Grigory Kabatiansky

Serge Vlǎdut

Cedric Tavernier
email: tavernier.cedric@gmail.com

On the Doubly Sparse Compressed Sensing Problem

A new variant of the Compressed Sensing problem is investigated when the number of measurements corrupted by errors is upper bounded by some value l but there are no more restrictions on errors. We prove that in this case it is enough to make 2(t + l) measurements, where t is the sparsity of original data. Moreover for this case a rather simple recovery algorithm is proposed. An analog of the Singleton bound from coding theory is derived what proves optimality of the corresponding measurement matrices.

Introduction and Definitions

A vector x = (x 1 , . . . , x n ) ∈ R n in n-dimensional vector space R n called tsparse if its Hamming weight wt(x) or equivalently its l 0 norm ||x|| 0 is at most t, where by the definition wt(x) = ||x|| 0 = |{i : x i = 0}|. Let us recall that the Compressed Sensing (CS) Problem [START_REF] Donoho | Compressed sensing[END_REF][START_REF] Candes | Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?[END_REF] is a problem of reconstructing of an n-dimensional t-sparse vector x by a few (r) linear measurements s i = h (i) , x (i.e. inner product of vectors x and h (i) ), assuming that measurements (h (i) , x) are known with some errors e i , for i = 1, . . . , r. Saying in other words, one needs to construct an r × n matrix H with minimal number of rows h (1) , . . . , h (r) , such that the following equation ŝ = Hx T + e,

has either a unique t-sparse solution or all such solutions are "almost equal". The compressed sensing problem was mainly investigated under the assumption that the vector e = (e 1 , . . . , e r ), is called the error vector, has relatively small Euclidean norm (length) ||e|| 2 . We consider another problem's statement assuming that the error vector e is also sparse but its Euclidean norm can be arbitrary large. In other words, we consider the doubly sparse CS problem when ||x|| 0 ≤ t and ||e|| 0 ≤ l. The assumption ||e|| 0 ≤ l was first time considered in [START_REF] Kabatiansky | What to do if syndromes are corrupted also[END_REF] as a proper replacement for discrete version of CS-problem of usual assumption that an error vector e has relatively small Euclidean norm.

Definition 1. A real r × n matrix H called a (t, l)-compressed sensing (CS) matrix if ||Hx T -Hy T || 0 ≥ 2l + 1 (2)
for any two distinct vectors x, y ∈ R n such that ||x|| 0 ≤ t and ||y|| 0 ≤ t.

This definition immediately leads (see [START_REF] Kabatiansky | What to do if syndromes are corrupted also[END_REF]) to the following

Proposition 1. A real r × n matrix H is a (t, l)-CS matrix iff ||Hz T || 0 ≥ 2l + 1 (3)
for any nonzero vector z ∈ R n such that ||z|| 0 ≤ 2t.

Our main result is an explicit and simple construction of (t, l)-CS matrices with r = 2(t + l) for any n. We show this value of r is the minimal possible for (t, l)-CS matrices by proving an analog of the well-known in coding theory Singleton bound for the compressed sensing problem. Besides that we propose an efficient recovery (decoding) algorithm for the considered double sparse CS-problem.

Optimal Matrices for Doubly Sparse Compressed Sensing Problem

We start with constructing of (t, l)-CS matrices. Let a real r × n matrix H be a parity-check matrix of an (n, n -r)-code code over R, correcting t errors, i.e. any 2t columns hi1 , . . . , hi2t of H are linearly independent. And let G be a generator matrix of an (r, r)-code over R of length r, correcting l errors. Let matrix H consists of the columns h 1 , . . . , h n , where

h T j = hT j G (4)
and transposition T means, that vectors h j and hj are considered in (4) as row vectors, i.e.

H = G T H (5)
In other words, we encode columns of parity-check matrix H, which is already capable to correct t errors, by a code, correcting l errors, in order to restore correctly the syndrom of H.

Theorem 1. Matrix H = G T H is a (t, l)-CS matrix.
Proof. According to Proposition 2 it is enough to prove that ||Hz T || 0 ≥ 2l + 1 for any nonzero vector z ∈ R n such that ||z|| 0 ≤ 2t. Indeed, u = Hz T = 0 since any 2t columns of H are linear independent. Then

Hz T = G T Hz T = G T u = (u T G) T and u T G is a nonzero vector of a code over R, correcting l errors. Hence ||Hz T || 0 = ||u T G|| 0 ≥ 2l + 1.
Now let us choose the well known Reed-Solomon (RS) codes (which are a particular case of evaluation codes construction) as both constituent codes. The length of the RS-code is restricted by the number of elements in the field so in the case of R the length of evaluation code can be arbitrary large. Indeed, consider the corresponding evaluation code RS (n,k) = {(f (a 1 , . . . , f (a n )) : deg f (x) < k}, where a 1 , . . . , a n ∈ R are n different real numbers. The distance of the RS (n,k) code d = n -k + 1 since the number of roots of a polynomial cannot exceed its degree and hence d ≥ n -k + 1,but, on the other hand, the Singleton bound states that d ≤ n -k + 1 for any code, see [START_REF] Macwilliams | The Theory of Error-Correcting Codes[END_REF]. Therefore the resulting matrix H is a (t, l)-CS matrix with r = 2(t + l). The next result, which is a generalization of the Singleton bound for the doubly sparse CS problem, shows these matrices are optimal in the sense having the minimal possible number r of linear measurements.

Theorem 2. For any (t, l)-CS r × n-matrix r ≥ 2(t + l). (6)
Proof. Let H be any (t, l)-CS matrix of size r × n, i.e., ||Hz T || 0 ≥ 2l + 1 for any nonzero vector z ∈ R n such that ||z|| 0 ≤ 2t. And let H 2t-1 be the (2t -1) × n matrix consisting of the first 2t -1 rows of H. There exists a nonzero vector ẑ = (ẑ 1 , . . . , ẑ2t , 0, 0, . . . , 0) ∈ R n such that H ẑT = 0 (a system of linear homogenious equations with the number of unknown variables larger than the number of equations has a nontrivial solution). Then ||H ẑT || 0 ≤ r -(2t -1) and finally r ≥ 2t + 2l since ||H ẑT || 0 ≥ 2l + 1.

Recovery Algorithm for Doubly Sparse Compressed Sensing Problem

Let us start from a simple remark that for e = 0 recovering of the original sparse vector x, i.e., solving the equation ( 1), is the same as syndrome decoding of some code (over R) defined by matrix H as a parity-check matrix. In general, syndrome s = Hx T is known with some error, namely, as ŝ = s + e and therefore we additionally encoded columns of H by some error-correcting code in order to recover the original syndrome s and then apply usual syndrome decoding algorithm. Therefore recovering, i.e., decoding algorithm for constructed in previous chapter optimal matrices is in some sense a "concatenation" of decoding algorithms of constituent codes.

Namely, first we decode vector ŝ = s + e by a decoding algorithm of the code with generator matrix G. Since ||e|| 0 ≤ l this algorithm outputs the correct syndrome s. After that we form the syndrome s by selecting first r coordinates of s and then apply syndrome decoding algorithm (of the first code with paritycheck matrix H) for the following syndrom equation s = Hx T .

Now let us discuss a right choice of constituent codes. It is very convenient to use the class of Reed-Solomon codes over R. There are well known algorithms of their decoding up to half of the code distance (bounded distance decoding, see [START_REF] Macwilliams | The Theory of Error-Correcting Codes[END_REF]), for instance, Berlekamp-Massey algorithm, which in our case (codes over R) is known also as Trench algorithm, see [START_REF] Trench | An algorithm for the inversion of finite Toeplitz matrices[END_REF]. Hence the total decoding complexity does not exceed O(n 2 ) operations over real numbers. Moreover we can even decode these codes over their half distances by application of Guruswami-Sudan list decoding algorithm [START_REF] Guruswami | Improved decoding of Reed-Solomon and algebraicgeometry codes[END_REF].

It is well known that encoding-decoding procedures of Reed-Solomon codes become more simple in the case of cyclic codes, when the set a 1 , . . . , a n is a cyclic group under multiplication. In order to do it let us consider a 1 , . . . , a n as complex roots of degree n and define our codes through their "roots", i.e. our codes consist of polynomials f (x) over R such that f (e 2πi m n ) = 0 for m ∈ {-s, . . . , -1, 0, +1, . . . , +s} with s = t for the first constituent code and s = l for the second. It easy to check that such codes achieve the Singleton bound with d = 2s + 2, so the corresponding doubly sparse code has redundancy r = 2(t + l + 1) what is slightly larger than the corresponding Singleton bound, but in return these codes can be decoded via FFT.

