# Depth- and Semantics-aware Multi-modal Domain Translation:
Generating 3D Panoramic Color Images from LiDAR Point Clouds*

Tiago Cortinhal1 and Eren Erdal Aksoy1 *This work was co-funded by the
European Union (grant no. 101069576) and supported by Innovate UK (contract
no. 10045139) and the Swiss State Secretariat for Education, Research and
Innovation (contract no. 22.00123). Views and opinions expressed are however
those of the author(s) only and do not necessarily reflect those of the
European Union or the European Climate, Infrastructure and Environment
Executive Agency (CINEA). Neither the European Union nor the granting
authority can be held responsible for them.1Halmstad University, School of
Information Technology, Center for Applied Intelligent Systems Research,
Halmstad, Sweden<EMAIL_ADDRESS>
###### Abstract

This work presents a new depth- and semantics-aware conditional generative
model, named TITAN-Next, for cross-domain image-to-image translation in a
multi-modal setup between LiDAR and camera sensors. The proposed model
leverages scene semantics as a mid-level representation and is able to
translate raw LiDAR point clouds to panoramic RGB-D camera images by solely
relying on semantic scene segments. We claim that this is the first framework
of its kind, and it has practical applications in autonomous vehicles, such as
providing a fail-safe mechanism and augmenting available data in the target
image domain. The proposed model is evaluated on the large-scale and
challenging Semantic-KITTI dataset, and experimental findings show that it
considerably outperforms the original TITAN-Net and other strong baselines by
23.7$\%$ margin in terms of IoU.

##  I INTRODUCTION

There are several attempts at converting images from one domain to another
type-similar domain, known as domain translation, including mapping sketches
or segmentation masks to RGB images [13, 50, 43]. These cross-domain image-to-
image translations have recently received significant attention in computer
vision and robotics. However, the cross-domain mapping in a multi-modal setup,
for instance, between LiDAR and camera sensors, is a challenge and still
remains in its infancy [26, 16, 15, 29]. This is mainly because LiDAR scans
return sparse, unstructured, and non-uniformly distributed 3D point clouds, in
contrast to structured 2D image counterparts acquired from RGB sensors. This,
as a result, makes the image synthesis from raw point clouds non-trivial.

In this work, we address the challenge of multi-modal domain translation by
leveraging the scene semantics as a mid-level representation between different
sensor modalities with unique data formats. To overcome this challenge, we
introduce a new depth- and semantics-aware conditional generative model, named
TITAN-Next (generaTive domaIn TrANslation Network), which adversarially learns
to translate LiDAR point clouds to RGB-D camera images by solely relying on
the semantic scene segments (see Fig. LABEL:fig:introSample).

TITAN-Next is built upon the model TITAN-Net [8], which is embedded in a
modular generative neural network framework. We, here, revise this modular
framework (see Fig. 2) to make the original TITAN-Net concurrently predict the
depth information while generating panoramic RGB camera images with higher
fidelity. More specifically, we have the following contributions:

  * •

We introduce a multi-scale feature pyramid module in the original TITAN-Net
encoder and decoder. To enforce stability and improve learning at different
scales, we inject a downsampled version of the LiDAR input into each level of
the encoder. Similarly, we extract the features from each decoder block,
upsample, and concatenate them with the final output. This multi-scale feature
aggregation yields a significant improvement in robustness, accuracy, and
convergence speed.

  * •

We further append a lightweight depth head to the TITAN-Net decoder to enrich
the generated scene image with the depth information derived directly from the
LiDAR input. This eventually leads to generating RGB-D images from given point
clouds.

  * •

We also report comprehensive quantitative and qualitative evaluations using
the large-scale, challenging Semantic-KITTI dataset [4]. Our experimental
findings show that our new framework considerably outperforms the original
TITAN-Net and other strong baselines by a large margin. Our code and trained
models are available at https://github.com/TiagoCortinhal/TITAN-Next to
encourage further research on the subject.

To the best of our knowledge, the here proposed multi-modal domain translation
framework is the first of its kind that can generate photo-realistic RGB-D
images from raw 3D point clouds by solely relying on the sensor agnostic scene
semantics. This has practical applications in autonomous vehicles. For
instance, an abrupt failure of RGB and depth sensors can be immediately
substituted by the synthesized RGB-D image streams to boost the subsequent
sensor fusion and planning modules. This introduces a natural fail-safe
mechanism to the vehicle to prevent a system collapse. Another potential
application is augmenting the available data in the target image domain. The
generated semantic scene segments can be employed to fabricate vast variants
of the original scene by feeding data from different distributions without any
additional effort. Furthermore, the framework can also be applied to colourize
point clouds without relying on real RGB camera readings.

##  II RELATED WORK

There exists a vast number of work on generating novel scene images [44, 24,
47] and rendering point clouds [32, 26, 3, 16, 15, 37]. Coupling both
addresses the multi-modal domain translation, which still remains an unsolved
challenging problem.

There is a large corpus of work in image-to-image translation, such as
converting an image from one domain (e.g., from a segmentation space) to a
different one (e.g., to an RGB space) [13, 50]. Image semantics has also been
injected to accelerate such a domain transfer process [19, 36, 39, 27].
Similarly, photo-realistic video-to-video translations, i.e., converting image
sequences by considering temporal cues, have been intensively studied [44,
24].

