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# Interface dynamics: Mechanisms of stabilization and destabilization and structure of flow fields

Edited by Jisoon Ihm, Pohang University of Science and Technology, Pohang, South Korea, and approved June 12, 2018 (received for review January 29, 2018)

## Abstract

Interfacial mixing and transport are nonequilibrium processes coupling kinetic to macroscopic scales. They occur in fluids, plasmas, and materials over celestial events to atoms. Grasping their fundamentals can advance a broad range of disciplines in science, mathematics, and engineering. This paper focuses on the long-standing classic problem of stability of a phase boundary—a fluid interface that has a mass flow across it. We briefly review the recent advances in theoretical and experimental studies, develop the general theoretical framework directly linking the microscopic interfacial transport to the macroscopic flow fields, discover mechanisms of interface stabilization and destabilization that have not been discussed before for both inertial and accelerated dynamics, and chart perspectives for future research.

- interface dynamics
- phase boundary
- Landau–Darrieus instability
- Rayleigh–Taylor instability
- interfacial mixing

Interfacial mixing and transport are nonequilibrium processes coupling kinetic to macroscopic scales (1). They commonly occur in fluids, plasmas, and materials over celestial events to atoms (2, 3). Their understanding has crucial importance for science, mathematics, and engineering as well as for technology, energy, and environment (1⇓⇓–4). In this paper, we focus on the long-standing problem of stability of a phase boundary (i.e., a fluid interface) (5, 6). By developing the general theoretical framework, we systematically study the interface stability and the flow fields’ structure; discover mechanisms of stabilization and destabilization of the inertial and accelerated dynamics that have not been discussed in the earlier studies (5, 6); elaborate diagnostics that have not been identified before and that directly link microscopic transport at the interface to macroscopic fields in the bulk; and chart perspectives for future research. For the readers’ convenience, technical details are given in *SI Appendix* for the corresponding sections.

Hydrodynamic instabilities and interfacial mixing control a broad range of processes in nature and technology at astrophysical and molecular scales under conditions of high- and low-energy densities (1⇓–3). Inertial confinement fusion and light–material interaction, supernovae and molecular clouds, stellar convection and ionospheric plasma, reactive fluids and material evaporation, fossil fuel production and nanoelectronics—these are some examples of processes to which nonequilibrium interfacial dynamics is directly relevant (7⇓⇓⇓⇓⇓⇓⇓⇓⇓–17). In realistic environments, the material transport is often characterized by sharply and rapidly changing flow fields and by relatively small effects of dissipation and diffusion. This leads to formation of discontinuities (interfaces) separating the flow nonuniformities (phases) at continuous (macroscopic) scales (1⇓–3, 18⇓–20).

For a far-field observer, two types of hydrodynamic discontinuities are usually considered—a front (with zero mass transport across it) and an interface (through which mass can be transported) (5). Their dynamics is analyzed in a continuous approximation and at length scales and timescales that are greater than characteristic scales induced by diffusion, dissipation, surface tension, and other stabilizing effects (5, 6, 11⇓⇓⇓–15). The fluid phases are broadly defined: These can be distinct materials or a material with distinct thermodynamic properties. The material(s) may also experience a phase transition, a change in chemical composition, be out of thermodynamic equilibrium, and/or have a nondiffusive interfacial mass transport (1⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–21). To describe the multiphase flow, a boundary value problem is solved for balancing at a freely evolving discontinuity the fluxes of mass and normal component of momentum in case of fronts and the fluxes of mass, momentum, and energy in case of interfaces (5). While solving the boundary value problem can be a challenge, this level of abstraction has a number of advantages. On the side of fundamentals, the problem is treated rigorously, powerful theoretical methods [e.g., group theory (2)] are applied, and essentials of the dynamics are explored. On the side of applications, this approach provides reliable macroscopic benchmarks for diagnostics, is free from adjustable parameters, and has high predictive capability in a broad parameter regime (2, 3, 5, 20).

