P Maugis

E Mouche

L Dewiere

INFLUENCE OF BOUNDARY CONDITIONS ON TRACER DISPERSION IN AN AQUIFER

Dispersivity is often assumed to be independent of the ow boundary conditions. As dispersion of a plume is a consequence of Darcy's velocity uctuations, boundary conditions should in uence this process and, possibly, a ect the dispersivities de ned far from the boundaries.

In this work, we analyze, by means of Monte-Carlo simulations, the in uence of di erent types of boundary conditions on ow and transport statistical quantities in a bidimensional aquifer. Results show that boundary conditions have a short range in uence on velocity mean and variances, and a long range in uence on transport. For instance, transverse dispersivity is very sensitive to the nature of lateral boundary conditions : for an imposed head the range of in uence is at least eight correlation lengths. On the basis of these results, dispersivity in bidimensional aquifers appears to be dependent of boundary conditions in a zone, along the boundaries, of rather large extent.

INTRODUCTION

Contaminant transport in heterogeneous aquifers is usually modeled by an advection di usion dispersion equation. The dispersivities are often assumed to be independent of the ow boundary conditions. Whatever the modeling approach, classical or stochastic, the dispersion process is a consequence of the Darcy velocity uctuations. Therefore, boundary conditions potentially in uence this process and, possibly, a ect the dispersivities de ned far from these boundaries.

Head boundaries set head uctuation to zero while ux boundaries annuls one component of Darcy's velocity. By constraining di erent variables, di erent type of boundary do not produce the same ows in heterogeneous media. For instance, a mean uniform ow can be generated by xed head up-and down-stream, and impervious lateral borders. These latter borders with xed linear head produce the same average ow, but are not impervious anymore. It is clear that transport of a plume or of a particle should also be a ected.

If this phenomenon is well understood and accepted, the question of its importance and its range in 2D-aquifers has been partially studied. Studies have shown analytically or by Monte-Carlo simulations that log-transmissivity-head cross-correlations C yh and head variogram hh are stationary beyond a few correlation lengths from the borders 7] 8] 6] 4], but head variance is strongly determined by the boundary conditions, all over the area they delimit, in a log-like variation 2] 4]. Head statistics can then never be considered stationary, except for very large areas. Incidence on velocity is then expected to have the same extent, but this topic is less richly described in the literature. Bellin 1] observed a very clear impact of xed-ux boundaries on velocity variance. This e ect, however, is located in the vicinity of the borders, no more than 3 away from them. The in uence of other types of boundary was not investigated, nor was the study extended to velocity correlations, which are the sources of dispersion. Therefore, it can not be concluded that there should be a "boundary free" zone where head, velocity or transport would behave, from the stochastic point of view, like in an in nite domain.

The purpose of this study is to understand and to investigate by Monte-Carlo simulations if and how dispersion in a uniform mean ow is a ected by boundaries, especially lateral boundaries. Simulations were made with the code CASTEM 2000. They rely on the resolution of the ow equations by Mixed-Hybrid Finite Element method 5]. Transmissivity realizations were generated by the matrix decomposition method for an exponential transmissivity correlation, and transport was simulated by particle tracking technique.

In this work, we analyze, by means of Monte-Carlo simulations, the in uence of di erent types of boundary conditions on ow and transport statistical quantities in a bidimensional aquifer. These are the means and variances of Darcy's velocity and particle position. The velocity rst two moments are rst analyzed analytically under the linearized model.

FLOW STATISTICS IN THE VICINITY OF BOUNDARIES 1.Theoretical framework

Let axis 1 be the direction of the uniform mean ow, and let us call Y the logtransmissivity, split into a mean <Y > and a random uctuation y. Y is supposed stationary so that the log-transmissivity correlation depends only on the separation r : C yy (r) =< y(x)y(x + r)>= exp( r= ). Head is noted H =<H> +h and Darcy's Velocity is written Ũ =< Ũ> +ũ.

There are four types of boundary : xed head or xed ux on lateral -parallel to the mean ow -or longitudinal -perpendicular to the mean ow -borders. In each case, only one variable uctuation is set to zero : either h = 0 or ũ ñ = 0 (where ñ is a unitary vector normal to the boundary) respectively.

The linearized model 3] gives expressions of Darcy's velocity mean and uctuation :

< Ũ>= T G h (1 + 2 Y =2) J0 + <y rh> + r <H (2) > i (1) ũ = T G h y J0 + rh i (2)
given the equations on rst order head uctuations h = H (1) and second order head H (2) : h = J0 ry with the proper boundary conditions

H (2) = rh ry ; H (2) equals zero on the boundaries (4)

The numbers in parenthesis are the y orders in the linearized development (written till order 2). The zeroth order head gradient has been written J0 , and the geometric mean of transmissivity T G . We de ne also the zeroth order mean velocity U 0 = T G J 0 . Each of these expressions takes di erent forms depending on the boundary conditions (from eq. ( 2)) : -a xed head longitudinal boundary sets h = 0 and @ 2 h = 0, so u 2 = 0, -a xed ux longitudinal boundary sets u 1 = 0, so @ 1 h = y J 0 , -a xed head lateral boundary sets h = 0 and @ 1 h = 0, so u 1 = y U 0 , -an impervious lateral boundary sets u 2 = 0, so @ 2 h = 0.

Mean Velocity

The analytical steps in the calculation of mean velocity are the determination of < y rh> from (3) and of r <H (2) > from (4), taking advantage of the symmetries generated by the boundaries. This has been made for lateral boundaries only. In the following, the origin is taken on the boundary itself.

Resolution of eq. ( 3) with head lateral boundary can be done using a composed Green function G(x; x0 ) = G 1 (x 1 ; x 2 ; x 0 1 ; x 0 2 ) G 1 (x 1 ; x 2 ; x 0 1 ; x 0 2 ), where G 1 is the Green function for the laplacian in an in nite 2D medium. This case can be described by a ctitious in nite domain where y would be anti-symmetrical in relation to the boundary. Multiplying (3) by h(x 0 ) and then taking ensemble mean gives :

<y(x)h( x0 )>= J 0 Z r2half plane G(x 0 ; x + r)@ r1 C yy (r)dr (5)
The particular symmetry of G associated to the parity of C yy (r) is such that this integral is null when x 1 = x 0 1 , whatever x 2 and x 0 2 , so that < y(x 1 ; x 2 ) @ x2 h(x 1 ; x 2 ) >= 0 too. Furthermore, due to the nature of the boundary, @ 1 h(x 1 ; 0) = 0. So : <y rh> (x 1 ; 0) = 0 (6) Moreover, H (2) being null on the boundary, @ x1 <H (2) >= 0. Mean longitudinal velocity in the vicinity of a head lateral boundary, expressed by (1), has then the following form : <U 1 > (x 1 ; 0) = (1 + 2 y = 2) U 0 (7) Mean velocity should then be increased by the proximity of head lateral boundaries, with a magnitude varying linearly with log-transmissivity variance. The range of the in uence, however, is not determined a priori.

Fixed-ux lateral boundary does not a ect mean velocity, because this situation corresponds to a ctitious in nite domain where y would be symmetrical in relation to the boundary. In this case, eq. ( 5) is identical to the expression of C yh in an unbounded domain. H (2) , <y rh> and nally < Ũ> are then unchanged.

Velocity variances

Velocity variances on boundaries are obtained directly from eq. (2). Some of them are expressed in relation to head derivative standard-deviation on the boundary. Those latter were not analytically calculated. It has been found that : -for xed head longitudinal boundary, u1 = T G @ x1 h (0; x 2 ) and u2 = 0 -for xed ux longitudinal boundary, u1 = 0 and u2 = y U 0 , -for impervious lateral boundary, u1 = T G @ x1 h (x 1 ; 0) and u2 = 0 -for head lateral boundary, u1 = y U 0 and u2 = T G @ x2 h (x 1 ; 0)

Monte-Carlo simulations

Four values of y have been investigated : 0.1, 0.5, 1.0 and 1.5. The number of realizations necessary to reach a satisfactory convergence were made. The grid re nement is =4 on a 21 13 wide area. Simulations have been performed for the two di erent lateral boundaries with xed head on the in-ow and the out-ow borders. Results are summarized in table 1 1: E ect and range of boundaries in a uniform ow Detailed values are not shown, but the simulated increase of <U 1 > in the vicinity of a xed-head lateral boundary is exactly the one predicted by eq. ( 7) except for y =1.5 (the linearized model usually requires y < 1). The t is not as good for longitudinal velocity variance increase near the same boundary. An overestimation of no more than 25% is observed. The results show that velocity variability tends to zero where expected, and increase signi cantly otherwise. Symmetrically, studies of Dagan 2] and Osnes 4] showed how head uctuations tend to zero on xed-head boundaries but increase near impervious lines. In any case, the range of the in uence of the boundaries on mean and variances of velocity does not exceed a few correlation lengths. To this point, nothing can be said on how, and how far, can velocity correlations be a ected by the boundaries.

