Steven Moran
email: steve.moran@lmu.de

Johann-Mattis List
email: listm@phil.hhu.de

A Python Library for Historical Comparative Linguistics

Keywords: Traditional Language Comparison Workflow Cognates Similarity, 3 Quantitative Language Comparison Workflow Issues QLC and LingPy, 4 Examples Preprocessing Alignments, 5 Conclusion

Historical Linguistics

Language History Similar to species in biology, languages also evolve. Words are lost, new words are gained, and also the pronunciation of all words changes slightly from day to day.

During its history, a language may split into two or more descendant languages when the speakers separate and their languages keep on changing independently.

To uncover, how the languages changed into their current shape is one of the major tasks of historical linguistics.

Uncovering Language History

There are only a few languages whose history is directly reflected in written sources.

For the majority of the 6909 languages spoken today (Lewis 2009), we would not know anything about their past if we didn't have methods to infer their history.

In order to uncover language history, the languages spoken today are manually searched for traces of common origin.

Finding these traces, however, is an extremely complicated task: it took scholars more than 50 years to prove that Armenian is an Indo-European language...

Uncovering language history

Speaker Language B how do you say "tree"? tìmá:

Manual inspection & comparison A B C D E

Cognate Detection

The most crucial part of language comparison is the detection of cognates.

Cognates are words from different languages that go back to a common ancestor word (compare German Hand and English hand).

Cognate words exhibit a specific kind of similarity which does not necessarily show up in form of surface resemblances of the sounds the words are made of, but rather in structural similarities of cognate words.

This kind of similarity is not easy to detect: German Zahn and English tooth, for example, are cognate, while Greek mati and Malay mata are not...

Sound Correspondences

Meaning Italian French "square" pjaʦːa plas "feather" pjuma plym "flat" pjano plã j = l

Sound Correspondences

Meaning Italian French "square" pjaʦːa plas "feather" pjuma plym "flat" pjano plãM eaning Italian French "tear" lakrima laʀm "tongue" liŋgwa lɑɡ

"moon" luna lyn j = l

Sound Correspondences

Meaning Italian French "square" pjaʦːa plas "feather" pjuma plym "flat" pjano plãM eaning Italian French "tear" lakrima laʀm "tongue" liŋgwa lɑɡ

"moon" luna lyn j = l l = l

Sound Correspondences

Meaning Italian French "square" pjaʦːa plas "feather" pjuma plym "flat" pjano plãM eaning Italian French "tear" lakrima laʀm "tongue" liŋgwa lɑɡ

"moon" luna lyn j = l l = l pj = pl , l = l

Sound Correspondences

Meaning Italian French "square" pjaʦːa plas "feather" pjuma plym "flat" pjano plãM eaning Italian French "tear" lakrima laʀm "tongue" liŋgwa lɑɡ

"moon" luna lyn j = l l = l pj = pl , l = l
Detecting regular sound correspondences can be compared to the task of finding a cipher that relates a source to a cipher text. The only problem is: It is much harder than that, since source and cipher text are badly maintained... her z --he a r t --c -o r d With the help of sound classes the large number of speech sounds can be reduced to a manageable size comparable to the number of characters used in biological algorithms.

h jä r t a -

Definition 1

An alignment of n sequences is an n-row matrix in which all sequences are aranged in such a way that all matching and mismatching segments occur in the same column, while empty cells, resulting from empty matches, are filled with gap symbols. (cf. Kruskal 1983) Alignment Analyses

Definition 1

An alignment of n sequences is an n-row matrix in which all sequences are aranged in such a way that all matching and mismatching segments occur in the same column, while empty cells, resulting from empty matches, are filled with gap symbols. (cf. Kruskal 1983) For reasons of computational complexity, the alignment problem is often split into a pairwise and a multiple alignment problem. Multiple sequence alignments are usually computed on the basis of previously computed pairwise alignments. most algorithms that have been developed for bioinformatics cannot be directly used to compare linguistic sequences, but have to be adapted to conform to the specific needs of historical linguistics (see List 2012 and the LingPy documentation for details).

Open ended language comparison problems How to distinguish between words that are similar because they are historically related and words that are similar by chance?

Greek théos' [θɛɔs] "god" and Spanish dios [diɔs] "god" are similar, yet they are not historically related, going back to different ancestor words.

How to distinguish between words that have a common origin and words that have been borrowed?

English has borrowed more than 40% of its lexicon, mostly from Romance languages.

How can we standardize the way we represent languages in our programs, how can we effectively pool our strengths in uncovering the unknown history of the languages of the world?

Standardization plays an important role in biology. In linguistics, it has been ignored so far.

"Bag of symbols" approach Alignment of sounds in lexical items Ignore linear structure of words (mostly)

Use parallel wordlist to estimate co-occurrences of n-grams N-grams that have a high probability of co-occurrence in parallel meaning are interested for historical linguistics

Bigram matching

Bora "two": mínjéékhɯɯ Muinane "two": míínokɨ

QLC

  be tokenized into Unicode code points Tokenization is required because sequences of code points can differ in their visual and logical orders Unicode normalization reorders code points into a canonical order Combining characters and space modifying letters are joined An orthography profile specifies graphemic sequences, e.

i hjärta herz heart cordis

	Issues Issues
	Workflow Biological Parallels -Caveats Biological Parallels -Caveats
	Problems in historical linguistics and biology are similar, but ... Problems in historical linguistics and biology are similar, but ...
	the amino acid alphabet for proteins has only 20 characters, while the amino acid alphabet for proteins has only 20 characters, while
	Speaker Language A there are more than 2000 different sounds in the languages of the there are more than 2000 different sounds in the languages of the
	world... world...
	biological sequences are very long while linguistic ones are very biological sequences are very long while linguistic ones are very
	Quantitative Language Comparison how do you say tímɛ́W short... short... "tree"? ordlists in biology, the alphabet remains stable during evolution, while it in biology, the alphabet remains stable during evolution, while it Spreadsheet oot wélú statistical parse Linguist(s) tree tímɛŕ Input for changes constantly in language history... changes constantly in language history...
	stem kùgɔlu	methods
	...
	how do you say Biological algorithms were designed for long sequences
	"tree"? drawn from small alphabets. In quantitative language com-tìmá: parison we need algorithms for short sequences drawn from
	Speaker large alphabets. While biologists can model their sequences
	Language B in ASCII characters, linguists cannot do without Unicode!

  g. <s> & <h> vs <sh> in <mis.hap> & <mish.mash>

			Examples	Preprocessing
		Computational orthographic levels Orthography profile example: Leach 1969 Orthography profile example: Leach 1969 Sound Classes Sound Classes Sound Classes Sound Classes Sound Classes Sound Classes
	. . . . . .	original string: tsʰo̰ ́shi code points (10) t characters (6) t graphemes (4) ts h # leach1969 orthography profile (interpretation by QLC) s h o ˜̰ ́s h i s h o̰ ́s h i o̰ ́sh i -, , a, a, á, a, an, ã, s, s, sh, ʃ, according to Aschmann it never occurs word initially t, t, Sound Classes . . . Sounds which often occur in correspondence relations in genetically related languages can be clustered into classes (types). It . Sound Classes . . Sounds which often occur in correspondence relations in genetically related languages can be clustered into classes (types). It k g p b ʧ ʤ f v . Sound Classes . . Sounds which often occur in correspondence relations in genetically related languages can be clustered into classes (types). It k g p b ʧ ʤ f v . Sound Classes . . Sounds which often occur in correspondence relations in genetically related languages can be clustered into classes (types). It k g p b ʧ ʤ f . Sound Classes . . Sounds which often occur in correspondence relations in genetically related languages can K P . Sound Classes . . Sounds which often occur in correspondence relations in genetically related languages can K P v be clustered into classes (types). It be clustered into classes (types). It án, ã, e, ɛ, according to Aschmann extremely rare é, ɛ, according to Aschmann extremely rare en, ɛ, according to Aschmann does not exist én, ɛ, according to Aschmann does not exist c, k, ts, ts, is assumed "that phonetic is assumed "that phonetic is assumed "that phonetic is assumed "that phonetic is assumed "that phonetic is assumed "that phonetic ty, t j , y, ʑ, not clear which sound it is, it never occurs word initially z, z, only in Spanish loan words # rules . . . . . correspondences inside a 'type' are more regular than those between different 'types'" (Dolgopolsky 1986: 35). . . . . . correspondences inside a 'type' are more regular than those between different 'types'" (Dolgopolsky 1986: 35). t d ʃ ʒ θ ð s z . . . . . correspondences inside a 'type' are more regular than those between different 'types'" (Dolgopolsky 1986: 35). t d ʃ ʒ θ ð s z . . . . . correspondences inside a 'type' are more regular than those between different 'types'" (Dolgopolsky 1986: 35). t d ʒ correspondences inside a 'type' are correspondences inside a 'type' are ʃ θ ð s . . . . . 35). . . . . . 35). z more regular than those between different 'types'" (Dolgopolsky 1986: T S more regular than those between different 'types'" (Dolgopolsky 1986: T S
		ch, ʧ, ([a|á|e|é|i|í|o|ó|u|ú])(n)(\s)([a|á|e|é|i|í|o|ó|u|ú]), \1 \2 \4
		d,	d, only in Spanish loan words
		..j, h, typo	1 1 1 1 1

/ 33

Sound Correspondences

Meaning Italian French "square" pjaʦːa plas "feather" pjuma plym "flat" pjano plã

Biological Parallels

In both biology and historical linguistics ... sequences, i.e. ordered collections drawn from a fixed set of characters, constitute the basic unit of replication.

sequence comparison is of great importance.

phylogenetic trees are a basic classification scheme.


M Guedda
email: guedda@u-picardie.fr

R Kersner

Non-Newtonian pseudoplastic fluids: analytical results and exact solutions

Keywords: Boundary-layers, Non-Newtonian power-law fluids, Pseudosimilarity, Similarity, Translation groups, Integral equation method

A one layer model of laminar non-Newtonian fluids (Ostwald-de Waele model) past a semi-infinite flat plate is revisited. The stretching and the suction/injection velocities are assumed to be proportional to x 1 1-2n and x -1 , respectively, where n is the power-law index which is taken in the interval (0, 1 2 ). It is shown that the boundary-layer equations display both similarity and pseudosimilarity reductions according to a parameter γ, which can be identified as suction/injection velocity. Interestingly, it is found that there is a unique similarity solution, which is given in a closed form, if and only if γ = 0 (impermeable surface). For γ = 0 (permeable surface) we obtain a unique pseudosimilarity solution for any 0 = γ ≥ -

. Moreover, we explicitly show that any pseudosimilarity solution exhibits similarity behavior and it is, in fact, similarity solution to a modified boundary-layer problem for an impermeable surface. In addition, the exact similarity solution of the original boundary-layer problem is used, via suitable transverse translations, to construct new explicit solutions describing boundary-layer flows induced by permeable surfaces.

