Delta-Shock Solution to the Eulerian Droplet Model by Variable Substitution Method

1 Introduction

We are concerned with the following one-dimensional Eulerian droplet model:

where α and u are the volume fraction and velocity of the particles (droplets), respectively, u_a is the velocity of the carrier fluid (air), and μ is the drag coefficient between the carrier fluid and the particles.

This model was proposed by Bourgault et al. [1] to compute the impingement of droplets on airfoils. In (1), the virtual mass force is neglected since the density of particles exceeds the air density by orders of magnitude. Other forces, such as lift force, gravity, and other interfacial effects, are also negligible when compared to the viscous drag force, though they may be important in some applications [1]. The Eulerian droplet model (1) corresponds to a dispersed phase subsystem in its simplest form: for instance, a multi-phase system for particles suspended in a carrier fluid. Currently, it is not only used to predict deposition patterns for high-speed external gas-particle flows [1], [2] but also works for low-speed gas-particle internal flows [3], [4].

It is clear that, as μ = 0, the system (1) is nothing but the pressureless Euler system, or the so-called the zero-pressure flow model, which can be used to describe the motion of free particles sticking together under collision [5] or to explain the formation of large-scale structures in the Universe [6], [7]. Since the delta-shock wave and vacuum are found in the solutions, the pressureless Euler system has been widely studied by a large number of scholars with strong interest since 1994 (see [8], [9], [10], [11] and the references therein). If ua=0 and μ = 1, the system (1) becomes a non-homogeneous, pressureless Euler system, which is derived from the Cucker-Smale model and can be used to describe the flocking phenomenon. The well posedness of entropy solution to this pressureless Euler system with flocking dissipation was systemically studied in [12], [13], [14], etc.

To our knowledge, investigations on the Eulerian droplet model (1) have mostly focused on the numerical level [1] and the practical level [3], [4], [15], [16], [17]. Recently, the theoretical arguments for (1) were completed by Keita and Bourgault [18]. They solved the Riemann problem for the Eulerian droplet model by going through the solution of the Riemann problems for the inviscid Burgers equation with a source term and the subsystem, respectively. Particularly, for the delta-shock solution, the generalised Rankine–Hugoniot condition, which is in the form of ordinary differential equations, was proposed. Nevertheless, as was pointed out in [18], “In general, it might be hard to find the analytical solution of this ordinary differential equations”. Thus, for the delta-shock solution of the Eulerian droplet model (1), with the help of the Cauchy-Peáno theorem, the Cauchy–Lipschitz existence theorem, and the Arzla–Ascoli theorem, they obtained the existence of a solution to the generalised Rankine–Hugoniot condition satisfying the Lax entropy condition. Finally, for the Riemann problem for (1), they were lucky to find an analytical solution for the generalised Rankine–Hugoniot condition. However, hydrodynamicists and engineers find it difficult to apply the theory of delta-shock waves conveniently in practice because of the lack of a recondite mathematical foundation. Therefore, one of the main objectives of this paper is to propose another effective and workable method to solve the Riemann problem for the Eulerian droplet model (1).

For this purpose, we consider the Riemann problem of (1) with the following initial data:

and aim at putting forward a new method to construct the Riemann solutions. Here, α_± and u_± are constants. As was done in [18], just for the convenience of the study, in the system (1), μ is assumed to be a positive constant, u_a is also supposed to be constant, and α > 0 is required.

Recall that, in 2016, Shen [19] studied the Riemann problem for the pressureless Euler system with the following source term:

where ρ and u denote the density and velocity of fluids, respectively, and β is a constant. The source term βρ in (3) is known as the Coulomb-like friction term, which was introduced by Savage and Hutter [20] to describe the granular flow behaviour. By skillfully performing the variable substitution v(x,t)=u(x,t)−βt, which was earlier used by Faccanoni and Mangeney [21] for the shallow water equations, they rewrote (3) in a conservative form that is linearly degenerate and whose solution contains a delta-shock wave in certain situations. Then they successfully constructed the explicit Riemann solutions of (3) with contact discontinuity, vacuum state, and delta-shock wave. Variable substitution is the common method used to discuss the balance law with the source term. For example, see [22] for the generalised zero-pressure flow model, [23] for the Suliciu relaxation system, and [24] for the perfect fluid model. Besides, one can also refer to [19] and [25], [26], [27], [28], [29], [30] for the Chaplygin and generalised Chaplygin gas equations with friction.

