# K-UniMorph: Korean Universal Morphology and its Feature Schema

Eunkyul Leah Jo1∗ Kyuwon Kim1,2∗ Xihan Wu1 KyungTae Lim3
Jungyeul Park1 Chulwoo Park4
1The University of British Columbia, Canada 2Seoul National University, South
Korea
3SeoulTech & Teddysum, South Korea 4Anyang University, South Korea
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Equally contributed authors.

###### Abstract

We present in this work a new Universal Morphology dataset for Korean.
Previously, the Korean language has been underrepresented in the field of
morphological paradigms amongst hundreds of diverse world languages. Hence, we
propose this Universal Morphological paradigms for the Korean language that
preserve its distinct characteristics. For our K-UniMorph dataset, we outline
each grammatical criterion in detail for the verbal endings, clarify how to
extract inflected forms, and demonstrate how we generate the morphological
schemata. This dataset adopts morphological feature schema from Sylak-Glassman
et al. (2015) and Sylak-Glassman (2016) for the Korean language as we extract
inflected verb forms from the Sejong morphologically analyzed corpus that is
one of the largest annotated corpora for Korean. During the data creation, our
methodology also includes investigating the correctness of the conversion from
the Sejong corpus. Furthermore, we carry out the inflection task using three
different Korean word forms: letters, syllables and morphemes. Finally, we
discuss and describe future perspectives on Korean morphological paradigms and
the dataset.

##  1 Introduction

The Universal Morphology (UniMorph) project is a collaborative effort
providing broad-coverage morphological paradigms for diverse world languages
(McCarthy et al., 2020; Kirov et al., 2018). UniMorph consists of a lemma and
bundle of morphological features related to a particular inflected word form
as follows, for example:

나서다naseoda 나섰다naseossda v;decl;pst

where 나서다naseoda is the lemma form and 나섰다naseossda (‘became’) is the
inflected form with v;decl;pst (verb, declarative, and past tense) as
morphological schema.

It started in 2016 as a SIGMORPHON shared task (Cotterell et al., 2016) for
the problem of morphological reinflection, and it introduced morphological
datasets for 10 languages. The inflection task, using the given lemma with its
part-of-speech to generate a target inflected form, has been continued through
the years: CoNLL–SIGMORPHON 2017 Shared Task (Cotterell et al., 2017),
CoNLL–SIGMORPHON 2018 Shared Task (Cotterell et al., 2018), SIGMORPHON 2019
Shared Task (McCarthy et al., 2019), SIGMORPHON 2020 Shared Task (Gorman et
al., 2020) and SIGMORPHON 2021 Shared Task (Pimentel et al., 2021). However,
the Korean language has not been a part of the shared task because of the lack
of the dataset.

Nonetheless, although rarely, morphological paradigms for Korean have been
explored in the context of computational linguistics. Yongkyoon (1993) defined
the inflectional classes for verbs in Korean using word-and-paradigm (WP)
(Hockett, 1954) approaches. His fifteen classes of the verb which can be
joined with seven different types of verbal endings, are based on inflected
forms of the verb. Seokjoon (1999) systematized the list of final endings and
their properties, which are also used as conjunctive endings in Korean.
Otherwise, properties of verbs such as mood, tense, voice, evidentiality,
interrogativity have been extensively studied in Korean linguistics
independently: for example, inter alia, tense (Byung-sun, 2003), grammatical
voice (Chulwoo, 2007), interaction of tense–aspect–mood marking with modality
(Jae Mog, 1998), evidentiality (Donghoon, 2008), and interrogativity
(Donghoon, 2011).

In continuation of the efforts, this paper proposes a new Universal Morphology
dataset for Korean. We adopt morphological feature schema from Sylak-Glassman
et al. (2015) and Sylak-Glassman (2016) for the Korean language and extract
inflected verb forms from the Sejong morphologically analyzed corpus over 0.6M
sentences with 9.5M words. We set the criteria in detail by explaining how to
extract inflected verbal forms (Section 2), and carry out the inflection task
using different Korean word forms such as letter, syllable and morpheme
(Section 3). Finally, we discuss future perspectives on a Korean UniMorph
dataset (Section 4).

##  2 UniMorph Features Schema

Verbal endings in the inflected forms of the predicate has been considered as
still being in the part of the word as proposed in several grammar formalisms
for Korean such as lexicalized tree adjoining grammars (Park, 2006), head
driven phrase structure grammars (Ko, 2010), and combinatory categorial
grammars (Kang, 2011) in contrast to government and binding (GB) theory
(Chomsky, 1981, 1982) for Korean in which the entire sentence depends on
separated verbal endings. This idea goes back to Maurice Gross’s lexicon
grammars (Gross, 1975), and his students who worked on a descriptive analysis
of Korean in which the number of predicates in Korean could be fixed by
generating possible inflection forms: e.g. Pak (1987); Nho (1992); Nam (1994);
Shin (1994); Park (1996); Chung (1998); Han (2000). However, we have separated
the postposition from the substantive such as noun phrases instead of keeping
themselves together. Therefore, with the current Korean dataset, we decide to
annotate morphological data for verbs (v).

Table 1 shows the morphological schema for Korean UniMorph where we adopt
features from Sylak-Glassman et al. (2015) and Sylak-Glassman (2016) for the
Korean language. In addition to the features schema, we consider following
these four different types of verbal endings, in which they convey grammatical
meanings for the predicate: sentence final ending (ef), non-final ending (ep),
conjunctive ending (ec), and modifier ending (etm).

Evidentiality | hrsy |  hearsay: 일il (‘work’)/NNB 이i (‘cop’)/VCP + 래lae (‘hrsy’)/EF (‘happen’)
---|---|---
| infer |  inferred: 괜찮gwaenchanh (‘fine’)/VA + 겠gess (‘infer’)/EP \+ 다da (‘decl’)/EF
Interrogativity | decl |  declarative: 모이moi (‘gather’)/VV + ㄴ다nda (‘decl’)/EF
| int |  interrogative: 배우baeu (‘study’)/VV + 는가neunga (‘int’)/EF
Mood | real |  realis: 얻eod (‘get’)/VV + 은eun (‘real’)/ETM
| irr |  irrealis: 잊ij (‘forget’)/VV + 을eul (‘irr’)/ETM
| purp |  general purposive: 달래dallae (‘appease’)/VV + 려고lyeogo (‘purp’)/EC
| oblig |  obligative: 이어지ieoji (‘connect’)/VV + 어야eoya (‘oblig’)/EC (‘should be connected’)
Tense | prs |  present: 들리deulli \+ (‘hear’)/VV + ㄴ다nda (‘prs,decl’)/EF
| pst |  past: 나타나natana (‘appear’)/VV + 았ass (‘pst’)/EP \+ 다da (‘decl’)/EF
Voice | caus |  causative: 보이boi (‘show’)/VV + 게ge (‘caus’)/EC
| pass |  passive: 잡히jabhi (‘be caught’)/VV \+ 었eoss (‘pat’)/EP + 다da (‘decl’)/EF
Table 1: Korean UniMorph schema for verbs: vv for verb, va for adjective, vcp
for copula, and nnb for bound noun,

#### Evidentiality

It is a grammatical category that reflects the source of information that a
speaker conveys in a proposition. It is often expressed through morphological
markers such as sentence final endings (ef) 대dae, 내nae, and 래lae bring in
hearsay (hrsy), and non-final endings (ep) 겠gess introduce inferred (infer).
Since the suffix for the quotative (quot) is denoted with a postposition (jkq)
in Korean instead of the verbal ending, it is excluded from the current set of
schemata.