Introduction and main results

Many problems in boundary-layer theory lead to particular exact solutions which are characterized by similarity velocity profiles and agree with experimental observations and numerical simulations. It is the purpose of this work to investigate possible conditions for similarity and pseudosimilarity solutions to a class of the boundary-layer flows of laminar non-Newtonian fluids. The range of non-Newtonian fluid behavior exhibited by industrial liquids is very large and the mathematical formulation is, in general, complex. A broad description of the behavior in both steady and unsteady flow situations, together with mathematical models, can be found for example, to mention a few, in Astin et al. [START_REF] Astin | Boundary layers in non-Newtonian fluids[END_REF], Astarita et al. [START_REF] Astarita | Principles of non-Newtonian Fluid Mechanics[END_REF], Barnes [3], Bird [START_REF] Bird | Non-Newtonian behavior of polymeric liquids[END_REF], Tanner [START_REF] Tanner | Engineering Rheology[END_REF], Schowalter [START_REF] Schowalter | Mechanics of Non-Newtonian Fluids[END_REF], Rajagopal et al. [START_REF] Rajagopal | On a boundary layer theory for non-Newtonian fluids[END_REF] and Rajacopal [START_REF]A note on unsteady unidirectional flows of a non-Newtonian fluid[END_REF], [START_REF]Mechanics of non-Newtonian fluids, Recent developments in Theoretical Fluid Mechanics[END_REF]. The most frequently used model in non-Newtonian fluid mechanics is the Ostwald-de Waele model, or the non-Newtonian power-law fluid, for which the shear stress τ is related to the strain rate u y via the expression [START_REF] Acrivos | Momentum and heat transfer in laminar boundary flow of non-Newtonian fluids past external surfaces[END_REF]- [START_REF] Guedda | Similarity and pseudosimilarity solutions of degenerate boundary-layer equations, Hand-Book of Differential Equations[END_REF] τ

= ν |u y | n-1 u y , (1)
where the subscript y denotes the partial derivative with respect to y, ν is a positive constant and n > 0 is the power-law index. The case n < 1 is referred to as pseudo-plastic or shear-thinning fluids, and the case n > 1 is known as dilatant or shear-thickening fluids. The Newtonian fluid is a special case where the power-law index n is one.

To begin with, we give a brief description of the problem. Consider a steady boundary-layer flow due to a moving plane surface in a quiescent fluid. The fluid can be injected or sucked. For the first approximation, the model is described by the Prandtl equations or the boundary-layer equations for non-Newtonian power-law fluids [START_REF] Acrivos | Momentum and heat transfer in laminar boundary flow of non-Newtonian fluids past external surfaces[END_REF]- [START_REF] Guedda | Similarity and pseudosimilarity solutions of degenerate boundary-layer equations, Hand-Book of Differential Equations[END_REF] ⎧ ⎨ ⎩ u x + v y = 0,

uu x + vu y = ν(|u y | n-1 u y ) y , (2)
with the boundary conditions ⎧ ⎨ ⎩ u(x, 0) = u w (x), v(x, 0) = v w (x), u(x, ∞) = lim y→∞ u(x, y) = 0.

(

) 3
The Cartesian coordinates (x, y) are such that the x ≥ 0 coordinate is along the plate and the y ≥ 0 coordinate is normal to it with y = 0 is the plate. Subscripts x and y denote partial derivatives with respect to those variables and u and v are the velocity components along the xand y-axes, respectively. The stretching and suction/injection velocities are assumed to be of the form:

u w (x) = U 0 x l m , v w (x) = V 0 x l p , p = m(2n -1) -n n + 1 , ( 4
)
where n = 1 2 and m = 1 1-2n (p = -1). U 0 and V 0 are constants and the number l (the characteristic length) is the distance x where the stretching velocity equals U 0 . It is assumed that the positive x-direction is that of the main stream, so that U 0 > 0. The positive and negative values of V 0 correspond there to the injection and suction, respectively. The case V 0 = 0 means that the plate is impermeable.

In two-dimensional incompressible flow can be characterized by the stream function ψ which is defined in the usual way as u = ψ y and v = -ψ x . Thereby, the first partial differential equation (PDE for short) in ( 2) is satisfied automatically, and the second one reduces to

ψ y ψ xy -ψ x ψ yy = ν(|ψ yy | n-1 ψ yy ) y , (5)
which has to be solved subject to the boundary conditions

ψ y (x, 0) = U 0 x l 1 1-2n , ψ x (x, 0) = -V 0 x l -1
, ψ y (x, ∞) = 0.

Let us note that if we rewrite [START_REF] Tanner | Engineering Rheology[END_REF] as

ψ y ψ xy -ψ x ψ yy = nν|ψ yy | n-1 ψ yyy ,
then we see for n = 1 that this PDE is degenerate or singular in the sense that the coefficient |ψ yy | n-1 of the highest-order derivative ψ yyy vanishes or blows up at the points where ψ yy vanishes. Hence, equation [START_REF] Tanner | Engineering Rheology[END_REF] needs not admit a classical solution. So, one of the important questions is to obtain a family of exact or explicit solutions (similarity solutions, pseudosimilarity solutions, traveling wave,...) which play a fundamental role in finding certain physical and mathematical properties of the general solutions.

A common question in boundary-layer problems is related to solutions having the usual similarity form

ψ(x, y) = x α f (η), η = yx -β , ( 7
)
where α and β are constants. Research on this subject has a long history, which dates back to the pioneering works by Blasius [START_REF] Blasius | Grenzchichten in Flussigkeiten mit kleiner Reibung[END_REF] and Falkner and Skan [START_REF] Falkner | Solutions of the boundary layer equations[END_REF] for the Newtonian case (n = 1). The first analysis of the boundary-layer equations for a power-law fluid is due to Schowalter [START_REF] Schowalter | The application of boundary-layer theory to power-law pseudoplastic fluids: Similar solutions[END_REF] and Acrivos et al. [START_REF] Acrivos | Momentum and heat transfer in laminar boundary flow of non-Newtonian fluids past external surfaces[END_REF] for the case of impermeable walls and to Thomson and Snyder [START_REF] Thomson | Drag reduction of a non-Newtonian fluid by fluid injection at the wall[END_REF], [START_REF] Thomson | Laminar boundary-layer flows of Newtonian fluids with non-Newtonian fluid injections[END_REF] for permeable walls. In the first paper the author derived the equations governing the similarity flow of a pseudoplastic fluid. In [START_REF] Acrivos | Momentum and heat transfer in laminar boundary flow of non-Newtonian fluids past external surfaces[END_REF], using numerics, similarity solutions for both shear-thinning and shear-thickening fluids are obtained. The results of [START_REF] Acrivos | Momentum and heat transfer in laminar boundary flow of non-Newtonian fluids past external surfaces[END_REF] showed that the boundary-layer is of finite-width for n > 1 (see also [START_REF] Teipel | Discontinuities in boundary-layer problems of power-law fluids[END_REF]). Using a phase-plane method similar results are obtained by Pavlov et al. [START_REF] Pavlov | Structure of laminar boundary-layer in non-Newtonian dilatant fluids[END_REF]. Physically and mathematically, this phenomenon has been reported in the literature and motivated some renewed interest in analytical and numerical studies (see the papers by Denier and Dabrowski [START_REF] Denier | On the boundary-layer equations for power-law fluids[END_REF], Filipussi et al. [START_REF] Filipussi | The self-similar laminar boundary-layer of power law non-Newtonian fluids[END_REF], Zheng et al. [START_REF] Zheng | Existence and estimate of positive solutions to nonlinear singular boundary value problem in the theory of dilatant non-Newtonian fluids[END_REF], Guedda [START_REF] Guedda | Similarity and pseudosimilarity solutions of degenerate boundary-layer equations, Hand-Book of Differential Equations[END_REF] and the references therein). In all the above papers the number m is assumed to satisfy the condition 1 + m(2n -1) > 0.

The present paper aims to study problem ( 5), [START_REF] Schowalter | Mechanics of Non-Newtonian Fluids[END_REF], under the condition 1 + m(2n -1) = 0. Our work is motivated by the recent results presented by Magyari et al. [START_REF] Magyari | The "missing" similarity boundary-layer flow over a moving plane surface[END_REF] for Newtonian fluids and by Guedda [START_REF] Guedda | Similarity and pseudosimilarity solutions of degenerate boundary-layer equations, Hand-Book of Differential Equations[END_REF] for non-Newtonian power-law fluids (n > 1 2 ). In their interesting paper Magyari et al., using a natural point-mechanical analogy, conjectured several results, most of them were supported by numerical calculations. The authors showed that there is no similarity solution in the usual form [START_REF] Rajagopal | On a boundary layer theory for non-Newtonian fluids[END_REF]. To solve the problem, they added the term γ ln(x) to expression [START_REF] Rajagopal | On a boundary layer theory for non-Newtonian fluids[END_REF] which supposes that the surface is permeable. In this way they showed that the boundary-layer equations are reduced to an ordinary differential equation, and obtained for (and only for) γ ≥ γ (=1.079131) new solutions called missing solutions or pseudosimilarity solutions (see Section 2 and Section 6). Moreover, the authors showed numerically that for the threshold value γ the solution is unique but above it multiple solutions are encountered for every given value of γ.

Although most of similarity and pseudosimilarity solutions were studied by numerical simulations in boundary-layer problems, it seems worthwhile to mathematically discuss some exact solutions, which may allow one to develop or test numerical methods. With this end in view problem ( 5), [START_REF] Schowalter | Mechanics of Non-Newtonian Fluids[END_REF], where m = 1

1-2n

and n > 1 2 , were treated analytically in [START_REF] Guedda | Similarity and pseudosimilarity solutions of degenerate boundary-layer equations, Hand-Book of Differential Equations[END_REF]. It is shown that γ ≥ n+1 3n(2n-1) n n+1 for any n > 1 2 and that the the pseudosimilarity solutions share for γ > γ the remarkable property of multiplicity.

Assuming merely that 0 < n < 1 2 the purpose of the present paper is to discus in detail the questions of existence and uniqueness of similarity and pseudosimilarity solutions to problem (5), [START_REF] Schowalter | Mechanics of Non-Newtonian Fluids[END_REF]. The essence of our main result is the conclusion that the boundary-layer problem under consideration has a solution in the usual similarity form [START_REF] Rajagopal | On a boundary layer theory for non-Newtonian fluids[END_REF] if and only if the surface is impermeable (γ = 0), and this solution is expressed in a closed form. For permeable surfaces pseudosimilarity solutions are obtained for all

0 = γ > - (n+1) 3n(1-2n) n n+1 .
In addition, the present boundary-layer problem serves as a motivation to unfold and study mathematical and physical connection between similarity and pseudosimilarity solutions. We show first that the pseudosimilarity reduction is invariant under arbitrary pseudosimilarity variables and, secondly, we demonstrate by an elementary argument that any possible pseudosimilarity solution (for permeable surfaces) is in fact similarity solution in the usual form of a modified boundary-layer problem for an impermeable surface for any n = 1 2 , including the Newtonian case. Moreover, using translation invariance groups, new explicit solutions describing boundary-layer flows induced by permeable surfaces are derived for 0 < n < 1 2 . To the best of our knowledge, these results have not been obtained before.