4 Discussion -no small errors case and slightly beyond Let us note that the initial papers on Compressed Sensing especially stated that this new technique (application of l 1 minimization instead of l 0 ) allows to recover information vector x ∈ R n in case when not many coordinates of x were affected by errors. For instance, "one can introduce errors of arbitrary large sizes and still recover the input vector exactly by solving a convenient linear program...", see in [START_REF] Candes | Decoding by linear programming[END_REF]. To achieve such performance some special restriction on matrix H was placed, called Restricted Isometry Property (RIP), as follows

(1 -δ D )||x|| 2 ≤ ||Hx T || 2 ≤ (1 + δ D )||x|| 2 , (8)
for any vector x ∈ R n : ||x|| 0 ≤ D, where 0 < δ D < 1. The smallest possible δ D called the isometry constant. Then typical result in [START_REF] Candes | Decoding by linear programming[END_REF] (Th. 1.1 ) is of the following form "if δ 3t + 3δ 4t < 2 then the solution of linear programming problem is unique and equal to x" Let us note that the condition δ 3t + 3δ 4t < 2 implies δ 4t < 2/3 (of course, it implies that δ 4t < 1/2, but for us enough to have δ 4t < 1). Hence Hx T = 0 for any nonzero x with wt(x) ≤ 4t, or in other words, an error-correcting code (over reals) corresponding to such parity-check matrix H has the minimal distance at least 4t + 1 and can correct 2t errors (instead of t). So we lost twice in error-correction capability but maybe linear programming provides more easier way for decoding ? In fact, NOT, since it is well known in coding theory that such problem can be solved rather easily (in complexity) over any infinite field by usage of the corresponding Reed-Solomon codes and known decoding algorithms. In case of real number or complex number fields one can use just an RS code with Fourier parity-check matrix, namely, h j,p = exp(2πi jp n ), where for complex numbers p ∈ {1, 2, ..., n}, j = a, a + d, a + 2d, . . . , a + (r -1)d, and "reversible" RS-matrix H for real numbers, where p ∈ {1, 2, ..., n}, j ∈ {-f, -f + 1, . . . , 0, 1, . . . , f } and r = 2f + 1. Fortunately, matrices with the RIP property allow to correct not only sparse errors but also additional errors with arbitrary support but relatively small (up to ε) Euclidean norm. Again, the RIP property is good for linear programming decoding but is too strong in general. Namely, it is enough to have the following property

λ 2t ||x|| 2 ≤ ||Hx T || 2 , (9)
for any vector x ∈ R n : ||x|| 0 ≤ 2t, where λ 2t > 0 and the largest such value we call extension constant. Indeed, then for any two solutions x and x of the Equation ( 1) we have that

||x -x|| 2 ≤ 2λ -1 2t ε, (10)
where ||e|| 2 , ||ẽ|| 2 ≤ ε. Hence [START_REF] Vlǎdut | On Error Correction with Errors in Both the Channel and Syndrome[END_REF] shows that all solutions of the Equation (1) are "almost equal" if λ 2t is large enough. Let us note that for RS-matrices any r columns are linear independent and hence λ 2t > 0, but λ 2t tends to zero when n grows and code rate is fixed. To find better class of codes over the field of real (or complex) numbers is an open problem.

Conclusion

In this paper we extends technique, which was developed in [START_REF] Vlǎdut | On Error Correction with Errors in Both the Channel and Syndrome[END_REF] for error correction with errors in both the channel and syndrome, to the Compressed Sensing problem. We hope this approach will help to find limits of the unrestricted (i.e. without LP usage) compressed sensing.

⋆ The work of Grigory Kabatiansky and Serge Vlǎdut ¸was carried out at the Institute for Information Transmission Problems of the Russian Academy of Sciences at the expense of the Russian Science Foundation, project no. 14-50-00150.


Khairi Sioud

Ali Abdennadher

Slim Kaddeche

Marie Catherine

Charrier Mojtabi
email: mojtabi@imft.fr

Abdelkader Mojtabi

Marie Catherine Charrier-Mojtabi

Soret-Driven Convection Inside Concentric Porous Spheres Saturated by Binary Fluid

Keywords: Heat and Mass Transfer, Porous Media, Species separation, Soret effect, Thermogravitational columns, Spherical cell, Binary mixture, Numerical simulations

come

Introduction

Species separation in binary or multicomponents mixture is a coupled heat and mass transfer phenomena. Pure thermo-diffusion leads to very weak species separation in binary mixtures. To increase the species separation in the presence of a gravity field, many authors use thermo-gravitational diffusion in vertical, fluid or porous, columns (TGC) ([1]- [START_REF] Melnikov | Separation of a binary liquid mixture in compound system: Fluid-porous-fluid[END_REF]). These TGC are parallelepipedic cavities consisting of two vertical plates facing each other. The plates are maintained at uniform temperatures, T h > T c . The spacing between these two parallel plates is millimetric. The other faces of the cavity are perfectly insulated. The combination between the convective flow, generated by the horizontal thermal gradient and the thermodiffusion between the two plates, leads to a stationary convective state and a constant vertical mass gradient. Often, one of the components of the binary mixture has particular industrial importance for pharmacy or perfumery, or has high added value components (rare earths). The Soret effect in liquid mixtures has been completely reviewed by Köhler et al. [START_REF] Köhler | The Soret effect in liquid mixtures[END_REF]. In 1959, Lorenz and Emery [START_REF] Lorenz | The packed thermal diffusion column[END_REF] considered columns (TGC) filled with porous packing saturated by a binary fluid to allow the use of columns with spacing higher than one millimetre. Until the 2000s, research on thermo-gravitational diffusion concerned mainly vertical columns. In 2003, Platten et al. [START_REF] Platten | Enhanced molecular separation in inclined thermogravitational columns[END_REF] performed an experimental study of the effect of the tilt with respect to the horizontal direction, of a rectangular cell heated only from below. Since 2007, various works aimed at improving species separation using different configurations and processes. Charrier-Mojtabi et al. [START_REF] Charrier-Mojtabi | Analytical and numerical stability analysis of Soret-driven convection in a horizontal porous layer[END_REF] showed that it was possible to obtain the species separation in a horizontal cell heated from below by studying the influence of the separation ratio and the normalized porosity on the stability of the equilibrium solution, theoretically and numerically. Elhajjar et al. [START_REF] Elhajjar | Separation of a binary fluid mixture in a porous horizontal cavity[END_REF] stablished that it was possible to achieve separation in the horizontal configuration when the separation factor was negative or greater than a positive value leading to the onset of a unicellular flow. They demonstrated that the maximum separation remained unchanged when the TGC was tilted from its vertical position to the horizontal position [START_REF] Elhajjar | Separation in an inclined porous thermogravitational cell[END_REF]. Khouzam et al. [START_REF] Khouzam | Species separation of a binary mixture in the presence of mixed convection[END_REF] used mixed convection in order to study the species separation in a binary mixture in a horizontal cell. Yacine et al. [START_REF] Yacine | Soret-driven convection and separation of binary mixtures in a horizontal porous cavity submitted to cross heat fluxes[END_REF] studied the case of a porous cavity saturated by a binary mixture that was subjected to cross heat fluxes. V.Yasnou et al. [START_REF] Yasnou | Shevtsova Role of a layer of porous medium in the thermodiffusion dynamics of a liquid mixture[END_REF] studied experimentally the transient mass transfer associated with a thermal gradient through a liquid mixture containing a layer of porous medium saturated with liquid. Mojtabi et al. [START_REF] Mojtabi | The effect of conducting bounding horizontal plates on species separation in porous cavity saturated by a binary mixture[END_REF] studied the influence of the thermophysical characteristics of the walls in the process for separating the constituents of a binary mixture. They showed that separation was underestimated when the effect of the thermal characteristics of the walls was not taken into account. Recently, Mojtabi et al. [START_REF] Mojtabi | Analytical and numerical study of Soret mixed convection in two sided lid-driven horizontal cavity: Optimal species separation[END_REF] considered a new configuration to separate the binary components mixture in a horizontal porous layer by moving the walls of the cavity. On the over hand, thermogravitational separation and time to reach the steady state were studied recently by Seta et al. [START_REF] Seta | Separation under thermogravitational effects in binary mixtures[END_REF].

In the present study, species separation in a porous cavity between two concentric spheres and a TGC column, saturated with a water-ethanol mixture, was compared. The separation corresponds to the difference in mass fractions between the two ends of the TGC and between the top and the bottom parts of the spherical cavity. The species separation and the time to reach the steady state leading to maximum separation were evaluated in both configurations. An analytical solution was obtained in the case of a TGC. The velocity field and the mass fraction gradient along the z-vertical axis were then calculated. The analytical results were corroborated by direct 2D and 3D numerical simulations. For the spherical configuration, an analytical solution of the axisymmetric laminar unicellular flow was obtained for the first time to our knowledge. 2D and 3D numerical studies were carried out using Comsol Multipysics software and a spectral code, in order to validate the analytical results. The results obtained with these different methods are in good agreement. The times to reach the stationary state and the separation in both configurations were compared.

Mathematical formulation

The two porous cavities studied are respectively the porous cavity saturated by a binary mixture between two concentric spheres and the vertical parallelepipedic column (TGC). These cavities have the same thickness, e = R o -R i , where R i and R o are the internal and external radii of the spherical cavity respectively. H = πR i is the height and the depth, L = 2R o of the TGC. This height corresponds to the length of the convective cell, which develops within the spherical cavity even for the lowest values of the temperature difference (T h -T c ). The two porous cavities are saturated with water (60.88 wt %)ethanol mixture. The cavities considered in this study are shown schematically in Fig. 1. The Boussinesq approximation is assumed to be valid; thus, the thermophysical properties of the binary fluid are constant, except for the density in the buoyancy term, which varies linearly with the local temperature, T, and the local mass fraction, C, of the denser component:

ρ = ρ 0 (1 -β T (T -T ref ) -β C (C -C ref )) (1)
where β T and β C are respectively the thermal and mass expansion coefficients of the binary fluid, defined as follows:

β T = -1 ρ0 ( ∂ρ ∂ T ) C > 0, β C = -1 ρ0 ( ∂ρ ∂ C ) T < 0.
In a binary fluid mixture subjected to a temperature gradient, a mass fraction gradient appears due to the thermodiffusion or Soret effect. In addition to the usual expression for the mass flux density J m given by Fick's law, a part due to the temperature gradient (namely the Soret effect) was used so that:

J m = -ρD * ∇ ∇ ∇C -ρD ′ T * ∇ ∇ ∇T (2
J m = -ρ(D * ∇ ∇ ∇C -C 0 (1 -C 0 )D T * ∇ ∇ ∇T ) (3)
The dimensional mathematical formulation of the physical problem is given by:

∇ • V ∇ • V ∇ • V = 0 V V V = - K µ (∇ ∇ ∇P -ρ 0 [1 -β T (T -T 0 ) -β C (C -C 0 )]g g g) (ρc) * ∂T ∂t + (ρc) f V • ∇ V • ∇ V • ∇T = λ * ∇ ∇ ∇ 2 T ϵ * ∂C ∂t + V • ∇ V • ∇ V • ∇C = D * ∇ ∇ ∇ 2 C + D T * C 0 (1 -C 0 )∇ ∇ ∇ 2 T (4)
In these equations, (ρc) * and λ * are respectively the effective heat capacity and the thermal conductivity of the porous medium, ϵ * its porosity and K its permeability. µ and ρ 0 are the dynamic viscosity and the density, at T = T 0 of the saturating fluid. The corresponding boundary conditions for the TGC and concentric spheres are respectively:

(5) for the TGC:

x = 0, x = e; V • x V • x V • x = 0, T (0, y, z) = T h , T (e, y, z) = T c , J m • x x x = 0 y = 0, y = L; V • y V • y V • y = 0, ∂T ∂y = ∂C ∂y = 0 z = 0, z = πR i ; V • z V • z V • z = 0, ∂T ∂z = ∂C ∂z = 0 (5)
and ( 6) for the concentric spheres (e r , e θ , e ϕ e r , e θ , e ϕ e r , e θ , e ϕ ):

r = R i ; T = T h , V • n V • n V • n = 0, J m • n n n = 0 r = R o ; T = T c , V • n V • n V • n = 0, J m • n n n = 0 θ = 0, π, ∀ϕ, ∂T ∂θ = ∂C ∂θ = 0, V • e θ V • e θ V • e θ = 0 (6)
where n denotes the unit normal vector to the surfaces of the spheres. The flow between the two concentric spheres is assumed to be axisymmetric.

3 Analytical resolution

For TGC column

To solve the system of equations ( 4) associated with the boundary conditions [START_REF] Melnikov | Separation of a binary liquid mixture in compound system: Fluid-porous-fluid[END_REF] analytically in the case of TGC, for e ≪ πR i , the forgotten effect hypothesis [START_REF] Platten | Enhanced molecular separation in inclined thermogravitational columns[END_REF] and parallel flow approximation [START_REF] Ouriemi | Natural convection in a horizontal layer of binary mixture[END_REF] are used. The parallel flow approximation leads to unicellular flow given by:

V u = W u (x)z z z, T u = f (x), C u = mz + h(x) (7)
where m is an unknown constant that represents the mass fraction gradient in the z direction. With these assumptions and for the steady state, the system of equations ( 4) with the boundary conditions ( 5) is reduced to the following set of equations, which can be solved using Maple software:

∂W u ∂x = K µ ρ 0 β T ∂T ∂x ∇ ∇ ∇ 2 T = 0 m W u -D * ∇ ∇ ∇ 2 C u -D * T C 0 (1 -C 0 )∇ ∇ ∇ 2 T u = 0 (8)
Using the boundary conditions ( 5), we deduce the expression for the temperature, the velocity and the mass fraction field, given by the following expressions:

T u = (T c -T h ) e x + T h W u = Kgβ T (T c -T h )(2x -e) 2eν
C u = mz + (T c -T h ) D * ( mKgβ T (e 3 -6ex 2 + 4x 3 ) 24eν -D * T C 0 (1 -C 0 )( x e - 1 2
))

The value of the mass fraction gradient m:

m = -10Kgνβ T D * T C 0 (1 -C 0 )(T c -T h ) 2 (Kegβ T (T c -T h )) 2 + 120(νD * ) 2 (10)
is obtained by specifying that the total mass flux of the component of mass fraction C, through any cross section of the TGC parallel to the x axis, is equal to zero:

e 0 (W u C u -(D * ∂C u ∂z + D * T C 0 (1 -C 0 ) ∂T u ∂z ))dx = 0 (11)
Application to the separation of a water-ethanol binary mixture: To illustrate the analytical results obtained in this study, we restrict ourselves to the experimental values of the thermophysical parameters of a water (60.88 wt%)-ethanol (39.12 wt%) solution studied Platten et al. [START_REF] Platten | Enhanced molecular separation in inclined thermogravitational columns[END_REF]. The values of the thermophysical properties of this binary solution, at the average temperature T 0 = 22.5 • C are given in Table 1.

Costesèque et al. [START_REF] Costesèque | Transient-state method for coupled evaluation of Soret and Fick coefficients, and related tortuosity factors, using free and porous packed thermodiffusion cells: Application to CuSO4 aqueous solution (0.25 M)[END_REF] showed that the Soret coefficients in free and porous

C D[m 2 s -1 ] D T [m 2 (s K) -1 ] β C β T [K -1 ] α[m 2 s -1 ] ρ 0 [kg m -3 ] ν[m 2 s -1 ]
4.32 10 -10 1.37 10 -12 -0.212 7.86 10 -4 10 -7 935.17

For the spherical configuration

Here, the assumptions used to determine the analytical solution in a TGC column are extended to the resolution of system of equations ( 4) associated with boundary conditions [START_REF] Köhler | The Soret effect in liquid mixtures[END_REF] in the case of concentric spheres with e ≪ R i .

The parallel flow approximation leads to the unicellular flow given by:

V u V u V u = V r (r, θ)e r e r e r + V θ (r, θ)e θ e θ e θ , (12)
With these assumptions and for the steady state, the system of equations ( 4) with boundary conditions ( 6) is reduced to the following set of equations [START_REF] Yasnou | Shevtsova Role of a layer of porous medium in the thermodiffusion dynamics of a liquid mixture[END_REF], which is solved using Maple software:

∇ ∇ ∇ • V u V u V u = 0 V u V u V u = K µ (-∇P + ρ 0 g[1 -β T (T u -T 0 )](cos(θ)e r e r e r -sin(θ)e θ e θ e θ )) ∇ ∇ ∇ 2 T u = 0 V r (r, θ) ∂C u ∂r + V θ (r, θ) ∂C u r∂θ = D * ∇ ∇ ∇ 2 C u D T * C 0 (1 -C 0 )∇ ∇ ∇ 2 T u (13)
Using the stream function, ψ, with V r = ∂ψ/(∂θ r 2 sin θ) and V θ = -∂ψ/(∂r r sin θ), the continuity equation is automatically checked.

∂ 2 ψ ∂r 2 + 1 r ∂ 2 ψ ∂θ 2 + cos(θ) r 2 sin θ ∂ψ ∂θ = - K µ ρ 0 gβ T ∂T u ∂r r sin 2 (θ) (14)
By setting ψ = φ(r) sin 2 (θ), in equation ( 14), we deduce the equation verified by the function φ:

∂ 2 φ ∂r 2 -2 φ r 2 = - K µ ρ 0 gβ T ∂T u ∂r (15)
Using boundary conditions (6), we deduce the expression for the temperature and the velocity given by the following expressions:

T u = 1 R o -R i (R i R o (T h -T c )/r + R o T c -R i T h ) V r = 2 cos(θ) r 2 φ(r) V θ = - sin(θ) r ∂φ(r) ∂r (16)
where:

φ(r) = K µ ρ 0 gβ T (T h -T c )(R o -r)(R i -r)((R o + r)R i + rR o )R i R o 2r(R o -R i )(R 2 o + R i R o + R 2 i ) (17)
It can thus be deduced that the velocity intensity is proportional to:

K µ ρ 0 gβ T (T h -T c ).
Knowing the temperature and velocity fields, given by equation ( 16), we deduce the mass fraction equation:

∂ 2 C u ∂r 2 + 1 r 2 ∂ 2 C u ∂θ 2 + 2 r ∂C u ∂r (1- cos(θ) r φ(r) D * )- 2 r 2 ∂C u ∂θ ( cos(θ) sin(θ) - sin(θ) D * ∂φ(r) ∂r ) = 0 (18)

Numerical results

The dimensionless set of equations ( 4) with boundary conditions ( 5) for (TGC) and ( 6) for concentric spheres, were solved numerically using a finite element code (Comsol Multiphysics) and a spectral collocation method [START_REF] Peyret | Spectral Methods for Incompressible Viscous Flow[END_REF] on Gauss-Lobatto-Chebyshev points. A time-dependent solver and the set of equations (incompressible Darcy equation, energy and mass diffusion equations) in transient form were used. The condition of conservation of the average mass fraction in the two different porous cavities saturated by a binary fluid was imposed on each iteration. For the spherical configuration, the spectral collocation method on Gauss-Lobatto-Chebyshev points in the radial direction was combined with the Fourier spectral method for the polar angle. Indeed the interpolated functions are periodic with respect to the polar angle due to the spherical geometry and the boundary conditions considered. Direct numerical simulations were performed for cavities of the same thickness, e = 6.10 -3 m, (e = R o -R i ), with R i = 0.05, 0.1, 0.2, 0.3 m. The height of the TGC was equal to H = πR i . This height was chosen so that the length of the stream function lines was almost equal in the two considered configurations. The physical mechanism inside the two configurations was the same, as well as the structure of the flow. Thus, the species separation would be expected to be of the same order of magnitude in the vertical column and the spherical cell. This result that was subsequently confirmed numerically. For numerical applications, constant temperatures, T h = 25 • C and T c = 15 • C,were imposed on the walls of the two configurations and for two values of the permeability, K = 2.5 10 -11 m 2 and K = 5.66 10 -11 m 2 . The initial mass fraction C 0 = C ref = 0.6088 was also kept constant in both configurations.

Comparison between analytical results and numerical simulations for TGC

In the vertical thermogravitational column, the thermal fields obtained analytically and numerically were almost identical. The maximal and minimal values of mass fraction, C max and C min , respectively, for four TGC of the same thickness, e = 6.10 -3 m, obtained using analytical and direct numerical simulations are given in Table 2. The results presented show very good agreement between the numerical results and the analytical ones.