INFLUENCE OF LATERAL BOUNDARIES ON TRANSPORT IN A UNIFORM FLOW

The in uence of the type of lateral boundaries on transport has been studied by Monte-Carlo simulations. We investigate here the particle displacement variances in a uniform ow generated in a 17 square medium discretized =4. From 800 up to 2000 realizations were performed for each value of y investigated. Figure 1: Normalized displacement variances functions of dimensionless abscissa s= for di erent boundary conditions : longitudinal head (thin) or ux (bold) conditions, and lateral head (small dashes) or ux (long dashes) boundaries.

We present the position variances as functions of mean travel distance s, of a particle released 3 from the up-stream border, on the symmetry axis of the medium. Results are quite similar for the four values of y . For the sake of clarity, X 11 is presented with y =1, and X 22 with y =0.1 (See Figure 2). Longitudinal dispersion seems not to be a ected by lateral boundaries, nor by head longitudinal boundaries ; but xed-ux longitudinal boundaries induce a signi cant, though light, decrease of X 11 . On the other hand, transverse dispersion is clearly dependent on the nature of all four boundaries. Lateral ux boundaries increase X 22 while lateral head boundaries lower it. After a travel distance of only 7 , the plume is no more than 0:3 wide, so still about 8 away from the lateral borders, but the di erence has already got up to 20%. Moreover, X 22 seems extremely sensitive to longitudinal ux boundaries.

These results suggest that velocity correlations are a ected on a very long range (more than 10 like other not included calculations showed), while velocity variances are not affected on more than a few . Thus, velocity correlation lengths for longitudinal separations seem to be a ected by the boundaries on long distances. Within this assumption, transverse velocity correlation length would decrease in the vicinity of a head lateral or ux longitudinal boundary, and increase in the vicinity of a ux lateral boundary. The explanation according to which e ective log-transmissivity correlation length doubles near a xed-ux boundary due to the ctitious symmetry of Y relatively to it, is not pertinent. If the boundary is parallel to the mean ow, then only correlation length for transverse separations is concerned ; and if the boundary is orthogonal to the mean ow, an increase of longitudinal position variance should be observed, which is not the case. The same remark applies to head boundaries. So the observed in uence of boundaries on particle position variances remains unexplained, yet indubitable.

It is commonly accepted that boundaries a ect ow and transport in their vicinity. That is why every modeler chooses, somewhat arbitrarily or on the strength of studies not plainly devoted to the topic, an area that he thinks is "boundary free". It has been showed here that boundaries determine ow and transport in a very wide area, at least 10 logtransmissivity correlation lengths away from them. For instance, with lateral boundaries 8 away from the release point, itself 3 from the in-ow boundary, the di erences in position variances go up to 30% after only 7 traveled. Longitudinal boundaries exert the more important in uence on transport. The in uence of transverse boundaries is also certain, though limited to transverse dispersion. This could not be expected on the basis of velocity mean and variances, which reach their in nite-domain values after a short distance. Velocity correlation lengths could be responsible of the e ect of boundaries on transport, by being a ected on very long ranges. Other explanations could be suggested, like highorder e ect not included in the linearized model used here. More analytical research are necessary to understand the long-range e ect of boundaries on transport.

From a practical point of view, it must be understood that, unless boundaries are very far apart, they take a full part in the modelization and should be subject to a careful choice. Real eld boundaries should be preferred, and if not available, boundary e ects should be kept in mind when interpreting results. Finally, the nature of boundaries might be of critical importance in other types of ow, like single or multi-pumping, where the nature ( xed draw-downs or xed pumping rates) of the wells must be chosen.

Table

  .

	Type of boundary Lateral, head Lateral, ux Long., head Range for all types E ect on h ! 0 high ! 0 everywhere
	E ect on <U 1 > E ect on u1 E ect on u2	(1 + 2 y = 2) strong increase strong increase	slight decrease -! 0	-strong increase ! 0	4 3 3


Valerie Hardouin Duparc

Rama M Shakaroun

Martine Slawinski

Jean-Francois Carpentier
email: jean-francois.carpentier@univ-rennes1.fr

Sophie M Guillaume
email: sophie.guillaume@univ-rennes1.fr

Valérie Hardouin Duparc

Martine Slawinsky

Jean-François Carpentier

Ring-opening (co)polymerization of six-membered substituted delta-valerolactones with alkali metal alkoxides

Keywords: δ-hexalactone, δ-valerolactone, ring-opening polymerization (ROP), copolymer, alkali metal catalyst

come   L'archive ouverte pluridisciplinaire

Introduction

Ring-opening polymerization (ROP) of cyclic esters (lactones, lactides or cyclic carbonates) is a commonly used method to synthesize polyesters with controlled molar masses, narrow dispersities and tailored macromolecular structures. [START_REF] Dove | Organic Catalysis for Polymerization[END_REF][START_REF] Hua | Macromolecular architectures through organocatalysis[END_REF][START_REF] Ottou | Update and challenges in organo-mediated polymerization reactions[END_REF]4,5,6,7,8,9,10,11,12] Homo-and co-polyesters have shown great potential for biomedical applications and as materials for the plastic industry as alternatives to regular polyolefins. [13,14,15] Metal-and organo-based ROP catalysts need to be appropriately chosen for each specific monomer type to achieve the best compromise between the rate and the control of the polymerization. The use of a bio-sourced lactone monomer, associated with a simple and cheap catalytic polymerization method, adds significantly to the interest in the corresponding polyester-based materials due to their renewability and sustainability.

Among the large variety of cyclic esters that have been investigated in ROP, the -Me substituted δ-valerolactone, often referred to as -hexalactone (HL) (Figure 1), is a biosourced sixmembered monomer that can be derived from pineapple and used as a food additive, [16,17] which has been less commonly explored then the related ubiquitous (unsubstituted) δ-valerolactone (VL).

The ROP of HL using "lanthanum trisisopropoxide" was reported in 2002; performed in dichloromethane/toluene (70:30 v/v) solution at 21 °C, it did not reach full conversion even after a long reaction time (77% in 18 h) and returned oligomers (M̅ n,SEC = 5200 g.mol 1 ) with a moderate control over molar mass distribution (Đ M = 1.3).[18] A better activity was reported recently with strontium bisisopropoxide as catalyst/initiator using n-dodecanol as co-initiator. [19] In toluene solution at 23 °C, poly(HL)s (PHLs) were obtained within short reaction times (82% conv. in 15 min, M̅ n,SEC < 5000 g.mol 1 ) and with a fair dispersity (Đ M = 1.161.30); noteworthy, the ROP hardly proceeded in the absence of the co-initiator (6% conv. in 15 min). Furthermore, the ROP of HL using the diphenyl phosphate (DPP) organic acid was reported independently in 2015-2016 by Hadjichristidis and Hillmyer.[20,21] At high monomer concentration ([HL] 0 = 3.5 M) and room temperature in toluene, DPP allowed a chain extension ROP with a high HL conversion (89%), a controlled molar mass (M̅ n,NMR = 11,400 g.mol 1 vs. M̅ n,theo = 10,900 g.mol 1 ) and a narrow dispersity (Đ M = 1.09) of the resulting PHL end-capped with DPP. [20] Thermodynamic investigations also revealed that HL shows, in bulk or in solution, a much higher reactivity (polymerization rate) and a slightly higher equilibrium conversion than analogous 5-alkyl-δvalerolactones with longer alkyl substituents (i.e., 5-butyl-δ-lactone and 5-pentyl-δ-valerolactone, also known as δ-nonalactone and δ-decalactone, respectively). [20,21,22] Besides, the ROP of HL (H°p = 19.3±0.5 kJ.mol 1 ; S°p = 62±2 J.mol 1 K 1 ) was shown to be more exothermic and entropically unfavored as compared to VL (H°p = 10.5 kJ.mol 1 ; S°p = 15 J.mol 1 K 1 ). [18,21,22] The ring-opening copolymerization (ROCOP) of HL with various comonomers has also been investigated. [23,24,25] The simultaneous copolymerization of HL with its structural isomer ε-caprolactone (CL) catalyzed by DPP at 100 °C in bulk can serve as an effective method to regulate the properties of poly(CL) (PCL); it typically returns random copolymers with reduced crystallinity and enhanced enzymatic degradability with HL content as low as 5mol%.

[23] Also, the ROCOP of HL with ethylene brassylate (EB) using BiPh 3 as catalyst gave copolyesters incorporating up to 90mol% of EB (bulk monomers, 3 to 6 days, 130 °C), with reduced crystallization capability, higher degradation rate, improved flexibility and good thermal stability in comparison to PCL.[24] Likewise, a new kind of low glass transition random copolyesters that showed a decreased crystalline fraction, a lower stiffness, and an improved biodegradability in comparison to PCL, was developed by copolymerization of ω-pentadecalactone (PDL) with HL using also BiPh 3 as catalyst under the same conditions.[25] The PDL content ranged from 39 to 82 mol%, and the resulting materials revealed non cytotoxic, and thus suitable for potential use in the biomedical field.