Motivated by the works in [19] and [22], [23], [24], [25], [26], [27], [28], [29], [30], here we introduce a new state variable v(x, t) and perform the following variable substitution for (1):

Then, system (1) is reduced to a modified system of conservation laws (see Section 2) and all the desired results on the Riemann problem (1) and (2) are obtained by the mathematical theory of hyperbolic conservation laws. Concretely, both the vacuum and delta-shock solutions for the modified system of conservation laws and (1) are obtained. For the delta-shock solution, we investigate and derive its position, propagation speed, and strength in detail under the generalised Rankine–Hugoniot relation and entropy condition. Interestingly, it is found that the generalised Rankine–Hugoniot relation proposed here can be reduced to a function equation. More precisely, it is a quadratic equation of one variable. Then, the existence and uniqueness of the delta-shock solution are solved completely under the entropy condition by studying the function equation, which is one of the outstanding advantages of our approach. Compared to the discussions in [18], this approach avoids using some analytical theorems that are complicated and not easy to understand for hydrodynamicists and engineers. Although we take a different method from [18], the results are just the same.

The novelty of this article comes from the following four aspects. First, compared with the method presented in [18], our method seems simpler and has the advantage of being easily workable. Second, the variable substitution (4) introduced in this paper is obviously different from that in previous works [19], [22], [23], [24], [25], [26], [27], [28], [29], [30]. Third, influenced by the source term, the Riemann solutions for (1) are not self-similar any more. All the characteristic curves, namely the curves of contact discontinuities and delta-shock waves, are bent into parabolic shapes. Fourth, we show that, as the drag coefficient μ→0+, the Riemann solution of the Eulerian droplet model (1) tends to the corresponding zero-pressure flow solution.

The rest of the paper is organised as follows. In Section 2, by the change of the state variable, we obtain a modified system of conservation laws with some new initial data, and constructively get its Riemann solutions containing delta-shock waves and vacuum states. In Section 3, we study the Riemann problem (1) and (2) and establish the existence and uniqueness of solutions under a suitable generalised Rankine–Hugoniot relation and entropy condition. Then, we rigorously prove that the delta-shock solution satisfies (1) in the sense of distributions. In addition, we show that the solutions of (1) and (2) converge to those of the pressureless Euler system with the same initial data when μ→0+. Finally, a brief conclusion is drawn in Section 4.

2 Riemann Problem to a Modified System of Conservation Laws

Under the transformation (4), the system (1) and initial data (2) are reduced into the following system of conservation laws:

with the new initial data

The system (5) can be rewritten in the quasi-linear form

where U=(α,v)T, and

From this form, we can calculate the eigenvalue of system (5) as

and the right eigenvector as r→=(1,0)T. Since ∇λ⋅r→=0, the system (5) is linear degenerate, and the elementary waves involve only contact discontinuities.

Noting that the parameter t appears in the flux functions of (5), it is quite different from the classical hyperbolic systems of conservation laws. However, the Rankine–Hugoniot conditions can also be derived via a standard technique as usual. For the bounded discontinuity x=x(t), we have the following Rankine–Hugoniot condition:

where [p]=p+−p− denotes the jump of p across the discontinuity, and σ(t)=x′(t).

If σ(t)≠0, from (7), one has

Simplifying (8) yields

Therefore, we can connect the two non-vacuum constant states (α−,v−) and (α+,v+) by a contact discontinuity J if and only if v−=v+. At this moment, the propagation speed of J is

The solutions to (5) and (6) can be constructed in two cases.