Recent works have, so far, focused on bridging the gap between the same sensor
modalities, i.e., either between RGB camera images [41, 46, 50] or between
LiDAR sensor readings [18, 35]. The literature is still lacking of translation
across different sensor modalities with distinct characteristics, which is of
particular interest in this work.

Most recent works on translating 3D point clouds to 2D RGB images are based on
conditional Generative Adversarial Networks (cGANs). This generative process
relies on a conditional constraint, ranging from a one-hot vector of a given
class set to any other type of data that conditions the generative process.
For instance, works like [32] exploit the informativeness of LiDAR point
clouds projected onto the RGB space to render a depiction of the RGB image,
where the conditions for the cGAN are real images. Following a similar
approach, predefined background image patches are employed in Milz et al. [26]
to bias a GAN model while exploiting the 3D representation in the bottleneck.
The recent method pc2pix introduced in [3] relies on auxiliary cGAN to render
a point cloud to an RGB image from the desired camera angle without employing
any surface reconstruction operation. This way, pc2pix can generate different
images by applying arithmetic operations in the latent space.

As an alternative to cGAN-based adversarial learning approaches, asymmetric
encoder-decoder networks [16] and U-NET models [15] are also introduced to
render point clouds in a rather supervised fashion. Likewise, the work in [37]
introduces an end-to-end framework with a point cloud encoder and an RGB
decoder enriched with a refinement network in order to enhance the image
quality generated from novel viewpoints in a point cloud.

The works from [26, 16, 15, 29] can be considered the most similar to our
method. Although these methods can, to some degree, generate scene images from
a given point cloud, they cannot process unstructured and sparse full-scan
LiDAR point clouds due to computational constraints. Furthermore, these
approaches neither incorporate scene semantics nor predict the depth map for
the generated image. In contrast, we employ scene semantics as a bridge to
convert a full-scan LiDAR point cloud to a panoramic scene image with the
corresponding depth information.

Regarding semantic scene segmentation, which is a well-studied topic, we have
a particular interest in projection-based models [9, 25, 2] to segment LiDAR
point clouds. This is because the spherical projection, for instance, yields
the native 2D view of the LiDAR scan, which allows for generating panoramic
RGB counterparts. There is also a large corpus of work on the semantic
segmentation of camera images [28, 10] with a surge of new papers focusing on
Vision Transformer approaches [38, 23]. Thanks to the modular structure of our
framework, any of these state-of-the-art segmentation models can be seamlessly
integrated into the proposed pipeline.

Figure 2: Our proposed modular framework has five neural networks, each of
which is depicted by a unique background colour. In the red box, a 3D LiDAR
point cloud $p$ is first projected onto the 2D range image plane $p^{\prime}$
to be segmented by SalsaNext [9], predicting $p^{\prime}_{s}$. Likewise, the
corresponding RGB camera image $y$ is processed by SD-Net [38] to predict
semantic segments $y_{s}$, as depicted by the blue box. The depth information
$y_{d}$ is meanwhile estimated by MiDaS [34] as shown in the purple box. The
green box highlights the TITAN-Next model, where the Generator is conditioned
on the concatenated $p^{\prime}_{s}$ and $p^{\prime}$ to generate the fake
camera segment map $\widehat{y}_{s}$ and the fake depth information
$\widehat{y}_{d}$. The TITAN-Next Discriminator is also conditioned on
$p^{\prime}_{s}$ while comparing $\widehat{y}_{s}$ and $\widehat{y}_{d}$ with
the expected $y_{s}$ and $y_{d}$, respectively. Finally, as shown in the
yellow box, the generated fake camera segment $\widehat{y}_{s}$ is processed
by Vid2Vid-Net [44] to synthesize the realistic RGB image $\widehat{y}$. Note
that the blue and purple boxes are employed during training only.

##  III Method

As depicted in Fig. 2, we propose a modular generative framework, which
involves five individually trained neural networks to translate a given LiDAR
point cloud to the corresponding RGB-D image. TITAN-Next shown in the green
box in Fig. 2 constitutes the main contribution of this work. We follow a cGAN
approach, where our model TITAN-Next is conditioned on the semantic segment
maps of the spherically projected LiDAR point clouds. More specifically, given
a LiDAR point cloud $p\in\mathbb{R}^{n\times 4}$, where $n$ is the cardinality
of the cloud storing $x$, $y$, $z$ point coordinates and $i$ intensity values,
TITAN-Next not only learns the mapping between the LiDAR and camera image
semantic segments but also estimates the corresponding depth information.
Estimated image segments and depth cues are further processed in order to
generate the corresponding four channel RGB-D image of the scene
$y\in\mathbb{R}^{w\times h\times 4}$, where $w$ and $h$ are the image width
and height. In the following subsections, we will detail the framework and
improvements.

###  III-A LiDAR Point Cloud Segmentation

Our framework starts with the semantic segmentation of LiDAR point clouds, as
shown in the red box in Fig. 2.

Following the works in [9, 25], we apply a spherical projection to a given
point cloud $p\in\mathbb{R}^{n\times 4}$ by mapping each point in $p$ into an
image coordinate. The final output of this projection is
$p^{\prime}\in\mathbb{R}^{\mathit{w^{\prime}}\times\mathit{h^{\prime}}\times
5}$, i.e., an image with the size of
$\mathit{w^{\prime}}\times\mathit{h^{\prime}}$. This way, the unstructured
point cloud $p$ is converted into a structured image with five channels
representing $(x,y,z)$ coordinates, intensity, and range values.