In addition, the related disubstituted δ-valerolactones, such as 2-ethylidene-6-hepten-5olide (EVL) and 2-ethylheptane-5-olide (EHO) (Figure 1), which can be readily produced by a telomerization reaction of CO 2 with 1,3-butadiene -an economical large-volume chemical that can also be derived from top biomass platform chemicals  and subsequent hydrogenation, respectively, have attracted a growing interest as new potential renewable monomers. [26,27,28,29,30] EVL was even produced at a miniscale plant.[31] However, the very few literature reports on the polymerizability of EVL indicate that, so far, only the radical homopolymerization of this monomer is operative. When initiated by 2,2′-azobis(isobutyronitrile) (AIBN) in the presence of ZnCl 2 (100 °C, 24 h), the polymerization provides a butadiene/CO 2 copolymer incorporating small amounts of alkenes (M̅ n,SEC up to 85,0000 g.mol 1 , Đ M = 1.11.5). [32,33,34,35] Alternatively, simple additive-/solvent-free heating of EVL in the presence of O 2 gave a full monomer conversion, thereby providing a convenient avenue toward CO 2 -based polymers. [36] Herein, we report new results on the polymerizability of these little-studied 6-membered δ-valerolactones, namely HL, EVL, EHO, and MH in comparison to VL. Various readily available organic and metal-based catalytic systems have been investigated for the (co)polymerization of these substituted δ-valerolactones.

Experimental section

Materials

All catalytic experiments were performed under an inert argon atmosphere using standard Schlenk line and glovebox techniques. KOtBu (Strem Chemicals, 98%), La(N(TMS) 2 ) 3 (Sigma Aldrich, 98%) and Y(OTf) 3 (Sigma Aldrich, 98%) were used as received. Benzyl alcohol (Acros, 98%) was distilled over Mg turnings under argon atmosphere and stored over 3-4 Å activated molecular sieves. NaOMe, NaOtBu and KOMe were prepared from the corresponding metal and alcohol, respectively.

2-tert-Butylimino-2-diethylamino-1,3-dimethylperhydro-1,
(CH 3 CH 2 CHCO).

Synthesis of 2-ethylidene-6-heptan-5-olide (MH).

In a typical procedure, EVL (1.50 g, 10.0 mmol), THF (20 mL) and Ni-Raney (75 mg) were added in a 50 mL stainless steel autoclave under argon. 20 bar of H 2 was introduced in the reactor and the suspension was stirred with a magnetic bar at room temperature for 1 h. After depressurization of the reactor, volatiles were removed under vacuum and the crude product was purified by column chromatography (Hexanes/AcOEt 4:1 v/v) to recover the desired product (MH) as a colorless liquid (1.15 g, 75%).

ESI-MS: 155.11 ([M+H] +

). 1 H NMR (CDCl 3 , 400 MHz, 23 °C) (Figure S14): δ 7.12 (q, 3 J = 8.0

Hz, 1H, CH General procedure for the ring-opening (co)polymerization of six-membered -valerolactones. In a typical experiment, a Schlenk flask placed under argon was charged with the Na (pre)catalyst/initiator (NaOMe, 4.75 mg, 0.088 mmol, 1 equiv.) (Table 2, entry 14). Then, the monomer (HL, 100 mg, 0.88 mmol, 10 equiv.) was added in using a syringe and the reaction mixture was magnetically stirred at the desired temperature using an oil bath. After a given period (reaction times were not systematically optimized), the reaction was quenched upon addition of acetic acid (a few drops of a 6 M toluene solution), and volatiles were removed under vacuum at low temperature (for the residual monomer not to evaporate). After determination of the monomer conversion by 1 H NMR analysis of the crude product, when some polymer was formed, the resulting crude residue was dissolved in CH 2 Cl 2 (0.5 mL) and the polymer was precipitated in cold hexane (1 mL, 0 °C), filtered and dried under vacuum. The recovered polymer was then analyzed by NMR and SEC. OMe: 1 H NMR (CDCl 3 ,400 MHz,23 °C) (Figure S1): δ 4.93 (m,1H

H-PHL-

Results and Discussion

Homopolymerization of HL

The polymerization performances of different ubiquitous catalytic systems regularly employed with success in the ROP of lactones were investigated towards HL (Scheme 1, Table 1).

Preliminary experiments, all performed in bulk (i.e., neat, solvent-free) conditions with the aim to enhance reaction rates and to reach high monomer conversions, established the complete inability of sebacic and decanoic Brönsted acids to produce polymers at temperatures as high as 230 °C (Table 1, entries 12).[37] Only a poor activity was observed with BEMP, a usually most effective ROP organocatalyst, especially with monomers that are reluctant to polymerize such as -lactones or -valerolactones: [START_REF] Ottou | Update and challenges in organo-mediated polymerization reactions[END_REF]38,39,40,41] at 60 °C, HL homopolymerization only reached 15% conversion in 2 h, without any further reaction overnight (Table 1, entries 34), while no reaction proceeded at room temperature. Similarly, reactions performed at room temperature using the strong Lewis acidic "Y(OTf) 3 " combined with benzyl alcohol (BnOH) as co-initiator, a catalytic system effective in the ROP of even touchy cyclic esters such as -lactones, [42,43] failed to consume any monomer (Table 1,entry 5). At 60 °C, progress of the reaction remained slow with only 27% conversion after 72 h. At 100 °C also, the reaction reached a plateau around 30% HL conversion (Table 1, entries 69), possibly suggesting a polymerization/depolymerization equilibrium. [18,44] The La(N(TMS) 2 ) 3 /BnOH catalytic system, which proved quite effective even towards ROP-reluctant 5-membered -lactones and at quite low temperatures to avoid depolymerisation, [45,46,47] showed also no activity towards HL at room temperature (Table 1, entry 10).[48] ROP of HL with this catalytic system proceeded at best, yet slowly, at 60 °C, giving a 44% maximum conversion after 72 h (Table 1, entries 11 and 12). At 100 °C, a plateau was reached within less than 2 h at ca. 30% HL conversion (Table 1, entries 13 and 14), a value comparable to that observed with "Y(OTf) 3 " under similar conditions. Raising further the temperature up to 160 °C proved ineffective, as no monomer consumption was then detected (Table Reaction temp.

[°C]

Reaction time [b] [h] HL conv. [c] [%] [a] All reactions were performed in bulk conditions (neat monomer). [b] Reaction times were not necessarily optimized. [c] As determined by 1 H NMR analysis of the crude reaction mixture.

In order to reach higher conversion of HL with an straightforward process and a cheap and easily available catalyst, different alkali metal alkoxides were then investigated as ROP initiator. [49,50,51,52,53,54] The activity of KOtBu (10 mol%) was first evaluated with HL, at room or higher temperature, in bulk or in toluene solution (Table 2, entries 110). In toluene solution, the HL conversion reached a plateau at ca. 30%, both at room temperature and at 60 °C (Table 2, entries 14). Under bulk conditions, as anticipated, the HL conversion reached higher values, which were slightly (and negatively) affected by the temperature: 78% at room temperature, 68% at 60 °C, and 6164% at 100 °C (Table 2, entries 510). Interestingly, the ROP of HL using NaOMe proved more efficient, as full conversions could be achieved under different conditions (Table 2, entries 1117). The reaction was rather sluggish at room temperature and required 10 mol% to reach full monomer conversion. Raising the temperature to 60 or 100 °C successfully enabled to convert > 90% of HL within 2 h with 10 mol% of NaOMe (Table 2, entries 1315) and within 18 h with "only" 3 mol% (Table 2, entries 1821).

Table 2. ROP of HL mediated by alkali metal alkoxides.

Entry Catalyst [HL] 0 /[Cat.] 0 [HL] 0 [a] Reaction temp.

[°C]

Reaction time [b] [h] HL conv. [c] [%] [a] All reactions were performed with [HL] 0 = 1 M in toluene or in bulk (neat monomer, no solvent). [b] Reaction times were not necessarily optimized. [c] As determined by 1 H NMR analysis of the crude reaction mixture.

To explore the influence of the alkali metal salt and of the alkoxide initiator on the reaction rate, comparative experiments were conducted with NaOtBu and KOMe (Table 2, entries 2225), while KOMe proved less effective than NaOMe (Table 2, entries 11,12 vs. 24,25). NaOtBu proved somehow more effective than KOtBu (Table 2, entries 5,6 vs. 22,23). Overall, the best reactivity was obtained with sodium salts and with the methoxide group. This can be tentatively rationalized by a higher Lewis acidity of the cationic metal center (Na + vs. K + ) and the lower steric hindrance of the initiator (OMe vs. OtBu).

NaOMe, as the most efficient initiator for the ROP of HL, was hence selected to further investigate the influence of the polymerization temperature and of the [HL] 0 /[NaOMe] 0 ratio on the ROP of HL (Table 3); in parallel, the microstructure and molar mass control of the PHL formed were studied. As revealed by 1 H and 13 C{ 1 H} NMR spectroscopy (Fig. S1S2) and as further supported by MALDI-ToF mass spectrometry (MS) analyses (Fig. S3), polymers with a major distribution of macromolecules end-capped by a methoxycarbonyl and a hydroxyl groups m/z calcd = 725.4088) was also observed. Hydrolytic work-up with acetic acid (refer to the Experimental section), adventitious presence of traces of water during the reaction or hydrolysis of cyclic oligomers with water under MS conditions, are possible explanations for the observation of these hydroxy end-capped polymers.

(
Table 3. Characteristics of PHL obtained from the ROP of HL initiated by NaOMe.

Entry

[HL] 0 / [NaOMe] 0 [a] Reaction temp.