Reformulation of the problem

Frequently, in the boundary-layer theory, one looks for general pseudosimilarity solutions of the form

ψ(x, y) = h(x)F (η, ζ), (8)
where η = yχ(x) is the similarity variable and ζ = ζ(x, y) is the pseudosimilarity variable which may be taken as independent of y. By a pseudosimilarity solution, we mean a solution to (5), ( 6) having the form [START_REF]A note on unsteady unidirectional flows of a non-Newtonian fluid[END_REF], such that in the new coordinates (η, ζ) the PDE satisfied by F is particularly simple. In a nutshell, the present discussion will be applied to all 1 2 = n > 0. This is pedagogically justified and can shed some light on non-similarity techniques.

Our approach is a slight reformulation of the analysis developed 45 years ago for the classical Falkner-Skan wedge flows for the Newtonian case [START_REF] Jones | Two dimensional boundary layers[END_REF]. In order not to unnecessarily treat h and χ as unknown functions we assume that χ = Bx -β and h ≡ A, where β = 1 2n-1 and A and B are real parameters such that νA n-2 B 2n-1 = 1 and AB = U 0 l -m [START_REF] Guedda | Similarity and pseudosimilarity solutions of degenerate boundary-layer equations, Hand-Book of Differential Equations[END_REF]; i.e.

A = ν 1 n+1 U 0 l 1/(2n-1) 2n-1 n+1 , B = ν -1 n+1 U 0 l 1/(2n-1) 2-n n+1 . ( 9
)
In terms of the new variables the boundary-layer equation becomes

|F ηη | n-1 F ηη η -m (F η ) 2 = x dζ dx F η F ζη -F ζ F ηη , ( 10
)
where we have assumed that ζ = ζ(x). Equation ( 10) must be a PDE for F, so that

x dζ dx = H(ζ), ( 11
)
where H ≡ 0 is an arbitrary continuous function. In this way

|F ηη | n-1 F ηη η -m (F η ) 2 = H(ζ) F η F ζη -F ζ F ηη . ( 12
)
The reduction of the pseudosimilarity PDE (12) will be performed with the aid of a modified first level of truncation [START_REF] Massoudi | Local non-similarity solutions for the flow of a non-newtonian fluid over a wedge[END_REF]. The main idea is to neglect only the term F ζη . This is plainly justified for (additive variables separation)

F (η, ζ) = f (η) + g(ζ), ( 13
)
where f and g are arbitrary smooth functions. Inserting ( 13) into (12) leads to

|f | n-1 f + βf 2 + γf = 0, (14)
where the primes denote derivatives with respect to the similarity variable η, and

H(ζ) dg dζ = - V 0 A l ≡ γ. ( 15
)
In view of the boundary conditions we have

f (0) = 1, f (∞) = 0. ( 16
)
The first surprising property of the above argument is found using the original variable x. The function g is expressed as

g(ζ) = γ ln(x) + constant, (17)
irrespective of the function H. Consequently, the stream function has an universal expression:

ψ(x, y) = A (f (η) + γ ln(x)) + σ, (18)
for some number σ, whenever f exists. This confirms the pseudosimilarity expression obtained by Magyari et al. [START_REF] Magyari | The "missing" similarity boundary-layer flow over a moving plane surface[END_REF]. The peculiarity of the pseudosimilarity expression ( 18) is that the condition on f (0) is unnecessary since ( 14) will be considered as a nonlinear second order differential equation for ϕ = f , i.e.

(|ϕ | n-1 ϕ ) + γϕ + βϕ 2 = 0. ( 19
)
The boundary conditions read

ϕ(0) = 1, ϕ(∞) = 0. ( 20
)
This is the problem we are going to deal with in the next sections, keeping in mind that if n > 1/2 the problem admits solutions only for large γ. In passing we note that an initial attempt to solve numerically [START_REF] Filipussi | The self-similar laminar boundary-layer of power law non-Newtonian fluids[END_REF], [START_REF] Zheng | Existence and estimate of positive solutions to nonlinear singular boundary value problem in the theory of dilatant non-Newtonian fluids[END_REF] via the shooting method failed because, as we shall see, solutions of the initial value problem for [START_REF] Filipussi | The self-similar laminar boundary-layer of power law non-Newtonian fluids[END_REF] are not always global.

Note that the above pseudosimilarity reduction and the nonintuitive expression ( 17) lead to a wide range of pseudosimilarity variables ζ, and to a family of pseudosimilarity solutions, for γ = 0, which have the property that a stream function can be obtained from another one by a simple translation and has the same shape ϕ or f. Important characteristics of the flow, namely the (non-dimensional) velocity v = (u, v, 0) and the wall shear stress τ w (x) are obtained from ϕ as follows:

u(x, y) = u w (x)ϕ(η), η = y x Re 1/(n+1) x , ( 21
) v(x, y) = u w (x) 2n -1 Re -1/(n+1) x ηϕ(η) + V 0 l x , (22)
or, equivalently,

v(x, y) = u w (x) 2n -1 Re -1/(n+1) x ηϕ(η) - γA x , ( 23
)
and

τ w (x) = u 2 w (x)Re -1/(n+1) x |ϕ (0)| n-1 ϕ (0). ( 24
)
So that the value ϕ (0) has to be given with sufficient accuracy. In the above Re x represents the local Reynolds number for non-Newtonian fluids;

Re x = u 2-n w (x)x n ν
. Note that the number γ can be written

as γ = -V 0 U (1-2n)/(n+1) 0 l n/(n+1) ν -1/(n+1)
. This parameter is usually referred to as the suction/injection parameter. The quantities ϕ and ϕ (0) are named the similar down stream velocity and similar shear stress, respectively. It should be noted here that the Prandtl velocity profile u/u w depends only on the similarity variable η, which is the essence of similarity reduction reported in a voluminous literature.

Finally, we stress once more that the present pseudosimilarity reduction is invariant under arbitrary pseudosimilarity change of variables. Moreover, as we have just seen, the velocity component u is still unchanged, but the velocity component v deviates from the usual similarity structure. We shall come back to this in the next two sections.

Pseudosimilarity or similarity reduction ?

As mentioned before, problem [START_REF] Filipussi | The self-similar laminar boundary-layer of power law non-Newtonian fluids[END_REF], [START_REF] Zheng | Existence and estimate of positive solutions to nonlinear singular boundary value problem in the theory of dilatant non-Newtonian fluids[END_REF] was considered in [START_REF] Magyari | The "missing" similarity boundary-layer flow over a moving plane surface[END_REF] for n = 1 and in [START_REF] Guedda | Similarity and pseudosimilarity solutions of degenerate boundary-layer equations, Hand-Book of Differential Equations[END_REF] for n > 1 2 , where it was found that the existence of solutions had required large positive values of parameter γ. This shows, in particular, the nonexistence of similarity solutions to the boundary-layer problem in its usual form. A more intuitive way to analyze this circumstance is to suppose that the suction/injection velocity v w is an unknown function. So the aim of this section is twofold: to identify v w ≡ 0 such that the physical problem may have similarity or pseudosimilarity solutions (a necessary condition) and establish a correspondence between similarity solutions and pseudosimilarity solutions. The starting point is to note that the continuity equation in system (2) is equivalent to

u x + (v -v w ) y = 0,
where the suction/injection velocity v w is unknown function to be determined. In terms of the new stream function Ψ, according to

u = Ψ y , v -v w = -Ψ x ,
we obtain the slightly modified PDE

Ψ y Ψ xy -Ψ x Ψ yy = ν(|Ψ yy | n-1 Ψ yy ) y -v w (x)Ψ yy , ( 25
)
accompanied by the following boundary conditions

Ψ y (x, 0) = U 0 x l 1 1-2n , Ψ x (x, 0) = Ψ y (x, ∞) = 0. ( 26
)
The only hypothesis in this brief analysis is to assume that the stream function has the usual similarity form; that is

Ψ(x, y) = Ax q h(η), η = Byx -r ,
where A and B are given by [START_REF]Mechanics of non-Newtonian fluids, Recent developments in Theoretical Fluid Mechanics[END_REF]. Repeating step by step the reduction argument we deduce that q = 0, r = 1 2n-1 and that the function

φ = h satisfies ⎧ ⎨ ⎩ (|φ | n-1 φ ) + γ 0 (x)φ + βφ 2 = 0, φ(0) = 1, φ(∞) = 0, (27)
where β = 1 2n-1 and γ 0 (x) = -vw(x)x A . Therefore, we may conclude that either γ 0 is a constant or a function of η. Since γ 0 does not contain y there exists a number γ(= -V0l A ) such that

γ 0 ≡ γ,
which leads to [START_REF] Filipussi | The self-similar laminar boundary-layer of power law non-Newtonian fluids[END_REF] and [START_REF] Zheng | Existence and estimate of positive solutions to nonlinear singular boundary value problem in the theory of dilatant non-Newtonian fluids[END_REF]. This line of analysis shows that a similarity reduction to the modified boundary-layer problem is possible only if the suction/injection velocity has the following exact expression

v w (x) = - Aγ x = V 0 x , ( 28
)
which is precisely the suction/injection velocity condition considered in [START_REF] Guedda | Similarity and pseudosimilarity solutions of degenerate boundary-layer equations, Hand-Book of Differential Equations[END_REF] and [START_REF] Magyari | The "missing" similarity boundary-layer flow over a moving plane surface[END_REF]. Note that ψ is connected to the new stream function Ψ via

ψ(x, y) = Ψ(x, y) + Aγ ln(x),
and that the analysis is not explicitly restricted to n < 1 2 . An interesting question is that what happens if [START_REF] Gilding | Travelling waves in nonlinear diffusion-convection-reaction[END_REF] is not satisfied ? In this case the parameter γ 0 becomes a function x; γ 0 = γ 0 (x). If γ 0 varies slowly with distance this quantity may be regarded as a constant parameter. Thus local pseudosimilarity solutions will be obtained by solving problem [START_REF] Magyari | Translation groups of the boundary-layer flows induced by continuous surfaces[END_REF] at different location x.