#### Interrogativity

It indicates either to express a statement (decl) or a question (int). We
consider all sentence final ending (ef) ended with 다da as declarative decl,
and sentence final ending (ef) included 가ga and 까kka as interrogative int.

#### Mood

The grammatical mood of a verb indicates modality on a verb by the
morphological marking. Realis (real) and irrealis (irr) are represented by a
verbal modifier ending (also known as an adnominal ending) (etm), ㄴn and ㄹl,
respectively. The usage of adnominal endings consists of (i) collocation such
as 인한inhan, 치면chimyeon, 대한daehan, (ii) modifiers and (iii) relative clauses.
Realis and irrealis are concerned with regardless of modifiers or relative
clauses. General purposive (purp) is decided by 려고lyeogo and 하러haleo, and
obligative (oblig) is introduced by 야ya. It is worthwhile to note that we do
not consider indicative (ind) because we specify declarative decl.

#### Tense

It refers to the time frame in which a verb’s action or state of being occurs.
Non-final endings (ep) such as 았ass and 었eoss and final endings (ef) such as
ㄴ다nda 는다neunda can represent the past (past) and the present (prs) tenses,
repectively. Since the future tense (fut) has been considered as irrealis
(irr) in Korean, we don’t annotate it here.

#### Voice

We deduce the passive (pass) from the verb stem instead of the verbal ending
such as jab-hi (‘be caught’). Whereas the verb jab (‘catch’) and the passive
suffix hi might be segmented, the current criteria of the Sejong corpus
combines them together as a single morpheme. 이히리기i, hi, li, gi are verbal
endings known for both the passive and the causative. If the verb has a verbal
ending 게ge such as verb stem+{이i|히hi|리li|기gi}+게ge {하ha|만들mandeul (‘make’)},
then it is causative (caus), otherwise passive (pass).

#### Other schema

For politeness, we introduce only polite (pol) using the non-final ending (ep)
시si as the direct encoding of the speaker-addressee relationship (Brown and
Levinson, 1987, p.276). Lastly, since we are not able to deduce the valency of
the verb from morphemes, we do not include intr (intransitive), tr
(transitive) and ditr (ditransitive). However, we leave them for future work
because the valency might still be valid morphological feature schemata for
Korean.

##  3 Experimental Results

###  3.1 Data creation

We prepare the data by extracting inflected verb forms from the Sejong
morphologically analyzed corpus (sjmorph) over 676,951 sentences with
7,835,239 eojeols (word units separated by space) which represent 9,537,029
tokens. We are using the same training/dev/test data split that Park and Tyers
(2019) proposed for Korean part of speech (POS) tagging. However, the current
sjmorph doesn’t contain POS labels for the eojeol (the word). Instead, it
contains the sequence of POS labels for morphemes as follows:

나섰다naseossda 나서naseo/VV+었eoss/EP+다da/EF

where it contains only each morpheme’s POS label: a verb 나서naseo (‘become’), a
non-final ending 었eoss (‘pst’), and a final ending 다da (‘decl’), and it does
not show whether the word 나섰다naseossda (‘became’) is a verb. Previous works
(Petrov et al., 2012; Park et al., 2016; Park and Tyers, 2019; Kim and
Colineau, 2020) propose a partial mapping table between Sejong POS (and the
sequence of Sejong POSs) (XPOS) and Universal POS (UPOS) labels where UPOS
represents the grammatical category of the word. However, no study has
presented the correctness of their conversion rules. Therefore, we utilize
UD_Korean-GSD (McDonald et al., 2013) in Universal Dependencies (Nivre et al.,
2016, 2020) that provides Sejong POS(s) and Universal POS labels for each
word. Nevertheless, we observed several critical POS annotation errors in
UD_Korean-GSD. For this reason, we proceeded to revise GSD’s Sejong POS(s) and
Universal POS to evaluate our criteria of getting verbs (inflected forms and
their lemmas) from sjmorph. This approach involved randomly selecting 300
sentences from the GSD and manually revising their POS labels based on the
Sejong POSs. For thorough verification, they were examined by our linguist for
over 60 hours over 3 weeks. The main places of error that we noticed were how
words for proper nouns were labeled as NOUN even with its XPOS of proper nouns
(NNP). They were corrected to the UPOS label of PROPN. Another common place of
error was how the dataset recognized and labeled words according to their
roles as constituent parts of the sentence they are in, instead of the word’s
own category. For example, the temporal nouns was usually annotated as ADV
instead of NOUN. We changed this mislabeling by acknowledging the word itself,
separate from the sentence. Again, the Sejong POS labels were revised based on
the criteria of the Sejong corpus. After correcting 738 words for Sejong POS
labels and 705 words for Universal POS labels from 300 sentences in the
development file, we trained the sequence of Sejong POS labels using semi-
supervised learning to predict the Universal POS label for each word. Among
3674 predictions, there were only 332 UPOS prediction errors, and an error
scarcely occurs for VERB labels, which we attempted to extract from sjmorph.
Therefore, we consider this current error rate for the verb to be negligible.
Finally, we extract 244,871 inflected verbal forms for 43,959 lemma types from
sjmorph. Then, we remove all duplicated items from train+dev datasets compared
to the test dataset. In Table 2 is the brief statistics of the current
dataset.

| train | dev | test
---|---|---|---
lemma | 41,631 | 7505 | 7595
inflected | 197,774 | 19,251 | 27,846
Table 2: Statistics of Korean UniMorph

###  3.2 Morphological reinflection

The goal of the morphological reinflection task creates the generative
function of morphological schema to produce the inflected form of the given
word. For Korean, we use 나서다naseoda and v;decl;pst to predict 나섰다naseossda by
using the composition of alphabet letters (l), syllables (s) and morphemes (m)
of the word as shown in Table 3. The word is decomposed into the sequence of
consonants and vowels by Letter, the sequence of units constructed with two or
three letters by syllable, and the sequence of morphological units by
morpheme. The conversion from the target form of each representation to the
surface form and vice versa are straightforward in technical terms.

| Source | Target
---|---|---
letter (l) | ㄴㅏㅅㅓㄷㅏ | ㄴㅏㅅㅓㅆㄷㅏ
syllable (s) | 나서다 | 나섰다
morpheme (m) | 나서다 | 나서었다
surface form | 나서다naseoda | 나섰다naseossda

Table 3: Example of the surface form and its different representation using
letters, syllables and morphemes.