[°C] HL conv. [b] [%]

,theo [a] All reactions were performed in bulk conditions (neat monomer) over 18 h; the reaction time was not optimized. [b] Experimental molar mas as determined by 1 H NMR analysis of the crude reaction mixture. [c] Theoretical molar mass calculated from the relation: [HL] 0 / [NaOMe] 0 × Conv. δ-HL × M δ-HL + M MeOH (M HL = 114 g.mol 1 ; M MeOH = 32 g.mol 1 ). [d] Experimental molar mass value determined by 1 H NMR analysis of the isolated polymer, from 1 H NMR resonances of the methoxy terminal group. [e] Experimental, uncorrected number-average molar mass value determined by SEC in THF at 20 °C versus polystyrene standards. [f] Dispersity (Đ M = M̅ w /M̅ n ) as determined by SEC using PS standards.

[c] M n [g mol 1 ] NMR [d] M n , [g mol 1 ] SEC [e] M n , [g mol 1 ] Đ M [f]
All HL oligomers produced using 1-5 mol% of NaOMe as initiator at room temperature (Table 3, entries 4, 6 and9) showed a fairly good match between the molar mass values measured by NMR analysis (on the basis of the methoxy chain end-group observed at  3.68 ppm) and the corresponding theoretical values calculated on the basis of HL conversion (M̅ n,theo ); noteworthy, the PHL produced at 21 °C with 10 mol% of NaOMe (Table 3, entry 1) showed a somewhat higher value than the theoretical one, possibly pointing out an incomplete initiation efficiency. Also, the dispersity values of all the polymers produced at room temperature were moderately narrow (Đ M = 1.17-1.21) hinting at some limited transesterification side reactions. Raising the temperature to 60 °C resulted, as anticipated, in slightly higher dispersity values (Đ M = 1.22-1.47). The experimental M̅ n,NMR values of PHL produced at this temperature with 5 or 10 mol% of initiator were in line with the ones calculated (Table 3, entries 2,5), but distinctly lower than the ones calculated when lower amounts of initiator were used (1 and 3 mol%; Table 3, entries 7,10). These trends were even more pronounced at 100 °C: the dispersity broadened in the range 1.35-2.03 and the M̅ n,NMR values, although in good line when using 3-10 mol% of initiator (Table 3, entries 3,8), significantly dropped below the theoretical value when only 1 mol% of NaOMe was used (Table 3, entry 11). Overall, these observations indicate a relatively fast and quantitative initiation and limited undesirable side reactions when operating at a relatively low temperature (21-60°C) with 1-5 mol% of NaOMe. On the other hand, only partial initiation takes place when excessive loading of NaOMe (10 mol%) is used and detrimental side reactions (especially transesterification i.e.

reshuffling, backbiting)  typical of cyclic esters ROP reactions , occur to a significant extent at 100 °C. Typically, oligomers with M̅ n,NMR < 3800 g.mol 1 were recovered under these operating conditions.

Copolymerization of HL with VL

Random ring-opening copolymerization (ROCOP) of HL with its unsubstituted parent monomer, VL, was targeted to explore the influence of the monomer substituent on the control of the polymerization and the relative monomer reactivity (Scheme 2). The reactions were performed using various monomer-to-initiator ratios under the above optimized conditions, namely at 60 °C with 1 or 2.5 mol% NaOMe, over a non-optimized 24 h time period (Table 4). Three different monomers feeds were used (i.e.

[HL] 0 /[VL] 0 = 1:1; 1:3 and 3:1), coupled to two different catalytic loadings ([NaOMe] 0 = 1 or 2.5 mol%). In each case, high conversion of both comonomers was observed, as monitored by 1 H NMR analysis: 72-87% for HL and 91-98% for VL, the latter lactone which revealed more reactive than its substituted analogue whatever the conditions, as expected from their respective homopolymerization behavior. As revealed by 1 H and 13 C NMR spectroscopy (Fig. S4,S6,S8), the random copolymers displayed, besides the repeating unit typical signals, a methoxycarbonyl chain-end group observed at ca. δ 3.65 ppm. The P(HLco-VL) molar mass values recorded up to M̅ n,NMR = 9700 g.mol 1 (Table 4) typically remained lower than those of the HL/CL, HL/PDL or HL/EB copolymers. [23,24,25] The copolymers recovered from the experiments conducted with 2.5 mol% of NaOMe showed M̅ n,NMR values that matched well the M̅ n,theo data based on HL and VL conversion, no matter of the initial ratio of comonomers (Table 4, entries 13). This was not necessarily the case in the polymerizations performed with 1 mol% of NaOMe ratio, with the aim to increase the copolymer molar masses (Table 4, entries 4-6). Indeed, with an equimolar mixture of comonomers or with an excess of HL, the copolymerization process was not well-controlled, that is M̅ n,NMR values << M̅ n,theo ; on the other hand, a copolymer with a predictable molar mass was obtained when using an excess of VL, the monomer most readily to polymerize. These results indicate that an excess of HL leads to a loss of control of the polymerization at a catalytic loading of 1 mol%. Long polymeric chains do not seem to be accessible for HL with this catalyst system.

[55] Chain-transfer or cyclization side-reactions could explain the reduced experimental average molar mass of the final copolymer. HL conv. [b] [%] VL conv. [b] [%]

,theo [a] Reactions conditions: bulk (neat monomers), 60 °C, 24 h (reaction time was not optimized). [b] As determined by 1 H NMR analysis of the crude reaction mixture. [c] Theoretical molar mass value calculated from the relation:

[c] M n [g mol 1 ] NMR [d] M n , [g mol 1 ] SEC [e] M n , [g mol 1 ] Đ M [f]
([HL] 0 /[NaOMe] 0 × Conv. HL × M HL ) + ([VL] 0 /[NaOMe] 0 × Conv. VL × M VL ) + M MeOH (M HL = 114 g.mol 1 ; M VL = 100 g.mol 1 ; M MeOH = 32 g.mol 1
). [d] Experimental molar mass value determined by 1 H NMR analysis of the isolated polymer, from 1 H resonances of the methoxy terminal group.

[e] Experimental, uncorrected number-average molar mass value determined by SEC in THF at 20 °C versus polystyrene standards. [f] Dispersity (M̅ w /M̅ n ) as determined by SEC.

Attempted ROP of disubstituted δ-valerolactones

Ethylidene-6-hepten-5-olide (EVL) is an interesting disubstituted δ-valerolactone that can be efficiently obtained from the telomerization of butadiene with CO 2 via palladium catalysis (selectivity over 90%) (Scheme 3). [26,27,28,29,30] As briefly reported by Behr et al.,[27] its hydrogenation, also performed by Pd catalysis, leads readily to 2-ethylheptane-5-olide (EHO); in our hands, this latter saturated disubstituted δ-valerolactone is typically recovered as a diastereomeric mixture (63:37, as determined by 13 C{ 1 H} NMR analysis using the methine signal at  40 ppm ; refer to the Experimental section and Fig. S12,S13). While optimizing the latter EHO synthesis, we observed that the previously unreported disubstituted, semi-hydrogenated δ-valerolactone MH can also be selectively and straightforwardly synthetized in high yields (up to 75%) using either Ni-Raney, Ru/C, Crabtree's catalyst or Wilkinson's catalyst under mild conditions (Table S1; Fig. S14,S15). We were therefore interested in exploring whether these six-membered disubstituted -lactones obtained from sustainable chemical reactants and processes can undergo ROP to access original polyester materials.

Disappointingly, all our attempts to ring-open polymerize EHO and MH with different catalytic systems (Chart S1) under a variety of reaction conditions, failed. None of these catalysts was effective in the 20180 °C temperature range, the monomer remains unreacted. [48] This may be due to a low ceiling temperature of EHO and MH. Note also that no polymerization occurred when using the La(N(TMS) 2 ) 3 /BnOH catalytic system effective towards ROP-reluctant 5membered -lactones (40 °C, toluene, 18 h). [45,46,47] Similarly, the attempted ring-opening copolymerization of EVL, MH or EHO with VL or HL all remained unsuccessful, even in the presence of an excess of VL or HL. These results are reminiscent of EVL for which no successful ROP has been reported in the literature to date.

Conclusion

Typical examples of Brönsted acids, organocatalysts or Lewis acidic metal complexes remain poorly active in the ROP of HL under a range of reaction conditions. On the other hand, alkali metal alkoxides successfully homopolymerize HL under optimized mild operating conditions (1 mol% NaOMe, solvent-free, 60 °C, 18 h). In particular, the readily available sodium methoxide promoted the ROP of HL to afford PHL oligomers (M̅ n , NMR up to ca. 4000 g.mol -1 , Ð M < 1.36). The results showed the occurrence of some -yet limited-transesterification side-reactions.