Note that the projection of $p$ to $p^{\prime}$ helps to solve the
correspondence problem between the two distinct sensor data readings without
leading to an enormous information loss since the spherical projection is the
native 2D representation of a 3D LiDAR point cloud and reduces issues like
shadows and distortions obtained in other projection types such as top-view
[2].

Next, the projected LiDAR data $\mathit{p^{\prime}}$ is fed to an off-the-
shelf segmentation network SalsaNext [9] to predict point-wise semantic labels
in a 2D image format,
$p^{\prime}_{s}\in\mathbb{R}^{\mathit{w^{\prime}}\times\mathit{h^{\prime}}}$.

###  III-B Camera Image Segmentation

As depicted in the blue box in Fig. 2, we also semantically segment the
corresponding RGB camera images of the scene, synchronized with the LiDAR
data. Note that this segmentation process is only called during training.

In this step, a state-of-the-art segmentation network SD-Net [38] is employed
to convert the original RGB images $\mathit{y}$ to one-channel segment masks,
$y_{s}\in\mathbb{R}^{\mathit{w}\times\mathit{h}}$, where $w$ and $h$ are the
width and height of the input RGB image $\mathit{y}$.

###  III-C Camera Image Depth Estimation

Our pipeline also incorporates the depth information of the scene image, as
shown in the purple box in Fig. 2.

The primary role of the depth information in our pipeline is not only to boost
the translation from LiDAR semantic segments to camera counterparts, but also
to estimate the depth cues for the generated final scene images. For this
purpose, we employ MiDaS [34] as an off-the-shelf depth estimator to produce
one-channel depth cues, $y_{d}\in\mathbb{R}^{\mathit{w}\times\mathit{h}}$,
where $w$ and $h$ are the width and height of the input RGB image
$\mathit{y}$. We also note that this depth estimation step is called during
training only.

###  III-D From LiDAR to RGB segments and Depth

The translation from LiDAR point clouds to the RGB-D space starts once the
semantic segments of both domains and the image depth information are
available, i.e., $p^{\prime}_{s}$, $y_{s}$, and $y_{d}$, as highlighted by the
green box in Fig. 2. We here note that this green box represents our core
contribution in this work, i.e., a new depth- and semantics-aware cGAN model,
named TITAN-Next.

TITAN-Next: is built upon the base TITAN-Net model [8] and has two core novel
contributions, yielding depth-aware and more accurate transfer of LiDAR
semantic segments to the image space. These novelties are the integration of
the feature pyramid module in the encoder and decoder, and introducing a new
depth branch to the TITAN-Net generator as described below.

Feature Pyramid Module introduces multiscale feature aggregation to the
original TITAN-Net encoder (see the green box in Fig. 2). We feed the
concatenated LiDAR input $p^{\prime}$ and $p^{\prime}_{s}$ into each encoder
layer to enforce meaningful bottleneck representations. Following the work of
[20], we hypothesize that injecting downsampled versions of the input at each
level would force the encoder to extract more descriptive and more robust
feature representations. This could be seen as one level of DenseNet [12],
where only the input is densely propagated in the encoder. In the decoder, we
extract the representation of each encoder layer, upsample them accordingly to
reach the original RGB input dimension, and finally concatenate them before
the final prediction layer.

The final output is composed of a merged version of each decoder level. The
main idea behind using each intermediary representation in the final
prediction is to boost the quality of the predicted segments for the less
frequent and smaller classes (e.g., traffic signs and poles) by ensuring that
the representations are expressive.

Depth Head is attached to the decoder as a new branch to concurrently predict
the depth information $\widehat{y}_{d}$ while generating the image segment
maps $\widehat{y}_{s}$. The depth head consists of 3 standard convolutional
layers and receives the logits of the last decoder layer as an input.

We here highlight the fact that TITAN-Next generator employs the scene depth
information ($y_{d}$ in Fig. 2) neither as an input nor as a condition. Due to
our new Feature Pyramid Module introduced above, TITAN-Next can concurrently
generate both the depth map and camera image segments by relying only on the
LiDAR data ($p^{\prime}$ and $p^{\prime}_{s}$). This new contribution makes
TITAN-Next independent of the camera depth cues during inference. Recall that
the purple box in Fig. 2 is called by the TITAN-Next discriminator during
training only.

In contrast to the original TITAN-Net, the TITAN-Next discriminator has a new
task of jointly distinguishing between semantic segmentation
($\widehat{y}_{s}$) and depth estimation ($\widehat{y}_{d}$) instead of having
a separate discriminator for each output type. The main reason is that both
outputs share enough similarities about the scene context, which can help the
discriminator learn the joint distribution.

TITAN-Next Loss Function: In contrast to TITAN-Net [8], we here use an
additional Weighted Cross-Entropy while generating the segmentation masks
($\widehat{y}_{s}$) and employ three different loss components for the depth
estimation ($\widehat{y}_{d}$).

To generate more accurate segmentation masks, particularly for the
underrepresented classes such as bicycles and traffic signs, we weight the
softmax cross-entropy loss with the inverse square root of the class frequency
as described in [9]. The overall segmentation loss is
$\mathcal{L}_{seg}=\mathcal{L}_{wce}+\mathcal{L}_{ls}$.

