# Process Knowledge-infused Learning for Suicidality Assessment on Social
Media

Kaushik Roy1111Contact Author Manas Gaur1 Qi Zhang1&Amit Sheth1
1University of South Carolina, Artificial Intelligence Institute, SC, USA
{kaushikr<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Improving the performance and natural language explanations of deep learning
algorithms is a priority for adoption by humans in the real world. In several
domains, such as healthcare, such technology has significant potential to
reduce the burden on humans by providing quality assistance at scale. However,
current methods rely on the traditional pipeline of predicting labels from
data, thus completely ignoring the process and guidelines used to obtain the
labels. Furthermore, post hoc explanations on the data to label prediction
using explainable AI (XAI) models, while satisfactory to computer scientists,
leave much to be desired to the end-users due to lacking explanations of the
process in terms of human-understandable concepts. We introduce, formalize,
and develop a novel Artificial Intelligence (A) paradigm - Process Knowledge-
infused Learning (PK-iL). PK-iL utilizes a structured process knowledge that
explicitly explains the underlying prediction process that makes sense to end-
users. The qualitative human evaluation con- firms through a annotator
agreement of 0.72, that humans are understand explanations for the
predictions. PK-iL also performs competitively with the state-of-the-art
(SOTA) baselines.

##  1 Introduction

A long-standing problem in adopting machine learning technologies to assist
humans in the real world has been the lack of a satisfactory explanation to
the end-users of the technology. In the traditional machine learning pipeline,
much attention is paid to fitting a function map from data points to labels.
However, during the annotation of data points in the ground truth dataset, a
guideline or process is often detailed by which the annotator can label the
dataset. For example, to label patients for degrees of suicidal tendencies in
a physical clinical setting, a well-known scale, the Columbia Suicide Severity
Rating Scale (CSSRS) Bjureberg et al. (2021), is used to determine the right
set of labels. Figure 1 shows this scale. Thus, it is clear to the patient how
a particular suicidal tendency is recognized once the clinician evaluates the
questions and patient responses.

Figure 1: Left: Columbia Suicide Severity Rating Scale. Right: The induced
process knowledge.

Similarly, when data points in a dataset are annotated in other domains, each
data point is evaluated against a process or guideline similar to the CSSRS by
several human annotators. The assumption is that the machine learning
algorithm will implicitly recover the underlying process or guideline used by
the annotators when learning a function map from data point to label. Popular
methods for XAI such as LIME and SHaP, are used to explain the function
learned, often through local approximations related to a single or sampled set
of data points Adadi and Berrada (2018); Ribeiro et al. (2016); Lundberg and
Lee (2017). However, due to the black-box nature of the function and the non-
convexity of the hypothesis function surfaces, it is challenging to evaluate
if the recovery of the underlying process or guideline was successful and is
meaningful to the end-users. Fundamentally, we might think of these XAI
methods as trying to roughly construct an explanation as saying, “This data
point is explainable using a simpler hypothesis function (a local
approximation) due to similar data points (data points in the local
neighborhood) also being classified correctly by the simpler hypothesis”.
Consequently, much depends on the choice of local approximation and the
machine learning models understanding of similar data points, on what is
already a highly non-convex gargantuan function such as a large SOTA language
model (LM) Vaswani et al. (2017). Also, while such an explanation may satisfy
the computer science community, “similarities” are hardly adequate for the
end-user (e.g., a psychotherapist). The pertinent questions include: Would the
human annotators consider the data points deemed similar by the LM also to be
similar to each other? Would the human annotators agree that the explanation
by the local approximation is aligned with the process or guideline used by
them to label those data points?

In our study, we ask the question, what if we were able to use not just the
annotator’s labels, but also the process or guidelines used to label them and
explicitly control the learning of a model to recover the process or guideline
(instead of implicitly). Such an algorithm would, by design, be explainable
and emulate the humans model of similarity between data points. This paper
takes the first step to answer this question grounded in the deep learning
task of suicidality assessment from social media data, where incorporating the
knowledge of medical processes and guidelines is of critical considerations.
To this end, we propose a novel class of algorithms Process Knowledge infused
Learning (PK-iL) for suicidality assessment from social media. We make the
following contributions:

  * •

Define Process Knowledge (PK) and create a dataset for suicidality assessment
task based on CSSRS with annotations to include both PK and labels.

  * •

Develop Process Knowledge infused Learning (PK-iL), an explainable algorithm
that explicitly controls the learning model to recover the process by
effective utilization of PK in the annotation and a globally optimal
optimization objective.

#### Can PK-iL utilize SOTA LMs?

We note that the notion of similar data points is the machine learning model’s
way of understanding the human annotator’s annotation process, i.e.,
fundamentally, the goal of the model is constructing a similarity space that
mimics the human annotator’s understanding. In many domains and applications,
large and SOTA LMs have excelled at capturing the similarity of some examples
exceptionally well. Hence, we believe that rather than try to implicitly learn
a similarity space as a model of the human annotator’s understanding over the
whole space of examples, we can leverage SOTA LMs to define the annotator’s
understanding at process-specific checkpoints in the PK. For instance, if a PK
has five questions (or guidelines) to go through, we can use the SOTA model as
a proxy to understand if a human annotator would have judged the guideline as
satisfied. Such finer-grained understanding can potentially leverage the
ability of SOTA models to learn similarity spaces while still maintaining the
explicitly explainable PK-iL structure.

####

We will see the data collection, examples and intuitions, and the
formalization of PK-iL in action through the following sections.

##  2 PK enhanced CSSRS dataset

Before the pandemic, suicidality was already a leading mental health issue
across the world. Since the pandemic, incidents of suicidality have increased
even further. Thus for both demonstrating high real-world impact through an
important use case on real data and users as well as for ease of exposition,
we explain our methods and experiments anchored around the application of
suicidal thought pattern detection. However, PK-iL is generalizable to any
domain that requires the integration of PK with data to derive high-quality
explanations.

To conduct our study in a physical, real-world experimentation setting, we
require responses from users physically present during the experiment.
Consider the clinical setting of suicidal thought assessments using the CSSRS
- obtaining access to a physical clinical setting presents many hurdles such
as ethics approval, incentives for honest responses, etc. The demand-supply
deficit in mental health already makes it hard to find a quality experimental
setting, and the recent COVID-19 pandemic has compounded this issue. The
significant number of persons turning to social media platforms presents an
exciting opportunity to leverage a large amount of data as a proxy for user
responses. Thus we utilize the dataset of Gaur et al., Gaur et al. (2021);
Alambo et al. (2019) which uses the CSSRS to label user posts from suicide-
related subreddits and thus provides a real-world test-bed to evaluate the
performance and explainability of PK-iL. Through the CSSRS, the domain experts
in the study annotated longitudinal data from 448 users for the following
labels: Suicide Ideation, Suicide Behavior, Suicide Attempt. High standards in
annotation were maintained with a substantial inter-rater agreement of 0.84.
Crucially, we expand this dataset to include the specific guideline (PK) for
annotation in addition to the label. Table 1 shows examples of the dataset
expanded with PK.

User text (x) |  Process Knowledge (PK) |  CSSRS Label
---|---|---
[…] a voice telling me to kill myself […] yes - I think I should do it […] |  1.2 (yes - a voice telling me to kill myself) |
2.2 (yes - I think I should do it) |
4 (yes - I think I should do it) |  Behavior or Attempt
[…] Rarely is a day where I dont suffer from thoughts of self-harm… |  1.1 (yes - Rarely is a day where I dont suffer from thoughts of self-harm) |
2 (no - no words indicating active suicidal thought) |  Ideation
Table 1: Examples of data set annotation expanded with PK. The […] collapses
the rest of the post for brevity. Each question (1-6) in the CSSRS has a main
question and sub questions 1.1, 1.2 etc, as can be seen in Figure 1. Thus the
PK denotes the main question or sub-question being answered in the user’s
Reddit post.

##  3 Tree Structure of PK

We see in Figure 1 that PK can be viewed as a decision tree. A process tree
(Process Knowledge(PK)) to to determine the probability of a label $y$ for a
user post can be written as a polynomial of the form

$y=\sum_{l\in
Leaves}p_{l}\prod_{i=1}^{N_{q}}\big{(}I_{yes}(q_{i}))\big{(}1-I_{no}(q_{i}))$
(1)

, where $N_{q}$ is the number of questions in the decision tree,
$I_{yes}(q_{i})$ and $I_{no}(q_{i})$ represents if the post follows a yes path
or a no path to the question $q_{i}$. $Leaves$ is a set of all leaves that
lead to the label $y$. For example, there are two paths in Figure 1 that lead
to $y=Ideation$. Here $p_{l}$ is computed as the ratio of the number of
annotators that chose that path for the example to the total number of
annotators - this in some sense captures the inter annotator agreement for
those set of examples. For example, considering a particular post, if among
three annotators, two annotators labeled the PK as the path $1.2\rightarrow
2.2\rightarrow 4$. Then the probability of $y=BehaviororAttempt$ for that post
is $0.66$. Note here that the sub questions aren’t stored in the tree leaves.
The path $1.2\rightarrow 2.2\rightarrow 4$ is equivalent to $1\rightarrow
2\rightarrow 4$. This is done for all the examples in the training set and the
final probability is an average of all the examples.

######  Assertion.

For any model $\mathcal{M}(y)$ that approximates the probability of label $y$
for a post according to Equation 1, let the inter-annotator for the post
labeled as $y$ be $\mathcal{A}(y)$. Then best approximation for the post,
$\mathcal{M}^{*}(y)\leq\mathcal{A}(y)$

We claim the above as an assertion instead of a theorem as it is trivial to
see that $\prod_{i=1}^{N_{q}}\big{(}I_{yes}(q_{i}))\big{(}1-I_{no}(q_{i}))\leq
1$ always, and therefore any approximation is upper bounded by the inter-
annotator agreement. This makes intuitive sense as what we are really
interested in capturing the annotator’s thought process while labeling and on
par accuracy. Improving upon the inter-annotator agreement may mean capturing
something that is not present in the ground truth. Thus, we are interested in
labeling unseen data as well as the human annotators would while explicitly
capturing their annotation process in the learned model.

How do we define mathematically that a post follows the yes path or the no
path to a question $q_{i}$, i.e., we need to define exactly what
$I_{yes}(q_{i})$ and $I_{no}(q_{i})$ means. Recalling our understanding of
similarity between question and answer as being a proxy to answered as yes or
no, we can use inner product based similarity between representations of
question and post to determine $I_{yes}(q_{i})$ and $I_{no}(q_{i})$. For
example, for a similarity model that takes as inputs representations for “Have
you thought about being dead or what it is like to be dead” and “Rarely is a
day where I don’t suffer from thoughts of self-harm”, the output is a value
indicative of high similarity relative to other input pairs. This is seen as
the question “Have you thought about being dead or what it is like to be dead”
being answered as yes by the response “Rarely is a day where I don’t suffer
from thoughts of self-harm”. There are several options in Natural Language
Processing (NLP) literature to construct representations of text

  * •

Count Vectorizer. Each sentence or text fragment is represented as counts of
the words in the fragment padded with zeros according to largest fragment.
Count vectorization however, does not consider the importance of words across
different parts of the post, for example stop words might occur most
frequently but provide little context.

  * •

TF-IDF. TF-IDF corrects the defficiency of the Count Vectorizer method by
adjusting counts by weighting for contextual importance across the post.
However, TF-IDF still relies on exact matches of words being present or absent
in the post.

  * •

Hashing Vectorizer. Each sentence or fragement of the text is simply passed
through a hash function. The idea is that similar fragments produce similar
hash codes. The crytpic nature of the hash function (this is by design) is not
amenable for interpretation or explainability analysis of the learned
function.

  * •

Text Embeddings. These are a set of neural network models that represent text
in a vector space. Models such as word2vec, Transformer LMs such as GPT-3 and
BERT are all examples of large neural networks that map the text to a vector
space such that contextually similar texts are placed close together in the
vector space while dissimilar texts are placed apart bert; gpt3; transformers.
Since, these are the state of the art and have shown remarkable effeciency and
performance in recent years, we will use Transformer based LM representations
for the text.

Note that the Text Embedding models provide vector representations of words.
To construct a representation for the text fragment one might average the word
representations contained in the fragment. However, this loses information
about the order of the words and phrases in the text and hence we use a
concatenation representation padded with zeros according to the longest text
fragement. Thus generally we will denote a similarity function by
$\mathcal{K}$ and representations of text $x$ and question $q_{i}$ using an
embedding model as $x^{R}$ and $q_{i}^{R}$ respectively. Thus

$\Bigg{(}\pm\mathcal{K}\bigg{(}\frac{x_{sub}^{R}}{|x_{sub}^{R}|},\frac{q_{i}^{R}}{|q_{i}^{R}|}\bigg{)}\geq\pm\theta_{i}\Bigg{)}$

denotes the similarity between the text and question where $\theta_{i}$ are
suitably chosen thresholds of accepted high similarity. The normalization of
the representations by size is what makes an inner product a valid similarity
measure in the range $-1~{}to~{}+1$. We will now formally develop the
algorithm for PK-iL

##  4 The PK-iL Algorithm

We define a function that predicts the probability of post label being $Y=y$
according to the PK as follows:

$\scriptsize P(Y=y\mid X=x)=\sum_{l\in
Leaves}p_{l}\prod_{i=1}^{N_{q}}\lor_{x_{sub}\in
x}\Bigg{(}\pm\mathcal{K}\bigg{(}\frac{x_{sub}^{R}}{|x_{sub}^{R}|},\frac{q_{i}^{R}}{|q_{i}^{R}|}\bigg{)}\geq\pm\theta_{i}\Bigg{)}$
(2)

, where $p_{l}$ is defined as detailed in Section 3, $x_{sub}\in x$ is a
fragment of the post $x$ (For example a sentence). $x_{sub}^{R}$, and
$q_{i}^{R}$ are representations of $x_{sub}$ and $q_{i}$ from an embedding
model, and $\mathcal{K}$ is a inner product function to measure similarity.

$\pm$ signifies if we are checking if the question $q_{i}$ is answered as yes
or no by fragment $x_{sub}$ in post $x$ with confidence $\theta_{i}$. Using
$\lor_{k=1}^{K}z_{k}=(\sum_{k=1}^{K}z_{k}\geq 0.5)$, we have:

$\scriptsize P(Y=y\mid X=x)=\sum_{l\in
Leaves}p_{l}\prod_{i=1}^{N_{q}}\sum_{x_{sub}\in
x}\Bigg{(}\pm\mathcal{K}\bigg{(}\frac{x_{sub}^{R}}{|x_{sub}^{R}|},\frac{q_{i}^{R}}{|q_{i}^{R}|}\bigg{)}\geq\pm\theta_{i}\Bigg{)}\geq
0.5$ (3)

We can then optimize the Bernoulli Loss $\mathcal{L}$ for an input post $X=x$
and label $Y=y$ is as follows:

$\mathcal{L}(\\{\theta_{i}\\}_{i=1}^{N_{q}})=P(Y=y|X=x)log(P(Y=y|X=x))$ $+$
$(1-P(Y=y|X=x))log(1-P(Y=y|X=x))$

We perform hyperparameter tuning to choose the embedding model, fragment size
$x_{sub}$, and $\mathcal{K}$ (see Section 5). Since
$\mathcal{L}(\\{\theta_{i}\\}_{i=1}^{N_{q}})$ is strongly convex, we use
Newton’s optimization method to learn the parameters of the model. The
algorithm for Process Knowledge infused Learning (PK-iL) is as follows:

Algorithm 1 Process Knowledge infused Learning (PK-iL)

1:Compute $p_{l}~{}\forall$ leaves $l$ from the ground truth

2:Choose Kernel $\mathcal{K}$, fragment size, and CE model for representation

3:Initialize $\theta_{i},~{}\forall i\leftarrow 1~{}to~{}N_{q}$

4:for $k\leftarrow 1~{}to~{}K$ do $\triangleright$ Begin Newton’s method

5: for $\theta_{i}$, where $i\leftarrow 1~{}to~{}N_{q}$ do

6: Compute
$\theta_{i}^{{}^{\prime}}=\nabla_{\theta_{i}}\mathcal{L}(\theta_{i})$

7: Compute
$\theta_{i}^{{}^{\prime\prime}}=\nabla\theta_{i}^{{}^{\prime}}=\nabla_{\theta_{i}}(\nabla_{\theta_{i}}\mathcal{L}(\theta_{i}))$

8: Set
$\theta_{i}=\theta_{i}-\frac{\theta_{i}^{{}^{\prime}}}{\theta_{i}^{{}^{\prime\prime}}+1}$
$\triangleright$ add $1$ to avoid divide by zero error

9:return $\theta_{i},~{}\forall i\leftarrow 1~{}to~{}N_{q}$

Here we see that PK-iL is general enough to allow embedding models suitable to
the task and PK suitable to any domain. However, in our experimental results
we will evaluate PK-iL both quantitatively and qualitatively evaluation using
the expanded PK enhanced CSSRS dataset (see Section 2.

####  Prediction:

Prediction is carried out by choosing the summand in Equation 3 that has the
highest value once normalized by dividing by the sum of the summands, in order
for it to be a probability.

##  5 Experimental Setup and Results

For the LM to understand language in the context of suicidal thought patterns
it needs to be fine-tuned on such a data. For this we word2vec representations
on corpus of suicide related subreddits as as well as fine-tune LMs during
training on the same corpus. Thus we obtain embeddings of the text
contextualized to suicidal conversation in order to accurately infer yes or no
from similarity. To implement the word2vec model, we use the gensim library
and the Continuous Bag of Words (CBOW) model Mikolov et al. (2013). Note that
in the word2vec model due to lack of tokenization coverage as in LMs, we chunk
the string one letter at a time and check against the list of words and their
vectors. The LMs we fine-tune are:

  * •

XLNET - An auto-regressive language model in which the training objective
calculates the probability of a token conditioned on all permutations of
tokens in a fragment. When trained on a very large data, the model achieves
SOTA performance across several tasks in the GLUE benchmark Yang et al.
(2019)Wang et al. (2018). We use the default hyperparameters during training.

  * •

RoBERTa - A BERT based model where the hyperparameters are further tuned for
superior performance over BERT on GLUE benchmark tasks. We use the default
hyperparameters during training RoBERTa Liu et al. (2019).

  * •

Google T5 - A transformer model trained on various text-to-text tasks such as
translation, summarization, etc. T5 demonstrate superior transfer capabilities
across tasks in the GLUE benchmark. We use the T5-small model and the default
parameters Raffel et al. (2019).

  * •

ERNIE - A transformer model trained on natural language corpora and large
knowledge graphs and is thus a suitable model to consider in a structured
knowledge intensive field like mental health. We use the default parameters
for ERNIE Zhang et al. (2019).

  * •

Longformer - A transformer model that excels at capturing long text inputs. As
some of the posts can be over $8000$ characters long, the longformer is a
suitable model to consider for our dataset. We use the default parameters for
the long former Beltagy et al. (2020).

We believe the wide range of transformer architectures above are sufficient to
test our approach. We train all our models on the Google Colab platform.

#### Inner Products

Bubeck et al., shown that $O(nd)$, where $n$ is the number of data points and
$d$ is the true underlying data dimension Bubeck and Sellke (2021). The
Transformer outputs are already high dimensional but Bubeck show that for
natural language the models still need to get larger! Thus we use a popular
trick to compute inner products in higher dimensions - the Kernel trick.
Polynomial Kernel can project the data to very high dimensions and the
Gaussian Kernel can project the data to an infinite number of dimensions. We
see the use of a Kernel significantly improves the performance over simple
cosine similarity (polynomial kernel of dimension $1$). In our experiments we
use the Gaussian Kernel to compute the inner product.

For the fragment size we found a span of 1-2 sentences to be the best
performing model for each transformer and kernel choice.

###  5.1 Quantitative Evaluation

For baseline accuracy we directly use the embedding models to predict the
label as in a traditional machine learning pipeline. For word2vec, we use the
representations of the post and pass it through a logistic regression model.
We make a slight modification where weights for all entries for a single word
vector are shared. Table 2 shows a comparison of accuracy for all the models
with their baseline, PK-iL with Cosine Similarity, and PK-iL with a Gaussian
Kernel.

#### Suicidality Context Capture

It is very interesting to note the word2vec, trained using the CBOW method, is
the best performing model in the Baseline, Cosine Similarity, and the Gaussian
Kernel case. We hypothesize upon inspection of the embeddings that word2vec,
since trained from scratch on the suicide related post corpus captures
contextual dependencies between suicidality tokens and phrases much better
than LMs. LMs need to be fine-tuned on very large amounts of data to adapt
against non suicidality term related contexts that they have trained on using
massive corpora.

From our analysis we note that for domain specific tasks such as mental health
related prediction, it is perhaps better to train contextual dependencies
between words and phrases from scratch as pretrained models are already
heavily biased towards the contextual dependencies on the corpora that they
are trained on.

#### Comparing Baselines

Across all the models we see, PK-iL improves upon the accuracy of the baseline
models by upto almost $15$% points (for Longformer). Although to confirm our
statement we have to rule out effects of collecting more data, adding/deleting
features etc, using neural representations and limited data alone, explicitly
controlling the learned model with process knowledge shows significant
performance gains.

#### High Dimensional Data

Our experiments indeed show that even for domain specific corpora such as
posts related to suicidality, the latent dimension of the text required to
learn metric spaces is indeed very high. Improving the dimensionality shows
little gain in this setting. But we hypothesize that for text from broader
domains (e.g. text related to mental health in general), the dimensionality
expansion will show more significant improvements.

Model | Baseline | Cosine Similarity | Gaussian Kernel
---|---|---|---
Word2Vec |  75% / 69% |  83% / 78% |  84% / 72%
XLNET | 70% / 65% | 80% / 69% |  84% / 71%
RoBERTa | 67% / 62% | 70% / 64% | 71% / 62%
T5 | 67% / 54% | 72% / 52% | 75% / 64%
ERNIE | 68% / 62% | 75% / 69% | 80% / 71%
Longformer | 50% / 38% | 65% / 49% | 67% / 48%
Table 2: The mean accuracy/AUC-ROC, rounded up, of all the models - Column one
shows the baseline where the model is directly used to predict the label,
Column two shows PK-iL with Cosine Similarity for Kernel choice for each
embedding model choice of representation, Column two shows PK-iL with a
Gaussian Kernel for Kernel choice for each embedding model choice of
representation

###  5.2 Qualitative Evaluation

As mentioned earlier the qualitative evaluations among three expert annotators
received a score of 0.7 agreement. Now, We will look at some of the
explanations generated for interesting examples that show cases where PK-iL
performed well and cases where it did not. We will also compare with
explanations of the word2vec model which is easy to visualize using the
weights of the word2vec vectors from the logistic regression model. We
highlight the phrases whose individual word sums are greater than a threshold.

#### Post Example 1.

We will compare Word2vec baseline and PK-iL with the Gaussian Kernel.

Prediction: Ideation
---
Ground Truth: Indication
Model: Word2Vec Baseline
’A book is usually what I do when Im getting down, but it doesnt work when I
start getting panicky. Ill try the carbs, the caffeine doesnt work because Ive
gotten it in a movie theater and had a soda with me…’, ’A few reasons. I feel
backed into a corner  mostly. And Im Tired of being Tired of everything. If
that makes sense.’, ’Thank you! I understand its a sad  thing. But I also want
people to realize that there can be humor in anything and its the best way to
deal with this. Its how I would do it. ’, ’I really dont want to ask for help.
Id rather not let anyone know Im having these kind of issues.’
Table 3: Example of attention visualization based explanations

From this example, we can clearly see word2vec associating phrases and words
that characterize a low mood with suicidal ideation. In real life such words
may raise triggers in the minds of a clinician and may benefit their analysis.
However, the human annotator seems to have labeled this as indication based on
the “there can be humor in everything” part of the post.

Prediction: Indication
---
Ground Truth: Indication
Model: PK-iL with Gaussian Kernel
’A book is usually what I do when Im getting down, but it doesnt work when I
start getting panicky. Ill try the carbs, the caffeine doesnt work because Ive
gotten it in a movie theater and had a soda with me…’, ’A few reasons. I feel
backed into a corner mostly. And Im Tired of being Tired of everything. If
that makes sense.’, ’Thank you! I understand its a sad thing. But I also want
people to realize that there can be humor in anything and its the best way to
deal with this. Its how I would do it. ’, ’I really dont want to ask for help.
Id rather not let anyone know Im having these kind of issues.’
Explanation: 1\. Wish to be dead (no) $\rightarrow$ indication
Table 4: Example of explanation based on PK-iL

Recall that PK-iL deals with whole fragements of text and can therefore never
highlight phrases as we experimented with fragment lenghts of 1-3 sentences.
The highest threshold among the similarity functions in Equation 3
corresponded to the fragment highlighted and the path 1. Wish to be dead no
and hence the model picks indication with probability equal to inter-annotator
agreement of $y=indication$ at that leaf. Such an explanation although subject
to annotator agreements is more informative to the clinician about the models
prediction. Although, the highlights from the Word2vec model provide important
cues as to the user’s suicidal thought patterns it is unclear to the clinician
why certain words were highlighted and certain others ignored. For example,
why just “panicky” and not the whole phrase “getting panicky”?.

#### Embeddings vs PK-iL explanations - Developer vs End-user Perspective:

Computer scientists with deep understanding of logistic regression weights and
biases may find the embedding model based visualization easier to understand
and replicate. They would clear understand the contextual dependencies between
tokens and phrases learned by the inner mechanism of Word2vec and could
therefore reasonably expect if the weights of a token or phrase are high in
logistic regression, contextually related words will also be high. Also, that
roughly statistically frequent tokens and the most frequent co-occurring
words, per class label, are most likely to be highlighted in the model
explanation. This makes perfect sense to the developer. However, the domain
expert will struggle to palate the idea of statistically likely words and
frequently co-occurring words as a valid explanation for the prediction.

#### Post Example 2.

We will see PK-iL with Gaussian Kernel outputs for a slightly more interesting
example.

Prediction: Behavior or Attempt
---
Ground Truth: Behavior or Attempt
Model: PK-iL with Gaussian Kernel
’I wish I could give a shit about what would make it to the front page. I have
been there and got nothing. Same as my life. I do have a gun.’, ’I thought I
was talking about it. I am not on a ledge or something, but I do have my gun
in my lap.’,  ’No. I made sure she got an education and she knows how to get a
job. I also have recently bought her clothes to make her more attractive. She
has told me she only loves me because I buy her things. ’
Explanation: 1\. Wish to be dead (yes) $\rightarrow$ 2\. Non-Specific Active
Suicidal Thoughts (yes) $\rightarrow$ Active Suicidal Ideation with Some
Intent to Act, without Specific Plan (yes) $\rightarrow$ Behavior or Attempt
Table 5: Example of explanation based on PK-iL for example 2

Here we can see how PK-iL highlights multiple sentences that satisfy its
explanation generated. Note that the Word2Vec models prediction was also
correct in this instance highlighting phrases such as “On a ledge”, “have a
gun” and “gun in my lap”.

#### Correctness of Prediction

For the example post considered, the correctness of the prediction is subject
to interpretation by human experts. This is why there is interannotator
disagreement. PK-iL is however theoretically capable of performing as well as
the annotators as per Assertion Assertion in the best case. The intutions
behind PK-iL focus more on understanding the experts thought process and
providing explanations that they can understand rather than on the correctness
of prediction.

Thus we believe fundamental algorithmic and data annotation changes like the
PK-iL paradigm will result in faster integration of assistive machine learning
technology in real-world applications.

##  6 Conclusion and Future Work

In this study we develop a novel paradigm PK-iL that introduces the need for
richer annotation and high performance explicit process guided explanation
models that the end-user can readily understand. The dataset contains a lot of
noisy and long posts. In such settings both PK-iL and embedding models
performed poorly. These inherent challenges of social media data will need to
be addressed in future work. Additionally, PK-iL also has the potential to
identify regions of the example space that the PK applies to with high inter-
annotator agreement. This can assist in soliciting more refined guidelines on
those cases where scales such as the CSSRS clearly do not work. While these
scales have been developed over decades of research, machine learning
techniques such as PK-iL have the potential to provide assistive refinement of
existing and established guidelines.

##  7 Acknowledgements

This research is support by National Science Foundation (NSF) Award # 2133842
“EAGER: Advancing Neuro-symbolic AI with Deep Knowledge-infused Learning,” .
Any opinions, findings, and conclusions or recommendations expressed in this
material are those of the author(s) and do not necessarily reflect the views
of the NSF.

## References

  * Adadi and Berrada (2018) Amina Adadi and Mohammed Berrada.  Peeking inside the black-box: a survey on explainable artificial intelligence (xai).  IEEE access, 6:52138–52160, 2018.
  * Alambo et al. (2019) Amanuel Alambo, Manas Gaur, Usha Lokala, Ugur Kursuncu, Krishnaprasad Thirunarayan, Amelie Gyrard, Amit Sheth, Randon S Welton, and Jyotishman Pathak.  Question answering for suicide risk assessment using reddit.  In 2019 IEEE 13th International Conference on Semantic Computing (ICSC), pages 468–473. IEEE, 2019.
  * Beltagy et al. (2020) Iz Beltagy, Matthew E Peters, and Arman Cohan.  Longformer: The long-document transformer.  arXiv preprint arXiv:2004.05150, 2020.
  * Bjureberg et al. (2021) Johan Bjureberg, Marie Dahlin, Andreas Carlborg, Hanna Edberg, Axel Haglund, and Bo Runeson.  Columbia-suicide severity rating scale screen version: initial screening for suicide risk in a psychiatric emergency department.  Psychological medicine, pages 1–9, 2021.
  * Bubeck and Sellke (2021) Sébastien Bubeck and Mark Sellke.  A universal law of robustness via isoperimetry.  Advances in Neural Information Processing Systems, 34, 2021.
  * Gaur et al. (2021) Manas Gaur, Vamsi Aribandi, Amanuel Alambo, Ugur Kursuncu, Krishnaprasad Thirunarayan, Jonathan Beich, Jyotishman Pathak, and Amit Sheth.  Characterization of time-variant and time-invariant assessment of suicidality on reddit using c-ssrs.  PloS one, 16(5):e0250448, 2021.
  * Liu et al. (2019) Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov.  Roberta: A robustly optimized bert pretraining approach.  arXiv preprint arXiv:1907.11692, 2019.
  * Lundberg and Lee (2017) Scott M Lundberg and Su-In Lee.  A unified approach to interpreting model predictions.  Advances in neural information processing systems, 30, 2017.
  * Mikolov et al. (2013) Tomas Mikolov, Kai Chen, Greg Corrado, and Jeffrey Dean.  Efficient estimation of word representations in vector space.  arXiv preprint arXiv:1301.3781, 2013.
  * Raffel et al. (2019) Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J Liu.  Exploring the limits of transfer learning with a unified text-to-text transformer.  arXiv preprint arXiv:1910.10683, 2019.
  * Ribeiro et al. (2016) Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin.  ” why should i trust you?” explaining the predictions of any classifier.  In Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining, pages 1135–1144, 2016.
  * Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin.  Attention is all you need.  Advances in neural information processing systems, 30, 2017.
  * Wang et al. (2018) Alex Wang, Amanpreet Singh, Julian Michael, Felix Hill, Omer Levy, and Samuel R Bowman.  Glue: A multi-task benchmark and analysis platform for natural language understanding.  arXiv preprint arXiv:1804.07461, 2018.
  * Yang et al. (2019) Zhilin Yang, Zihang Dai, Yiming Yang, Jaime Carbonell, Russ R Salakhutdinov, and Quoc V Le.  Xlnet: Generalized autoregressive pretraining for language understanding.  Advances in neural information processing systems, 32, 2019.
  * Zhang et al. (2019) Zhengyan Zhang, Xu Han, Zhiyuan Liu, Xin Jiang, Maosong Sun, and Qun Liu.  Ernie: Enhanced language representation with informative entities.  arXiv preprint arXiv:1905.07129, 2019.


xxx–yyy

# Fully developed and transient concentration profiles of particulate
suspensions sheared in a cylindrical Couette

Mohammad Sarabian1 Mohammadhossein Firouznia1
Bloen Metzger2 & Sarah Hormozi1 Email address for correspondence:
<EMAIL_ADDRESS>1Department of Mechanical Engineering, Ohio University, 251
Stocker Center, Athens, Ohio, USA, 45701.
2Aix-Marseille Universite, CNRS, IUSTI UMR 7343, 13453 Marseille, France.

(xxx; ?? and in revised form ??)

###### Abstract

We experimentally investigate particle migration in a non-Brownian suspension
sheared in a Taylor-Couette configuration and in the limit of vanishing
Reynolds number. Highly resolved index-matching techniques are used to measure
the local particulate volume fraction. In this wide-gap Taylor-Couette
configuration, we find that for a large range of bulk volume fraction,
$\phi_{b}\in[20\%-50\%]$, the fully developed concentration profiles are well
predicted by the Suspension Balance Model of Nott & Brady (J. Fluid Mech.,
vol. 275, 1994, pp. 157–199). Moreover, we provide systematic measurements of
the migration strain scale and of the migration amplitude which highlight the
limits of the suspension balance model predictions.
Key words: suspensions

††volume: xxx

##  1 Introduction

Shear-induced migration occurs in suspensions of particles sheared even at low
Reynolds number when the flow configuration presents shear-rate gradient such
as in pipe flows or in large-gap Taylor-Couette flows. In such geometries,
particles migrate towards regions of low shear rate producing an inhomogeneous
concentration field. This phenomenology can be particularly problematic in the
industrial context as it strongly affects the rheological response of
suspensions. It can induce clogging or give rise to inhomogeneous
preparations.

Two main descriptions have been proposed to model shear-induced migration. The
earliest comprehensive work dates back to the phenomenological approach of
Acrivos and his co-workers. This approach adds diffusive terms to the
continuity equation for solid-phase continuum ([Leighton & Acrivos (1987),
Phillips _et al._ (1992)]). These diffusion terms include the effects of
particle collisions, gradient in the relative viscosity of the suspension, and
Brownian motion. Another approach is the Suspension Balance Model (SBM) of
[Nott & Brady (1994)], which is a continuum description accounting for the
relative motion between the fluid and the particle phases. Within this latter
model, the particle flux is driven by the presence of particle normal stress
gradients. In viscous suspensions, the particle normal stresses scale linearly
with the local shear rate. Thus, in inhomogeneous flows where the shear rate
varies with space, particle normal stress gradients spontaneously arise,
thereby leading to particle migration. The closures for stresses in
suspensions are based on rheological laws such as those of [Morris & Boulay
(1999)] and of [Boyer _et al._ (2011)] which involve many constants and
constitutive laws that were determined from computational studies and
experimental measurements.

Shear-induced migration has also been investigated computationally via
discrete particle simulation methods such as: Stokesian Dynamics (SD), force
coupling method (FCM) and immersed boundary method (IBM). We refer the reader
to the recent comprehensive review of [Maxey (2017)].

On the experimental front, measurements were carried out in different
geometries such as cylindrical Couette cells ([Leighton & Acrivos (1987)],
[Abbott _et al._ (1991)], [Graham _et al._ (1991)], [Phillips _et al._
(1992)], [Chow _et al._ (1994)], [Corbett _et al._ (1995)], [Ovarlez _et al._
(2006)], [Gholami _et al._ (2018)]), pipes ([Karnis _et al._ (1966)], [Hampton
_et al._ (1997)], [Oh _et al._ (2015)], [Snook _et al._ (2016)]), channels
([Koh _et al._ (1994)], [Lyon & Leal (1998)]) and parallel disks ([Chow _et
al._ (1994)], [Deshpande _et al._ (2010)]). In these works, different
experimental techniques have been employed such as Nuclear Magnetic Resonance
(NMR) imaging, laser Doppler velocimetry (LDV), refractive index matching
(RIM), X-ray radiography and Computed Tomography (CT). In cylindrical
geometries, SBM is found to provide satisfactory predictions for the fully
developed volume fraction and velocity profiles, where the diffusive flux
model encounters difficulties ([Stickel & Powell (2005)]).

However, in such geometries, the data characterizing the migration dynamics
are sparse (only available for specific concentrations). Conversely, in pipe
geometries, the recent experimental work of [Snook _et al._ (2016)] provides
substantial information for both the steady-state profiles and the transient
evolution of the migration process. They show that the volume fraction at the
pipe centreline (i.e. where the shear rate vanishes) decreases with decreasing
bulk volume fraction while the SBM predicts a centreline volume fraction
reaching maximum packing. Moreover, the SBM predicts a significantly faster
rate of migration than what is observed experimentally.

This short summary shows that the description of migration of non-Brownian
suspensions remains incomplete. In the present paper, we investigate shear-
induced migration in a wide-gap Taylor-Couette geometry providing systematic
results, i.e. for a large range of bulk volume fraction, for both the fully
developed and the transient evolution of the concentration profiles. This
geometry differs from the Poiseuille configuration of [Snook _et al._ (2016)]
in that it contains no region flowing at zero shear rate. We find that, in
such a case, SBM provides more accurate predictions for both the fully
developed and the transient evolution of the concentration profiles. The
highly converged and detailed experimental results provided here allow us to
highlight the limits of the SBM and discuss the relevance of the different
rheological laws on which this model is grounded. These experimental results
could represent a benchmark for fully resolved numerical simulations of
curvilinear flows. After presenting the experimental procedure in section 2,
the results are detailed in section 3. Conclusions are drawn in section 4.

##  2 Experimental set-up and protocols

Figure 1: Schematic of the experimental Taylor Couette cell.

###  2.1 Experimental Apparatus

The experimental set-up is sketched in figure 1. The inner cylinder and outer
cylinder of this Couette device have a radius of $R_{i}=40~{}mm$ and
$R_{o}=60~{}mm$ respectively, and their height is 120 mm. The transparent and
fixed outer cylinder is machined from a full block of poly(methyl
methacrylate) (PMMA) and is embedded in a square water-bath jacket connected
to a recirculator (Isotemp 4100R20 Bath-115V), enabling one to control the
temperature of the whole set-up ($\pm 0.1^{\circ}C$). The inner moving
cylinder is driven by a precision rotating stage (M-061.PD PI piezo-nano
positioning) with an angular resolution of $3\times 10^{-5}$ rad.

We use a semiconductor green laser diode of $200$ mW power and wavelength of
$532$ nm as the light source to illuminate a vertical plane ($r,z$) of the
suspension. The light beam, after reflection on the rotating mirror, is
focused down to a thin ($\approx 60$ $\mu$m) vertical light sheet using a
plano-convex cylindrical lens. The suspending fluid is doped with rhodamine 6G
which fluoresces under the laser diode illumination.

###  2.2 Particles and fluid

Particles are rigid PMMA spheres of radius $a=0.79\pm{0.01}\hskip 4.30554pt$
mm and density of $1.19\hskip 4.30554ptg/cm^{3}$. The gap to particle size
ratio is thus $\ell_{\rm gap}/a=(R_{o}-R_{i})/a\approx 25$. The suspending
fluid is a viscous mixture, similar to that used by [Pham (2016)], composed of
Triton X-100 ($76.242\hskip 4.30554ptwt\%$), Zinc Chloride ($14.561\hskip
4.30554ptwt\%$), Water ($9.112\hskip 4.30554ptwt\%$) and Hydrochloric Acid
(HCl) ($0.085\hskip 4.30554ptwt\%$), having at room temperature a viscosity of
$\eta=4.64$ Pa s. This large viscosity of the suspending fluid ensures that
experiments are performed in the limit of vanishing Reynolds number
($Re<10^{-3}$). A small amount of hydrochloric acid (HCl) is added to the
preparation to improve its transparency. The composition of the suspending
fluid is chosen to match both the refractive index and the density of the PMMA
particles. This makes the suspension transparent and prevents sedimentation of
the particles.

After preparation, the bubble-free suspension is delicately poured into the
Couette cell. Further mixing inside the cell is performed using a thin rod to
make sure that the initial particle distribution is homogeneous throughout the
gap.

###  2.3 Flow visualization

In order to distinguish particles from the fluid phase while imaging,
rhodamine 6G is added to the suspending fluid with a concentration of
$0.1\hskip 4.30554ptmg/L$. This amount of dye is chosen to maximize the
contrast between the bright fluorescent fluid and the dark particles, while
allowing the laser light to penetrate deep enough into the measurement volume.
The temperature of the set-up is set to $20^{\circ}C\pm 0.1^{\circ}C$ in order
to optimize index matching between the suspending fluid and the particles
(within $\pm 10^{-4}$). This temperature adjustment improves index matching
(down to the fourth digit) and thereby the imaging quality. Image acquisition
is performed by using a digital camera (Basler Ace acA2000-165um USB3
Monochromatic) and a high-quality magnification lens (Sigma APO-Macro-180
mm-F3.5-DG). A $550$ nm long-pass coloured glass filter is placed before the
lens to remove all direct light reflections and only keep the light re-emitted
by the fluorescent fluid.

###  2.4 Experimental protocol and data analysis

Figure 2: Image processing steps for (top) $\phi_{b}=20\%$ and (bottom)
$\phi_{b}=50\%$: a) Raw image, b) Thresholding, c) Gaussian blur filtering, d)
Hough transform. Figure 3: Evolution of the concentration field $\phi(r,z,t)$
for $\phi_{b}=50\%$ at a) $\gamma=0$, b) $\gamma=50$, c) $\gamma=75$ and d)
$\gamma=550$ (steady state).

Experiments are performed for six different bulk volume fractions, (i.e
$20\%\leq\phi_{b}\leq 50\%$). For each experiment, once the suspension is
poured and homogenized into the cell, the suspension is sheared and image
acquisition is started. In all of these experiments the rotating speed of the
inner cylinder is $\Omega_{i}=0.058~{}rad/sec$ resulting in a shear rate at
the inner cylinder of $\dot{\gamma}=0.2$ s-1. Images are acquired every strain
unit for a total accumulated strain of $4500$ for the lowest volume fraction.
To provide a measure of accuracy and error bars, four experiments were
performed for each bulk volume fraction following exactly the same procedure.

To obtain the particle concentration field $\phi(r,z,t)$, where $r$ and $z$
correspond to the radial and vorticity direction respectively, we use an image
processing method similar to that of Snook et al. (2016). As shown in figure
2, the raw images are successively thresholded, blurred and finally, a
circular Hough transform is applied to detect each particle centre and
apparent radius in the plane of visualization ([Duda & Hart (1972)]). It is
noteworthy to mention that in figure 2, all particles have the same size;
their apparently different radii arise from their intersection with the light
sheet. Each image is divided into 125x235 overlapping horizontal and vertical
sub-images of width $0.1d_{p}$. The concentration field is obtained by
integrating the area covered by the particles within each sub-image. This
information can easily be inferred from the knowledge of the particle centre
locations and their apparent radii within the visualization sheet. The above
technique was found to be more accurate than directly measuring the local
volume fraction from the thresholded image, the latter method being biased by
unavoidable horizontal streaks on the images arising from dust or air bubbles.

Since the concentration field evolves slowly, concentration fields obtained
for successive images are time averaged over 10 successive images (10 strain
units) to reduce fluctuations. Figure 3 shows a typical example of the
concentration field measured as a suspension of bulk volume fraction
$\phi_{b}=50\%$ is sheared. The suspension is initially rather homogeneous
and, as it is sheared, the volume fraction decreases near the inner rotating
cylinder (small $r$) while it increases near the outer fixed cylinder (large
$r$). Note that the last concentration field, i.e for figure 3 .d, corresponds
to an accumulated strain of $550$ at which the concentration field reached
steady state. We find that the concentration averaged over the whole domain
$\phi_{\rm average}$ (i.e. averaged over $r$ and $z$) is constant in time
remaining equal to $\phi_{b}$, the initial bulk volume fraction; see Figure 5
(Green square). This result indicates that migration in the $z$-direction is
negligible, as can also been observed from the invariance along $z$ of the
concentration field in Figure 3.d. We thus spatially average the concentration
fields $\phi(r,z,t)$ along $z$ to obtain the concentration profile
$\phi(r,t)$. In the following, these radial concentration profiles are used to
characterize the migration process.

##  3 Results

Figure 4: a) Initial and b) steady state concentration profiles averaged over
four experimental runs. Black solid and dashed lines correspond to SBM
predictions using Morris $\&$ Boulay and Boyer $\&$ co-workers rheology
respectively.

###  3.1 Initial and steady-state profiles

Figure 4 shows the initial and steady-state (fully developed) concentration
profiles (averaged over four experimental runs) across the gap of the cell for
different bulk volume fractions. The initial particle concentration profiles
are homogeneous across the gap for all the bulk volume fractions. Some
particle layering can be observed in the vicinity of both the inner and outer
cylinders especially as the bulk volume fraction is increased. For all the
bulk volume fractions investigated, the fully developed concentration profiles
clearly indicate a migration of the particles from the inner cylinder (high-
shear-rate-region) towards the outer cylinder (low-shear-rate-region), see
figure 4.b. Moreover, particle layering is more pronounced as the bulk solid
volume fraction increases with a stronger layering close to the outer
cylinder.

In figure 4.b, we compare our experimental results to the predictions of the
SBM obtained using the rheological laws of [Morris & Boulay (1999)] and [Boyer
_et al._ (2011)]. The SBM framework, the rheological laws and the numerical
scheme implemented are presented in Appendix A. We find that the SBM
predictions capture well the average migration process and provide very
similar results when using [Morris & Boulay (1999)] or [Boyer _et al._ (2011)]
laws. Obviously, this model, which is based on a continuum approach, does not
capture the observed particle layering which is a discrete effect induced by
the finite particle size and confinement (e.g. [Alghalibi _et al._ (2018)]).

###  3.2 Migration dynamics

Figure 5: Evolution of the concentration, averaged over the inner half (red
triangle), the outer half (blue circle) and the whole gap (green square) for
different initial bulk volume fractions: a) $\phi_{b}=20\%$ b)
$\phi_{b}=30\%$, c) $\phi_{b}=35\%$ d) $\phi_{b}=40\%$, e) $\phi_{b}=45\%$ f)
$\phi_{b}=50\%$. Comparison with SBM using (full line) Morris $\&$ Boulay and
(dashed line) Boyer rheology.

To investigate the dynamics of the migration process, we compute the average
value of the concentration over the inner half of the gap
$\phi_{\rm\scriptsize inner-half}$ (from the inner cylinder to the middle of
the gap) and over the outer half $\phi_{\rm\scriptsize outer-half}$ (from the
middle of the gap to the outer cylinder). Figure 5 shows the evolution of
these quantities averaged over four experimental runs as a function of the
accumulated strain $\gamma$ for experiments performed with different bulk
volume fractions. Starting from $\phi_{b}$, the volume fraction averaged over
the inner half of the gap (where the shear rate is the largest) decreases
while that of the outer half (where the shear rate is the lowest) increases
before eventually reaching a steady-state value. Again we compare these
experimental measurements to the predictions obtained from the SBM solved
using both rheological laws of [Morris & Boulay (1999)] and [Boyer _et al._
(2011)]. A striking result is that, conversely to what was found in the
experiments of [Snook _et al._ (2016)] performed in a Poiseuille
configuration, here in a Taylor-Couette configuration, the SBM predictions of
the transient evolution seem to show a better agreement with the experimental
results. The inspection of these curves indicates that the SBM migration
strain scale is in rather good agreement at low volume fraction. However the
SBM predicts a faster migration at large volume fraction than that observed
experimentally.

To provide a more quantitative comparison between the SBM predictions and our
experimental results, we investigate the migration dynamics by fitting
exponential relaxations of the form
$\phi=\phi_{b}\pm\Delta\phi(1-e^{-\gamma/\gamma_{m}})$ to the transient
evolution of the volume fraction in the outer-half part of the gap; see Figure
6.a. Here, as depicted in Figure 6.a, $\Delta\phi$ corresponds to the
variation of the volume fraction within the outer-half part of the gap and
$\gamma_{m}$ is the typical strain scale for migration to proceed.

Interestingly, as shown in Figure 6.b, we find that the strength of the
migration process, here characterized by $\Delta\phi_{\rm\scriptsize outer-
half}$, is a non-monotonous function of $\phi_{b}$. Note that the same trend
is found for $\Delta\phi_{\rm\scriptsize inner-half}$ (yet not plotted for the
sake of clarity). This non-monotonic behaviour can easily be understood since,
in both limits where $\phi_{b}\rightarrow 0$ and $\phi_{b}\rightarrow\phi_{c}$
($\phi_{c}$ corresponding to the critical volume fraction at which the
suspension jams) no migration should occur, i.e. $\Delta\phi\rightarrow 0$. We
thus expect a maximum $\Delta\phi$ in between. In our geometry, we find
migration is the strongest for $\phi_{b}\approx 35\%$.

Figure 6: a) Inner and outer mean volume fractions versus strain fitted (red
lines) with $\phi_{b}-\Delta\phi_{\rm\scriptsize inner-
half}(1-e^{-\gamma/\gamma_{m}})$ and $\phi_{b}+\Delta\phi_{\rm\scriptsize
outer-half}(1-e^{-\gamma/\gamma_{m}})$ respectively, for different initial
bulk volume fractions $\phi_{b}=20,30,35,40,45,50\%$. b) Amplitude of
migration $\Delta\phi_{\rm\scriptsize outer-half}$ versus initial bulk volume
fraction $\phi_{b}$ (The same trends are observed with
$\Delta\phi_{\rm\scriptsize inner-half}$ ). c) Migration strain scale
$\gamma_{m}$ to reach steady state versus bulk volume fraction $\phi_{b}$.
Comparison with SBM using (full line) Morris $\&$ Boulay and (dashed line)
Boyer rheology.

The above measurement of the amplitude of migration $\Delta\phi$ is used to
quantitatively test the SBM predictions. As shown in Figure 6.b, both the
rheological laws of [Morris & Boulay (1999)] and of [Boyer _et al._ (2011)]
predict a non-monotonic behaviour of $\Delta\phi$. However, that of [Morris &
Boulay (1999)] provides better agreement with the experimental results for the
amplitude of migration. It also captures more accurately the bulk volume
fraction at which migration is the strongest, here $\phi_{b}\approx 35\%$.

The fitting by an exponential relaxation also gives us a measure of the
migration strain scale $\gamma_{m}$. Figure 6.c shows that $\gamma_{m}$
decreases when the bulk volume fraction is increased. Again, we find that the
rheological law of [Morris & Boulay (1999)] provides better agreement
especially at low volume fractions (when $\phi_{b}<40\%$), whereas the
migration strain scale obtained with [Boyer _et al._ (2011)] is too small by
approximately a factor 2 to 3. Interestingly, when $\phi_{b}>30\%$, the
critical strain measured experimentally seems to saturate to a value of
$\gamma_{m}\approx 200$, while the SBM predictions keep decreasing. As
discussed in the following, this behaviour may be the results of the strong
particle layering observed at large volume fractions in our experiments.

##  4 Conclusion

The fully developed and transient particulate volume fraction profiles of non-
Brownian suspensions were investigated experimentally in a Taylor-Couette
device using refractive index matching and fluorescence imaging techniques.
Systematic experiments were performed for a large range of bulk volume
fractions ranging from $20\%$ to $50\%$. Steady-state concentration profiles
as well as their dynamics were analysed and compared with the predictions of
the suspension balance model (SBM) using two different rheological laws
introduced by [Morris & Boulay (1999)] and [Boyer _et al._ (2011)]. The
present experimental data may be used as a benchmark for refining the
continuum SBM framework or discrete-particle simulations.

The comparison of the steady-state profiles obtained experimentally and
numerically using the SBM framework provides rather good agreement regardless
of the chosen rheological law [Morris & Boulay (1999)] or [Boyer _et al._
(2011)]. However, by performing more systematic characterization of the
migration dynamics, we could highlight significant differences, in particular
through the measurement of the strain scale $\gamma_{m}$ needed to reach
steady state and of the migration amplitude $\Delta\phi$. We showed that the
rheological law of [Morris & Boulay (1999)] captures more accurately the
amplitude and the non-monotonic trend of the migration amplitude $\Delta\phi$.
In our geometry, migration is found to be the strongest for $\phi_{b}\approx
35\%$. Moreover, for moderate volume fractions ($\phi_{b}<40\%$), the latter
rheological law provides quantitative predictions for the migration strain
scale, $\gamma_{m}$, while that predicted by [Boyer _et al._ (2011)] is too
small by approximately a factor 3. The two rheological laws investigated here
mostly differ in the contact contribution to the particle stress term (see
appendix A). In Boyer’s rheology, this term is approximately $7$ times larger
(mostly because of the small value of $I_{0}$), thereby yielding a faster
migration dynamics. It is important to recall that the rheology of [Boyer _et
al._ (2011)] was obtained experimentally for volume fractions $\phi_{b}>40\%$
and that of [Morris & Boulay (1999)] was developed to describe migration also
at large volume fraction. Thus, extending these laws to low volume fraction
might require some modifications. However, since our aim here is not to
provide new rheological laws but to contribute to the understanding of shear-
induced migration, we have plotted the SBM predictions based on these laws
using their original form, i.e. without adjusting any parameter. Following
this approach, our result shows that the rheological law of [Morris & Boulay
(1999)] captures more accurately the migration process.

The highly converged and detailed experimental results provided here should
help further to improve the continuum modelling for migration. Indeed, it
appears that the SBM framework is globally more successful when applied to
cylindrical Couette configurations than to Poiseuille configurations. For
instance, the SBM predicts complete jamming at the centreline of a pipe
($\phi\rightarrow\phi_{c}$) while this trend is usually not observed in
experiments ([Snook _et al._ (2016)]). In the same geometry, the predicted
migration strain scales are significantly smaller (more than an order of
magnitude) than those measured experimentally ([Snook _et al._ (2016)]).
Conversely, in the present cylindrical Couette configuration, the steady-state
volume fraction profiles and the migration strain scale are found to be
captured more accurately. One reason that may explain this difference is that
the above two systems differ in the fact that in Poiseuille configurations the
shear rate goes to zero at the centreline. In such a region, as a result of
migration, the suspension viscosity is expected to diverge as
$\phi\rightarrow\phi_{c}$. The absence of such a singular region in the
cylindrical Couette configuration may explain the relative success of SBM in
predicting the migration process.

It is also important to discuss two related effects, namely those of
_confinement_ and _layering_. The present experiments were performed for
$\ell_{\rm gap}/a\approx 25$. Such confinement is motivated by several points.
In practice, index matching and direct optical visualization require small
systems. Moreover, providing experimental results for such confinement is
useful to compare with numerical simulations which, limited by computation
time, also investigate systems having similar sizes ([Gallier (2016)]). Most
importantly, for $\ell_{\rm gap}/a>20$, previous experiments and numerical
simulations both already showed that confinement has a negligible impact on
macroscopic quantities such as the suspension viscosity ([Maxey (2017)]), but
also on the fully developed concentration profiles ([Snook _et al._ (2016)]).
We thus performed experiments for only one confinement $\ell_{\rm
gap}/a\approx 25$ for which no systematic bias is expected. Yet, it would be
interesting in future studies to vary this parameter and investigate its
impact on the migration strain scale.

The impact of layering is less clear. Layering is inherent to monodisperse
suspensions sheared at large volume fraction and it is found to be more
prominent in highly confined systems. When increasing $\ell_{\rm gap}/a$
(which damps the layering), [Snook _et al._ (2016)] found similar fully
developed concentration profiles. We also find quite an accurate match between
our steady-state concentration profiles and the SBM prediction, even at large
volume fraction where layering becomes very strong. Thus, layering seems not
to influence the fully developed concentration profiles. However, the impact
of layering on the migration dynamics still needs to be clarified. The work of
[Metzger (2013)] shows that at large volume fraction, the structuration of the
suspension into layers strongly damps the particle shear-induced diffusion
coefficient. As particles organize into layers, fewer collisions per unit of
strain occur between particles thereby reducing the amount of fluctuations.
According to this scenario, layering should slow down the migration process,
as we observed in the present study where, above $\phi_{b}=40\%$, the
migration strain scale no longer decreases. Conversely, the numerical
simulations of [Gallier (2016)] indicate that the particle stress (whose
divergence in inhomogeneous flow drives the migration) is only slightly
affected by the structuration of the suspension. From the latter observation,
one would expect the migration dynamics to be unaffected by layering. Whether
layering affects the migration dynamics thus remains an open question.
Addressing this point would deserve a dedicated investigation as disrupting
the structuration of the suspension requires using polydisperse suspensions.
In such a case, one should account for changes in the critical volume fraction
of the suspension.

## Acknowledgments

This work was supported by NSF (Grant No. CBET-1554044-CAREER) and ACS PRF
(Grant No. 55661-DNI9) via the research awards (S.H.) and by ANR JCJC SIMI 9,
the Labex MEC ANR-10-LABX-0092 and ANR-11-IDEX-0001-02.

##  Appendix A Suspension Balance Model (SBM)

###  A.1 Mathematical Modeling

In the SBM of [Nott & Brady (1994)], particle diffusive fluxes arise from
gradients in the particle normal stress, which enter the solid-phase
continuity equation directly by considering the particle drag closure. The
transport equation for the solid volume fraction is

$\displaystyle\frac{\partial\phi}{\partial{t}}+{\mathbf{\nabla}}\cdot[\phi{\mathbf{u}}]$
$\displaystyle=$
$\displaystyle-{\mathbf{\nabla}}\cdot[\phi(1-\phi){M}~{}{\mathbf{\nabla}}\cdot{\Sigma}^{p}],~{}M=\frac{d_{p}^{2}}{18\eta_{f}\phi(1-\phi)}f(\phi).$
(1)

Where $u$ is the volume average velocity of the suspension and ${\Sigma}^{p}$
is the particle phase stress. Here, $M$ denotes the particle mobility and
$f(\phi)=(1-\phi)^{\alpha}$ is the sedimentation hindrance function
([Richardson & Zaki (1954)]). The viscosity of the suspending fluid is denoted
by $\eta_{f}$. The above transport equation for $\phi$ is solved coupled with
the equations of continuity and momentum for the suspension given respectively
as follows:

$\nabla.\textbf{u}=0,~{}~{}~{}~{}\nabla.\Sigma=0,$ (2)

where $\Sigma$ is the bulk average suspension stress, which is composed of the
particle-phase ($\Sigma^{p}$) and fluid-phase stresses ($\Sigma^{f}$):

$\Sigma^{f}=-p_{f}\textbf{I}+2\eta_{f}\textbf{E},~{}~{}~{}~{}\Sigma^{p}=-\Sigma^{p}_{nn}+2\eta_{f}\eta_{p}\textbf{E}.$
(3)

Here, $p_{f}$ is the fluid-phase pressure, I is the identity tensor, E is the
rate-of-strain tensor, and $\Sigma^{p}_{nn}$ is the diagonal component of
$\Sigma^{p}$ representing the particle-phase normal stress given by

$\Sigma^{p}_{nn}=\eta_{f}\eta_{N}|\dot{\gamma}|\textbf{Q},~{}~{}~{}~{}~{}~{}\textbf{Q}=\begin{pmatrix}\lambda_{2}&0&0\\\
0&1&0\\\ 0&0&\lambda_{3}\end{pmatrix},$ (4)

with $\lambda_{2}=\frac{\Sigma^{p}_{22}}{\Sigma^{p}_{11}}$ and
$\lambda_{3}=\frac{\Sigma^{p}_{33}}{\Sigma^{p}_{11}}$ ([Dbouk _et al._
(2013)]). Combining equations (3) and (4), the bulk average suspension stress
can be written as
$\Sigma=\Sigma^{f}+\Sigma^{p}=-p_{f}\textbf{I}-\eta_{f}\eta_{N}|\dot{\gamma}|\textbf{Q}+2\eta_{f}\eta_{s}\textbf{E}$,
Where $\eta_{s}=\eta_{p}+1$ and $\eta_{N}$ are the relative shear and normal
viscosity of the suspension respectively. We use the following rheological
laws introduced by [Morris & Boulay (1999)] and [Boyer _et al._ (2011)] to
close the system of equations (1 and 2). We then solve the equations following
the numerical scheme of the two-stage, two-level finite difference method
proposed by [Meek & Norbury (1982)].
[Morris & Boulay (1999)] rheological law:
$\eta_{N}=K_{n}(\frac{\phi}{\phi_{m}})^{2}(1-\frac{\phi}{\phi_{m}})^{-2}$,
$\eta_{s}=\frac{\eta}{\eta_{f}}=1+2.5\phi(1-\frac{\phi}{\phi_{m}})^{-1}+K_{s}(\frac{\phi}{\phi_{m}})^{2}(1-\frac{\phi}{\phi_{m}})^{-2}$,
$K_{n}=0.75$, $K_{s}=0.1$, $\alpha=4$, $\lambda_{2}=0.8$, $\lambda_{3}=0.5$
and $\phi_{m}=0.585$.
[Boyer _et al._ (2011)] rheological law:
$\eta_{N}=(\frac{\phi}{\phi_{m}})^{2}(1-\frac{\phi}{\phi_{m}})^{-2}$,
$\eta_{s}=\frac{\eta}{\eta_{f}}=1+2.5\phi(1-\frac{\phi}{\phi_{m}})^{-1}+\mu^{c}(\phi)(\frac{\phi}{\phi_{m}})^{2}(1-\frac{\phi}{\phi_{m}})^{-2}$,
$\mu^{c}(\phi)=\mu_{1}+\frac{\mu_{2}-\mu_{1}}{[1+I_{0}\phi^{2}(\phi_{m}-\phi)^{-2}]}$,
$\mu_{1}=0.32$, $\mu_{2}=0.7$, $I_{0}=0.005$, $\phi_{m}=0.585$, $\alpha=5$,
$\lambda_{2}=0.95$ and $\lambda_{3}=0.6$.

###  A.2 Numerical Algorithm

We solve the system of equations (1) and (2) coupled with the rheological laws
of [Morris & Boulay (1999)] and [Boyer _et al._ (2011)] following the
numerical scheme of the two-stage, two-level finite difference method proposed
by [Meek & Norbury (1982)] . We have used the simplified Newton modified
predict-and-correct scheme (MPC). It was shown that this scheme is consistent
and second-order-accurate in space. This scheme basically linearizes the
standard Crank-Nicolson discretization equation and solves the system of
linear algebraic equations at each time step. For more information about the
details of the MPC scheme, we refer the reader to [Meek & Norbury (1982)].

## References

  * [Abbott _et al._ (1991)] Abbott, J. R., Tetlow, N., Graham, A. L., Altobelli, S. A., Fukushima, E., Mondy, L. A. & Stephens, T. S Experimental observations of particle migration in concentrated suspensions: Couette flow. Journal of Rheology 35.5 (1991): 773-795.
  * [Alghalibi _et al._ (2018)] Alghalibi, D., Lashgari, I., Brandt, L., & Hormozi, S.  Interface-resolved simulations of particle suspensions in Newtonian, shear thinning and shear thickening carrier fluids. Journal of Fluid Mechanics 852 (2018): 329-357.
  * [Boyer _et al._ (2011)] Boyer, F., Guazzelli, É. & Pouliquen, O. Unifying suspension and granular rheology. Physical Review Letters 107.18 (2011): 188301.
  * [Chow _et al._ (1994)] Chow, A. W., Sinton, S. W., Iwamiya, J. H. & Stephens, T. S.  Shear-induced particle migration in Couette and parallel-plate viscometers: NMR imaging and stress measurements. Physics of Fluids 6.8 (1994): 2561-2576.
  * [Corbett _et al._ (1995)] Corbett, A. M., Phillips, R. J., Kauten, R. J., & McCarthy, K. L.  Magnetic resonance imaging of concentration and velocity profiles of pure fluids and solid suspensions in rotating geometries. Journal of Rheology 39.5 (1995): 907-924.
  * [Deshpande _et al._ (2010)] Deshpande, K. V. & Shapley, N. C. Particle migration in oscillatory torsional flows of concentrated suspensions. Journal of Rheology 54.3 (2010): 663-686.
  * [Duda & Hart (1972)] Duda, R. O. & Hart, P. E. Use of the Hough transformation to detect lines and curves in pictures. Communications of the ACM 15.1 (1972): 11-15.
  * [Dbouk _et al._ (2013)] Dbouk, T., Lemaire, E., Lobry, L. & Moukalled, F. Shear-induced particle migration: Predictions from experimental evaluation of the particle stress tensor. Journal of Non-Newtonian Fluid Mechanics 198 (2013): 78-95.
  * [Gallier (2016)] Gallier, S., Lemaire, E., Lobry, L., & Peters, F.  Effect of confinement in wall-bounded non-colloidal suspensions. Journal of Fluid Mechanics 799 (2016): 100-127.
  * [Gholami _et al._ (2018)] Gholami, M., Rashedi, A., Lenoir, N., Hautemayou, D., Ovarlez, G. & Hormozi, S.  Time-resolved 2D concentration maps in flowing suspensions using X-ray. Journal of Rheology 62.4 (2018): 955-974.
  * [Graham _et al._ (1991)] Graham, A. L., Altobelli, S. A., Fukushima, E., Mondy, L. A., & Stephens, T. S.  Note: NMR imaging of shear-induced diffusion and structure in concentrated suspensions undergoing Couette flow. Journal of Rheology 35.1 (1991): 191-201.
  * [Hampton _et al._ (1997)] Hampton, R. E., Mammoli, A. A., Graham, A. L., Tetlow, N. & Altobelli, S. A. Migration of particles undergoing pressure-driven flow in a circular conduit. Journal of Rheology 41.3 (1997): 621-640.
  * [Karnis _et al._ (1966)] Karnis, A., Goldsmith, H. L. & Mason, S. G.  The kinetics of flowing dispersions: I. Concentrated suspensions of rigid particles. Journal of Colloid and Interface Science 22.6 (1966): 531-553.
  * [Koh _et al._ (1994)] Koh, C. J., Hookham, P. & Leal, L. G.  An experimental investigation of concentrated suspension flows in a rectangular channel. Journal of Fluid Mechanics 266 (1994): 1-32.
  * [Leighton & Acrivos (1987)] Leighton, D. & Acrivos, A. The shear-induced migration of particles in concentrated suspensions. Journal of Fluid Mechanics 181 (1987): 415-439.
  * [Lyon & Leal (1998)] Lyon, M. K. & Leal, L. G.  An experimental study of the motion of concentrated suspensions in two-dimensional channel flow. Part 1. Monodisperse systems. Journal of Fluid Mechanics 363 (1998): 25-56.
  * [Maxey (2017)] Maxey, M. R. Simulation methods for particulate flows and concentrated suspensions. Annual Review of Fluid Mechanics 49 (2017): 171-193.
  * [Metzger (2013)] Metzger, B., Rahli, O., & Yin, X. Heat transfer across sheared suspensions: role of the shear-induced diffusion. Journal of Fluid Mechanics 724 (2013): 527-552.
  * [Morris & Boulay (1999)] Morris, J. F. & Boulay, F. Curvilinear flows of noncolloidal suspensions: The role of normal stresses. Journal of Rheology 43.5 (1999): 1213-1237.
  * [Meek & Norbury (1982)] Meek, P. C. & Norbury, J. Two-stage, two-level finite difference schemes for non-linear parabolic equations. IMA Journal of Numerical Analysis 2.3 (1982): 335-356.
  * [Nott & Brady (1994)] Nott, P. R. & Brady, J. F. Pressure-driven flow of suspensions: simulation and theory. Journal of Fluid Mechanics 275 (1994): 157-199.
  * [Ovarlez _et al._ (2006)] Ovarlez, G., Bertrand, F. & Rodts, S.  Local determination of the constitutive law of a dense suspension of noncolloidal particles through magnetic resonance imaging. Journal of Rheology 50.3 (2006): 259-292.
  * [Oh _et al._ (2015)] Oh, S., Song, Y. Q., Garagash, D. I., Lecampion, B. & Desroches, J. Pressure-driven suspension flow near jamming. Physical Review Letters 114.8 (2015): 088301.
  * [Phillips _et al._ (1992)] Phillips, R. J., Armstrong, R. C., Brown, R. A., Graham, A. L. & Abbott, J. R. A constitutive equation for concentrated suspensions that accounts for shear‐induced particle migration. Physics of Fluids A: Fluid Dynamics 4.1 (1992): 30-40.
  * [Pham (2016)] Pham, P. Origin of shear-induced diffusion in particulate suspensions: Crucial role of solid contacts between particles. Doctoral dissertation, Aix-Marseille, (2016).
  * [Richardson & Zaki (1954)] Richardson, J. & Zaki, W. Fluidization and sedimentation–Part I. Transactions of the American Institute of Chemical Engineers 32 (1954): 38-58.
  * [Snook _et al._ (2016)] Snook, B., Butler, J. E. & Guazzelli, É. Dynamics of shear-induced migration of spherical particles in oscillatory pipe flow. Journal of Fluid Mechanics 786 (2016): 128-153.
  * [Stickel & Powell (2005)] Stickel, J. J. & Powell, R. L.  Fluid mechanics and rheology of dense suspensions. Annual Review of Fluid Mechanics 37 (2005): 129-149.