1 Introduction
Abstract

We study the Vlasov-Stokes equations which macroscopically model the sedimentation of a cloud of particles in a fluid, where particle inertia are taken into account but fluid inertia are assumed to be negligible. We consider the limit when the inertia of the particles tends to zero, and obtain convergence of the dynamics to the solution of an associated inertialess system of equations. This system coincides with the model that can be derived as the homogenization limit of the microscopic inertialess dynamics.

The inertialess limit of particle sedimentation modeled by the Vlasov-Stokes equations

Richard M. Höfer111University of Bonn, Institute For Applied Mathematics. Endenicher Allee 60, 53115 Bonn, Germany.
Email: hoefer@iam.uni-bonn.de, Phone: +49 228 735602


July 16, 2019

1 Introduction

We consider the sedimentation of a cloud of identical spherical particles suspended in a fluid subject to gravitation. It is assumed that the suspension is sufficiently dilute such that collisions of particles do not play a role. Furthermore, we neglect inertial forces of the fluid, i.e., the fluid is modeled by a Stokes equation, but particle inertia are taken into account. These assumptions are justified if the Reynolds number is much smaller than the Stokes numbers which is the case for very small particles in gases. We refer to [Koc90] for the details of the microscopic model and a discussion about the regime of validity.

Let a nonnegative function describe the number density of particles at time and position with velocity . We denote the position density and current by

(1)
(2)

Here, the mean velocity is defined to be zero in the set . As a model for the macroscopic dynamics, we consider the so-called Vlasov-Stokes equations, a Vlasov equation for the particles coupled with Brinkman equations for the fluid,

(3)

Here, and are the fluid velocity and pressure respectively, with being the gravitational acceleration, and and are constants that will be discussed below. The first equation expresses that the forces acting on the particles are the gravitation and the drag exerted by the fluid. The Brinkman equations are Stokes equations with a force term that arises from the same drag.

A rigorous derivation of these macroscopic equations from the microscopic dynamics has not been achieved yet, a formal derivation can be found in [Koc90]. In the quasi-static case, the Brinkman equations have been established in [Al90a], [DGR08]. Using this, the Vlasov-Stokes equations (3) can be formally derived from the microscopic dynamics after non-dimensionalizing. The constants and are given by

(4)

where is the fluid viscosity, and are the particle and fluid mass density respectively, is the volume fraction of the particles, is the diameter of the cloud of particles, and the radius of the particles. The constant determines the interaction strength between fluid and particles. The quantity is known as the Stokes number and determines the strength of the inertial forces. For definiteness, we assume such that . Then, the larger , the less important inertial effects become. For a more detailed discussion of these parameters as well as a formal derivation of the system (3), we refer to [Hof16].

For similar equations as (3), global well-posedness has been proven in [Ham98] and [BDGM09]. In [Jab00], the author considers the inertialess limit of the system, where the fluid velocity in (5) is replaced by a force term that is given by a convolution operator which is more regular than the Stokes convolution operator. In [Gou01], similar limits are studied for a one dimensional model without gravity and including inertial forces on the fluid. In [GP04], the authors consider limits of high and low inertia of the system of a Vlasov equation without gravity and with a given random fluid velocity field. Similar systems that include Brownian motion of the particles and their limits have been studied among others in [CP83], [GJV04a], [GJV04b] [CG06], and [GHMZ10].

1.1 Main result

We are interested in the limit , which corresponds to inertialess particles. For the ease of notation we drop all the other constants and consider the system

(5)

For inertialess particles, the following macroscopic equation has been proven in [Hof16] to be the homogenization limit of many small particles.

(6)

Moreover, well-posedness of this system has been proven in [Hof16].

In these equations, particles are described by their position density only, because their velocity is the sum of the fluid velocity and the constant which is the direct effect due to gravitation.

The main result of this paper is the following theorem.

Theorem 1.1.

Assume is compactly supported. Then, for , there exists a unique solution to (5). Let be the unique solution to (6). Then, for all , and all

(7)
(8)

Formally, for large values of , the first equation in (5) forces the particle to attain the velocity , i.e., the density concentrates around . Using that and integrating the first equation in (5) in leads to the first equation in (6). Moreover, in the fluid equation in (5) can formally be replaced by , which leads to the fluid equation in (6).

Formally, the adjustment of the particle velocities described above happens in times of order . In fact, the process is more complicated as the fluid velocity changes very fast in this time scale as well. In other words, there is a boundary layer of width at time zero for the convergence of the fluid (and particle) velocity. This is the reason, why the convergence can only hold uniformly on time intervals for as stated in the theorem. The particles, however, do not move significantly in times of order . Thus, there is no boundary layer in the convergence .

1.2 Idea of the proof

We introduce the kinetic energy of the particles

Using the Vlasov-Stokes equations (5) yields the following energy identities for the fluid velocity and the particle energy (cf. Lemma 2.1 and Lemma 2.2).

