On Fluid-Particle Interaction

Transcription

1 Complex Fluids On Fluid-Particle Interaction Women in Applied Mathematics University of Crete, May 2-5, 2011 Konstantina Trivisa

2 Complex Fluids 1 Model 1. On the Doi Model: Rod-like Molecules Colaborator: Hantaek Bae 2 Model 2. On Fluid-Particle Interaction: The Bubbling Regime Collaborators: Jose Carrillo and Trygve Karper

3 On the Doi Model for the Suspensions of Rod-Like Molecules MODEL 1 On the Doi Model for the Suspensions of Rod-Like Molecules May, 2011

4 On the Doi Model for the Suspensions of Rod-Like Molecules OUTLINE: 1 Description of the Model. Motivation 2 Notion of Weak Solutions 3 Main Result. Global-in-Time Existence 4 Quasi-compressible Approximation 5 Propagation of Compactness Method 6 Future Program

5 On the Doi Model for the Suspensions of Rod-Like Molecules Modeling Polymeric fluids arise in many practical applications: biotechnology, medicine, fuel droplets in combustion, sprays etc. Here, we focus on the Doi model for suspensions of rod-like molecules in a dilute regime. The Doi model describes the interaction between the orientation of rod-like polymer molecules at the microscopic scale and the macroscopic properties of the fluid in which these molecules are contained (cf. Doi and Edward [5]).

6 On the Doi Model for the Suspensions of Rod-Like Molecules Modeling The macroscopic flow leads to a change of the orientation and, in the case of flexible particles, to a change in shape of the suspended microstructure. This process, in turn yields the production of a fluid stress. As a first approximation, we view the identical liquid crystal molecules as inflexible rods of a thickness b, which is much smaller than their length L. In the dilute regime the rods are well separated, as expressed by b L 3.

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8 On the Doi Model for the Suspensions of Rod-Like Molecules Modeling The orientation distribution of the rods f = f (t, x, τ) is described by a Fokker-Planck-type equation, with t f + u f + τ (P τ ( x uτ)f ) D r τ f D x f = 0 S d 1 f (t, x, τ)dτ = 1, f 0, describing the time-depended probability that a rod with a center mass at x has an axis τ in the area element dτ.

9 On the Doi Model for the Suspensions of Rod-Like Molecules Modeling u = u(t, x) represents the velocity field { the change of f due to the displacement u x f of the center of the mass of the rods by advection τ (P τ ( x uτ)f ) the shear-forces acting on the rods.

10 On the Doi Model for the Suspensions of Rod-Like Molecules Modeling { the Brownian effects: translatational diffusion (D x f, D r τ f ) and rotational diffusion respectively; D, D r denote the diffusivity parameters Diffusion can be seen as a gradient flow of the entropy functional E[f ] := νk B T f ln fdndx. Ω S 2

11 On the Doi Model for the Suspensions of Rod-Like Molecules Modeling The fluid flow is given by the Navier-Stokes equation, which is now enhanced by an additional macroscopic stress reflecting the orientation of the rods on the molecular level, { NS t u + u u u + p = σ, u = 0. Here, p denotes the pressure and σ the macroscopic stress tensor derived from the orientation of the rods at the molecular level and is given by σ = (3τ τ Id)fdτ. S 2

12 On the Doi Model for the Suspensions of Rod-Like Molecules Modeling The system of equations now reads f t + u x f + τ (P τ ( x uτ)f ) τ f x f = 0 Doi-FPNS σ = (3τ τ Id)fdτ S 2 u t + u u u + p = σ u = 0 f (0, x, τ) = f 0 (x, τ), u(0, x) = u 0 (x),

13 On the Doi Model for the Suspensions of Rod-Like Molecules Modeling where P τ ( x uτ) = x uτ (τ x uτ)τ is the projection of x uτ on the tangent space of the (d 1)-dimensional space S d 1 at τ S d 1. τ, τ the gradient and the Laplace-operator on the unit sphere, x the gradient in R d, d = 2, 3 σ = 3 j=1 σ ij x j.