For our task, we use the baseline system from The CoNLL–SIGMORPHON 2018 Shared
Task (Cotterell et al., 2018).111https://github.com/sigmorphon/conll2018 The
system uses alignment, span merging and rule extraction to predict the set of
all inflected forms of a lexical item (Durrett and DeNero, 2013). We also
build a basic neural model using
fairseq222https://github.com/facebookresearch/fairseq (Ott et al., 2019) and
Transformer (Vaswani et al., 2017). Table 4 shows the experimental results for
Korean UniMorph using the three different representation forms. It is notable
that the morpheme forms outperform the other surface representation forms such
as by letters and syllables of the word. This is because morpheme forms imply
lemma forms for both source and target data. While the average number of
inflected forms per lemma is 8.285, there are 22 verb lemmas that have more
than 400 different inflected forms. The average number of inflected forms per
lemma and morphological feature pair is also 5.634, and this makes Korean
difficult to predict the inflected form.

| l | s | m
---|---|---|---
baseline | 26.88 | 27.75 | 31.29
neural | 51.97 | 49.72 | 54.26
Table 4: Experimental results (accuracy)

| UniMorph 4.0 Korean | K-UniMorph
---|---|---
Evide. | - |  hrs, infer
Finit. |  fin, nfin | -
Inter. |  decl, int, imp |  decl, int
Mood |  cond, purp |  real, irr, purp, oblig
Tense |  prs, pst, fut |  prs, pst
Voice | caus |  caus, pass
Polit. |  form, inform, pol elev | pol
Per. |  1, 2 | -
Num. | pl | -

Table 5: Feature schema comparison between UniMorph 4.0 Korean K-UniMorph.  Core case | nom |  nominative which marks the subject of a verb: 병원byeongwon (‘hospital’)/NNG + 이i (‘nom’)/JKS
---|---|---
| acc |  accusative which marks the object of a verb: 원인wonin (‘cause’)/NNG + 을eul (‘acc’)/JKO
Non-core, non-local case | dat |  dative which marks the indirect object: 국민gugmin (‘people’)/NNG + 에게ege (‘dat’)/JKB
| gen |  genitive which marks the possessor: 사회sahoe (‘society’)/NNG + 의ui (‘gen’)/JKG
| ins |  instrumental which marks means by which an action occurred: 대리석daeliseog (‘marble’)/NNG + 으로eulo (‘ins’)/JKB
| com |  comitative which marks the accompaniment: 망치mangchi (‘hammer’)/NNG + 와wa (‘com’)/JC
| voc |  vocative which indicate the direct form of address: 달dal (‘moon’)/NNG + 아a (‘voc’)/JKV
Local case | all |  allative which marks a type of locative grammatical case: 길gil (‘road’)/NNG + 로lo (‘all’)/JKB
| abl |  ablative which expresses motion away from something: 밑mit (‘bottom’)/NNG + 에서부터eseobuteo (‘abl’)/JKB
Comparison | cmpr |  comparative: 예상yesang (‘expectation’)/NNG + 보다boda (‘cmpr’)/JKB
Information structure | top |  topic which is what is being talked about: 사람salam (‘people’)/NNG + 은eun (‘top’)/JX
Table 6: Korean UniMorph schema for nouns.

###  3.3 Comparison with UniMorph 4.0 Korean

UniMorph 4.0 (Batsuren et al., 2022) includes a Korean dataset, which provides
2686 lemma and 241,323 inflected forms that are automatically extracted from
Wiktionary. It is mainly comprised of adjectives and verbs with totals of
52,387 and 188,821, respectively.333The counts are short of some numbers
because the errors, 92 forms without morphological schema, are excluded.
Thoroughly, we inspected the verbs in UniMorph 4.0 Korean to compare with
K-UniMorph: Among the 152,454 inflected forms of verbs in UniMorph 4.0 Korean,
there are only 16,489 forms that appear in 9.5M words of the Sejong corpus,
and 135,965 forms (89.18%) that never occur. UniMorph 4.0 Korean annotated all
verbs (v) as fin and all participles (v.cptp) as nfin. We can consider adding
fin for all verbs endings with ef (final verbal endings) and nfin for all
verbs ending with etm (adnominal endings, which are utilized for relative
clauses, modifiers, and a part of collocations). To inspect this, UniMorph 4.0
Korean provides the imperative-jussive modality imp which consists of 1;pl and
2, but it seems that Number (pl) occurs only with 1 (Person). While K-UniMorph
considers only 시si (an honorific for the agent) as pol, UniMorph 4.0 Korean
uses elev for 시si, and pol comes from verbal endings 요yo and 습니다seubnida with
either form or infm. However, form.elev is to elevate the referent. Therefore,
it should be with imp;2$|$3 and instead, form.humb can be introduced with
imp;1 for 습니다seubnida, and infm.elev$|$infn.humb for 요yo. Hence, K-UniMorph
provides a richer feature schema based on linguistics analysis. Table 5
summarises the different usage of the feature schema between UniMorph 4.0
Korean K-UniMorph.

##  4 Discussion and Future Perspectives

We have dealt with UniMorph schema for verbs, and obtained experimental
results for the morphological reinflection task using the different
representation forms of the word. Nouns in Korean have been considered by
separating postposition from the lemma of the noun instead of keeping
themselves together (e.g. 프랑스peulangseu (‘France’) and 의ui (‘gen’) instead of
프랑스의peulangseuui) in several grammar formalisms for Korean. However, in
addition to exogenously given interests such as inflection in
context,444https://sigmorphon.github.io/sharedtasks/2018/task2/ recent studies
insist the functional morphemes including both verbal endings and
postpositions in Korean should be treated as part of a word, with the result
that their categories do not require to be assigned individually in a
syntactic level (Park and Kim, 2023). Accordingly, it would be more efficient
to assign the syntactic categories on the fully inflected lexical word derived
by the lexical rule of the morphological processes in the lexicon. Therefore,
we will investigate how we adopt features for nouns such as cases including
non-core and local cases such as nom (nominative), acc (accusative),
comparison (cmpr), and information structure top (topic) (Table 6). It will
also include a typology of jkb (adverbial marker), which raises ambiguities.
An adverbial marker can represent ‘dative’ which marks the indirect object,
‘instrumental’ which marks means by which an action occurred, ‘allative’ which
marks a type of locative grammatical case, ‘ablative’ which expresses motion
away from something, or ‘comparative’ (cmpr, 예상yesang. We leave a detailed
study on nouns and other grammatical categories for future work. All datasets
of K-UniMorph are available at https://github.com/jungyeul/K-UniMorph to
reproduce the results.

## Acknowledgement

We would like to thank Ekaterina Vylomova and Omer Goldman at the UniMorph
project for their help and support. We also wish to thank three anonymous
reviewers for providing us with helpful feedback. This research was based upon
work partially supported by the Students as Partners Course Design Grants
through the Office of the Provost & Vice-President Academic at the University
of British Columbia to Eunkyul Leah Jo, and by Basic Science Research Program
through the National Research Foundation of Korea (NRF) funded by the Ministry
of Education (2021R1F1A1063474) to KyungTae Lim. This research was also
supported in part through computational resources and services provided by
Advanced Research Computing at the University of British Columbia.

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##  Appendix A Neural Experiment Description

We use the default setting of fairseq for the neural experiment for the Table
4 in $\S$3.2 as described in Table 7.

fairseq

fairseq-preprocess, fairseq-train and fairseq-interactive.

GPU

around 1 hour of GPU has been consumed for the training step for each
experiment.

Total runtime

It takes about 2 to 3 hours for completing one experiment including all steps
(preprocessing, training and evaluation).