The issue of the pseudosimilarity approach, which has a remarkable degree of simplicity, is clarified by showing that pseudosimilarity solutions are in fact similarity solutions to the modified boundarylayer equations, whenever solutions to [START_REF] Filipussi | The self-similar laminar boundary-layer of power law non-Newtonian fluids[END_REF] and [START_REF] Zheng | Existence and estimate of positive solutions to nonlinear singular boundary value problem in the theory of dilatant non-Newtonian fluids[END_REF] exist, and that the additive variables separation hypothesis [START_REF] Thomson | Laminar boundary-layer flows of Newtonian fluids with non-Newtonian fluid injections[END_REF] emerges naturally from the expression of the new stream function Ψ, for any n = 1 2 . Those pseudosimilarity solutions can be referred to as similarity solutions of the second kind of the boundary-layer problem (2)-(4). In 1963 Jones and Watson [25, p. 243] mentioned that solutions given by [START_REF] Denier | On the boundary-layer equations for power-law fluids[END_REF] or by a stream function of the general form

ψ(x, y) = h(x)f (η) + ψ 0 (x) (29)
are similarity solutions in the sense that the velocity component u has the same shape across any transverse section of the layer. Let us return to the modified boundary-layer problem. Clearly, if we put v w ≡ 0 problem ( 25), ( 26) is exactly the original boundary-layer problem for impermeable surfaces. Note here that the boundary conditions [START_REF] Massoudi | Local non-similarity solutions for the flow of a non-newtonian fluid over a wedge[END_REF] are preserved, except the transpiration velocity distribution which equals now to zero (impermeable surface). Conversely, if Ψ is a solution to the modified PDE [START_REF] Jones | Two dimensional boundary layers[END_REF] the new function ψ(x, y) = Ψ(x, y)ψ 0 (x), where ψ 0 an arbitrary function, satisfies

ψ y ψ xy -ψ x ψ yy = ν(|ψ yy | n-1 ψ yy ) y + (ψ 0 (x) -v w (x))ψ yy . ( 30
)
So that if ψ 0 (x) = v w (x) the above PDE reduces to [START_REF] Tanner | Engineering Rheology[END_REF]. In this way the function Ψ, which corresponds to the stretching velocity u w for the impermeable surface, generates a solution describing a flow induced by a permeable surface. The stretching velocity u w still invariant, but the boundary layer equation is changed. The solution Ψ may be referred to as a primary solution according to the recent work by Magyari [START_REF] Magyari | Translation groups of the boundary-layer flows induced by continuous surfaces[END_REF] (see Section 5). Let us conclude this section with the following observation. In contrast to the case n > 1 2 , here the number β is negative. Consequently, as mentioned before, a local solution with an arbitrary shear stress parameter ϕ (0) = ω is not necessary global, even for positive values of γ (see below).

Similarity solutions for impermeable surfaces

In this section we are mainly interested in finding similarity solutions for problem [START_REF] Tanner | Engineering Rheology[END_REF], [START_REF] Schowalter | Mechanics of Non-Newtonian Fluids[END_REF] in the case of impermeable surfaces. This situation will be nicely illustrated by a simple explicit solution. Since v w ≡ 0 (γ = 0) we are concerned with solutions to

⎧ ⎨ ⎩ |ϕ | n-1 ϕ + βϕ 2 = 0, ϕ(0) = 1, ϕ(∞) = 0. ( 31
)
To solve the above problem we consider the initial value problem

⎧ ⎨ ⎩ |ϕ | n-1 ϕ + βϕ 2 = 0, ϕ(0) = 1, ϕ (0) = ω, ( 32
)
where the number ω is the shooting (shear stress) parameter. Varying ω one expects to find a solution which is global and goes to 0 at infinity. Note that if we look for solutions under the form

ϕ 0 (η) = k(η 0 -η) -q ,
where k, η 0 and q are positive constants, we find that

q = 1 + n 2 -n , k n-2 = |β| nq n (1 + q)
and ω = qη -q-1 0 , give rise to blowing-up solution.

Consequently, if ω > 0 any solution to (32) is not global. So, the task is to obtain, by an explicit construction, ω such that the (unique) local solution to (32) is global and satisfies the desired condition at infinity. Note that any local solution satisfies

|ϕ (η)| n-1 ϕ (η) = |ω| n-1 ω + |β| η 0 ϕ 2 (s)ds, (33)
on (0, η ω ), where (0, η ω ) being the maximal interval of existence. It follows immediately that if ϕ is a global (bounded) solution to (32) the following property

|ω| n-1 ω = -|β| ∞ 0 ϕ 2 (s)ds,
holds. Therefore, the shouting parameter has to be negative. On the other hand, the ordinary differential equation in (32) has the first integral E = constant, where the function E is given by

E = n n + 1 |ϕ | n+1 + β 3 ϕ 3 . ( 34
)
We may conclude, therefore, that the global structure of solution curves (in the phase-plane) are simply given by and then

ϕ = ± n + 1 n c + |β| 3 ϕ 3 1 n+1 , c = constant, ( 35
)
ϕ = - n + 1 n |β| 3 ϕ 3 1 n+1 . ( 36
)
The necessary condition c = 0 is obtained from the boundary (physical) condition at infinity; ϕ(∞) = 0. This leads in particular to ω = ω c , where

ω c = - (n + 1)|β| 3n 1 n+1 . ( 37
)
The meaning of the above analysis is that (31) has a unique solution and this solution can be found by a simple integration of (36). More precisely the unique algebraic solution ϕ is given by

ϕ(η) = 1 + 2 -n n + 1 (n + 1)|β| 3n 1 n+1 η -n+1 2-n , ( 38
)
or

ϕ(η) = 1 + 2 -n n + 1 |ω c |η -n+1 2-n . ( 39
)
Returning to our original physical problem, we may conclude that if n < 1 2 and m = 1 1-2n the boundary-layer problem (5), (6) (or (2)-( 4)) has exactly one similarity solution for impermeable surfaces which is expressed in term of the stream function The (non-dimensional) velocity v = (u, v) is obtained as follows

ψ(x, y) = U 2n-1 0 lν 1 n+1 (3n) 1 n+1 n + 1 1 -2n n n+1 1 + 2 -n n + 1 n + 1 3n(1 -2n) 1 n+1 η 1-2n 2-n , η = Re 1 n+1 x y x .
u(x, y) = u w (x) 1 + 2 -n n + 1 n + 1 3n(1 -2n) 1 n+1 η -n+1 2-n , ( 40
) v(x, y) = u w (x) 2n -1 Re -1 n+1 x η 1 + 2 n n + 1 n + 1 3n(1 -2n) 1 n+1 η -n+1 2-n . ( 41
)
The resulting formula for the wall shear stress is

τ w (x) = -u 2 w (x)Re -1/(n+1) x n + 1 3n(1 -2n) n n+1 , ( 42
)
which is proportional to x 2n 1-2n . The boundary-layer thickness δ(x), which is defined as the value of y when u reaches 0.01u w , has the explicit form

δ(x) = 10 2(2-n) n+1 -1 2 -n n + 1 n + 1 3n(1 -2n) -1/(n+1) xRe -1/(n+1) x .
This being derived either from (40) or from the equation ϕ(η δ ) = 0.01, where η δ = Re x δ(x)

x . The boundary-layer thickness is therefore a decreasing function in proportion to x 1 2n-1 . Figures 2 and3 illustrate the power-law index effect. It can be noted that the profile ϕ(η), η > 0, tends to 0 as n increases to 1/2, or decreases to 0. This result can be seen from (39) and from the variation of vs n. There exists n c (≈ 0.2552) such that ω c increases for (0, n c ) and decreases for (n c , 1/2). Moreover, ω c tends to -∞ as n approaches 0 and 1/2.

The translation invariance

In this section we consider, when 0 < n < 1 2 , certain exact solutions describing boundary-layer flows induced by permeable surfaces. The key is the invariance property or the translation groups (TG) introduced very recently by Magyari for the Newtonian case [START_REF] Magyari | Translation groups of the boundary-layer flows induced by continuous surfaces[END_REF]. It is proposed here to extend the TG method to non-Newtonian power-law flows. We briefly describe this approach. The idea is based on the fact that equation ( 5) is invariant under smooth translation

y → y + y 0 (x) (43)
of the transverse coordinate y. By this we mean that, thanks to the special structure of the left-hand-side of ( 5), the function

ψ(x, y) ≡ ψ(x, y + y 0 (x)) (44)
is a solution of equation ( 5) if it is satisfied by ψ. In this way the velocity components u and v are transformed into

u(x, y) = u(x, y + y 0 (x)), v(x, y) = v(x, y + y 0 (x)) -y 0 (x) u(x, y), ( 45
)
where y 0 = d dx y 0 . Thereby the wall conditions read

u(x, 0) ≡ u w (x) = u(x, y 0 (x)), v(x, 0) ≡ v w (x) = v(x, y 0 (x)) -y 0 (x) u w (x). ( 46
)
The TG transformation will not change the boundary condition at infinity. A remarkable consequence of this approach is that a solution to the boundary-layer flow induced by an impermeable surface (primary solution) generates a set of solutions induced by permeable surfaces. Conversely, a primary solution (u, v) can be obtained from a given solution ( u, v) by the displacement y 0 , which is obtained by solving the following ODE y 0 (x) u(x, y 0 (x))v(x, y 0 (x)) = 0. (47) Note that the above ODE admits a trivial solution y 0 ≡ 0 if and only if ( u, v) is a primary solution.

To illustrate the use of the Magyari approach, we return to problem ( 5), ( 6), with V 0 = 0. From ( 40) and ( 41) we find that the transformed counterpart of the primary velocity components are

u(x, y) = u w (x) 1 + 2 -n n + 1 n + 1 3n(1 -2n) 1 n+1 (η + η 0 (x)) -n+1 2-n , v(x, y) = u w (x) 2n -1 Re -1 n+1 x (η + η 0 (x)) 1 + 2 -n n + 1 n + 1 3n(1 -2n) 1 n+1 (η + η 0 (x)) -n+1 2-n -y 0 (x) u(x, y), ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ( 48
)
where

η 0 (x) = Re 1 n+1 x y 0 (x) x = Re 1 n+1 x 1 1-2n y 0 (x), ( 49
)
is the displacement of the similarity variable η corresponding to the displacement y 0 (x). In (49) Re stands for the Reynolds number for non-Newtonian fluids. The transformed wall conditions are

u w (x) = u w (x) 1 + 2 -n n + 1 n + 1 3n(1 -2n) 1 n+1 η 0 (x) -n+1 2-n , v w (x) = u w (x) 2n -1 Re -1 n+1 x η 0 (x) 1 + 2 -n n + 1 n + 1 3n(1 -2n) 1 n+1 η 0 (x) -n+1 2-n -y 0 (x) u w (x), ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (50) or v w (x) = u w (x)x -1 2n -1 y 0 (x) -(2n -1)xy 0 (x) 1 + 2 -n n + 1 n + 1 3n(1 -2n) 1 n+1 η 0 (x) -n+1 2-n . ( 51
)
Here, we shall deal with with two particular cases: η 0 = constant and y 0 = constant. These cases follow from the general solution of the following ODE (2n -1)xy 0y 0 (x) = -κ = constant.

