Fluid-Particles Interaction Models Asymptotics, Theory and Numerics I

Fluid-Particles Interaction Models Asymptotics, Theory and Numerics I J. A. Carrillo collaborators: T. Goudon (Lille), P. Lafitte (Lille) and F. Vecil (UAB) (CPDE 2005), (JCP, 2008), (JSC, 2008) ICREA - Universitat Autònoma de Barcelona Summer School in IMS-NUS, 04/08/09

Outline 1 Motivations: Aerosols:Fumes/Dusts/Sprays 2 Micro/Meso/Macroscopic Modelling 3 Modelling Fluid-Particles Interaction

Atmospheric Pollution Modelling Pollution in Los Ángeles, Madrid, Beijing and Singapore.

Formation of Aerosols in the atmosphere Pandis & Seinfeld (1998), Madelaine (1982)

Automotive Industry: Diesel Engines KIVA Code: Los Alamos National Laboratory, Amsden - O Rourke

Aerosols in Medecine Boudin, Grandmont, Baranger, Desvillettes (INRIA)

Statistical description Particles Description: impossible due to their huge number. Kinetic Description: f (t, x, ξ) represents the number density of particles at time t in position x with velocity ξ. Hydrodynamic Description: Continuum mechanics approach based on balance equations for density, momentum and temperature.

Macroscopic Quantities: Moments Particle density: ρ(t, x) = Momentum: Z J(t, x) = ρu(t, x) = R 3 f (t, x, ξ) dξ Z R 3 ξ f (t, x, ξ) dξ Temperature: 3ρθ(t, x) = ξ U(t, x) 2 f (t, x, ξ) dξ R 3 Z

Models: Continuum versus Kinetic Full Continuum models: fluid equations for two phase flows involving the density ρ i(t, x), the velocity v i(t, x), the temperature θ i(t, x) for each phase and the pressure p(t, x) (usually common). If needed, one has to consider the volume or mass fraction occupied by the phases m f (t, x). Kinetic-Fluid coupling: the dense phase is assume to follow certain fluid equations: isentropic compressible Euler, compressible Euler with energy equation, compressible or incompressible Navier-Stokes system. The dispersed phase is modeled at the mesoscopic level by a number density of particles f (t, x, ξ) in phase-space (x, ξ) per infinitesimal volume at time t 0.

Assumptions of the model Two phase flow: Dense phase Fluid: continuum mechanics description in terms of density of the fluid n(t, x) and velocity field u(t, x). Let ρ F a typical value of the fluid mass per unit volume. Fluid Equations 1: Compressible Euler: ρ F tn + div x(nu) = 0, t(nu) + Div x(nu u) nf F + xp(n) = 0 with p = p(n) a general pressure law and F F the force acting per unit mass on the fluid. Fluid Equations 2: Incompressible Navier-Stokes with uniform density ρ F : div x(u) = 0, tu + Div x(u u) + xp = ν u + F F with the kinematic viscosity ν = µ/ρ F with µ the dynamic viscosity of the fluid.

Assumptions of the model Two phase flow: Dispersed phase Particles: kinetic description in terms of the number density of particles f (t, x, ξ) in phase space (x, ξ) to compute velocity fluctuations around the fluid velocity u(t, x). Particles are spheres of radius a > 0 with mass given by m P = 4 3 ρ Pπa 3, ρ P being the particle mass per unit volume. Particles are assumed to follow Newton s law: x = F P(t, x, x ) with F P the force per unit mass on the particles. Collisions (depending on application): Fluid-particles: Brownian motion. Particle-Particle: inelasticity, Boltzmann type terms. Coagulation-Break up: it needs to enter a new variable, the radius of the droplet. Reactive flows: it needs to enter internal energy as a new variable. Particles Equation: Vlasov-Fokker-Planck or Vlasov-Boltzmann

Vlasov kinetic equation: Sparse/Thin aerosol Assumptions: The volume fraction occupied by the aerosol is negligible but not the mass. The particles are assumed to be incompressible (solid particles) and collisions are neglected. The characteristic system associated to the particles subjected to the force F P(t, x, ξ) is given by: 8>< >: d X (t; t0, x, ξ) = Ξ dt, X (t0; t0, x, ξ)) = x d Ξ(t; t0, x, ξ) = F P(t, X ) dt, Ξ(t0; t0, x, ξ)) = ξ

Vlasov kinetic equation: Sparse/Thin aerosol Z Assume that the number density of particles is "transported" through the characteristics: f (t, x, ξ) dx dξ = f 0(x, ξ) dx dξ Ω t Ω 0 for any set Ω 0 with Ω t = (X, Ξ)(t; 0, Ω 0). Z Vlasov Equation: 8 > < > : (Exercise) f t + ξ xf + div ξ(f Pf ) = 0 f (0, x, ξ) = f 0(x, ξ)

Forces to be considered Friction or Drag Forces: The fluid produces a friction force on the particles 6πµa u(t, x) ξ, with µ > 0 being the dynamic viscosity of the fluid. This is given by Stokes law used for moderate Reynolds number Re = LU. µ Rayleigh s law assume that the drag force is proportional to u(t, x) ξ u(t, x) ξ. One can also assume a density dependent viscosity. Accordingly, by action-reaction the force exerted by the particles on the fluid is given by the sum 6πµa Z R 3 ξ u(t, x) f dξ.

Forces to be considered Gravity+Buoyancy: External forces per unit volume and mass acting on the particles xφ and on the fluid αρ F xφ. α R is a dimensionless parameter which measures the ratio of the strength of the external force on each phase: by Archimedes rule. Φ(x) = (1 ρ F /ρ P )gx 3 α = 1 1 ρ F /ρ P,

Forces to be considered Stokes Settling velocity: Given a particle dropped in a fluid at rest, having viscosity µ and mass per unit volume ρ F, the motion of the particle is described by: 4 3 πρ Pa 3 d 2 dt X(t) = 6πµa d 2 dt X(t) 4 3 πρ Pa 3 g 1 ρ F. Then, as t, the speed of the particle d X(t) has a limit given by dt ρ P with V S = T S g 1 ρ F ρ P T S =. m P 6πµa = 2ρ Pa 2 9µ.

Fluid-Particle Collisions Brownian motion: Particles are assumed to follow a Brownian motion: x F P(t, x, x ) = Γ(t), where Γ(t) is a Wiener process with final diffusion coefficient given by Einstein formula: D ξ = kθ0 6πµa = kθ0 9µ m P m P m P 2a 2 ρ P where k stands for the Boltzmann constant, and θ 0 > 0 controls the noise strength. This translates onto a diffusion term on the kinetic equation of the form D ξ ξ f on the right-hand side. (Check in Risken or Gardiner).

Disperse Phase Vlasov-Fokker-Planck equation: tf + ξ xf xφ ξ f = 9µ 2a 2 ρ P Fokker-Planck term: div ξ (ξ u)f + kθ0 ξ f. m P The Fokker-Planck term implies a relaxation in velocity towards equilibrium densities of the form 3/2 o ρ(t, x)m(ξ) = ρ(t, x) 2π kθ0 exp n m m P ξ u(t, x) 2 /2kθ 0, P with typical Stokes relaxation time given by (Exercise) T S = m P 6πµa = 2ρ Pa 2 9µ.

Final PDE Model Vlasov-Euler-Fokker-Planck system: We arrive at the system: tf + ξ xf xφ ξ f = 9µ div 2a 2 ξ (ξ u)f + kθ0 ξ f, (1) ρ P m P Z ρ F t(nu) + Div x(nu u) + αn xφ + xp(n) = 6πµa tn + div x(nu) = 0, (2) R 3 (ξ u)f dξ. (3) where k stands for the Boltzmann constant, and θ 0 > 0 controls the noise strength and p(n) is a general pressure law, for instance p(n) = C γ n γ, γ 1, C γ > 0.

Vlasov-Boltzmann kinetic equation: Moderately Dense aerosols Moderately Dense Aerosols: Particle-Particle interaction is no longer negligible: Solid Particles: particles are assumed to be incompressible, then elastic/inelastic collisions can be included in the modelling given rise to Vlasov-Boltzmann equations: coupled to: ρ F f t + ξ xf + div ξ(f Pf ) = Q(f, f ) tn + div x(nu) = 0, t(nu) + Div x(nu u) nf F + xp(n) = 0 by the frictional forces included in F P and F F. Liquid Particles (droplets): in this case the statistics of particles need an additional degree of freedom, the radius of the particles r > 0, f (t, x, ξ, r). Operators with fragmentation and coagulation has to be taken into account in Q(f, f ).

Vlasov-Boltzmann kinetic equation: Dense aerosols Dense Aerosols: Volume or mass fraction of the dispersed phase is no longer negligible. The equation are of the form: f t + ξ xf + div ξ(f Pf ) = Q(f, f ) coupled to: ρ F t(m f n) + div x(m f nu) = 0, t(m f nu) + Div x(m f nu u) m f nf F + xp(n) = 0 by the frictional forces included in F P and F F with 1 m f = Z R 3 Z 0 4 3 πρ Pr 3 f (t, x, ξ, r) dr dξ.