Jérémy Danna
email: jeremy.danna@cnrs.fr

Frédéric Puyjarinet
email: frederic.puyjarinet@umontpellier.fr

Caroline Jolly
email: caroline.jolly@univ-grenoble-alpes.fr

Tools and Methods for Diagnosing Developmental Dysgraphia in the Digital Age: A Brief State-of-Art

Keywords: handwriting, developmental dysgraphia, product, process, diagnosis handwriting, developmental dysgraphia, product, process, diagnosis 1

published or not. The documents may come

Handwriting, considered as language by hand [START_REF] Berninger | Writing and reading: Connections between language by hand and language by eye[END_REF], is a complex perceptual-motor task involving attentional, perceptual, linguistic, and fine motor skills. Handwriting occupies a large proportion of children's daily activities at school [START_REF] Mchale | Fine motor activities in elementary school: Preliminary findings and provisional implications for children with fine motor problems[END_REF][START_REF] Mc Master | Handwriting in 2015: A main occupation for primary school-aged children in the classroom?[END_REF] and is the basis, together with reading, for the acquisition of higher-level skills such as spelling, grammar, syntax, and text composition. A relationship between the mastery of handwriting movement and the quality of writing content has been established, both at the semantic level in text production [START_REF] Jones | Relationship between automaticity in handwriting and studentsʹ ability to generate written text[END_REF] and at the orthographic level, in word formation [START_REF] Danna | Interaction between orthographic and graphomotor constraints in learning to write[END_REF]. If children pay too much attention to handwriting movements, they may have difficulties in the allocation of cognitive resources to higher-level processes [START_REF] Olive | Concurrent activation of high-and low-level production processes in written composition[END_REF].

From a developmental perspective, handwriting originates from drawing, from which it slowly differentiates as the child grows. In younger children, the quality of drawings is correlated to the quality of handwriting [START_REF] Marr | Handwriting readiness: Locatives and visuomotor skills in the kindergarten year[END_REF]. Then, with the acquisition of handwriting, this relationship between drawing and writing quality decreases to eventually disappear [START_REF] Karlsdottir | Problems in developing functional handwriting[END_REF]. The formal acquisition of handwriting begins around the age of 5 at preschool, and its mastering requires about 10 years of practice and training. The automation of handwriting is partial at the age of 10 (5th-grade) and is considered almost complete around the age of 14 (9th-grade) (for a review, see [START_REF] Palmis | Motor control of handwriting in the developing brain: A review[END_REF]). During acquisition, handwriting evolves first in terms of quality (primarily between 1st-and 5th-grade), then in terms of speed (from 4th-grade essentially). Efficient, fully automated handwriting relies on a balance between speed and quality: it should be fast enough to allow the retranscription of a course or the transcription of ideas, and of sufficient quality to be readable by the writer and by others.

Handwriting deficits

Despite correct learning and practice of handwriting, some children never master this skill to a sufficient level of automation (reviewed in [START_REF] Chung | Disorder of written expression and dysgraphia: definition, diagnosis, and management[END_REF][START_REF] Kalenjuk | A scoping review to map research on children with dysgraphia, their carers, and educators[END_REF][START_REF] Aiswarya | The modularity of dysgraphia[END_REF]. These handwriting deficits, referred to as developmental dysgraphia in children, have been defined as a written-language disorder that concerns mechanical writing skills in children of average intelligence and with no distinct neurological or perceptual-motor deficits [START_REF] Hamstra-Bletz | A longitudinal study on dysgraphic handwriting in primary school[END_REF]. Currently, dysgraphia is not recognised as a disorder per se by the Diagnostic and Statistical Manual of Mental Disorders, fifth edition (DSM-5) [START_REF]Diagnostic and statistical manual of mental disorders (DSM-5®)[END_REF] or the International Classification of Diseases 11th edition (ICD-11). The DSM-5 only mentions « deficits in the fine motricity required for handwriting », in the chapter dedicated to the development and evolution of learning disorders. Due to the diversity of methodological approaches and the absence of a consensual definition, the exact prevalence of dysgraphia is not known, and probably differs between countries and writing systems.

Dygraphia is generally found in association with neurodevelopmental disorders, namely dyslexia (DL), Developmental Coordination Disorder (DCD) and Attention Deficit Disorder/Hyperactivity Disorder (ADHD) [START_REF] Landau | ADHD and dysgraphia: underlying mechanisms[END_REF][START_REF] Barnett | Handwriting Difficulties in Developmental Coordination Disorder (DCD)[END_REF][START_REF] Biotteau | Developmental coordination disorder and dysgraphia: signs and symptoms, diagnosis, and rehabilitation[END_REF][START_REF] Capodieci | Handwriting difficulties in children with attention deficit hyperactivity disorder (ADHD)[END_REF][START_REF] Cohen | Handwriting in children with Attention Deficient Hyperactive Disorder: role of graphology[END_REF][START_REF] Di Brina | Handwriting in children with developmental coordination disorder: Is legibility the only indicator of a poor performance?[END_REF]. Dysgraphia preferentially affects boys (3:1 ratio), most likely because of the prevalence of the associated disorders in boys [START_REF] Chung | Disorder of written expression and dysgraphia: definition, diagnosis, and management[END_REF][START_REF] Berninger | Evidence-based diagnosis and treatment for specific learning disabilities involving impairments in written and/or oral language[END_REF]. Many studies have shown differences handwriting deficits depending on the associated disorder [START_REF] Afonso | Writing impairments in Spanish children with developmental dyslexia[END_REF][START_REF] Alamargot | Handwriting delay in dyslexia: children at the end of primary school still make numerous short pauses when producing letters[END_REF][START_REF] Huau | Graphomotor skills in children with developmental coordination disorder (DCD): Handwriting and learning a new letter[END_REF][START_REF] Johnson | Aquantitative comparison of handwriting in children with high-functioning autism andattention deficit hyperactivity disorder[END_REF][START_REF] Jolly | Dysgraphia differs between children with developmental coordination disorder and/or reading disorder[END_REF][START_REF] Sandler | Neurodevelopmental study of writing disorders in middle childhood[END_REF][START_REF] Sumner | Children with dyslexia are slow writers because they pause more often and not because they are slow at handwriting execution[END_REF]. DCD primarily affects handwriting quality [START_REF] Huau | Graphomotor skills in children with developmental coordination disorder (DCD): Handwriting and learning a new letter[END_REF][START_REF] Prunty | Accuracy and consistency of letter formation in children with developmental coordination disorder[END_REF][START_REF] Richards | Contrasting brain patterns of writing-relatedDTI parameters, fMRI connectivity, and DTI-fMRI connectivity correlations in children with and without dysgraphia or dyslexia[END_REF] while DL affects both speed and, to a lesser extent, handwriting quality [START_REF] Sumner | Children with dyslexia are slow writers because they pause more often and not because they are slow at handwriting execution[END_REF][START_REF] Gosse | Do children with dyslexia present a handwriting deficit? Impact of word orthographic and graphic complexity on handwriting and spelling performance[END_REF]. Children with comorbid DL and DCD present nearly the same profile of difficulties as children with DL, although with a much higher within-group variability. Comorbidity seems to lead to the addition of DCD and DL writing difficulties but without aggravation of the deficits in each of the two dimensions [START_REF] Jolly | Dysgraphia differs between children with developmental coordination disorder and/or reading disorder[END_REF] Given the central role of handwriting in the acquisition of other skills, these deficits can seriously hamper the acquisition of other skills [START_REF] Berninger | Relationships of attention and executive functions to oral language, reading, and writing skills and systems in middle childhood and early adolescence[END_REF][START_REF] Graham | The role of self-regulation and transcription skills in writing and writing development[END_REF][START_REF] Nielsen | Occupational therapy interventions for children with attention deficit disorder: A systematic review[END_REF]. It has been shown that, at equal content, worst quotes are attributed to less legible school works [START_REF] Markham | Influences of handwriting quality on teacher evaluation of written work[END_REF], resulting in a decrease in the child's self-esteem. Dysgraphia may thus impact the academic success of the child if it is not diagnosed and handled early [START_REF] Engel-Yeger | Handwriting performance, selfreports, and perceived self-efficacy among children with dysgraphia[END_REF][START_REF] Graham | Writing characteristics of students with attention deficit hyperactive disorder: A meta-analysis[END_REF]. To this end, different tools are available to allow researchers and clinicians to analyze the two dimensions of handwriting: the final product of handwriting and the dynamic process of handwriting that generates the trace [START_REF] Rosenblum | Product and process evaluation of handwriting difficulties[END_REF].

Evaluating the handwriting product refers to the static, spatial features of the written trace. This kind of analysis is performed afterwards. This is the principle of many tests used in different countries (for a review see [START_REF] Chung | Disorder of written expression and dysgraphia: definition, diagnosis, and management[END_REF]). The quality of the trace is evaluated based on different features such as letter size and form, spatial organization of handwriting on the paper sheet, margin, etc.

Evaluating the handwriting process refers to the analysis of dynamic, kinematic and temporal features of handwriting. Several type of variables can be analyzed, depending on the tools used for the evaluation: posture, finger and arm movements, pen grip and finger pressure on the pen, in-air and on-paper durations, pen velocity, pen pressure, etc. The increasing number of publications on the analysis of the handwriting process over the past years attests for the growing interest of researchers for this field (e.g., [START_REF] Asselborn | Extending the spectrum of dysgraphia: A data driven strategy to estimate handwriting quality[END_REF][START_REF] Drotár | Dysgraphia detection through machine learning[END_REF][START_REF] Guilbert | Handwriting on a tablet screen: Role of visual and proprioceptive feedback in the control of movement by children and adults[END_REF][START_REF] Rosenblum | Identifying developmental dysgraphia characteristics utilizing handwriting classification methods[END_REF].

