From a Mesoscopic to a Macroscopic Description of Fluid-Particle Interaction

Transcription

1 From a Mesoscopic to a Macroscopic Description of Fluid-Particle Interaction Carnegie Mellon University Center for Nonlinear Analysis Working Group, October 2016

2 Outline 1 Physical Framework 2 3 Free Energy Stationary Solutions 4 Flowing Regime Bubbling Regime

3 Assumptions on Fluid and Particles Fluid is inviscid and compressible Only one type of particle in the fluid Particles are uniform spheres with density ϱ P and radius a Fixed spatial domain Ω R 3 System is at a fixed, constant temperature θ 0 > 0. Fluid is described by density ϱ(x, t) [0, ) and velocity field u(x, t) R 3.

4 Description of Particles Particle distribution in the fluid is described by density function f (x, ξ, t) [0, ) where ξ is the microscopic velocity fluctuation. Particles subject to Brownian motion, leading to diffusion in ξ, with diffusion constant kθ 0 6πµa = kθ 0 9µ m p m p m p 2a 2 ϱ P where k is the Boltzmann constant and µ the dynamic viscosity of the fluid. Macroscopic particle density η(x, t) given by η(x, t) := f (x, ξ, t) dξ. (1) R 3

5 Coupling of Fluid and Particles Coupling of the system is due to the friction between the particles and the fluid following Stokes Law Definition (Stokes Law) Consider a uniform, spherical particle of radius a. The friction force exerted on a particle by the fluid is F (x, ξ, t) = 6πµa[u(x, t) ξ] (2) Thus, the force exerted on the fluid by the particles is 6πµa [ξ u(x, t)]f (x, ξ, t) dξ R 3 by Newton s Third Law.

6 External Force Physical Assumptions Both fluid and particles are influenced by an external force with a time independent potential Φ(x). Force exerted on a particle: m P x Φ. Force exerted per unit volume on fluid: αϱ F x Φ. α: dimensionless constant measuring ratio of external force s strength on fluid and particles ϱ F : typical value of fluid mass per volume Measures settling phenomena such as gravity, buoyancy, and centrifugal forces.

7 Vlassov-Euler System t ϱ + div x (ϱu) = 0 (3) t (ϱu) + div x (ϱu u) + 1 ϱ F x p(ϱ) t f + ξ x f x Φ ξ f = αϱ x Φ + 6πµa ϱ F = 9µ 2a 2 ϱ P div ξ R 3 (ξ u)f dξ (4) [ (ξ u)f + kθ ] 0 ξ f m P We assume a pressure of the form p(ϱ) = κϱ γ where κ > 0 and γ > 1. (5)

8 Unitless Parameters I In order to find a macroscopic model, we transform (3)-(5) to a unitless model with dimensionless parameters, and scale these parameters, then take the appropriate limit. Stokes settling time T S = Thermal speed kθ 0 V th = m P m P 6πµa = 2ϱ Pa 2 9µ

9 Unitless Parameters II We also define characteristic time T, length L, and velocity U = L/T. We define a pressure unit P and associate a velocity V S to the external potential Φ. Thus, the physical values in relation to the dimensionless parameters are ( indicates unitless quantity) t = Tt x = Lx ξ = V th ξ ϱ(lx, Tt ) = ϱ (x, t ) u(lx, Tt ) = Uu (x, t) p(lx, Tt ) = Pp (x, t ) f (x, ξ, t ) = 4 3 πa3 Vth 3 f (Lx, V th ξ, Tt ) Φ(Lx ) = V S L T S Φ (x )

10 Unitless Parameters III We also define the unitless constants β = T L V th = V th U, 1 ε = T T S, n = V S T V th T S, χ = PT ϱ F LU = P ϱ F U 2

11 Unitless Vlassov-Euler System Using the previous relations in (3)-(5) yields after dropping primes t ϱ + div x (ϱu) = 0 (6) t (ϱu) + div x (ϱu u) + x (χp(ϱ)) = αβnϱ x Φ + 1 ϱ P (βξ u)f dξ (7) ε ϱ F R 3 t f +βξ x f n x Φ ξ f = 1 ε div ξ [(ξ 1β ) ] u f + ξ f (8)

12 Free Energy Stationary Solutions Pressure and Internal Energy The enthalpy is defined as h(ϱ) := ϱ 1 p (s) s ds and is in L 1 loc (0, ). The internal energy is defined as Π(ϱ) := ϱ 0 h(s) ds

13 Free Energy Stationary Solutions Free Energy Assume that ϱ P = 1 and n = β. ϱ F β2 The free energies are The fluid free energy 1 F F (ϱ, u) = 2 ϱ u 2 + χπ(ϱ) + αβ 2 ϱφ dx (9) Ω The particle free energy R3 F P (f ) = f ln f + ξ 2 f + f Φ dξ dx (10) 2 Ω The total free energy F(ϱ, u, f ) = F F (ϱ, u) + F P (f ) (11)

14 Free Energy Stationary Solutions Energy Dissipation Theorem Assuming the scaling above, we have the following dissipation. d dt F + 1 (ξ β 1 u) 2 f + 2 ξ f dξ dx 0. (12) ε Ω R 3 This result follows formally from integration by parts.

15 Free Energy Stationary Solutions External Force Mathematical Assumptions exp( Φ) L 1 (Ω), Φ exp( Φ) L 1 (Ω). Φ W 1,1 (Ω) for bounded Ω; Φ W 1,1 loc (Ω) for unbounded Ω. αω is bounded below on Ω. The sub-level sets of αφ are bounded, that is {x Ω αφ k} is bounded for any k R.