14 On the Doi Model for the Suspensions of Rod-Like Molecules Modeling { Boundary Conditions u n = 0, u tan + (D(u)n) tan = 0, σ = 0 on Ω, with tan - the tangential component of the vector field at the boundary, n outer normal vector to the boundary Ω. Initial Conditions u 0 L 2 n, u 0 = 0, f 0 L 2 xwτ s,2, ρ 0 (x) = f 0 (x, τ)dτ L 2 S 2 such that s > 5 2 and Ω S 2 (f 0 log f 0 f 0 + 1)dτdx C

15 On the Doi Model for the Suspensions of Rod-Like Molecules Modeling W 1,r n = {u; u W 1,r (Ω), tru n = 0 on Ω}, where tr means the trace operator onto the boundary. Since the velocity field is incompressible, we define a subspace of W 1,r n,div such that W 1,r 1,r n,div = {u Wn ; u = 0}.

16 On the Doi Model for the Suspensions of Rod-Like Molecules Modeling L 2 n = {u W 1,2 n,div } L 2 subspace of L 2 W 1,r n = (W 1,r n ), W 1,r n,div = (W 1,r n,div ), where r is the conjugate of r. Helmholtz decomposition of a vector field Let v Wn 1,q. Let g be a solution of the following elliptic problem. g = v in Ω, g n = 0 on Ω, gdx = 0. Then, we can define the divergence free part of v as v div = v g. Ω

17 On the Doi Model for the Suspensions of Rod-Like Molecules Notion of weak solutions Weak solutions We say that (u, p, f ) is a weak solution to (Doi-FPNS) if u C([0, T ]; L 2 weak (Ω)) L2 (0, T ; W 1,2 n,div ), u t L 5 3 (0, T ; W 1, 5 3 n ) In addition, = T 0 T 0 p L 5 3 (0, T ; L 5 3 (Ω)) Ω C 1,1 Rf L (0, T ; L 2 (K)) L 2 (0, T ; W 1,2 (K)) ( < u t, ψ > (u u, ψ) + (D(u), D(ψ)) + (u, ψ) Ω ) dt ( ) (p, ψ)+ < σ, ψ > dt for all ψ L 2 (0, T ; Wn 1,2 )

18 On the Doi Model for the Suspensions of Rod-Like Molecules Notion of weak solutions T 0 < Rf t, Ψ > < R(f )u, Ψ > < R(P τ uτf ), τ Ψ >dt T + 0 where R = (1 τ ) s. < R τ f, τ Ψ > + < R f, Ψ > dt = 0 for all Ψ L (0, T ; W 1,2 (K))

19 On the Doi Model for the Suspensions of Rod-Like Molecules Notion of weak solutions Remarks The operator R = (1 τ ) s 2, s > 5 2, satisfies the following properties: [R, x ] = 0, (1) R τ : L 1 (S 2 ) L p (S 2 ) bounded for any p > 1. (2) R : L 2 (S 2 ) H s (S 2 ) is bounded. (3)

20 On the Doi Model for the Suspensions of Rod-Like Molecules Notion of weak solutions As one can see from the definition of weak solution of the density term f, we apply the operator R on f and deal with Rf instead of f. Note, the nonlinear term ( ) R(P τ uτf ), τ Ψ Ω Ω S 2 i u j [ R ( i u j τ j f ) ] τi Ψ dτdx S 2 ( τ j fr ( τi Ψ )) dτdx Ω i u j S 2 i u j Ω S 2 Ω Ω S 2 i u j K = [ R ( τ i i u j τ j f ) ] τ τ Ψ dτdx= S 2 ( RfR 1( τ j R( τi Ψ) )) dτdx= ( RfR 1( τ i τ j R(τ τ Ψ) )) dτdx. ( τ i τ j fr ( τ τ Ψ )) dτdx=