Further copolymerization of HL with the unsubstituted alike VL mediated by NaOMe showed a good-to-high conversion (72-98%) of both comonomers, the latter unsubstituted lactone showing a slightly higher reactivity. Random copolymers with a predictable molar mass (M̅ n,NMR up to ca. 10,000 g.mol -1 , Ð M < 1.40) were thus recovered. Overall, the formation of high molar mass copolymers of HL/VL seems to be impeded by chain-transfer and/or cyclization side-reactions. This is the first report of the successful RO(CO)P of HL promoted by the readily available sodium methoxide. Finally, HL showed a significantly better reactivity than the analogous 1,5-alkyl 6membered δ-valerolactones EHO, EVL and MH, which unfortunately remained inaccessible to ROP under the experimental conditions we investigated and regardless of the catalyst/initiator used. These results shine new lights on the ring-opening polymerizability of substituted δ-valerolactones, of which the alkyl substituent on the 1-and/or 5-position on the cyclic ester ring play a major detrimental role in their ROP.

Figure 1 .

 1 Figure 1. Six-membered ring (substituted) -valerolactones explored in the present study.

Scheme 2 .

 2 Scheme 2. Simultaneous ring-opening copolymerization of HL with VL.

Scheme 3 .

 3 Scheme 3. Access to the disubstituted δ-valerolactones EVL, EHO and MH, from CO 2 and butadiene and subsequent (semi)hydrogenation.

Figure S1. 1 HFigure S3 .

 1S3 Figure S1. 1 H NMR spectrum (CDCl 3 , 400 MHz, 23 °C) of a PHL sample obtained from the ROP of HL using NaOMe at 60 °C (Table 3, entry 3).

Figure S4. 1 HFigure S6. 1 H

 11 Figure S4. 1 H NMR spectrum (CDCl 3 , 400 MHz, 23 °C) of a P(VL-co-HL) sample obtained from the ROCOP of VL and HL (1:1) using NaOMe at 60 °C (Table4, entry 1). *stands for residual monomers trapped in the oligomers.

Figure S8. 1 H

 1 Figure S8. 1 H NMR spectrum (CDCl 3 , 400 MHz, 23 °C) of a P(VL-co-HL) sample obtained from the ROCOP of VL and HL (3:1) using NaOMe at 60 °C (Table 4, entry 3). *stands for residual monomers trapped in the oligomers

Figure S10. 1 H

 1 Figure S10.1 H NMR spectrum (CDCl 3 , 400 MHz, 23 °C) of EVL(2-ethylidene-6-hepten-5-olide).

Figure S11 .

 S11 Figure S11. 13 C{ 1 H} NMR spectrum (CDCl 3 , 100 MHz, 23 °C) of EVL (2-ethylidene-6-hepten-5-olide).

Figure S12. 1 H

 1 Figure S12.1 H NMR spectrum (CDCl 3 , 400 MHz, 23 °C) of EHO (2-ethylheptane-5-olide) (as a 63:37 diastereomeric mixture).

Figure S13 .

 S13 Figure S13. 13 C{ 1 H} NMR spectrum (CDCl 3 , 100 MHz, 23 °C) of EHO (2-ethylheptane-5-olide) (as a 63:37 diastereomeric mixture).

Figure S14. 1 H

 1 Figure S14.1 H NMR spectrum (CDCl 3 , 400 MHz, 23 °C) of MH(2-ethylidene-6-heptan-5-olide).

Figure S15 .

 S15 Figure S15. 13 C{ 1 H} NMR spectrum (CDCl 3 , 100 MHz, 23 °C) of MH (2-ethylidene-6-heptan-5-olide).

  Various catalysts assessed in the attempted ROP of EHO (all proved unsuccessful).

Synthesis of 2-ethylheptane-5-olide (EHO). EVL

  CH 3 CH 2 CHCH 2 CH 2 CHCH 2 CH 3 ), 1.01 (m, 6H, CH 3 ). 13 C{ 1 H} NMR (CDCl 3 , 100 MHz, 23 °C) (Figure S13) (data in parentheses refer to the second minor diastereomer): δ 175.69 (173.84) (COO), 79.29 (82.53) (COOCH), 39.70 (42.10) (CHCOO), 28.29 (29.15) (CH 3 CH 2 CHCO), 26.24

	Instrumentation and measurements	(1.50 g, 10.0 mmol), THF (20 mL) and
	1 H (400 MHz) and 13 C{ 1 H} (100 MHz) NMR spectra were recorded on Bruker Avance AM Ascend Pd/C (37.5 mg) were added in a 50 mL stainless steel autoclave under argon. 20 bar of H 2 was
	400 spectrometer in CDCl 3 at 25 ˚C, and referenced internally relative to SiMe 4 (δ 0 ppm) using introduced in the reactor and then the suspension was stirred magnetically at 60 °C for 1 h. After
	the residual solvent resonances. depressurization of the reactor at room temperature, volatiles were removed under vacuum and the
	crude product was purified by fractional distillation to recover 2-ethylheptane-5-olide (EHO) as a
	Number-average molar mass (M̅ n,SEC ) and dispersity (Đ M = M̅ w /M̅ n ) values of the colorless liquid (1.30 g, 85%). ESI-MS: 157.12 ([M+H] + ). 1 H NMR (CDCl 3 , 400 MHz, 23 °C)
	(co)polymers were determined by size exclusion chromatography (SEC) in THF at 30 ˚C (flow rate = 1.0 mL min 1 ) on a Polymer Laboratories PL50 apparatus equipped with a refractive index (Figure S12): δ 4.22 (m, 1H, CHCOO), 2.39 (m, 1H, COOCH), 1.52.2 (m, 8H,
	detector and a set of two ResiPore PLgel 3 μm MIXED-D 300 × 7.5 mm columns. The polymer
	samples were dissolved in THF (2 mg mL 1 ). All elution curves were calibrated with polystyrene
	standards; the reported M̅ n,SEC values of the copolymers are uncorrected for the difference in
	hydrodynamic radius vs. polystyrene.
	Monomer conversions were calculated from 1 H NMR spectra of the crude polymer samples
	in CDCl 3 by using the integration (Int.) ratio Int. Polymer /[Int. Polymer + Int. Monomer ] of the COO-CH 2
	methylene (δ PVL 4.11 ppm, δ VL 4.37 ppm) or COO-CH methine (δ PHL 4.93 ppm, δ HL 4.45 ppm), in
	the polymer and residual monomer, respectively. The molar mass values of copolymer samples
	were determined by 1 H NMR analysis in CDCl 3 , from the relative intensities of the signals of the
	main-chain methylene OC(O)CH 2 ( CH2 2.30 ppm) relative to the chain-end OMe signals ( CH3
	3.68 ppm).
	Mass spectra were recorded at CRMPO-ScanMAT (Rennes, France). ESI mass spectra
	were recorded on an orbitrap type Thermo Fisher Scientific Q-Exactive instrument with an ESI
	source in positive or negative mode by direct introduction at 5-10 µg mL -1 . Samples were prepared
	3,2-diazaphosphorine in CH 2 Cl 2 at 10 µg mL -1 . High resolution MALDI-ToF mass spectra were recorded using an
	(BEMP; Aldrich, 98%), δ-valerolactone (VL; TCI, 98%) and 5-Me substituted δ-valerolactone (δ-ULTRAFLEX III TOF/TOF spectrometer (Bruker Daltonik Gmbh, Bremen, Germany) in positive
	hexalactone (HL); Alfa Aesar, 98%) were distilled from CaH 2 and stored under argon. 2-and/or negative ionization mode. Spectra were recorded using reflectron mode and an accelerating
	Ethylidene-6-hepten-5-olide (EVL) was synthesized from 1,3-butadiene and CO 2 according to the voltage of 25 kV. A mixture of a freshly prepared solution of the polymer in THF or CH 2 Cl 2 (HPLC
	literature procedure, and similarly distilled from CaH 2 and stored under argon.[26,27,28,29,30] grade, 10 mg mL -1 ) and DCTB (trans-2-(3-(4-tert-butylphenyl)-2methyl-2-propenylidene)
	malononitrile, and a MeOH solution of the cationizing agent (NaI, 10 mg mL -1 ) were prepared.
	The solutions were combined in a 1:1:1 v/v/v ratio of matrix-to-sample-to-cationizing agent -if
	added. The resulting solution (0.25-0.5 mL) was deposited onto the sample target (Prespotted
	AnchorChip PAC II 384 / 96 HCCA) and air or vacuum dried.

(24.87) (CH 2 CH), 23.85 (22.88) (CHCH 2 ), 11.59 (11.05) (CH 3 CH 2 CHO), 9.58 (9.24)

(HL-co-VL)-OMe: 1

  , COOCH), 3.68 (s, OMe terminal group), 2.30 (m, 2H, CHCH 2 ), 1.64 (3, 4H, CHCH 2 CH 2 CH 2 ), 1.42 (d, 3H, J = 7 Hz, CH 3). 13 C{ 1 H} NMR (CDCl 3 , 100 MHz, 23 °C) (Figure S2): δ 172.91 (COO), 70.41 (COOCH), 51.55 (OMe terminal group), 35.24 (CHCH 2 CH 2 CH 2 ), 34.21 (CHCH 2 CH 2 CH 2 ), 20.84 (CHCH 2 CH 2 CH 2 ), 19.91 (CH 3 ). H NMR (CDCl 3 , 400 MHz, 23 °C) (Figures S4,S6,S8): δ 4.89 (COOCH HL), 4.09 (COOCH 2 VL), 3.69 (s, OMe terminal group), 2.35 (CHCH 2 HL and COOCH 2 CH 2 VL), 1.69 (COOCH 2 CH 2 CH 2 HL and VL), 1.22 (CH 3 HL). 13 C{ 1 H} NMR (CDCl 3 , 100 MHz, 23 °C) (Figures S5,S7,S9): δ 173.23 (COO VL), 172.83 (COO HL), 70.38 (COOCH HL), 63.90 (COOCH 2 VL), 35.23 (CHCH 2 CH 2 CH 2 HL), 34.03 (CHCH 2 CH 2 CH 2 HL), 33.66 (COOCH 2 CH 2 VL), 28.08 (COOCH 2 CH 2 CH 2 VL), 21.42 (COOCH 2 CH 2 CH 2 CH 2 VL), 20.77 (CHCH 2 CH 2 CH 2 HL), 19.00 (CH 3 HL).

