# Multiple Classifiers Based Maximum Classifier Discrepancy for Unsupervised
Domain Adaptation

Yiju Yangl†, Taejoon Kim†, Guanghui Wang‡
†Department of Electrical Engineering and Computer Science, University of
Kansas, Lawrence KS, USA, 66045
‡Department of Computer Science, Ryerson University, Toronto ON, Canada, M5B
2K3

###### Abstract

Adversarial training based on the maximum classifier discrepancy between the
two classifier structures has achieved great success in unsupervised domain
adaptation tasks for image classification. The approach adopts the structure
of two classifiers, though simple and intuitive, the learned classification
boundary may not well represent the data property in the new domain. In this
paper, we propose to extend the structure to multiple classifiers to further
boost its performance. To this end, we propose a very straightforward approach
to adding more classifiers. We employ the principle that the classifiers are
different from each other to construct a discrepancy loss function for
multiple classifiers. Through the loss function construction method, we make
it possible to add any number of classifiers to the original framework. The
proposed approach is validated through extensive experimental evaluations. We
demonstrate that, on average, adopting the structure of three classifiers
normally yields the best performance as a trade-off between the accuracy and
efficiency. With minimum extra computational costs, the proposed approach can
significantly improve the original algorithm.

##  I INTRODUCTION

Thanks to the powerful capability of convolutional neural networks (CNN) in
extracting image features, CNN-based deep learning models have achieved
significant breakthroughs in various computer vision applications like
classification [6, 22], segmentation [16], detection [30], tracking [49],
image translation [46], and counting [39]. However, most of the current neural
network models rely heavily on accurately labeled data. When the trained
models are applied to unfamiliar scenes, the performance of these deep
learning models will be degraded significantly. To address this issue, one
commonly used approach is based on transfer learning to fine-tune the existing
models using the labeled data in the new environment by fixing the parameters
of the first few layers of the network and using new data to fine-tune the
parameters of the last few layers of the network. The reason why this method
can be widely used is that the amount of labeled data it needs is much less
than that of supervised learning and it is easier to implement. However, the
performance is normally inferior to supervised learning. In practice, data
collection is much easier than data annotation, Thus, how to use the unlabeled
data for unsupervised learning has attracted a lot of attention in recent
years.

In order to solve this problem, many researchers try to make deep learning
models learn the common features (i.e., domain invariant features) in
different scenarios. If the models can discover common features among
different scenes, they will effectively classify these scenes. In the study
[10], RevGrad is proposed to use domain classifiers to perform image
classification through unsupervised domain adaptation methods. This method can
effectively merge the two domains and allows the feature extractor to extract
the domain-invariant features. Inspired by this study, more and more
researchers have begun to explore how to use deep-learning models to extract
the domain invariant features, which is one of the main challenges in
unsupervised domain adaptation.

Recently, Maximum Classifier Discrepancy (MCD) [38] was proposed as an
adversarial training framework. In this framework, the two classifiers are
trained by using the maximum classifier discrepancy, and the feature extractor
parameters are adjusted inversely through the decision boundary of the two
classifiers so that the source domain and the target domain are fused
together. In MCD, only two classifiers are used. Is it possible to employ more
classifiers and will the performance benefit from multiple classifiers? The
paper will answer this question and explore an approach to adding multiple
classifiers to the system.

In this study, we propose a very straightforward method to train multiple
classifiers based on the MCD [38] framework. We also compare the performance
difference between the multi-classifier structure and the 2-classifier
structure.

The main contributions of this paper include:

  1. 1.

We propose a new approach to construct a multi-difference loss function based
on the principle of different classifiers, making it possible to add any
number of classifiers into the structure.

  2. 2.

We empirically prove that multi-classifier structures have obvious performance
advantages compared to the 2-classifier structure, however, adding the number
of classifiers could not increase its performance indefinitely.

  3. 3.

Based on our experimental study, we recommend the structure of using three
classifiers as a trade-off between the classification accuracy and the
computational complexity. The performance outperforms the 2-classifier
counterpart by a large margin.

The source code of the proposed approach can be downloaded from this page
https://github.com/rucv/MMCD_DA.

##  II Related Work

Domain adaptation is a commonly used technique in many computer vision tasks
to improve the generalization ability of a model trained on a single domain.
In this section, we describe some existing domain adaptation methods.

Learn domain invariant features. Recently, Chen et al. [8] explored what
enables the contrastive prediction tasks to learn useful representations. [5]
proposed to learn the semantic labels in a supervised fashion, and broadens
its understanding of the data by learning from self-supervised signals how to
solve a jigsaw puzzle on the same images. Yang et al. [47] proposed a dual-
module network to learn more domain invariant features.

Distance-based methods. Aligning the distribution between the source domain
and the target domain is a very common method in solving unsupervised domain
adaptation problems. Maximum Mean discrepancy (MMD) is a method of measuring
the difference between two distributions [13, 28, 44, 26]. DAN [26] explored
the multi-core version of MMD to define the distance between two
distributions. JAN [28] learned a transfer network by aligning the joint
distributions of multiple domain-specific layers across the domains based on a
joint maximum mean discrepancy (JMMD) criterion. [27] enabled the classifier
adaptation by plugging several layers into the deep network to explicitly
learn the residual function with reference to the target classifier. CMD [48]
defined a metric on the set of probability distributions on a compact
interval.

To solve the problem of unbalanced datasets, Deep Asymmetric Transfer Network
(DATN) [45] proposed to learn a transfer function from the target domain to
the source domain and meanwhile adapting the source domain classifier with
more discriminative power to the target domain. DeepCORAL [42] proposed to
build a specific deep neural network by aligning the distribution of second-
order statistics to limit the invariant domain of the top layer. [7] proposed
a Higher-order Moment Matching (HoMM) method to minimize the domain
discrepancy.

Adversarial methods. Adversarial training is another very effective method to
transfer domain information. Inspired by the work of gradient reversal layer
[10], a group of domain adaptation methods has been proposed based on
adversarial learning. RevGrad [10] proposed to learn the global invariant
feature by using a discriminator that is used to reduce the discriminative
features in the domain. Deep Reconstruction-Classification Networks (DRCN)
[11] jointly learned a shared encoding representation for supervised
classification of the labeled source data, and unsupervised reconstruction of
the unlabeled target data. [3] extracted image representations that are
partitioned into two subspaces. Adversarial Discriminative Domain Adaptation
(ADDA) [43] trained two feature extractors for the source and target domains
respectively, to generate embeddings to fool the discriminator.

Maximum Classifier Discrepancy (MCD) [38] was proposed by exploring task-
specific decision boundaries. CyCADA [17] introduced a cycle-consistency loss
to match the pixel-level distribution. SimeNet [36] solved this problem by
learning the domain invariant features and the categorical prototype
representations. CAN [19] optimized the network by considering the discrepancy
between the intra-class domain and the inter-class domain. Graph Convolutional
Adversarial Network (GCAN) [31] realized the unsupervised domain adaptation by
jointly modeling data structure, domain label, and class label in a unified
deep model. [12] proposed a domain flow generation (DLOW) model to bridge two
different domains by generating a continuous sequence of intermediate domains
flowing from one domain to the other. [4] employed a variational auto-encoder
to reconstruct the semantic latent variables and domain latent variables
behind the data. Drop to Adapt (DTA) [21] leveraged adversarial dropout to
learn strongly discriminative features by enforcing the cluster assumption.
Instead of representing the classifier as a weight vector, [29] modeled it as
a Gaussian distribution with its variance representing the inter-classifier
discrepancy.

##  III Proposed Method

In this work, we only consider the close-set unsupervised domain adaptation
problem. Suppose we have a source domain
$D_{s}={\\{(X_{s},Y_{s})\\}}={\\{(x_{s}^{i},y_{s}^{i})\\}}_{i=1}^{n_{s}}$ with
$n_{s}$ labeled samples and a target domain
$D_{t}={\\{(X_{t})\\}}={\\{(x_{t}^{i})\\}}_{i=1}^{n_{t}}$ with $n_{t}$
unlabeled samples. The two domains share the same label space
$Y=\\{1,2,3,...,K\\}$, where $K$ is the number of categories. We assume that
the source sample $x_{s}$ belongs to the source distribution $P_{s}$, and the
target sample $x_{t}$ belongs to the target distribution $P_{t}$, where
$P_{s}\neq P_{t}$. Our goal is to train a classifier $f_{\theta}(x)$ that can
minimize the target risk $\epsilon_{t}=E_{x\in D_{t}}[f_{\theta}(x)\neq
y_{t}]$, where $f_{\theta}(x)$ represents the output of the deep neural
network, and $\theta$ represents the model parameters to be learned.

###  III-A Discrepancy Loss for 2-classifiers

We follow the discrepancy loss in [38] and use the absolute value of the
difference between the probability outputs of the two classifiers as the
discrepancy loss:

$dis(p^{1},p^{2})=\frac{1}{K}\sum^{K}_{k=1}|p^{1}_{k}-p^{2}_{k}|$ (1)

where $p^{1}$ and $p^{2}$ are the probability outputs of the two classifiers
respectively, which are the prediction scores for all the categories, and $K$
is the number of categories, and $p^{1}_{k}$ and $p^{2}_{k}$ are the specific
values of their $k_{th}$ category.

###  III-B Discrepancy Loss for multi-classifiers

We conduct the Discrepancy Loss of the multi-classifiers $Dis()$ based on the
principle that the classifiers are different from each other. In our training
step, we need to maximize the difference between the classifiers. If our
classifiers cannot maintain the principle of mutual difference, then there
will be a situation where the overall discrepancy is maximized but some local
discrepancy is close to zero. Some classifier parameters will become the same
as some classifiers in this process. If the parameters of multiple classifiers
tend to be the same when training multi-classifiers, this will lead to the
collapse of the model training. This conflict will make the performance of the
model even worse.

The following three Discrepancy Losses are designed for the cases of 3
classifiers, 4 classifiers, and $n$ classifiers.

For 3 classifiers:

$Dis(p^{1},p^{2},p^{3})=dis(p^{1},p^{2})+dis(p^{1},p^{3})+dis(p^{2},p^{3})$

For 4 classifiers:

$\displaystyle
Dis(p^{1},p^{2},p^{3},p^{4})=dis(p^{1},p^{2})+dis(p^{1},p^{3})+dis(p^{1},p^{4})$
$\displaystyle+dis(p^{2},p^{3})+dis(p^{2},p^{4})$
$\displaystyle+dis(p^{3},p^{4})$

For $n$ classifiers:

$\displaystyle
Dis(p^{1},\ldots,p^{n})\\!=\\!dis(p^{1},p^{2})\\!+\\!dis(p^{1},p^{3})\\!+\\!\cdots\\!+\\!dis(p^{1},p^{n})$
$\displaystyle\\!+\\!dis(p^{2},p^{3})\\!+\\!\cdots\\!+\\!dis(p^{2},p^{n})$
$\displaystyle\\!+\\!dis(p^{3},p^{4})\\!+\\!\cdots\\!+\\!dis(p^{3},p^{n})$
$\displaystyle\\!+\\!\cdots\\!+\\!dis(p^{n-1},p^{n})$

where $n$ is the number of classifiers, and $p^{1}$, $p^{2}$,$\ldots$, $p^{n}$
are the probability outputs of the $n$ classifiers respectively.

###  III-C Training Steps

Our proposed improvement is based on the framework of two classifiers.
Therefore, when we use multiple classifiers, we replace the original
discrepancy loss with the multi-classifier discrepancy loss under the original
framework.

Step 1: We directly use the source domain data to train the model once so that
the model can initially get some source domain information (features). After
this step, the decision boundary from each classifier can classify the
features extracted from the source domain.

$\displaystyle Loss_{1}=L_{C1}(f_{\theta 1}(X_{s}),Y_{s})+L_{C2}(f_{\theta
2}(X_{s}),Y_{s})+\cdots$ (2) $\displaystyle+L_{Cn}(f_{\theta n}(X_{s}),Y_{s})$

where $C1,C2,\ldots,Cn$ denote the $1$st, $2$nd, and $n$th classifiers,
respectively. $L()$ is the CrossEntropy Loss function.

Step 2: In this step, we use both the source and target domain data to train
our model. During the training, we fix the parameters of the feature extractor
and only adjust the parameters of the classifiers. When we train using the
source domain data, we follow the same way as the first step. When we train
using the target domain data, we cannot perform supervised training since they
have no labels. We employ the discrepancy loss for the target domain training
at this step. We use the $n$-classifier discrepancy loss to maximizes the
discrepancy between the results of the classifiers predicting the target
domain. We maximize the discrepancy by reverse the gradient.

For the source data:

$\displaystyle Loss_{s}=L_{C1}(f_{\theta 1}(X_{s}),Y_{s})+L_{C2}(f_{\theta
2}(X_{s}),Y_{s})+\cdots$ (3) $\displaystyle+L_{Cn}(f_{\theta n}(X_{s}),Y_{s})$

For the target data:

$\displaystyle Loss_{t}=Dis(f_{\theta 1}(X_{t}),\ldots,f_{\theta n}(X_{t}))$
(4)

For joint training:

$\displaystyle Loss_{2}=Loss_{s}-Loss_{t}$ (5)

Step 3: In this step, we minimize the discrepancy in the prediction results of
the features of the target domain extracted by the feature extractor. During
this process, we fix the $n$-classifier parameters and update the CNN’s
parameters by back-propagation. Since the parameters of the classifiers are
fixed, in order to minimize the prediction discrepancy of the $n$-classifier,
the CNN must adjust the parameters so that the features extracted from the
target domain are consistent with the features obtained from the source
domain, then they can achieve similar prediction results with totally
different parameters.

$\displaystyle Loss_{3}=Dis(f_{\theta 1}(X_{t}),...,f_{\theta n}(X_{t}))$ (6)

The source domain is trained by the labeled data, even when the classifier has
the maximum discrepancy to the target domain, it will not have much impact
when detecting the source domain.

##  IV Experiments

###  IV-A Experiments on the Toy Dataset

In this experiment, we compared our proposed method with MCD [29] and the
Source Only method. The purpose is to compare the decision boundaries of
different methods under the same task. We followed the same experimental
setting as MCD [29]. We generated 300 source and target domain data and only
gave labels to the source domain data. In Fig 1, it is very obvious that the
decision boundary under multi-classifiers has better unsupervised
classification capabilities than MCD with a 2-classifier structure and the
Source Only method. The decision boundaries are drawn considering both the
source and target samples. The outputs of three or multiple classifiers make
nearly the same prediction for the target samples, and they classified most
target samples correctly, so we randomly pick one of them for illustration in
Fig 1.

a) Source Only

b) Adapted: MCD

c) Adapted: Ours

Figure 1: Experiments on the toy dataset. The red points and the green points
are the two classes of source domain data, and the blue points are the target
domain data. The dividing line in the middle is the decision boundary. We can
clearly see that the method we proposed is more effective than others. a) is
the model trained only on source samples. b) is the original MCD [29] with the
2-classifier structure. c) is our proposed method with a 3-classifier
structure.

###  IV-B Experiments on Digits and Signed Datasets

| SVHN | MNIST | USPS | SYNSIG
---|---|---|---|---
Method | to | to | to | to
| MNIST | USPS | MNIST | GTSRB
Source only | 67.1 | 79.4 | 63.4 | 85.1
DANN [10] | 71.1 | 85.1 | 73.0$\,\pm\,$0.2 | 88.7
ADDA [43] | 76.0$\,\pm\,$1.8 | - | 90.1$\,\pm\,$0.8 | -
CoGAN [25] | - | - | 89.1$\,\pm\,$0.8 | -
PixelDA [2] | - | 95.9 | - | -
ASSC [14] | 95.7$\,\pm\,$1.5 | - | - | 82.8$\,\pm\,$1.3
UNIT [24] | 90.5 | 96.0 | 93.6 | -
CyCADA [17] | 90.4$\,\pm\,$0.4 | 95.6$\,\pm\,$0.2 | 96.5$\,\pm\,$0.1 | -
GTA [40] | 92.4$\,\pm\,$0.9 | 95.3$\,\pm\,$0.7 | 90.8$\,\pm\,$1.3 | -
DeepJDOT [1] | 96.7 | 95.7 | 96.4 | -
SimNet [36] | - | 96.4 | 95.6 | -
GICT [37] | 98.7 | 96.2 | 96.6 | -
MCD [38] | 96.2$\,\pm\,$0.4 | 96.5$\,\pm\,$0.3 | 94.1$\,\pm\,$0.3 | 94.4$\,\pm\,$0.3
ours (n = 3) | 98.2$\,\pm\,$0.1 | 98.5$\,\pm\,$0.2 | 97.0$\,\pm\,$0.1 | 95.0$\,\pm\,$0.2
ours (n = 4) | 98.6$\,\pm\,$0.1 | 98.4$\,\pm\,$0.3 | 97.1$\,\pm\,$0.1 | 95.1$\,\pm\,$0.1
ours (n = 5) | 98.8$\,\pm\,$0.2 | 98.1$\,\pm\,$0.1 | 96.1$\,\pm\,$0.3 | 95.5$\,\pm\,$0.2
ours (n = 6) | 98.9$\,\pm\,$0.1 | 98.0$\,\pm\,$0.1 | 96.6$\,\pm\,$0.3 | 95.3$\,\pm\,$0.3
TABLE I: The performance on digit classification and sign classification. The
variable $n$ refers to the number of classifiers. We report the mean and the
standard deviation of the accuracy obtained over 5 trials.

In this section, we evaluate our model using the following five datasets:
MNIST [20], Street View House Numbers (SVHN) [33], USPS [18], Synthetic
Traffic Signs (SYN SIGNS) [32], and the German Traffic Signs Recognition
Benchmark (GTSRB) [41].

MNIST: The dataset contains images of digits 0 to 9 in different styles. It is
composed of $60,000$ training and $10,000$ testing images.

USPS: This is also a digit dataset with $7,291$ training and $2,007$ testing
images.

SVHN: Another digit dataset with $73,257$ training, $26,032$ testing, and
$53,1131$ extra training images.

SYN SIGNS: This is a synthetic traffic sign dataset, which contains $100,000$
labeled images, and 43 classes.

GTSRB: A dataset for German traffic signs recognition benchmark. The training
set contains $39,209$ labeled images and the test set contains $12,630$
images. It also contains 43 classes.

We evaluate the unsupervised domain adaptation model on the following four
transfer scenarios:

  * •

SVHN $\xrightarrow{}$ MNIST

  * •

USPS $\xrightarrow{}$ MNIST

  * •

MNIST $\xrightarrow{}$USPS

  * •

SYNSIG $\xrightarrow{}$ GTSRB

Results: The experimental results are shown in Table I. We can clearly see
that our proposed method has a significant improvement over the original MCD
model. In the SVHN $\xrightarrow{}$ MNIST task, our method has improved the
performance by a margin of $2.7\%$. In the MNIST $\xrightarrow{}$USPS task,
our method improved by $2.0\%$. In the task of USPS $\xrightarrow{}$ MNIST,
our method improved by $3.0\%$. In the SYNSIG $\xrightarrow{}$ GTSRB task, our
method improved by $1.1\%$.

It can also be seen from Table I that blindly adding classifiers does not
increase the model performance indefinitely. As the number of classifiers
increases, the growth rate of the model’s performance will decrease and the
model’s performance will approach a peak range. When the model reaches this
peak range, adding more classifiers will cause a decrease in its performance.
The most significant increase happens when we increase the number of
classifiers from 2 to 3. For example, for USPS-MNIST, the accuracy is
increased by 2.0% when 3 classifiers are employed, however, the performance is
only improved by 0.1% when the number of classifiers is increased from 3 to 4.

Method | plane | bcycl | bus | car | horse | knife | mcycl | person | plant | sktbrd | train | truck | mean
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Source Only | 55.1 | 53.3 | 61.9 | 59.1 | 80.6 | 17.9 | 79.7 | 31.2 | 81.0 | 26.5 | 73.5 | 8.5 | 52.4
MMD | 72.3 | 53.1 | 64.7 | 31.8 | 58.2 | 14.3 | 80.7 | 60.0 | 70.0 | 41.4 | 89.7 | 20.7 | 55.9
DANN [10] | 81.9 | 77.7 | 82.8 | 44.3 | 81.2 | 29.5 | 65.1 | 28.6 | 51.9 | 54.6 | 82.8 | 7.8 | 57.4
MCD [38] | 87.0 | 60.9 | 83.7 | 64.0 | 88.9 | 79.6 | 84.7 | 76.9 | 88.6 | 40.3 | 83.0 | 25.8 | 71.9
OURS (n = 3) | 91.1 | 70.0 | 84.1 | 68.8 | 93.5 | 85.3 | 88.2 | 76.7 | 89.2 | 75.0 | 86.2 | 31.2 | 78.3
OURS (n = 4) | 93.3 | 77.6 | 85.3 | 77.4 | 91.8 | 90.2 | 86.4 | 79.3 | 89.5 | 80.4 | 88.5 | 33.1 | 81.1
OURS (n = 5) | 95.2 | 81.4 | 82.9 | 82.7 | 91.9 | 89.6 | 86.7 | 79.3 | 91.0 | 66.6 | 88.1 | 35.7 | 80.9
OURS (n = 6) | 95.0 | 80.1 | 85.5 | 83.1 | 90.5 | 89.9 | 85.9 | 79.2 | 92.8 | 85.3 | 88.3 | 33.9 | 82.5
TABLE II: Results of unsupervised domain adaptation on VisDA2017 [35] image
classification task. The accuracy is obtained by fine-tuning ResNet-101 [15]
model pre-trained on ImageNet [9]. This task evaluates the adaptation
capability from synthetic CAD model images to real-world MS COCO [23] images.
Our model achieves the best performance in most categories.

###  IV-C Experiments on VisDA Classification Dataset

We further evaluate our method on the large VisDA-2017 dataset [35]. The
VisDA-2017 image classification is a 12-class domain adaptation dataset used
to evaluate the adaptation from synthetic-object to real-object images. The
source domain consists of 152,397 synthetic images, where 3D CAD models are
rendered from various conditions. The target domain consists of 55,388 real
images taken from the MS-COCO dataset [23].

In this experiment, we employ Resnet-101 [15] as our feature extractor, and
the parameters are adopted from the ImageNet pre-trained model. The pre-
trained model of our Resnet-18 comes from Pytorch [34] and all experimental
implementations are based on Pytorch. The input images are of the size
$224\times 224$. First, we resize the input image to 256, and then crop the
image to $224\times 224$ in the center. When we train the model using only the
source domain, we just modify the output size of the original last fully
connected layer to a size that conforms to VisDA-2017 [35]. In other tasks, we
utilize a three-layer fully connected layer structure to replace the one-layer
fully connected layer structure of the original classifier. In order to
eliminate the interference factors, except for the source only, all other
algorithms use the same classifier. We uniformly use SGD as the optimizer for
training, and use $1\times 10^{-3}$ for the learning rate of all methods. We
use 16 as the batch size for training.

Results: The experimental results of this part are shown in Table II. In this
experiment, we achieve similar experimental results with the digits dataset.
Compared with the original MCD [29] method, our proposed multi-classifier
implementation method significantly improves the performance in all
experiments in comparison with the 2-classifier structure. The most prominent
increase happens when the number of classifiers is increased from 2 to 3 with
the mean accuracy promoted from 71.9% to 78.3%.

###  IV-D Ablation Study

Figure 2: Result of the ablation study

We conducted the ablation study based on the digital datasets (SVHN
$\xrightarrow{}$ MNIST) with the same unsupervised domain adaptation setting
as Section IV-B. Since our proposed method can add any number of classifiers,
in order to simplify the work, we only use the structure with 3 classifiers as
an example in this experiment. We compare our proposed method under three
different situations: (i) the original MCD; (ii) arbitrarily remove a pair of
discrepancy loss from our Loss; and (iii) arbitrarily replace a discrepancy
loss with any existing discrepancy loss. The latter two cases simulate the
situation where there is a gap in the loss closed loop we proposed. The first
one is caused by the missing, and the second is caused by duplication. In this
regard, we hope to prove the necessity of the closed structure of the loss
function.

The results of the experiment are shown in Figure 2. It can be seen from the
figure that whether it is missing or duplicated, both cases will cause a
significant drop in the performance of our proposed method, even lower than
the baseline MCD. In the case of repetition, the repeated sub-loss conflicts
between the training of the two classifiers, which makes the function
difficult to fit, and therefore all classifiers are affected.

###  IV-E Convergence and Efficiency Analysis

In this section, we make further analysis on the convergence and efficiency of
the models. First, we explore the effect of the number of classifiers on the
convergence speed of the model using the USPS-MNIST dataset as an example. We
record the changes in the model’s average discrepancy loss with iterations and
presented them in Fig 3. From this experiment, we find that the convergence
speeds of different models are roughly close to each other. As the number of
classifiers increases, the convergence speed of the model will slightly
increase.

Second, we compare the average time required to train an epoch with different
numbers of classifier structures. We record the training time complexity in
the USPS-MNIST experiment and show the result in Fig 4. From this figure, we
can see that, as the number of classifiers increases, the time complexity will
increase accordingly. However, the time increase is not too prominent when the
number of classifiers is increased from 2 to 3.

Based on the above experiments, we can see that the classification accuracy is
boosted dramatically for all datasets when we increase the number of
classifiers from 2 to 3, while the computation time only increases slightly.
Therefore, as a trade-off between performance and efficiency, we believe the
structure with 3 classifiers is the best choice in practice for most
unsupervised domain adaptation tasks.

Figure 3: The convergence of different numbers of classifier structures for
the USPS-MNIST task. The convergence speed of discrepancy loss will be a
little bit faster for more classifiers. Figure 4:  The time complexity of our
proposed method for the task of USPS-MNIST. The experiment is conducted using
a RTX 2070 single GPU, and set the batch size to 256, and the other settings
are the same as Section IV-B. From the data, we can see that the time
complexity increases with the increase of the number of classifiers, however,
the time it takes under the structure of three-classifier is close to that of
two-classifier.

##  V Conclusion

In this paper, we have proposed a straightforward approach of adding
classifiers for the adversarial training framework of the maximum classifier
discrepancy of the multi-classifier structure. In order to prevent conflicts
during the training of multiple classifiers, we have proposed a discrepancy
loss function based on the principle that the classifiers are different from
each other. This loss function allows us to add any number of classifiers
under the original maximum classifier discrepancy framework. Extensive
experiments show that the multi-classifier structure has distinct performance
advantages over the original 2-classifier structure. In all classification
tasks, the performance will improve dramatically when we adopt the structure
with three classifiers.

## ACKNOWLEDGMENT

The work was supported in part by The National Aeronautics and Space
Administration (NASA) under grant no. 80NSSC20M0160, and the Natural Sciences
and Engineering Research Council of Canada (NSERC) under grant
RGPIN-2021-04244.

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# Rheology and structure of a suspension of deformable particles in plane
Poiseuille flow

Vassanti Audemar  Université Grenoble Alpes, CNRS, LIPhy, F-38000 Grenoble,
France  Université Paris Saclay, I2BC, Paris, France  Thomas Podgorski
Université Grenoble Alpes, CNRS, LIPhy, F-38000 Grenoble, France  Université
Grenoble Alpes, CNRS, Grenoble INP, LRP, F-38000 Grenoble, France  Gwennou
Coupier<EMAIL_ADDRESS>Université Grenoble Alpes,
CNRS, LIPhy, F-38000 Grenoble, France<EMAIL_ADDRESS>
###### Abstract

We present an experimental study of the rheology and structure of a confined
suspension of deformable particles flowing in a quasi-two-dimensional
Poiseuille flow. Thanks to a precise microfluidic viscosimetry technique
combined with measurements of concentration profiles, our study provides the
first experimental confirmation with three-dimensional particles of a strong
relationship between structuring effects and rheology, previously only
reported in numerical simulations of purely two-dimensional systems. In
conditions where strong structuring effects take place due to confinement, the
evolution of the effective viscosity with particle concentration (here, red
blood cells) shows a remarkable succession of ranges of rapid growth and
plateaus that are associated to qualitative transitions in the structure of
the suspension.

##  I Introduction

The rheological properties of complex materials may differ drastically
depending on the scale at which they are probed. When a specific length scale
is imposed by the flow geometry, such as when confining the material, the bulk
properties of the material are often insufficient to properly describe the
flow behaviour when this length scale becomes comparable to the characteristic
length of the microstructure. A well-known example is the Fåhræus-Lindqvist
effect in blood flows, where the apparent viscosity in tube flows decreases
with the tube diameter due to structuring of the suspension and the formation
of a cell-free layer near walls Lindqvist31 .

Even for simple molecular fluids with a Newtonian behaviour at the continuum
scale, the molecular scale sets a threshold below which the fluid cannot be
considered any more as a continuous material: layering transitions marked by
jumps in the normal force have been observedhorn81 , or dependency of the
effective shear viscosity with the gap size bureau10 ; Maali06 .

Three orders of magnitude above, suspensions of micron-size particles have
more recently been shown to exhibit such connections between layering effects
and rheology. The actual picture is still sparse ; while there seems to be no
clear correlation between rheology and structure for colloidal suspensions
xu13 , non-colloidal suspensions of hard spheres exhibit jumps in shear
viscosity associated with varying the gap of the Couette geometry: simulations
aerov15 ; fornari16  show that for gaps of order a few particle diameters, the
effective viscosity oscillates with the gap size, with a period equal to
particle diameter, and viscosity minima are found for gaps that are multiple
of this particle diameter. In a less confined situation yeo10 , a decrease in
viscosity is also associated with ordering transitions, when they come to
occur. Other studies have highlighted some ordering in the plane of shear,
with no clear signature on the dissipation blanc2013 ; deboeuf2018 .

The above mentioned systems of rigid particles exhibit layering transitions
when the particle concentration is such that steric effects trigger ordering.
Deformable particles influence each other over larger distances through
hydrodynamic interactions loewenberg97 ; singh09 ; lac08 ; pranay10 ; gires14
, thus opening the possibility for suspension structuring at relatively lower
concentrations.

Two-dimensional (2D) numerical simulations of a suspension of vesicles (drops
encapsulated by an incompressible membrane) have highlighted such structuring
effects under imposed simple shear flow. Plateauing of the effective viscosity
is observed periodically as a function of cell concentration at fixed gap size
thiebaud14 ; shen17 ; Naitouhra2019 . This feature coincides with the self-
organization of the particles into discrete layers separated by layers of
suspending fluid, until a new layer is created at a critical value of the
concentration. 2D simulations of capsules in Poiseuille flow have recently
exhibited similar phenomenon: a marked increase in viscosity is associated to
the transition from single file to double filefeng21 . Another study on a
similar system has also highlighted, yet on a sparse set of data, the
influence of cell organization on effective rheology lazaro19 .Since the 2D
nature of these configurations leads to strong topological constraints and
interactions that favor the alignment of particles and their organization into
regular layers, one can ask to what extent these structures and their impact
on rheology are significant in three-dimensional (3D) suspensions, simulated
or experimental. Due to the relatively high computational cost of 3D
simulations, systematic studies of the structure-rheology relationship in 3D
channel flows of suspensions are still scarce fedosov14b . Experimentally, the
simultaneous measurement of subtle changes in the structural organization of
suspensions in channels and of related viscosity variations in ranges of
particle volume fractions between 0 and 50 % is a considerable challenge.

In this paper, we contribute to this important structure-rheology link in
confined suspensions, which has important consequences on flow and particle
transport in confined channels and networks, by exploring experimentally the
rheological behaviour of a confined model suspension of deformable particles.
To that end, we consider red blood cells, which are conveniently monodisperse
enough to be a model system and whose mechanical properties and dynamics have
received a lot of attention in recent experimental, theoretical and
computational works. Red blood cells exhibit collective dynamics that can lead
to the expected layering behaviour in confined flow, a phenomenon that we
control by varying the viscosity of the external fluid. Through measurements
of the effective viscosity of the suspension and of concentration profiles and
by varying RBC volume fraction, we show correlations between viscosity
variations and layering of the suspension when the viscosity of the suspending
fluid favors the formation of coherent structures through hydrodynamic
interactions, that are reminiscent of observations made in 2D simulations in
Poiseuille flow feng21 .

##  II Experimental set-up

Figure 1: (a) Principle of the microfluidic viscosimeter. The channel is very
flat, with $w=50h$. A fluid of known viscosity $\eta_{0}$ spans over a width
$w_{0}$ while the other fluid (here, the suspension of red blood cells) spans
over a width $w_{1}$. An example of (i) a raw image and (ii) a time-averaged
image, viewed along the $z$ axis, are shown inside the channel. The intensity
profile along the $y$ axis is calculated by averaging also in the flow
direction $x$. Inset shows such a profile, which has been normalized by its
extreme values. (b) A different set-up allows for visualisation of the
associated structure in the shear direction. The width of the channel is
$w^{\prime}\ll w$, but a flat configuration is preserved:
$w^{\prime}=3.25\times h$. Examples of time-averaged pictures are shown below
for (i) $\eta_{0}=1.6$ mPa.s and $\Phi_{T}=0.204$, (ii) $\eta_{0}=5.8$ mPa.s
and $\Phi_{T}=0.137$ (contrast enhanced for illustrative purposes).

Red blood cell suspensions are prepared following standard techniques for in-
vitro studies. Blood was supplied by the Etablissement Français du Sang (EFS).
It was taken from healthy volunteers and stored in tubes containing citrate to
prevent coagulation. Blood samples were first washed 3 times in a solution of
phosphate-buffered saline solution (PBS : 9 mM Na2HPO4, 1.3 mM NaH2PO4, 140 mM
NaCl, pH 7.4) and red blood cells were separated from other components of
blood (platelets, white cells, plasma, citrate) by centrifugation. From the
pellet, successive dilutions were performed using 3 suspending solutions with
different viscosities. Following an initial suggestion by Roman et al.roman12
, which has been fruitfully exploited in several subsequent studies roman16 ;
Shen16 ; losserand19 ; minetti19 ; merlo22 , the density of the suspending
solutions was adjusted with Optiprep (iodixanol solution, Axis Shield) to
match the average density of red blood cells and thus avoid sedimentation
during the experiments. Three solutions where prepared, respectively with (a)
water 68.5 %, Optiprep 31.5 %, PBS, BSA 0.1 %, of viscosity 1.6 mPa.s at 20∘C
; (b) water 72.5 %, Optiprep 27.5 %, Dextran 100 kDa 6.5 %, PBS, BSA 0.1 %, of
viscosity 5.8 mPa.s at 20∘C ; (c) water 75 %, Optiprep 25 %, Dextran 100 kDa
10 %, PBS, BSA 0.1 %, of viscosity 8.1 mPa.s at 20∘C. BSA (Bovine Serum
Albumin) aims at maintaining an initial coating of the walls of the
microfluidic system to prevent cell adhesion. The concentration of Dextran 100
kDa (from Leuconostoc mesenteroides, Roth) is such that it does not induce RBC
agregationneu2002 .

Bulk shear viscosity measurements were performed in a cone-plate rheometer
(Anton Paar MCR 301) at fixed shear rate 1000 s-1, in a water-saturated
atmosphere to prevent evaporation. The cone-plate geometry was cleaned between
each sample.

Shear rheology of highly confined liquids is a difficult task as it requires
to maintain a constant gap thickness while measuring potentially low stresses.
An attempt to use an elaborate small-gap rheometer that had proven useful in
other systems baik11 ; jofore15  showed that low viscosities of our
suspensions and their small variations reach the sensitivity limit of the
instrument. Visualisation of the structure of the suspension is also
problematic in such a shear flow chamber where side view cannot be
implemented. Using confocal microscopy to scan the sample in planes parallel
to the walls is tricky as soon as the RBC volume fraction reaches a few
percent, due to the high refraction of light by red blood cells.

We turned to a simple yet powerful microfluidic design proposed by Galambos
and Forster Galambos1998ANOM , and developed for the study of different
systems guillot06 ; solomon14 ; solomon16 . It allows viscosity measurements
of a fluid under a quasi-2D Poiseuille flow using co-flowing laminar streams.
Such a set-up has already been used to measure the viscosity of low stress
fluids such as an active suspension of E. coli bacteria gachelin13 , or blood
Kang16 ; KIM18 . In these latter studies, the confinement of the suspension
was not as important as here (50 $\mu$m Vs 20 $\mu$m). More precisely, we
consider a Y-shaped Hele-Shaw cell, built on demand in glass by Klearia. Each
inlet is fed by two different fluids, a Newtonian one of known viscosity
$\eta_{0}$ and another one whose viscosity $\eta_{1}$ is to be measured (see
Fig. 1(a)). Both flows are driven at an identical flow rate $Q=10\,\mu$L/min,
corresponding to a maximum wall shear rate of 5000 s-1, thanks to a two-
channel syringe pump (KDS Legato 180). The two flows meet in a very flat
channel of height $h=20\,\mu$m , width $w=1$ mm and length $L=15$ mm, such
that shear is mainly in the direction $z$ of the thickness of the channel. The
choice of glass as material is dictated by the necessity to avoid uncontrolled
swelling of the channels under pressuregervais06 , which would be an issue for
both the deduction of the viscosity of the unknown sample from the image
analysis, and the analysis in term of suspension structuring.

Thanks to the low Reynolds number (of order 0.1), the position of the
interface between the two fluids reaches a steady position along the flow axis
$x$ at a distance of order $h$, after which the width $w_{i}$, $(i=0,1)$
occupied by each fluid is measured. When the interface between the two fluids
is far from the lateral walls, solving the Stokes equation with the no slip
condition at the channel walls, the velocity continuity between the two fluids
streams and the low aspect ratio, leads to a simple relationship between
viscosities and the widths of the two parallel streamsGalambos1998ANOM :
$\eta_{1}/\eta_{0}=w_{1}/w_{0}$. The viscosity of the tested fluid is
therefore deduced from the measurement of the position of the interface. The
precision in the measurement is associated with the precision in the
determination of the positions of the lateral walls and of the interface
between the two liquids. Here, we measure this position before shear-induced-
diffusion blurs the interface grandchamp13 . Observations are made around 5 mm
after the two flows merge. Since the cells have already flown in the inlet
branch of length at least 5 mm, this is highly enough to ensure the cell
structure is steady: two different studies zhou2020 ; losserand20  show that
for a confinement of 20 microns, 2 mm are sufficient to establish a steady
structure. For each sample, a time of 10 minutes was respected between the
beginning of the injection and the time of measurement to allow stabilization
of the flow. This relaxation time is a well-known direct consequence of
driving a flow by imposing its rate through channels with a high hydraulic
resistance Lake2017  and has been reduced as much as possible by an adequate
choice of rigid materials such as small glass syringes (Hamilton 1002TLL 2.5
mL) and PTFE tubing. The chip was placed on an inverted microscope (Olympus IX
71) for imaging.

Before each experiment, the channel was filled with distilled water and placed
under vacuum to remove any trapped bubble that would strongly increase
equilibration times. The channel was then filled with a PBS solution with 1%
of BSA and kept in a refigerator overnight. This allowed albumin to coat the
surfaces in such a way that no cell adhesion would occur during experiments by
preventing contact between RBCs and glass. Syringes were also treated. Then,
the channel was filled with the reference fluid of viscosity $\eta_{0}$, which
is the fluid (a), (b) or (c) that was used to suspend cells and a reference
image was taken. The suspension to be studied was then injected through the
other inlet and both fluids were driven at the same flow rate. The channel was
rinsed with the same reference fluid when switching between different samples
prepared with the same considered medium but a different red blood cell
concentration. When switching to a different suspending medium, the channel
was completely cleaned and refilled.

For each sample, a short movie was recorded once the interface reaches a
steady position, and the time-averaged image of the stack was calculated using
ImageJ in order to reduce noise and small fluctuations. The background
(reference) image was then subtracted. To detect the position of the interface
between the test fluid and the reference fluid, the intensity profile across
the channel ($y$ direction) was averaged over the dimension of the image along
the flow direction $x$ (see Fig. 1(a)), and fitted with an error function with
IDL software. The position of the interface was chosen at 85% of the height
separating the two plateaus of the error function, yielding the width $w_{0}$
and $w_{1}$ of each stream. The viscosity ratio
$\eta_{1}/\eta_{0}=w_{1}/w_{0}$ was then deduced. The inaccuracy in the
detection of the interface and side walls is estimated to about $\pm$2 pixels,
which leads to a maximum error on $w_{1}/w_{0}$ of about $\pm 1\%$, a value
that is used for error bars on the graphs.

For each sample, its volume fraction, called reservoir volume fraction
$\phi_{R}$, was measured by centrifugation of the samples in glass haematocrit
capillary tubes. The tube volume fraction $\phi_{T}$ is the local volume
fraction of the suspension in the flow. In confined flows, it generally
differs from the reservoir volume fraction, a phenomenon known as Fåhræus
effect for bloodpries92 . This is a consequence of the non homogeneous
distribution of cells across the channel thickness and the formation of a
cell-free layer near walls, leading to an average velocity of red blood cells
higher than the average fluid velocity. Tube volume fraction was determined by
an effective absorbance measurement of flowing red blood cells (Beer-Lambert
law), a method that has proven to be reliable up to rather high volume
fractions levels pries83 ; roman16 ; mantegazza20 ; merlo22 . The effective
absorption coefficient was calibrated on diluted samples of cells, that could
be counted individually. The conversion into cell volume fraction is then done
by considering 90 $\mu$m3 as the average red blood cell volumeMcLaren87 .

The micro-viscosimeter does not allow for an easy evaluation of the structure
of the suspension in the shear direction $z$. Direct determination requires to
observe the flow in the $y$ direction, which is not possible due to the very
high aspect ratio of 20 $\mu$m $\times$ 1 mm of this channel. As a compromise,
we injected the same suspensions of cell in straight channels made in a PDMS
chip of rectangular cross section with the same thickness $h=20\,\mu$m in the
$z$ direction and a width $w^{\prime}=65\mu$m in the $y$ direction. Its
orientation in the microscope is such that the optical axis is now the $y$
axis (see Fig. 1(b)). Observation is made at a distance of at least 5 mm from
the inlet, to ensure a steady profile is observed. With a PDMS technology, a
higher aspect ratio $w^{\prime}/h$ would imply non straight walls upon bonding
of the PDMS matrix on the sealing plate, and in addition a too high aspect
ratio would lead to optical aberrations due to reflections and refraction of
incident light by the walls, whatever their quality and material. In our case,
while we are not exactly in a situation where $w^{\prime}\gg h$, we still have
$w^{\prime}/h=3.25$, meaning that the flow and the structure of the suspension
are dominated by the smallest dimension $h$ that sets the main direction of
velocity gradients. The particle distribution and layering effects are
therefore expected to be very similar, though probably smoother, to the flow
structure taking place in the microfluidic viscometer. Note that while
structuration in the vorticity direction has been reported for highly confined
plane Poiseuille flow of RBC suspensions iss2019 , it does not seem to take
place in our case where the thickness is twice as large: no structure is
visible from the direct vizualisation in the viscosimeter (Fig. 1(a)). The
relationship between the volume fraction $\Phi_{T}$ within the observation
channel and its value $\Phi_{R}$ in the reservoir at the entrance is assumed
to be the same as in the micro-viscosimeter. This is justified by the fact
that in the low viscosity suspending medium, and at least until volume
fractions of $\Phi_{T}\sim 25\%$, the relationship between $\Phi_{R}$ and
$\Phi_{T}$ in the viscosimeter is close to that proposed by Pries et al. in a
cylindrical tube of diameter $h$ (Fig. 2) — which is a result similar to that
on viscosities, shown in Fig. 3 and discussed later on. As the observation
channel, of aspect ratio 3.25, can be considered as an intermediate situation
beween the Hele-Shaw cell and the cylinder, we can estimate that the
relationship between both volume fractions should be similar. As the logarithm
of the intensity profiles in the channel yields only a profile that is
proportional to the local volume fraction profiles (Beer-Lambert law), the
values $\Phi_{T}$ estimated from this relationship are used to rescale these
profiles. Noteworthy, the relationship between reservoir and tube volume
fractions does not seem to depend much on the viscosity of the external medium
while RBC dynamics does minetti19 , a fact for which we have no clear
explanation.

Figure 2: Tube volume fraction $\Phi_{T}$ in the micro-viscosimeter as a
function of reservoir volume fraction $\Phi_{R}$, for three different initial
viscosities of the suspending fluid. Full line corresponds to the relationship
given by Pries et al. pries92 , for blood in cylindrical channels of 20
$\mu$m, characterizing the Fåhræus effect Fahraeus1929 .

##  III Results and discussion

###  III.1 Viscosity

In discussions on the rheology of suspensions, a convenient and widely used
parameter is the relative viscosity, that evaluates the contribution of the
particles to the dissipative mechanisms. Recalling that $\eta_{1}(\Phi_{T})$
is the viscosity of the suspension with tube volume fraction $\Phi_{T}$, the
relative viscosity is defined as
$\hat{\eta}(\Phi_{T})=\eta(\Phi_{T})/\eta^{\prime}_{0}(\Phi_{T})$ where
$\eta^{\prime}_{0}(\Phi_{T})$ is the viscosity of the suspending medium of the
suspension. It differs from the initial viscosity $\eta_{0}$ of the reference
fluid in which the cells were suspended since there is always a residual
volume of washing fluid (of viscosity $\eta_{w}\sim$ 1 mPa.s) in the
centrifuged red blood cells. It has been estimated to be around 3% for similar
centrifugation speeds by Chaplin et al chaplin52 . Assuming linear
contribution of this residual fluid, the corrected viscosity for a suspension
where a volume $\Phi_{R}$ has been mixed with a volume $1-\Phi_{R}$ of
suspending fluid is then
$\eta^{\prime}_{0}=((1-\Phi_{R})\eta_{0}+0.03\Phi_{R}\eta_{w})/(1-0.97\Phi_{R})$.
This leads to corrections in viscosity of up to 3% for the more viscous fluid.
The impact of the residual fluid after centrifugation on the final value of
the volume fraction is, in contrast, much below the precision of haematocrit
measurements.

In principle, the microfluidic rheometer directly yields the ratio between the
unknown viscosity of the suspension and that of the reference fluid. However,
the finite thickness of the interface between the two fluids (the intensity
profile necessarily varies on a scale of a few cell radii, i.e. around 20
microns) makes this measurement sensitive to the choice of a threshold level
to determine the position of this interface. This means that the absolute
values of the relative viscosity that are found here are subjected to an
offset of a few % depending on the choice of the threshold for the position of
the interface between the two fluids to be compared. Importantly, this offset
does not vary much with $w_{1}/w_{0}$, meaning that this choice has no impact
on the discussion about relative variations of viscosities, which is the goal
of this paper.

For a suspending fluid of viscosity similar to plasma in blood, a marked
difference between confined and bulk rheology can be seen (Fig. 3). The
decrease of viscosity with decreasing tube diameter (until the latter
approaches RBC diameter, that is about 7.9 $\pm$ 0.7 $\mu$m linderkamp82 ) is
known as the Fåhræus-Lindquist effect Lindqvist31  and has been thoroughly
studied in the literature. The decrease is essentially explained by the
existence of a cell depleted layer near the walls, acting as a lubrication
film, and whose relative contribution to the overall dissipation becomes more
and more important as the tube diameter decreases. An ad-hoc model accounting
for the dependency of the viscosity with the cell volume fraction and the tube
diameter has been proposed by Pries et al.pries92 , which is now recognized as
a correct description of blood apparent viscosity in tube flow. While our
geometry is confined only in one dimension, we show in Fig. 3 that this
empirical law describes well our experimental data, indicating that a 2D
structuring has a similar effect on rheology as in a 3D axi-symmetric
situation.

Figure 3: Relative viscosity $\hat{\eta}$ for $\eta_{0}=1.6$ mPa.s, as a
function of reservoir volume fraction $\Phi_{R}$ in two geometries : 20 $\mu$m
channel and cone-plate rheometer measuring the bulk viscosity. Lines
correspond to the blood viscosity model given by Pries et al. pries92 , for
cylindrical channels of 20 $\mu$m and infinitely large diameters,
respectively, with no adjustable parameter.

If we now consider the three considered suspending media, Fig. 4 shows that
the contribution of cells to bulk rheology may depend on the viscosity of the
suspending fluid; more precisely, the roughly linear behaviour
$\hat{\eta}=1+[\hat{\eta}]\Phi_{T}$ of the relative viscosity with volume
fraction is characterized by an intrinsic viscosity $[\hat{\eta}]$ which is
about 3 times smaller when $\eta_{0}$ goes from 1.6 to 5.8 mPa.s. By contrast,
this coefficient is almost unchanged when further increasing $\eta_{0}$ to 8.1
mPa.s. This is in overall agreement with the exhaustive study of Vitkova et
al.vitkova08  where the authors discuss the link between this intrinsic
viscosity and the dynamics of red blood cells, which itself strongly depends
on the viscosity contrast between their cytosol and the suspending
mediumfischer13 ; minetti19 . They showed that when this viscosity contrast is
varied, the intrinsic viscosity exhibits a minimum when the viscosity of the
suspending medium is close to the viscosity of the cytosol, corresponding to a
transition between flipping and tank-treading dynamics of the cells. In our
experiments, the last two cases where $\eta_{0}=5.8$ and $8.1$ mPa.s are close
to this minimum where RBCs are aligned with the flow, while for $\eta_{0}=1.6$
mPa.s they undergo a more dissipative tumbling dynamics.

Figure 4:  Relative bulk viscosity $\hat{\eta}$ as a function of tube volume
fraction $\Phi_{T}$ (here equal to reservoir volume fraction $\Phi_{R}$), for
three different initial viscosities of the suspending fluid. Figure 5:
Relative viscosity $\hat{\eta}$ in the 20 $\mu$m channel as a function of tube
volume fraction $\Phi_{T}$, for three different initial viscosities of the
suspending fluid.

In the narrow channel, the evolution of viscosity with tube volume fraction is
more complex (see Fig. 5). In the $\eta_{0}=1.6$ mPa.s suspending medium a
marked, steady increase of viscosity with volume fraction is observed. By
contrast, in the $\eta=8.1$ mPa.s suspending medium, there is no significant
variation of the viscosity over a large range of volume fraction $\Phi_{T}$,
also in marked contrast to the scenario in bulk: remarkably, adding more
particles to the suspension up to a volume fraction of 30 % does not increase
viscous dissipation and the effective viscosity by more than a few %. In the
$\eta_{0}=5.6$ mPa.s fluid, an intermediate scenario arises, with successions
of increases with slopes similar to that in the less viscous fluid, followed
by plateauing and even slight decreases within experimental errors, in
particular around $\Phi_{T}=10\%$.

These features are all the more significant as they are highlighted through
$\hat{\eta}$, which compares two viscosities measured with the same method.
The existence of a local minimum of viscosity around $\Phi_{T}=0.12$ is in
line with the unpublished results of a recent PhD thesis hogan19 , where 2D
simulations of cells having the same cytosol viscosity as that of the
suspending fluid were run. In channels of width respectively 9, 10 and 12
$\mu$m, a local minimum in the relative viscosity is found for volume
fractions around 0.3, 0.25 and 0.2, respectively. No minimum was observed for
larger channels, but only a few volume fractions were considered, leaving the
possibility for the minimum to be missed.

For a better understanding of this non-monotonic behaviour, we have observed
and compared the structure of the suspension in the cases $\eta_{0}=1.6$ mPa.s
and 5.8 mPa.s.

###  III.2 Structure

Figure 6: Concentration profiles in the $\eta_{0}=1.6$ mPa.s suspending fluid
for different tube volume fractions $\Phi_{T}$. Figure 7: Concentration
profiles in the $\eta_{0}=5.8$ mPa.s suspending fluid for different tube
volume fractions $\Phi_{T}$.

In the low viscosity case, while a slight double-peak structure is visible at
very low volume fractions, as in other workstsvirkun17 , a single peak, blunt
profile quickly emerges as the volume fraction goes beyond 7% (Fig. 6). By
contrast, in the $\eta_{=}5.8$ mPa.s suspending medium, a marked double peak
structure is present for almost all volume fractions (Fig. 7). Note that in
all cases, the concentration between the two peaks is never 0, as can be
sometimes seen in numerical simulations thiebaud14 ; shen17 ; feng21 . This
can be related to limits of these numerical simulations, namely (i) they are
2D, a situation in which strong topological constraints impose that layers of
particles, when they form, completely exclude each other while there are
additional degrees of freedom in 3D simulations or experiments, in which
particles do not need to be aligned in the same $(x,z)$ plane and (ii) most
simulation works consider that all cells are identical (regarding their
geometry and mechanical properties), while it is now well known that the
dispersion in these properties (due in particular to ageing) leads to a
dispersion of behaviour under flow fischer13 ; minetti19 , including the
intensity of their migration velocity far from the channel walls losserand19 .
Obtaining less pronounced structures in experiments was therefore expected.

We can characterize the importance of the double-peak structure by analysing
the difference between the height of the peak and the concentration in the
middle of the channel ($z=0$), rescaled by the peak value. This relative
amplitude $A_{peak}$ of the peaks is plotted in Fig. 8 as a function of tube
volume fraction.

We observe that in the 0.02-0.075 range of $\Phi_{T}$, the net increase in
viscosity coincides with the formation of a marked double-peak structure (Fig.
8) and a significant decrease of the thickness of the cell-free layer near top
and bottom walls seen in Fig. 7, as roughly evaluated by considering the
position of the mid-height point. Overall, the formation of this double-peak
structure widens the distribution in comparison to what is seen in Fig. 6.
While the relative dissipation at the particle level should decrease when
going from $\eta_{0}=1.6$ mPa.s to $\eta_{0}=5.8$ mPa.s due to different
dynamics, as seen in bulk rheology (Fig. 4), the structuring effects (double
peak and decreasing cell-free layer) actually lead to relative viscosities
that are comparabe for $\eta_{0}=5.8$mPa.s in this range of $\Phi_{T}$ as seen
in Fig. 5. For $0.075<\Phi_{T}<0.15$, the relative attenuation of the double
peak structure is accompanied by a plateau or apparent small decrease in the
measured viscosity. One can indeed suspect that adding more cells in the
central part, where the velocity gradients are probably small, has only a
small effect on viscous dissipation. In the $0.15<\Phi_{T}<0.25$ range, while
it is more difficult to reach a conclusion as the trends are less marked, as
also the case in simulations in simple shear flow thiebaud14 , one can still
notice, again, that in Fig. 7 a significant increase of the width of the
distribution or decrease of the cell free layer takes place between
$\Phi_{T}=0.137$ and $\Phi_{T}=0.192$, corresponding to a sharp increase of
the viscosity, while between $\Phi_{T}=0.192$ and $\Phi_{T}=0.258$ only the
height of the profile varies, corresponding to a slower increase of the
viscosity. One can suspect that in this range, the velocity profile is
essentially a blunt one in the central part, with little contribution to the
overall dissipation.

While it would be interesting to extend and confirm these results, especially
by investigating the structure-rheology relationship at different values of
the channel thickness $h$, the trends observed here are in line with recent
simulations in 2D Poiseuille flowfeng21  and show marked changes in the
evolution of the effective viscosity with particle volume fraction. These are
associated to structural transitions from one layer to two layers, the
relative amplitude of these two peaks and the width of the overall
distribution.

Figure 8: Blue dots, left axis: relative amplitude of the two peaks of the
concentration profiles in the $\eta_{0}=5.8$ mPa.s suspending fluid. Faded
points correspond to imprecise results at very low volume fractions for which
the variations of concentration lead to fluctuations of only a few gray-levels
in the images. The relative viscosity of the same suspension, similar to that
of Fig. 5, is also plotted in full green line (right axis).

##  IV Conclusions

We performed an experimental study of the rheology and structure of a
suspension of 3D deformable particles - red blood cells - flowing in a
quasi-2D Poiseuille flow in a narrow channel. Combining a precise microfluidic
viscosimetry technique and measurements of the particle concentration profiles
in the thickness of the channel, we highlighted the evolution of the structure
and viscosity of the suspension when varying a) the volume concentration of
particles and b) their dynamics (and subsequent ability to organize into
layers) by varying the viscosity of the suspending fluid.

While a general feature is that the effective viscosity of suspensions is
significantly lower in confined channel flow than in bulk shear flow due to
the well-know existence of a particle-free lubricating layer near walls,
noticeable qualitative and quantitative differences are revealed when the
viscosity of the suspending medium is increased.

At low suspending medium viscosity, a situation close to physiological
conditions in which the particles are effectively less deformed by the flow
and exhibit a tumbling dynamics in an unbounded shear flow, a simple structure
takes place. Particle migration towards the centerline and shear-induced
diffusion due to particle-particle interactions lead to a bell-shaped
concentration profile with a maximum at the center and cell-free layers near
walls, this profile being almost self-similar when the average concentration
is increased. This is associated to a nearly steady, monotonic increase of the
relative viscosity when the particle volume fraction is increased, with an
average slope of about 1.

When the viscosity of the suspending fluid is increased, the situation becomes
more complex and nontrivial. At an intermediate value of this viscosity (and
thus of particle deformability), strong structuring effects take place in
which the system shows a transition between a 1-peak concentration profile (as
in the previous case) to a 2-peak structure in which hydrodynamic repulsion
between particles leads to a local minimum of the concentration along the
centerline. This qualitative change in the structure of the suspension as well
as irregular variations of the thickness of the particle-free layer near walls
lead to strong fluctuations of the evolution of effective viscosity with
concentration, with a succession of ranges of rapid growth and plateaus (that
may even suggest small decreases of the viscosity). At even higher suspending
fluid viscosity, the impact of particle dynamics is even more drastic and a
variation of only 3% of the effective viscosity is measured for particle
volume fractions up to 30%.

While a few predictions on the relationship between strong structuring effects
and rheology have been made in confined simple shear flow thiebaud14 ; shen17
or more recently in Poiseuille flow feng21 , these numerical simulations were
made in pure 2D situations where topological constraints intuitively lead to
stronger structuring effects due to the smaller number of degrees of freedom.
To our knowledge, our experimental study provides an important confirmation
that such effects exist and are measurable with 3D particles confined in only
one direction and qualitatively confirms the predictions made on capsules in
2D-Poiseuille flow feng21 . This study, which certainly deserves to be
extended to a wider range of parameters and possibly to other physical systems
of deformable particles, provides interesting perspectives on the structure-
rheology relationship of suspensions in channel flows.

###### Acknowledgements.

V. A. acknowledges a PhD fellowship from ”Ecole Doctorale de Physique de
Grenoble”. T. P. and G. C. acknowledge funding from CNES on blood rheology and
flows.The authors’ team is part of LabEx Tec 21 (Investissements d’Avenir -
grant agreement ANR-11-LABX-0030).

## Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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