Comparison between analytical results and numerical simulations for a porous cavity between two concentric spheres

In concentric spheres, the values of thermal field according to r, T u (r, θ = π/2), obtained analytically and numerically are in very good agreement (cf. Fig. 4). The plot of the tangential velocity, V θ (r, θ = π/2), versus r in Fig. 5 (a) and the plot of the radial velocity V r (r, θ = 3π/4), in Fig. 5 (b) also show very good agreement between the analytical and the numerical results. Below we present the streamlines in the upper part of the spherical configuration Fig. 6 (b). The comparison between the values of the stream function ψ(r, θ = π/2), obtained analytically and numerically are in good agreement (Fig. 6 (a)). The values of maximal, C max , and minimal, C min , mass fraction for 4 spherical configurations, with the same thickness, e = 6.10 -3 m, obtained using two direct numerical simulation approaches, are given in Table 3. The result presented shows a very good agreement between the two numerical approaches used in the present work. The values obtained for C max and C min , in Table 2 for the vertical column and in Table 3 for the spherical cavity, showed that the maximum separation obtained (C max -C min ) for each of the four values of the radii Ri studied is slightly greater in the spherical cell. For the spherical cells corresponding to Ri = 0.05, 0.1, 0.2, 0.3m, (C max -C min ) = 0.086, 0.160, 0.308, 0.456 while for TGC: (C max -C min ) = 0.074, 0.143, 0.282, 0.421. The respective ratios of the species separation in these two configurations (1.16, 1.12, 1.09, 1.08) show that the species separation in the spherical cavity is of the order of 1.1 times greater than the separation obtained in the vertical column.

Flow stability in the upper part of spherical configuration or around its symmetrical part

The direct numerical simulations obtained ( §4.2), in the case where the porous medium is saturated by a binary fluid with a positive Soret coefficient showed that the flows obtained are stable. The instability in this configuration may happen either in the upper part or around the its symmetrical part. Indeed, as the inner sphere is maintained at T h > T c , the temperature of the outer sphere, and the difference of the radii of the two spheres verifies R 0 -R i = e = 6 × 10 -3 m ≪ R i then locally the top and the bottom of the spherical configuration can be assimilated respectively to porous plane layers saturated by a binary fluid. The top layer is heated from below and the bottom one is heated from above. In the upper part, the heaviest constituent migrates towards the upper wall, which is colder, leading to a loss of flow stability for a Rayleigh number higher than Ra c = 12 Leφ , where Le is the Lewis number of the binary fluid in the porous layer and φ , the separation factor, Cf. [START_REF] Charrier-Mojtabi | Analytical and numerical stability analysis of Soret-driven convection in a horizontal porous layer[END_REF]. For a water-ethanol binary mixture saturating the porous medium, the Lewis number is Le = 654, 86 and φ = 0.406 which gives Ra c = 0.045. For this experimental configuration, the Rayleigh number, Ra = gβ(T h -Tc) αν Ke, is equal to 0.0143 since Ra is proportional to the product of the thickness, e = 6×10 -3 m, by the permeability of the medium which is of the order of 10 -10 . To conclude with, for our configuration Ra < Ra c and the flow is stable. For the bottom layer, it is locally a plane layer heated from above, the heaviest constituent which migrates towards the bottom wall leads to an infinitely stable layer. For a negative Soret coefficient, the heaviest constituent migrates upwards from the hot inner sphere. Locally, the plane layer is heated from below and the binary fluid being denser at the bottom of that layer, the critical Rayleigh number is then greater than 4π 2 , see [START_REF] Charrier-Mojtabi | Analytical and numerical stability analysis of Soret-driven convection in a horizontal porous layer[END_REF]. In the symmetrical part, there is locally a plane layer heated from above, but the binary fluid is denser along the hot wall, a situation that could become unstable.

5 Time to reach the stationary state in the TGC and in the spherical configuration

The numerical simulations, using the spectral collocation method and the Comsol Multipysics software, allow us to calculate the time necessary to reach the stationary state. The calculations were performed for two values of R i = 0.1, 0.2 m and H = πR i .

In Table 4 for the TGC and Table 5 for spherical cell, the values of C max and C min corresponding to four physical times during the evolution of C towards the stationary state are presented. It emerges from Table 4 that the values of C max and C min calculated from the Comsol software and from the spectral code are very close with a maximal discrepancy of 10 -4 . The mass fractions C max and C min , as a function of time in the vertical column and the spherical configuration, are presented, in Fig. 7 and Fig. 8 respectively. It emerges from the analysis of these two figures that the species separation is slightly higher in the spherical configuration (C max = 0.623) than for the TCC column (C max = 0.613) In addition, in the spherical cavity, the stationary state in the lower part is reached after a time twice as long as that in the upper part (Fig. 8). On the other hand, the stationary state is reached simultaneously at the top and at the bottom of the TGC (Fig. 7).

A qualitative interpretation of this result is understandable. In the spherical configuration, all the streamlines ensuring the species separation converge at the same places (the upper and lower parts of the spherical configuration around θ = π and θ = 0 and for all vertical planes ϕ = cst.

In the TGC column, the contributions of the streamlines occur in distinct places (z = 0 and z = H for all vertical planes y = cst). Even if the flow is The amount of species separated at the bottom or top of the cells is about one and a half times larger for the spherical configuration than for the TGC.

Conclusion

Up to now, species separation was mainly obtained in vertical thermogravitational columns (TGC). At our knowledge, for flows between two spheres, in porous or fluid media, the works focused only on natural convection. In this study, numerical 2D and 3D simulations, were carried out in TGC and spherical configurations, for 4 values of R i and 4 heights H and a moderate temperature difference between the walls. They showed that the flows remained 2D axisymmetric for the spherical configuration and invariant by translation in the horizontal direction, y, for the parallelepipedic vertical column. This result is due to the fact that in porous media the hydrodynamic boundary conditions are of the slip type on the walls. Moreover, the analytical and numerical studies carried out in each of these two configurations showed that the differences in mass fractions between the top part and the bottom part of the spherical cavity are slightly higher than those obtained between the two ends of the TGC for the same thickness, e = R o -R i and for a height H = πR i . At our knowledge, this is the first time that an analytical solution has been obtained in the spherical configuration.

In addition, we showed that the amount of species separated at the bottom or top of the cells is about one and a half times larger for the spherical configuration than for TGC column having similar geometric dimensions in two spatial directions.

Declarations

Funding: NO FUNDING

Fig. 1

 1 Fig. 1 Schematic diagram of the concentric spheres (a) and the TGC column (b)

  ) where D * and D ′ T * are respectively the mass diffusion coefficient and the thermodiffusion coefficient of the denser component in porous media, ρ is the mixture density, T the temperature and C the mass fraction of the constituent of interest D ′ T * is often written as: D ′ T * = F (C)D * T where F (C) is a particular function satisfying F (C = 0) = 0 and F (C = 1) = 0. In many studies, the authors consider that F (C) = C(1 -C) ≃ C 0 (1 -C 0 ), where C 0 = C ref is initial mass fraction. The previous assumptions lead to a new form of equation (2):

2 .

 2 716 10 -6 media can be considered as equal. On the other hand, the values of the mass diffusion coefficients and of the thermal diffusion coefficients are different for these two media. The mass and thermal diffusion coefficients of the component of interest are deduced from the coefficients D and DT in a fluid medium by means of the following relations D * = D/2.30 and D T * = D T /2.30.

Fig. 2 (

 2 a) presents the values of the velocity, W u , as a function of x for z = H/2, since the velocity is an independent function of z, except in the neighbourhood of z = 0 and H. The plot of the mass fraction C u versus z for x = e/2 in Fig. 2 (b) shows very good agreement between the analytical and numerical results.

Fig. 2

 2 Fig. 2 (a) (on left) velocity, Wu(x, z = H/2) and (b) (on right) mass fraction Cu(x = e/2, z), obtained analytically (continuous line) and using direct numerical simulation (dotted line)

Fig. 3

 3 Fig. 3 (a) (on left) mass fraction Cu(x, z = 0), and (b) (on right) mass fraction Cu(x, z = πR i = 0.157), obtained analytically

Fig. 4

 4 Fig. 4 (a) (on left) temperature T (r, θ = π/2), obtained analytically (continuous line) and using direct numerical simulation (dotted line), (b) (on right), isothermal lines for 0.05 ≤ z ≤ 0.08 m

Fig. 6

 6 Fig. 6 (a) (on left) stream function ψ(r, θ = π/2), obtained analytically (continuous line) and numerically (dotted line), (b) (on right), streamlines obtained numerically for 5π 6 ≤ θ ≤ π and for every azimuthal plane ϕ = ϕ 0 considered

Fig. 7 Fig. 8

 78 Fig. 7 Variation of Cmax (in red) and C min (in blue) as a function of time, in TGC, from the initial uniform state, C 0 to the stationary non uniform state for H = πR i , R i = 0.1 m and e = 6.10 -3 m, obtained using the spectral method

Fig. 9 2 Fig. 10

 9210 Fig. 9 (a) (on left) the amount of separation in a spherical cell, (b) (on right) Iso-C, streamlines in a spherical cell, R i = 0.2 m, e = 6.10 -3 m, δT = 10 K and K = 2.5 10 -11 m 2

Table 1

 1 Properties for a water (60.88 wt%)ethanol (39.12 wt%) mixture at a mean temperature of 22.5 •

Table 2

 2 Maximal and minimal mass fraction in TGC for e = 6.10 -3 m, for different values

	of H(Ri = 0.05; 0.1; 0.2; 0.3 m), obtained
	analytically and by direct numerical methods
	Mass Fraction of TGC		Analytical results Spectral method Comsol
	H=0.157 m	Cmax	0.646	0.646	0.646
		C min	0.572	0.572	0.572
	H=0.314 m	Cmax	0.681	0.680	0.680
		C min	0.537	0.537	0.537
	H=0.628 m	Cmax	0.752	0.750	0.750
		C min	0.466	0.468	0.468
	H=0.942 m	Cmax	0.822	0.819	0.819
		C min	0.395	0.398	0.398

