Synthetic chemistry has drastically progressed during the last few decades. Like in DNA, it is now possible to synthesize abiotic macromolecules with precisely-controlled sequences of monomers.

Artificial systems allowing hybridization and information transfer have been described. Simple metabolism models have also been prepared and investigated. Thus, the creation of minimal artificial life forms appears as a (still relatively far) but possibly reachable objective. In this context, the aim of the present essay is to discuss recent achievement and future challenges towards man-induced abiogenesis. In terms of polymer design, a synthetic macromolecule that would support Life as DNA do would require at least all the following properties: (i) information-storage capacity, (ii) information-transfer ability and (iii) the possibility to evolve through molecular mutations. Recent advances in that direction are summarized herein. However, Life is not only a matter of macromolecular design. In order to operate, a self-replicator shall co-exist with other molecules in non-equilibrium conditions. Hence, systems chemistry aspects are also discussed in this essay.

Introduction

About 3.5-4 billion years ago, Life has emerged on Earth (Figure 1). Biological Life is primarily operated by sequence-defined polymers, which are nucleic acids (DNA and RNA) and proteins. [1] Although a large amount of research has been dedicated to prebiotic chemistry and abiogenesis, scenarios leading to the formation of these polymers and ultimately to the emergence of the first unicellular marine organisms are still actively debated. The reader is therefore directed to classical reviews for detailed information on this broad topic. [2] In brief, since the Miller experiment that has evidenced the formation of some biological monomers in laboratory conditions mimicking prebiotic Earth environment, [3] it is believed that Life has emerged from a primordial chemical soup. [START_REF] Oparin | Proiskhozhdenie zhizni, Moskovskii Rabochii[END_REF] Alternatives theories such as panspermia, which suggests that Earth was seeded through collisions with asteroids, meteors and comets, are nowadays less prominent.

Even if the emergence of Life is not fully explained, molecular biology is very well understood. [START_REF] Berg | [END_REF] DNA is a self-replicating polymer that stores genetic information in the form of a controlled sequence of monomers and enables information transfer from generation to generation. It is also prone to rare but possible sequence variations (i.e. mutations), which allow Evolution through natural selection to take place. [6] This molecular interpretation of Darwinian Evolution is supported by numerous works including Dawkins' popular theory of the selfish gene. [START_REF] Dawkins | The selfish gene[END_REF] Through Evolution and self-organization, [START_REF] Lehn | [END_REF] complex living forms have gradually emerged (Figure 1). Over very long timescales (typically a few billion years in the case of Earth), this process may lead to the emergence of thinking species such as Humans. Interestingly, such species may start to perform intentional Chemistry and to understand how to transform their environment. This creates a rupture point in the chemical evolution continuum of the Universe that I coined as "thinking-species chemistry" in an earlier essay. [9] Indeed, thinking species can synthesize and produce molecules that have not been generated by prebiotic chemistry and biology. They are therefore chemodiversity accelerators. For instance, Humans synthesize actively non-biological polymers for about a hundred years. [START_REF] Staudinger | This Rosarium Philosophorum special issue of Israel Journal of Chemistry coincides with the 100 th anniversary of the coinage of the term "Polymerization[END_REF] In general, man-made macromolecules are less structurally-defined than biological polymers. Most of them are polydisperse; i.e. they have non-uniform molecular structures. Yet, as already pointed out decades ago by Staudinger, synthetic polymers are interesting models to understand Life and biology. [11] In fact, these models have progressively become more and more refined over the last thirty years. [12] Indeed, it is nowadays possible to control accurately the chain-length and monomer sequence of synthetic macromolecules. [1] Furthermore, synthetic hybridization [13] as well rudimentary replication and translation processes have been described. [14] In this context, it is worth asking the following question: will synthetic macromolecules ever lead to Life? [15] In other words, is it possible to conceive non-DNA self-replicators leading to rudimentary life forms? [16] Considering recent progress in Chemistry, this is conceivable. Life has already emerged from polymers and it might be possible to make it artificially. However, creating a viable metabolism is not only a matter of polymer synthesis but also of non-equilibrium thermodynamics and systems chemistry.

All these aspects are critically discussed in the present essay.

Figure 1. Simplified summary of the emergence and evolution of Life on Earth. The advent of thinking-species that can perform intentional chemistry is a breakpoint in the chemical evolution continuum of the planet. It increases chemodiversity and may lead to an induced abiogenesis (denoted as 2 nd-abiogenesis to distinguish it from primordial abiogenesis). The picture of the rock including fossils of multicellular organisms is reprinted with permission from reference [17]. The portraits show from left to right Antoine de Lavoisier (1743-1794), Hermann Staudinger (1881-1965) as well as James Watson (1928-) and Francis Crick (1916-2004).

Man-induced chemodiversity

Thinking species

Multicellular organisms

Standard Evolutionary Scale

Human acceleration today -4. Before discussing the Life-supporting ability of synthetic polymers, it is mandatory to understand which molecular features allow a macromolecule such as DNA to carry Life. First of all, it shall be reminded that the terms "Life", "alive" and "living" are prone to different interpretations and, therefore, the reader is directed to prior literature discussing the definition of Life. [18] It shall also be clarified that polymers are probably not the only options for creating artificial Life. Other chemical, biological or technological systems can be conceived. However, since terrestrial Life is operated by a polymer, which is DNA, it is plausible that other types of polymers could lead to Life. This reasoning is similar to the one used in exoplanets search. Extra-terrestrial Life is primarily expected on water-based planets (by analogy with Earth), although it could theoretically occur in other chemical environments. [16] Table 1. Summary of the important features that make DNA a Life-supporting macromolecule. Depurination, deamination, dimerization [b] Repair Mostly enzymatic Epigenetics Methylation, hydroxymethylation [b] [a] The content of this table is somehow subjective and is not meant to be absolute . [b] The listed terms are only indicative examples of more complex mechanisms.

Table 1 lists eight important features that make DNA a unique Life-supporting macromolecule:

(i) information storage in a controlled monomer sequence, (ii) duplex formation, which is a mandatory for replication (iii) topological accessibility, which permits to expose or not certain regions of the genome for replication and transcription, (iv) self-replication, (v) translation into another type of messenger polymer, (vi) random mutations, which allow Evolution, (vii) possibility of repairing molecular damages, and (viii) dormant and active mode through epigenetics. It is anticipated that a synthetic polymer that would carry Life shall possess at least some (if not all) of these properties. For instance, to attain a rudimentary form of molecular Life, the possibility to store and transfer information seems to be the minimum required. Yet, such a minimal system shall also proceed in steady state conditions, which require constant chemical input and output. [19] Another crucial feature would be, of course, the possibility for such a system to evolve. This implies that a synthetic self-replicator would be able to undergo random or controlled molecular variations through generations. In the following sections, the possibility to fulfil (or not) these requirements with synthetic polymers is discussed. [START_REF] Benner | There are different ways to interpret the terms Synthetic Biology as discussed[END_REF] An obvious path towards artificial Life is to synthesize and manipulate biopolymers such as DNA in the laboratory. In such approaches, the principles of molecular biology are used as such or are slightly modified. In other words, self-replicating macromolecules are not designed from the scratch but are derived from existing natural models. Thus, such studies are usually more related to biology than chemistry. However, these approaches permit to understand what kind of molecular variations are tolerated by living systems. Synthetic biology is a very broad field of research and only selected aspects are discussed in the following sub-sections.

Synthetic biology

Natural nucleic acids

Laboratory synthesis of DNA

Sequence-defined DNA and RNA can be synthesized in the laboratory using chemical [21] or enzymatic [22] protocols. DNA Chemical synthesis has begun in the 1960s and has been drastically facilitated by the discovery of phosphoramidite chemistry, [23] a process than can be easily automatized [24] and photocontrolled. [25] It has been widely demonstrated that chemically-synthesized DNA segments can be incorporated into bacteria-expressible plasmids using recombinant DNA technology. In 2012, it was even demonstrated that a bacterial cell can be operated by an artificial genome that was entirely synthesized in a chemical laboratory. [26] These example clearly demonstrate than Life can be partly or fully operated by lab-made macromolecules. However, these results have been obtained with man-made polymers that have the same molecular structure as biological DNA.

In vitro replication

Nucleic acids replication and transcription can be performed in vitro. [27] For instance, the widespread polymerase chain reaction (PCR) is a biological DNA amplification method that is routinely performed in the laboratory. [28] Non-enzymatic DNA replication and DNA→RNA transcription have also been widely studied, as described in details in recent reviews. [1,29] The complementarity between RNA and peptide nucleic acids (PNA) was also reported. [30] Even if they are polyamides, PNA oligomers can act as templates for RNA synthesis and vice versa. It was also recently demonstrated that DNA can act as a template for the synthesis of non-natural sequence-defined polymers (i.e. artificial translation). [14d] In this example, PNA adapters were used as intermediates to translate the sequence information of DNA into non-natural sequences. All these studies indicate that the essential information transfer mechanisms of Life can be recreated in a chemical laboratory.