Solving this ODE we find

y 0 (x) = κ + κ 0 x -1 1-2n , ( 52
)
where κ 0 is a constant, and then

η 0 (x) = Re 1 n+1 κx 1 1-2n + κ 0 . ( 53
)
From relation (50) we then see for η 0 = constant (κ = 0) that the transformed velocity u w (x) is proportional to x 1 1-2n and the transformed transpiration velocity v w = 0. In such situation, it is known that the velocity field is solved in a closed form in a full agreement with (48) (see Section 4). The transformed similar down stream velocity and the transformed boundary-layer thickness are given by

ϕ(η) = ϕ(η + η 0 ) ϕ(η 0 ) , δ(x) = [1 + η 0 N ] δ(x), ( 54
)
where

N = 2-n n+1 n+1 3n(1-2n) 1 n+1 .
For the case y 0 (x) ≡ κ, we have η 0 (x) = κRe

1 n+1 x 1 1-2n . The transformed solution is u(x, y) = u w (x) 1 + 2 -n n + 1 n + 1 3n(1 -2n) 1 n+1 (η + η 0 (x)) -n+1 2-n , v(x, y) = κ u w (x)x -1 2n -1 1 + 2 -n n + 1 n + 1 3n(1 -2n) 1 n+1 (η + η 0 (x)) -n+1 2-n , ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (55)
and corresponds to the wall functions

u w (x) = u w (x) 1 + 2 -n n + 1 n + 1 3n(1 -2n) 1 n+1 η 0 (x) -n+1 2-n , ( 56
)
v w (x) = κ u w (x)x -1 2n -1 1 + 2 -n n + 1 n + 1 3n(1 -2n) 1 n+1 η 0 (x) -n+1 2-n . ( 57
)
On the basis of equation ( 56), the transformed stretching velocity u w approaches the initial one u w for small values of x, but deviates it far downstream. Note that v w (x) ∼ x 2n 1-2n as x approaches 0, and that the similarity structure of the solution is

It is worthwhile to observe that we obtain similar conclusions if both η 0 and y 0 vary with x in accordance with (52) and (53).

Clearly, as mentioned in [START_REF] Magyari | Translation groups of the boundary-layer flows induced by continuous surfaces[END_REF], the TG approach opens new possibilities for constructing several solutions with a regime prescribed on the wall. Those solutions are generated from a given solution by displacement y 0 = y 0 (x). Consequently, if the displacement of the transverse coordinate y 0 , or the displacement of the similarity variable η 0 is known, expressions (48), (50) give the velocity filed of the boundary-layer problem corresponding to the stretching and transpiration velocities ( u w , v w ) given by (50). Conversely, for a given ( u w , v w ) the TG approach allows us to determine the range of admissible transverse coordinates associated to our primary solution via (50), to construct the solution ( u, v) from (48), and to find the features that distinguish the primary velocity (u, v) from ( u, v), as precisely as possible. Then, under what conditions on u w = 0 and v w y 0 exists ? An immediate answer is formulated in the following compatibility condition

(2n -1)xχ -χ = -(2n -1) x v w u w , ( 58
)
where the function χ(= y 0 ) is given by

χ(x) = M -1 x 1 2n-1 u w u w n-2 n+1 -1 , (59)
with

M = 2 -n n + 1 n + 1 3n(1 -2n) R e 1 n+1
. Note that if u w = Cu w , for some constant C, we deduce from the compatibility condition, as has been shown above, that v w = 0.

In the main context of this paper we are interested particularly in finding a solution ( u, v), from the primary solution, which corresponds to the stretching and suction/injection velocities u w , v w being proportional to x 1 1-2n and x -1 respectively. Clearly, in this circumstance the compatibility condition is violated. In [START_REF] Magyari | Translation groups of the boundary-layer flows induced by continuous surfaces[END_REF], it is indicated that for Newtonian flows the case of inverse linear stretching velocity requires a special approach.

Pseudosimilarity solutions

For the post part we will be concerned with a detailed analysis of the solutions for the modified boundary layer problem ( 25), [START_REF] Massoudi | Local non-similarity solutions for the flow of a non-newtonian fluid over a wedge[END_REF]. Hence the problem of interest is the eigenvalue problem ( 19), [START_REF] Zheng | Existence and estimate of positive solutions to nonlinear singular boundary value problem in the theory of dilatant non-Newtonian fluids[END_REF]; that is (for the reader convenience)

(|ϕ | n-1 ϕ ) + γϕ + βϕ 2 = 0, η > 0, (60)
ϕ(0) = 1, ϕ(∞) = 0, (61)
where β = -1 1-2n , 0 < n < 1/2 and γ = 0. In the light of the analysis of the previous section with γ = 0, it seems natural to expect that problem (60), (61) admits unique solutions for γ = 0 (at least for small |γ|). A mathematical question is to provide a range of spectrumfor which problem (60), (61) has a solution.

To solve the above problem we use the Crocco variables approach with the help of the theory of the integral equation developed by Gilding and Kersner [START_REF] Gilding | Travelling waves in nonlinear diffusion-convection-reaction[END_REF]. Firstly, it is easily checked, by noting that β is negative and using the function E (see (34)), that any solution to (60), (61) is monotonic (strictly) decreasing. We introduce a new variable θ by

θ(s) = -ϕ (η), s = ϕ, ( 62
)
which satisfies the first order degenerate ODE

nθ n θ = γθ -βs 2 , 0 < s < 1, (63)
or the following integral equation

θ n (s) = γs + |β| s 0 r 2 dr θ(r) , ( 64
)
obtained by a simple integration. Thus, the study of solutions to (60), ( 61) is reduced to that of solutions to equation (64) supplemented by the initial condition θ(0) = 0. (65)

Eigensolutions for positive γ

It may be noted that the integral equation ( 64) can be used to obtain the explicit solution to [START_REF] Filipussi | The self-similar laminar boundary-layer of power law non-Newtonian fluids[END_REF], [START_REF] Zheng | Existence and estimate of positive solutions to nonlinear singular boundary value problem in the theory of dilatant non-Newtonian fluids[END_REF] for γ = 0. So, when looking for solutions under the form θ 0 (s) = as χ , it can be shown (γ = 0)

χ = 3 n + 1 , a = |β| n + 1 3n 1 n+1 . Hence θ 0 (s) = |β| n + 1 3n 1 n+1 s 3 n+1 . ( 66
)
In view of (62), the ODE in (31) holds which leads to the explicit solution (38). Consequently, for any γ > 0 problem (65), (64) has a positive solution θ γ defined on (0, 1) [START_REF] Gilding | Travelling waves in nonlinear diffusion-convection-reaction[END_REF]. Moreover, one sees from (63)

θ γ (s) ≥ γ 1 n s 1 n and θ γ (s) ≥ θ 0 (s), (67)
for all 0 ≤ s ≤ 1. Together with the integral equation we deduce

θ n γ (s) ≤ γs + |β| s 0 r 2 dr θ 0 (r) ≡ γs + θ n 0 (s), (68)
and then max γs, |β| n + 1 3n

n n+1 s 3n n+1 ≤ θ n γ (s) ≤ γs + |β| n + 1 3n n n+1 s 3n n+1 . ( 69
)
The above estimates show that θ γ tends to θ 0 as γ → 0. Moreover

θ γ (s) ∼ |β| n + 1 3n 1 n+1 s 3 n+1
(70) as s → 0. Next, we shall refine estimate (70). We assume that θ γ can be written as

θ γ (s) = s 3 n+1 (a + bs k + ...), (71)
where the constants b, k are to be determined and a = |β| n+1 3n 1 n+1 (see (70)). Substituting (71) into (63) confirms that

k = 1 -2n n + 1 , b = γ n |β| n + 1 3n 1-n n+1 . Thus θ γ (s) ∼ |β| n + 1 3n 1 n+1 s 3 n+1 + γ n |β| n + 1 3n 1-n n+1 s 2(2-n) n+1 ( 72
)
as s → 0.

Returning to problem (60), (61), we deduce that, for any γ ≥ 0 and 0 < n < 1 2 , all solutions ϕ decay algebraically according to

ϕ(η) ∼ 2 -n n + 1 (n + 1)|β| 3n 1 n+1 η -n+1 2-n , ( 73
)
as η goes to infinity, irrespective of γ, showing that the pseudosimilarity solutions behave like the unique similarity solution obtained in Section 4. Note that estimate (69) can also be used to exhibit an estimate of the shear stress ϕ (0). Clearly, we infer from (69)

-γ + |β| n + 1 3n n n+1 1/n ≤ ϕ (0) ≤ -max γ 1/n , |β| n + 1 3 1 n+1 . ( 74
)
The next analysis deals with the uniqueness result. Let γ > 0 be fixed and θ 1 γ and θ 2 γ satisfy (65), (64). Assume that θ 1 γ > θ 2 γ on some [0, s γ ], 0 < s γ < 1. One subtracts the integral equations for θ 1 γ and θ 2 γ , one gets

(θ 1 γ ) n (s) -(θ 2 γ ) n (s) = |β| s 0 r 2 θ 2 γ (r) -θ 1 γ (r) θ 1 γ (r)θ 2 γ (r) dr, ( 75
)
for all 0 ≤ s ≤ s γ . The LHS of the latter is positive for 0 ≤ s ≤ s γ , while the RHS is negative. A contradiction. Thus θ 1 γ ≡ θ 2 γ on [0, s γ ]. Hence, there cannot be two different solutions satisfying (65), (64). Consequently, for any γ ≥ 0 problem (60), (61) has a unique solution for n < 1 2 , and there exists a unique ω c = ω c (γ) such that ϕ (0) = ω c . In addition, the estimate

ϕ(η) ≤ 1 + 2 -n n + 1 (n + 1)|β| 3n 1 n+1 η -n+1 2-n , ( 76
)
holds in (0, ∞), for all γ ≥ 0. In particular, this indicates that the solution with γ = 0 gives a limiting form for any solution with γ, and that the profile ϕ(η) tends to 0 as the power-law index approaches 0 and 1/2. The γ effect is visible in the following estimate

ϕ(η) ≤ 1 + 1 -n n γ 1/n η -n 1-n , (77)
which shows that ϕ(η) decreases as γ increases and goes to 0 as γ tends to infinity (strong suction velocity). Another property obtained from (77) is the following estimate of the boundary-layer thickness:

δ(x) ≤ n 1 -n 10 2(1-n) n -1 γ -1/n xRe -1/(n+1) x , γ > 0.
The next γ-effect is found from the large γ-behavior of the shear stress parameter. Using (69) one sees that |ω c | increases with γ as

ω c ∼ -γ 1/n , ( 78
)
as γ approaches infinity. Note also that the shear stress is connected to γ via the equation

|ϕ (0)| n-1 ϕ (0) = -γ + β ∞ 0 ϕ 2 (η)dη. (79)
This indicates, in particular, that the integral term becomes negligible compared to γ. A similar estimate (for ϕ (0)) and conclusion are given in [START_REF] Magyari | The "missing" similarity boundary-layer flow over a moving plane surface[END_REF] for the Newtonian case. Lastly, we point out, as mentioned before, that our results do differ in one essential way from the earlier works by Magyari et al. [START_REF] Magyari | The "missing" similarity boundary-layer flow over a moving plane surface[END_REF] for the Newtonian case and by Guedda [START_REF] Guedda | Similarity and pseudosimilarity solutions of degenerate boundary-layer equations, Hand-Book of Differential Equations[END_REF] for n > 1 2 . It is shown that problem [START_REF] Filipussi | The self-similar laminar boundary-layer of power law non-Newtonian fluids[END_REF], [START_REF] Zheng | Existence and estimate of positive solutions to nonlinear singular boundary value problem in the theory of dilatant non-Newtonian fluids[END_REF], white n > 1 2 , has a one parameter family of solutions if (and only if) the suction parameter γ > γ ; one solution for each specified ϕ (0). More precisely, for any γ > γ there exist a smallest negative initial shear stress ϕ 0,min (γ) and a maximal positive initial shear stress ϕ 0,max (γ), such that the multiple solutions are parametrized by ϕ (0) in the interval ϕ 0,min (γ) ≤ ϕ (0) ≤ ϕ 0,max (γ).

For γ = 2 it is found that ϕ 0,min = -1.725126 and ϕ 0,max = +9.212868 [START_REF] Magyari | The "missing" similarity boundary-layer flow over a moving plane surface[END_REF]. The solution of ( 19), [START_REF] Zheng | Existence and estimate of positive solutions to nonlinear singular boundary value problem in the theory of dilatant non-Newtonian fluids[END_REF] is only unique for γ = γ . On the other hand, all these solutions decay algebraically according to γ(2n -1)η -1 , in contrast to (73), as η → ∞, for γ > γ . This result has been proved numerically in [START_REF] Magyari | The "missing" similarity boundary-layer flow over a moving plane surface[END_REF] for the Newtonian case (see Figure 5 or Figure 3 of [START_REF] Magyari | The "missing" similarity boundary-layer flow over a moving plane surface[END_REF]).

Figure 5: Plots of ϕ versus η when n = 1 and γ = 2, according to Ref [START_REF] Magyari | The "missing" similarity boundary-layer flow over a moving plane surface[END_REF]. This shows multiple solutions corresponding to different values of ϕ (0)(= φ 0 ).

Eigensolutions for negative γ

We close our analysis with the case γ < 0. We can ask, naturally, if the above results still true if γ is negative. The uniqueness result still holds. As for the existence of solutions we note that problem (65), (64) may be viewed as a particular case of the problem

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ |θ(s)| n-1 θ(s) = γs + |β| s 0 r 2 dr θ(r) , 0 < s < 1, θ(0) = 0, (80)
which can be written as

⎧ ⎨ ⎩ n|θ| n-1 θθ = γθ + |β|s 2 , θ(0) = 0. (81)
To prevent any ambiguity, we shall consider only positive solutions. If γ = 0, there are two explicit solutions θ ± = ±θ 0 , where θ 0 is given by (66). So problem (80) have both positive and negative solutions for small |γ|.

To obtain positive solutions, we look for an approximate solution in the polynomial form

θ ap (s) = s χ (a + bs k + ...), (82)
where w, k, a, b are some some appropriate constants. By substituting (82) into (81) it is found that

χ = 3 n + 1 , a = ± |β| n + 1 3n 1 n+1 , k = 1 -2n n + 1 , b = γ n |β| n + 1 3n 1-n n+1
.

Since the approximate solution must to be positive, the number a is given by a = |β| n+1 3n 1 n+1 . From this we may conclude that for sufficiently small ε > 0, problem (81) (or (80)) has a positive solution θ γ in (0, ε). This solution is unique and satisfies (72) for small s and

θ n 0 (s) + γs < θ n γ (s) < θ n 0 (s), (83)
as long as θ γ is positive. Below we will show that θ γ is the positive solution to (65), (64) under a favorable condition on γ < 0. Define Since s 0 < 1 and n < 1 2 , we deduce |γ| > |γ n |, which contradicts (85). Hence θ γ is the unique positive solution to (65), (64). Consequently, for any γ ≥ -γ n problem (65), (64) has a unique positive solution.

Summary and Conclusions

We have investigated the boundary-layer behavior of a non-Newtonian pseudoplastic fluid over a continuous (permeable or impermeable) plane surface. The stretching velocity is given by u w (x) = U 0 (x/l) m , where m = 1 1-2n , 0 < n < 1 2 , and the suction/injection velocity satisfies v w (x) = V 0 l/x. The analysis reveals that there is a transparent difference between the case n < 1 2 and the case n > 1 2 . For n < 1 2 the boundary-layer problem is solved in a closed similarity for (and only for) impermeable surfaces. In term of the stream function, this solution is given by ψ 0 (x, y) = U 2n-1 0 lν 1 n+1 (3n)

1 n+1 n + 1 1 -2n n n+1 1 + 2 -n n + 1 n + 1 3n(1 -2n) 1 n+1 η 1-2n 2-n , η = Re 1 n+1 x y x .
The second part of the analysis deals with pseudosimilarity solutions for permeable surfaces. In this situation it is shown that the boundary-layer problem in its usual form has no similarity solutions. A modified boundary-layer equation is proposed and pseudosimilarity solutions are presented for any V 0 = 0 such that -V 0 l/A = γ ≥ -γ n = - x y

x , irrespective of γ > 0. The effect of the suction velocity γ is also investigated. It is found analytically that the stream velocity decreases as γ increases. The asymptotic behavior of the wall shear stress parameter ϕ (0) at large γ is analyzed. It is established that ϕ (0) ∼ -γ 1/n , showing that the wall shear stress τ w (x) behaves like τ w (x) ∼ -u 2 w (x)Re -1/(n+1) x γ as γ → ∞. The above similarity stream function ψ 0 (x, y) is used to find new explicit solutions describing boundary-layer flows induced by permeable surfaces by adapting the TG approach of Magyari [START_REF] Magyari | Translation groups of the boundary-layer flows induced by continuous surfaces[END_REF].

The existence or the nonexistence of positive solutions for γ < -γ n is an open question. However, it is reasonable to expect that the solutions exist for all negative values of γ. Our investigation may be hampered by the lack of numerical solutions for the integral equation. However, we believe that the results presented here will be used for testing numerical schemata.

> In this study we examine a class of non-Newtonian pseudoplastic fluids. > The power-law index is taken in the interval (0,1/2). > Exact similarity solutions are obtained in a closed form.> New explicit solutions are also obtained using the translation invariance method.

Figure 1 :

 1 Figure 1: Numerical solutions of the initial value problem (32), with n = 1/3. Profiles showing the effect of the shooting parameter ω.

Figure 2 :

 2 Figure 2: Numerical profiles showing the effect of the power-law index n. It is observed that ϕ(η) decreases as n increases in the interval [0.2, 0.5[. For small values of n the velocity profiles ϕ(η) decreases with n (see Figure 3).

Figure 3 :

 3 Figure 3: Numerical solutions (full lines) to problem (31) for n = 0.01 and n = 0.46. Numerical result is found to be in a full agreement with the exact solution (dotted line).

Figure 4 :

 4 Figure 4: Profiles showing the effect of the parameter γ.

  is to show that θ γ still positive on (0, 1) for |γ| ≤ γ n .(85)Let us suppose that there exists ε ≤ s 0 < 1 such that θ γ is positive on (0, s 0 ) and vanishes at s = s 0 . Setting s = s 0 in (83) we arrive at the inequality

.

  Moreover, we have demonstrated, in particular, that those pseudosimilarity solutions are in fact similarity solutions to a modified boundary-layer problem. In addition, it is found that the stream function behaves like ψ(x, y) ∼ ψ 0 (x, y) for large Re1 n+1

Acknowledgement. The authors thank the referee for the careful reading of the original manuscript and for making constructive suggestions, which have improved the presentation of this paper. E. Magyari kindly supplied us with a copy of [27]. The results of Section 5 are based on his ideas. The present work was partially done during a visit of the authors to the LaMUSE, University of Saint-Etienne. The authors wish to thank M. Boukrouche and L. Paoli for the invitation and their kind hospitality. Supported in part by Direction des Affaires Internationales, UPJV, Amiens France. To the memory of Najia Ait Bella.


S Daggumati

W Van Paepegem

J Degrieck

T Praet

B Verhegghe

J Xu

S V Lomov

I Verpoest

LOCAL STRAIN IN A 5 -HARNESS SATIN WEAVE COMPOSITE UNDER STATIC TENSION: PART II -MESO-FE ANALYSIS

Keywords: A. Textile composites, D. Local strain analysis, D. Meso-FE modeling, D. embedded sensors, D. Digital image correlation

This paper presents the local strain analysis in a thermoplastic 5-harness satin weave composite under uni-axial static tensile load using meso-FE simulations. In order to predict the local strain profiles as observed in the experiments (Part I) at various locations of the composite, different unit cell stacking models with appropriate boundary conditions are used for the FE analysis. Apart from the calculation of local strain values at different locations (inside / traction free surface) of the composite laminate, the aim of the numerical simulations is to understand the 'shadowing' effects of the internal ply shifting on the surface strain behaviour of a 5-harness satin weave composite.

Comparison of the experimental local strain values (Part I) at various locations of the satin weave composite reveals that the effects of local yarn constraints are negligible on the local longitudinal strain behavior of the composite.

Introduction

From the analysis of the heterogeneous strain fields in textile composites, recent publications [START_REF] Potluri | Meso-scale strain mapping in UD woven composites[END_REF][START_REF] Lomov | Full-field strain measurements for validation of meso-FE analysis of textile composites[END_REF][START_REF] Summerscales | Observations on the fibre distribution and fibre strain in a woven fabric reinforcement[END_REF] underscore the importance of the accurate prediction of local strain profiles and the corresponding spatial strain distribution. Based on the evaluation of local strain profiles and gradients, reliable predictions of the damage initiation and strength can be accomplished. In this regard, developing a unit cell computational model which can predict the local strain profiles as observed in the experiments can be a challenging task.