Table 3

 3 Comparison of the maximal and minimal mass fractions, Cmax and C min , respectively, in concentric spheres for e = Ro -R i = 6.10 -3 m and for different values of Ri, obtained using the spectral collocation method and Comsol software

	Mass Fraction		Spectral method Comsol Multiphysics
	R i = 0.05 m	Cmax	0.651	0.651
		C min	0.565	0.565
	R i = 0.1 m	Cmax	0.687	0.687
		C min	0.527	0.527
	R i = 0.2 m	Cmax	0.759	0.760
		C min	0.452	0.452
	R i = 0.3 m	Cmax	0.832	0.833
		C min	0.376	0.377

Table 4

 4 Variation of Cmax and C min , in TGC, from the initial uniform state C 0 , to the stationary non uniform state for H = 0.314 m, H = 0.628 m, and e = 6.10 -3 m obtained using the spectral method and Comsol software

	TGC (H=0.314m)		2.10 4 (s) 4.10 4 (s) 6.10 4 (s) 8.10 4 (s)
	Spectral code	Cmax	0.6129	0.6130	0.6129	0.6129
		C min	0.6047	0.6046	0.6047	0.6047
	Comsol code	Cmax	0.6133	0.6134	0.6132	0.6132
		C min	0.6043	0.6042	0.6044	0.6044
	TGC (H=0.628m)		5.10 4 (s) 10 5 (s)	2.10 5 (s) 4.10 5 (s)
	Spectral code	Cmax	0.6149	0.6157	0.6159	0.6159
		C min	0.6027	0.6019	0.6017	0.6017
	Comsol code	Cmax	0.6149	0.6159	0.6164	0.6165
		C min	0.6027	0.6017	0.6012	0.6011

Table 5

 5 Variaton of Cmax and C min , in the spherical cavity, from the initial uniform state, C 0 to the stationary non uniform state for R i = 0.1 m, R i = 0.2 m, e = Ro -R i = 6.10 -3 m obtained using the spectral method and Comsol software

	Spherical cell (Ri=0.1m)		2.10 4 (s) 4.10 4 (s) 6.10 4 (s) 8.10 4 (s)
	Spectral code	Cmax	0.6162	0.6208	0.6224	0.6229
		C min	0.6001	0.5969	0.5973	0.5972
	Comsol code	Cmax	0.6147	0.6184	0.6203	0.6213
		C min	0.6025	0.5994	0.5985	0.5983
	Spherical cell (Ri=0.2m)		5.10 4 (s) 10 5 (s)	2.10 5 (s) 4.10 5 (s)
	Spectral code	Cmax	0.6212	0.6279	0.6324	0.6345
		C min	0.5949	0.5876	0.5855	0.5860
	Comsol code	Cmax	0.6148	0.6206	0.6272	0.6312
		C min	0.6053	0.6019	0.5976	0.5935

Availability of data and materials : YES Code availability : YES Authors' contributions: ALL AUTHORS CONTRIBUTED EQUALLY TO THIS WORK


Khairi Sioud

Ali Abdennadher

Alain Bergeon

Slim Kaddeche

Marie Catherine

Charrier Mojtabi

Abdelkader Mojtabi

Marie Catherine Charrier-Mojtabi

Thermogravitational separation in porous vertical and horizontal cylindrical annular cells saturated by a binary mixture

Thermogravitational separation in porous vertical and horizontal cylindrical annular cells saturated by a binary mixture 1 Introduction

If a temperature gradient is applied to a binary or multi-component mixture, a mass fraction gradient appears for each species. Thus the expression of the mass flux density vector J ′ m is given by Fick's law with ana e-mail: mojtabi@imft.fr other term proportional to the temperature gradient namely Soret contribution:

J ′ m = -ρ ′ D * ∇C -ρ ′ D T * ∇T ′ (1)
Where ρ ′ , D * are respectively the density of the binary fluid, the mass diffusion coefficient of the denser component and D T * its thermodiffusion coefficient in the porous medium. C and T ′ are respectively the mass fraction of the denser component and the temperature. The species separation due to thermodiffusion or Soret effect are very weak. So if we consider the coupling between thermodiffusion and natural convection called thermo-gravitational diffusion, the species separation may increase. Thermo-gravitational diffusion has been widely studied due to its numerous fundamental and industrial applications. A summary of these applications were detailed by Nield and Bejan [START_REF] Nield | Convection in Porous Media[END_REF], Vafai [START_REF] Vafai | Handbook of Porous Media second editions[END_REF] and Legros et al. [START_REF] Legros | [END_REF]. The Soret effect in liquid mixtures has been reviewed by Khöhler et al. [START_REF] Köhler | The Soret effect in liquid mixtures[END_REF]. The first thermogravitational columns (TGC) were filled with bi-nary fluids. Furry, Jones and Onsager, developed the theory of thermodiffusion to interpret the experimental processes of isotope separation [START_REF] Furry | On the theory of isotope separation by thermal diffusion[END_REF]. In their study the authors assumed that the density of the binary fluid depends only on the temperature and not on the mass fraction, in gravity forces, this was commonly called the forgotten effect. In 1959, Lorenz and Emery [START_REF] Lorenz | The packed thermal diffusion column[END_REF] considered vertical columns (TGC) filled with porous packing saturated by a binary fluid in order to increase the thickness of the cell leading to maximum species separation. Research on species separation in vertical thermogravitational columns (TGC) has been performed for a long time. In 2003, Platten et al. [START_REF] Platten | Enhanced molecular separation in inclined thermogravitational columns[END_REF] studied experimentally the effect of the tilt, relative to the horizontal direction, of a rectangular cell heated only from below. Then several papers were devoted to the horizontal porous configuration saturated by a binary fluid, in order to improve the species separation. Charrier-Mojtabi et al. [START_REF] Charrier-Mojtabi | Analytical and numerical stability analysis of Soret driven in a horizontal porous layer[END_REF] showed that it was possible to obtain the species separation in a horizontal cell heated from below. The authors studied the influence of the separation ratio,

ψ = -β C β T D T * D * C 0 (1 -C 0 ), (β T , β C and C 0 are defined in §2)
and the normalized porosity on the stability of the equilibrium solution, theoretically and numerically. Seta et al. [START_REF] Seta | Gravitational stability analysis on double diffusion convection in ternary mixtures[END_REF] studied instabilities in DCMIX1 ternary mixtures when two liquid layers with different concentrations are superimposed. El Hajjar et al. [START_REF] Elhajjar | Separation of a binary fluid mixture in a porous horizontal cavity[END_REF] studied the species separation in a porous horizontal cell for negative or positive values of the separation ratio leading to the onset of a unicellular flow. El Hajjar et al. [START_REF] Elhajjar | Separation in an inclined porous thermogravitational cell[END_REF] considered a shallow porous cavity filled with a binary fluid. The cavity was heated from below or from above and was inclined with respect to the vertical axis. The authors showed analytically and numerically that the species separation can be increased for an optimal value of the tilt angle of the cavity. Further studies have been carried out for porous horizontal layers saturated by a binary fluid with various boundary conditions. Khouzam et al. [START_REF] Khouzam | Species separation of a binary mixture in the presence of mixed convection[END_REF] used mixed convection in order to study the species separation in a horizontal porous cell. Yacine et al. [START_REF] Yacine | Soretdriven convection and separation of binary mixtures in a horizontal porous cavity submitted to cross heat fluxes[END_REF] considered the case of a porous cavity that was subjected to cross heat fluxes. Yasnou et al. [START_REF] Yasnou | Role of a layer of porous medium in the thermodiffusion dynamics of a liquid mixture[END_REF] studied experimentally the thermodiffusion dynamics in a compound system when a layer of a porous medium is surrounded by free liquids. Mojtabi et al. [START_REF] Mojtabi | Analytical and numerical study of Soret mixed convection in two sided lid-driven horizontal cavity: Optimal species separation[END_REF] studied a new configuration of the horizontal porous layer by moving the cavity walls at a constant and opposite velocity. Abahri et al. [START_REF] Abahri | Thermogravitational separation in horizontal annular porous cell[END_REF] analyzed, for the first time, the species separation in a horizontal porous annulus. An analytical resolution using a perturbation method till order two and function of the Rayleigh number was presented. The analytical results were corroborated by numerical results obtained with a finite element method (Comsol Mutiphysics). Thermogravitational separation and time to reach the steady state have been studied recently by Seta et al. [START_REF] Seta | Separation under thermogravitational effects in binary mixtures[END_REF]. Nasrabadi et al. studied, experimentally, the influence of permeability on species separation and time to reach steady state in a thermogravitational column filled with a porous medium [START_REF] Nasrabadi | An analysis of species separation in a thermogravitational column filled with a porous medium[END_REF]. First, there has never been before a comparative study dealing with the species separation in a horizontal and in a vertical porous annular column. In the case of the horizontal column, article [START_REF] Abahri | Thermogravitational separation in horizontal annular porous cell[END_REF], only an approximate analytical solution using the perturbation method, developed at order two, was obtained. We performed in the present paper 3D numerical simulations which verified the validity of the 2D results presented in the article [START_REF] Mojtabi | Numerical and Analytical Studies of Soret-Driven Convection Flow Inside an Annular Horizontal Porous Cavity[END_REF]. We also checked that the results of the 3D numerical simulations were in good agreement with those of the 2D numerical simulations. The authors examine, analytically and numerically, the species separation in horizontal and vertical porous cylindrical annular cells saturated by a binary water-ethanol mixture. The species separations at steady state were compared as well as the time needed to reach this steady state.