Protocells including minimal metabolisms

Although a self-replicating macromolecule such as DNA is essential to Life, the molecular machinery of living cells is by far more complex than the simplified mechanisms described in the previous sections. [31] Indeed, cell membranes confine a large number of different molecules that are in constant dynamic interaction. Such a "society" of molecules is best described by the theory of complex systems. In this context, there are a large of number of studies dealing with the inclusion of simplified self-replicating mixtures (e.g. PCR) or other minimal metabolic models (e.g. enzymatic cascades) in the confined interior of vesicles, microcapsules or microgels. Such protocell models have mainly been built using preformed lipid-based or polymer-based vesicles, namely liposomes and polymersomes. [32] However, the formation of coacervates is also believed to have played a role in the origin of Life and therefore interesting examples of membrane-free protocells have been reported. [33] The in situ self-organization of lipids (i.e. membrane formation) on the surface of coacervates droplets has also been observed. [34] All these examples indicate that in vitro replication and transcription can be performed in artificial protocells.

Nucleic acids analogues

Main-chain modifications

A significant amount of research has been dedicated to the synthesis and the assessment of the biological properties of nucleic acids that slightly differ from natural ones (i.e. so-called nucleic acids analogues). [29] For instance, xeno nucleic acids (XNA) have a main chain structure that differ from the ribophosphate backbone of natural DNA and RNA. Threose nucleic acids (TNA), locked nucleic acids (LNA) and morpholinos are typical examples of XNA. Among all reported examples, PNA, initially described by Nielsen and coworkers, [35] have been shown to have a remarkable biological activity [36] and are therefore used in various medical applications. More recently, Holliger and coworkers have evidenced that Darwinian evolution and heredity are not restricted to DNA and RNA but can also be attained with different types of XNA. [37] It was also reported that some of these polymers can catalyze some chemical transformations like ribozymes do. [38]

Side-chain modifications

Besides main-chain modifications, numerous studies have been devoted to the synthesis and properties of nucleic acids containing non-natural nucleobases (i.e. side-chain modifications). [29,39] In particular, various types of non-canonical nucleotides have been studied in in vitro enzymatic replication. For instance, Romersberg and coworkers have reported that non-natural base-pairs can be recognized by the biological replication machinery. [40] More recently, an expanded genetic alphabet containing 8 symbols (instead of 4 in DNA and RNA) has been described by Benner and coworkers (Figure 2). [41] Remarkably, it has been shown that such 8-bases DNA still exhibit a biological behavior and can be for instance transcribed into a corresponding RNA structure. These examples evidence that the basic features of Life (i.e. information transfer, mutability, evolvability) can be preserved after performing main-chain or sidechain modifications in nucleic acids.

Figure 2. Expanded genetic alphabet containing eight different nucleobases, forming four orthogonal base pairs. The atoms highlighted in red and blue indicate hydrogen -bond acceptor and donor sites, respectively. Redrawn from reference [41] .

Synthetic chemistry

Addressing synthetic Life from a non-biological point of view is by far more challenging than working with genuine-or slightly-modified biological systems. Indeed, taking into account all available building blocks and reactions in synthetic chemistry, the search space is extremely large.

However, over the last decades, significant progress has been made for the synthesis of abiotic polymers with controlled primary structure (i.e. monomer sequence) and secondary structure (i.e.

single chain folding, strand hybridization). Furthermore, self-replication has been obtained with small abiotic molecules. These advances are described in the following sub -sections.

Controlled information sequences

Monomer sequence control is the solution found by Nature to store information at the molecular scale. [1] Most of man-made synthetic copolymers (i.e. composed of two or more different monomers) do not have controlled primary structures because they are synthesized by statistical copolymerization methods such as step-growth or chain-growth polymerizations. [12] However, with the emergence of synthetic

A T G C Z P S B
biochemistry in the 1950s and 1960s, efficient multistep synthetic approaches have been introduced for the preparation of sequence-defined proteins and nucleic acids. [42] In such multistep-growth approaches, [12] monomers are usually attached one-by-one to a growing chain, which is time-consuming. Thus, these syntheses are most of the time performed on a solid-phase support, [1] which simplifies and shortens experimental procedures. Starting about thirty years ago, multistep synthesis has also been used to prepare non-natural sequence-defined bio-oligomers, such as peptidomimetics and peptoids. [1,43] Afterwards, a wide variety of fully-synthetic (i.e. structurally-unrelated to natural polymers) sequence-defined polymers have been synthesized and reported. [44] In particular, starting about five years ago, [45] it has been demonstrated that abiotic informational polymers can be prepared by solid-phase iterative chemistry. [46] Hence, this essential particular property of Life-supporting polymers (Table 1) can now be easily recreated in synthetic systems. Yet, for the moment the chain-length of such macromolecules is limited to about100 information residues, [47] which is much shorter than in natural polymers. To give an idea of the gap between biological and non-biological polymers, the human chromosome 1 is, for example, composed of two hybridized DNA chains, each of which containing a perfectly defined sequence of 248956422 monomers (i.e. more than 248 million). Nevertheless, medium-size manmade informational polymers can be synthesized and routinely utilized.

Synthetic hybridization

As listed in Table 1, strand hybridization is a key property in molecular biology. Several research groups have described the formation of abiotic supramolecular duplexes. Some of the earliest examples have been reported by Lehn and coworkers using either metal-ligand or hydrogen-bonding supramolecular interactions. [13a, 48] Afterwards, other interesting studies have been reported by the groups of Gong,[13b] Yashima, [49] Hunter, [13c] Scott [50] and others. [51] However, in most reported examples, the hybridization of two homopolymer strands was studied, which is the simplest situation that can be achieved. It can be obtained by either hybridizing the same strand (i.e. self-complementary homopolymer) or using two complementary homopolymers. Therefore, these systems do not allow orthogonal strand recognition. In order to achieve this, sequence-specific hybridization is mandatory. Sequence-specific duplexes can be obtained using defined sequences of two, four or eventually more complementary monomers. Yet, there are currently very few examples of synthetic oligomers that can undergo sequence-specific hybridization. [13b, 52]

Self-replication and information transfer

The ability of a macromolecule to copy itself and to transfer its information is crucial to Life. There is currently no man-made macromolecule that copy itself in the way DNA does. However, during the last thirty years, there has been significant research on the design of minimal replicating systems. [14a, 53] These pioneer studies, involving in most cases small model molecules, have been described in details in a recent review. [54] Although rudimentary self-replication behaviors have been demonstrated, most of the reported systems do not allow transfer of coded information sequences as achieved by nucleic acids. [27] In these context, there was earlier this year two interesting examples of sequence information transfer using dynamic [14b] or covalent [14c] bonds between coded template and substrates. Yet, these examples were achieved with rather small information sequences (i.e. dimer or trimer) and much remains to be done to extend these concepts to long information sequences.

B B B B A

A A A

Synthetic mutations

Mutability allows evolution. Indeed, in a self-replicating system, discrete molecular variations may lead to information content changes from generation to generation and therefore to an adaptive behavior. Since the self-replication of abiotic informational polymers has not been achieved yet, the mutability of synthetic polymers has not been investigated. However, it is already possible to perform discrete monomer modifications in synthetic information-containing macromolecules. For instance, our group has recently reported that light can be used as a simple trigger to induce molecular changes in sequence-coded poly(phosphodiester)s. [55] Such a concept is inspired by irradiation-mediated mutagenesis and enables sitedirected mutations in synthetic polymers. In the future, such modifications could be used in conjunction with a self-replicating system to generate an evolutionary behavior in synthetic polymers. Yet, in a real evolvable polymer, such mutations shall not be site-directed but shall occur randomly (and rarely).

Emergence vs. Intention

As described in the previous pages, artificial macromolecular Life can be envisaged from two different standpoints: either tweaking existing biopolymers (synthetic biology) or preparing new types of polymers using non-natural building-blocks and reactions (synthetic chemistry).

Addressing Life from the latter approach is much more challenging, taking into account the number of building blocks and reactions that are available to chemists. Moreover, given the numerous molecular parameters (e.g. interdistance between coded monomer units, backbone stiffness, stereochemistry, secondary structure, affinities of different part of the molecule with the solvent… etc…) that could influence the behavior and physico-chemical properties of informational macromolecules, it is unreasonable to think that all aspects can be solved with a single lucky design, even with the help of computational chemistry. In fact, an artificial Life-supporting macromolecule shall probably not be entirely conceived. As learned from Nature, it shall better emerge as a viable and adapted solution from a component mixture under selective pressure. In the paragraph, recent efforts for preparing artificial metabolisms that proceed in non-equilibrium thermodynamic conditions are briefly discussed.