Results

A single run with a seed number

task | translation
---|---
arch | transformer
dropout | 0.3
learning rate | 0.0001
lr-scheduler | inverse_sqrt
attention-dropout | 0.3
activation-dropout | 0.3
activation-fn | relu
encoder-embed-dim | 256
encoder-ffn-embed-dim | 1024
encoder-layers | 4
encoder-attention-heads | 4
decoder-embed-dim | 256
decoder-ffn-embed-dim | 1024
decoder-layers | 4
decoder-attention-heads | 4
optimizer | adam
adam-betas | (0.9, 0.98)
clip-norm | 1.0
warmup-updates | 4000
label-smoothing | 0.1
batch-size | 400
max-update | 20000
Table 7: Hyperparameter


Beneitez et al. (Phys. Rev. Fluids, 8, L101901, 2023) have recently discovered a new linear “polymer diffusive instability” (PDI) in inertialess rectilinear viscoelastic shear flow using the FENE-P model when polymer stress diffusion is present. Here, we examine the impact of inertia on the PDI for both plane Couette (PCF) and plane Poiseuille (PPF) flows under varying Weissenberg number $\textit{W}$, polymer stress diffusivity $\varepsilon$, solvent-to-total viscosity ratio $\beta$, and Reynolds number $\textit{Re}$, considering the FENE-P and simpler Oldroyd-B constitutive relations.  Both the prevalence of the instability in parameter space and the associated growth rates are found to significantly increase with $\textit{Re}$. For instance, as $Re$ increases with $\beta$ fixed, the instability emerges at progressively lower values of $W$ and $\varepsilon$ than in the inertialess limit, and the associated growth rates increase linearly with $Re$ when all other parameters are fixed. For finite $Re$, it is also demonstrated that the Schmidt number $Sc=1/(\varepsilon Re)$ collapses curves of neutral stability obtained across various $Re$ and $\varepsilon$. The observed strengthening of PDI with inertia and the fact that stress diffusion is always present in time-stepping algorithms, either implicitly as part of the scheme or explicitly as a stabiliser, implies that the instability is likely operative in computational work using the popular Oldroyd-B and FENE-P constitutive models. The fundamental question now is whether PDI is physical and observable in experiments, or is instead an artifact of the constitutive models that must be suppressed.

§ INTRODUCTION

The addition of polymers to a Newtonian solvent can induce dramatically different flow behaviours compared to those observed in the Newtonian fluid alone [Datta et al., 2022, Sánchez et al., 2022]. In industrial processes, for instance, viscous polymer melts are susceptible to instabilities which constrain the maximum extrusion rate [Petrie & Denn, 1976], while polymer additives are used in oil pipelines to reduce turbulent wall drag [Virk, 1975]. Two particularly important viscoelastic phenomena are the existence of `elastic turbulence' (ET), a chaotic flow state sustained in the absence of inertia [Groisman & Steinberg, 2000, Steinberg, 2021], and  `elasto-inertial turbulence' (EIT), an inherently two-dimensional state arising when both inertia and elasticity are present [Samanta et al., 2013, Sid et al., 2018, Choueiri et al., 2021]. While the initial pathway to ET in curvilinear geometries is understood [Larson et al., 1990, Pakdel & McKinley, 1996, Shaqfeh, 1996, Datta et al., 2022], relatively little is known about what happens in rectilinear viscoelastic flows.

Initial progress in characterizing ET in rectilinear situations arose through consideration of Kolmogorov flow over a 2-torus, where [Boffetta et al., 2005] found a linear instability leading to ET [Berti & Boffetta, 2010]. [Garg et al., 2018] subsequently discovered a centre-mode instability in viscoelastic pipe flow at finite Reynolds number $Re$, which was later also identified in plane Poiseuille flow (PPF, hereafter referred to as channel flow) [Khalid et al., 2021] but notably not in plane Couette flow (PCF). Interestingly, this instability could only be traced down to $Re=0$ in channel flow [Khalid et al., 2021, Buza et al., 2022]. The finite-amplitude state resulting from this instability is an `arrowhead' travelling wave [Page et al., 2020, Buza et al., 2022, Morozov, 2022] which has been observed in channel flow EIT [Dubief et al., 2022] and, in retrospect, ET in 2D Kolmogorov  flow [Berti & Boffetta, 2010]. In channel flow, efforts have began to establish a dynamical link between the arrowhead solution and both ET [Lellep et al., 2023] and EIT [Beneitez et al., 2023]. In the latter case in two-dimensions, there does not appear to be a simple dynamical pathway between these arrowhead solutions, where the dynamics is concentrated near the midplane, and EIT [Beneitez et al., 2023], which seems more dependent on a near-wall mechanism [Shekar et al., 2019, Shekar et al., 2021, Dubief et al., 2022].

The very recent discovery of a new wall-mode “Polymer Diffusive Instability” (PDI) in plane Couette flow at $Re=0$ [Beneitez et al., 2023], however, has added another intriguing possibility for the origin of ET. This instability is dependent on the existence of small but non-vanishing  polymer stress diffusion which is invariably present in any time-stepping algorithm, whether added explicitly to stabilise a numerical scheme like a spectral method (see e.g. [Dubief et al., 2023]) or arising implicitly such as through a finite difference formulation (see e.g. [Zhang et al., 2015, Pimenta & Alves, 2017]). The PDI wall mode is primarily confined to a boundary layer of thickness $\sqrt{\varepsilon}$, where $\varepsilon$ is the (small) diffusion coefficient, traveling at the wall speed with a streamwise wavelength on the order of the boundary layer thickness. The instability is robust to the choice of boundary conditions applied to the polymer conformation equation, and has growth rates which remain $O(1)$ as $\varepsilon \rightarrow 0$. Direct numerical simulations have demonstrated that PDI can lead to a sustained three-dimensional turbulent state, thus providing a potential mechanism for the origin of an ET-like state in the popular finitely-extensible nonlinear elastic constitutive model of Peterlin (FENE-P) [Beneitez et al., 2023].

While PDI has the potential to be a viscoelastic instability of significant importance, there is an important caveat: the instability emerges at small length scales which can, depending on the parameters, approach the order of the polymer gyration radius [Beneitez et al., 2023], violating the continuum approximation. There is thus a question of whether the instability is physical or actually an artifact of the Oldroyd-B and FENE-P models. Either possibility has important implications: if the instability is a physical phenomenon then it provides a pathway to ET and EIT, albeit one which will likely be challenging to establish experimentally due to the small length scales involved; or, it is an artificial feature of the popular FENE-P model, which can compromise its predictions. It thus appears important to now establish the prevalence of PDI across a much wider region of parameter space than was considered in the initial study of [Beneitez et al., 2023]. Therefore, we here map out the regions where PDI is operative at finite $Re$, considering both plane Couette flow (`PCF') and the more experimentally-relevant channel flow scenarios. Both the prevalence of PDI and the associated growth rates are found to significantly increase at finite $Re$ and are relatively insensitive to the bulk flow geometry. PDI is therefore a candidate to trigger both ET and EIT in simulations using the FENE-P model.