Mathematical formulation

The vertical and horizontal porous cylindrical annular cells are presented in fig. 1. The binary fluid saturating the porous medium is a water-ethanol mixture (60.88% water). The thickness of the horizontal and the vertical column is: e = R o -R i , where R i and R o are the internal and the external radii respectively. H = πR i is the height of the vertical column, and the length of the horizontal cell. Constant temperatures, T hot and T cold were imposed on the inner and outer cylinders. The fluid flow within the porous medium is assumed to be incompressible and governed by Darcy's law. It is also assumed that viscous dissipation and Dufour effect are neglected since their influence is very weak in the considered liquid mixtures. Also, the Boussinesq approximation is considered for binary fluid properties which are considered as constant except the density of the fluid in the buoyancy contribution which varies linearly with both local temperature and local mass fraction:

ρ ′ = ρ 0 (1 -β T (T ′ -T 0 ) -β C (C -C 0 )) (2)
where, ρ 0 , β T , β C are respectively the density of the binary fluid at the reference state T 0 and C 0 , the thermal and solutal expansion coefficients. Under these assumptions, the continuity equation, the Darcy's law, and the conservation equations of energy and chemical species are written in dimensional form as follows:

∇ • V ′ = 0 V ′ = - K µ (∇P ′ -ρ 0 [1 -β T (T ′ -T 0 ) -β C (C -C 0 )]g) (ρc) * ∂T ′ ∂t ′ + (ρc) f V ′ • ∇T ′ = λ * ∇ 2 T ′ ϵ * ∂C ∂t ′ + V ′ • ∇C = D * ∇ 2 C + D T * C 0 (1 -C 0 )∇ 2 T ′ (3)
V ′ and P ′ are respectively the velocity of the flow and the pressure. In these equations λ * , (ρc) * are respectively the effective thermal conductivity and the effective volumetric heat capacity of the porous mediummixture system and (ρc) f is the volumetric heat capacity of the mixture. ϵ * is the porosity of the porous medium.

If Ω ′ represents the union of surfaces S ′ i and S ′ o which are respectively the inner and the outer surfaces of the cylinders, the boundary conditions associated with system (3) are system (4).

V ′ (M ∈ Ω ′ ) • n = 0 T ′ (M ∈ S ′ i ) = T hot T ′ (M ∈ S ′ o ) = T cold For(M ∈ S ′ i ∪ S ′ o ); (D * ∇C + D T * C 0 (1 -C 0 )∇T ′ ) • n = 0 (4)
The base surfaces are insulated and impermeable in each configuration:

∇C • n = 0; ∇T ′ • n = 0 (5)
The height H = πR i was chosen so that the length of the streamlines would be almost equal in the two configurations. The length of the horizontal cell is chosen equal to H, in order to compare the results obtained for the two configurations.

In order to simplify parametric representation of the physical problem, a dimensionless formulation of system (3) is considered. The reference scales are R i for the length, λ * /(R i (ρc) f ) for the velocity, ((ρc) * R 2 i )/λ * for time, (λ * µ)/(K(ρc) f ) for the pression and δT for the temperature. The dimensionless mathematical formulation of the problem is given by :

∇ • V = 0 V + ∇P = -(Ra T T + R S (C -C 0 ))e z ∂T ∂t + V • ∇T = ∇ 2 T ϵ ∂C ∂t + V • ∇C = 1 Le (∇ 2 C + ARa T ∇ 2 T ) (6)
The problem under consideration depends on five nondimensional parameters: the thermal Rayleigh number

V(M ∈ Ω) • n = 0 T (M ∈ S i ) = 1 T (M ∈ S o ) = 0 For(M ∈ S i ∪ S o ); (∇C + ARa T ∇T ) • n = 0 (7)
The dimensionless boundary conditions at base surfaces would be as following:

∇C • n = 0; ∇T • n = 0 (8)
3 Analytical resolution

Vertical porous cell

For the vertical porous cylindrical cell, the following approximations were taken into account in order to simplify the resolution:

   T = f (r) V = V (r)e z C = mz + h(r) (9)
When we take into consideration the boundary conditions [START_REF] Platten | Enhanced molecular separation in inclined thermogravitational columns[END_REF][START_REF] Charrier-Mojtabi | Analytical and numerical stability analysis of Soret driven in a horizontal porous layer[END_REF] for the resolution of the set of equations ( 6), the temperature is written as:

T = 1 - ln(r) ln(R) (10)
In our case e << R i and the flow is axisymmetric with respect to e z which allow us to assume that the vertical annular cell can be considered as a cell between two vertical planes heated differentially. Thus the velocity expression of V (r) noted V 2 (r) can be written as:

V 2 = Ra T (R ln(R) -(ln(r) + 1)(R -1)) (R -1) ln(R) (11)
and the associated mass fraction is noted

C 2 = m 2 z + h 2 (r) ( 12
)
where the expressions of m 2 and h 2 (r) are given in Appendix A). If the curvature effect is taken into account for the annular cylindrical column, we obtain for the velocity V (r) an expression different from V 2 (r) noted V 3 (r) given by:

V 3 = Ra T (2R 2 ln(R) -2 ln(r)R 2 -R 2 + 2 ln(r) + 1) 2(R 2 -1) ln(R) (13)
Fig. 2 shows a good accuracy between velocities V 2 (r) and V 3 (r) although their expressions are different. We also consider that, for the cylindrical cell, the continuous sum of the mass fraction is conserved in the cylindrical cell and the total mass flux through any annular horizontal surface is equal to zero which lead us to the expression of mass fraction C 3 :

C 3 = m 3 z + h 3 (r) (14)
where the expressions for m and h 3 (r) are given in Appendix A.

Horizontal porous cell

In this paragraph, the main analytical results published in the article by Mojtabi et al. [START_REF] Mojtabi | Numerical and Analytical Studies of Soret-Driven Convection Flow Inside an Annular Horizontal Porous Cavity[END_REF] are reported in order to compare the importance of species separation in both horizontal and vertical TGC. Natural convection flow between two isothermal cylinders, maintained at constant and different temperatures T hot on the inner cylinder and T cold on the outer cylinder was characterized by a symmetrical flow with respect to the vertical plane containing the axis of the two cylinders. This flow is also invariant in any vertical plane orthogonal to the horizontal axis of the two cylinders. The binary fluid rises along the hot inner cylinder and descends along the cold outer cylinder. This convective regime is all the more stable as (R o -R i ) and (T hot -T cold ) are small, Cf.[ [START_REF] Mojtabi | Energy stability of a natural convective flow in a horizontal annular space[END_REF]]. The thermal field obtained in horizontal configuration is purely conductive. It is therefore identical to the one obtained in a vertical configuration, i.e.:

T = 1 - ln(r) ln(R) (15)
The analytical solution of this problem has been carried out in terms of the stream function, ϕ:

ϕ = Ra T sin(θ)F (r) ( 16
)
where F (r) is:

F (r) = R 2 (r 2 -1) ln(R) -r 2 (R 2 -1) ln(r) 2r(R 2 -1) ln(R) ( 17
)
Only the species balance equation has been solved numerically from a partial differential equation with the mass fraction, C, as the only unknown:

( ∂(r ∂C ∂r ) ∂r + 1 r ∂ 2 C ∂θ 2 ) = Le( ∂ϕ ∂r ∂C ∂θ - ∂ϕ ∂θ ∂C ∂r ) ( 18
)
4

Results and discussion

For a binary fluid with a positive thermodiffusion coefficient, D T > 0, the lightest component migrates to the top of the annular duct and the heaviest to the bottom of the cell (As seen in Fig. 8 below): the steady state is stable. However, for negative D T , it is the opposite situation: the steady state may be instable. This situation for D T < 0, has not been the subject of a stability analysis.

Comparison between analytical and numerical results

To illustrate the analytical and 2D and 3D numerical results obtained in this study, the authors only considered the experimental values of the thermophysics parametersthe water-ethanol solution (60.88% water -39.12% ethanol) studied by Platten et al. [START_REF] Platten | Enhanced molecular separation in inclined thermogravitational columns[END_REF]. The values of the thermophysical properties of this binary solution for an average temperature T = 22.5 • C are detailed in Table .1

. Table.2 shows the values of the coeffi-

Table 1 Properties of a water (60.88 wt%) -ethanol (39.12 wt%) mixture at a mean temperature of 22.5

• C. D[m 2 s -1 ] D T [m 2 (sK) -1 ] β C β T [K -1 ]
4.32 × 10 -10 1.37 × 10 -12 -0.212

7.86 × 10 -4 α[m 2 s -1 ] ρ0[kg m -3 ] ν[m 2 s -1 ]
2.23 × 10 -7 935.17 cients of thermodiffusion, D * T and mass diffusion, D * of the water-ethanol binary solution saturating the porous medium are detailed in Table 2. The two values of the permeability K 1 and K 2 of the porous medium and its porosity ϵ * are also reported.

Table 2 Properties of the porous medium at a mean temperature of 22.5

• C. D * [m 2 s -1 ] DT * [m 2 (sK) -1 ] ϵ * K1[m 2 ] K2[m 2 ]
1.878 × 10 -10 5.96 × 10 -13 0.47 6.58 × 10 -10 2.5 × 10 -11

The values of dimensionless numbers: thermal Rayleigh number Ra T , radius ratio R, Lewis number Le, solutal Rayleigh number R S and A used in this study, for a temperature difference of 10 • C, are detailed in Table 3.

For both configurations, temperature T is the same function of r. Fig. 4 shows the good agreement between the numerical (points) and analytical (solid line) results. The hypothesis leading to V = V (r)e z is per- However, for the horizontal cells, both radial and angular velocities are not equal to zero. The analytical results obtained for low values of R and moderate values of Ra T show that V r ≪ V θ . The tangential velocity V θ given in fig. 6 reaches the maximum at θ = π/2, when V r is equal to 0. Near the values, θ = 0, π, V r tends to its maximum.