The objective of this quasi-systematic review is to make a concise listing of the tools and methods which are the most reported in the literature for the analysis of handwriting and the diagnosis of dysgraphia. Tools focusing on both the final handwriting product and the handwriting process will be considered. We will then discuss the pros and cons of the existing tools, and the perspectives for the development of future tools.

Handwriting tools based on the product

In order to list the diagnosis tools based on the analysis of the handwriting product, we searched in the two scientific browsers PubMed and Google Scholar using the following keywords: Handwriting, Assessment, Test, Tool, Quality, Evaluation, Battery, Children, Students, and Questionnaire.

The tools meeting our search criteria are listed in Table 1. We included only tools for which the following data were available: norms or age class, type of task, subdomain analyzed, criteria evaluated. Although mainly designed for a developmental population (from the age of 5), some diagnosis tools can also be used in adults up to the age of 80 (QNST-3; [START_REF] Mutti | QNST-3R: Quick Neurological Screening Test[END_REF]). The test duration is variable, from a few minutes up to 30 minutes. This parameter is interesting because deficits may not be visible during the first few minutes of handwriting, but may appear during a continuous handwriting task, as it is the case in the classroom. The tasks used in the tests are of three main types: copying a text or a sentence, writing under dictation (letters, digits, words, or text), and spontaneous writing. These complementary tasks explore different aspects of handwriting. The copy task is the easiest, and can be used with beginner writers. Moreover, it resembles the condition of the classroom, where children are often asked to copy texts. However, the reading component can pose problems for children with dyslexia, introducing a possible bias in the interpretation of the test results. The dictation task is ecologic too, without the reading component, but the spelling processes and the orthographic components may again pose problems for children with dyslexia. Finally, the spontaneous writing task is likely to be the most relevant task. The difficulty here is the establishment of norms, since the texts produced are all unique. General criteria of legibility and quality are thus used in this case, which may provide a less fine-grained analysis of handwriting.

It should be noted that one test includes an analysis of texts produced at school, the TOLH (Test Of Legible Handwriting, [START_REF] Larsen | Test of legible handwriting: An ecological approach to holistic assessment[END_REF]). Two others include writing from memory: the ETCH-M (Evaluation Tool of Children's Handwriting -Manuscript, [START_REF] Amundson | Evaluation Tool of Children's Handwriting; OT KIDS[END_REF]) and the MMHAP (Mac Master Handwriting Assessment Protocol, [START_REF] Pollock | The McMAster Handwriting Assessment Protocol -2 nd[END_REF]). Two tests also add another level of analysis, thanks to two conditions in the copy task: normal speed and maximum speed (the DASH, [START_REF] Barnett | Development and standardization of a new handwriting speed test: The Detailed Assessment of Speed of Handwriting[END_REF]). This approach is particularly interesting since it mimicks certain classroom conditions, and it is well known that adding constraints (temporal or spatial) during handwriting helps revealing handwriting deficits [START_REF] Chartrel | The impact of spatio-temporal constraints on cursive letter handwriting in children[END_REF][START_REF] Fitzpatrick | The influence of instructional variables and task constraints on handwriting performance[END_REF]. Combining different tasks and/or conditions can provide a fine and detailed analysis of handwriting. It is worth noting that although these tasks are complementary, only three tests involve all three types: the BVSCO-3 [START_REF] Cornoldi | BVSCO-3 Batteria per la Valutazione clinica della Scrittura e della Competenza Ortografica-3 [BVSCO-3, Battery for the assessment of writing and spelling accuracy[END_REF], the ETCH-M [START_REF] Amundson | Evaluation Tool of Children's Handwriting; OT KIDS[END_REF], and the MMHAP [START_REF] Pollock | The McMAster Handwriting Assessment Protocol -2 nd[END_REF].

The majority of the tests listed in Table 1 analyze handwriting quality using different criteria such as legibility, letter form, spatial organization of letters or words, alignment, etc. Some tests also measure handwriting speed by evaluating the number of characters or letters (BHK, [START_REF] Hamstra-Bletz | Beknopte beoordelingsmethode voor kinderhandschriften/ Concise evaluation scale for children's handwriting[END_REF]; French adaptation, [START_REF] Charles | BHK -Echelle d'évaluation rapide de l'écriture chez l'enfant[END_REF]; BHK-ado, [START_REF] Soppelsa | Editions du Centre de Psychologie Appliquée[END_REF]; BVSCO-3, [START_REF] Cornoldi | BVSCO-3 Batteria per la Valutazione clinica della Scrittura e della Competenza Ortografica-3 [BVSCO-3, Battery for the assessment of writing and spelling accuracy[END_REF] ; CHES-M, [START_REF] Phelps | The Children's Handwriting Evaluation Scale for manuscript writing[END_REF]; ETCH-M, [START_REF] Amundson | Evaluation Tool of Children's Handwriting; OT KIDS[END_REF]; EVEDP, [START_REF] Pouhet | L'évaluation de la vitesse d'écriture manuelle à l'aide d'une dictée de niveau progressif: l'EVEDP[END_REF]; MMHAP, [START_REF] Pollock | The McMAster Handwriting Assessment Protocol -2 nd[END_REF]; MHA, [START_REF] Reisman | Development and reliability of the research version of the Minnesota Handwriting Test[END_REF][START_REF] Reisman | Minnesota Handwriting Assessment[END_REF]) or the number of words (DASH, [START_REF] Barnett | Development and standardization of a new handwriting speed test: The Detailed Assessment of Speed of Handwriting[END_REF]; EVEDP, [START_REF] Pouhet | L'évaluation de la vitesse d'écriture manuelle à l'aide d'une dictée de niveau progressif: l'EVEDP[END_REF]) produced in a fixed period of time. Since a universal, gold standard test for the diagnosis of dysgraphia is not available, it is sometimes necessary to combine several tests to perform an optimal clinical assessment. The DASH test appears to be the most complete one, since it includes various types of tasks, different constraints of writing and it requires about 15 minutes of writing. Its weakness is that it only evaluates handwriting speed.

Finally, we also mention in Table 1 a couple of questionnaires, which can be interesting to use in complement to the other tests (HPSQ, [START_REF] Rosenblum | Development, reliability, and validity of the Handwriting Proficiency Screening Questionnaire (HPSQ)[END_REF]; « questionnaire for children », [START_REF] Santamaria | Troubles graphomoteurs chez les enfants d'intelligence supérieure[END_REF]). Indeed, these questionnaires provide information about the evaluation of handwriting quality by the teacher or the child himself, which can be useful in the perspective of a rehabilitation program.

Another important point to consider when choosing which test to use is the existence of standards. The Table 2 presents the psychometric properties of the main tests used both in research and in clinical practice. A number of tests have relatively inter-rater and test-retest reliabilities (the French adaptation of the BHK for example, [START_REF] Charles | BHK -Echelle d'évaluation rapide de l'écriture chez l'enfant[END_REF]), while others reach high validity-related standards (the MHA, [START_REF] Reisman | Development and reliability of the research version of the Minnesota Handwriting Test[END_REF], and the TOLH [START_REF] Larsen | Test of legible handwriting: An ecological approach to holistic assessment[END_REF] for example). More recently, a few computerized diagnostic tools based on the analysis of the final product of handwriting have also been developed. They are listed in Table 3.

These algorithms are all based on pattern recognition methods using images of letters, digits, words or sentences. They use a large database of images, from which characteristic features of « poor writing » are extracted and analyzed using machine learning approaches. The performances of computer tools are evaluated using a series of criteria. The precision, also called positive predictive value, is defined as the number of correct classifications of dysgraphic children divided by the total number of classifications. The sensitivity represents the true-positive detection rate (correct classification of children with dysgraphia). The specificity represents the true-negative detection rate (correct classification of typically developing children). As shown in Table 3, the performance of these classification tools is below that of the paper-andpen tools listed above (73% for [START_REF] Isa | Automated detection of dyslexia symptom based on handwriting image for primary school children[END_REF], 79.7% for [START_REF] Skunda | Method for Dysgraphia Disorder Detection using Convolutional Neural Network[END_REF]). The only exception is TestGraphia, the algorithm developed by Dimauro et al. [START_REF] Dimauro | a software system for the early diagnosis of dysgraphia[END_REF], with good performances very closed to that of the original BHK test. It analyses the same criteria as the original BHK test [START_REF] Di Brina | Test BHK-Scala sintetica per la valutazione della scrittura in età evolutiva[END_REF], but using scanned images of the BHK texts. The sensitivity of TestGraphia is 83%, and its specificity is 98%. This algorithm seems thus very promising for the future development of computerized diagnostic tools.

Handwriting tools based on the process

Collecting the spatio-temporal characteristics of a written trace has become possible thanks to the development of digital tablets. The principle is simple: the tablet records the x, y and sometimes z (up to 2 cm) positions of the pen with a high frequency (every 5 or 10 milliseconds), as well as the time, the pen pressure, and the angle of the pen with the tablet. From these data, a large variety of static (size, alignment…), kinematic (speed, acceleration, jerk…) and dynamic (pen pressure, pen tilt…) features can be calculated. To avoid the undesirable effects of loss of surface roughness (e.g., [START_REF] Drotár | Dysgraphia detection through machine learning[END_REF]), a sheet of paper must be attached to the digital tablet and an ink pen compatible with the tablet must be used.

Over the last decades, a growing number of studies have focused on the development of tools for the diagnosis of dysgraphia using digital tablets. In this review, we present a non-exhaustive overview of these tools, which are not yet available to clinicians (Table 4). The different digital tools for the diagnosis of dysgraphia presented in Table 4 combine dynamic, kinematic, and static features extracted from the handwritten tracks. These features are then analyzed mainly using machine learning approaches to classify the data (i.e. classifiers). These tools differ by the nature of the tasks analyzed (handwriting or graphomotor tasks), the size of the dataset, and the computational approach used to analyze the data.

Of the 22 studies reported here, four use graphomotor tasks, the others use handwriting alone or a combination of handwriting and drawings. It is interesting to mention that several studies use tasks that have been validated in clinical practice, such as the BHK [START_REF] Asselborn | Extending the spectrum of dysgraphia: A data driven strategy to estimate handwriting quality[END_REF][START_REF] Asselborn | Automated Human-Level Diagnosis of Dysgraphia Using a Consumer Tablet[END_REF][START_REF] Deschamps | Development of a pre-diagnosis tool based on machine learning Algorithms on the BHK test to improve the diagnosis of dysgraphia[END_REF][START_REF] Gargot | Acquisition of handwriting in children with and without dysgraphia: A computational approach[END_REF], the BVSCO2 [START_REF] Dui | Digital tools for handwriting proficiency evaluation in children[END_REF], or the Minnesota Handwriting Assessment (MHA, [START_REF] Falk | On the development of a computer-based handwriting assessment tool to objectively quantify handwriting proficiency in children[END_REF]).

The size of the dataset used varies between 35 and 580 participants, and the children included in the different studies are between 5 and 15 years of age.

Nine studies use classical statistical comparisons to identify discriminative features between groups (in blue in Table 4; [START_REF] Asselborn | Extending the spectrum of dysgraphia: A data driven strategy to estimate handwriting quality[END_REF][START_REF] Dui | A tablet app for handwriting skill screening at the preliteracy stage: Instrument validation study[END_REF][START_REF] Dui | Digital tools for handwriting proficiency evaluation in children[END_REF][START_REF] Falk | On the development of a computer-based handwriting assessment tool to objectively quantify handwriting proficiency in children[END_REF][START_REF] Galaz | Advanced parametrization of graphomotor difficulties in school-aged children[END_REF][START_REF] Gargot | Acquisition of handwriting in children with and without dysgraphia: A computational approach[END_REF][START_REF] Herstic | Validity of an automated handwriting assessment in occupational therapy settings[END_REF][START_REF] Rosenblum | Automatic segmentation as a tool for examining the handwriting process of children with dysgraphic and proficient handwriting[END_REF][START_REF] Zvoncak | Fractional order derivatives evaluation in computerized assessment of handwriting difficulties in school-aged children[END_REF]. The others (in black in Table 4) use different algorithms of machine learning (Random Forest, Support Vector Machine, Convolutional Neuron Network, etc.) to classify the children into different groups. These methods are called « supervised learning approaches » since the algorithm is trained to identify groups which were previously labeled. Most of the studies reported here present a simplistic classification of children in two groups: with or without dysgraphia. Only one study classifies the children into four groups: typically developing, with mild dysgraphia, with mean dysgraphia, and with severe dysgraphia [START_REF] Sihwi | Dysgraphia identification from handwriting with Support Vector Machine method[END_REF]. This approach is interesting since it considers dysgraphia as a continuum of severity. This is probably closer to the reality than a dichotomic classification, as recently suggested by Lopez & Vaivre-Douret [START_REF] Lopez | Exploratory investigation of handwriting disorders in school-aged children from first to fifth grade[END_REF], who described 3 levels of handwriting disorders in children from 1 st -to 5 th -grade: mild disorder, moderate disorder, and dysgraphia.

The tools based on the analysis of handwriting samples obtained the best classification performance. For example, Asselborn et al. [START_REF] Asselborn | Automated Human-Level Diagnosis of Dysgraphia Using a Consumer Tablet[END_REF] reached a sensitivity of 96.6% and a specificity of 99%, and Mekyska et al. [START_REF] Mekyska | Identification and rating of developmental dysgraphia by handwriting analysis[END_REF] reached a sensitivity of 96%. It is worth noting, however, that the excellent performances obtained in [START_REF] Asselborn | Automated Human-Level Diagnosis of Dysgraphia Using a Consumer Tablet[END_REF] must be considered with caution since they may be biased by the fact that the authors only included participants with severe dysgraphia [START_REF] Deschamps | Methodological issues in the creation of a diagnosis tool for dysgraphia[END_REF]. The most discriminative features between children with and without dysgraphia vary among the studies, but generally include a larger size in dysgraphic handwriting, numerous velocity variations, a lower mean speed, increased lift and stop duration, and variations in the pen angle with the tablet.

The tools based on the analysis of drawing samples appear promising too, although their performances are slightly lower than those based on handwriting. For instance, the algorithm developed by Mekyska et al. [START_REF] Mekyska | Computerised assessment of graphomotor difficulties in a cohort of school-aged children[END_REF] obtained a sensitivity of 90%. The idea that a dysgraphia can be identified based on graphomotor tasks suggests that dysgraphia can be independent from higher order processes, namely linguistic ones. Developing diagnostic tools based on drawings is interesting for two reasons: these tools would be more universal since they are independent of the language and the alphabet, and they can be used with younger children to identify « at-risk » children, which could be handled earlier.

Developing a computer tool for the diagnosis of dysgraphia is not trivial, as attested by the variability in the performance of the tools presented in Table 4. Several reasons can explain these differences. First, the variety of the tasks used and the number of participants leads to large differences in the size of the databases, which is a critical determinant in a classifier's performances. Second, a large panel of machine learning approaches were used, with a different number of features analyzed among studies. Although certain classification methods appear better than others (Random Forest for example), none reached excellent performances nowadays. Since the interest of researchers for these tools is growing, it seems obvious that their efficiency will rapidly be improved. To do so, however, a number of key elements will be important to consider. First, it will require the constitution of large databases of handwriting and drawing samples from children which are perfectly characterized from a clinical point of view. It will also be necessary to estimate the severity of dysgraphia, and not only provide a dichotomic classification of children with or without dysgraphia, as proposed by Sihwi et al. [START_REF] Sihwi | Dysgraphia identification from handwriting with Support Vector Machine method[END_REF]. Moreover, other processes involved in handwriting, such as visuomotor aspects, which are currently investigated [START_REF] Lambert | Analysis of Eye Movements in Children with Developmental Coordination Disorder During a Handwriting Copy Task[END_REF], would be interesting to include in future diagnostic tools. Finally, it is also worth noting that diagnostic tools fully integrated into the pen and using machine learning approaches are also under investigation [START_REF] Lopez | Validity of kinematics measures to assess handwriting development and disorders with a graphomotor task[END_REF][START_REF] Bublin | Handwriting evaluation using deep-leraning with SensoGrip[END_REF][START_REF] Lopez | Concurrent and predictive validity of a cycloid loops copy task to assess handwriting disorders in children[END_REF].

Perspectives: Towards a universal, standardized test of dysgraphia?

Although very promising, none of the computer and paper-and-pen tools presented above is fully satisfactory and sufficient to provide a completely reliable diagnosis of dysgraphia. In addition, most of the tools available to clinicians today do not give precise information about the specific handwriting difficulties of each child.

In this context, it appears interesting to think about developing a reliable, comprehensive and universal diagnostic tool for dysgraphia. Several important points need to be considered for the development of such an instrument. First, an "ideal" diagnostic tool will probably combine computer and paper-and-pen approaches, since they are complementary and provide distinct information on the writing process and product, respectively. A fully computerized tool could also be envisaged, provided that it is complemented by the assessment of the clinician, who must remain the reference assessor. Indeed, the spreading of tablets and the rapidity of computerized analyses could allow the collection of written samples in school or at the children's house, which could then be sent to a clinician. Standard pen-and-paper tools could subsequently be used in case the computer tools detected a risk for dysgraphia in the child's handwritten productions, in order to firmly confirm the diagnosis. In this perspective, the goal of the computer tools is thus not to replace the clinician and the existing, validated tests, but to help in screening larger populations of children and in facilitating the clinician diagnosis (Figure 1). In addition, these tools provide valuable information on the process of handwriting himself, by identifying dynamic or kinematic features which may be altered in each particular child. This information would be very relevant for the clinician, since it would offer cues for an individualized rehabilitation of handwriting. Second, using a combination of tasks targeting different skills seems crucial to provide more information about handwriting difficulties. Indeed, some children with dysgraphia may succeed certain tasks and thus be undiagnosed if only a single one is used. Combining different tasks in a unique test would thus greatly increase its efficacy, as previously suggested by Safarova et al. [START_REF] Safarova | Developmental dysgraphia: A new approach to diagnosis[END_REF]. Namely, the test should include spontaneous handwriting, copying of words and/or sentences, writing to dictation, digits writing, writing under speed and accuracy constraints, and drawing and/or graphomotor tasks. Temporal (i.e. speed) or spatial (i.e. size) constraints add a cognitive load and are known to increase handwriting difficulties [START_REF] Capodieci | Handwriting difficulties in children with attention deficit hyperactivity disorder (ADHD)[END_REF][START_REF] Chartrel | The impact of spatio-temporal constraints on cursive letter handwriting in children[END_REF][START_REF] Fitzpatrick | The influence of instructional variables and task constraints on handwriting performance[END_REF]. With regards to the spontaneous production task, we could, for example, ask to write a 7-sentence text corresponding to the writer's ideal weekly schedule. This would enable a specific analysis to me made of the days of the week, which would be common to all texts produced. As mentioned above, the addition of graphomotor and/or drawings tasks, which are language-independent, will enable to target younger children than with the existing tests and thus detect and handle earlier children "at-risk" of dysgraphia. In addition, it would provide a universal test, allowing comparisons between countries and alphabetic systems. In addition, the test needs to last at least 20 minutes in order to enhance the difficulty of the task and induce fatigue. Finally, completing the test by a self-questionnaire will enable the clinician to better characterize the difficulties experienced by the writer.

Thirdly, the choice of the cohort of participants will be crucial. A large developmental window ranging from 5 to at least 15 years old should be included, and the content of the test should be adapted depending on the age and/or class of the child, and the level of handwriting automation. The number of participants should be important enough to allow machine learning approaches. It would also be important to include children presenting a dysgraphia in various clinical contexts and precisely characterized from a clinical perspective. This would enable to evaluate the severity of dysgraphia, which could eventually be an additional evaluation criterion provided by the diagnostic tool. Finally, participants should be recruited in multiple sites representative of different socioeconomic and educational statuses.

Developing such a complete diagnostic tool implies to collect large databases of handwriting and drawing samples in different places around the world. This would be possible by the implication of a consortium of laboratories and clinicians. Besides the diagnostic tool itself, the benefits of these developments would be twice: (i) from a clinical perspective, it would allow to estimate the prevalence of dysgraphia in different countries and it would further tailor rehabilitation programs to the characteristics of handwriting difficulties, and (ii) from a research perspective, it would provide large annotated databases that could be freely available to researchers working in the fields graphonomics, whether in educational, clinical, or human movement sciences.

Figure 1 .

 1 Figure 1. Examples of the complementary use of paper-and-pen tests and computerized tools for the diagnosis and rehabilitation of handwriting deficits.

Table 1 .

 1 List of the most commonly used tools for the diagnosis of dysgraphia in children based on the analysis of the handwriting product (presented in alphabetical order). N.A.: not available.

	Test name	Reference	Ages / Classes	Duration of test	Language	Task(s)	Subdomains
	BHK: Brave Handwriting Kinder	[43]	1st-to 5th-grade	5 mn	Multi-language	Copy	Quality Speed
	BHK Ado: Rapid writing
	evaluation scale for adolescents [Echelle d'évaluation rapide de	[44]	6th-to 9th-grade	5 mn	French	Copy	Quality Speed
	l'écriture chez l'adolescent]
	BVSCO-3 -Test for the Evaluation of Writing and Orthographic Ability, 3 rd ed.	[45]	6-14 y	Variable	Multi-language	Copy production Dictation Spontaneous	Speed % of errors
	CHES -Childrenʹs Handwriting Evaluation Scale	[46]	3rd-to 8th-grade	2 mn	English	Copy	Quality Fluency
	CHES-M -Childrenʹs
	Handwriting Evaluation Scale -Manuscript Writing	[47]	1st-to 2 nd -grade	2 mn	English	Copy	Quality Fluency
						Alphabet copy at
	DASH -Detailed Assessment of Speed of Handwriting	[48]	9-16 y	14 mn	English	normal and high Spontaneous speed	Speed
						production
	DRHP -Diagnosis and Remediation of Handwriting Problems	[49]	From 3rd-grade	Variable	English	Spontaneous images observation production from	Quality
						Copy
	ETCH-M -Evaluation Tool of					Dictation
	Childrenʹs Handwriting -Manuscript	[50]	1st-to 2 nd -grade	15-20 mn	English	Spontaneous production	Quality Speed
						Handwriting from
						memory
	EVEDP -Evaluation de la Vitesse d'Ecriture -Dictée Progressive	[51]	2 nd -to 5th-grade	Variable	French	Dictation	Speed
	HHE -Hebrew Handwriting Evaluation	[52]	6-18 y	5 min	Hebrew	Alphabet Copy of a text	Quality Speed
	HLS -Handwriting Legibility Scale	[53]	9-14 y	10 mn	English	Spontaneous production	Quality
						Copy
	MMHAP -Mac Master Handwriting Assessment Protocol	[54]	Preschool to 6th-grade	Variable	English	Dictation Handwriting from Spontaneous production	Quality Speed
						memory
	MHA -Minnesota Handwriting Assessment	[55,56]	1st-to 2 nd -grade	2 mn ½	English	Alphabet Copy	Quality Speed
	QNST-3 Revised -Quick Neurological Screening Test, 3 rd	[57]	5-80 y	30 mn	English	Copy	Quality
	ed. Revised
	SCRIPT-Scale of Children's Readiness in PrinTing	[58]	N.A.	3-8 mn	English	Copy	Quality
						Spontaneous
	TOLH -Test Of Legible Handwriting	[59]	2 nd -to 12th-grade	Variable	English	Text composition at production	Quality
						school
	THS-R	[60]	6-18 y	N.A.	English	Alphabet Copy	Quality
	HPSQ -Handwriting Proficiency Screening Questionnaire	[61]	7-14 y	N.A.	Multi-language	Questionnaire	Quality
	Questionnaire for children	[62]	1st-to 5th-grade	2-3 mn	French	Auto-questionnaire	Self-report on handwriting

Table 2 .

 2 Psychometric properties of the main diagnosis tools for handwriting assessment. BHK: Brave Handwriting Kinder; CHES-M: Children's Handwriting Evaluation Scale for Manuscript writing; DASH: Detailed Assessment of Speed of Handwriting; DRHP: Diagnosis and Remediation of Handwriting problems; ETCH-M: Evaluation Tool for Children's Handwriting-Manuscript; HHE: Hebrew Handwriting Evaluation; HPSQ: Handwriting Proficiency Screening Questionnaire; THS-R: Test of Handwriting; TOLH: Test Of Legible Handwriting; N.A.: not available.

	Test name [ref]	Number of participants	Country of validation	Inter-rater validity	Test-retest validity	Internal consistancy
	BHK [13,43]	121	Netherlands	.71 to .89	.74 to .86	N.A.
	BHK -French adaptation [65]	837	France	.68 to .90	.80 to .92	N.A.
	BHK Ado [44]	471	France	.24 to .66	N.A.	N.A.
	BHK -Italian adaptation [66,67]	562	Italy	.82 to .93 for speed .42 to .63 for quality	N.A.	N.A.
	CHES-M [47]	643	USA	.85 to .93	N.A.	N.A.
	DASH [48]	1163	Netherlands	.85 to .99	.50 to .92 .81 .88 to .94
	DRHP [49]	300	UK	.61 to .65	N.A.	N.A.
	ETCH-M [50]	N.A.	N.A.	.75 to .92	.63 to .77	N.A.
	HHE [52]	N.A.	Israël	.75 to .79	N.A.	N.A.
	HPSQ [61]	230	Israël	.92	.84	.90
	MHA [55,56]	N.A.	USA	.87 to .98	.58 to .94	N.A.
	THS-R [60]	N.A.	USA	N.A.	.82	N.A.
	TOLH [59]	1723	USA	.95	.90	.86

Table 3 .

 3 List of the computerized diagnosis tools in children based on the analysis of the handwriting product. CNN: Convolutional Neural Network; RF: Random Forest; SVM: Support Vecotr Machine; ANN: Artificial Neural Network.

	Ref	Ages / Classes	Characteristics of participants	Task(s)	Language	Approach	Performances
	[68] 7-10 y	Dysgraphic	BHK (5 lines)	Italian	Algorithms for document analysis	Sensitivity: 83 % Specificity: 98 % Precision: 96 %
	[69] 7-12 y	Dyslexic	Letters and digits writing	Malaysian Machine Learning (ANN)	Sensitivity: 73 %
	[70] 8-15 y	Typically Developing and Dysgraphic	Letters, syllables, words, pseudowords, and sentences	Slovak	Machine Learning (CNN, RF, SVM, AdaBoost)	Precision: 79.7 %

Table 4 .

 4 List of the algorithms and computer tools for the diagnosis of handwriting deficits. In blue: tools based on statistical approaches, in black: tools based on machine learning approaches. ADHD: Attention-Deficit/Hyperactivity Disorder. BEM: Beta-elliptic Model; BVSCO-2 -Test for the Evaluation of Writing and Orthographic Ability, 2 nd ed.; CNN: Convolutional Neural Network; DG: dysgraphic children; FDM: Fourier Descriptor Model; FOD: Fractional Order Derivatives; KNN: K-nearest Neighbors; MHA: Minnesota Handwriting Assessment; MLP: Mumti Layers Perceptron; P: precision; RBF: Radial Basis Function; RF: Random forest; SE: sensitivity; SP: specificity; TD: Typically Developing children.

	Reference	Ages / Classes	n	Tasks	Langage / alphabet	Approach	Criteria analysed Performances
	[71]	6-10 y	242 TD 56 DG	Copy of a (BHK)	French	RF	Static Kinematic Pressure Pen tilt	SE: 96,6 % SP: 99,2 % P: 97,98 %
	[39]	5-12 y	390 TD 58 DG	Letters, words, sentences French	PCA + K-means clustering	Static Kinematic	SE: 91 % SP: 90 %
	[72]	10-13 y	39 TD 39 DG	Letters, words, sentences Slovak	SVM	Kinematic	SE: 75,5 %
	[73]	7-11 y	262 TD 63 DG	Copy of graphic shapes	N.A.	SVM, RF, MLP, extra trees, AdaBoost, Gaussian Naive Bayes	Kinematic	SE: 75,1 % (RF) SP: 72,1 % (MLP) P: 73 % (extra trees), 73,4 % (RF)
			458 TD					SE: 91 %
	[74]	7-11 y	122 DG Copy of a text (BHK)	French	SVM	Kinematic Spatial	SP: 81 %
								P: 86 %
			63 TD	Letters, syllables, words,				SE: 79,7 %
	[40]	8-15 y	57 DG	pseudowords, sentences	Slovak	AdaBoost, RF, SVM	Kinematic	SP: 76,7 %
				with speed constraints				P: 80 %
	[75]	5-8 y	76 TD 28 DG	Copy of words (8y), graphic shapes (5 and 8y)	Italian	Statistical comparison between groups	Kinematic Pressure	N.A.
				Subtest of the BVSCO-2
	[76]	7-8 y 52 TD	(digits, sequence of small	Italian Statistical comparisons	Kinematic	N.A.
				and large loops, words)
			241 «at-					SE: 75 %
	[77]	5 y	risk of	Copy of graphic shapes	N.A.	1 dimension CNN	Kinematic	SP: 77 %
			DG»					P: 76 %
	[78]	6-7 y	26 TD 9 DG	MHA	English Statistical comparisons	Static Kinematic	N.A.
	[79]	8-12 y 26 TD Copy of graphic shapes	Czech		Static	P: 84 %

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 17 November 2023 doi:10.20944/preprints202311.1132.v1

								7
						Q factor wavelet	Kinematic
			27 DG			transform + statistical
						comparisons
	[80]	7-10 y	218 TD 62 DG	Copy of a text (BHK)	French	Statistical comparisons between groups (linear regression), clustering	Static Pen tilt Kinematic Pressure	N.A.
	[81]	6-11 y	5 TD 9 ADHD	Dictation of letters and digits MHA	English	Statistical correlations between manual and digital data	Static Kinematic	N.A.
	[82]	7-12 y	60	Copy of words, sentences and graphic shapes	Latin	RF, Decision tree, SVM	Kinematic	SE: 92,8 % P: 92,6 %
	[83]	8-15 y	63 TD 57 DG	Letters, words, sentences Slovak	KNN, SVM, RF, AdaBoost	Kinematic (on-surface and in-air)	SE: 78,5 % P: 80,8 %
	[84]	8 y	27 TD 27 DG	Letters	Hebrew	RF, linear discriminant analysis	Kinematic	SE: 96 %
	[85]	8-9 y	61 TD 15 DG	Copy of patterns and figures	Czech	XG-Boost	Kinematic	SE: 90 %
	[86]	8-9 y	14 TD 14 DG	Copy of a text	Hebrew	Statistical comparisons between groups	Static Kinematic	N.A.
	[42]	8-9 y	50 TD 49 DG	Copy of letters and sentences	Hebrew	SVM	Static Kinematic	SE: 90 % SP: 90 % P : 89,9 %
	[87]	8-11 y 32 TD	Spontaneous writing (sentences), drawings	Indonesian SVM & RBF Kernel	Kinematic	P : 82,5 %
			33 TD			Tunable Q-factor		SE: 88,7 %
	[88]	8-9 y	32 DG	Copy of a text	Czech	wavelet transfom, RF	Kinematic	SP: 83 %
						and SVM classifiers		P: 84,7 %
	[89]	8-9 y	30 TD 25 DG	Spontaneous writing of letters	Czech	Correlation between and the HPSQ-C the kinematic features	Kinematic	N.A.

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 17 November 2023 doi:10.20944/preprints202311.1132.v1

Funding: This research received no external funding.

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Author Contributions: Writing-original draft preparation, C.J.; writing-review and editing, C.J., F.P., J.D. All authors have read and agreed to the published version of the manuscript.

Conflicts of Interest:

The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.


Simona Dahdah
email: simona.dahdah@insa-lyon.fr

Nans Biboulet

Antonius Adrianus Lubrecht

Pierre Charles

Scuffing initiation caused by local starvation in a piston ring cylinder liner contact

Keywords: Scuffing, PRCL contact, Lubrication, Marangoni effect

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

1

Scuffing initiation caused by local starvation in a piston ring cylinder liner contact 1 Introduction

The automotive industry is facing a customer demand for powerful vehicles and the environmental protection requirements for lower pollutant emissions. One possible way to reduce emissions is to reduce engine power losses. The frictional loss occurring in the Piston-Ring/Cylinder-Liner (PRCL) contact is estimated to be about half of the total engine friction loss [START_REF] Tung | Automotive tribology overview of current advances and challenges for the future[END_REF]. In addition, this contact is exposed to several failure modes that might affect the overall engine performance and lifetime. For these reasons, the understanding of the phenomena acting in this contact as well as optimizing its lubrication mechanism have attracted the attention of researchers.

Wear is a common failure that can take place in the contact, it is a slowly progressing mechanism [START_REF] Ludema | A review of scuffing and running-in of lubricated surfaces, with asperities and oxides in perspective[END_REF]. Scuffing is an uncommon failure process and its occurrence in the internal combustion engine is rare, but once it initiates, the damage is catastrophic. Hence, the importance of studying and predicting its initiation. Scuffing is the least understood tribological failure because of the large number of variables involved in its initiation and the diversity of machine operating conditions. It occurs primarily in a lubricated contact with sliding bodies [START_REF] Saeidi | Automatic detection of scuffing using acoustic emission[END_REF]. It is commonly accompanied by a sudden increase in friction, temperature and vibration [START_REF] Tas | Roles of mirror-like surface finish and dlc coated piston rings on increasing scuffing resistance of cast iron cylinder liners[END_REF]. Numerous definitions are proposed to describe this failure such as: "gross damage characterized by formation of local welds between sliding surfaces" [START_REF] Dyson | The failure of elastohydrodynamic lubrication of circumferentially ground discs[END_REF], "surface roughening by plastic flow whether or not there is material loss or transfer" [START_REF] Ludema | A review of scuffing and running-in of lubricated surfaces, with asperities and oxides in perspective[END_REF] and "localized damage caused by solid-phase welding between sliding surfaces" [START_REF] De Gee | Glossary of terms and definitions in the field of friction, wear and lubrication tribology[END_REF]. An additional description of scuffing is provided by the ASTM committee: "a form of wear occurring in inadequately lubricated tribo-systems that is characterized by macroscopically-observable changes in surface texture" [START_REF] Astm | Standard terminology relating to wear and erosion[END_REF].

Dyson [START_REF] Dyson | The failure of elastohydrodynamic lubrication of circumferentially ground discs[END_REF] considered that the failure of the lubricating film is the main reason for scuffing initiation. Shuster et al. [START_REF] Shuster | Piston ring cylinder liner scuffing phenomenon studies using acoustic emission technique[END_REF] and Obert et al. [START_REF] Obert | The influence of oil supply and cylinder liner temperature on friction, wear and scuffing behavior of piston ring cylinder liner contacts-a new model test[END_REF] showed in their experimental studies that the oil quantity and the temperature are vital parameters.

Another approach of scuffing is studied by Shen et al. [START_REF] Shen | Scuffing failure in heavily loaded slow speed conformal sliding contacts[END_REF], they concluded that the lubricant desorption leads to an increase in friction and then accelerates scuffing initiation. The rate of lubricant desorption increases with the temperature as shown by Li et al. [START_REF] Li | Kinetics of lubricant desorption and decomposition under heat treatment: a molecular dynamics study[END_REF] when the lubricant decomposition under heating test is studied. Qu et al. and Van et al. [START_REF] Qu | Scuffing transition diagrams for heavy duty diesel fuel injector materials in ultra low-sulfur fuellubricated environment[END_REF][START_REF] Van Dam | Observations from cylinder liner wear studies in heavy duty diesel engines and the evolution towards lower viscosity heavy duty engine lubricants[END_REF] showed that scuffing initiates in the vicinity of the top dead center where the lubrication conditions are the worst. Barunovis et al. [START_REF] Barunovic | Sliding wear of 100cr6 in a diesel-lubricated flat-flat contact under realistic loads[END_REF] reported wear maps and scuffing limits for a pin on disc model test as a function of the contact load and the sliding speed. From these maps, one can conclude that scuffing is reached sooner when the speed increases. Based on that, exists a probability that scuffing may occur at mid-stroke where the velocity is high. Another approach for scuffing prediction is based on the shear stress. Scuffing initiates when the shear stress exceeds the critical one, this is observed when plastic deformation occurs [START_REF] Ajayi | Analytical predictive modeling of scuffing initiation in metallic materials in sliding contact[END_REF]. The shear stress was identified to depend on the lubricant additives. [START_REF] Yu | Influence of lubricant additives on friction in a disc machine[END_REF].

The time to scuff depends on the surface material and its wear resistance as shown by Wan et al [START_REF] Wan | Comparison of the scuffing behaviour and wear resistance of candidate engineered coatings for automotive piston rings[END_REF]. Surface coatings such as Carbon-based, diamond-like are used for components operating in a solid-to-solid contact [START_REF] Kennedy | Lower friction losses with new piston ring coating[END_REF][START_REF] Etsion | The effect of various surface treatments on piston pin scuffing resistance[END_REF][START_REF] Riahi | Scuffing resistances of al-si alloys: effects of etching condition, surface roughness and particle morphology[END_REF]. They increase the surface scuffing resistance and delay the scuffing initiation [START_REF] Tas | Roles of mirror-like surface finish and dlc coated piston rings on increasing scuffing resistance of cast iron cylinder liners[END_REF][START_REF] Ye | An experimental investigation of piston skirt scuffing: a piston scuffing apparatus, experiments, and scuffing mechanism analyses[END_REF][START_REF] Wang | Scuffing resistance of coated piston skirts run against cylinder bores[END_REF][START_REF] Wang | Wear and scuffing characteristics of composite polymer and nickel/ceramic composite coated piston skirts against aluminum and cast iron cylinder bores[END_REF]. However, when oil starvation occurs, friction and wear increase progressively which can lead to a total removal of the coating. Subsequently, the metal-to-metal contact increases the probability of scuffing initiation. Moreover, the insufficient oil supply to the PRCL contact might cause it to operate under mixed lubrication conditions [START_REF] Bewsher | An investigation into the oil transport and starvation of piston ring pack[END_REF][START_REF] Sherrington | Measurement techniques for piston-ring tribology[END_REF][START_REF] Gore | Direct measurement of piston friction of internal-combustion engines using the floatingliner principle[END_REF][START_REF] Mishra | Tribology of compression ring-tocylinder contact at reversal[END_REF][START_REF] Mishra | Tribology of the ring bore conjunction subject to a mixed regime of lubrication[END_REF] therefore increasing the friction. This local increase in friction generates a local heat flux which in turn generates a temperature gradient. The latter engenders a surface tension gradient that causes an oil displacement, hence, a possible oil film rupture exists as mentioned by Fantom et al. [START_REF] Fanton | Spreading and instabilities induced by a solutal marangoni effect[END_REF] and Monti et al. [START_REF] Monti | Wetting prevention by thermal marangoni effect. experimental and numerical simulation[END_REF]. This effect is known as the thermal Marangoni effect, which relates the fluid motion to the surface tension gradient generated by the temperature gradient, see Gugliotti et al. [START_REF] Gugliotti | Surface tension gradients induced by temperature: The thermal marangoni effect[END_REF].

The oil redistribution effect caused by the ring displacement along the liner is a counteracting phenomenon, see Organisciak et al. [START_REF] Organisciak | Variable lubricant supply of a starved hydrodynamic linear contact: lubricant lateral flow for smooth and laser textured surfaces[END_REF]. The oil redistribution is beneficial for the PRCL contact lubrication performance since it helps to avoid the local starvation thus the scuffing initiation [START_REF] Hu | A new efficient flow continuity lubrication model for the piston ring-pack with consideration of oil storage of the cross-hatched texture[END_REF]. These two opposing mechanisms affect the lubricant distribution in the PRCL contact: the thermal Marangoni process tends to create a localized starved zone and the ring displacement spreads oil along the contact and helps to avoid starvation. The current paper highlights the competition occurring between the Marangoni and the ring displacement effects which is the basis of a computational model for scuffing initiation prediction in the piston ring cylinder liner contact of engines.

Model

The temperature of the liner and the ring surfaces is initially considered constant and equal to T0. However, when considering the occurrence of a temporarily local lack of lubricant due to a non-uniform oil distribution in the contact, the coefficient of friction increases (see figure 1). Then, an additional heat flux is locally generated. The additional heat flux generated (qheated-zone) is given in the equation below:

q heated-zone = ζµp contact u m ( 1
)
where µ is the coefficient of friction, pcontact is the contact pressure (equation 2) , um is the mean velocity (equation 3) of the ring and ζ is the ratio of the heat exchange occurring between the liner and the ring. Its value ranges from 0 to 1. 0 meaning that there is no heat transferred to the surface whilst 1 means that the heat is totally transferred to the surface. The coefficient of friction is defined via the Stribeck curve (figure 1) as a function of the ratio of the oil film thickness (h0) divided by the surface roughness (σ). From the curve, it can be concluded that the lower the film thickness, the higher the coefficient of friction. Note that σ is considered constant in the model and that the values given in the curve are generic ones. The mean contact pressure (pcontact) applied by the ring itself is calculated from the ring tension (Fring) via equation 2.

p contact 2F ring = φ t ( 2
)
axial
where φ is the nominal ring diameter and taxial is the ring axial thickness. As the piston ring velocity varies along the stroke (low at the top and bottom dead centers and high at mid-stroke), the mean ring velocity (um) along the stroke is used in the model. It depends on the engine frequency in rpm (N ) and the stroke length (C) as shown in equation 3. It increases with the engine speed. As the engine speed and the stroke distance are both considered constants in the model, the piston velocity is also constant. the material is considered to be isotropic; the conductivity is the same in all directions, there is no heat stored in the lubricant, the heat exchange occurring between the liner and the ring surfaces with their surroundings is not studied in the current model, Neumann boundary conditions are used to simulate the heat flux generated in the starved/heated zone (q imposed), whilst Dirichlet boundary conditions (T0 imposed) are used elsewhere, where T0 is the nominal liner temperature at which minimum friction occurs, its upper bound is approximately 140 • C [START_REF] Rahmani | The effect of cylinder liner operating temperature on frictional loss and engine emissions in piston ring conjunction[END_REF]. This is the simplest boundary condition to be considered to simplify the model.

um = NC (3) 30
The ring is moving along the liner in the x-direction as shown in figure 2. Thus, a transient 3D model is required to calculate the liner temperature distribution. -

q mean = ζµp contact u m ∆t ∆t passage = ζ2µp contact t axial f (4)
The liner surface exposed to the mean heat flux (qmean) is presented in figure 3. As the heat flux and the temperature-increase are constant along the liner, and the surface is considered infinitely long in the x-direction, the transient 3D model

Equations

The stationary 2D Fourier heat equation without considering an internal heat source is:

∂y y ∂y ∂z z ∂z ∂ α ∂T + ∂ α ∂T = 0 (5)
where αy, αz are the conductivity in respectively y and z directions. As the material is isotropic hence, αy=αz=α.

The dimensionless 2D Fourier heat equation is:

∂ 2 T ¯ ∂ 2 T ¯ ∂Y 2 + ∂Z 2 = 0 (6)
The thermal equation is solved numerically to obtain the temperature variation required for the study. More details about the equation discretization and resolution are given in appendix 5. Equations 7 and 8 show respectively the lubricant surface tension and viscosity as a function of the temperature.

γ = γ0 T n 1 -T (7) log (log (η + 0.7)) = a log (T ) + b (8)
The dimensionless surface tension and viscosity are respectively γ ¯ η ¯ = η/η0. = γ/γ0 and Where γ0 and η0 are the surface tension and the viscosity of the lubricant at the reference temperature T0. Tc is the critical temperature of the lubricant and a and b are constants depending on the lubricant. Note that the lubricant used in the model is a typical, commercial car engine oil.

Marangoni effect

The lubricant displacement due to the Marangoni effect is shown in figure 5. Note that the lubricant curvature is exaggerated in the figure and that the forces due to that curvature are negligible. The oil flows from the high temperature zone where the surface tension is low (low γ zone) to the low temperature zone where the surface tension is high (high γ zone) and it is submitted to the following stresses: the Marangoni stress:

∂γ τ Marangoni = ∂y (9) c ∂t = -∂y
the viscous stress:

v τviscous = η h ( 10
)
The variation of the oil film thickness as a result of the Marangoni effect is given by equation 13. It is the result of the combination of two equations: the equilibrium of the Marangoni and the viscous stresses (τMarangoni=τviscous), from where the equation of the oil velocity (v) is obtained:

h ∂γ v = η ∂y ( 11
)
the equilibrium of the variation of the oil film over time (∂h/∂t) and the Couette flux (∂φc/∂y) along the y direction (∂h/∂t=∂φc/∂y),

where

φc = vh (12) 2
The velocity term in equation 12 is replaced by equation 11 and the second equilibrium is solved to obtain the Marangoni equation:

∂h ∂ h 2 ∂γ
The dimensionless Marangoni equation is:

∂H ∂t ¯ ∂ = -∂Y H 2 ∂γ ¯ 2η ¯ ∂Y ( 14
)
Appendix 5 details the Marangoni equation discretization and resolution.

Oil redistribution effect 2.3.1 Hydrodynamic assumptions

The hydrodynamic assumptions considered to obtain a simple and stationary model are:

2η ∂y (13)

Figure 5: Thermal Marangoni model the PRCL contact is considered as a hydrodynamic lubricated contact, meaning that the contacting surfaces are rigid, the ring velocity is constant along the x-direction and equal to the mean speed, the surface roughness is considered constant and surface texturing is not taken into consideration to decrease the number of the parameters, the lubricant viscosity is considered constant in the contact and independent of the local temperature and pressure, the squeeze effect is negligible, which describes the variation of the film thickness as a function of time.

Equations

The iso-viscous, constant density and steady-state Reynolds equation describing the flow of a thin film lubricant between two surfaces for a fully flooded lubrication regime is presented below:

1 ∂ (h 3 ∂p ) + 1 ∂ (h 3 ∂p ) = u ∂h (15) 12η ∂x ∂x 12η ∂y ∂y m ∂x
The steady-state assumption is valid as the liner surface is considered to be smooth and the oil film variation is negligible. In the starved lubrication regime, a limited amount of oil is available in the contact and the pressurized zone is narrower than in the fully flooded regime. To analyze the fluid film under starved conditions, a parameter θ is introduced which is the void fraction (1-ratio between the oil film

≤ ≤ ∂x 12η ∂y ∂y ∂X ∂X ∂Y ∂Y ∂X
thickness (hoil) and the geometry gap between the surfaces (h)). Its value ranges from 0 to 1 (0 θ 1). The Reynolds equation is then modified to take into consideration θ and describes the starved lubrication regime.

1 ∂ h 3 ∂p + 1 ∂ h 3 ∂p = u ∂ ((1 -θ) h) (16)
The dimensionless equation is:

∂ H 3 ∂P + ∂ H 3 ∂P - ∂ ((1 -θ) H) = 0 ( 17
)
In order to study the ring displacement impact on the oil distribution, a circumferentially non-uniform oil geometry is imposed at the inlet of the contact as shown in figure 6. The oil distribution is obtained by equation 18. A mass-conserving solver is used for the solution of the Reynolds equation, see Biboulet and Lubrecht [START_REF] Biboulet | Efficient solver implementation for reynolds equation with mass-conserving cavitation[END_REF]. The calculation output are the pressure and the circumferential oil distribution after the passage of the ring. The redistribution is studied and quantified by comparing the oil geometry imposed and the one obtained after the ring passage. hvar is the oil variation with respect to hmean, it defines the maximum and the minimum thickness of the oil film at the inlet of the contact, λ is the wavelength.