Dynamics of fronts separating fluids of different densities is usually neutrally stable and can be destabilized by accelerations and shocks, leading to the Rayleigh–Taylor (RT) and Richtmyer–Meshkov (RM) instabilities, respectively, and the ensuing interfacial mixing of the fluids (22⇓⇓⇓–26). Rigorous theoretical approaches have been developed to describe RT and RM flows with account for the nonlocal anisotropic, heterogeneous, and statistically unsteady character of their dynamics (2, 3, 27⇓⇓⇓–31). These approaches have captured the fundamental properties of the instabilities and mixing (including the multiscale RT/RM dynamics and the order in RT mixing) and have explained the observations (2, 3, 26, 30).

Dynamics of interfaces separating fluids of different densities and having an interfacial mass flux is a long-standing problem with a broad range of applications (5). It is studied in plasmas (stability of ablation fronts in inertial confinement fusion), astrophysics (thermonuclear flashes on the surface of compact stars), material science (material melting and evaporation), gas dynamics (shocks and explosions), combustion (stability of flames), and industry (scramjets) (6⇓⇓⇓⇓⇓⇓⇓⇓⇓–16, 32⇓⇓⇓⇓⇓⇓⇓–40). The classic theoretical framework for the problem was developed by Landau (40). It analyzes the dynamics of a discontinuous interface separating ideal incompressible fluids of different densities. By balancing at the interface the fluxes of mass and momentum and by implementing a special condition for the perturbed mass flux, this analysis finds the interface to be unconditionally unstable, leading to the Landau–Darrieus instability (LDI) (5, 6, 40).

To connect the classic framework (5, 40) to realistic environments, several approaches have been developed. In high-energy density plasmas, significant departures have been detected between the ablative RT and accelerated Landau–Darrieus (LD) instabilities as well as between the ablative RM and LD instabilities (32⇓⇓–35, 41⇓–43). It has been recently shown that the interface stability is sensitive to the flux of energy fluctuations produced by the perturbed interface (20). In reactive and supercritical fluids, the stabilizing influences have been found of dissipation, diffusion, surface tension, and finite interface thickness on the dynamics at small scales (6, 11, 37⇓–39). Significant progress has been achieved in the understanding of nonlinear stages of the LDI and in modeling of turbulent combustion (11, 44, 45). These theories and models have successfully expanded the classic framework (5, 40) to explain the observations (6, 11, 32⇓⇓⇓⇓⇓⇓–39, 41⇓⇓⇓⇓–46). However, some fundamental challenges remain.

First, the classic framework (5, 40) describes the evolution of a phase boundary and is relevant to a range of phenomena far beyond processes with gradually changing flow fields (1⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–16, 32⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–45). We still need to understand whether the interface is stable when the flow quantities experience sharp changes, the effect of dissipation and diffusion is negligible, and the interfacial transport is nondiffusive. Second, the flow evolution is usually observed from a far field and at timescales and length scales that are substantially greater that those induced by interfacial processes at small scales (6⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–20). We still need to quantify what the flow sensitivity is to the boundary conditions at the interface. Third, direct diagnostics of various physical effects on dynamics of a multiphase flow require detailed information of the interface structure. Such information is often a challenge to obtain directly in experiments and simulations (6⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–18, 32⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–46). We need to better comprehend what the qualitative and quantitative influence is of the interfacial transport at microscopic scales on volumetric flow fields at macroscopic scales and elaborate reliable benchmarks. This knowledge is necessary to identify the mechanisms of stabilization and destabilization of interfacial dynamics; to improve the diagnostics of complex processes in plasmas, fluids, and materials; and to better understand a broad range of phenomena in nature and technology (1⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–16).