§ FORMULATION

We consider the following dimensionless equations governing the flow of an incompressible viscoelastic fluid:

Re\left(\frac{\partial\mathbf{u}}{\partial t}+\left(\mathbf{u}\cdot\nabla\right)\mathbf{u}\right)=-\nabla p+\beta\nabla^{2}\mathbf{u}+\left(1-\beta\right)\nabla\cdot\boldsymbol{\tau},
\quad
\nabla\cdot\mathbf{u}=0,
\eqno{(\theequation{\mathit{a},\mathit{b}})}
\label{eq:Gov1_NS}

\begin{equation}
\frac{\partial\mathbf{\mathbf{c}}}{\partial t}+\left(\mathbf{u}\cdot\nabla\right)\mathbf{c}+\boldsymbol{\tau}=\mathbf{c}\cdot\nabla\mathbf{u}+\left(\nabla\mathbf{u}\right)^{T}\cdot\mathbf{\mathbf{c}}+\varepsilon\nabla^{2}\mathbf{\mathbf{c}}, \label{eq:Gov2_Conf}
\end{equation}

where $\mathbf{u}=(u,v,w)$ and $p$ denote the velocity and pressure fields, respectively, and $\boldsymbol{\tau}$ denotes the polymeric contribution to the stress tensor. Following [Beneitez et al., 2023], the equations have been nondimensionalized using the channel half-width $H$ and a characteristic flow speed $U_{0}$, taken to be the wall speed in plane Couette flow or the centerline velocity of the base channel flow. In (<ref>a), the Reynolds number $Re:=U_{0} H/\nu_T$ describes the ratio of inertial to viscous forces (with $\nu_T = \nu_S + \nu_P$ denoting the total kinematic viscosity comprised of solvent and polymer components), and $\beta := \nu_S/\nu_T$ denotes the ratio of solvent to total viscosity. The polymeric stress tensor $\boldsymbol{\tau}$ may be described in terms of the polymer orientation through the conformation tensor $\mathbf{c}$ as in (<ref>). We emphasize that the inclusion of a polymer stress diffusion term $\varepsilon\nabla^{2}\mathbf{\mathbf{c}}$ (associated with diffusivity $\varepsilon := \left(ReSc\right)^{-1}$, where $Sc$ denotes the Schmidt number) is the crucial ingredient for the polymer diffusive instability (PDI) identified by [Beneitez et al., 2023]. In the inertialess limit, $\varepsilon=\textit{Sc}^{-1}$ when the governing equations are non-dimensionalized using viscous scales.

To close equations (<ref>-<ref>), the FENE-P constitutive model is used to relate $\boldsymbol{\tau}$ and $\mathbf{c}$:

\begin{equation}
\boldsymbol{\tau}:=\frac{f\left(\mathrm{tr}\,\mathbf{c}\right)\mathbf{c}-\mathbf{I}}{W},\ \ f\left(s\right):=\left(1-\frac{s-3}{L^{2}}\right)^{-1}, \label{eq:FENE-P}
\end{equation}

where $\mathbf{I}$ is the identity matrix, $L$ denotes the maximum extensibility of the polymer chains and $W:=U_{0} \lambda /H$, the Weissenberg number, describes the ratio of time-scales for polymer relaxation ($\lambda$) to the flow. In the limit $L\rightarrow \infty$, the simpler Oldroyd-B model is obtained. Inspection of (<ref>-<ref>) reveals five parameters of interest governing the flow dynamics: $Re$, $W$, $\beta$, $\varepsilon$, $L$.

We analyze the linear stability of (<ref>-<ref>) by perturbing them about their base state: $\mathbf{u}=\mathbf{U}+\mathbf{u}^{*}$, $p=P+p^{*}$, $\boldsymbol{\tau}=\mathbf{T}+\boldsymbol{\tau}^{*}$, $\mathbf{c}=\mathbf{C}+\mathbf{c}^*$. Our coordinate system ($x$, $y$, $z$) is aligned with the streamwise, wall-normal, and spanwise directions of the channel, respectively. The base state  $\left(U(y),C_{xx}(y),C_{yy}(y),C_{zz}(y),C_{xy}(y)\right)$ satisfies

f(tr𝐂)𝐂-εW𝐂”-2WU'C_xy+1     WU'C_yy     0

WU'C_yy     1     0

0     0     1

where primes indicates derivatives in the wall-normal ($y$) direction. We use pressure gradients $\partial_{x} P = 0$ and $\partial_{x} P = -2$ for plane Couette and channel flow, respectively. While we here compute the base flow accounting for the presence of a finite diffusivity $\varepsilon$ in (<ref>b), it is worth noting that the presence of diffusion in the base flow is not required to induce PDI; the instability still arises if $\varepsilon=0$ in (<ref>b) as the inclusion of $\varepsilon \neq 0$ does not significantly change the base state except in the limit of large $\varepsilon = \mathcal{O}(1)$.

Eigenvalue spectra, plotted in terms of the real $c_r$ and imaginary $c_i$ components of the complex wavespeed, obtained by solving the system (<ref>a-g) using the Oldroyd-B constitutive model at $Re=1000$, $\varepsilon=10^{-5}$ and $\beta=0.9$ for a) plane Couette flow ($W=44.7$, $k=37$) and b) channel flow ($W=22.1$, $k=52$). These parameters are chosen to roughly lie on the corresponding neutral curves plotted in Figure <ref>a. Results using bases of $N=300$ and $400$ Chebyshev polynomials are depicted by blue and red markers, respectively, to demonstrate convergence of the discrete unstable PDI modes highlighted in the zoomed inset above each panel.

Normal mode solutions of the perturbed flow are sought using the ansatz $\phi^{*}\left(x,y,t\right)=\tilde{\phi}\left(y\right)e^{ik\left(x-ct\right)}$, where real-valued $k$ denotes the streamwise wavenumber and $c=c_{r}+ic_{i}$ is a complex wavespeed, with instability arising if $c_i > 0$. The perturbed state is governed by the following system of seven equations for $(\tilde{u},\tilde{v},\tilde{p},\tilde{c}_{xx},\tilde{c}_{yy},\tilde{c}_{zz},\tilde{c}_{xy})$:

ikũ+ṽ' = 0,

Re(-ikcṽ+ikUṽ)+  p̃'-β(-k^2ṽ+ṽ”)-(1-β)(ikτ̃_xy+τ̃'_yy)=0,

[εk^2+ik(U-c)]c̃_yy+ṽC'_yy+τ̃_yy-εc̃”_yy               -2(ikC_xyṽ+C_yyṽ')=0,

[εk^2+ik(U-c)]c̃_zz+ṽC'_zz+τ̃_zz-εc̃”_zz                                                   =0,

[εk^2+ik(U-c)]c̃_xy+ṽC'_xy+τ̃_xy-εc̃”_xy      -ikC_xxṽ-U'c̃_yy-C_yyũ'=0.

By expanding the variables in (<ref>) in terms of a basis of Chebyshev polynomials, an eigenvalue problem is obtained which may be solved using standard Python libraries [Beneitez et al., 2023]. A basis of $N=300$ polynomials was used here to target the eigenvalues associated with PDI via an inverse iterations algorithm, which yielded sufficient convergence. Representative eigenvalue spectra are shown in Figure <ref> for plane Couette (PCF) and channel (PPF) geometries, illustrating the results using bases of both $N=300$ and $N=400$ Chebyshev polynomials to demonstrate convergence. The unstable PDI mode emerges with characteristic wavespeeds in the vicinity of $\left|c_{r}\right|\approx1$ and $c_r \approx 0$ for PCF and PPF geometries, respectively.