(9)
(10)

Here and in the following, the weighted -norm is defined by

As expected, equation (10) shows that there is loss of energy due to friction (friction between the particles and the fluid as well as friction inside of the fluid), but the gravity pumps energy into the system (if we assume , which at least after some time should be the case). Note that the Vlasov-Stokes equations (5) also imply that the mass of the particles is conserved.

To analyze solutions to the Vlasov equation in (5), we look at the characteristic curves starting at time at position , where denotes the value of the solution along the characteristic curve.

(11)

By the standard theory, any solution with is of the form

(12)

Using the characteristics as well as estimates based on the energy identities (9) and (10) and regularity theory of Stokes equations, we prove global well-posedness of the Vlasov-Stokes equations (5) for compactly supported initial data . A similar approach based on an analysis of the characteristics has been used to prove existence of solutions to the Vlasov-Poisson equations in [BD85], [Pfa92], and [Sch91] (see also [Gla96]). From the PDE point of view, the electrostatic potential appearing in the Vlasov-Poisson equation is similar to the fluid velocity in the Vlasov-Stokes equations. However, in the Vlasov-Poisson equations, the force acting on the particles is the gradient of the electrostatic potential. whereas in the Vlasov-Stokes equations, only the fluid velocity itself contributes. This makes it possible to prove existence (and also uniqueness) in a much simpler way for the Vlasov-Stokes equations.

In order to prove the convergence in Theorem 1.1, the starting point is integrating the characteristics which yields

(13)

Thus,

(14)

Therefore, provided the speed of the particles does not blow up, we see that for large values of the particles are almost transported by the fluid plus the gravity. Clearly, this is also what happens for solutions to the limit inertialess equations (6).

In order to show that is close to , we introduce a fluid velocity which can be viewed as intermediate between and by

(15)

In order to prove smallness of , one needs estimates on and that are uniform in , which are more difficult to obtain than those that we use in the proof of well-posedness. Indeed, in view of the energy identity for the particles (10), any naive estimate based on that equation will blow up as . However, as the first term is linear in the velocity and the other terms (which have a good sign) are quadratic, the energy cannot exceed a certain value as long as the particle density is not too concentrated (cf. Lemma 3.2). In other words, if the energy is high enough, the quadratic friction terms will prevail over the linear gravitation terms and therefore prevent the energy from increasing further. However, if concentrations of the particle density occur, the particles essentially fall down like one small and heavy particle, leading to large velocities. Indeed, the terminal velocity of a spherical particle of radius in a Stokes fluid at rest is

In order to rule out such concentration effects, we use again the representation of in (12) obtained from the characteristics. Indeed, computing by taking the integral over in (12), we can show that the prefactor in that formula is canceled due to concentration of in velocity space in regions of size as long as we control in a suitable way (cf. Lemma 3.4). As is controlled by due to the energy identity (9), this enables us to get uniform estimates for both , , and for small times.

It turns out that also estimates on derivatives of are needed to prove smallness of . These are provided by a more detailed analysis of the characteristics.

1.3 Plan of the paper

The rest of the paper is organized as follows.

In Section 2, we prove global well-posedness of the Vlasov Stokes equations (5), based on energy estimates, analysis of the characteristics, and a fixed point argument.

In Section 3, we derive a priori estimates that are uniform in for small times by analyzing the characteristics more carefully. In particular we prove and use that the supports of the solutions concentrate in the space of velocities.

In Section 4.1, we use those a priori proven in Section 4 to show that the fluid velocity is close to the intermediate fluid velocity defined in (15) as . In Section 4.2, we prove the assertion of the main result, Theorem 1.1, up to times where we have uniform a priori estimates. This follows from compactness due to the a priori estimates and convergence of averages of on small cubes, which we prove using again the characteristic equations. In Section 4.3, we finish the proof of the main result, Theorem 1.1, by extending the a priori estimates from Section 3 to arbitrary times. This is done by using both the a priori estimates and the convergence for small times.

2 Global well-posedness of the Vlasov-Stokes equations

In this section, we write for any constant that depends only on the initial datum. Any additional dependencies are denoted by arguments of , e.g. is a constant that depends only on and the initial datum. We use the convention that is monotone in all its arguments.

2.1 Estimates for the fluid velocity

Lemma 2.1.

Let be nonnegative, and assume is such that . Let

(16)
(17)
(18)

Then there exists a unique weak solution to the Brinkman equation

Moreover,

(20)
(21)
(22)
Proof.

Existence and uniqueness of weak solutions in follows from the Lax-Milgram theorem.