In this paper, we consider from a far field the inertial and accelerated dynamics of a hydrodynamic discontinuity that separates ideal incompressible fluids of different densities and is accompanied by the interfacial mass flux. By generalizing the classic framework (5, 40), we link directly the interface stability to the flow fields’ structure. Mechanisms are identified of the interface stabilization and destabilization that have not been discussed before. We find that the inertial dynamics is stable when it conserves the fluxes of mass, momentum, and energy; the stabilization is due to the inertial effect, leading to small oscillations of the velocity of the interface as a whole. An energy imbalance can destabilize the dynamics (20), which is fully consistent with the classic results (5, 40). In reactive fluids, the energy imbalance can be due to chemical reactions (4, 46). For accelerated dynamics, the interface stability is determined by the interplay of the effects of inertia and buoyancy. A hydrodynamic instability is found that has not been identified in earlier studies (5, 6, 11, 32⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–46) and that develops when the gravity value exceeds a threshold. This unstable dynamics conserves the fluxes of mass, momentum, and energy; has potential velocity fields in the bulk; and is shear-free at the interface. The qualitative, quantitative, and formal properties of this instability differ dramatically from those of the accelerated LDI and the Rayleigh–Taylor instability (RTI) (5, 22, 23, 40).

## Theoretical Approaches

### Governing Equations.

Dynamics of ideal fluids is governed by the conservation of mass, momentum, and energy. In an inertial reference frame,

We introduce a continuous local scalar function

Inertial reference frame can be referred to the reference frame moving with a constant velocity

### Linearized Dynamics.

We define

To the leading order,

To the first order, boundary conditions at the interface are

### Solution Structure.

We seek solution such that the velocity of the heavy fluid is potential in accordance with the Kelvin theorem and the velocity of the light fluid can be a superposition of the potential and vortical components (5, 6):

### Fundamental Solutions.

The governing equations are reduced to a linear system

## Results—Inertial Dynamics

For inertial dynamics, gravity is zero,

### Conservative Dynamics.

Conservative dynamics balances the fluxes of mass, momentum, and energy at the interface. For this dynamics, the matrix Μ is *A*). There are four fundamental solutions. The solutions

Fundamental solutions

Fundamental solution

### Classic LD Dynamics.

Classic Landau dynamics balances the fluxes of mass and normal and tangential components of momentum and uses the continuity of normal component of the perturbed velocity *B*). There are three fundamental solutions:

Fundamental solution

For fundamental solution

The classic Landau dynamics has a smaller number of fundamental solutions than the degrees of freedom. Eliminating this degeneracy may lead to appearance of a (fourth) neutrally stable solution with a seed vortical field triggering the LDI.

### Comparative Study.

Conservative dynamics

### Mechanisms of Stabilization and Destabilization.

To better understand the mechanism(s) of the interface (de-)stabilization, we consider the interface velocity (11, 38⇓–40). In the laboratory reference frame, the interface velocity is

This suggests the inertial effect as the stabilization mechanism of the conservative dynamics

For the LD dynamics

Remarkably, for ideal incompressible fluids, the solution

The enthalpy perturbations are

In realistic fluids, this energy imbalance can be induced by energy fluctuations. The effect can be self-consistently derived from entropy conditions with account for chemical reactions. In ideal fluids, to quantify the effect of energy imbalance on the interface stability, we can introduce an additional artificial energy flux; study a transition from stable to unstable dynamics with increase of the fluctuations’ strength; and find that, for strong (weak) fluctuations, the eigenvalues and the flow fields are similar to those in the classic Landau (conservative) dynamics (20) (*SI Appendix*).

### Chemistry-Induced Instabilities.

In reactive fluids, it is generally well-understood that chemically reactive systems can be hydrodynamically unstable. However, it is a challenge to construct a model experiment or a simulation that cleanly displays significant chemical reaction instability at a simple interface. The paper in ref. 46 reports atomistic simulations for studying the energy transport in a reactive system and the effect of chemical reaction on the interface stability. Additional investigations are required to fully understand the properties of chemistry-induced instabilities at atomistic and continuous scales.

## Results—Accelerated Dynamics

For accelerated dynamics, gravity g is directed from the heavy to the light fluid,

### Accelerated Conservative Dynamics.

For conservative dynamics balancing the mass, momentum, and energy at the interface, matrix Μ is *A*).