Following [Beneitez et al., 2023], we choose boundary conditions that set $\varepsilon=0$ at the walls in the governing equations to match the typical configuration used in direct numerical simulations (DNS). While $Sc\sim\mathcal{O}\left(10^{6}\right)$ is characteristic of polymer diffusion in a typical solvent like water (yielding $\varepsilon = 1/(ReSc) \approx 10^{-9}-10^{-6}$ depending on $Re$), in DNS much larger values of $\varepsilon \approx\mathcal{O}(10^{-3})$ are often employed to achieve numerical stability (see e.g. $\S$2.2 of Dubief et al., 2023). Thus, it is commonplace to set $\varepsilon=0$ at the walls in order to recover the idealised limit $\varepsilon\to 0$ [Sureshkumar et al., 1997, Samanta et al., 2013, Dubief et al., 2022].

§ RESULTS OF LINEAR STABILITY ANALYSIS

The results of our linear stability analysis are presented for both the Oldroyd-B ($\S$<ref>) and FENE-P ($\S$<ref>) constitutive relations, delineating the trajectory of neutral curves (characterized by growth rate $\sigma := kc_i=0$) associated with PDI in the four-dimensional parameter space spanned by $(Re, W, \beta,\varepsilon)$. For a given point in parameter space, we perform a sweep over wavenumbers $k$ to ensure that we are tracking the most unstable mode. The results for plane-Couette (PCF) and channel (PPF) geometries are overlaid on the same axes using blue and red colouring, respectively. To promote a collapse of the curves for both geometries, the Weissenberg ($W$) axis is scaled by the respective wall shear rates: $U_{\textrm{wall}}'=\left\{ 1\ (\textrm{PCF}),\ 2\ (\textrm{PPF})\right\}$. As described in $\S$<ref>, the PDI is a wall mode confined to a boundary layer of thickness $\mathcal{O}(\sqrt{\varepsilon})$ and so its behaviour will primarily be influenced by the wall shear. However, we will demonstrate that this collapse of geometries fails in certain scenarios (such as for large $\varepsilon$), when the boundary layer grows sufficiently large to be influenced by the non-uniform shear profile away from the wall. In the case of finite $Re$, we also demonstrate that curves associated with variable $Re$ and $\varepsilon$ may be collapsed based on $Sc = 1/(\varepsilon Re)$.

§.§ Oldroyd-B

Curves of neutral stability for plane Couette (`PCF', blue) and channel (`PPF', red) geometries, using the Oldroyd-B constitutive relation for five values of $\beta \in [0.7,0.98]$. a) The $Re$-$W$ plane for fixed $\varepsilon = 10^{-5}$, noting that the PCF and PPF curves are virtually indistinguishable. b) The $\varepsilon$-$W$ plane at three fixed $Re=\left\{ 0,1000,5000\right\}$. In panels a-b, the $W$ axis is scaled by the wall shear rate: $U_{\textrm{wall}}'=\left\{ 1\ (\textrm{PCF}),\ 2\ (\textrm{PPF})\right\} $. c) The streamwise wavenumber $k$ of the PDI, as a function of $\varepsilon$, along each of the neutral curves in panel b. d) A collapse of the $Re=\left\{ 1000,5000\right\}$ curves for both geometries from panel b) based on the inverse Schmidt number $1/Sc = \varepsilon Re$. By plotting $1/Sc$ rather than $Sc$, small $\varepsilon$ occurs at the bottom of panel d) facilitating an easier comparison with panel b). e) Colormap of the trace of the polymer conformation tensor $\mathrm{tr}(\mathbf{c})$ (red and blue denote positive and negative values, respectively), with contours of the streamfunction superimposed, for PPF eigenfunctions in the upper half channel with $\beta=0.9$, at locations indicated by the square and triangular markers in panels b-c. One wavelength $\lambda = 2\pi / k$ of each eigenfunction is shown.

The trajectory of neutral curves in the $Re$-$W$-$\varepsilon$ volume are presented in Figure <ref> for various $\beta\in\left[0.7,0.98\right]$. Figure <ref>a considers the $Re$-$W$ plane at a fixed $\varepsilon = 10^{-5}$, while Figure <ref>b considers the $\varepsilon$-$W$ plane at three fixed $Re\in\left\{ 0,1000,5000\right\}$. Regions of stability and instability are found to the left and right of each curve, respectively. For a given $\beta$, increasing $Re>0$ is found to promote instability at progressively lower values of both $W$ (Figure <ref>a) and $\varepsilon$ (Figure <ref>b). Notably, in the inertialess limit ($Re=0$), [Beneitez et al., 2023] found that the neutral curves tracked a roughly fixed $W \sim \beta / (1-\beta)$ for $\varepsilon \lesssim 10^{-2}$ (see their Figure 2b) leading to the disappearance of PDI at finite $W$ in the limit $\beta \rightarrow 1$. Here, for $Re>0$, our Figure <ref>b demonstrates that while for small $\varepsilon=\mathcal{O}(10^{-7})$ the neutral curves do indeed still track a constant $W$, they then begin to significantly deviate to lower $W$ beyond $\varepsilon \gtrsim 10^{-6}$, with the $Re=5000$ case deviating at lower $\varepsilon$ than the $Re=1000$ case. Inertial effects thus play a significant role in promoting a greater prevalence of PDI in the parameter space. In Figure <ref>d, we demonstrate that the $Re=\left\{ 1000,5000\right\}$ curves plotted in Figure <ref>b may be collapsed for both geometries based on the Schmidt number $Sc = 1/(\varepsilon Re)$.

The streamwise wavenumbers $k$ associated with the neutral curves in Figure <ref>b are plotted in Figure <ref>c as a function of $\varepsilon$. At $Re=0$, $k$ follows the $1/\sqrt{\varepsilon}$ scaling reported by [Beneitez et al., 2023] for all $\varepsilon$, whereas for $Re>0$, $k$ deviates significantly from this scaling to plateau to a roughly constant value for $\varepsilon \gtrsim 10^{-5}$, with the unstable modes at higher $Re$ being more tightly confined to the wall (as indicated by a larger $k$). This deviation from the $1/\sqrt{\varepsilon}$ scaling for $Re>0$ corresponds to the previously noted deviation of the neutral curves away from a constant $W$ in Figure <ref>b, where the curves begin turning to lower $W$ in the vicinity of $\varepsilon \approx 10^{-6}-10^{-5}$. The behaviour of $k$ in Figure <ref>c also explains the mismatch in the collapse of the PCF and PPF curves in Figure <ref>b at higher $\varepsilon$ for $Re=0$ (see solid lines). Specifically, at higher $\varepsilon$, $k$ becomes sufficiently small at $Re=0$ such that the instability is no longer strongly confined to the wall (see square eigenfunction, Figure <ref>e) and so the local wall shear, $U'_\textrm{wall}$, does not accurately describe the non-uniform shear profile influencing the instability. Conversely, the curves for $Re=\left\{ 1000,5000\right\} $ do remain collapsed at high $\varepsilon$, as the wavenumbers $k$ in Figure <ref>c plateau to sufficiently large values such that the instability remains confined to the wall (see triangle eigenfunction, Figure <ref>e). In Figure <ref>c, it is also worth noting that the prefactor of the PPF scaling (1/5) is roughly twice that of the PCF scaling (1/8), thus indicating that PDI is more tightly confined to the wall in the channel geometry.

