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, ua 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.  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 . 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 ,  but also works for low-speed gas-particle internal flows , .
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  or to explain the formation of large-scale structures in the Universe , . 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 , , ,  and the references therein). If
To our knowledge, investigations on the Eulerian droplet model (1) have mostly focused on the numerical level  and the practical level , , , , . Recently, the theoretical arguments for (1) were completed by Keita and Bourgault . 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 , “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 , just for the convenience of the study, in the system (1), μ is assumed to be a positive constant, ua is also supposed to be constant, and α > 0 is required.
Recall that, in 2016, Shen  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  to describe the granular flow behaviour. By skillfully performing the variable substitution
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 , 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 , the results are just the same.
The novelty of this article comes from the following four aspects. First, compared with the method presented in , 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 , , , , , , , , , . 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
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
2 Riemann Problem to a Modified System of Conservation Laws
with the new initial data
The system (5) can be rewritten in the quasi-linear form
From this form, we can calculate the eigenvalue of system (5) as
and the right eigenvector as
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
Simplifying (8) yields
Therefore, we can connect the two non-vacuum constant states
Then we consider (11) with sufficiently smooth initial data
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
Differentiating the second equation of (11) with respect to x gives
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
This implies that α and vx 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:
A two-dimensional weighted delta function
for all test functions
A pair (α, v) is called a delta-shock solution of (5) in the sense of distributions if there exist a smooth curve S and a function w(t) such that α and v are represented in the following form:
for all test functions
We conclude that if (22) satisfies the generalised Rankine–Hugoniot condition
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 , so we only deliver the second equality in (21) for completeness. Actually, for any test function
By using Green’s formula and integrating by parts, one has
Moreover, the entropy condition
should be assumed to guarantee the uniqueness.
Now we solve the Riemann problem (5) and (6) when
From (23), we can compute that
Under the entropy condition (24), we have
Owing to (23), one has
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
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+,
A pair (α, u) is called a delta-shock solution of (1) in the sense of distributions if there exist a smooth curve S and a function w(t) such that α and u are represented in the following form:
for all test functions
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:
From the second equation of (37), we have
Then, we have
from which one has
By the entropy condition (39), we choose
as the admissible solution. Furthermore, we can compute that
As was done in  and , in what follows, we need to check that the delta-shock solution satisfies Definition 3 in the sense of distributions. We only prove the second equation in (35), because the proof for the other one is similar. In fact, one can deduce that
Here, without loss of generality, we suppose that
By using the change of variables and exchanging the ordering of the integrals again, we have
Substituting vδ into
from which we immediately get
It is easily observed that the solutions obtained here are completely coincident with those in . In other words, the transformation (4) is an effective way to obtain the solutions of the Euler droplet model (1). It also shows that the method used in  is not the only one to solve the Riemann problem of (1).
In view of
The Riemann problem for the one-dimensional Eulerian droplet model was solved. Compared to the discussions in , we introduced a special kind of non-linear variable substitution to rewrite the original Eulerian droplet model into a conservative one. Then, the delta-shock and vacuum solutions for both systems were constructively obtained. This method adopted here is easy to understand and workable. Furthermore, the variable substitution introduced here is different from that in , , , , , , , , , , etc. It is also shown that, because of the effect of the external force, the contact discontinuities and delta-shock waves are curved, and then the solutions are not self-similar. Finally, this work can provide a fundamental method of exploration for the study of the Riemann problem of the Eulerian droplet model with the initial data containing the Dirac measure, namely the Radon measure initial data problem. We leave it for a future study.
This work was supported by the National Natural Science Foundation of China (Funder Id: http://dx.doi.org/10.13039/501100001809, 11501488, 11801490), the Scientific Research Foundation of Xinyang Normal University (No. 0201318), the Nan Hu Young Scholar Supporting Program of XYNU, the Yunnan Applied Basic Research Projects (Funder Id: http://dx.doi.org/10.13039/100007471, 2018FD015), and the Scientific Research Foundation Project of Yunnan Education Department (Funder Id: http://dx.doi.org/10.13039/501100007846, 2018JS150).
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