Systems Chemistry

Life does not occur at thermodynamic equilibrium. [19] Indeed, living organisms are constantly exchanging energy and matter with their environment. Therefore, as a matter of fact, a Lifesupporting macromolecule is by itself not sufficient to spark Life. In other words, having a polymer that fulfil all criteria listed in Table 1 is not enough to obtain living matter. For instance, DNA can be stored as inanimate matter in archaeological remains or in a laboratory flask. To trigger and support Life, DNA shall be involved in a metabolic process. In this context, there was recently an increased interest for the burgeoning field of systems chemistry, which focuses on the comprehension of complex molecular networks (e.g. dynamic combinatorial libraries, oscillating reactions, autocatalytic compounds). [56] For instance, several minimal self-replicating systems (see section 4.3) but also more complex networks of synthetic replicators have been studied. [57] However, in most cases, these systems end up in thermodynamically or kinetically-trapped products. An elegant example of chemically-fuelled self-replicating system was reported earlier this year by Fletcher and coworkers. [58] This system involves disulfide low molecular weight surfactants and does not allow information transfer. Yet, although far from macromolecular science, it demonstrates that simple metabolic cycles can be attained with abiotic components.

Toward evolutionary systems

As described in the previous section, minimal chemical systems that self-replicate have been reported. However, in order to exhibit a true Darwinian evolutionary behavior, a molecule shall not only copy itself but also be able to mutate, as summarized in Table 1. Hence, a rudimentary proofof-principle of artificial Chemical Life would require the design of a macromolecular informational system, in which both information transfer and mutations are achievable. Although technically conceivable, such a system has not been reported to date with abiological macromolecules.

Nevertheless, it would permit to derive from initial intentional design toward the emergence of adapted informational sequences and properties. In other words, such a macromolecular system would become truly adaptive. [59]

Accelerated progress through Human selection

Sequence-controlled polymers constitute a particularly vast domain of investigation for chemists because the number of possible molecules can be dramatically high and exceed Human experimental and calculation capabilities. [1] For proteins (which is only one type of macromolecule), it has already been pointed out long time ago by Staudinger that the number of isomers of a moderate-size sequence exceeds the number of water molecules in the seas of Earth. [60] Actually, it even exceeds the number of available atoms in the observable Universe. [16] Including man-made sequence-defined polymers in the search space will increase even further the number of possibilities. Thus, progress and discovery shall be somehow guided. For a polymer capable of Darwinian evolution (vide supra), change can operate under the selective pressure of environmental conditions. However, such processes are usually very slow, even though they can be artificially accelerated. [61] The selection of promising molecules by Human chemists is another important parameter for guiding discovery and narrowing down the search space. With the help of intuition and reasoning, Humans may: (i)

first synthesize and screen a restricted number of promising molecules, (ii) select the most promising structures and (iii) involve them in a metabolic process leading to further evolution (vide supra). In other words, a subtle combination of macromolecular design and systems chemistry could be the key toward the development of artificial Life-supporting macromolecules.

Artificial mimics of other biological functions

The core discussion of the present Essay is about the realization of artificial Life-supporting macromolecules (i.e. polymers that store information, transfer information and evolve). Yet, as pointed out by one of the reviewers of this manuscript, polymer science offers many other possibilities to mimic biological behaviors. For instance, lipid cell-membranes can be replaced by block copolymer membranes, as discussed in section 3.1.3. Moreover, the compartmentalization of such polymer-based membranes leading to the formation of artificial organelles has been described. [62] The catalytic activity of enzymes has also been mimicked with synthetic polymers, such as single-chain nanoparticles. [63] There is also an abundant literature, which will not be discussed herein, about synthetic polymers the properties of polysaccharides and other types of proteins than enzymes. Thus, fully artificial cell-mimics could be envisioned, although it is probably not a realistic aim for this century. Indeed, preparing an artificial cell, entirely operated by man-made molecular components, appears improbable based on the current state-of-the-art. Thus, the first steps toward polymer-based artificial Life shall probably involve minimal model systems, as discussed in the next section.

How far is polymer-based artificial Life?

Significant research efforts have already been made to synthesize abiotic (macro)molecules exhibiting some of the Life-relevant properties listed in Table 1. Therefore, it is already possible to store information in the comonomer sequence of a synthetic polymer, to form duplexes from nonnatural polymer strands and to replicate to some degree small molecules. However, none of the examples discussed in previous pages are describing molecules that could successfully support Life.

There are several reasons for that. As previously mentioned, the first main reason is that, in terms of possible building blocks and chemical reactions, the search space is ex tremely wide. For instance, all the systems described in sections 4 and 5 were somehow obtained with different types of molecules. Consequently, there is currently no non-nucleic acid molecule that combine several (i.e.

at least two or three) properties listed in Table 1. Another important drawback is that most of the examples listed in section 4 have been obtained with small molecules. In fact, the transition from low molecular weight models to true informational macromolecules (i.e. oligomers or polymers containing at least 5-10 monomer units) is far from being trivial. Therefore, the answer the question written in the title of this section is: polymer-based artificial Life was probably never that close but still relatively far. In order to achieve it, progress remains to be done regarding the synthesis and mutability of informational macromolecules and their implementation in out -of-equilibrium metabolic networks.

Scientific and philosophical outlook

As discussed in the present text and in an earlier essay, [9] Humans are polymer-operated thinking species that have emerged through many years of Darwinian natural selection. Interestingly, and maybe ironically, these "polymer-operated machines" may succeed someday in creating new types of self-replicating polymers (i.e. different from those of Earth's biology) that may lead to new Life forms. This is, of course, not for tomorrow. As summarized in sections 5 and 6, there are still many hurdles ahead. The will to create a Life-supporting macromolecule by complete chemical design is maybe one of them. Synthetic Life may well emerge from a mixture of carefully-designed components but may also happen from a random pool of chemicals, as learned from the History of our Universe. Here, interesting experiments may be imagined for the next generations and centuries. For instance, if Humans succeed someday in creating efficient evolvable self-replicators, directed panspermia could be tried. [64] If the search for water-based exoplanets becomes routine someday, it might be conceivable to send seeding spaceships to sterile planets. These spaceships could contain cocktails of basic biological synthons (i.e. nucleotides, lipids, amino acids) but the same experiment could also be performed with man-made molecules. In other words, a sterile isolated planet could be used as a "closed flask" for artificial Life experiments (in planeto chemistry). Although all the above considerations sound today as pure science-fiction, recent NASA (as well as other private companies) efforts for studying and sending DNA to space show that such goals are not so unrealistic on a long term. There are, of course, major ethical issues regarding the creation of artificial Life systems and the fact to spread them on Earth or through the Universe. Such ideas are definitively provocative in the early 21 st century where conservationist and preservationist ideas currently dominate. However, we shall ask ourselves is the faith of the Universe is to be preserved and or to Evolve? If the latter answer is the right one, polymer-based artificial Life may become someday a reality.

Figure 3 .

 3 Figure 3. Examples of self-replicating cyclic molecules containing dynamic imine bonds. The red and blue spheres highlight the information sequences that are copied during the replication process. Redrawn from reference [14b].

Acknowledgements

J.-F.L. thanks the CNRS and the excellence network Chemistry of Complex Systems (LabEx CSC) for recurrent financial support.


Lucien Vienne

Simon Marié
email: simon.marie@lecnam.net

Francesco Grasso
email: francesco.grasso@lecnam.net

Simulation of Viscous Fingering Instability by the Lattice Boltzmann Method

The viscous fingering instability is successfully simulated within a lattice Boltzmann framework. Each species of the mixture is governed by its own kinetic equation and a force takes into account the diffusion between species. The influence of the porous medium is mimicked by using the gray lattice Boltzmann model or the Brinkman force model. In this study, both representations of the porous medium yield equivalent results. Then a physical analysis of the instability is performed and two different dynamical behaviour are stated and discussed. Finally, it is observed that a high Péclet number intensify the instability and the viscous dissipation stemming from the Darcy-Brinkman equations delay the development of the fingers in the case of large effective viscosity.

