noise m x F ( x ) dx ∫ = − − 1 f C exp{ [ ]} 0 kT 2 m = π δ − 0 n ( t ) n ( t ) 2 S ( t t ) Probability Probability 1 2 nn 1 2 Density Density 2 & 1 m x ! Choose a time step ( ∆ t<< τ ) = − + f C exp{ [ V ( x )]} 0 kT 2 ! Generate a sequence of uniform random numbers (0 <U<1) 2 − & Gravitational Gravitational m x mg ( x x ) − − 0 = ! Transform to Gaussian random f C e 2 kT e kT Field Field 0 numbers. ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi 3

= − π ! G 2 ln U cos 2 U 1 1 2 n i = − π G 2 ln U sin 2 U 2 1 2 ! ! Amplitude of the Brownian force t ∆ t π is given by S U ∆ t = nn n ( t ) G ∆ i i t ! The generated sample of Brownian U ∆ t force need to be shifted by ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi Brownian Force Brownian Force n i t ∆ t U ∆ t ME 437/537 G. Ahmadi ME 437/537 Ounis, Ahmadi and McLaughlin (1991) G. Ahmadi 4

= x 2 ( t ) 2 Dt Simulations Mean + σ Mean - σ ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi Mean + σ Mean + σ Mean - σ Mean - σ ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi 5

z t = o Equation of Motion N N erfc ( ) Equation of Motion Equation of Motion o 4 Dt Simulations p d 1 u = − + + f p ( ) ( t ) u u g n τ dt σ = 2 Variance ( t ) 2 Dt Variance Variance y ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi ME 437/537 G. Ahmadi 6