$\mathcal{L}_{wce}$ is the Weighted Cross Entropy loss defined as:

$\displaystyle\mathcal{L}_{wce}(y,\hat{y})=-\sum_{i}\alpha_{i}p(y_{i})log(p(\hat{y}_{i})~{},~{}~{}with~{}~{}~{}\alpha_{i}=1/\sqrt{f_{i}}\quad,$

(1)

where $y_{i}$ and $\hat{y}_{i}$ are the true and predicted class labels and
$f_{i}$ stands for the frequency of appearance in the dataset.

$\mathcal{L}_{ls}$ is the Lovász-Softmax loss [5] described as:

$\displaystyle\mathcal{L}_{ls}=\frac{1}{|C|}\sum_{c\in
C}\overline{\Delta_{J_{c}}}(m(c))~{},~{}~{}and~{}~{}~{}m_{i}(c)=\left\\{\begin{array}[]{l
l}1-x_{i}(c)&\text{if ~{} $c=y_{i}(c)$ }\\\ x_{i}(c)&\text{otherwise}\\\
\end{array}\right.~{},$

(2)

where $|C|$ represents the class number, $\overline{\Delta_{J_{c}}}$ defines
the Lovász extension of the Jaccard index, $x_{i}(c)\in[0,1]$ and
$y_{i}(c)\in\\{-1,1\\}$ hold the predicted probability and ground truth label
of the pixel $i$ for class $c$, respectively.

When it comes to the depth estimation, TITAN-Next combines three losses:
$\mathcal{L}_{depth}=\mathcal{L}_{ids}+\mathcal{L}_{mae}+\mathcal{L}_{ssim}.$

Inverse Depth Smoothness loss, $\mathcal{L}_{ids}$, is first introduced in
[42]. The main idea behind this loss term is to enforce flat slopes between
different objects at different depth levels. A smoothness cost is implemented
using the Laplacian of the image intensity, where higher intensity means that
the pixel is most likely on an edge or at a corner. In such regions, a lower
cost is imposed:

$\mathcal{L}_{ids}(d_{i})=e^{-\nabla^{2}\mathcal{I}(x_{i})(|\partial_{xx}d_{i}|+|\partial_{xy}d_{i}|+|\partial_{yy}d_{i}|)}~{},$
(3)

where $\nabla^{2}\mathcal{I}(x_{i})$ represents the Laplacian of the image
intensity, and $\partial$ the partial derivatives of the depth estimation.

Mean Absolute Error (MAE) loss, $\mathcal{L}_{mae}$ is computed as:

$\mathcal{L}_{mae}=\frac{1}{N}\sum_{i\in N}|y_{i}-\hat{y}_{i}|~{},$ (4)

where $N$ is the total number of pixels in the image, $y_{i}$ is the ground-
truth, and $\hat{y}_{i}$ is the network prediction.

Structural Similarity Index Measure (SSIM) [45] measures the similarity
between two images regarding their luminance, contrast, structure, and
covariance matrix. The loss form of this metric, $\mathcal{L}_{ssim}$, can be
computed as:

$\mathcal{L}_{ssim}(y,\hat{y})=\frac{1-SSIM(y,\hat{y})}{2}~{},$ (5)

where $y$ and $\hat{y}$ represent the ground truth and prediction,
respectively.

In addition to these supervised losses, a Wasserstein Gradient Penalty [11] is
used for the adversarial training. The loss can be described as:

$\displaystyle\mathcal{L}_{D}^{WGANGP}=\mathbb{E}_{(\tilde{x},\tilde{y})\sim\mathbb{P}_{g}}[D(\tilde{x},\tilde{y})]-\mathbb{E}_{(x,y)\sim\mathbb{P}_{r}}[D(x,y)]+$
(6)
$\displaystyle\lambda\mathbb{E}_{(\hat{x},\hat{y})\sim\mathbb{P}_{(\hat{x},\hat{y})}}[(||\nabla_{(\hat{x},\hat{y})}D((\hat{x},\hat{y}))||_{2}-1)^{2}]~{}\quad,$
$\displaystyle\qquad\mathcal{L}_{G}^{WGANGP}=-\mathbb{E}_{(\tilde{x},\tilde{y})\sim\mathbb{P}_{g}}[D((\tilde{x},\tilde{y}))]~{},$
(7)

where $\mathcal{P}_{r}$,$\mathcal{P}_{g}$,$\mathcal{P}_{\hat{x}}$ represent
the real, generated, and sampling probabilities and $x$ defines the segments
and $y$ is the depth information. The sampling probabilities are uniformly
sampled along straight lines between $\mathcal{P}_{r}$ and $\mathcal{P}_{g}$
as in [11] and correspond to the gradient penalty, $\lambda$ is set to 10
following the original implementation.

The final TITAN-Next loss is the linear combination of the above losses,
$\mathcal{L}=\mathcal{L}_{seg}+\mathcal{L}_{depth}+\mathcal{L}_{wgan\\_gp}$.

###  III-E From RGB segments and Depth to RGB-D Images

Finally, the generated camera segments $\widehat{y}_{s}$ are converted to RGB
scene images, $\widehat{y}$, as highlighted in the yellow box in Fig. 2. This
translation of $\widehat{y}_{s}$ to $\widehat{y}$ is achieved by another off-
the-shelf state-of-the-art cGAN model Vid2Vid-Net [44]. By combining
$\widehat{y}$ with the predicted depth maps $\widehat{y}_{d}$, we then
generate the RGB-D scene image, which is the final output of our modular
pipeline in Fig. 2.