Species separation and time to reach the steady state

The system of equations ( 3) with the boundary conditions ( 4) and ( 5), was solved numerically using two different softwares: Comsol Multiphysics with finite elements in 2D and 3D (fig. 8) and code written on MAT-LAB based on a method of spectral collocation, using Gauss Lobatto Chebyshev points. The flow in the vertical porous cell has a vertical symmetry axis e z . The flow in the horizontal porous annular cell is invariant under translation in any vertical plane perpendicular to the axis of the cell. The study was conducted with a porosity, ϵ * and two permeabilities, K 1 and K 2 . The binary fluid is a water (60.88 wt%)-ethanol (39.12 wt%) solution studied by Platten et al. [START_REF] Platten | Enhanced molecular separation in inclined thermogravitational columns[END_REF]. The values of the thermophysical properties of this binary solution are mentionned in Table 1 and Table 2. The imposed temperature difference δT and the permeability K in both vertical and horizontal cells were investigated. The analytical result of mass fraction at steady state is found numerically, fig. 7, without resorting to the simplifying assumptions used in paragraph 3.1. Species separation S is defined by: S = C max -C min . In all the calculations, the thickness is supposed to be constant: e = 6 × 10 -3 m. The respective influence on the species separation, of the internal radius R i , of the imposed temperature difference δT and of the permeability K in both vertical and horizontal cells, was presented in Table 4, 5, 6 and 7 where comparison between the two methods results are detailed in Table 4 and Table 5. It appears from these various results that the separation S = C max -C min : 1°) increases as the value of R i increases 2°) increases as δT increases 3°) decreases as K increases As already mentioned, the species separations in horizontal cells are always slightly greater than in vertical cells (for fixed values of R i and δT ).When K increases, the convective velocity increases and becomes higher than the velocity leading to the optimal separation. However the time to reach the steady state when K decreases, increases.Therefore a good compromise must be found between permeability and time to reach steady state. Furthermore, the influence of the value of the inner cylinder is slightly greater than the influence of the temperature difference, Table 5 and7. On fig. 9 and fig. 10, we present the variations of C max and C min as a function of time in order to study the time to reach the steady state. The study shows that the steady state was reached for a time of about 6 × 10 4 s for the vertical annular cell and about 12 × 10 4 s for the horizontal porous annular layer. Moreover the steady state was reached in the lower part of the horizontal annular cell two times more quickly than in the upper part (fig. 10). However, the steady state is reached simultaneously at the top and at the bottom of the vertical annular cell (fig. 9). These results obtained numerically only, fig. 9 and fig. 10, led us to look for the physical reason why the steady state in the vertical configuration is reached faster than in the horizontal configuration. It emerges from the analysis of figures ( 5) and ( 6) that the velocity V of the fluid along the vertical axis e z is of the same order of magnitude as the tangential velocity V θ calculated for θ = π/2 where it is maximum. As the time to reach the stationary state depends on the number of cycles performed by the fluid particles, we can deduce that the stationary state will be reached first in the vertical column. Fig. 9 shows the presence of a maximum and a minimum, in the vertical annular column, just before the stationary state is reached. This result has also been measured experimentally by Seta et al. [START_REF] Mojtabi | Analytical and numerical study of Soret mixed convection in two sided lid-driven horizontal cavity: Optimal species separation[END_REF] in a parallelepipedic vertical column. Indeed, the process of thermogravitational separation continues and stops as soon as the reverse process of pure mass diffusion balances the thermogravitational contribution to lead to the stationary state.

Verification of the hypothesis

C(1 -C) = C 0 (1 -C 0 )
In order to verify the validity of the hypothesis introduced in the mathematical formulation of the problem, direct numerical simulations were performed when

C(1 -C) ̸ = C 0 (1 -C 0 )
. Table 8 and Table 9 show that the hypothesis can no longer be adopted for R i > 0.3m because the difference between mass fraction values, for C(1 -C) ̸ = C 0 (1 -C 0 ), would be relatively important. For this reason, all the cases studies were limited to configurations where R i ≤ 0.3 m. The two previously mentioned configurations (fig. 11 and12) were considered. The difference between these two configurations is mainly due to the amount of species separated at the top or the bottom of the cells.The useful volume in the vertical cell where the denser component with C > 0.7 can be collected is obtained by: V ol 1 = e(0.1075)π(R i + R o ) = 8.2 × 10 -4 m 3 (cf. Fig. 13).

R i C(1 -C) ̸ = C 0 (1 -C 0 ) C(1 -C) = C 0 (1 -C 0 ) C max C min C max C min 0
R i C(1 -C) ̸ = C 0 (1 -C 0 ) C(1 -C) = C 0 (1 -C 0 ) C max C min C max C min 0.
In the horizontal annular cell, the volume where the denser component with C > 0.7 can be collected is approximately: V ol 2 = (π

(R 2 o -R 2 i ) 4
πR i ) = 12 × 10 -4 m 3 (cf. Fig. 14). The amount of species separated at the bottom or top of the cells is about one and a half times bigger for the horizontal annular cell than for the vertical annular cell. This paper focuses on the species separation in a vertical and in a horizontal porous cylindrical annular cell. The cells are filled with a porous medium saturated by a water-ethanol mixture (60.88% water). The thickness of the two cells is e = R o -R i ≪ R i where R i and R o are respectively the inner and outer radius of the cylinder. For the vertical thermogravitational column, we succeeded in finding the analytical solution of the thermal, velocity and mass fraction fields. Two alternative solutions were obtained in the case where e ≪ Ri. The first one (V 2 ) corresponds to the case where the annular geometry approaches the parallelipipedic thermogravitational column while the second one (V 3 ) takes into account the curvature of the annular geometry. A good agreement was observed between these two solutions. The governing equations and the associated boundary conditions were solved numerically using a finite element method with Comsol Multiphysics sofware. For R i = 0.005, 0.1, 0.2 and 0.3 m, e = 0.006 m, δT = 10 • C, K 2 = 2.5 × 10 -11 m 2 , the results were compared with those obtained by a spectral collocation method with Gauss-Lobatto-Chebyshev points. The results obtained by these two methods were in good agreement. The influence of R i and δT on the species separation were investigated as well as the permeability of the porous medium. For two cylindrical annular columns of the similar geometric dimensions, the amount of species separated from the same binary mixture is 1.5 higher in the horizontal column than in the vertical column. On the other hand the time necessary to reach the stationary state in the vertical column is almost twice as short as the one necessary to reach the stationary state in the horizontal column.

Fig. 1

 1 Fig. 1 Annular vertical and horizontal porous cells

Fig. 2

 2 Fig. 2 Comparison between V 2 (blue line) and V 3 (black dots) as a function of r for R = 1.06.

Fig. 3

 3 Fig. 3 Comparison between C 2 (blue line) and C 3 (black dots) as a function of z for R i = 0.1 m.

2 .

 2 716 × 10 -6

Fig. 4

 4 Fig. 4 Numerical (dots) and analytical (continuous line) dimensional temperature T as a function of r.

Fig. 5

 5 Fig. 5 Numerical(dots) and analytical (line) values of dimensional velocity V in vertical cell as a function of r for z = H/2.

Fig. 6

 6 Fig. 6 Numerical(dots) and analytical (line) values of dimensional angular velocity V θ in horizontal cell as a function of r for θ = π/2.

Fig. 7

 7 Fig. 7 Numerical (dots) and analytical values of mass fraction in vertical cell as a function of z for r = (R i + R o )/2, K 2 = 2.5 × 10 -11 m 2 and δT = 10 • C.

Fig. 8

 8 Fig. 8 Annular vertical and horizontal mass fraction fields in 3D porous cylindrical cells for K 2 = 2.5 × 10 -11 m 2 and R = 1.06.

Fig. 9 Fig. 10

 910 Fig. 9 Variation of C max (in red) and C min (in blue) as a function of time in vertical annular cell for: R i = 0.1 m e = 6 × 10 -3 m and δT = 10 • C for K 1 = 6.58 × 10 -10 m 2 .

Table 8

 8 Comparison of maximum and minimum mass fractions values in vertical annular cell between considering or not the hypothesis C(1 -C) = C 0 (1 -C 0 ) for R i = 0.05 m, 0.1 m, 0.2 m, 0.3 m and 0.4 m for fixed δT = 10 • C and for K 2 = 2.5 × 10 -11 m 2 .

Table 9

 9 Comparison of maximum and minimum mass fractions values in horizontal annular cell between considering or not the hypothesis C(1 -C) = C 0 (1 -C 0 ) for R i = 0.05 m, 0.1 m, 0.2 m, 0.3 m and 0.4 m for fixed δT = 10 • C and for K 2 = 2.5 × 10 -11 m 2 .

4. 4

 4 Mass fraction fields and streamlines in the two configurationsIn fig.11and fig.12, the steady state mass fraction field and the streamlines in the horizontal annular cell and in the vertical annular cell are presented. It can be observed for the horizontal configuration that the velocity field is almost tangential except near the azimuth angles θ = 0 or π. For the vertical configuration the velocity field is almost vertical except near the two extremities of the cell.

Fig. 11

 11 Fig. 11 Iso-C and streamlines in an annular horizontal cell*R i = 0.2 m, e = 6 × 10 -3 m and δT = 10 • C and K 2 = 2.5 × 10 -11 m 2 . *the scales are not respected in order to clearly visualize the mass fraction field and the streamlines.

Fig. 12

 12 Fig. 12 Iso-C and streamlines in an annular vertical cell*R i = 0.2 m, e = 6 × 10 -3 m and δT = 10 • C and K 2 = 2.5 × 10 -11 m 2 . *the scales are not respected in order to clearly visualize the mass fraction field and the streamlines.

