Transport: Two limits: Collisionless and hydrodynamic Mean-free path: σ - scattering cross-section → σ or cross-section λ - contain one collision center on average. λ = 1 / (nσ) n: density of collision center Scattering cross-section of two hard balls: σ = π (2r)² = 4πr² [Diagram of two hard balls colliding] [Diagram illustrating diffusion and hydrodynamic flow] Effusion distribution of momentum f(p)d³p = density of particles with momentum in d³p of p f(p) = C * e^(-p²/2m) Boltzmann distribution ∫ f(p)d³p = n total density of particles => C = n * (2πmkT)^(-3/2) Flux of effusion (of particle per area per second) n*0 I = ∫₀ᵤₓ f(p)d³p = C * ∫₀ᵤₓ ∫ p/m * e^(-p₁²+p₂²+p₃²)/2mkT dp₁dp₂dp₃ = n * √(kT/2πm) I α n * T^(1/2) * m^(-1/2) Why? Physical picture Navier-Stokes equation ΔV (mass) $\left(\frac{\partial}{\partial t} + \vec{u}\cdot\nabla\right)u_{i} = $ force in i-direction = A∏ j (P ij (x+ex j )+P ij (x)) ⇒ ( ∂t ∂ + u ⋅∇)u i + ∂x j ∂P ij =0 ⇒ ( ∂t ∂ + u ⋅∇) u +∇⋅(P− 3 γ ∇⋅ u )−ν∇ 2 u =0 Any quantity can be set to 1 by changing the unit. ∂t ∂L +v⋅(ρL)=0 ( ∂t ∂ + u ⋅∇)(pρ −γ )=0 The above three equations determine the characters of flow in the pipe quantities The effect of viscosity: L F/V = length force/area = L m = t 2 L = L 2 ν = L 2 ν = L 2 t m Dimensionless Reynolds number R= μ ρLu 0

IX. Transport

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