I. Introduction

Mixing in porous media is particularly difficult due to the absence of inertia but it plays a key role in carbon sequestration, oil recovery, mantle convection, microfluidic device, and chromatographic columns. High Reynolds number flows can be actively stirred by turbulence but other mechanisms are needed in a porous medium. The generation of an interfacial instability is one mechanism among others that can improve the mixing. This instability can be driven by chemical reactions, density stratification (Rayleigh-Taylor instability), or difference in viscosity. Here, we focus on the latter, which is called viscous fingering or Saffman-Taylor instability. This interfacial instability occurs when a less viscous fluid displaces a more viscous fluid in a porous medium [START_REF] Homsy | Viscous Fingering in Porous Media[END_REF]. Finger-like patterns emerge and grow, exhibiting complex dynamics.

Since the pioneer work of Hill [START_REF] Hill | Channeling in packed columns[END_REF] in 1952 and Saffman and Taylor [START_REF] Saffman | The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid[END_REF] in 1958, extensive investigations has been performed on both immiscible and miscible fluids. A distinctive characteristic of miscible fluids is the lack of a sharp interface. Mixing produces changes to the fluid since the properties of the mixture vary continuously while diffusion occurs. Usually, the viscous fingering is modeled by using Darcy's law and a scalar transport equation for the concentration. A relation between the viscosity and the concentration of the components must be defined. Most articles follow the standard literature rather than a physical rigor and assume an exponential dependence. In this paper, the viscous fingering instability is simulated with a lattice Boltzmann framework. The interactions between the species of the mixtures are taken into account by means of a forcing term. This model for miscible fluids has been recently introduced in Ref. [START_REF] Vienne | Lattice Boltzmann model for miscible gases: a forcing term approach[END_REF]. One of the important feature of this model lies in the computation of the viscosity coefficient which has a more rigorous physical foundation, stemming from approximations of the kinetic theory of gases extended to mixtures. Adopting a lattice Boltzmann approach, we investigate two strategies so as to mimic the porous medium effects: the gray lattice Boltzmann model assuming a partial reflection condition and the Brinkman force model adding a drag force.

This paper is laid out as follows. In the first section, we present the lattice Boltzmann model for miscible mixture, and how the porous medium is modeled. In the second section, the viscous fingering dynamic is highlighted. The development of the instability is studied globally through the evolution of the mixing length. Then, we focus on the two main stages of the instability, the first one in the early times when the fingers grow linearly and the second one when the fingers interact with strong non-linearity. The influence of different parameters such as the ratio of the rate of advection to the rate of diffusion (Péclet number), the porous representation, and the effective viscosity are underlined.

II. Model

A. Simplified kinetic model for miscible mixtures

The lattice Boltzmann method is an alternative method for simulating fluid flows by solving a simplified formulation of the kinetic model. A mixture is composed of multiple species and each species is defined by its own distribution function, which is governed by its own kinetic equation. The present lattice Boltzmann model for miscible mixtures is based on a forcing term approach. We summarize here its main features and a more detailed presentation can be found in Ref. [START_REF] Vienne | Lattice Boltzmann model for miscible gases: a forcing term approach[END_REF].

Let m and n denote different species (m, n = 1, 2, ..., N; N being the total number of species). The distribution function of species m, f m α , obeys the following discrete kinetic equation

f m α (x + e α δ t , t + δ t ) = f m α (x, t) - δ t τ m f m α (x, t) -f m(eq) α (x, t) + δ t S m α (x, t) (1)
where x, t, α, and τ m are, respectively, the spatial coordinate, the time, the number of discrete kinetic velocities e α , and the relaxation time of each species. Usually, the left and right hand sides of Eq. ( 1) are processed separately into two steps: streaming and collision, respectively. The equilibrium distribution functions, f m(eq) α

, are given by the standard polynomial formulation,

f m(eq) α = ρ m ω α 1 + u m • e α c 2 s + (u m • e α ) 2 2c 4 s - u m • u m 2c 2 s . (2)
S m α is the source term from Guo's forcing scheme [START_REF] Guo | Discrete lattice effects on the forcing term in the lattice Boltzmann method[END_REF], widely used in order to include forces in the lattice Boltzmann algorithm,

S m α = (1 - δ t 2τ m )ω α e α -u m c 2 s + (e α • u m )e α c 4 s • F m (3)
F m being the force acting on the m-th species which is derived in the following to take into account the species interactions.

In this study, we use the so-called D2Q9 isothermal, two-dimensional and nine-velocity discretisation. Extension to the three-dimensional formulation (D3Q19 or D3Q27) is straightforward. The pseudo-sound velocity is c 2 s = 1 3 , the kinetic velocities are expressed as

e α = 0 1 0 -1 0 1 -1 -1 1 0 0 1 0 -1 1 1 -1 -1 T 1 ≤ α ≤ 9 (4)
and the lattice weights are equals to

ω α = 4 9 1 9 1 9 1 9 1 9 1 36 1 36 1 36 1 36 T 1 ≤ α ≤ 9. (5)
The macroscopic quantities, namely the density and momentum of each species, are obtained by computing the different moments of the distribution functions.

ρ m = α f m α , ρ m u m = α f m α e α + δ t 2 F m . (6)
The resulting macroscopic equations are the conservation equations for low Mach number flows subjected to a body force (e.g. gravity). In order to take into account the interaction of miscible species, we introduce diffusion forces. These forces are derived from kinetic theory by Kerkhof and Geboers [START_REF] Kerkhof | Toward a unified theory of isotropic molecular transport phenomena[END_REF],

F D,m = -p N n=1 x m x n D mn (u m -u n ), (7)
and

F m becomes F m = F D,m + F B,m (8)
where F B,m is a body force and p is the total pressure given by Dalton's law p = m p m . As a result, the discrete kinetic equations Eq. ( 1) for the various species are coupled through F m . Since the diffusion force F D,m depends on the velocity, total pressure p, molar fractions x m , and (Maxwell-Stefan or binary) diffusion coefficients D mn , a linear system must be solved at each time step in order to compute the species momentum by means of Eq. ( 6).

In the standard lattice Boltzmann equation, the pseudo (isothermal) speed of sound (c s ) is fixed by the lattice. For D2Q9 velocity set, c s is equal to c 2 s = 1/3 for all species which is not the case for mixture of species having different molecular masses. Recalling that the partial pressure obeys the ideal gas law, p m = ρ m c 2 s,m and c 2 s,m = RT/M m , where R is the universal gas constant, T is the temperature, and M m is the species molecular mass; we introduce the following body force in order to take into account species having different molecular masses

F B,m = (1 -β m )c 2 s ∇ρ m , (9)
where the gradient term is computed by means of a compact scheme

∇ρ m (x) = 1 c 2 s α ω α e α ρ m (x + e α ). (10)
The partial pressure then becomes

p m = β m c 2 s ρ m (11)
where β m = c 2 s,m /c 2 s is the ratio between the species and the standard LBM pseudo speeds of sound. The lattice Boltzmann scheme still retains a connection with the macroscopic scale through the moments of the distribution functions (Eqs. 6), and the relation between the relaxation time and the viscosity stemming from the Chapman-Enskog expansion (see V.A)

µ m = ρ m c 2 s (τ m - δ t 2 ). (12)
The viscosity of a species has a similar formulation as Wilke's law [START_REF] Vienne | Lattice Boltzmann model for miscible gases: a forcing term approach[END_REF]

µ m = x m µ 0,m N n x n Φ mn (13)
where µ 0,m is the viscosity of the pure species m, x m is the molar fraction of species m, and

Φ mn = 1 2 √ 2 1 + M m M n -1 2 1 + µ 0,m µ 0,n 1 2 M n M m 1 4 2 . ( 14
)
Complex diffusion between species can successfully be simulated by the present approach [START_REF] Vienne | Lattice Boltzmann model for miscible gases: a forcing term approach[END_REF]. For purely diffusive flow, the dynamic predicted by Maxwell-Stefan's equations is recovered and non-Fickian behaviors such as reverse, osmotic or barrier diffusion may occur.

B. Porous medium

Porous media, such as natural soils or sedimentary rocks, are incredibly difficult to model due to the heterogeneous structure and composition of the pores at many different scales. Moreover, the simulation of a large-scale flow at the pore scale is often impractical because of long CPU time and excessive memory consumption. Hence, a volume average approach is usually used to overcome these difficulties. In this case, Darcy or Darcy-Brinkman semi-empirical equations are solved and porosity, permeability, and other measurements are defined. In the case of a single fluid, the Brinkman equation adds a drag force to the Stokes equations,

∇p = - µ K u + ∇ • µ e ∇u + (∇u) T ( 15
)
where p is the pressure, K is the permeability of the medium, u is the fluid velocity, and µ e is the effective viscosity which may not be equal the fluid viscosity. The Brinkman equations are a convenient transition model between Darcy regime (K → 0 or ∇ • µ e ∇u + (∇u) T → 0) and Stokes regimes (K → +∞).

