Two-phase flows: Non-smooth well posedness and the compressible to incompressible limit
Introduction
In a 1D setting, a droplet of a compressible inviscid fluid fills the segment and is surrounded by another compressible fluid filling the rest of the real line. The fluids are assumed to be immiscible. For simplicity, we refer to the fluid forming the droplet as to a liquid, while its complement is labeled as gas. In the isentropic (or isothermal) approximation, this whole system can be described by Here, is the density of liquid in the droplet, is its speed and is the pressure law, while , and denote the analogous quantities for the gas. The kinetic relations state that along the boundary of the droplet the liquid and the gas have the same speed. This prevents any exchange of matter between the two fluids. In particular, they do not mix. Note also that these conditions obviously ensure the energy conservation at the interfaces, while energy is dissipated in the interior of the 2 fluids. Indeed, energy plays here the role of the mathematical entropy.
As it is physically necessary, we also assume that the total mass and the total linear momentum of the liquid and of the gas are conserved, that is Below, we show that (1.1)–(1.2) is well posed. This proof fits neither in the existing results concerning perturbations of large shocks, as in [1], nor in those concerning phase transitions, such as [2]. Nevertheless, we prove that the analytical framework therein can be adapted also to (1.1)–(1.2).
In this connection, we recall [3], where the existence of solutions to a different model for immiscible gas flow is proved. Therein, the choice of the pressure law is restricted to -laws but may vary continuously, so that several different gases can be considered at once. A similar model, with only two fluids but admitting a variable relative mass density between the two fluids, is considered in [4], where a global existence result is proved. Other multi-phase models with sharp interface representation based on viscous flow can for example be found in [5], [6]. An overview of another class of multi-phase models based on diffuse interface representations can be found in [7], some existence results in [8], [9], [10], [11]. Here, we consider two different compressible inviscid fluids with minimal requirements on the pressure law, see below, and show that well posedness is obtained for this model.
As it is well known, in several instances the liquid forming the droplet can be considered incompressible. In this approximation, on is lead to substitute the compressible equations in the liquid by the incompressible formulation. This is equivalent to substituting (1.1)–(1.2) with the system consisting of a conservation law describing the compressible gas coupled with an ordinary differential equation for the incompressible droplet. We recall that (1.3) is known to be well posed, see [12, Proposition 3.1].
It is then natural to expect that the solutions to (1.1)–(1.2) converge to solutions to (1.3), as the Mach number in the liquid phase vanishes. To our knowledge, a detailed rigorous treatment of this issue is still an open question, at least in the present compressible and inviscid framework for non smooth, albeit 1D, solutions. In the case of smooth solutions a wide literature is available, we refer for instance to [13], [14], [15], [16] and to the references therein.
Consider a shock wave coming from the gas phase and approaching the droplet at rest. Solving the resulting Riemann problem at the gas–liquid boundary shows that, as the liquid becomes incompressible, the shock is reflected, no wave is refracted into the liquid and the droplet does not move. In other words, this interface apparently turns into a solid wall, which is physically hardly acceptable. Below, a more careful limiting procedure shows that the acceleration of the droplet due to the impinging shock has a non zero limit as the Mach number tends to 0, coherently with (1.3).
More generally, we conjecture that (1.1)–(1.2) converges to (1.3) in the incompressible limit. We expect that, as the Mach number vanishes, weak entropy solutions to the former system of conservation laws yield solutions to the mixed p.d.e.–o.d.e. system (1.3). Below, we verify analytically this conjecture for particular initial data. Moreover, numerical experiments support this conjecture in cases not covered by the theoretical treatment.
We recall here that the compressible to incompressible limit, since the classical work [13], is still being widely studied in the current literature under various conditions. In the case of the Navier–Stokes equations we recall [17], [18], [15], [16] and the references therein. The magnetohydrodynamic setting has been considered, for instance, in [14], [19]. Immiscible flows are described in [20], [3], [21], [10], [22] deal with ad hoc numerical methods.
The paper is organized as follows. Section 2 deals with the well posedness of the models above: in the case of (1.1), a new theorem is stated while in the case of (1.3) a known result from the literature is recalled. Section 3 considers the compressible incompressible limit in the case of a solution containing a shock. All technical details are deferred to Section 4.
Section snippets
Well posedness of the model (1.1)–(1.2)
We denote and is the open ball centered at with radius . Throughout, our general reference for the theory of conservation laws is [23], see also [24]. Wherever necessary, we identify the state of the fluid both as and , with . For brevity, we call -system, respectively -system, the equations The explicit expression of the relevant quantities related to
Analytical approach
Consider a non smooth solution to (1.1). This section is devoted to the incompressible limit of . More precisely, we consider a shock in the gas phase hitting the gas–liquid interface. As the compressibility of the liquid vanishes, solving the limiting Riemann problem at the phase boundary apparently shows that the interface turns into a solid fixed wall reflecting the shock, (see (2.12) for ). Below, through a detailed wave front tracking technique, we show that the interaction of
Technical details
For the sake of completeness, we state here without proof the following relations about the -system. The sound speed is given in (2.2): The speeds of -shock waves between and the state at density are The (forward) 1, 2-Lax curves have the expressions
Acknowledgments
The supports of the 2009 Vigoni project Non-Local Transport Processes Modeling, Analysis, Numerics and Optimal Control and of the 2010 GNAMPA project Non Standard Applications of Conservation Laws are acknowledged.
The second author gratefully acknowledges the support by the German Research Foundation (DFG) in the framework of the Collaborative Research Center Transregio 75 Droplet Dynamics under Extreme Ambient Conditions.
An anonymous referee is kindly acknowledged for having suggested related
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