16 Free Energy Stationary Solutions Stationary Solutions I Provided the total fluid mass is conserved in time and finite, the system (6)-(8) has a stationary solution (ϱ s, u s, f s ) such that u(x) 0 The stationary particle density function is f s (x, ξ) = Z P e Φ(x) e ξ 2 /2 where ( Z P = Ω (2π) 3/2 ) ( 1 f 0 dξ dx e dx) Φ(x). R 3 Ω

17 Free Energy Stationary Solutions Stationary Solutions II The stationary fluid density is given by ( ϱ s (x) = σ Z F αβn ) χ Φ(x) where ( ) ( Z F = ϱ 0 dx e Φ(x) dx Ω Ω and σ is the generalized inverse of h. ) 1

18 Flowing Regime Bubbling Regime Estimates I Assuming that ) (1 + ln(f 0 ) + ξ Φ dξ dx Ω R 3 f 0 and Ω ϱ 0 + ϱ 0 u Π(ϱ 0 ) + ϱ 0 βn αφ dx are both bounded, we can use the free energy inequality and the hypotheses on Φ to obtain the following uniform bounds

19 Flowing Regime Bubbling Regime Estimates II f (1 + ξ 2 + Φ + ln f ) is bounded in L (R + ; L 1 (Ω R 3 )) ϱ, Π(ϱ) and βnϱ αφ are bounded in L (R + ; L 1 (Ω)) ϱu is bounded in L (R + ; L 2 (Ω)) 1 ε [ (ξ β 1 u ) f + 2 ξ f ] is bounded in L 2 (R + Ω R 3 ).

20 Flowing Regime Bubbling Regime Expansions of Macroscopic Particle Quantities We define the unitless first moment of f as J(x, t) = β ξf (x, ξ, t) dξ R 3 and unitless second moment P(x, t) = ξ ξ f (x, ξ, t) dξ. R 3 Using the uniform bounds, these quantities can be expanded as J = uη + β εk and P = ηi + β 2 J u + εk where the components of K and K are bounded in L 2 (R + ; L 1 (Ω)).

21 Flowing Regime Bubbling Regime Flowing Regime Scaling We are interested when the settling time scale is much smaller than the observational time scale, that is T S T so ε is small. We are interested then in the limit ε 0. We take β 2 = ϱ F /ϱ P to be a constant and n = β a fixed positive constant. Thus, V S U = V th. ϱ F and ϱ P are of the same order.

22 Flowing Regime Bubbling Regime Flowing Regime Vlassov-Euler System t ϱ ε + div x (ϱ ε u ε ) = 0 (13) t (ϱ ε u ε ) + div x (ϱ ε u ε u ε ) + x (χp(ϱ ε )) = αβ 2 ϱ ε x Φ + 1 εβ 2 (ξ u ε )f ε dξ (14) R 3 t f ε + β(ξ x f ε x Φ ξ f ε ) = 1 [( ε div ξ ξ 1 ) ] β u ε f ε + ξ f ε (15)

23 Flowing Regime Bubbling Regime Macroscopic Limit of Flowing Regime Assuming that the limits of the various unknown quantities and their non-linear combinations involved in the system exist, the limits ϱ, u, and η as ε 0 are t ϱ + div x (ϱu) = 0 (16) t [(ϱ + β 2 η)u] + div x [(ϱ + β 2 η)u u] + x (χp(ϱ) + η) = (αβ 2 ϱ + η) x Φ (17) t η + div x (ηu) = 0 (18)

24 Flowing Regime Bubbling Regime Bubbling Regime Scaling Again, the settling time scale is much smaller than the observational time scale, that is T S T so ε is small. We are interested then in the limit ε 0. Again, β 2 = ϱ F /ϱ P and n = β, but β = ε 1/2 and α = sgn(α)ε. Physically, V S = U V th.

25 Flowing Regime Bubbling Regime Bubbling Regime Vlassov-Euler System t ϱ ε + div x (ϱ ε u ε ) = 0 (19) t (ϱ ε u ε ) + div x (ϱ ε u ε u ε ) + x (χp(ϱ ε )) ( ) ξ = sgn(α)ϱ ε x Φ + ε u ε f ε dξ (20) R 3 t f ε + 1 ε (ξ x f ε + x Φ ξ f ε ) = 1 ε div ξ[(ξ εu ε )f ε + ξ f ε ] (21)

26 Flowing Regime Bubbling Regime Macroscopic Limit of Bubbling Regime Assuming that the limits pass as ε 0, t ϱ + div x (ϱu) = 0 (22) t (ϱu) + div x (ϱu u) + x (χp(ϱ) + η) = (sgn(α)ϱ + η) x Φ (23) t η + div x (ηu η x Φ) = x η (24)

27 Flowing Regime Bubbling Regime Remarks We assume that f ε, ϱ ε, and ε converge to f, ϱ, and u in each regime, as well as any non-linear terms converge. While we have weak compactness for the sequences (f ε ), (ϱ ε ), (u ε ), and ( ϱ ε u ε ) from the energy inequality, this is not the case for the non-linear terms. Such rigor can be shown in the case of a viscous fluid. Even in the case of no external force, the evolution of the fluid still depends on the evolution of the particle density.

28 J. A. Carrillo and T. Goudon. Stability and asymptotic analysis of a fluid-particle interaction model. Comm. Partial Differential Equations, 31(7-9): , A. Mellet and A. Vasseur. Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations. Math. Models Methods Appl. Sci., 17(7): , A. Mellet and A. Vasseur. Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations. Comm. Math. Phys., 281(3): , 2008.