It is to be noted that the dynamics of the viscous fingering is mainly governed by the Darcy equation [START_REF] Homsy | Viscous Fingering in Porous Media[END_REF] thereby the Brinkman term, ∇ • µ e ∇u + (∇u) T , is not of primary importance and the effective viscosity is taken equal to the fluid viscosity unless otherwise stated. Indeed, the literature [START_REF] Homsy | Viscous Fingering in Porous Media[END_REF][START_REF] Tan | Stability of miscible displacements in porous media: Rectilinear flow[END_REF][START_REF] Pramanik | Effect of Péclet number on miscible rectilinear displacement in a Hele-Shaw cell[END_REF][START_REF] Hota | Nonmodal linear stability analysis of miscible viscous fingering in porous media[END_REF][START_REF] Nijjer | The dynamics of miscible viscous fingering from onset to shutdown[END_REF] only considers Darcy's equations. The latter equations correctly described flows in Hele-Shaw cells and in simple porous media whereas Darcy-Brinkman's equations are often used as a transition model between porous flow and Stokes flow such as in the case of fractured porous media. In order to give some insight on the influence of the effective viscosity, three simulations are also performed with different ratios of viscosity and effective viscosity .

Furthermore, the porous medium is assumed to be isotropic and the permeability is given by a scalar value. Viscous fingering in anisotropic media is studied in Ref. [START_REF] Norouzi | A numerical study on miscible viscous fingering instability in anisotropic porous media[END_REF]. The originality of this work is the use of the aforementioned lattice Boltzmann model for miscible mixtures. Neither transport equation for the concentration, nor Darcy equation with exponential viscosity dependency are solved. Each species dynamic is governed by its own kinetic equation (Eq. ( 1)) and the viscosity of species stemmed from the kinetic theory of gases (Eq. ( 13)). In this section, we present how the porous medium effects are introduced in the lattice Boltzmann framework.

Gray lattice Boltzmann

Taking the advantage of the mesoscopic view of the lattice Boltzmann method, a porous medium can be modeled by different means. We first present the so-called gray lattice Boltzmann scheme where a special condition is applied on the whole domain: in each cell during the streaming step, the distribution functions are bounced back with a certain amount. Various schemes are proposed in the literature. In Ref. [START_REF] Walsh | A new partial-bounceback lattice-Boltzmann method for fluid flow through heterogeneous media[END_REF], the local pre-collision distribution functions are reflected whereas in Ref. [START_REF] Zhu | An improved gray lattice Boltzmann model for simulating fluid flow in multi-scale porous media[END_REF], the local post-collision distribution functions are taken as reflected distribution functions. Yoshida and Hayashi [START_REF] Yoshida | Transmission-Reflection Coefficient in the Lattice Boltzmann Method[END_REF] also suggest a gray lattice Boltzmann model based on Ref. [START_REF] Thorne | Lattice Boltzmann model for the elder problem[END_REF] but the neighboring post-collision distribution functions are reflected. Despite some differences, all the previous gray lattice Boltzmann schemes [START_REF] Walsh | A new partial-bounceback lattice-Boltzmann method for fluid flow through heterogeneous media[END_REF][START_REF] Zhu | An improved gray lattice Boltzmann model for simulating fluid flow in multi-scale porous media[END_REF][START_REF] Yoshida | Transmission-Reflection Coefficient in the Lattice Boltzmann Method[END_REF] recover the Darcy-Brinkman equation [START_REF] Chen | Chapman-Enskog Analyses on the Gray Lattice Boltzmann Equation Method for Fluid Flow in Porous Media[END_REF]. In this study, the use of the model proposed in Ref. [START_REF] Yoshida | Transmission-Reflection Coefficient in the Lattice Boltzmann Method[END_REF] is motivated by two main reasons. First, a redefinition of the macroscopic velocity is not required in this latter scheme. This redefinition would lead to complicated expressions especially on the boundary nodes since in the present approach, velocity and forces are coupled (Eqs. (6-7)). Furthermore, the classical bounce-back boundary condition [START_REF] Krüger | The Lattice Boltzmann Method[END_REF] is obtained as a limit expression. If we use the notation Γ m α to substitute for the right-hand side of Eq. ( 1) (post-collision part), the following gray lattice Boltzmann scheme is obtained,

f m α (x + e α δ t , t + δ t ) = (1 -θ m )Γ m α (x, t) + θ m Γ m α (x + e α δ t , t) , (16)
where Γ m α (x + e α δ t , t) are the post-collision distribution functions from the opposite direction of α velocity located at the neighboring node and θ m is the amount of reflection. When θ m = 0, the usual streaming step occurs and θ m = 1 is similar to the classical bounce-back (no-slip) condition [START_REF] Krüger | The Lattice Boltzmann Method[END_REF]. For 0 < θ m < 1, the scheme mimics the dynamics of a porous medium and this approach should be related to a subgrid model for porous medium. The parameter θ m depends on the viscosity of the fluid and the permeability of the considered porous medium and the following relation can be obtained [START_REF] Yoshida | Transmission-Reflection Coefficient in the Lattice Boltzmann Method[END_REF][START_REF] Chen | Chapman-Enskog Analyses on the Gray Lattice Boltzmann Equation Method for Fluid Flow in Porous Media[END_REF]

µ m K ρ m = 2θ m (1 -θ m )δ t (17)
Here, θ is defined on the link between nodes instead of being a node based value. In each direction α, θ is computed by the means of an interpolation between the two nodes. This ensures that the mass is conserved strictly by the scheme.

Brinkman drag force

In the second approach, referred as Brinkman force model, the porous medium resistance is included explicitly through a drag force to the lattice Boltzmann equation, e.g. [START_REF] Ginzburg | Consistent lattice Boltzmann schemes for the Brinkman model of porous flow and infinite Chapman-Enskog expansion[END_REF][START_REF] Guo | Lattice Boltzmann model for incompressible flows through porous media[END_REF][START_REF] Martys | Improved approximation of the Brinkman equation using a lattice Boltzmann method[END_REF][START_REF] Spaid | Lattice Boltzmann methods for modeling microscale flow in fibrous porous media[END_REF]. Here, we follow a similar approach as in Ref. [START_REF] Grosfils | Structural and dynamical characterization of Hele-Shaw viscous fingering[END_REF] and the forcing term is written as

F por ous,m = - µ m K u m (18)
As mentioned in Refs. [START_REF] Talon | Assessment of the two relaxation time Lattice-Boltzmann scheme to simulate Stokes flow in porous media[END_REF][START_REF] Khirevich | Coarse-and fine-grid numerical behavior of MRT/TRT lattice-Boltzmann schemes in regular and random sphere packings[END_REF][START_REF] Ginzburg | Comment on An improved gray Lattice Boltzmann model for simulating fluid flow in multi-scale porous media: Intrinsic links between LBE Brinkman schemes[END_REF], the numerical permeability depends lightly on the viscosity for both Brinkman force and gray lattice Boltzmann models. This nonphysical variation can be removed by keeping the so-called magic number Λ constant (see [START_REF] Ginzburg | Consistent lattice Boltzmann schemes for the Brinkman model of porous flow and infinite Chapman-Enskog expansion[END_REF][START_REF] Talon | Assessment of the two relaxation time Lattice-Boltzmann scheme to simulate Stokes flow in porous media[END_REF][START_REF] Khirevich | Coarse-and fine-grid numerical behavior of MRT/TRT lattice-Boltzmann schemes in regular and random sphere packings[END_REF][START_REF] Ginzburg | Comment on An improved gray Lattice Boltzmann model for simulating fluid flow in multi-scale porous media: Intrinsic links between LBE Brinkman schemes[END_REF]). We use the multiple relaxation times (MRT) collision model [START_REF] Coveney | Multi-ple–relaxation–time lattice Boltzmann models in three dimensions[END_REF] with Λ = 3/16 corresponding to s q = 8 2-1/τ m 8-1/τ m [START_REF] Krüger | The Lattice Boltzmann Method[END_REF][START_REF] Coveney | Multi-ple–relaxation–time lattice Boltzmann models in three dimensions[END_REF].

III. Results and discussions

A. Darcy law

In order to show that Darcy's law is recovered by both of the proposed schemes, a flow composed of two identical species (m = 1, 2 and β m = 1) through a homogeneous porous medium is simulated. Domain size is set to (n x = 101, n y = 20). Periodic conditions are applied on top and bottom of the domain and a constant velocity profile (u x,m = U, u y,m = 0) is imposed at left and right side of the domain [START_REF] Zou | On pressure and velocity boundary conditions for the lattice Boltzmann BGK model[END_REF]. Distribution functions are initialized at their equilibrium values f m α (t = 0) = f m(eq) α (ρ m , u x,m = U, u y,m = 0) with an uniform initial density field ρ 1 = ρ 2 . Simulation is stopped when steady state is achieved. In Fig 1, the pressure drop between the left and right side of the domain is plotted against the horizontal inlet velocity. Both gray lattice Boltzmann and Brinkman force models give equivalent results and show the expected Darcy behavior of the simulated system. This first result shows that the present model is able to accurately reproduce the standard behaviour of a porous medium for different values of K and for both porous modeling strategies.