Previous research work [START_REF] Anzelotti | Mesomechanic strain analysis of twill-weave composite lamina under unidirectional in-plane tension[END_REF][START_REF] Nicoletto | Mesoscopic strain fields in woven composites: Experiments vs. finite element modeling[END_REF][START_REF] Lomov | Meso-FE modelling of textile composites: Road map, data flow and algorithms[END_REF] in this area dealt with the FE analysis of a single unit cell using infinite laminate boundary conditions. However, comparison of the experimental strain profiles with the meso-FE strain profiles [START_REF] Potluri | Meso-scale strain mapping in UD woven composites[END_REF][START_REF] Lomov | Full-field strain measurements for validation of meso-FE analysis of textile composites[END_REF][START_REF] Ivanov | Strain mapping analysis of textile composites[END_REF][START_REF] Hale | An experimental investigation into strain distribution in 2D and 3D textile composites[END_REF][START_REF] Carvalho | Compressive failure of 2D woven composites[END_REF] leads to the conclusion that it is difficult to achieve an exact correspondence between the ideal unit cell computational stress-strain profiles compared to the shifted (nested) and more complicated composite laminates used in the experimental analysis.

In the process of improving the traditional unit cell FE simulations, Owens et al. [START_REF] Owens | Effect of Finite Thickness and Free Edges on Stresses in Plain Weave Composites[END_REF] accomplished the unit cell FE analysis using finite and infinite laminate conditions. The above mentioned work concludes that there is a substantial difference in the stress distribution between the laminate outer and inner layers. Moreover, finite and infinite laminate unit cell FE analysis by Ivanov et al. [START_REF] Ivanov | Stress distribution in outer and inner plies of textile laminates and novel boundary conditions for unit cell analysis[END_REF] leads to the conclusion that local stress-strain patterns in a textile composite unit cell depend on: 1) yarn placement in the unit cell [START_REF] Anzelotti | Mesomechanic strain analysis of twill-weave composite lamina under unidirectional in-plane tension[END_REF]; 2) number of plies in the laminate; 3) stacking sequence (internal yarn nesting). In addition, conventional meso-FE analysis employs boundary conditions to the unit cell of a textile composite on the assumption of periodicity in the thickness direction.

In this case, the stress (strain) concentration can be drastically underestimated, especially in the outer plies. From the experimental local strain analysis of a controlled stacked (inphase) plain weave laminate, Lee et al. [START_REF] Lee | Digital phase-shifting grating shearography for experimental analysis of fabric composites under tension[END_REF] concluded that the yarn undulation (yarn crimp) effects are smoothened for the laminate inner layers compared to the surface layers, indicating the possible difference in strain profiles inside the laminate compared to the traction free surface.

Based on the above observations from the literature, detailed local strain analysis in the plies of a 5-harness satin weave composite is accomplished using the unit cell FE simulations as outlined in the current section: 1) computation of the local strain profiles inside the laminate is accomplished with the well established meso-FE procedure i.e.

single unit cell with 3D PBCs (Periodic Boundary Conditions).

Later, the computed numerical strain profiles are compared to the fibre optic local strains obtained from the laminate inner layers (Part I); 2) in order to compute the surface strain profiles, initially, single unit cell with in-plane PBCs is used for the FE analysis.

However, comparison of the single unit cell FE strain profiles and contours to the surface strain measurements obtained from the digital image correlation (DIC) reveals that neither 3D PBCs nor in-plane PBCs applied to the single unit cell are capable of capturing the local surface strain behaviour both qualitatively (spatial strain distribution)

and quantitatively (local strain values) comparable to the DIC strain results.

In order to capture the 'shadowing' effects caused by internal ply shift of the beneath layers on the surface strain pattern, different unit cell stacks were created as explained in [START_REF] Ito | Elastic moduli and stress field of plain-weave composites under tensile loading[END_REF]. The aforementioned study shows that the increase in tension / bending coupling effects at the yarn crimp location for different ply stacking configurations will reduce the effective elastic modulus. And the order of the calculated numerical effective elastic constants for different unit cell stacking configurations in the ascending order is as follows: "cross-ply > random phase stacking > out-of-phase stacking > iso-phase stacking > single lamina". Based on the above conclusions, instead of computing the surface strain profile using infinite laminate conditions, different unit cell stacks were created with various local ply configurations.

The underlying principle in the FE analysis of the unit cell stacks with various ply configurations relies on the fact that, by allowing or constraining the yarn undulation in different degrees at the yarn crimp location, FE analysis of the different unit cell stacks combined with appropriate boundary conditions provides an insight into the variation of the local strain values in a carbon-PPS 5-harness satin weave composite. In addition, variations in maximum and minimum strain locations can be determined at different locations of the unit cell.

FE local strain analysis and comparison with the experimental results

Local strain analysis inside the laminate

In order to start the meso-FE simulations, construction of the unit cell geometrical model and translation into FE mesh is accomplished as explained in [START_REF] Daggumati | Local damage in a 5-harness satin weave composite under static tension: Part II -Meso-FE modelling[END_REF] and briefly described as follows : 1) variation of the internal yarn dimensions in the 5-satin weave composite is quantified using the micro-CT analysis; 2) along with the carbon fibre data, the obtained textile parameters from the micro-CT measurements (Table 1) are used as an input to the unit cell geometrical modeler software 'WiseTex' [START_REF] Lomov | Meso-FE modelling of textile composites: Road map, data flow and algorithms[END_REF][START_REF] Verpoest | Virtual textile composites software WiseTex: Integration with micro-mechanical, permeability and structural analysis[END_REF]; 3) finally, the 'WiseTex' generated unit cell geometrical model is translated into FE mesh and filled with the matrix mesh in the 'MeshTex' software [START_REF] Lomov | Meso-FE modelling of textile composites: Road map, data flow and algorithms[END_REF].

The 'MeshTex' generated unit cell FE model in the form of nodal and element information is transferred to the commercial FE software ABAQUS, and assigned with the linear brick element (C3D8) for the numerical simulations. The entire unit cell model used for the FE simulations consists of 53200 solid elements (Figure 1a).

In order to accomplish the FE simulations, the micro-mechanical (UD) material properties of the carbon-PPS representing the homogenized material properties of the impregnated yarn (Table 2) are derived from the individual elastic properties of the carbon fibre and PPS matrix using the Chamis [START_REF] Chamis | Mechanics of Composite Materials Past, Present, and Future[END_REF] analytical homogenization formulas.

The obtained homogenized micro-mechanical material properties are assigned to the yarn cross-sections in their local coordinate system to account for the yarn crimp in the FE model. In order to simulate the local strain behavior in the laminate inner layers, 3D

PBCs [START_REF] Lomov | Meso-FE modelling of textile composites: Road map, data flow and algorithms[END_REF][START_REF] Whitcomb | Derivation of Boundary Conditions for Micromechanics Analyses of Plain and Satin Weave Composites[END_REF] are applied to the unit cell FE mesh. As explained in Part I, in order to calculate the local strains in the elastic regime and to compare with experimental fibre optic sensor results, an average tensile strain of 0.2% is applied to the unit cell in the warp yarn direction (Figure 1a) and the results are discussed in the current section.

The local strain pattern on the scale of a single unit cell with the matrix as well as on the reinforcement is shown in Figure 1(a,b). Figure 1c depicts the local longitudinal strain profile plotted over the length of the unit cell in the load carrying direction from one yarn crimp to the other as marked with the dotted line in Figure 1a.

From Figure 1(a, c), it is evident that the maximum tensile strain (0.27%) in the unit cell occurs at the edges of the weft yarn at the yarn crimp location. The minimum local strain (0.17%) in the entire unit cell is detected in the resin pockets at the geometrical transition location between the yarn crimp (weft) to the straight yarn location (warp). Finally, along the straight portion of the load carrying warp yarn, the mean strain around 0.18-0.19% is observed. Comparison of the FE local strains with the FOS (Fibre Optic Sensor) results measured in the laminate inner layers (Table 3) shows good correlation starting from the maximum to the minimum values.

Based on the heuristic approach, it is assumed that the local strain values measured from the optical fibre sensor inside the laminate (maximum, minimum) belong to the same locations as observed in the unit cell FE analysis. Moreover, comparison of the experimental (surface) and numerical (3D PBCs-inside the laminate) local strain profiles (Figure 1c) shows that, quantitatively, the predicted FE local strain values are within the close limits of the experimental surface strain values. However, comparison of the experimental (Figure 3) and numerical strain spatial distribution (Figure 1a) indicates that there is a difference in the maximum and minimum strain locations. The obtained difference in the spatial strain distribution between the experiment and numerical simulations is manifested by a change in the local strain profiles (Figure 1c). The above obtained qualitative difference underscores the necessity for improvement in the unit cell FE procedure used for the surface local strain analysis.

In order to understand the difference between computed (inner) local strain profile and the experimental local surface strain profiles (DIC), a unit cell stack should be developed with the internal yarn shifting (nesting) that is present in a textile laminate. Consequently, BC's used for the FEA should be changed according to the BC's that are prevailing on the composite laminate under the experimental tensile test. However, producing a unit cell stack with all shifting (nesting) details that are present in a textile laminate is practically not feasible. The above mentioned problem can be circumvented by creating different unit cell stacking configurations for the FE analysis as explained in [START_REF] Ito | Elastic moduli and stress field of plain-weave composites under tensile loading[END_REF][START_REF] Page | Finite element simulation of woven fabric composites[END_REF].

Apart from the computation of the local strain profiles at various locations of the composite, different unit cell stack FE analyses provide an insight into the effect of internal yarn shifting (nesting) on the local surface strain pattern.

Local strain analysis on the laminate traction free surface

As explained in the previous section, for the calculation of local strain profiles in the elastic strain regime on the unit cell traction free surface, an average tensile strain of 0.5% is applied to the unit cell in the warp yarn direction. The results are discussed in the current section.

Initially, for the simulation of the surface strain profile, FE analysis is started with a single unit cell using in-plane periodic boundary conditions. Comparison of the FE local strain values with the DIC local strains at the average tensile strain of 0.5% (Figure 2b)

shows that the single unit cell FE analysis predicts 50% higher local strain at the yarn crimp location compared to the DIC results. The difference in the local strain values between the numerical and experimental results can be attributed to the large tension/bending coupling effects caused by the unrestricted yarn undulation in a single ply lamina [START_REF] Lee | Digital phase-shifting grating shearography for experimental analysis of fabric composites under tension[END_REF]. However, FE (Figure 2a) and DIC (Figure 3) strain contours show that the maximum local strain occurs at the centre of the weft yarn at the yarn crimp location.