Solutions

For accelerated conservative dynamics

For solution

### Accelerated LD Dynamics.

For accelerated LDI, the dynamics balances the fluxes of mass, normal and tangential components of momentum, and the normal component of the perturbed velocity *B*). Solutions

Fundamental solution

For solution

### Accelerated RT Dynamics.

For the RTI (22, 23), there is no mass flux at the interface, and there is no fluid motion far from the interface. This leads to the boundary conditions

For any

### Comparative Study.

While large gravity values destabilize the interface, the accelerated conservative dynamics

For solution

Solutions

Solutions

### Mechanisms of Stabilization and Destabilization.

The stabilization mechanism of the accelerated conservative dynamics is revealed by the qualitative dependence of the solution

While velocity fields in accelerated conservative dynamics

These results are consistent with the dependence of the interface velocity on the gravity value. In the laboratory reference frame, the interface velocity is

For large accelerations,

In dimensional units, for given values

### Effect of Surface Tension.

Our general framework enables the systematic study of interfacial dynamics influenced by surface tension, thermal conductivity, compressibility, viscosity, mass ablation, and flow geometry and dimensionality, upon the corresponding modification of the governing equations. Here we briefly consider the effect of surface tension. It is important in multiphase flows, and is straightforward to account for. The outline of results is given in the *SI Appendix*. Note that stabilizations of the conservative dynamics by the inertial effect and by the surface tension are distinct mechanisms.

## Discussion

Interfacial mixing is a nonequilibrium process coupling kinetic to macroscopic scales (1). Grasping fundamentals of the interfacial dynamics is crucial for science, mathematics, and engineering (2⇓⇓⇓–6). This work is focused on the classic problem of stability of a fluid interface (phase boundary) that has a mass flux across it (5, 6, 40). We have briefly reviewed the recent advances in theoretical and experimental studies, have developed the general theoretical framework to systematically study the interface stability and the flow fields, and have identified the mechanisms of interface stabilization and destabilization in the inertial and accelerated flows that have not been discussed in earlier studies (5, 6, 32⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–45) (Figs. 1–7).

Our theoretical framework is consistent with and generalizes the classic approach (5, 40); directly links the flow fields to the boundary conditions at the interface; and assumes sharp changes of the flow quantities at the interface and negligible effects of dissipation, diffusion, compressibility, and interface thickness. By examining the interface from a far field in a sample case of a 2D flow, we found extreme sensitivity of the dynamics to the interfacial boundary conditions and discovered properties that have not been identified before (Figs. 1–7).

The inertial conservative dynamics is stable and is stabilized by the reactive force. The flow is a superposition of two motions—the background motion of the fluids following the interface with slightly oscillating velocity and stable oscillations of the interface perturbations (Figs. 1–3). For classic Landau dynamics, the interface velocity is constant, the reactive force is absent, and the dynamics is unstable. An energy imbalance may enable the LDI to occur. In reactive fluids, the energy imbalance can be due to chemical reactions (46). The LD unstable dynamics is a superposition of two motions—the background motion of the fluids following the interface with the constant velocity and the growth of the interface perturbations (Figs. 1–3).

For accelerated conservative dynamics, the flow stability depends on the interplay of inertia and buoyancy (i.e., reactive force and gravity) (Figs. 4–7). For gravity values smaller than a threshold, the dynamics is stabilized by inertial effect and reactive force. For large gravity values, buoyancy effect dominates, and gravity destabilizes the flow. The dynamics is a superposition of two motions. Below the threshold, these are the background motion of the fluids following the interface with slightly oscillating velocity and stable oscillations of the interface perturbations. Above the threshold, the interface perturbations grow and therefore, it is the interface velocity. For strong accelerations, this instability grows faster than the accelerated LDI and RTI; for weak acceleration, the LDI has the largest growth rate.

For unstable conservative dynamics in a gravity field, the flow is potential in the fluids’ bulk, similar to the RTI and in contrast to the accelerated LDI; it is shear-free at the interface, similar to the LDI and in contrast to the RTI. The conservative dynamics is nondegenerate in contrast to the LDI and the RTI. The degeneracy suggests a singular (ill-posed) character of the RT and LD dynamics requiring the (neutrally stable) seed vortical field in the bulk for the LDI or the seed interfacial shear for the RTI.