In the following, we write instead of and instead of . Testing the Brinkman equation with itself yields

(23)

By the Cauchy-Schwarz inequality

(24)

Hence,

Using again (23) yields (20). Using the critical Sobolev embedding, we have

(25)

Moreover, we can use this Sobolev inequality in (20) to get

Using the definition of yields and therefore

(26)

Standard regularity theory for the Stokes equation (see [Ga11]) implies

(27)

for all . In order to prove (22), we use (27) and (25) to get

Hence, by Sobolev embedding and (26)

and similarly for .

It remains to prove (21). Let . Then,

(28)

We choose

Thus,

and therefore,

(29)

Moreover, by definition of , (29) implies for all ,

(30)

Sobolev and Hölder’s inequality imply

where we used (25), (29), and (30). Now, we can repeat the argument, using this improved estimate for in (27). This yields

Using again (27) yields

As , we can apply Sobolev embedding to get

which finishes the proof of (21). ∎

2.2 A priori estimates for the particle density

Lemma 2.2.

Let and and let be minimal such that . Assume is a solution to (5) with . Then, is compactly supported on . Let be minimal such that . Furthermore, define

(31)

Then,

(32)
(33)
(34)
(35)
(36)
(37)
Proof.

By the regularity assumptions on and , the characteristics in (11) are well defined and (12) holds. This shows that the support of remains uniformly bounded on compact time intervals.

The exponential growth of the -norm of (32) follows from the characteristic equations as we have seen in (12).

Mass conservation (33) follows directly from integrating the Vlasov equation (5).

We multiply the Vlasov equation by and integrate to find

(38)

This yields the identity (34). By the Cauchy-Schwarz inequality

(39)

Moreover, by definition of in (2)

(40)

Using (39) and (40) shows (35).

In particular

This proves (36) by a comparison principle for ODEs.

The characteristic equation for in (11) implies

Thus, for all , we get by Lemma 2.1, (32), (33), and (36)

(41)

By the equation for , we get for all

(42)

Hence,

Gronwall’s equation yields (37). ∎

2.3 Well-posedness by the Banach fixed point theorem

Proposition 2.3.

Let with compact support. Then, for all , there exists a unique solution to (5) with .

Proof.

We want to prove existence of solutions using the Banach fixed point theorem. Let . We define the metric space, where we want to prove contractiveness,

(43)
(44)

Then, is a complete metric space. Let and . For , we define to be the solution to

We define the characteristics analogously to (11) by

(45)
(46)

Then, the solutions to the equation

(47)

with initial datum is given by

(48)

and . We estimate

(49)

Furthermore, writing instead of and similar for , we have for all

(50)
(51)
(52)
(53)

where we used Lemma 2.1. Gronwall’s inequality implies

Inserting this in (49) yields

(54)

For , consider . Then, for all , equation (54) implies that there exists such that the mapping is contractive. We have to check that implies . First,

(55)

follows from the equation. Moreover, for any , equation (48) implies that we can choose sufficiently small such that

(56)

Furthermore, we have

(57)
(58)

Hence, using mass conservation, equation (55),

Therefore, Lemma 2.1 and Gronwall’s inequality imply

(59)

Thus, for any , we can choose small enough such that for all .

Finally, we need to control the support of . To do this, we follow the same argument as in the last part of the proof of Lemma 2.2 to get

Again, for any , we can choose small enough such that for all .

Therefore, by the Banach fixed point theorem, we get local in time existence of solutions to (5). Global existence follows directly from the a priori estimates in Lemma 2.2, since these ensure that all the relevant quantities for the fixed point argument do not blow up in finite time.

Since with uniform compact support, higher regularity of follows from taking derivatives in the Brinkman equations in (5) and using regularity theory for Stokes equations similar as in the proof of Lemma 2.1. ∎

3 Uniform estimates on and

In the following, we assume that is the solution to the Vlasov-Stokes equations (5) for some and some compactly supported initial datum . In this section we want to derive a priori estimates for these solutions that do not depend on . This is why we cannot use the a priori estimates derived in Lemma 2.2. However, the drawback of the estimates that we prove in this section is that they allow for blow-up in finite time. This is also why they are not suitable in the proof of global well-posedness, that we showed in the previous section. Later, we will use the limit equation in order to show that the estimates derived here allow for uniform estimates for arbitrary times.

Again, we denote by any constant, which only depends on and may change from line to line.

3.1 Estimates for the fluid velocity

In this subsection we show that the fluid velocity as well as the particle velocity is controlled by , uniformly in , which means that high velocities can only occur if particles concentrate in position space. This also implies control on the particle positions and velocities

The proof is based on the energy identity from Lemma 2.2, equation (34), and the subsequent estimate (35). The idea is to estimate the sum of the quadratic terms in that expression, which have a negative sign, by from below. The following Lemma, which is a general observation on weighted -spaces, shows why such an estimate is true if is not too large.

Having shown this estimate, the quadratic terms in (35) dominate the linear term, which has been estimated by . This leads to control of uniformly in .

Lemma 3.1.

There exists a constant , such that for all nonnegative , and ,