21 On the Doi Model for the Suspensions of Rod-Like Molecules Notion of weak solutions In the sequel, we require to take the limit to { Rf (m)} and not { f (m) }. Next, we obtain uniform bounds of { Rf (m)} in L ( 0, T ; L 2 (K) ) L 2( 0, T ; W 1,2 (Ω S 2 ) ), which is enough to pass to the limit to the nonlinear term ( ) R(P τ uτf ), τ Ψ. By the Navier boundary condition, all integrals in the definition of weak solution are finite. Moreover, K tru L 2 (0, T ; L 2 ( Ω))

22 On the Doi Model for the Suspensions of Rod-Like Molecules Notion of weak solutions Lemma Let q 1 1 and r, q 2 (1, ). Let S be defined as S = { v L (0, T ; L 2 (Ω)) L r (0, T ; Wn 1,r (Ω)), } v t L q 1 (0, T : W 1,q 2 n,div (Ω)). If {v i } is bounded in S and r ( 2d d+2, 2), then {trv i } is precompact in L p (0, T : L s ( Ω)), where ( 2d 2 s, d r(d 1) ) dr + 2r 2d, p < s d r sd 2d + 2. Lemma Let {v i } be bounded in S with d = 3 and r = 2. Then, trv i is precompact in L 2 (0, T ; L 2 ( Ω)) and also in L q (0, T ; L 4 3 ( Ω)) for all q [1, ).

23 On the Doi Model for the Suspensions of Rod-Like Molecules Notion of weak solutions Theorem Let Ω be a three dimensional bounded domain. Suppose that the initial data satisfy the initial conditions. Then, there is a weak solution to (Doi-FPNS). Moreover, a solution satisfies the following entropy inequality d ( u 2 L + dt 2 R 3 S 2 (f log f f + 1)dτdx + D(u) 2 L 2 + α u 2 L 2 ( Ω) 0 ) + 4 τ f 2 dτdx R 3 S 2

24 On the Doi Model for the Suspensions of Rod-Like Molecules Notion of weak solutions Definition of (ɛ, η) approximations Quasi-compressible approximation to divu = 0. ɛ p = u in Ω, p n = 0 on Ω, Ω pdx = 0. Next we define a divergenceless, smooth η-approximation of the velocity field u η = ((λ η u) w η ) div, where w η denotes the standard mollification with kernel w η having the support in a ball of radius η and λ η a cut-off function.

25 On the Doi Model for the Suspensions of Rod-Like Molecules Notion of weak solutions f t + u η x f + τ (P τ ( x uτ)f ) τ f = 0 σ = (3τ τ Id)fdτ S 2 (FPNS) ɛ,η u t + u η u u + p = σ ɛ p = u f (x, τ, 0) = f 0 (x, τ), u(x, 0) = u 0 (x)

26 On the Doi Model for the Suspensions of Rod-Like Molecules Notion of weak solutions Definition We say a triple (u (ɛ,η), p (ɛ,η), f (ɛ,η) ) = (u, p, f ) is a weak solution to (FPNS) ɛ,η if u C([0, T ]; L 2 weak (Ω)) L2 (0, T ; W 1,2 n,div ), u t L 2 (0, T ; W 1,2 n ) p L 2 (0, T ; W 1,2 (Ω)) Rf L (0, T ; L 2 (K)) L 2 (0, T ; W 1,2 (K)) and the following integral relations hold ɛ( p, π) = (π, u) for all π W 1,2 for a.e. t [0, T ] ) T 0 ((u t, ψ) (u η u, ψ) + (D(u), D(ψ)) + (u, ψ) Ω dt ( ) (p, ψ) + ( σ, ψ) dt for all ψ L 2 (0, T ; Wn 1,2 ) = T 0 [ T 0 (Rf t, Ψ) (R(f )u η, Ψ) (R(P τ uτf ), τ Ψ) ] +(R τ f, τ Ψ) + (R f, Ψ) dt = 0, Ψ L (0, T ; W 1,2 (K))

27 On the Doi Model for the Suspensions of Rod-Like Molecules Notion of weak solutions Uniform estimates of (u, p) Uniform estimates of p and u t To obtain uniform estimates on the pressure p, we consider the following Neumann problem. h = p β 2 p 1 p β 2 pdx in Ω Ω Ω h n = 0 on Ω, hdx = 0 Ω