§.§ FENE-P

We now consider how a finite polymer extensibility $L$ modifies the behaviour of the instability as compared to the Oldroyd-B case ($L\rightarrow \infty$) presented in $\S$<ref>. Focusing first on $L=200$, Figure <ref> illustrates the much richer behaviour of the neutral curves for the FENE-P case, presented in an analogous manner to Figures <ref>a-d. In the $Re$-$W$ plane (Figure <ref>a), a finite $L$ introduces two notable differences compared to Oldroyd-B. First, the neutral curves for a given $\beta$ now have a left and right branch and so the range of instability is bounded by an upper value of $W$. Second, there is now a critical value of $\beta$ ($\approx 0.865$ for both geometries) above which the neutral curves no longer intersect the $Re=0$ axis, at fixed $\varepsilon=10^{-5}$. Therefore, inertial effects are once again demonstrated to promote PDI, generating instability at finite $Re$ for ultra-dilute polymer solutions ($\beta \rightarrow 1$) that would otherwise remain stable at $Re=0$.

Neutral curves in the $\varepsilon$-$W$ plane for the inertialess case $Re=0$ are shown in Figure <ref>b, exhibiting notable differences to the Oldroyd-B $Re=0$ curves presented in Figure <ref>b. While for Oldroyd-B the plane Couette and channel curves adopt roughly the same shape, the FENE-P curves display dramatically different behaviours at large $\varepsilon$. Specifically, the plane Couette curves have an inverted `U'-shape, highlighting that the instability ceases to exist at large $\varepsilon$ for certain $\beta$ (e.g. see the $\beta=0.86$ curve which reaches a maximum value at $\varepsilon \approx 6\times 10^{-3}$). Conversely, the channel curves are roughly `U'-shaped, and the range of instability only increases with increasing $\varepsilon$.

Curves of neutral stability using the FENE-P constitutive relation with a fixed extensibility $L=200$, presented analogously to the Oldroyd-B curves in Figure <ref> for various $\beta$. Curves are shown in a) the $Re$-$W$ plane with a fixed $\varepsilon=10^{-5}$, b) the $\varepsilon$-$W$ plane for fixed $Re=0$, and c) the $1/Sc = \varepsilon Re$ versus $W$ plane, which collapses curves obtained at $Re=1000$ and $5000$ for both geometries. By plotting $1/Sc$ rather than $Sc$, small $\varepsilon$ occurs at the bottom of panel c) facilitating an easier comparison with panel b). Panel d) illustrates the dependence of the optimal streamwise wavenumber $k$ on $\varepsilon$ for the PPF curves in panel c) at $Re=1000$. Left and right branches of the curves in panel c) are distinguished here using solid and dashed lines, respectively. Comparison with the Oldroyd-B curves in Figure <ref>c reveals that the left branches behave similarly to the single Oldroyd-B branch, deviating from the $1/\sqrt{\varepsilon}$ scaling at large $\varepsilon$, while the right branches retain this scaling to the highest $\varepsilon$ considered.

At finite $Re$, as for the Oldroyd-B curves in Figure <ref>d, the FENE-P curves shown in Figure <ref>c at $Re=\left\{ 1000,5000\right\}$ are again found to collapse for both geometries based on the Schmidt number $Sc=1/(\varepsilon Re)$. Comparing Figures <ref>d and <ref>c reveals two main differences in the structure of the FENE-P and Oldroyd-B curves. First, for $\beta \approx 0.925$, in both geometries a pinch-off phenomenon occurs for the FENE-P case where the neutral curves form a “bubble” of instability within the $\varepsilon-W$ plane in an otherwise stable region. Second, in the limit of $\varepsilon \rightarrow 0$ ($Sc \rightarrow \infty$), the neutral curves behave differently as compared to the $Re=0$ case reported by [Beneitez et al., 2023]. In particular, there now appears to be a critical value of $\beta \approx 0.865$ (note the similarity to the critical $\beta$ in Figure <ref>a corresponding to lift-off from the $Re=0$ axis) at which the two branches of the neutral curve asymptotically approach each other in the limit of small $\varepsilon$. Curves associated with $\beta$ greater than this critical value are thus observed to turn back at higher $\varepsilon$ (see e.g. the $\beta=0.9$ curves, Figures <ref>c), suggesting that, at finite $Re$, PDI will not exist in the limit $\varepsilon \rightarrow 0$ for all $\beta$. We note another possibility, however, which is that an `hourglass'-like pinch-off behaviour occurs, in which the critical curves touch at some finite $\varepsilon$ (appearing here to be around $Sc \approx 10^{4}$), but then separate again for lower $\varepsilon$. The concave-up neutral curves (e.g. $\beta=0.9$) seen here, might then be reflected as concave-down branches at much lower $\varepsilon$ and instabilities for such $\beta$ could then still exist in the $\varepsilon \rightarrow 0$ limit. As $\varepsilon \rightarrow 0$ isolates the instability to an increasingly thin boundary layer at the wall, significantly increased computational power is required to resolve the neutral curves for $Sc > 10^{4}$ and so we do not further consider this behaviour here. It is also worth noting in Figures <ref>c that while increasing $Re$ does not significantly influence the horizontal position of the neutral curves along the $W$ axis, the increase in inertial effects does induce a significant downward shift of the curves in $\varepsilon$ (by a factor proportional to $Re$), thus increasing the prevalence of PDI in parameter space.

a) Dependence of the $\beta=0.87$ neutral curve presented in Figure <ref>a on variable $\varepsilon$, with fixed $L=200$. b) A collapse of the curves in panel a) based on the inverse Schmidt number $1/Sc = \varepsilon Re$, plotted analogously to Figures <ref>d and <ref>c such that small $\varepsilon$ appears at the bottom of the plot. The plane Couette curves (PCF) exhibit a perfect collapse and are virtually indistinguishable, whereas the $\varepsilon = 10^{-3}$ and $10^{-4}$ channel curves intersect the $Re=0$ axis in panel a) and thus diverge to infinite $Sc$ in panel b). c) Dependence of the $\beta=0.87$ neutral curve presented in Figure <ref>a on variable $L$, with fixed $\varepsilon = 10^{-5}$. d) Neutral curves in the $\beta$-$W$ plane for fixed $\varepsilon=10^{-5}$ and $Re=1000$.

The streamwise wavenumbers $k$ associated with the FENE-P channel (PPF) neutral curves at $Re=1000$ in Figure <ref>c are presented in Figure <ref>d as a function of $\varepsilon$, to compare with the scaling of the Oldroyd-B curves in Figure <ref>c. As for Oldroyd-B, the left branches follow the $1/\sqrt{\varepsilon}$ scaling reported by [Beneitez et al., 2023] for $\varepsilon \lesssim 10^{-5}$, when the neutral curves are roughly independent of $W$ in Figure <ref>c. In contrast, the right branches follow this $k$ scaling for the entire range of $\varepsilon$ considered here, thus explaining the slight mismatch in the collapse of the right branches of the two geometries at low $\varepsilon$ in Figure <ref>c (see e.g. the right branches of the $\beta=0.7$ and 0.8 curves); in this regime, $k$ has become sufficiently small such that the instability is no longer confined to the wall and thus the wall shear $U_{\textrm{wall}}'$ is not entirely suitable for scaling the $W$ axis. In Figure <ref>d, we also note that for $\beta \gtrsim 0.864$, the pinchoff value seen in Figure <ref>c at $Sc = \mathcal{O}(10^{4})$, the neutral curves turn back to higher $\varepsilon$ before the $1/\sqrt{\varepsilon}$ scaling regime is reached.