	H-P

Table 1 .

 1 ROP of HL mediated by organo-or metal-based catalyst systems.

	1, entry 15). These observations also suggest a possible polymerization/depolymerization
	issue.[18]
	O	O	bulk or solution catalyst/initiator	H	O	O n
	HL			PHL
	Scheme 1. Ring-opening polymerization of HL towards PHL.
	Entry (Pre)catalyst	[HL] 0 /[Cat.] 0 /[BnOH] 0 [a]

Table 4 .

 4 Simultaneous ring-opening copolymerization of HL and VL.

	Entry	[HL] [NaOMe] 0 /[VL] 0 / 0 [a]

Table 3 ,

 3 entry 2). +, X and * refer to methoxy-terminated chains (C 6 H 10 O 2 ) n (CH 4 O), hydroxy-terminated chains (C 6 H 10 O 2 )

n (H 2 O), and cyclic oligomers (C 6 H 10 O 2 ) n , respectively.

(C

6 H 10 O 2 ) n (C 6 H 10 O 2 ) n (H 2 O) (C 6 H 10 O 2 ) n (CH 4 O)

4

Acknowledgements

Financial support of this research by Total TPRF (support to V. H.D.) is gratefully acknowledged.

Data availability

The raw/processed data required to reproduce these findings cannot be shared at this time due to legal or ethical reasons.

Supporting Information

Figure S1. 1 H NMR spectrum (CDCl 3 , 400 MHz, 23 °C) of a PHL sample obtained from the ROP of HL using NaOMe at 60 °C (Table 3,entry 3).

Figure S2

. 13 C{ 1 H} NMR spectrum (CDCl 3 , 100 MHz, 23 °C) of a PHL sample obtained from the ROP of HL using NaOMe at 60 °C (Table 3,entry 3).

Figure S3

. MALDI-ToF mass spectrum of a PHL sample obtained from the ROP of HL using NaOMe as initiator (Table 3, entry 2).

Highlights

The ring-opening polymerization (ROP) of some original bio-derived six-membered (substituted) δ -valerolactones is investigated Alkali metal alkoxides successfully homopolymerize for the first time the 5-Me substituted δvalerolactone (aka δ-hexalactone (HL)) Well-defined PHL oligomers and P(HL-co-δ-valerolactone) random copolymers are thus recovered HL shows a significantly better reactivity than the analogous 1,5-alkyl 6-membered δ valerolactones

The alkyl substituent at the 1-and/or 5-position on the cyclic ester ring significantly affects its ring-opening polymerizability of substituted δ-valerolactones

The authors declare no conflict of interest.


M Guedda

Z Hammouch
email: zakia.hammouch@u-picardie.fr

On similarity and pseudo-similarity solutions of Falkner-Skan boundary layers

Keywords: Similarity solutions, Pseudo-similarity solutions, Falkner-Skan problem, Boundary Layer, Stretching surfaces PACS numbers: 47.15, 47.27 Te

The present work deals with the two-dimensional incompressible, laminar, steady-state boundary layer equations.

First, we determine a family of velocity distributions outside the boundary layer such that these problems may have similarity solutions. Then, we examen in detail new exact solutions, called Pseudo-similarity, where the external velocity varies inversely-linear with the distance along the surface (ue(x) = u∞x -1 ). The analysis shows that solutions exist only for a lateral suction. For specified conditions, we establish the existence of an infinite number of solutions, including monotonic solutions and solutions which oscillate an infinite number of times and tend to a certain limit. The properties of solutions depend on the suction parameter. Furthermore, making use of the fourth-order Runge-Kutta scheme together with the shooting method, numerical solutions are obtained.

Introduction

In this paper we are concerned with the classical two-dimensional laminar incompressible boundary layer flow past a wedge or a flat plate [START_REF] Schlichting | Boundary layer theory, 8th revised and Enlarged Edition[END_REF]. For the first approximation, the model is described by the Prandlt equations or the boundary layer equations u∂ x u + v∂ y u = u e ∂ x u e + ν∂ 2 yy u, ∂ x u + ∂ y v = 0, (1.1) where (x, y) denote the usual orthogonal Cartesian coordinates parallel and normal to the boundary y = 0 (the wall), u and v are the corresponding velocity components, and the constant ν > 0 is the kinematic-viscosity. The function u e is a given exterior streaming velocity flow which is assumed throughout the paper to be nonnegative function of the single variable x; u e = u e (x), and is such that u(x, y) tends to u e (x) as y → ∞. Equations (1.1) can be written in the form This problem with appropriate external velocity flow has been the main focus of studies of particular exact solutions. Research on this subject has a long history, which dates to the pioneering works by Blasius [START_REF] Blasius | Grenzschichten in Flüssigkeiten mit kleiner Reibung[END_REF] and Falkner and Skan [START_REF] Falkner V | Solutions of the boundary layer equations[END_REF] in which the external velocity is given by u e (x) = u ∞ x m (u ∞ > 0). (1.3) Their investigations lead to solutions to (1.2) in the form ψ(x, y) = ax α f (byx -β ), a, b > 0, (1.4) where

α = m + 1 2 and β = 1 -m 2 . (1.5)
The equation for f is the well-known equation obtained by Falkner-Skan [START_REF] Falkner V | Solutions of the boundary layer equations[END_REF]

f ′′′ + m + 1 2 f f ′′ + m(1 -f ′ 2 ) = 0 on (0, ∞), (1.6) or, if m > -1, f ′′′ + f f ′′ + σ(1 -f ′ 2 ) = 0 on (0, ∞), (1.7) where σ = 2m m + 1 .
Such equations occur at wedge flows [28, p. 170]. These equations have received considerable attention in the literature. We refer the reader to the works of Rosenhead [START_REF] Rosenhead | Laminar Boundary layers[END_REF], Schlichting and Gersten [START_REF] Schlichting | Boundary layer theory, 8th revised and Enlarged Edition[END_REF], Weyl [START_REF] Weyl | On the differential equations of the simplest boundary-layer problems[END_REF] and Coppel [START_REF] Coppel | On a differential equation of boundary layer theory[END_REF] and the references therein. Note that from (1.4) it is easily verified that,

ψ(x, y) = x x 0 α ψ x 0 , y x x 0 -β .
This means that a solution ψ(x, y) for y fixed is similar to the solution ψ(x 0 , y) at a certain x 0 . This solution is called invariant or similarity solution and the function f is called the shape function or the dimensionless stream function.

The broad goals of this paper is to study equation (1.7) when taking the limit σ → -∞. This limit case, corresponding to m = -1, is that of flow in a two-dimensional divergent channel (or sink flow) [28, p. 170]. We prove that transformation (1.4) is much too restrictive; that is problem (1.1) has no similarity solution and we shall see, by rigouros arguments, that the term γlog(x) must be added to the expression (1.4) and the surface must be permeable with suction to obtain new exact solutions which are not similarity. In addition, we shall prove that there is an infinite number of solutions. These results are given in Section 3. Before this analysis, we shall identify, in Section 2, external flows, such that problem (1.2) may have similarity solutions. The main result of this section indicates that problem (1.2) has solutions under the form (1.4) if the external flow is of the power-law type (1.3).

Similarity solutions

In this section we shall obtain external flows such that the partial differential equation (1.2) accompanied by the boundary condition

lim y→∞ ∂ y ψ(x, y) = ∂ y ψ(x, ∞) = u e (x), (2.1)
has a solution under the form (1.4), where α + β = 1.