28 On the Doi Model for the Suspensions of Rod-Like Molecules Notion of weak solutions By the elliptic regularity theory of the Neumann problem, h β W 1,β (Ω) p β L β Taking ψ = h in (2.18) leads to T 0 p β L β dt = I 1 + I 2 + I 3 + I 4 + I 5

29 On the Doi Model for the Suspensions of Rod-Like Molecules Notion of weak solutions The drift term in τ Rf R τ (P τ ( x uτ)f )dτ x u Rf L 2 R τ (P τ ( x uτ)f ) L 2 S 2 x u Rf L 2 f L 1 x u Rf L 2. Rf L (0, T ; L 2 (K)) L 2 (0, T ; W 1,2 (K)).

30 On the Doi Model for the Suspensions of Rod-Like Molecules Notion of weak solutions D=0 t f + u f + τ (P τ ( x uτ)f ) D r τ f = 0 The proof of the global existence of solutions is based on propagation of compactness, namely if we take a sequence of weak solutions which converges weakly and such that the initial data converges strongly then the weak limit is also a solution. Evolution of defect measures R(f (m) f ) 2 ν, u (m) u 2 β, σ (m) σ 2 α.

31 On the Doi Model for the Suspensions of Rod-Like Molecules Notion of weak solutions After some analysis we deduce that β α νdτ. S 2 Since { Rf (m j ) } converges strongly to Rf in L 2( 0, T ; L 2 (K) ), νdτ = 0, s 2 and thus β = α = 0.

32 On the Doi Model for the Suspensions of Rod-Like Molecules Notion of weak solutions References Bae, Trivisa (2010) On the Doi Model for the Suspension of Rod-like Molecules: Global-in-time-Existence. Bulcek, Feireisl, Malek (2009) A N-S-F system for incompressible fluids with temperature dependent material coefficients. Constantin (2005). Nonlinear Fokker-Planck NS systems. Doi, Edwards (1986). The theory of polymer dynamics. Lions, Masmoudi (2007). Global solutions of weak solutions to some micro-macro models. Masmoudi (2009). Global existence of weak solutions to the FENE dumbell model of polymeric flows. Otto, Tzavaras (2008). Continuity of velocity gradients in suspensions of rod-like molecules.

33 Fluid-Particle Interaction: The Bubbling Regime MODEL 2 Fluid-Particle-Interaction: The Bubbling Regime May, 2011

34 Fluid-Particle Interaction: The Bubbling Regime Modeling Fluid-particle interactions arise in many practical applications: biotechnology, medicine, fuel droplets in combustion, sprays etc. Here, we focus on a particular system derived by formal asymptotics from a mesoscopic description. This is based on a kinetic equation for the particle distribution of Fokker-Planck type coupled to fluid equations. The coupling between the kinetic and the fluid equations: friction forces that the fluid and the particles exert mutually.

35 Fluid-Particle Interaction: The Bubbling Regime Modeling The cloud of particles is described by its distribution function f ε (t, x, ξ) on phase space, which is the solution to the dimensionless Vlasov-Fokker-Planck equation t f ε + 1 ( ) ξ x f ε x Φ ξ f ε = 1 ( (ξ ε ε div ) ) ξ εuε f + ξ f ε. The friction force is assumed to follow Stokes law and thus is proportional to the relative velocity vector, i.e., is proportional to the fluctuations of the microscopic velocity ξ R 3 around the fluid velocity field u.