##  IV Experimental Setup

###  IV-A Implementation Details

For a fair comparison, the same training protocols introduced [8] are followed
here. We use Adam [17] with a learning rate of $1\times 10^{-4}$ and
$(0.5,0.999)$ as $(\beta_{1},\beta_{2})$. The batch size is fixed at 6, while
the dropout probability is set to 0.2 throughout the network. The spherical
projection size is fixed at $512\times 64$ pixels, being the corresponding
area of the RGB space. The camera segmentation output is fixed at $1241\times
376$ pixels.

We also performed data augmentation by flipping randomly around the $y$-axis
and by randomly dropping points before creating the projection. Both data
augmentation techniques are set at $0.5$ probability.

For more details about the implementation, we refer to the publicly available
source code https://github.com/TiagoCortinhal/TITAN-Next.

###  IV-B Dataset

We use the large-scale challenging Semantic-KITTI dataset [4] for the
experimental evaluation. Semantic-KITTI has over 43K point-wise annotated full
LiDAR scans with 19 distinct classes. We follow the same training, validation,
and test splits used in [8].

Since Semantic-KITTI does not provide annotated camera images, we exploit the
semantic segmentation labelling of the Cityscapes dataset by inferring pseudo-
labels via SD-Net [31]. We follow the same mapping between Cityscapes and
Semantic-KITTI as in [8]. Note that due to this mapping between the two
datasets, the total class number is reduced to 14 as described in [8].

###  IV-C Evaluation Metrics

The following metrics have been used to evaluate the quality of the generated
segmentation masks, depth maps, and rendered images.

Intersection over Union (IoU): measures the overlap between the real and
predicted segments and is primarily used for evaluating the accuracy of the
generated semantic segmentation maps. The higher the IoU score, the better.

Fréchet Inception Distance (FID): measures the visual quality of the generated
images and is sensitive to artifacts and noise [45]. A low FID score indicates
that the generated image is closer to the expected real image.

Sliced Wasserstein Distance (SWD): compares structural similarities between
the generated and real images at different scales [14] by computing a
Laplacian Pyramid [1] starting with a resolution of $16\times 16$ that is
doubled and repeated until reaching the full image size. A lower SWD value
represents better image quality.

Absolute Relative Error (AbsRel): represents the magnitude of the differences
between ground-truth and prediction. The lower, the better.

Squared Relative Difference (SqRel): is the squared difference between ground-
truth and prediction. The lower the SqRel score, the better.

Root Mean Square Error (RMS): measures how far off each pixel is from ground-
truth values. A low RMS score indicates a better prediction.

Root Mean Square Error Log scale (RMSlog10): is similar to RMSE but in log
scale. The lower, the better.

Threshold Accuracy ($\delta_{r}$): indicates the accuracy of the prediction by
setting a threshold for different ratios (r) between ground-truth and
prediction. The higher, the better.

For the depth evaluation, we follow the previous methods [6, 48, 49] and use
the same evaluation protocols.

Approach |  Car |  Bicycle |  Motorcycle |  Truck |  Other-Vehicle |  Person |  Road |  Sidewalk |  Building |  Fence |  Vegetation |  Terrain |  Pole |  Traffic-Sign | mIoU $\uparrow$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Pix2Pix [13] | 8.8 | 0 | 0 | 0 | 0 | 0 | 57.7 | 15.7 | 32.8 | 12.5 | 32.7 | 14.8 | 0.5 | 0 | 12.5
GauGAN [33] | 42.9 | 0 | 0 | 6.4 | 0 | 0 | 55.5 | 31.3 | 36.8 | 11.7 | 38.7 | 18.7 | 0.07 | 0.01 | 17.3
TITAN-Net [8] | 68.2 | 9.9 | 7.6 | 7.6 | 0 | 7.3 | 75.4 | 48.3 | 62.9 | 33.6 | 60.1 | 49.9 | 2 | 3.7 | 31.1
TITAN-Next w/o Depth | 85.0 | 38.7 | 18.3 | 16.6 | 0 | 29.3 | 89.3 | 67.2 | 70.0 | 52.6 | 74.3 | 66.9 | 21.4 | 21.7 | 46.5
TITAN-Next (Ours) | 86.6 | 49.1 | 45.9 | 35.7 | 33.1 | 38.7 | 89.9 | 68.5 | 71.9 | 53.2 | 74.2 | 70.0 | 20.5 | 30.2 | 54.8

TABLE I: Quantitative results for the generated camera segments on the test
sequences. $\uparrow$ denotes that higher is better.