It is commonly believed that an interface separating nearly ideal incompressible fluids and having an interfacial mass flux is subject to the LDI at large scales and that, in realistic environment, the LDI is a challenge to implement, because the effects of dissipation, diffusion, and finite interface thickness stabilize the small scales (5, 6, 11, 12, 37⇓–39). Our far-field analysis is fully consistent with these results: For fluids with similar densities, **1**–**8**, Figs. 1–7, and *SI Appendix*).

One such experiment can be a study of the dynamics of fluids with very different densities, with diagnostics of the flow fields near the interface and in the fluids’ bulk, and with the measurements of the interface evolution, including the interface velocity as a whole and the interface perturbation growth rate. By comparing the observations with our benchmarks, one can further identify the fundamentals of the interfacial dynamics in realistic environments and elaborate approaches for the flow control (Figs. 1–7) (1⇓–3, 20).

Several questions may frame these perspective studies (20, 46). Can the LDI unconditionally develop at the large scales (20) (Eqs. **4** and **5** and Figs. 1–3)? How can the dynamics be stabilized—by inertial effect, by dissipation and diffusion, or by their combination? How strong should energy fluctuations be to destabilize the flow (20) (*SI Appendix*)? Can these fluctuations be induced by chemical reactions (46)? If so, how are the properties of chemistry-induced instabilities compared with those of the LDI (46)?

For accelerated dynamics, our results suggest that the conservative dynamics is driven by the interplay of the effects of inertia and buoyancy (reactive force and gravity). These results are consistent with recent studies of ablative Rayleigh–Taylor and Richtmyer–Meshkov (RMI) instabilities in compressible fluids (32⇓⇓–35, 41⇓–43). Our analysis yields the properties of the accelerated interfacial dynamics that have not been discussed before (Eq. **6** and Figs. 4–7).

According to our results, the conservative dynamics can be stable even for ideal incompressible fluids at any density ratio when the gravity value is smaller than a threshold (Fig. 7). In the stable regime, the interface velocity experiences stable oscillations, whereas in the unstable regime, along with the growth of interface perturbations, the interface velocity may also increase. The latter qualitatively explains the intensive material mixing observed in experiments in inertial confinement fusion (7, 33, 35). Thus, our analysis can self-consistently resolve this long-standing puzzle. Note that our conservative dynamics instability is the fastest (compared with the LDI and the RTI) in the extreme regime of strong accelerations that are common in high-energy density plasmas (32⇓–34).

For our accelerated conservative dynamics with the growth rate

Existing experimental and numerical studies of the interface stability are focused on the growth of the perturbation amplitude (5⇓–7, 11, 33⇓⇓⇓⇓⇓–39). Our analysis derives the amplitude growth rate and finds that the dynamics is highly sensitive to interfacial boundary conditions. According to our theory, by measuring at macroscopic scales the flow fields in the bulk and at the interface, one can capture the transport properties at microscopic scales at the interface (Figs. 1–7). This information is especially important for systems where experimental data are a challenge to obtain (7, 8, 13, 15, 16, 32⇓⇓⇓⇓⇓⇓–39).

Our approach can systematically incorporate the effects of dissipation, diffusion, compressibility, radiation transport, stratification, finite interface thickness, and nonlocal forces that are important for material dynamics in realistic environments (5, 7⇓⇓⇓⇓⇓⇓⇓⇓–16). According to our results, at small length scales, the interface dynamics can be stabilized by surface tension that influences critical (maximum) wave vector

## Acknowledgments

We thank the University of Western Australia, the National Science Foundation, and the Summer Undergraduate Research Fellowship Program at the California Institute of Technology.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: snezhana.abarzhi{at}gmail.com.

Author contributions: S. I. Abarzhi designed research; S. I. Abarzhi and D.V.I. performed research; S. I. Abarzhi, D.V.I., W.A.G., and S. I. Anisimov analyzed the results and data; and S. I. Abarzhi, D.V.I., W.A.G., and S. I. Anisimov wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1714500115/-/DCSupplemental.

Published under the PNAS license.

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