In Figure <ref>, we further consider the behaviour of the neutral curves presented in Figure <ref> by varying parameters that were previously held fixed. While the $Re$-$W$ plane was considered with a fixed $\varepsilon = 10^{-5}$ in Figure <ref>a, in Figure <ref>a we focus on the $\beta=0.87$ neutral curve and demonstrate its dependence on variable $\varepsilon$, where it is found that decreasing $\varepsilon$ pushes the region of PDI to progressively higher $Re$. In Figure <ref>b, we collapse the curves from Figure <ref>a based on the Schmidt number $Sc = 1/(\varepsilon Re)$. A near perfect collapse is observed for the plane Couette (PCF) curves, while some channel (PPF) curves diverge to $Sc \rightarrow \infty$ due to their intersection with the $Re=0$ axis in Figure <ref>a. Additionally, having only considered a fixed polymer extensibility $L=200$ in Figure <ref>, in Figures <ref>c,d we consider the influence of variable $L$ on the neutral curves in the $Re$-$W$ and $\beta$-$W$ planes, respectively. In Figure <ref>c, decreasing $L$ induces a turn-back of the neutral curves at progressively higher values of $Re$, decreasing the extent of PDI in the $Re$-$W$ plane. Figure <ref>d, at fixed $Re=1000$, displays the same qualitative features as those reported in the inertialess limit by [Beneitez et al., 2023] (see their Figure 2c), again demonstrating that decreasing $L$ decreases the prevalence of PDI in the $\beta$-$W$ plane.

§.§ Growth Rates

Growth rates $\sigma := kc_{i}$ of the most unstable mode as a function of Reynolds number $Re$ obtained at a fixed $W$ corresponding to either the intersection of the neutral curve with the $Re=0$ axis or the minimum of the neutral curve in $Re$ for cases in which there is no intersection with $Re=0$. Growth rates are shown for neutral curves spanning various $\beta$ in a) figure <ref>a (Oldroyd-B, fixed $\varepsilon=10^{-5}$) and b) figure <ref>a (fixed $L=200$, $\varepsilon=10^{-5}$), and neutral curves at fixed $\beta=0.87$ spanning c) various $\varepsilon$ in figure <ref>b (fixed $L=200$) and d) various $L$ in figure <ref>c (fixed $\varepsilon=10^{-5}$).

While we have thus far considered the behaviour of the neutral curves associated with PDI, it is also informative to consider how the growth rate of PDI evolves with $Re$ in regions of instability. Starting with each $\beta$ curve in Figure <ref>a (Oldroyd-B), we first fix the value of $W$ at which the neutral curve intersects the $Re=0$ axis, and where the growth rate is thus zero by definition. At this fixed $W$, we then increase $Re$ incrementally and track the growth rate of the most unstable PDI mode, as shown in Figure <ref>a. In Figure <ref>b, we repeat this process for the FENE-P curves presented in Figure <ref>a. As some FENE-P curves do not intersect the $Re=0$ axis, in these cases we fix $W$ to correspond to the minimum $Re$ of the neutral curve, and then increase $Re$ from there. In Figures <ref>c-d, we again repeat this process for the neutral curves plotted in Figures <ref>c-d, respectively.

In all cases, for neutral curves that intersect the $Re=0$ axis, the growth rate is observed to grow linearly with $Re$ as one moves away from the neutral curve, emphasizing the intensification of PDI due to the presence of inertia. Notably, the streamwise wavenumber $k$ remains virtually constant during this scaling, until $Re \gtrsim 10^3$ at which point the most unstable $k$ beings to vary and the linear scaling breaks down. For the subset of FENE-P curves that do not intersect the $Re=0$ axis (e.g. $\beta=0.87$, 0.88, 0.90 in Figure <ref>b), it is also intriguing that the growth rates display a dramatic increase with $Re$ to quickly join the linear $Re$ scaling of the curves that do intersect the $Re=0$ axis. We note that the relative vertical translation of the various curves in Figure <ref> is due to differences in slope between the neutral curves at their intersection with the $Re=0$ axis; at a fixed $W$, these differences in slope will govern the rate at which one moves away from the neutral curve as $Re$ is increased, and hence the observed difference in the growth rate magnitudes.

§ CONCLUSIONS

In this study, we have demonstrated that the polymer diffusive instability (PDI) is active in both plane Couette and channel flows with or without inertial effects, and that the instability intensifies with increasing Reynolds number $Re$. Through exploration of a variety of dimensionless parameters, we have found that PDI is operational across large regions of the parameter space including those relevant to many prior experiments [Choueiri et al., 2018, Qin et al., 2019, Choueiri et al., 2021, Jha & Steinberg, 2021]. In particular, increasing $Re$ enhances the prevalence of the instability, promoting instability at progressively smaller values of both $W$ and $\varepsilon$ than in the inertialess limit. Our results therefore significantly extend the conclusion of Beneitez et al., 2023 that PDI could also present a possible transition mechanism to EIT as well as ET in FENE-P fluids.

The eigenfunction for PDI is a wall mode, confined to a boundary layer of thickness $\sqrt{\varepsilon}$. As a result, the neutral curves for Couette and channel flow are found to nearly overlap in most regions of parameter space when $W$ is scaled by the wall-shear rate. This collapse breaks down when the streamwise wavenumber $k$ approaches $\mathcal{O}(1)$, as the instability is no longer confined to the wall and thus feels a non-monotonic shear profile in the channel's interior, as occurs for large $\varepsilon$ at $Re=0$ (but notably not at higher $Re$, see Figure <ref>c), and small $\varepsilon$ at high $W$ in FENE-P fluids. The finite extensibility of the polymer chains ($L$) is also found to have a significant impact on the prevalence of PDI, as compared to that predicted by the Oldroyd-B model. Considering $L=200$, we found that for sufficiently high $\beta$, the instability is suppressed at $Re=0$ and only appears at progressively larger $Re$ (Figure <ref>a). Similarly, beyond a critical value of $\beta$, the instability may also be suppressed at small values of $\varepsilon$ (Figure <ref>c). Given that PDI emerges as a wall mode, we have also confirmed its presence in cylindrical pipe flow as well as Taylor-Couette flow.

The lengthscale associated with PDI raises some intriguing questions. As indicated by Beneitez et al., 2023, PDI can emerge at a lengthscale roughly on the order of the polymer gyration radius, where the continuum assumptions of the FENE-P model may not hold. It is thus possible that PDI is an unphysical feature of the widely-used FENE-P model, which would have significant implications for computational studies of ET, EIT and polymer drag reduction using this model. Until now, such studies may have been unknowingly influenced by PDI, due to the ubiquity of stress diffusion in numerical schemes, either introduced explicitly as a regularisation term or arising implicitly through the discretization scheme. Assessing the relevance of PDI to real viscoelastic fluids is now a key challenge to be confronted.

[Funding]We are grateful for the support of EPSRC under grant EP/V027247/1.

[Declaration of interests]The authors report no conflict of interest.

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