| SWD $\times 10^{3}$ $\downarrow$
---|---
Approach | FID $\downarrow$ | 1024$\times$1024 | 512$\times$512 | 256$\times$256 | 128$\times$128 | 64$\times$64 | 32$\times$32 | 16$\times$16 | avg
SC-UNET [15] | 261.28 | 2.65 | 2.56 | 2.36 | 2.20 | 2.14 | 2.14 | 4.07 | 2.59
Pix2Pix [13] $\rightarrow$ Vid2Vid | 73.47 | 2.29 | 2.22 | 2.18 | 2.15 | 2.11 | 2.13 | 3.95 | 2.59
GauGAN [33] $\rightarrow$Vid2Vid | 94.35 | 2.25 | 2.18 | 2.14 | 2.14 | 2.11 | 2.14 | 4.07 | 2.43
TITAN-Net [8] $\rightarrow$ Vid2Vid | 61.91 | 2.22 | 2.18 | 2.15 | 2.11 | 2.08 | 2.13 | 3.78 | 2.38
TITAN-Next (ours) $\rightarrow$Vid2Vid | 29.56 | 2.18 | 1.88 | 1.90 | 1.86 | 1.76 | 1.57 | 1.46 | 1.82
Pix2Pix [13]$-$ w/o SegMap | 209.15 | 2.43 | 2.35 | 2.31 | 2.31 | 2.35 | 2.42 | 4.76 | 2.71
GauGAN [33]$-$ w/o SegMap | 251.1 | 2.24 | 2.15 | 2.10 | 2.10 | 2.01 | 1.99 | 3.50 | 2.08
TITAN-Net [8] $-$ w/o SegMap | 326.29 | 3.24 | 3.14 | 2.98 | 2.61 | 2.23 | 2.10 | 3.94 | 2.89
TITAN-Next (ours)$-$ w/o SegMap | 300.3 | 1.99 | 1.94 | 1.89 | 1.84 | 1.79 | 1.79 | 3.22 | 2.27
TITAN-Next (ours)$-$ w/o Depth | 312.13 | 4.38 | 4.26 | 4.16 | 4.04 | 3.93 | 3.93 | 7.03 | 4.53
SD-Net [38] $\rightarrow$ Vid2Vid | 20.3694 | 2.10 | 1.99 | 1.86 | 1.71 | 1.56 | 1.41 | 1.70 | 1.76

TABLE II: Quantitative results for the synthesized RGB images using the test
sequences. Each level of the Laplacian Pyramid corresponds to a given
resolution. The distances are shown per level, and the average gives us the
overall distance between both distributions. Due to the nature of SWD, both
images are resized to $1024\times 1024$ before calculating the distance.
$\downarrow$ denotes that lower scores are better.

###  IV-D Baselines

In addition to the original TITAN-Net [8], various baselines are used by
following the same training protocols on the same dataset for a fair
performance evaluation. When it comes to Image-to-Image translation, to the
best of our knowledge, only [13, 21, 22, 30, 33, 50] perform the mapping from
segmentation maps to RGB Images. We restrict our baselines to Pix2Pix [13] and
GauGAN [33] since their source codes are available.

Pix2Pix [13] is a well-known generative model dedicated to image-to-image
translation. To measure the quality of the generated segments and images, we
replace TITAN-Next with Pix2Pix in the framework shown in Fig. 2.

GauGAN [33] is another GAN model for image-to-image translation. We follow the
same strategy used for Pix2Pix.

SC-UNET [15] is a U-Net based generative model which is not relying on
adversarial training. SC-UNET generates RGB scene images directly from raw
point clouds without incorporating the semantic segments. Since the source
code and data split are not released for public use, we here use our SC-UNET
implementation to have a baseline for RGB image synthesis directly from LiDAR
point clouds. Note that the here provided scores substantially differ from
these reported in [15] since the original work in [15] uses a very constricted
subset of the dataset, which is not publicly available.

Figure 3: Sample synthesized images (at the top) with the corresponding
generated segment masks and depth maps. From left to right, we have the
ground-truth images and predictions from GauGAN [33], Pix2Pix [13], TITAN-Net
[8], and TITAN-Next. Note that all these models are combined with Vid2Vid [44]
to translate segments to RGB images.

##  V Quantitative and Qualitative Results

The RGB-D image synthesis in our framework solely relies on the generated
semantic scene segments. Therefore, we first start with quantitatively
evaluating the segment maps $\widehat{y}_{s}$ generated from $p^{\prime}$ and
$p^{\prime}_{s}$ as shown in Fig. 2.

Table I reports the mean IoU scores for the generated camera segmentation maps
on the Semantic-KITTI test sequences. We here note that the ground truth
segments are those generated by the SD-Net [38], i.e., $y_{s}$, which are also
used by all the baseline and TITAN-Next discriminators. Table I clearly shows
the substantial performance boost that our model TITAN-Next leads to. In
contrast to the original TITAN-Net base model, we obtain a massive accuracy
enhancement, i.e., $23.7\%$ improvement in the mean IoU score. There is a
certain improvement for each individual segment class. In particular, a
significant boost is obtained for rare classes such as bicycle, motorcycle,
person, pole, among others. When it comes to other baseline models Pix2Pix
[13] and GauGAN [33], the segmentation performance gap gets even larger, i.e.,
TITAN-Next considerably outperforms the others.

Table II presents quantitative evaluations for the synthesized RGB images,
$\widehat{y}$, on the Semantic-KITTI test sequences. As shown in Fig. 2,
TITAN-Next is combined with Vid2Vid (i.e., TITAN-Next $\rightarrow$ Vid2Vid)
to render the RGB scene images. Likewise, all the other baseline models are
coupled with Vid2Vid, e.g., GauGAN $\rightarrow$ Vid2Vid, for a fair
comparison. This, however, does not apply to the baseline SC-UNET [15], since
this model can generate scene images directly from the raw point cloud. The
results in Table II indicate that our TITAN-Next model (i.e., TITAN-Next
$\rightarrow$ Vid2Vid) substantially reduces the FID and SWD scores, thus,
considerably outperforms the others. At the bottom of Table II, we also
provide the obtained scores for SD-Net [38] $\rightarrow$ Vid2Vid, since SD-
Net [38] is mainly employed to generate the ground truth semantic segments.
This way, we can clearly measure how much the synthesized images by our model
(TITAN-Next $\rightarrow$ Vid2Vid) deviate from the expected image quality. In
contrast to other baselines, our TITAN-Next model returns the closest scores
to these reference scores.