4. 5

 5 Amount of species separated at the bottom and the top of the two cells

Fig. 13

 13 Fig. 13 Calculation of V ol 1 .

Fig. 14

 14 Fig. 14 Calculation of V ol 2

Table 3

 3 Dimensionless parameters for δT = 10 • C, for K 2 = 2.5 × 10 -11 m 2 .

	R	Ra T	R S	A	Le
	1.06 0.3182 -8.5844 0.0237 1187.4
	1.03 0.6365 -17.169 0.0118 1187.4

Table 4

 4 Comparison of maximum and minimum mass fractions values in vertical annular cell between spectral method and Comsol Multiphysics for R i = 0.05 m, 0.1 m, 0.2 m, 0.3 m for fixed δT = 10 • C and for K 2 = 2.5 × 10 -11 m 2 .

	R i	Comsol	Spectral method
		C max	C min	C max	C min
	0.05 0.6475 0.5719	0.646	0.572
	0.1	0.6804 0.5372	0.680	0.537
	0.2	0.7498 0.4678	0.750	0.468
	0.3	0.8191 0.3985	0.819	0.398

Table 5

 5 Comparison of maximum and minimum mass fractions values in horizontal annular cell between spectral method and Comsol Multiphysics for R i = 0.05 m, 0.1 m, 0.2 m, 0.3 m for fixed δT = 10 • C and for K 2 = 2.5 × 10 -11 m 2 .

	R i	Comsol	Spectral method
		C max	C min	C max	C min
	0.05 0.6508 0.5652	0.651	0.565
	0.1	0.6871 0.5273	0.687	0.527
	0.2	0.7595 0.4517	0.759	0.452
	0.3	0.8320 0.3762	0.832	0.376

Table 6

 6 Maximum and minimum mass fractions in vertical and horizontal annular cells for R i = 0.1 m, 0.2 m, and 0.3 m for fixed δT = 10 • C and for K 1 = 6.58 × 10 -10 m 2 .

	R i	Vertical cell	Horizontal cell
		C max	C min	C max	C min
	0.1 0.6133 0.6043 0.6236 0.5982
	0.2 0.6165 0.6010 0.6338 0.5898
	0.3 0.6200 0.5980 0.6444 0.5813

Table 7

 7 Maximum and minimum mass fractions in vertical and horizontal annular cells for δT = 10, 15, 20, 25 • C for R i = 0.1 m, K 2 = 2.5 × 10 -11 m 2 .

	δT	Vertical cell	Horizontal cell
		C max	C min	C max	C min
	10	0.6804 0.5372 0.6871 0.5273
	15	0.6900 0.5276 0.7134 0.5009
	20	0.6945 0.5231	07328	0.4822
	25	0.6971 0.5205 0.7483 0.4680

Acknowledgements This work was supported by CNES, the French National Space Agency.

Appendix A: Expression of m 2 , h 2 (r), m 3 and h 3 (r)


HAL is

m6A RNA modification as a new player in R-loop regulation

Aline Marnef and Gaëlle Legube * LBCMCP, Centre de Biologie Intégrative (CBI), CNRS, Université de Toulouse, UT3 * corresponding author: gaelle.legube@univ-tlse3.fr RNA:DNA hybrids that form across genomes control a wide range of biological processes.

A new study shows that N6-methyladenosine (m6A) modification on the RNA moiety of these hybrids regulates their formation and genome integrity. This opens up a new avenue of research where the RNA modifications (the "epitranscriptome") can help control genome maintenance. R-loops are tripartite structures that form co-transcriptionally, with the nascent RNA hybridizing back to the DNA template (RNA:DNA hybrid), leaving a displaced non-hybridized single-stranded DNA (ssDNA) (Fig. 1). Besides their prevalence in highly transcribed genes, R-loops also accumulate in repeats such as centromeres, telomeres and retrotransposons. Rloops are beneficial for various biological processes such as transcription initiation and termination, replication, telomere maintenance and immunoglobulin class switch recombination 1 . Yet, they are also potent inducers of DNA damage 2 and roadblocks for repair 3 , thereby making R-loops a source of genome instability. Tight regulation of R-loop levels is therefore critical. This is achieved by both preventing their co-transcriptional formation and favoring their removal via specific nucleases (e.g. RNAse H) and helicases (e.g. Senataxin, FANCM) 1,2 . In this issue, Abakir et al. propose an unexpected new regulatory pathway where N6-methyladenosine deposition on the RNA moiety of the RNA:DNA hybrid contributes to Rloop removal 4 .

N6-methyladenosine (or m6A) is thought to be the most prevalent internal mRNA modification in eukaryotes. This reversible mark plays key roles in the post-transcriptional control of gene expression by affecting nearly every steps of RNA metabolism such as splicing, export, translation, and degradation (Fig. 1). m6A is added co-transcriptionally by "writers" (e.g. METTL3), recognized by specific proteins called "readers" (e.g.: YTHDF1-3, YTHDC1-2) and removed by "erasers" (e.g. FTO) 5 . Importantly, the diverse regulatory functions of m6A are mainly determined by the identity of the m6A-bound reader. The list of physiological processes affected by m6A is extending and includes response to DNA damage 6 , virus infection, cell differentiation, brain development as well as cell growth and tumor progression 5,7,8 .

m6A regulates R-loop stability

In their study, Abakir et al. adapted the antibody-based DRIP technique (that immunoprecipitates RNA:DNA hybrids) to purify m6A-modified RNA:DNA hybrids with the available m6A antibody (m6A-DNA immunoprecipitation, or m6A-DIP) in human pluripotent stem cells 4 . m6A-DIP-seq and DRIP-seq datasets largely overlapped and displayed preferential accumulation on the same genomic features (transcribed regions, repeated sequences), suggesting that most RNA:DNA hybrids are m6A-modified. Importantly, DRIP followed by m6A-DIP-qPCR, also retrieved m6A-modified RNA:DNA hybrids, albeit less efficiently, further attesting that hybrids can be m6A-modified. Furthermore, a recent study by Yang et al. also corroborated the physical presence of m6A-modified RNA within R-loops even though their results rather indicate that only a subset of R-loops are m6A-modified 9 . The definitive answer regarding the prevalence of m6A RNA modification in RNA:DNA hybrids will therefore require the development of alternative reagents and strategies, since efficiency of the m6A-DIP-seq procedure relies greatly on the specificity of the antibody and includes a denaturation step which could in theory selectively affect the integrity of a subset of R-loops.

In this study, Abakir et al. also identified YTH-domain family member 2 (YTHDF2) to be the main reader of m6A-modified R-loops. One of their key observation is that YTHDF2 functions by preventing the accumulation of m6A-modified R-loops. It remains to be determined whether YTHDF2 destabilizes R-loop via the recruitment of known RNA:DNA hybrids nuclease/helicase or by specifically targeting the endoribonuclease RNaseP/MRP, which has been shown to promote the cytoplasmic decay of specific m6A-containing mRNAs 10 . This hypothesis is appealing since RNAseP depletion triggers replication stress and genome instability in flies 11 , two processes also involving R-loops. By contrast, Yang et al. recently proposed that instead of downregulating R-loop stability, m6A rather promotes R-loop formation at transcription termination sites 9 . The discrepancy between these studies calls for additional work to characterize in which instance m6A modifications destabilize or promote Rloop formation.

Consequences on genome stability, and beyond

One of the most remarkable finding of this study is that depletion of YTHDF2 and METTL3 (the writer that deposits m6A) triggered increased H2AX level, a marker of DNA doublestrand breaks, suggesting that pathological R-loops accumulation in the absence of the m6A RNA methylation pathway challenges genomic integrity. This is in line with a large number of studies that clearly points towards the potential of R-loops to induce DNA double-strand breaks 2 . Moreover, dysregulation of R-loops is emerging as a critical factor driving genome instability in a large variety of pathological contexts, including following oncogenic stress 12 , in Kaposi's sarcoma-associated herpesvirus infected cells, in neurological disorders associated with trinucleotide repeat expansion (such as Huntington's disease and Fragile X syndrome) and in multiples other inherited ataxias (e.g. ataxia with oculomotor apraxia 2) 2 . The use of existing drugs targeting the m6A pathway (e.g.: FTO inhibitors) could therefore be envisaged as a new therapeutic approach to treat R-loops-related diseases 13 .

Beyond genome stability, the finding that m6A methylation controls R-loop levels across the genome considerably expands the biological functions of the m6A modification pathway to potentially all R-loop-related functions. These include DNA topology, as R-loops were recently proposed to relieve superhelical stress, immunoglobulin class switch recombination (and therefore the immune response), replication initiation and transcription 1,2,14 . Additionally m6A also accumulates at sites of UV-induced DNA damage 6 , raising the interesting possibility that this modification also regulates R-loops during DNA repair and thus the frequency of chromosomal translocations 15 .

In summary, the results presented by Abakir et al. unveil an unexpected interplay between RNA modifications (the epitranscriptome) and the maintenance of genome integrity. Re-analyzing the pathological contexts implicating dysregulation of the m6A RNA pathway through the prism of genome instability will therefore warrant further investigation (Fig. 1). m6A is the most abundant internal modification of eukaryotic mRNA. This co-transcriptionally added epitranscriptomic mark plays multiple roles in mRNA metabolism and expectedly, m6A has been associated with a wide variety of pathologies. Notably, nascent RNA can also hybridize back to the DNA template to form the so-called "R-loop". Abakir et al. propose that R-loops are m6A-modified on their RNA moiety. The m6A-RNA reader YTHDF2 is recruited to destabilize RNA:DNA hybrids through a mechanism that remains to be determined (action of nucleases, helicases, RNaseP). The tight regulation of R-loops level is critical to ensure genome stability, and accordingly, stabilization of m6A-modified R-loops (upon YTHDF2 depletion) leads to increased DNA double-strand break formation. This broadens the previously acknowledged functions of m6A pathway in RNA metabolism to the maintenance of genome integrity and opens up new therapeutic strategies for R-loops related diseases.

Figure 1 :

 1 Figure 1: Expanding the functions of the m6A RNA pathway.