Therefore, in order to keep the surface layer under traction free condition, and at the same time to provide the support in the form of adjacent layers in the unit cell thickness direction, investigation of the local surface strain analysis is accomplished by using the following unit cell stacks [START_REF] Ito | Elastic moduli and stress field of plain-weave composites under tensile loading[END_REF][START_REF] Woo | Phase shift effect on the stress distribution for satin weave composites[END_REF].

• Unit cell stack with in-phase stacking

• Unit cell stack with step stacking

• Unit cell stack with out-of-phase stacking

In the in-phase stacking, unit cells are placed in such a way that the anticipated displacement at the yarn crimp location is compatible and in-phase [START_REF] Carvalho | Compressive failure of 2D woven composites[END_REF] as shown by the arrows in Figure 4a. In contrast, for the out-of-phase stacking (Figure 4b) the anticipated displacement at the yarn crimp is not compatible and will act to reduce displacement of the weft yarn at the yarn crimp location. In case of step stacking, the yarn crimps of the unit cells are placed to form steps as shown in Figure 4c.

In the above mentioned three unit cell stacks, in-phase and step stacking unit cell models are created using the 'WiseTex' software [START_REF] Verpoest | Virtual textile composites software WiseTex: Integration with micro-mechanical, permeability and structural analysis[END_REF][START_REF] Lomov | Nesting in textile laminates: geometrical modelling of the laminate[END_REF]. Where as the third model (out-ofphase stacking) is developed using python script ('Pyformex' [START_REF] Bagger | Modeling plate shell structures using pyFormex[END_REF]) by performing series of translational and rotational operations on the single unit cell FE mesh obtained from 'MeshTex'. Moreover, nodes at the joining faces of the unit cells are tied to each other so that the perfect bonding is assured between the unit cells.

Choosing four unit cells in the stack (Figure 4a-c) from the laminate of eight layers, FE analysis of the unit cell stack is accomplished using in-plane periodic boundary conditions [START_REF] Zako | Finite element analysis of damaged woven fabric composite materials[END_REF]. In addition, symmetry boundary conditions are applied at the bottom xzsurface and the top xz surface is allowed to deform in the out-of-plane direction (Figure 4a). Using different unit cell stacks along with the above specified boundary conditions, FE simulation of the unit cell stacks is accomplished at 0.5% of the average tensile strain and results are presented here.

Comparison of the experimental (Figure 3) and numerical surface strain contours (Figure 4a-c) show that, in correlation with the DIC and the single unit cell (2D PBC) FE strain contours, in-phase unit cell stack (Figure 4a) predicts the maximum strain at the centre of the weft yarn at the yarn crimp location. Moreover, it is also evident that the change in the unit cell stack and the applied BC's causes the shift in the maximum strain location from centre to the edge of the weft yarn for the other two unit cell stacks (Figure 4 b,c).

Figure 5 (a, b) compares the experimental strain profiles with the numerical longitudinal surface strain profiles obtained from the different unit cell stacks. As shown in Figure 5a, the variation in the numerical local strains on the unit cell traction free surface is summarized as follows:

• Depending on the unit cell stacking and boundary conditions used for the FE simulation, the maximum strain location on the weft yarn at the yarn crimp location changes from the centre to the edge. Comparison of local longitudinal strain profiles computed from different unit cell stacks (Table 3) gives the following order of local strain at the yarn crimp location (first part): single unit cell (2D PBC) > in-phase stacking > step stacking > single unit cell with 3D PBC > out-of-phase stacking.

• The second part of the strain profile is associated with the minimum strain value adjacent to the yarn crimp location. The obtained minimum strain values from the different unit cell stacks as well the experimental techniques show very good correlation.

• The final part of the strain profile belongs to the straight portion of the load carrying warp yarn, whose value from all the measurements varies in between 0.48-0.53%.

Figure 5c presents the comparison between the experimental and numerical local transverse strain, which shows good correlation for all unit cell stacks. As observed in the experiments, meso-FE simulations predict the positive transverse strain on the weft yarn at the yarn crimp location. And the resin pockets as well as the straight position of the warp yarns sustain the compressive transverse strain due to the Poisson's effect.

Moreover, it should be noted that each yarn in the above mentioned unit cell stacks contains four or more elements over its cross section. In-phase and step stacking unit cell models contain 78500 elements, where as the out-of-phase model contains 319200 elements. Regarding the effect of mesh density on the local strain behavior, Lomov et al.

[2] concludes that the mesh density shows minor influence on the local strain behavior.

Also, from the investigation of the effect of mesh density on the unit cell local stress behavior [START_REF] Daggumati | Local damage in a 5-harness satin weave composite under static tension: Part II -Meso-FE modelling[END_REF] it appears that when the yarn contains four or more elements over its crosssection, the local stress behaviour converges to the same value.

In the process of investigating the local stress components that are responsible for the sequential weft yarn damage in the carbon-PPS 5-harness satin weave composite, local stress analysis is conducted. From the cumulative AE energy curve and the microscopic damage analysis of the carbon-PPS satin weave composite, it was concluded that the weft yarn damage in the plies of a satin weave composite is a sequential process [START_REF] Daggumati | Local damage in a 5-harness satin weave composite under static tension: Part I -Experimental analysis[END_REF].

However, experimental local strain analysis (Part I) concludes that there is hardly any influence of the internal yarn shifting (nesting) on the local longitudinal strain behavior.

In order to investigate the stress component that causes the sequential damage in the satin weave composite, local stress analysis is conducted anond the results are discussed in the current sect. In correlation with the experimental observations of the local longitudinal strain, the local longitudinal stress analysis in the laminate inner layers as well as on the unit cell traction free surface shows very good correlation (Figure 6a). However, at the applied average tensile strain of 0.5%, there is a difference of 23% in the local transverse stress for weft yarns at the yarn crimp location between the infinite laminate and the finite laminate with out-of-phase stacking(Figure 6b).

CONCLUSION

In the previous paper (Part I), an experimental procedure has been introduced for the analysis of local strain behavior in the plies of a thermo-plastic carbon-PPS satin weave composite under uni-axial static tensile load. In order to understand the influence of ply stacking (nesting) on the local strain behavior of the satin weave composite, meso-FE simulations are performed using different unit cell stacks. Comparison of the experimental and numerical results leads to the following conclusions:

• Local strain profiles are sensitive to the unit cell geometrical model as well as the applied boundary conditions to the unit cell.

• Comparison of the experimental and numerical surface strain results implies that, by restricting the yarn undulation effects using 3D PBCs or shifting of the beneath layers in the unit cell stack, quantitatively, the numerical surface strain results approach to the experimental results. However, the maximum strain locations obtained from FE simulations differ from those of the experimental strain contours.

• From the local stress analysis on different unit cell models, it is evident that the weft yarn transverse stress is sensitive to the ply position in the laminate along with the out-of-plane shear stress [START_REF] Daggumati | Local damage in a 5-harness satin weave composite under static tension: Part II -Meso-FE modelling[END_REF]. The variation in the local transverse stress at the yarn crimp location causes the sequential weft yarn damage according to the ply position in the laminate.

As a summary, the present communication emphasizes the effect of internal yarn shifting (nesting) on the local strain behavior of a satin weave composite. To predict the FE local surface strains both qualitatively and quantitatively comparable to experimental results, unit cell geometry with maximum ply shift should be developed. However, it is a tedious process to develop a full scale unit cell stack with the detailed internal yarn nesting for the computational purpose. As a consequence, it is imperative to develop the above explained unit cell stacks to obtain an insight into the maximum and minimum limits for the surface local strain profiles as well as the variation in local strain contours.

Figure 1 .Figure 2 .Figure 4 .Figure 5 .Figure 6 .

 12456 Figure 1. Single unit cell FE analysis with infinite laminate boundary conditions: a) longitudinal strain contours on the matrix surface; b) longitudinal strain contours on the reinforcement; c) comparison of the experimental (surface) and numerical (inner) local strain profiles at <0.2%>.

Table 1 . Input data for the unit cell geometrical modeler software 'WiseTex'.

 1

	WiseTex input data at fabric level
	Property	5 harness satin weave warp 5 harness satin weave weft
	Yarn	198 tex
	Number of measurements	20	20
	Yarn width (mm)	1.31 ± 0.01	1.32 ± 0.08
	Yarn thickness (mm)	0.162 ± 0.01	0.161 ± 0.06
	Yarn spacing (mm)	1.50 ± 0.04	1.49 ± 0.07
	WiseTex input data at fibre level
	TEX (g/km)	198
	Yarn filament count	3000
	Filament diameter (mm)	0.007
	Carbon fibre density (g/cm 3 )	1.75

Table 2 . T300 JB carbon fibre, PPS elastic and strength properties.

 2

	T300 JB carbon fibre		PPS (PolyPhenelyene Sulphide)
	Elastic properties
	Longitudinal modulus 11 E , GPa	231	Modulus of elasticity E , GPa	3.8
	Transverse modulus 22 E , GPa	28	Shear modulus G , GPa	1.38
	In-plane shear modulus 12 G , GPa	24	Poisson's ratio ν	0.37
	Transverse shear modulus 23 G , GPa	10.7
	In-plane Poisson's ratio 12 ν	0.26
	Transverse Poisson's ratio 23 ν	0.39

Homogenized elastic properties of the carbon-PPS impregnated yarn

	φ	f	=	0	.	7	11 E , [GPa]	, [GPa 22 E ]	12 G , [GPa]	23 G , [GPa]	ν	12	ν	23
	Chamis micro
	mechanical	162.60	13.70	6.50	5.07	0.29	0.35
	homogenization

Table 3 Comparison of the local strain values -experiment vs. numerical simulations.

 3

		Inside the laminate	Laminate surface
		Max,%	Min,%	Max,%	Min,%
	FBG's -strain inside the laminate -0.2%	0.25	0.16	-NA-	-NA-
	Single unit cell -3D PBC 0.2%	0.27	0.17	-NA-	-NA-
	Single unit cell -3D PBC 0.5%	0.65	0.47	-NA-	-NA-
	LIMESS (DIC) -Surface strain 0.5%	-NA-	-NA-	0.58-0.62	0.43-0.46
	Single unit cell -2D PBC 0.5%	-NA-	-NA-	1.4%	0.1%
	4 Unit cells -out-of-phase stacking 0.5%	-NA-	-NA-	0.65	0.44
	4 Unit cells -step stacking 0.5%	-NA-	-NA-	0.65-0.0.69	0.32-0.44
	4 Unit cells -in-phase stacking 0.5%	-NA-	-NA-	0.73	0.35

ACKNOWLEDGEMENTS

The authors would like to acknowledge the FWO -Vlaanderen for the financial support through the project G.0233.06H. The help of laboratory staff of the department of the MTM -Kris van der Staey and Johan Vanhuist is gratefully acknowledged. The authors also express their gratitude to 'Ten Cate' for supplying the composite plates. The authors also want to acknowledge the assistance of Dmitry Ivanov.