36 Fluid-Particle Interaction: The Bubbling Regime Modeling The RHS of the moment equation in the Navier-Stokes system takes into account the action of the cloud of particles on the fluid through the forcing term ( ) ξ F ε = ε u ε(t, x) f (t, x, ξ) dξ. R 3 The density of the particles η ε (t, x) is related to the probability distribution function f ε (t, x, ξ) through the relation η ε (t, x) = f ε (t, x, ξ) dξ. R 3

37 Fluid-Particle Interaction: The Bubbling Regime The model t ϱ + div x (ϱu) = 0 t (ϱu) + div x (ϱ(u u) + x (p(ϱ) + η) µ u λ x div x u = (η + βϱ) x Φ, t η + div x (η(u x Φ)) η = 0. ϱ = ϱ(t, x) total mass density t time, x Ω R 3 u = u(t, x) velocity field η = η(t, x) the density of the particles

38 Fluid-Particle Interaction: The Bubbling Regime The model p(ϱ) = aϱ γ a > 0, γ > 1, β 0 Φ external potential µ > 0, λ + 2 µ 0 viscosity parameters 3 β > 0 if Ω is unbounded

39 Fluid-Particle Interaction: The Bubbling Regime The model Another macroscopic effect is that the total pressure function in the momentum equation depends on both the particle and the fluid densities Pressure = P(ϱ, η) = p(ϱ) + η

40 Fluid-Particle Interaction: The Bubbling Regime The model Boundary Conditions u Ω = x η ν + η x Φ ν = 0 on (0, T ) Ω with ν denoting the outer normal vector to the boundary Ω. Initial Conditions (ϱ 0, m 0, η 0 ) such that ϱ(0, x) = ϱ 0 L γ (Ω) L 1 +(Ω), (ϱu)(0, x) = m 0 L 6 5 (Ω) L 1 (Ω), η(0, x) = η 0 L 2 (Ω) L 1 +(Ω).

41 Fluid-Particle Interaction: The Bubbling Regime The model Total Energy Ω E(η, ϱ, u)(t) := [ 1 2 ϱ(t) u(t) 2 + a ] γ 1 ϱγ (t) + (η log η)(t) + (βϱ + η)(t)φ At the formal level, the total energy can be viewed as a Lyapunov function satisfying the energy inequality de [ dt + µ x u 2 + λ div x u x η + η x Φ 2] 0. Ω

42 Fluid-Particle Interaction: The Bubbling Regime The model Confinement hypothesis Given a domain Ω C 2,ν, ν > 0, Ω R 3, and given a bounded from below external potential Φ : Ω R + 0 satisfying Φ(x) = 0 we will say that (Ω, Φ) verifies the confinement inf x Ω hypotheses (HC) for the two-phase flow system coupled with no-flux boundary conditions whenever: If Ω is bounded, Φ is bounded and Lipschitz continuous in Ω and the sub-level sets [Φ < k] are connected in Ω for any k > 0. If Ω is unbounded, we assume that Φ W 1, loc (Ω), β > 0, the sub-level sets [Φ < k] are connected in Ω for any k > 0, and e Φ/2 L 1 (Ω), Φ(x) c 1 x Φ(x) c 2 Φ(x), x > R > 0

43 Fluid-Particle Interaction: The Bubbling Regime The model Examples The confinement assumption (HC) has physical relevance in our setting as it is verified for several domains Ω with Φ being the gravitational potential. For instance, 1 when Ω = {x R 3 (x 1, x 2 ) [a, b] 2, x 3 [0, H]} and Φ(x) = gx 3, where β = 1 ϱ F ϱ P. 2 when Ω = {x R 3 (x 1, x 2 ) [a, b] 2, x 3 > 0} and Φ(x) = gx 3, where β = 1 ϱ F ϱ P and ϱ F < ϱ P. 3 when Ω = R 3 \ B(0, R) and Φ(x) = g x, where B(0, R) is the ball centered at the origin with radius R and β > 0. Here, ϱ F and ϱ P are the typical mass density of fluid and particles, respectively. Remark that 1. corresponds to the standard bubbling case in which particles move upwards due to buoyancy.

44 Fluid-Particle Interaction: The Bubbling Regime The model Problem D. t ϱ + div x (ϱu) = 0 t (ϱu) + div x (ϱ(u u) + x (p(ϱ) + η) µ u λ x div x u = (η + βϱ) x Φ, t η + div x (η(u x Φ)) η = 0. B.C. u Ω = x η ν + η x Φ ν = 0 on (0, T ) Ω

45 Fluid-Particle Interaction: The Bubbling Regime Strategy Strategy Variational Formulation Derivatives in the sense of distributions Equations family of integral identities Approach collect all available a priori estimates construct a sequence of approximate problems whose solutions satisfy these estimate show that the sequence of approximate slns converges to solution of the original problem.

46 Fluid-Particle Interaction: The Bubbling Regime Notion of weak solutions Weak solutions {ϱ, u, η} is an admissible free energy solution of Problem D, supplemented with the initial data {ϱ 0, u 0, η 0 } provided that ϱ 0, u is a renormalized solution of the continuity equation,that is, T (ϱb(ϱ) t ϕ + ϱb(ϱ)u x ϕ b(ϱ)div x uϕ) dxdt 0 Ω = Ω ϱ 0 B(ϱ 0 )ϕ(0, )dx holds for any test function ϕ D([0, T ) Ω) and suitable b and B. The balance of momentum holds in distributional sense. The velocity field u belongs to the space L 2 (0, T ; W 1,2 (Ω; R 3 )), therefore it is legitimate to require u to satisfy the boundary conditions in the sense of traces.

47 Fluid-Particle Interaction: The Bubbling Regime Notion of weak solutions η 0 is a weak solution of the Smoluchowski equation. That is, η t ϕ + ηu x ϕ η x Φ x ϕ x η x ϕdxdt 0 Ω = η 0 ϕ(0, )dx Ω is satisfied for test functions ϕ D([0, T ) Ω) and any T > 0. In particular, η L 2 ([0, T ]; L 3 (Ω)) L 1 (0, T ; W 1, 3 2 (Ω))

48 Fluid-Particle Interaction: The Bubbling Regime Notion of weak solutions Given the total free-energy of the system by ( 1 E(ϱ, u, η)(t) := 2 ϱ u 2 + a ) γ 1 ϱγ + η log η + (βϱ + η)φ, Ω then E(ϱ, u, η)(t) is finite and bounded by the initial energy of the system E(ϱ, u, η)(t) E(ϱ 0, u 0, η 0 ) a.e. t > 0 Moreover, the following free energy-dissipation inequality holds ( µ x u 2 + λ div x u x η + η x Φ 2) dt 0 Ω E(ϱ 0, u 0, η 0 )

49 Fluid-Particle Interaction: The Bubbling Regime Existence Result Theorem Let Ω R 3 bounded domain and (Ω, Φ) satisfy the confinement hypotheses (HC). Then, Problem D admits a weak solution {ϱ, u, η} on (0, ) Ω. In addition, i) the total fluid mass and particle mass given by M ϱ (t) = ϱ(t, ) dx and M η (t) = Ω respectively, are constants of motion. ii) the density satisfies the higher integrability result ϱ L γ+θ ((0, T ) Ω), for any T > 0, where Θ = min{ 2 3 γ 1, 1 4 }. Ω η(t, ) dx,

50 Fluid-Particle Interaction: The Bubbling Regime Large-time Asymptotics Theorem Let us assume that (Ω, Φ) satisfy the confinement hypotheses (HC). Then, for any free-energy solution (ϱ, u, η) of the Problem D, there exist universal stationary states ϱ s (x), η s (x), such that ϱ(t) ϱ s strongly in L γ (Ω), ess sup ϱ(τ) u(τ) 2 dx 0, τ>t Ω η(t) η s strongly in L p 2 (Ω) for p 2 > 1, as t, where (η s, ϱ s ) are characterized as the unique free-energy solution of the stationary state problem:

51 Fluid-Particle Interaction: The Bubbling Regime Large-time Asymptotics { x p(ϱ s ) = βϱ s x Φ, x η s = η s x Φ, (ϱ s, η s ) dx = Ω Ω (ϱ 0, η 0 ) dx, given by the formulas ϱ s = ( ) 1 γ 1 aγ [ βφ + C ϱ)] + γ 1 η s = C η exp( Φ), where C η and C ϱ are uniquely given by the initial masses.