Regarding the evaluation of the generated depth cues, Table III reports the
obtained quantitative depth estimation results on the Semantic-KITTI test set.
As clearly shown in Table III, there is a substantial improvement in all
metrics in contrast to the baseline models, combined with an additional depth
branch for the sake of fairness.

Fig. 3 depicts sample synthesized images together with the corresponding
generated segmentation masks and depth maps. This figure ensures that the
reconstructed semantic segments and depth maps by our TITAN-Next model are
more accurate and, thus, the synthesized RGB images are more realistic
compared to other baselines. TITAN-Next has a better reconstruction
capability, particularly for rare classes such as sidewalk, pole, and traffic
sign which are generated with high fidelity. Note that for a fair comparison,
in this figure, we exclude results for SC-UNET [15] since it bypasses the
segmentation step.

Figure 4: Sample synthesized images with the corresponding images trained on
SC-UNET[15], Pix2Pix[13], TITAN-Net[8] and TITAN-Next (ours), without the
semantic maps as an intermediary representation. First images of each row are
the ground-truth.

| AbsRel $\downarrow$ | SqRel $\downarrow$ | RMS $\downarrow$ | RMSlog10 $\downarrow$ |  $\delta_{1}$ $\uparrow$ |  $\delta_{2}$ $\uparrow$ |  $\delta_{3}$ $\uparrow$
---|---|---|---|---|---|---|---
Pix2Pix + Depth | 123.73 | 1119.50 | 14.24 | 2.19 | 0.16 | 0.40 | 0.40
GauGAN + Depth | 110.06 | 637.50 | 3.72 | 1.42 | 0.63 | 0.87 | 0.87
TITAN-Next (ours) | 43.14 | 323.22 | 3.08 | 1.27 | 0.75 | 0.92 | 0.92

TABLE III: Quantitative results for the depth estimation. $\downarrow$ denotes
lower is better while $\uparrow$ denotes higher is better.

###  V-A Ablation Study

In this section, we ablate each of our contributions and investigate the
individual improvements over the original TITAN-Net model. Table IV reports
the mean IoU scores on the Semantic-KITTI test set together with the total
parameter numbers in the generator network. This table clearly shows that each
contribution to TITAN-Net leads to a certain improvement in the model
performance.

Network |  | Number of
---
Generator Parameters
mIoU$\uparrow$ | FLOPs
TITAN-Net [8] | 6.73M | 31.1 | 20.58G
\+ Encoder Feature Pyramid | 7.22M | 46.0 | 64.58G
\+ Decoder Feature Pyramid | 7.22M | 54.8 | 67.88G

TABLE IV: Ablative analysis. $\uparrow$ means higher the better.

Furthermore, we pose the following questions to better study the network
performance:

What if we exclude the semantic segmentation step? This question is crucial to
explore the role of semantic segments in the RGB image synthesis. To assess
this role, we measure the performance of TITAN-Next and other baseline models
once the RGB images ($\widehat{y}$) are directly generated from the raw LiDAR
point cloud projections ($p^{\prime}$) without incorporating the semantic
segments, as in the case of SC-UNET [15].

Table II presents the obtained results, which clearly indicate that without
the intermediate segmentation cues (w/o SegMap) the image reconstruction
performance for all models dramatically drops. Fig.4 shows sample synthesized
images where the quality is drastically diminished.

What if we exclude the depth head? Depth cues can be an important factor in
helping the network gain a more complete understanding of the scene. To
examine the impact of depth cues, we trained TITAN-Next without the depth head
(w/o Depth) and evaluated the results, which are presented in Table I and
Table II.

Our results show that including the depth head leads to a $8.3\%$ improvement
in semantic segmentation performance (see Table I) and a substantial
improvement in the synthesized images quality (see Table II).

###  V-B Runtime

The overall training time on two Quadros RTX 6000 GPUs is around five days for
144 epochs and 19K samples. On a single Quadros RTX 6000 GPU, the inference
runtime for point cloud segmentation, image segmentation together with depth
estimation, translation between segmentation maps, and synthesizing a 376
$\times$ 1241 image takes 40.89, 198.75, 41.20, and 123.72 msec, respectively.

##  VI Limitations and Discussion

At the core of this work, we hypothesize that the scene semantics can bridge
the gap across different domains with distinct data structures, such as
unorganized sparse 3D point clouds and structured 2D image data. Our
experimental findings reported in Table II and Fig. 3 reveal the fact that our
hypothesis holds true that semantic scene segments, to a great extent,
overcome the domain translation problem between the LiDAR and RGB camera
modalities.

In contrast to other relevant works [26, 16, 15, 29], our framework is the
only work in the literature that has the capacity to process the full scan
LiDAR data. This plays an important role to synthesize panoramic RGB-D scene
images from the projected point clouds. Fig. 5 depicts two sample panoramic
RGB images together with the corresponding depth maps, synthesized by our
proposed TITAN-Next model on the Semantic-KITTI test split. We here underline
the fact that it is non-trivial to evaluate the quality of these generated
panoramic RGB-D images, since there exists no corresponding ground truth data.
The results reported so far in Figs. 3-4 show images synthesized only from a
restricted region in the LiDAR projection, which approximates the original
camera view as the only available ground truth.