B. Viscous fingering

Global description of the instability

The viscous fingering instability occurs when a less viscous fluid invade a more viscous fluid. In the case of miscible fluids, if the interface between fluids is sharp enough, finger-like patterns emerge and grow, exhibiting complex dynamics.

In the following simulations, the fluid consists of two species having equal molar masses (m = 1, 2 β m = 1). The resting mixture is composed of molar fractions x 1 = 0.1 and x 2 = 0.9 and these are swapped for the injected mixture (x 1 = 0.9, x 2 = 0.1). Following the literature, dimensionless parameters are defined such as the log-viscosity ratio and the Péclet number (ratio of the advective transport rate and the diffusive transport rate)

R = ln µ 0,2 µ 0,1 , P e = U L r e f D 12 ( 19
)
where U is the injected velocity and L r e f is the reference length. Here, the characteristic scales are based on the convective length L r e f = n y and time t r e f = n y /U with n y the height of the domain. If R < 0, the invading fluid is more viscous than the resting fluid, and the interface is always stable. On the other hand, if R > 0, the invading fluid is less viscous than the resting fluid, and the interface between the two fluids may be unstable. The number of fingers increases with both the Péclet number and the ratio R. We recall that the relation between the pure viscosity and the species is given by Eq. ( 13). The influence of different parameters on the instability is studied. First, we compare the results obtained from the gray lattice Boltzmann and the Brinkman force models. Afterward, the effects of the Péclet number on the viscous fingering are highlighted. Finally, we present the differences on the emergence of the fingers caused by the effective viscosity. The domain considered is two-dimensional, (n x = 1024, n y = 512) unless otherwise specified, and we impose periodicity at the top and bottom boundaries. At the left and right boundaries, a constant velocity condition equal to U is applied for both species. In order to compare our results, we keep the reference pressure constant p r e f = µ 0,1 Un y /K = 1 and the permeability of the porous medium is set to K = 1. Each simulation is initialized by an almost sharp interface with a small perturbation so as to trigger the instability, x 1 (x, y, t = 0) = 0.9 + (0.9 -0.1) 0.5 + 0.5 × erf

x -50

√ t 0 + r(y) × 10 -3 exp -(x -50) 2 √ t 0 , (20)
x 2 (x, y, t = 0) = 1 -x 1 (x, y, t = 0), (21)
where the function r returns a random number which is uniformly distributed in the interval [-1, 1]. In each simulation, the same seed is used for the pseudo-random generator. We set t 0 = 10 -6 to avoid strong gradients. The initial total pressure is computed from Darcy's law for the mixture

p(x, y, t = 0) = p r e f + (1 - x n x ) µ 1 + µ 2 K U, (22)
and the distribution functions are initialized at their equilibrium values f m α (t = 0) = f m(eq) α (ρ m , u x,m = U, u y,m = 0). As the instability evolves, fingers develop first linearly then interact non-linearly, and the dynamics become chaotic. Before exploring theses different regimes, we examine the global dynamics of the viscous fingering. The average molar fraction over the transverse direction is a classical measurement

x(x, t) = 1 n y ∫ x 1 (x, y, t)dy, (23)
as well as the length of the mixing zone, which is defined here as

l mix (t) = 1 L r e f x| x(x,t)=0.9-10 -3 -x| x(x,t)=0.1+10 -3 (24)
In Figure 2, the mixing length is represented in logarithmic scale according to the non-dimensional time t * = t/t r e f for various Péclet number at R = 3. Two different regimes are clearly visible. In the first regime up to t * = 0.13, the diffusion dominates the growth of the mixing length, which is proportional to √ t * as in the case of no viscosity contrast (R = 0). Afterward, a transition regime occurs and depends on the Péclet number. The second regime can be seen in Fig. 3. The mixing length is represented in linear scale to better highlight the longtime behavior and the linear evolution of the fingers. The growth of the mixing length is linear in time suggesting that the advection now dominates. It is interesting to note that both regimes exist even when the initial molar fractions are not perturbed and no finger emerges. In this case, the development of the mixing length is reduced in the second regime. For instance in Fig. 3, the approximate slope is 1 with no finger, whereas the slope is around 2.2 with fingering. Compared to the pure diffusive case, the mixing length is heavily affected by the viscous fingering instability. Moreover, Fig. 3 shows no significant difference between the gray lattice Boltzmann and the Brinkman force model. The small deviation for t * > 6.5 is probably due to the influence of the boundary condition close to the location of the fingers. Same dynamic results independently of how the porous media is modeled confirming that the gray lattice Boltzmann or the Brinkman force adequately reproduce the viscous fingering instability.

The development of the instability is shown in Fig. 4. The onset of fingers is observed on the diffusive regime. Then, the fingers interact via highly non-linear mechanisms such as merging, splitting, etc (more details are given in section III.B.3) during the transition between diffusive and advective regimes (0.13 < t * < 3). A better measurement of the transition between linear and non-linear behavior is given by a sudden growth of the interfacial length

I(t) = 1 L r e f ∬ [(∂ x 1 /∂ x) 2 + (∂ x 1 /∂ y) 2 ]
1 2 dxdy. Finally for t * > 3, the few remaining fingers grows at constant rate. At very late time (not observed here), the fingers should mix with the ambient fluid and are convected at a given speed close to the injected velocity. The length of the fingers should stay roughly the same since the interface is diffuse enough to smooth out the viscosity and molar gradients. In order to give some insight on the influence of the effective viscosity, three simulations are performed with µ e,m = 0.1µ m , µ e,m = µ m , µ e,m = 10µ m . As expected, the diffusion caused by the effective viscosity affects the instability as shown in Fig. 5. The development of fingers is delayed in the case of a high effective viscosity. The dynamic when no finger emerges does not dependent on the effective viscosity. The influence of the fingers on the mixing length becomes only apparent at t * ≈ 0.6 for µ e,m = 10µ m instead of t * ≈ 0.3 for µ e,m = µ m . In contrast, the differences between µ e,m = 0.1µ m and µ e,m = µ m are small. Thus, when the effective viscosity is sufficiently small, the Darcy drag is the dominant term in Eq. ( 15) and the development of fingers is not altered.

In this section, the global development of the instability has been discussed. We now focus on two specific parts of the viscous fingering, namely the very beginning of the onset of the instability when the fingers grow linearly and the intermediate times when strong non-linear interactions occur.

Early times: growth rate of the perturbation

At early times, the interface between the two mixtures spread because of diffusion and fingers develop. At the very beginning, the flow is linearly unstable and perturbations grow exponentially. The onset of fingers is explored by many authors using linear stability analysis [START_REF] Homsy | Viscous Fingering in Porous Media[END_REF]. Because of diffusion, the base flow is unsteady. Tan and Homsy [START_REF] Tan | Stability of miscible displacements in porous media: Rectilinear flow[END_REF] use a quasi-steady state approximation in which the base flow is frozen and modal analysis is performed at successive frozen time. This method is commonly used to study the viscous fingering [START_REF] Pramanik | Effect of Péclet number on miscible rectilinear displacement in a Hele-Shaw cell[END_REF][START_REF] Norouzi | A numerical study on miscible viscous fingering instability in anisotropic porous media[END_REF]. Non modal analysis is carried out in Ref. [START_REF] Hota | Nonmodal linear stability analysis of miscible viscous fingering in porous media[END_REF] and shows some discrepancy with linear stability analysis.