When u−<u+, one can construct the Riemann solution of (5) and (6) by constant, vacuum, and contact discontinuity, which is

in which

When u−>u+, the characteristic lines from the x-axis will overlap in the domain Ω={(t,x)|uat+(u+−ua)(1−e−μt)μ≤x≤uat+(u−−ua)(1−e−μt)μ,t>0}, so the singularity of solutions must occur. We affirm that there exist no solutions in the bounded variation space. Actually, for a smooth solution, (5) is equivalent to

Then we consider (11) with sufficiently smooth initial data (α,v)(0,x)=(α0(x),v0(x)), in which v0′(x)<0.

The characteristic equations of the system (5) are

For any given point (0, b) on the x-axis, the characteristic curve passing through this point is

on which v takes the constant value v=v0(b).

Differentiating the second equation of (11) with respect to x gives

that is

which is a standard type of the Riccati equation. As a result, along the characteristic curve (13), we can obtain

which when combined with the third equation of (12) yields

Noting that μ > 0 and v0′(b)<0, there exists b such that μ<−v0′(b). Let T*=infμ<−v0′(b){−ln(1+μv0′(b))μ}, so along the characteristic curve (13), one can easily observe that

This implies that α and v_x must blow up simultaneously at a finite time, which leads to unboundedness and discontinuities in the solution.

We will construct the solution using a delta-shock wave for this case. In order to define the delta-shock solution, we give the following two definitions:

With these two definitions, for the Riemann problem (5) and (6) we seek the solution in the form

where x(t)∈C1 is the position of the discontinuity, w(t) denotes the strength of the delta-shock wave, and v_δ is the corresponding value of v on the delta-shock wave curve.

We conclude that if (22) satisfies the generalised Rankine–Hugoniot condition

then the solution (α,v)(t,x) defined in (22) satisfies (5) in the sense of distributions.

Now we check that the solution obtained from solving (23) satisfies Definition 2 in the sense of distributions. The proof is similar to that of Theorem 4.2 in [18], so we only deliver the second equality in (21) for completeness. Actually, for any test function φ∈C0∞([0,+∞)×(−∞,+∞)), if (23) holds, we have

By using Green’s formula and integrating by parts, one has

It means that the second equation of (21) holds. The first one of (21) can be proved similarly.

Moreover, the entropy condition

that is

should be assumed to guarantee the uniqueness.

Now we solve the Riemann problem (5) and (6) when u−>u+. Under the entropy condition (24), the Riemann problem is reduced to solving the ordinary differential equations (23) with the initial data x(0) = 0, w(0) = 0.

From (23), we can compute that

and

Then, we immediately obtain from (26) and (27)

If α+−α−=0, from (28) we get

If α+−α−≠0, by solving the quadratic equation of vδ−ua in (28), one can obtain

Under the entropy condition (24), we have

Owing to (23), one has

So far, we have obtained the solution of the Riemann problem (6) for the system of conservation laws (5).

3 Riemann Solutions for the Eulerian Droplet Model (1)

In this section, we pay attention to the Riemann solutions for the original Eulerian droplet model (1). Based on the results in Section 2, when u−<u+, as shown in Figure 1, the Riemann solution of (1) and (2) can be expressed as

Figure 1:

Riemann solution of (1) and (2) in the (x, t)-plane when u−<u+.

in which

which is a vacuum-state solution.

For convenience, here we only give the structure of the Riemann solution in the (x, t)-plane for the cases where u₋, u₊, ua>0. At this moment, the two curves J₁ and J₂ are monotonously increasing.

When u−>u+, the delta-shock solution for (1) and (2) in the sense of distributions can be introduced as follows:

Then, we give the solution to the Riemann problem (1) and (2) in the following form:

where uδ(t) is the value of u on the delta-shock wave curve x=x(t), and (uδ(t)−ua)eμt is assumed to be a constant based on the result in Section 2.

The delta-shock solution of the Riemann problem (1) and (2) defined above should satisfy the following generalised Rankine–Hugoniot relation:

in which the discontinuity becomes

To ensure the uniqueness of the solution, we add the entropy condition

In this case, the solution can be described by the following theorem: