On a Fluid-Particle Interaction Model: Global in Time Weak Solutions Within a Moving Domain in $\mathbb{R}^{3}$

Abstract

A fluid-particle interaction model is presented for the evolution of particles dispersed in a fluid. The fluid flow is governed by the Navier-Stokes equations for a compressible fluid while the evolution of the particle densities is given by the Smoluchowski equation. The coupling between the dispersed and dense phases is obtained through the drag forces that the fluid and particles exert mutually. In the present context, the flow occupies a physical domain Ω _t with boundary Γ _t both of which vary in time. Global-in-time weak solutions are obtained using an approach based on penalization of the boundary behavior and viscosity in the weak formulation.

Dedicated to Walter Craig, mentor and friend, in honor of his 60th birthday.

Appendix

1.1 Free Energy Solutions: Global in Time Existence Within Bounded and Unbounded Domains $\varOmega \subset \mathbb{R}^{3}$

In this section, we present for completeness the main results on global existence of free energy solutions to (2a)–(2c) for both bounded and unbounded fixed domains. We impose the no-slip boundary condition for the velocity and no-flux for the particle density, which in the present context has the form

$$\displaystyle{ \mathbf{u}\big\vert _{\partial \varOmega } = (\nabla _{x}\eta +\eta \nabla _{x}\varPhi ) \cdot \nu \big\vert _{\partial \varOmega } = 0\,\,\,\mbox{ for a.a.}\,\,\,t \in (0,T), }$$

(59)

with ν denoting the outer normal vector to the boundary ∂ Ω. The usual pressure and stress tensor, (3), (4), are imposed.

Our problem is supplemented with the initial data {η ₀, ρ ₀, u ₀} such that

$$\displaystyle{ \begin{array}{ll} \eta (0,x) & =\eta _{0} \in L^{2}(\varOmega ) \cap L_{+}^{1}(\varOmega ), \\ \rho (0,x) & =\rho _{0} \in L^{\gamma }(\varOmega ) \cap L_{+}^{1}(\varOmega ), \\ (\rho \mathbf{u})(0,x)& = \mathbf{m}_{0} \in L^{\frac{6} {5} }(\varOmega ) \cap L^{1}(\varOmega ). \end{array} }$$

(60)

The total energy of the mixture is given by

$$\displaystyle{ E(\eta,\rho,\mathbf{u})(t):=\int _{\varOmega }\frac{1} {2}\rho (t)\vert \mathbf{u}(t)\vert ^{2} + \frac{a} {\gamma -1}\rho ^{\gamma }(t) + (\eta \log \eta )(t) + (\beta \rho +\eta )(t)\varPhi \ \mathrm{d}x. }$$

(61)

At the formal level, the total energy can be viewed as a Lyapunov function satisfying the energy inequality

$$\displaystyle{ \frac{dE} {dt} +\int _{\varOmega }\mu \vert \nabla _{x}\mathbf{u}\vert ^{2} +\lambda \vert \mathrm{div}_{ x}\mathbf{u}\vert ^{2} + \vert 2\nabla _{ x}\sqrt{\eta } + \sqrt{\eta }\nabla _{x}\varPhi \vert ^{2}\ \mathrm{d}x \leq 0. }$$

(62)

Definition 4.

Let us assume that (Ω, Φ) satisfy the confinement hypotheses (HC). We say that {ρ, u, η} is a free-energy solution of problem (2a)–(2c) with boundary conditions (59) and initial data satisfying (60) provided that the following hold:

• The fluid density ρ ≥ 0 represents a renormalized solution of Eq. (2a) on a time-space cylinder $(0,\infty )\times \varOmega$, that is, for any test function $\varphi \in \mathcal{D}([0,T) \times \overline{\varOmega })$, any T > 0, and any b such that

  $$\displaystyle{b \in L^{\infty }\cap C[0,\infty ),\ B(\rho ) =\rho B(1) +\rho \int _{ 1}^{\rho }\frac{b(z)} {z^{2}} \ \mathrm{d}z,}$$

  the following integral identity holds:

$$\displaystyle{ \int _{0}^{\infty }\int _{ \varOmega }\Big(B(\rho )\partial _{t}\varphi + B(\rho )\mathbf{u} \cdot \nabla _{x}\varphi - b(\rho )\mathrm{div}_{x}\mathbf{u}\varphi \Big)\ \mathrm{d}x\ \mathrm{d}t = -\int _{\varOmega }B(\rho _{0})\varphi (0,\cdot )\ \mathrm{d}x. }$$

(63)

• The balance of momentum holds in a distributional sense, namely

  $$\displaystyle\begin{array}{rcl} & & \int _{0}^{\infty }\int _{ \varOmega }\Big(\rho \mathbf{u} \cdot \partial _{t}\varphi +\rho \mathbf{u} \otimes \mathbf{u}: \nabla _{x}\varphi + (p(\rho )+\eta )\mathrm{div}_{x}\varphi \Big)\ \mathrm{d}x\ \mathrm{d}t \\ & & \quad =\int _{ 0}^{\infty }\int _{ \varOmega }- (\mu \nabla _{x}\mathbf{u}\ +\lambda \mathrm{div}_{x}\mathbf{u}\mathbb{I}): \nabla _{x}\varphi + (\eta +\beta \rho )\nabla _{x}\varPhi \cdot \varphi \ \mathrm{d}x\ \mathrm{d}t \\ & & \quad \ -\int _{\varOmega }(\rho \mathbf{u})_{0} \cdot \varphi (0,\cdot )\ \mathrm{d}x {}\end{array}$$

  (64)

  for any test function $\varphi \in \mathcal{D}([0,T);\mathcal{D}(\overline{\varOmega }; \mathbb{R}^{3}))$ and any T > 0 satisfying $\varphi \vert _{\partial \varOmega } = 0$.

  All quantities appearing in (64) are supposed to be at least integrable. In particular, the velocity field u belongs to the space $L^{2}(0,T;W^{1,2}(\varOmega; \mathbb{R}^{3}))$, therefore it is legitimate to require u to satisfy the boundary conditions (59) in the sense of traces.

• The particle density η ≥ 0 is a weak solution of (2c). In particular, the integral identity

  $$\displaystyle{ \int _{0}^{\infty }\!\!\!\int _{ \varOmega }\eta \partial _{t}\varphi +\eta \mathbf{u} \cdot \nabla _{x}\varphi -\eta \nabla _{x}\varPhi \cdot \nabla _{x}\varphi -\nabla _{x}\eta \cdot \nabla _{x}\varphi \ \mathrm{d}x\mathrm{d}t = -\int _{\varOmega }\eta _{0}\varphi (0,\cdot )\ \mathrm{d}x }$$

  (65)

  is satisfied for test functions $\varphi \in \mathcal{D}([0,T) \times \overline{\varOmega })$ and any T > 0.

  All quantities appearing in (65) must be at least integrable on (0, T) ×Ω. In particular, η belongs $L^{2}(0,T;L^{3}(\varOmega )) \cap L^{1}(0,T;W^{1,\frac{3} {2} }(\varOmega )).$

• Given the total free-energy of the system by

  $$\displaystyle{E(\rho,\mathbf{u},\eta )(t):=\int _{\varOmega }\left (\frac{1} {2}\rho \vert \mathbf{u}\vert ^{2} + \frac{a} {\gamma -1}\rho ^{\gamma } +\eta \log \eta +(\beta \rho +\eta )\varPhi \right )\ \mathrm{d}x,}$$

  then E(ρ, u, η)(t) is finite and bounded by the initial energy of the system, i.e., E(ρ, u, η)(t) ≤ E(ρ ₀, u ₀, η ₀) a.e. t > 0. Moreover, the following free energy-dissipation inequality holds

$$\displaystyle{ \int _{0}^{\infty }\!\!\!\int \left (\mu \vert \nabla _{ x}\mathbf{u}\vert ^{2} +\lambda \vert \mathrm{div}_{ x}\mathbf{u}\vert ^{2} + \vert 2\nabla _{ x}\sqrt{\eta } + \sqrt{\eta }\nabla _{x}\varPhi \vert ^{2}\right )\,\mathrm{d}x\mathrm{d}t \leq E(\rho _{ 0},\mathbf{u}_{0},\eta _{0}). }$$

(66)

We can now state the main result on global existence. For the details of the proof we refer the reader to Carrillo, et al. [10].

Theorem 2.

Global Existence Let us assume that (Ω,Φ) satisfy the confinement hypotheses (HC) . Then, the problem  (2a)–(2c) supplemented with boundary conditions  (59) and initial data satisfying  (60) admits a weak solution {ρ, u ,η} on $(0,\infty )\times \varOmega$ in the sense of Definition  4 . In addition,

• i) the total fluid mass and particle mass given by

  $$\displaystyle{M_{\rho }(t) =\int _{\varOmega }\rho (t,\cdot )\ dx\qquad \mbox{ and}\qquad M_{\eta }(t) =\int _{\varOmega }\eta (t,\cdot )\ dx,}$$

  respectively, are constants of motion.

• ii) the density satisfies the higher integrability result

  $$\displaystyle{\rho \in L^{\gamma +\varTheta }((0,T)\times \varOmega ),}$$

  for any T > 0, where $\varTheta =\min \{ \frac{2} {3}\gamma - 1, \frac{1} {4}\}$ .

We can also completely characterize the large time behavior of free-energy solutions.

Theorem 3.

Large-time Asymptotics: Let us assume that (Ω,Φ) satisfy the confinement hypotheses (HC) . Then, for any free-energy solution (ρ, u ,η) of the problem  (2a)–(2c) , in the sense of Definition  4 , there exist universal stationary states ρ _s = ρ _s (x), η _s = η _s (x), such that

$$\displaystyle{\left \{\begin{array}{ll} &\rho (t) \rightarrow \rho _{s}\ \mbox{ strongly in}\ \ L^{\gamma }(\varOmega ), \\ &\mathop{\mathrm{ess\ sup}}\nolimits \limits _{\tau >t}\int _{\varOmega }\rho (\tau )\vert \mathbf{u}(\tau )\vert ^{2}\,dx \rightarrow 0, \\ &\eta (t) \rightarrow \eta _{s}\ \mbox{ strongly in}\ \ L^{p_{2}}(\varOmega )\,\,\mbox{ for}\,\,p_{2} > 1.\\ \end{array} \right.}$$