Linear stability analysis can also be directly performed in the lattice Boltzmann framework (see. [START_REF] Vergnault | A lattice Boltzmann method for nonlinear disturbances around an arbitrary base flow[END_REF][START_REF] Pérez | Lattice Boltzmann methods for global linear instability analysis[END_REF]) however this approach is still in its infancy and is only applied to simple single fluid flow. In this study we carry out non linear simulation with the aforementioned lattice Boltzmann model and compute the growth rate of the perturbations. The molar fractions can be decomposed into a base state x 0 m and a perturbative component x m ,

x m (x, t) = x 0 m (x, t) + x m (x, t). (25)
If we assume that the perturbation can be expressed in the form of

x m (x, t) = x m (x) exp (σt) (26)
where σ is the growth rate of the perturbation. The dispersion curves are obtained by applying a fast Fourier transform (FFT),

x m (x, t) = x 0 m (x, t) -x m (x, t) (27) x(x, k, t) = FFT y x m (x, t) (28)
a(k, t) = || x(x, k, t)|| 2 = ∫ x • xdx (29) σ(k, t) = d ln (a(k, t)) dt ( 30
)
where k is the wave number and assuming that the growth rate is constant between two derivation steps. The perturbative components are computed by subtracting the components from the non-linear perturbed solution to the components from the unperturbed base state. No finger emerge in this latter case. First, we compare in Fig. 6 the results obtained from the gray lattice Boltzmann and the Brinkman force models. Both approaches lead to equivalent growth rates. Thus, at early times, the linearly development of the instability is well captured by the two models. The growth rate decreases in time. The most dangerous wave number corresponding to the largest growth rate as well as the threshold and the cutoff wave numbers are reduced as the instability progresses resulting in widening fingers. Diffusion acts as a stabilization factor by smoothing perturbations and heading to an homogeneous mixture. Hence, as shown in Fig. 7, the instability is more dominant with higher Péclet numbers. A lower fluid dispersion expands the range of unstable wave numbers and increases the growth rate. The cutoff wave number increases with the Péclet number whereas it has a very limited influence on the threshold wave number. These results are in qualitative agreement with the growth rates obtained by linear stability in Ref. [START_REF] Pramanik | Effect of Péclet number on miscible rectilinear displacement in a Hele-Shaw cell[END_REF]. Remark that the choice of scaling is not neutral for the linear stability analysis. As noted by the previous authors, taking the reference length and time as the diffusive length l re f = D/U and t r e f = L r e f /U leads to dispersion equations independent of the Péclet number in the linear stability analysis [START_REF] Tan | Stability of miscible displacements in porous media: Rectilinear flow[END_REF] whereas in the case of the convective length l r e f = n y and t r e f = L r e f /U, dispersion curves depends on the Péclet number [START_REF] Pramanik | Effect of Péclet number on miscible rectilinear displacement in a Hele-Shaw cell[END_REF]. The Péclet number also influences the transition from linear to non-linear instability. At t * = 0.1, for P e = 5000, fingers starts to interact non-linearly whereas for lower Péclet numbers, the perturbation is still well-described by Eq. [START_REF] Coveney | Multi-ple–relaxation–time lattice Boltzmann models in three dimensions[END_REF]. The present non-linear simulations confirm the important role of the Péclet number in the development of the instability at early times. As observed in Fig. [START_REF] Guo | Discrete lattice effects on the forcing term in the lattice Boltzmann method[END_REF], the value of the effective viscosity changes the beginning of the instability. The linear growth of the fingers is also affected. At early times, the dissipation due to the effective viscosity reduces the growth rate of the perturbation and the cutoff wave number also decreases as shown in Fig. 8. However, the time evolution of the growth rate and the cutoff wave number is different from the effect of the diffusion caused by a low Péclet number. The growth rate for µ e,m = 10µ m increases slightly from t * = 0.01 to t * = 0.05 and then keeps constant from t * = 0.05 to t * = 0.1. The cutoff wave number stays at k ≈ 200. For µ e,m = 0.1µ m and µ e,m = µ m , the growth rate and the cutoff wave number decrease in time. As the instability progresses, the differences resulting from the effective viscosity are less visible.

Intermediate times: non-linear interactions

At early times, the sharp interface diffuses and begins to deform. This stage is called the linearly unstable regime and linear stability analysis tools are suitable to describe the instability. The fingers then start to interact via various nonlinear mechanisms. Many thin fingers develop, and strong linear interactions take place, which lead globally to a coarsening of the fingers in the transverse direction and a growth of the fingers in the streamwise direction. The number of fingers and the intensity of these interactions increase with the Péclet number and the log-viscosity ratio. The influence of the effective viscosity is less significant as long as the interface is not overly diffuse after the delay in the emergence of fingers. The interactions are also dependent of the noise of the initial perturbation and may vary from simulation to simulation. Results obtained from the Gray lattice Boltzmann model and the Brinkman force model are identical if the same initial perturbation is applied.

Figure 9 shows a sequence of snapshots of a simulation performed with the Brinkman force model at R = 3, P e = 16000. This high Péclet number is chosen to assure intense interactions between fingers for better identification of the main non-linear mechanisms. These include spreading, shielding, fading, and coalescence. By spreading, we refer to the process whereby one finger is slightly ahead of others. The finger grows very quick and the gradient steepens between the finger and the surrounding more viscous fluid. The finger may then widens at the tip and shields the growth of the smaller neighbouring fingers. Few thin fingers fade and diffuse in the ambient fluid resulting in a region of intermediate viscosity. Coalescence phenomenon describes the merging of two or more fingers together. The present dynamics of the simulation is rich displaying all major mechanisms, and exhibits also more complex behaviors such as tip splitting, when a finger splits into two at the tip and side branching when a finger splits at its side. These fascinating interaction mechanisms occurs in the nonlinear regime.

IV. Conclusion

In this study, the lattice Boltzmann simulation of the viscous fingering instability is discussed. The global dynamic is presented, and we especially focus on two stages of the development of the instability. First, the very beginning of the instability, when the fingers grow linearly, is studied by computing the growth rate associated to the perturbation. Secondly, we describe qualitatively intermediate times when strong non-linear interactions take place.

For these two stages, we show that both of the gray lattice Boltzmann and Brinkman force porous models give the same result despite of being based on different formulations. Both result in Darcy-Brinkman equations that is why the influence of the viscous dissipation on the instability is then explored by varying the value of the effective viscosity. A high value of the effective viscosity causes a delay in the emergence of fingers whereas a low value of the effective viscosity has not a significant effect on the instability. Moreover, the differences are reduced when the instability is growing up. Indeed, contrary to the Darcy drag, the terms related to the effective viscosity are not the driving force of the instability. We also observed that a high Péclet number leads to a more intense instability. In the linear regime, the growth rate as well as the most dangerous and the cutoff numbers increase with the Péclet number. In addition, more fine-scale fingers develop and interact considerably in the non-linear regime at high Péclet number.

V. Appendix

A. Macroscopic limit of the Lattice Boltzmann model for miscible mixtures

The macroscopic limit of a lattice Boltzmann scheme can be obtained via the Chapman-Enskog analysis. This multiple scale expansion provides a relation between the mesoscopic scale of the Boltzmann equation and the macroscopic scale of the Navier-Stokes equation. This derivation is straightforward and is similar to the standard lattice Boltzmann model with a force arising from Guo's forcing scheme (see Ref. [START_REF] Guo | Discrete lattice effects on the forcing term in the lattice Boltzmann method[END_REF]). Therefore in the low Mach and continuum limit, the kinetic equation Eq. ( 1) and its moments Eq. ( 6) are equivalent to the following hydrodynamics equations, where the partial pressure is equal to p m = ρ m c 2 s,m , and the total pressure is given by Dalton's law p = N m=1 p m . The dynamic viscosity of species m is expressed in terms of the relaxation time according to

∂ t ρ m + ∇ • (ρ m u m ) = 0, (31)
µ m = ρ m c 2 s (τ m - δ t 2 ). (33)

Fig. 1

 1 Fig. 1 Pressure drop according to the velocity for different permeability at fixed viscosity µ m = 2 × 10 -1 . Values are given in lattice units. Squares and stars stand for numerical results obtained respectively with the gray lattice Boltzmann and the Brinkman force schemes. Lines are the analytical solution -∇p m = µ m /KU.

Fig. 2

 2 Fig. 2 Mixing length multiplied by √ Pe at various Péclet for R = 3 with the gray lattice Boltzmann model. Dashed line shows a power law with exponent 1/2.

Fig. 3 Fig. 4

 34 Fig. 3 Mixing length for R = 3, Pe = 2000. Simulations are performed with the lattice Boltzmann and Brinkman force models on a larger domain (n x = 8192, n y = 512). Blue dots stand for a simulation carried out with no initial perturbation.

Fig. 5

 5 Fig. 5 Mixing length for different values of the effective viscosity and R = 3, P e = 2000 with the gray lattice Boltzmann model.

Fig. 6

 6 Fig. 6 Dispersion curves for R = 3, P e = 2000 at various times. • gray lattice Boltzmann, + Brinkman force model.

Fig. 7

 7 Fig. 7 Dispersion curves for R = 3 at t * 0.01, 0.05, 0.1 with different Péclet values. Simulations are performed with the gray lattice Boltzmann model.

Fig. 8

 8 Fig. 8 Dispersion curves for R = 3, P e = 2000 at t * = 0.01, 0.05, 0.1 with different effective viscosity values. Simulations are performed with the gray lattice Boltzmann model.