52 Fluid-Particle Interaction: The Bubbling Regime Difficulties What is new? A new approximating scheme based on a 2-level approximating procedure: 1. artificial pressure approximation 2. time discretization scheme. The analysis treats both the case of a bounded physical domain as well as the case of an unbounded domain. (HC) on (Ω, Φ) control of {η log η} in the free-energy bounds for unbounded domains.

53 Fluid-Particle Interaction: The Bubbling Regime Difficulties High integrability properties for the density need to be established for the limit passage in the family of approximate solutions and in particular in taking the vanishing artificial pressure limit. In the present context, the potential Φ is not integrable on unbounded domains. To deal with this new difficulty we employ Fourier multipliers. Both the total fluid mass and the total particle mass are constants of motion. In particular, we are able to conserve the total masses also in the large-time limit allowing us to uniquely determine the long time asymptotics.

54 Fluid-Particle Interaction: The Bubbling Regime Approximating scheme The approximation scheme Let δ > 0 be fixed. Given a time step h > 0, we discretize the time interval [0, T ] in terms of the points t k = kh, k = 0,..., M, with Mh = T. Now, we sequentially determine functions such that: {ϱ k δ,h, uk δ,h, ηk δ,h } W(Ω), k = 1,..., M, The time discretized continuity equation, d h t [ϱ k δ,h ] + div x(ϱ k δ,h uk δ,h ) = 0, holds in the sense of distributions on Ω.

55 Fluid-Particle Interaction: The Bubbling Regime Approximating scheme The time discretized momentum equation with artificial pressure, dt h [ϱ k δ,h uk δ,h ] + div x(ϱ k δ,h uk δ,h uk δ,h ) µ uk δ,h λ xdiv x u k δ,h ) + x (p δ (ϱ k δ,h ) + ηk δ,h = (βϱ k δ,h + ηk δ,h ) xφ holds in the sense of distributions on Ω

56 Fluid-Particle Interaction: The Bubbling Regime Approximating scheme The time discretized particle density equation, ( ) dt h [ηδ,h k ] + div x ηδ,h k (uk δ,h xφ) ηδ,h k = 0, holds in the sense of distributions on Ω. In the above equations, dt h [φ k ] = φk φ k 1 h denotes implicit time stepping.

57 Fluid-Particle Interaction: The Bubbling Regime Approximating scheme In order to prove strong convergence of the density we prove the weak sequential continuity of the effective viscous flux. That is, [(λ + µ)div x u n p δ (ϱ n )] ψϱ n dx = Ω lim n Ω [ ] (λ + µ)div x u p δ (ϱ) ψϱ dx, ψ Cc (Ω).

58 Fluid-Particle Interaction: The Bubbling Regime Approximating scheme Given an unbounded domain Ω and an external potential Φ satisfying the assumptions (HC), we show that find an increasing sequence of domains Ω r, with r > 0 such that Ω r are bounded and (Ω r, Φ) satisfies (HC) approximating Ω in the sense r>0 Ω r = Ω. For any r > 0, there is a solution on Ω r. Next we let r to obtain a solution in Ω.

59 Fluid-Particle Interaction: The Bubbling Regime Approximating scheme There exists an artificial pressure solution (h 0). Vanishing artificial pressure limit (δ 0).

60 Fluid-Particle Interaction: The Bubbling Regime Approximating scheme References Carrillo, Karper, Trivisa On the Dynamics of a Fluid-Particle Interaction Model: The Bubbling Regime. Submitted (2010) Donatelli - Trivisa Combustion models (CMP (2006), ARMA (2007)) Feireisl, Petzeltová, Trivisa Multicomponent reactive flows: Global-in-time existence for large data (CPMA (2008)) Trivisa Liquid - vapor phase transition (SIMA (2008)) Feireisl Dynamics of viscous compressible fluids. (Oxford University Press (2003)) P.-L. Lions Mathematical topics in fluid dynamics Vol.2, Compressible models. (Oxford Science Publication (1998))