Lubricant distribution model

Figure 7 presents a schematic of the lubricant distribution in the contact. Assuming that under certain operating conditions, a local lack of lubricant might temporarily occur due to an oil displacement and a non-uniform oil distribution in the contact. As a result, the coefficient of friction increases which in turn leads to a temperature increase of the liner surface (∆T ) compared to the initial surface temperature (T0). The surface tension and viscosity gradients depending on the temperature are obtained (equations 7 and 8). The oil film thickness (h0) is then calculated using equation 19 [START_REF] Moes | Lubrication and beyond[END_REF].

h = 2.45 ηu m R x 0 w ( 19
)
1
where Rx is the radius of curvature and w1 is the load per unit length. A Marangoni calculation is then carried out to predict the oil variation (hvar) caused by the surface tension gradient compared to the mean film thickness (hmean) which is considered to be equal to the oil film thickness (hmean = h0).

The last part quantifies the following surfaces:

the surface displaced due to the Marangoni effect (SMarangoni) as a function of the temperature increase (∆T ), the oil film thickness (h0), the lubricant surface tension and viscosity gradients. In fact, the volume flow rate of the displaced oil is presented as a moving surface in order to compare it to the one moved due to the ring passage, the surface moved due to the ring passage (Sredistributed) as a function of the oil film thickness and the oil geometry distribution parameters (hmean, hvar and λ).

If S Marangoni is larger than S redistributed (S Marangoni >S redistributed ), which means that the lubricant starvation caused by the Marangoni effect is more pronounced than the lubricant redistribution caused by the ring passage, scuffing might initiate. One has to note that these two opposing effects are repetitive and occur at every engine cycle, meaning that the imbalance can be accumulated then leading to a localized lack of lubricant after a sufficient number of cycles. Else, when starvation is balanced by the oil redistribution, the local lack of lubricant is avoided and no risk of scuffing exists. The purpose of this model is to study the contact performance when it is exposed to local heating yet the lubricant is re-supplied. The question to explore is that despite the local heating, will the contact perform normally when it is re-lubricated?

Results

The values of the input parameters used in this section are given in tables 3 and 4. Figure 3 shows the temperature distribution when the average heat flux is qmean = 18 * 10 3 W/m 2 . The temperature increase due to qmean is ∆T = 0.1815K (when considering that T0 = 363K, ∆T = 1.0005T0 T0). This temperature increase is negligible and insufficient to generate the Marangoni effect and to consider the probability of failure risk. Thus, in the following study, the temperature increase compared to the nominal temperature is imposed as an input parameter, instead of the heat flux. Then, the heat flux needed to be injected to cause the chosen temperature increase is known. Therefore, the temperature distribution required for the Marangoni calculation is obtained. Note that the relation between the heat flux and the temperature increase is linear (q = c∆T ), where c is a constant.

Lubricant displacement

Temperature gradient

Assuming in this example that the width of the starved/heated zone occurring on the liner surface is s = 4.4mm, the nominal surface temperature is T0 = 363K and the temperature increase is ∆T = 20K. Then, the heat flux (q) to be injected is computed and it is imposed as a Neumann condition over s. The 2D Fourrier equation (equation 6) is solved to calculate the temperature distribution. The latter is shown in figure 8. Note that the temperature increase is ∆T =(1.055 1) (T0). As shown in the figure, the highest temperature is located in the middle of the contact and the temperature gradient extends from Y = 1 to Y = 1, this is the starved/heated zone. As a consequence of the temperature gradient, a lubricant surface tension gradient is generated as shown in figure 9. The increase in temperature leads to a decrease in the lubricant surface tension (equation 7), so the surface tension has its lowest value in the heated zone. In addition, the temperature gradient leads to a lubricant viscosity gradient as shown in figure 10, and the increase in the temperature decreases the lubricant viscosity (equation 8).

Influence of the temperature gradient

Starting with the hypothesis that an increase in the temperature occurs in the contact due to a localized lack of lubricant, the temperature, surface tension and viscosity distributions are defined and a Marangoni calculation is carried out (equation 13).

Figure 8: Dimensionless temperature distribution generated on the liner surface Figure 11 shows the lubricant film thickness variation in the circumferential direction caused by the Marangoni effect for different temperature increases. One can conclude that the maximum lubricant thickness variation is located in the heated zone. This is caused by the decrease in the lubricant surface tension. In fact, the lubricant tends to flow from the warmer zone (where the surface tension is at its lowest value) to the colder zone (where the surface tension is at its highest value). Note that the displacement caused by the Marangoni effect is continuous and lasts for a period of a passage (tpassage = C/um). Furthermore, the film thickness decrease is proportional to the temperature increase, and more precisely to the temperature gradient since the source width is kept constant in these examples (s = 4.4mm). The surface difference (∆S) between the initial oil distribution for ∆T = 0 and the one corresponding to ∆T /= 0 represents the oil flux displaced as a result of the Marangoni effect and i t is called SMarangoni. To quantify the oil flow, the surface difference is calculated as a function of the model parameters.

Analytic oil flow

In order to quantify the displacement caused by the Marangoni effect, the differential equation 13 is analytically developed so the surface moved (SMarangoni) can be obtained by equation 20.

S Marangoni

= s h 2 ∂ (1/η) ∂γ 1 ∂ 2 γ + t passage (20) 2
∂y ∂y The development is detailed below:

η ∂y 2 SMarangoni = ∆S = s∆h (21)
The expression of ∆h is obtained from equation 20: As the equation shows, the displacement depends on the oil film thickness, the local temperature, the temperature increase, the surface tension and viscosity gradients and the time of passage.

h 2  ∂ 1 ∂γ 1 ∂ 2 γ  ∆h = 2 ∂y ∂y + η ∂y 2  t passage (22)

Influence of the engine speed

The influence of the engine speed on the surface moved due to the Marangoni effect is studied. The temperature increase is 20K, the width of the heated zone is s = 4.4mm and the oil supply regime is moderately starved. Note that if hff is the oil film thickness in a fully flooded supply regime, the oil film thickness in a moderately starved regime is hff /3.

The increase in engine speed leads to:

an increase in the mean velocity (um), hence an increase in the oil film thickness (h0) (equations 3 and 19), As shown in equation 20, the Marangoni effect depends on the oil quantity and on the time of passage. For an imposed temperature increase, it is more pronounced when the oil quantity and/or the passage time increases. Figure 12 shows that SMarangoni increases with the engine speed. This result indicates that the sensibility of the oil quantity on the surface moved is more important than that of the time passage.

Lubricant redistribution

In order to study the influence of the ring passage on the oil distribution, a lubricant geometry distribution is imposed at the inlet of the contact. Figure 13 is an example of an oil distribution where the contact occurs between the liner and a parabolic ring centered at zero. The oil geometry distribution imposed at the inlet of the contact is shown in figure 17 as the continuous-blue curve.

Using a mass-conserving solver, the pressure and oil distributions after the ring passage are obtained, see Biboulet et al. [START_REF] Biboulet | Efficient solver implementation for reynolds equation with mass-conserving cavitation[END_REF]. The pressure distribution obtained after the first passage of the ring is shown in figure 14. It is divided into two zones: the pressurized and the non-pressurized zones.

the pressurized zones are obtained where the oil quantity available is sufficient to generate pressure (the white zones on figure 15), Figure 13: Lubricant geometry distribution before the ring passage along the liner the non-pressurized zones are obtained where the pressure can not be generated as the lack of lubricant is important. They are called the cavitated zones (the grey zones on figure 15).

As shown in figure 16, presenting the pressure distribution after ten passages of the ring along the liner, the pressurized zones are almost touching. This result indicates that the ring passage along the liner redistributes the lubricant and helps to avoid starvation. The ring passage is beneficial for the lubricant distribution hence for the contact performance.

Lubricant redistribution prediction

Figure 17 presents the imposed lubricant geometry (see equation 18) before the ring passage (continuous-blue curve) and the one obtained after the ring passage (dotdash-purple curve). The lubricant redistributed as a result of the ring passage is quantified as the surface difference between the two curves (Sredistributed=∆S1+∆S2). The parameters of the imposed geometry in this example are: The oil redistribution (Sredistributed) caused by the ring passage along the liner is studied as a function of the minimum oil film thickness (h0), the wavelength of the initial oil geometry distribution (λ) and the number of passages (passage). The numerical results have then been fitted in order to predict the redistribution as a function of the problem parameters. The obtained curve fit equation is:

Hmean=1, Hvar=0.075,
S redistributed = ξ (h mean -h 0 + h var ) 1.125 λ 0.875 passage 2 (23)
where ξ is the curve fit parameter which is a constant.

Influence of the oil film thickness: Figure 18 shows that the redistribution increases with the oil film thickness. The redistribution depends on the oil quantity supplied to the contact, it is more important when the oil supply is more pronounced. In fact, the redistribution is a result of the pressure gradient generated by the lubricant.

Figure 15: Cavitated zones after one passage of the ring along the liner

Influence of the wavelength:

The wavelength of the oil distribution geometry impacts the lubricant redistribution. When the wavelength decreases, the number of cavitated zones increases. In this study, the redistribution occurring at each cavitated zone is quantified separately. Results show that the redistribution increases with the wavelength (figure 19).

Influence of the number of passages:

The redistribution is a repetitive process that takes place at each passage of the ring along the liner. Figure 20 shows that the redistribution slows down with the number of passages or with time since the oil quantity available in the non cavitated zones decreases after each passage. Note that as the lubricant distribution occurring between two passages is the main interest of this study, the number of passages is limited to 1.

Scuffing initiation limit

The algorithm presented in paragraph 2.4 quantifies the oil displaced due to the Marangoni effect (SMarangoni) and the oil redistributed due to the ring displacement effect (Sredistributed) as a function of the operating parameters. Then, the surfaces moved are compared in order to study the scuffing initiation limit. When the two surfaces are balanced, further lack of lubricant is prevented, hence, starvation is limited and scuffing is avoided. Therefore, the contact performs normally, this is called the safe zone. Whilst, when the Marangoni effect predominates the ring passage effect, starvation is not limited and scuffing might initiate. In this case, the contact no longer operates in the safe zone.

The scuffing initiation limit is presented as a curve. The safe zone where scuffing is prevented is located below this curve. Whilst above it, scuffing can occur. The limit is studied as a function of the model parameters such as the contact load (ring tension), the engine speed (N ) and the thermal parameters as the heated zone width (s) and the local temperature increase (∆T ). This section intends to present the dependency of the scuffing initiation limit as a function of these parameters. From this chart, two vital conclusions can be drawn:

Influence of the thermal parameters

for the same ring tension and surface temperature increase, the scuffing initiation limit shifts up by increasing the width of the heated zone. Thus, the safe zone becomes wider, for the same width of the heated zone and by increasing the ring tension, scuffing might be reached sooner. In fact, the ring tension has an influence on the oil film thickness (the higher the ring tension, the lower the oil film thickness). The latter in turn affects the moved surfaces as shown in equations 20 and 23.

The temperature increase is a vital variable impacting the oil displacement caused by the Marangoni effect. The higher the temperature increase, the larger the surface Figure 22: Scuffing initiation limit as a function of ring tension and surface temperature increase for different engine speeds moved due to the Marangoni effect. Based on this conclusion, the influence of the Marangoni effect is more important when the temperature gradient increases.

Influence of the engine speed

Figure 22 presents the scuffing initiation limit for different engine speeds. It shows that the scuffing initiation limit is shifted up when the engine speed increases. In fact, the surface displaced due to the Marangoni effect increases with the engine speed due to the increase in the oil film thickness as shown previously in figure 12. Moreover, the surface redistributed due to the ring displacement effect increases also with the oil film thickness which increases with the engine speed as shown in figure 18. The balance between the moving surfaces seems to be more pronounced when the engine speed increases thus scuffing initiation is delayed.

Conclusion

This paper presents the influence of two opposing effects on the lubricant distribution and their complementary contribution to scuffing initiation risk in a PRCL contact. The first one is the Marangoni effect, which tends to create a local lack of lubricant due to the temperature gradient generated on the surface. The lubricant tends to flow from the high-temperature zones to the low-temperature zones because of the lubricant surface tension gradient. This lubrication condition might be one of the causes that accelerates scuffing initiation. The second effect is the ring passage along the liner. It redistributes the oil in the contact. This redistribution helps to avoid local starvation and might have an important role in delaying scuffing initiation. This work shows that when the contact is exposed to heating, its lubrication conditions are affected and temporarily starvation can occur. After re-lubricating the contact, two cases can occur. If starvation is balanced with the re-lubrication, the heating is avoided and the contact operates in the safe zone, it performs normally. Else, if starvation is not balanced by the re-lubrication, meaning that the heating is not avoided, the contact no longer operates in the safe zone and a risk of scuffing initiation exists. A computational model was developed and charts were generated illustrating the scuffing limit as curves as a function of the problem parameters. These charts provide new insights for scuffing initiation prediction in the PRCL contact of engines.

Appendix

Discretization and resolution

The equations are solved numerically using MATLAB software.

Thermal Fourier equation

The thermal system to be solved (see figure 4) for the Marangoni calculation is:

[A]{T ¯} = {f } ( 24
)
where A is a matrix, T ¯ is the unknown temperature field vector to be calculated and f is the right hand side depending on the boundaries conditions. The heat equation and the Neumann boundary condition (surface flux) are discretized using a second order centered scheme as shown in the equation below: 2 (∆Z) 2 where j and k are Y and Z indices respectively, ∆Y and ∆Z are the space mesh sizes. The equation below gives the discretization of the boundary condition:

T ¯j+1,k -2T ¯j,k + T ¯j-1,k + T ¯j,k+1 -2T ¯j,k + T ¯j,k-1 = 0 (25) (∆Y )
Q j,k = T ¯j,k+1 -T ¯j,k-1 2∆Z (26)
As the flux boundary condition is imposed on the surface (k=0), the ghost points method is applied to the discretization, where the points at k=-1 are virtual.

Q j,0 = T ¯j,1 -T ¯j,-1 2∆Z (27)
Once the system is defined, the temperature field is calculated by a simple matrix calculation:

{T ¯} = [A] -1 {f } (28)

Marangoni equation

The Marangoni equation is a 1D parabolic equation. The discretization of the equation in space and time is given by the equation below:

H n+1 = H n -∆t ¯γ ¯j+1 -γ ¯j (Hj+1 + Hj) 2 ∆t - γ ¯j-1 -γ ¯j (Hj-1 + Hj) 2 j j ∆Y 2 4 (η ¯j+1 + η ¯j) ∆Y 2 4 (η ¯j-1 + η ¯j) (29
) where n and j are time and space indices respectively, ∆Y is the space mesh size and ∆t ¯ is the time step.

Dimensionless variables

The thermal and hydrodynamic dimensionless variables are detailed respectively in table 1

Numerical model input parameters

Figure 1 :

 1 Figure 1: Stribeck curve: coefficient of friction as a function of the oil film thickness divided by the surface roughness

Figure 2 :

 2 Figure 2: Liner surface with a moving heat flux

  is simplified to a stationary 2D model. Note that in this example qmean = ζ2µpcontacttaxialf = 1 * 2 * 0.1 * 10 6 * 1.5e -3 * 60 = 18 * 10 3 W/m 2 .

Figure 3 :

 3 Figure 3: Liner surface with mean heat flux

Figure 4 :

 4 Figure 4: 2D liner thermal model

Figure 6 :

 6 Figure 6: Oil distribution geometry

Figure 7 :

 7 Figure 7: Lubricant distribution model

Figure 9 :

 9 Figure 9: Dimensionless lubricant surface tension distribution

Figure 10 :

 10 Figure 10: Dimensionless lubricant viscosity distribution

Figure 11 :

 11 Figure 11: Dimensionless film thickness variation along the circumferential direction for different temperature increase values

Figure 12 :

 12 Figure 12: The surface moved due to the Marangoni effect as a function of the engine speed (N )

Figure 14 :

 14 Figure 14: Pressure distribution in the contact after one passage of the ring along the liner

Figure 16 :Figure 17 :Figure 18 :Figure 19 :Figure 20 :

 1617181920 Figure 16: Pressure distribution in the contact after ten passages of the ring along the liner

Figure 21

 21 Figure 21 is a chart giving the scuffing initiation limit as a function of the ring tension and the local temperature increase for different widths of the heated zone and a moderately starved oil supply regime.

Figure 21 :

 21 Figure 21: Scuffing initiation limit as a function of ring tension (Fring) and surface temperature increase (∆T )

depth direction coordinate 6 Acknowledgments

 6 Notation α material conductivity [W/m/K] η ¯ dimensionless viscosity (η ¯ = η/η0) [-] γ ¯ dimensionless surface tension (γ ¯ = γ/γ0) [-] T ¯dimensionless temperature (T ¯ = T/T0) t ¯ dimensionless time (t ¯ = t/t0) Hmean dimensionless mean film thickness (Hmean = hmean/h0) hmean mean film thickness [m] Hvar dimensionless film variation (Hvar = hvar/h0) hvar film variation with respect to hmean [m] N engine frequency [rpm] P dimensionless pressure p pressure [Pa] pcontact contact pressure [Pa] phi nominal ring diameter temperature [m] phic Couette flux [m 2 /s] Q dimensionless heat flux (Q = q/q0) qheated-zone heat flux in the starved zone [W/m 2 ] qmean averaged heat flux [W/m 2 ] Rx ring radius of curvature [m] s width of the heated zone [m] taxial axial ring thickness [m] tauMarangoni Marangoni stress [N/m 2 ] tauviscous viscous stress [N/m 2 ] um mean ring velocity [m/s] v lubricant velocity [m/s] w1 contact load per unit length [N/m] X dimensionless axial coordinate (X = x/x0) x axial direction coordinate Y dimensionless circumferential coordinate (Y = y/y0) y circumferential direction coordinate Z dimensionless depth coordinate (Z = z/z0) z The Armand Peugeot Chair in cooperation with Stellantis (Fusion between Groupe PSA and FCA Group), National Institute of Applied Sciences of Lyon (INSA-Lyon), LaMCoS laboratory, fund Dahdah's thesis under the 'ANRT-CIFRE' contract n 2018/0816.

Table 2 :

 2 and table 2. Hydrodynamic dimensionless parameters

	x0 y0 z0 T0 γ0 η0 t0	q0
	s s s T0 γT 0 ηT 0 s 2 η0/h0γ0 αT0/s
	Table 1: Thermal dimensionless parameters
	x0 y0 λ0 h0 p0
	a a a h0 12ηuma/h 2 0

Table 3

 3 and table4give respectively the chosen geometric and operating parameters. The values of the ring geometry parameters are typical ones. Likewise for the operating parameters. Concerning the heated zone width, its value is small enough to model a localized heated zone and wide enough to cause a temperature gradient that perturbs the operating conditions of the contact.

	Variable	Symbol Unit Value
	Nominal ring diameter	φ	mm 75.4
	Axial ring thickness	t axial	mm 1.5
	Ring radius of curvature	Rx	mm 10
	Heated zone width	s	mm [1; 6.25]
	Calculationnal domain width	sd	mm 50
	Calculationnal domain thickness ed	mm 2.5
	Surface roughness	σ	µm 0.2

Table 3 :

 3 Geometric parameters

	Variable	Symbol Unit Value
	Stroke length	C	cm	10
	Engine speed	N	rpm [1000; 4000]
	Initial surface tension γ0	N/m 0.38
	Initial viscosity	η0	Pa.s 0.0081
	Initial temperature	T0	K	363
	Ring tension	F tension N	[5; 25]

Table 4 :

 4 Operating parameters