Fig. 9

 9 Fig. 9 Instability development for R = 3, P e = 16000 at t * ≈ 0.91, 1.22, 1.52. The whole domain (n x = 4096, n y = 1024) is plotted. The color-map (black-red-yellow-white) represents the variation of the molar fraction from x 1 = 0.9 to x 1 = 0.1. Same scaling is used for both x and y-axis.

∂

  t (ρ m u m ) + ∇ • (ρ m u m ⊗ u m ) = -∇p m + ∇ • µ m ∇u m + (∇u m ) Tp N n=1 x m x n D mn (u mu n ) (32)

Acknowledgments

This work was performed using HPC resources from GENCI-CINES (Grant 2018-A0052A10636)


Mireille Laforge

Carole Elbim

Corinne Frère

Miryana Hémadi

Charbel Massaad

Philippe Nuss

Jean-Jacques Benoliel

Chrystel Becker
email: christel.becker@parisdescartes.fr

Corinne Frere

Massaad Charbel

Tissue damage from neutrophil-induced oxidative stress in COVID-19

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COVID-19 is caused by the betacoronavirus SARS-CoV-2 and has several unique features compared with other coronavirus infections. In the most vulnerable individuals (for example,, older, obese or diabetic individuals), the virus sometimes triggers a cascade of acute biological events that can, unfortunately, lead to patients being ventilated and even dying. A far from negligible number of patients require intensive care, and although most hospital stays are short in duration, this places a huge strain on health systems. It is therefore urgent to gain an indepth understanding of the critical activators of disease severity, in order to reduce mortality and hospitalization rates.

The high neutrophil to lymphocyte ratio reported in critically ill patients with COVID-19 has been found to predict in-hospital mortality [START_REF] Fu | The clinical implication of dynamic neutrophil to lymphocyte ratio and D-dimer in COVID-19: A retrospective study in Suzhou China[END_REF]. Lung autopsies of deceased patients have revealed neutrophil infiltration in pulmonary capillaries, their extravasation into the alveolar spaces, and neutrophilic mucositis [START_REF] Barnes | Targeting potential drivers of COVID-19: Neutrophil extracellular traps[END_REF]. Increased levels of circulating neutrophil extracellular traps (NETs), which are indicative of neutrophil activation, have also been described in patients [START_REF] Golonka | Harnessing innate immunity to eliminate SARS-CoV-2 and ameliorate COVID-19 disease[END_REF]. Oxidative stress is the result of an imbalance between oxidant production and antioxidant mechanisms that leads to oxidative damage, including lipid peroxidation and DNA oxidation [START_REF] Bergamini | Oxygen, reactive oxygen species and tissue damage[END_REF]. In addition to the neutrophil infiltration and release of reactive oxygen species (ROS), viral infections are associated with a decrease in antioxidant defences [START_REF] Khomich | Redox Biology of Respiratory Viral Infections[END_REF]. Exposure to pro-oxidants usually leads to nuclear translocation of the master redox-sensitive transcription factor NRF2, which activates antioxidant defences; however, respiratory viral infections have been associated with inhibition of NRF2-mediated pathways and NF-kB signalling activation, which can promote inflammation and oxidative damage during these infections [START_REF] Komaravelli | Respiratory Viral Infections and Subversion of Cellular Antioxidant Defenses[END_REF].

Furthermore, there is evidence of a link between decreased expression of the antioxidant enzyme superoxide dismutase 3 (SOD3) in the lungs of elderly patients with COVID-19 and disease severity [START_REF] Abouhashem | Is Low Alveolar Type II Cell SOD3 in the Lungs of Elderly Linked to the Observed Severity of COVID-19?[END_REF]. Interestingly, children -whose neutrophils are less reactive and adherent, with no alteration of redox balance -are less prone to developing severe forms of COVID-19 [START_REF] Anderson | Diminished lectin-, epidermal growth factor-, complement binding domain-cell adhesion molecule-1 on neonatal neutrophils underlies their impaired CD18-independent adhesion to endothelial cells in vitro[END_REF]. The cascade of events triggered by the oxidative stress state in SARS-CoV-2 infection undoubtedly contributes to the severity of host disease, and needs to be further explored.

We postulate that, particularly in vulnerable individuals, neutrophilia generates an excess of ROS that exacerbates the host immunopathological response, resulting in more severe disease. The deleterious action of ROS on the functions of both pulmonary celsl and red blood cells (RBCs) can be seen as a major contributor to the hypoxic respiratory failure observed in the most severe cases of COVID-19. Thus, in addition to its damaging effects on alveolar epithelial and endothelial cells with procoagulative endotheliitis [START_REF] Teuwen | COVID-19: the vasculature unleashed[END_REF], an excess of ROS can also affect the RBC membrane and haeme group functionality.

Neutrophils usually initiate aggressive responses upon encountering danger signals, which leads to their rapid migration to the targeted tissue, release of NETs, and their production and release of ROS in an oxidative burst. It had been assumed that neutrophil migration from the vascular lumen into extravascular tissues is unidirectional, but recent studies have demonstrated that neutrophils can migrate back into the bloodstream, in a process referred to as neutrophil reverse transendothelial migration (rTEM) [START_REF] De Oliveira | Neutrophil migration in infection and wound repair: going forward in reverse[END_REF]. rTEM neutrophils are relatively rigid cells, and this physical characteristic may delay their passage through the tissue's microvasculature and prolong contact with the sinusoids. They may become mechanically entrapped in the microvasculature of major organs, thus contributing to distant organ damage and multiorgan failure. By producing excessive ROS, deregulated neutrophils can spread a local inflammatory response so that it becomes systemic [START_REF] Nemeth | The role of neutrophils in autoimmune diseases[END_REF], which explains why they have been involved in whole-body processes such as atherosclerosis and thrombosis [START_REF] Darbousset | Tissue factor-positive neutrophils bind to injured endothelial wall and initiate thrombus formation[END_REF]. Improper activation of neutrophils is also a potential explanation for the diffuse microvascular thrombosis and capillary leak syndrome observed in critically ill patients with COVID-19 [START_REF] Klok | Confirmation of the high cumulative incidence of thrombotic complications in critically ill ICU patients with COVID-19: An updated analysis[END_REF]. In addition, excessive ROS production may affect membrane lipids, integrins and cytoplasmic proteins in various circulating cells. These effects are particularly critical for RBCs, which may become dysfunctional. First, excess ROS can cause oxidation of polyunsaturated fatty acids in the RBC membrane, bringing about a profound modification of the membrane lipids' lateral and transversal distribution and organization at the nanoscale level. This results in biophysical and biomechanical changes in the RBC membrane that affect both the diffusion of oxygen and carbon dioxide and the deformability capability of RBCs in the capillary vessels, thereby favouring thrombocytosis. Reactivation of neutrophils in response to modification of the RBC membrane further fuels this vicious circle [START_REF] Tyurina | Nitrosative stress inhibits the aminophospholipid translocase resulting in phosphatidylserine externalization and macrophage engulfment: implications for the resolution of inflammation[END_REF]. In addition, this modification affects the release of ATP and nitric oxide, both necessary for adequate oxygen transport and vasodilatation between metabolizing tissues and respiratory surfaces. Second, ROS excess may upset the Fe 2+ / Fe 3+ balance and disturb iron homeostasis for which iron must be kept in the Fe 2+ state to bind oxygen. The protonation of superoxide ion associated to Fe 3+ within the haemoglobin haeme keeps the iron in its higher oxidation state and incapable of binding oxygen, resulting in less efficient oxygen transport, despite a high oxygen supply.In conclusion, the presence of oxidative stress markers (for example, lipid peroxidation, rTEM and a high neutrophil to lymphocyte ratio) in patients with COVID-19 may help to identify high-risk individuals early in the course of the disease, and prevent their sudden deterioration. This approach may also pave the way to new therapeutic approaches. We propose that antioxidants in combination with elastase inhibitors could be used to target rTEM neutrophils in patients with severe COVID-19.

Figure 1

Covid-19 infection leads to neutrophilia-induced ROS release. In non-at-risk individuals (A), ROS excess is counterbalanced by an increased antioxidant defense. In subjects with impaired redox balance (B), ROS production is not properly controlled leading to erythrocyte membrane peroxidation which in turn perpetuates neutrophil activation. This vicious circle, leading to disproportionate oxidative stress, might be responsible for the alveolar epithelial and endothelial cell damage, thrombosis and heme homeostasis dysregulation.

Acknowledgements

We thank Martin Deotto for his valuable help in the creation of the graphic summary.

Competing interests

The authors declare no competing interests.