The main problems, arising in the study of similarity solutions, are related to the existence of the exponents α and β and to the rigorous study of the ordinary differential equation satisfied by the profile f which is, in general, nonlinear. For the layer equation (1.2), the classical approach for identifying α and β is the scaling and transformation group [START_REF] Barenblatt | Scaling, selfsimilar, and intermediate asymptotics[END_REF]. The essential idea is to seek a and b such that if ψ satisfies (1.2) the new function ψ κ (x, y) = κ a ψ(κ b x, κy) is also a solution. Let ψ be a stream-function to (1.2) defined by (1.4) where α + β = 1. Assume first that β = 0. We choose a = -α β , b = 1 β , and define ψ κ (x, y) = κ a ψ κ b x, κy . Hence a + b = 1, ψ ≡ ψ κ and

L(ψ κ )(x, y) = κ a+3 L(ψ)(κ b x, κy)
for any κ > 0, where L is the operator defined by

L(ψ) = ∂ y ψ∂ 2 xy ψ -∂ x ψ∂ 2 yy ψ -ν∂ 3 yyy ψ.
According to equation (1.2) we deduce

h(x) = κ a+3 h(κ b x),
where h(x) = u e (x)∂ x u e (x). In particular, for fixed

x 0 > 0 h(κ b x 0 ) = κ -(a+3) h(x 0 ). Setting x = κ b x 0 we infer h(x) = x -a+3 b x a+3 b 0 h(x 0 ).
Solving the equation

u e du e dx = x -a+3 b x a+3 b 0 h(x 0 ) (2.2) yields us u 2 e (x) = c 1 x 2m + c 2 , (2.3)
for all x > 0, where m = α -β and c 1 and c 2 are constants, for β = 0, since

-a+3 b + 1 = 2(α -β). For β = 0, hence α = m = 1, the new function ψ κ (x, y) = κ a ψ(κ -a x, y),
for any fixed a = 0, is equivalent to ψ and satisfies

L(ψ κ )(x, y) = κ a L(ψ)(κ -a x, y)
for any κ > 0. Arguing as in the case β = 0 one arrives at (2.3) with m = 1. Next, because lim y→∞ x 2m f ′ (yx -β ) 2 = c 1 x 2m + c 2 , the function f ′ 2 has a finite limit at infinity, which is unique and is given by c 1 + c 2 x -2m . This is acceptable only for c 2 = 0.

The above result indicates, in particular, that for a prescribed external velocity satisfying (2.3) the real numbers α and β are given by (1.5) and condition (2.3) is necessary and sufficient to obtain similarity solution under the form (1.4) where α + β = 1. However, for a general external velocity, it is possible to obtain an exact solution which is not similarity solution as it is seen in [START_REF] Ludlow | New similarity solutions of the unsteady incompressible boundary-layer equations[END_REF]. In this paper the authors considered

u e (x) = c 1 x 1/3 + c 2 x -1/3 .
A stream-function ψ is given by

ψ(x, y) = x 2/3 f (yx -1/3 ) + cx -1 3 y, c = const, (2.4)
where f is a solution of

νf ′′′ + αf f ′′ -(α -β)f ′ 2 = ξ, ξ = const. (2.5)

The pseudo-similarity solutions

In the present section we restrict the attention to the case m = -1 and get new solutions to the problem

∂ y ψ∂ 2 xy ψ -∂ x ψ∂ 2 yy ψ = ν∂ 3 yyy ψ -u 2 ∞ x -3 , (3.1)
subject to the boundary conditions

∂ y ψ(x, 0) = u w x -1 , ∂ x ψ(x, 0) = -v w x -1 , ∂ y ψ(x, ∞) = u ∞ x -1 , (3.2)
where v w is a real number ( v w > 0 for suction and v w < 0 for injection), u w and u ∞ are nonnegative and satisfy u w < u ∞ . The velocity distribution u e = u∞

x is found in the case of divergent channel (or sink flow) [28, p. 170]. The analysis of this section is motivated by the work by Magyari, Pop and Keller [START_REF] Magyari | The missing similarity boundary-layer flow over a moving plane[END_REF] concerning a boundary-layer flow, over a permeable continuous plane surface, where the x-component velocity tends to zero for y large (u ∞ = 0). In [START_REF] Magyari | The missing similarity boundary-layer flow over a moving plane[END_REF] the authors showed that if m = -1 problem (1.2),(2.1) has no solution in the usual form (1.4). For u ∞ = 0 and according to Section 2, the function ψ can be written as

ψ(x, y) = √ νu ∞ f u ∞ ν yx -1 . Since v w = m+1 2 √
νu ∞ f (0) (see the Appendix) we deduce v w = 0 for m = -1 and the following oridinary differential equation for f

   f ′′′ + f ′ 2 -1 = 0, f ′ (0) = ζ, f ′ (∞) = 1, (3.3) where ζ = uw
u∞ is in the interval [0, 1) and f (0) can be any real number. Since this problem does not contain f it is convenient to study the equation satisfied by θ = f ′ ; that is

   θ ′′ + θ 2 -1 = 0, θ(0) = ζ, θ(∞) = 1, (3.4)
The stability of equilibrium point (1, 0) of (3.4) cannot be determined from the linearization. To analyze the behavior of the nonlinear equation (3.4) 1 , we observe that

E ′ (t) = 0,
where E is the Liapunov function given by

E(t) = 1 2 θ ′ (t) 2 + 1 3 θ(t) 3 -θ(t).
Then, for some constant c, the following

θ ′ = ± √ 2 c + θ - 1 3 θ 3 1/2 ,
holds. The analysis of the algebraic equation of the phase path in the phase plane reveals that the equilibrium point (1, 0) is a center. Hence, Problem (3.4) has no solution for any ζ > -1 except the trivial one θ = 1 (see Fig. 3.1).

Note that if we impose the condition θ(∞) = -1 instead of θ(∞) = 1, which is also of physical interest, it is easy to see that, for any ζ ≤ 2, there exists a unique solution up to translation. This solution satisfies

1 2 θ ′ (t) 2 + 1 3 θ(t) 3 -θ(t) = 2 3 ,
and we find that To obtain exact solutions to (3.1),(3.2), we look for "pseudo-similarity " solutions under the form ψ(x, y) = aF (x, byx -1 ), (3.5) where a = √ νu ∞ and b = u∞ ν . Assuming F (x, t) = f (t) + H(x) one sees that (cf. Appendix)

θ(t) = 2 -3 tanh 2 ±t/ √ 2 + arctanh (2 -ζ)/3) 1/2 . (1,0) (-1,0) θ θ ′ (1,0) (-1,0) -2 -1.5 -1 -0.5 0 0.5 1
   f ′′′ + γf ′′ + f ′ 2 -1 = 0, f ′ (0) = ζ, f ′ (∞) = 1, (3.6)
and

H(x) = γ log x + c,
where γ = v w (νu ∞ ) -1/2 and c is constant. Without loss of generality we may take c = 0, since

ψ(x, y) = √ νu ∞ f (t) + v w log(x) + √ νu ∞ c satisfies (3.1) -(3.
2) for any real number c. Note that γ plays the role of suction/injection parameter.

To study (3.6) it is more convenience to consider the second ordinary differential equation

   θ ′′ + γθ ′ + θ 2 -1 = 0, θ(0) = ζ, θ(∞) = 1, (3.7)
where 0 ≤ ζ < 1 and γ = 0. In fact, the real number γ will be taken in (0, ∞). The existence of solutions to (3.7) will be proved by means of shooting method. Hence, the boundary condition at infinity is replaced by the condition θ ′ (0) = d, where d is a real number. For any d the new initialvalue problem has a unique local solution θ d defined in the maximal interval of the existence, say (0, T d ), T d ≤ ∞. We shall see that for an appropriate d the solution θ d is global and satisfies

θ d (∞) = 1. (3.8)
A simple analysis in the phase plane shows that problem (3.6) may have solutions only for γ > 0. In fact, the ordinary differential equation in (3.6) is considered as a nonlinear autonomous system in R 2 , with the unknown (θ, θ ′ ), mainly

   θ ′ = ϕ, ϕ ′ = -γϕ + 1 -θ 2 , (3.9)
subject to the boundary condition

θ(0) = ζ, ϕ(0) = d. (3.10)
The linear part of the above system at (1, 0) is the matrix

J =   0 1 -2 -γ   .
The eigenvalues of J are

λ 1 = -γ -γ 2 -8 2 , λ 2 = -γ + γ 2 -8 2 , if γ ≥ √ 8 and for |γ| < √ 8, λ 3 = -γ -i 8 -γ 2 2 , λ 4 = -γ + i 8 -γ 2 2 .
Therefore, the hyperbolic equilibrium point (1, 0) is asymptotically stable if γ is positive and unstable for negative γ. In particular problem (3.5)

d 2 ≤ 2ζ(1 - ζ 2 3
), (3.11) where 0 ≤ ζ ≤ √ 3. We shall see that any local solution of (3.9),(3.10) is global and satisfies (3.8). To this end we consider again the Liapunov function E(θ(t), ϕ(t)) = 1 2 ϕ(t) 2 + 1 3 θ 3 -θ. Along an orbit we have

d dt E(θ(t), ϕ(t)) = -γϕ(t) 2 ≤ 0. Hence E(θ d (t), θ ′ d (t)) ≤ E(ζ, d
), for any t < T d . On the other hand, from (3.9) and (3.10), there exists t 0 > 0, small, such that θ d is positive on (0, t 0 ). Assume that θ d vanishes at some t 1 > t 0 and suppose that θ ′ d (t 1 ) = 0. Because

E(ζ, d) ≥ E(θ d (t), θ ′ d (t)) ≥ 1 2 θ ′ d (t 1 ) 2 , for all 0 ≤ t ≤ t 1 . we deduce 1 2 d 2 > ζ(1 -1 3 ζ 2 )
, which contradicts (3.11). Therefore θ ′ d (t 1 ) = 0. In this case we deduce from the equation of θ d that θ ′′ d (t 1 ) = 1 and then θ d is nonnegative on a some neighbourhood of t 1 . Consequently the local solution is nonnegative as long as there exists. To show that θ d is global we note that

E(ζ, d) ≥ 1 2 θ ′ d (t) 2 + 1 3 θ 3 d (t) -θ d (t) ≥ - 2 3
, for all t ≤ T d , since θ d is nonnegative. Hence θ d and (then) θ ′ d are bounded. Consequently θ d is global. It remains to show that θ goes to unity at infinity. To this end we use the Bendixon Criterion. Let T be the trajectory of (θ d , θ ′ d ) in the phase plane (0, ∞) × R for t ≥ 0 and let Γ + (T ) be its w-limit set at infinity. From the boudedness of T it follows that Γ + (T ) is a nonempty connected and compact subset of (0, ∞) × R ( see, for example [1, p 226]). Moreover (-1, 0) ∈ Γ + (T ), since θ d is nonnegative. Note that if Γ + (T ) contains the equilibrium point (1, 0) then Γ + (T ) = {(1, 0)} , since (1, 0) is asymptotically stable. Assume that (1, 0) ∈ Γ + (T ), Applying the Poincaré-Bendixon Theorem [14, p 44] we deduce that Γ + (T ) is a cycle, surrounding (1, 0). To finish, we shall prove the nonexistence of such a cycle. We define

P (θ, ϕ) = ϕ, Q(θ, ϕ) = -γϕ + 1 -θ 2 , ϕ = θ ′ d and θ = θ d .
The function (θ, ϕ) satisfies the system θ ′ = P (θ, ϕ), ϕ ′ = Q(θ, ϕ). Let D be the bounded domain of the (θ, ϕ)-plane with boundary Γ + . As P and Q are regular we deduce, via the Green-Riemann Theorem,

D (∂ ϕ Q + ∂ θ P ) dϕdθ = Γ + (Qdθ -P dϕ) = 0, (3.12
) thanks to the system satisfied by (θ, ϕ). But ∂ ϕ Q + ∂ θ P = γ which is positive. We get a contradiction. To complete our analysis, we shall determine a basin of the critical point (1, 0). Let

P = (ζ, d) ∈ R 2 : ζ > -1, 1 2 d 2 + ζ 1 3 ζ 2 -1 < 2 3 .
Let us consider a one-parameter of family of curves defined by

E(θ, ϕ) = 1 2 ϕ 2 + 1 3 θ 3 -θ = C,
where C is a real parameter. Note that, in the phase plane, this family is solution curves of system (3.4). The curve ϕ 2 = 2θ -2 3 θ 3 + 4 3 , corresponding to C = 2 3 , goes through the point (2, 0) and has the saddle (-1, 0) (γ = 0) as its α and w-limit sets. We note this solution curve by H, which is, in fact, an homoclinic orbit and define a separatrix cycle for (3.4). We shall see that the bounded open domain with the boundary H is an attractor set for (1, 0) of system (3.9) where γ > 0. This domain is given by E(θ, ϕ) = C, θ > -1, for all -2 3 ≤ C < 2 3 , which is P. As d dt E ≤ 0 any solution, with initial data in P cannot leave P. By LaSalle invariance principle we deduce that for any (ζ, d) in P the w-limit set, Γ + (ζ, d) is a nonempty, connected subset of P ∩ {ϕ = 0} , (see [1, p. 234]). However, if θ = 1, ϕ = 0 is a transversal of the phase-flow, so the w-limit set is {(1, 0)} . This means that P is a basin of the critical point (1, 0) of (3.9).

Numerical results

In this section numerical solutions of the boundary-value problem (3.7) are obtained by using the fourth-order Runge-Kutta scheme with the shooting method. We plot the dimensionless velocity θ in term of the similarity variable t, for various value of the shooting parameter d.

Conclusion

In this work the laminar two-dimensional steady incompressible, boundary layer flow past a moving plane is considered. It has been shown that the problem has solutions having a similarity form if the velocity distribution outside the boundary layer is proportional to x m , for some real number m. In the second part of this paper, we are interested in question of existence of solutions in the case where the external velocity is the inverse-linear function; m = -1. This situation occurs in the case of sink flow. To obtain exact solutions the stream function ψ is written under the form

ψ(x, y) = √ νu ∞ f (t) + v w log(x). (5.1)
It is shown that the ordinary differential equation satisfied by f has multiple solutions for any v w positive and no solution can exist if v w ≤ 0. A sufficient condition for the existence is derived:

ζ > -1, 1 2 f ′′ (0) 2 + ζ ζ 2 3 -1 < 2 3 . (5.2)
We have obtained two family of solutions according to γ = v w (µu ∞ ) -1/2 . If γ ≥ √ 8, f ′ is monotonic and goes to 1 at infinity, but if 0 < γ < √ 8, we have a stable spiral. The function f ′ oscillates an infinite number of times and goes to 1. So if we are interested in solutions to (3.7) such that

-1 < f ′ < 1 we must take u w , v w and u ∞ > 0 satisfying -u ∞ < u w < u ∞ and v w > (8νu ∞ ) 1/2 .
Condition (4.1) indicates also that for the same positive value of the suction parameter the permeable wall stretching with velocity u w x -1 , u w > 0 has multiple boundary-layer flows. Every flow is uniquely determined by the dimensionless skin friction f ′′ (0) which can be any real number in the interval

-4 3 + 2ζ (1 -ζ 2 /3), 4 3 + 2ζ (1 -ζ 2 /3) , where ζ = u w u -1 ∞ .
The case u ∞ = 0 was considered by Magyari, Pop and Keller [START_REF] Magyari | The missing similarity boundary-layer flow over a moving plane[END_REF]. The authors showed, by numerical solutions, that the boundary layer flow exists only for a large suction parameter (γ ≥ 1.079131). The existence of exact solutions of the Falkner-Skan equation under the present condition was discussed by Rosenhead [27, who mentioned that these results may be obtained by rigorous arguments which, in fact, motivated the present work. We note, in passing, that it is possible to obtain solutions if the the skin friction satisfies |f ′′ (0)| > 4 3 + 2ζ (1 -ζ 2 /3).

where the primes denote partial differential with respect to t = u∞ ν yx m-1 2 . By writing F (x, t) = f (t) + H(x), we find

         f ′′′ + 1+m 2 f f ′′ -m f ′ 2 -1 + f ′′ xH ′ + 1+m 2 H = 0, 1 + m 2 f (0) + 1 + m 2 H(x) + xH ′ (x) = v w √ νu ∞ , f ′ (0) = ζ ∈ [0, 1), f ′ (∞) = 1.
(5.7)

Hence, there exists a real number γ such that

                 f ′′′ + 1+m 2 f f ′′ -m f ′ 2 -1 + γf ′′ = 0, t > 0, xH ′ + 1+m 2 H = γ, x > 0, 1 + m 2 f (0) + γ = v w √ νu ∞ , f ′ (0) = ζ, f ′ (∞) = 1.
(5.8)

First, let us assume that m = -1. The solution H is given by

H(x) = cx -1+m 2 + 2γ 1 + m ,
where c is a constant, and then ψ(x, y) = ac + ax

∂ y ψ∂ 2

 2 xy ψ -∂ x ψ∂ 2 yy ψ = u e ∂ x u e + ν∂ 3 yyy ψ, (1.2)where ψ is the well-known stream function defined by u = ∂ y ψ, v = -∂ x ψ.

Fig. 3 . 1

 31 Fig. 3.1 Classification of solutions of θ ′′ + θ 2-1 = 0 according to θ(0) and θ ′ (0).

Fig. 3 . 2 A

 32 Fig. 3.2 A basin of attraction P of the critical point (1, 0).

Fig. 4 . 1 Fig. 4 . 2

 4142 Fig. 4.1 Velocity profiles in terms of d = θ ′ (0) for fixed ζ = 0.2 and γ = 0.5

1+m 2 ff

 2 (t) + 2γ 1+m . The new function g = f + 2γ 1+msatisfies the Falkner-Skan equation. Thereafter, we will assume that m = -1 and this leads to′′′ + γf ′′ + f ′ 2 -1 = 0, f ′ (0) = ζ, f ′ (∞) = 1, γ = vw ) = γ log x + c, c = const.

  has no nontrivial solutions if γ < 0. If γ > 0 we deduce from the above that there exists δ > 0 such that for any d and ζ satisfying d 2 + (ζ -1) 2 < δ 2 the local solution θ d is global and satisfies (3.8). In the following we construct solution to(3.6) where the condition d 2 + (ζ -1) 2 < δ 2 is not necessarily required. For a mathematical consideration the parameter ζ will be taken in (-1, √ 3]. The following result deals with nonnegative values of ζ. Let us consider a real number d such that

Acknowledgments. The authors would like to thank R. Kersner for stimulating discussions and the referees for their careful reading of the original manuscript and for making constructive suggestions, which have improved the presentation of this work. This work was partially supported by Direction des Affaires Internationales (UPJV) Amiens, France and by PAI No MA/05/116.

Appendix

Let us now derive problem (3.5). We assume that the external velocity is given by u e (x) = u ∞ x m , where m is not necessary equal to -1. We recall that the stream function satisfies the following equation

with the boundary conditions

To obtain exact solutions to (5.3),(5.4), we look for "pseudo-similarity " solutions under the form ψ(x, y) = ax α F (x, byx -β ). (5.5) where