Figure 5: Two sample panoramic images together with the corresponding depth
estimates, synthesized by our proposed TITAN-Next model on the Semantic-KITTI
test set.

Figure 6: A sample scene with a black car. From top to bottom, we have the
original (ground-truth) RGB image, LiDAR segments superimposed on the
predicted depth map, and the synthesized image, respectively. The red
rectangle highlights the dense depth estimate for the region where the LiDAR
data becomes sparse.

There exist potential applications of such depth- and semantics-aware domain
translations, particularly in the field of autonomous vehicles. For instance,
failed modality readings (e.g., camera images) can be substituted by the data
predicted from other functional sensors (e.g., LiDAR) to feed the subsequent
sensor fusion and planning modules. These predictions can form the initial
beliefs about the scene to make intelligent decisions, instead of naively
triggering an emergency stop.

In addition, this framework can also be employed to enrich the scene
perception in the source domain, i.e., in the LiDAR space. For instance,
visual object detection could be concurrently performed on photo-realistic
RGB-D image streams reconstructed from LiDAR point clouds. This way, black
vehicles can be detected with higher accuracy, which is otherwise non-trivial
to detect directly from LiDAR point clouds due to the low LiDAR reflectivity
as shown in [29]. Fig 6 shows a sample scene including a black vehicle. As
highlighted by the red rectangle in this figure, the predicted depth map of
the black car is quite dense, although the LiDAR point cloud becomes
relatively sparse around the same vehicle.

Furthermore, such multimodal domain translation can help augment the already
collected data, which is of utmost importance in regularizing the network
training. By simply replacing the Vid2Vid head with models trained on other
datasets, such as Cityscapes [7], a different variant of the same scene can be
fabricated at no extra cost. Last but not least, this framework can also be
employed to colorize point clouds without relying on camera readings. The work
in [40] is one good application of such a point painting operation.

The main limitation of the proposed framework is the visual artifacts, such as
broken vehicle boundaries, that emerge in the generated RGB images (see Fig.
5). There are two main reasons for this. First, the semantic segmentation
models (SalsaNext and SD-Net) in our framework are not instance-aware
approaches, thus, object boundaries between the same instances are not
distinguishable. Second, our framework works frame-wise without incorporating
the temporal coherence between adjacent frames. We believe that relying on
individual segment instances while maintaining temporal consistency between
frames can largely mitigate these visual artifacts in synthesized images.

We refer the readers to the supplementary videos111Panoramic Test Scene
Video,222Test Scene Video,333Test Scene Video,444Test Scene Video,555Depth
Comparison Video to show the TITAN-Next performance on the validation and test
splits.

##  VII Conclusion

We present a new modular generative framework to address the multi-modal
domain translation problem. More specifically, we introduce a depth- and
semantics-aware conditional GAN model, named TITAN-Next which converts LiDAR
point cloud segments to the camera space to synthesize panoramic RGB-D camera
images. TITAN-Next builds up on the TITAN-Net [8] and can achieve 23.7% more
accuracy for translating the semantic segments (see Table I). TITAN-Next
differs in that it can construct the scene image enriched with semantic
segments and depth information using the raw LiDAR point cloud only. Our work
can be found on GitHub.

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# Instability and morphology of polymer solutions coating a fiber111Accepted
in Journal of Fluid Mechanics.

F. Boulogne, L. Pauchard, F. Giorgiutti-Dauphiné222Univ Pierre et Marie Curie-
Paris 6, Univ Paris-Sud, CNRS, F-91405, Lab FAST, Bat 502, Campus Univ, Orsay,
F-91405, France.

###### Abstract

We report an experimental study on the dynamics of a thin film of polymer
solution coating a vertical fiber. The liquid film has first a constant
thickness and then undergoes the Rayleigh-Plateau instability which leads to
the formation of sequences of drops, separated by a thin film, moving down at
a constant velocity. Different polymer solutions are used, i.e. xanthan
solutions and polyacrylamide (PAAm) solutions. These solutions both exhibit
shear-rate dependence of the viscosity, but for PAAm solutions, there are
strong normal stresses in addition of the shear-thinning effect. We
characterize experimentally and separately the effects of these two non-
Newtonian properties on the flow on the fiber. Thus, in the flat film observed
before the emergence of the drops, only shear-thinning effect plays a role and
tends to thin the film compared to the Newtonian case. The effect of the non-
Newtonian rheology on the Rayleigh-Plateau instability is then investigated
through the measurements of the growth rate and the wavelength of the
instability. Results are in good agreement with linear stability analysis for
a shear-thinning fluid. The effect of normal stress can be taken into account
by considering an effective surface tension which tends to decrease the growth
rate of the instability. Finally, the dependence of the morphology of the
drops with the normal stress is investigated and a simplified model including
the normal stress within the lubrication approximation provides good
quantitative results on the shape of the drops.

###### Contents

  1. 1 Introduction
  2. 2 Materials and characterizations
    1. 2.1 Samples preparation
      1. 2.1.1 Polyacrylamide
      2. 2.1.2 xanthan
    2. 2.2 Rheological characterization
      1. 2.2.1 Polyacrylamide
      2. 2.2.2 xanthan
    3. 2.3 Experimental setup
  3. 3 Flat film
  4. 4 Instability growth rate
  5. 5 Drop morphologies: normal stress effect
  6. 6 Conclusion

##  1 Introduction

The fiber coating process is widespread in numerous industrial applications
such as the manufacture of glass, polymeric and optical fibers, conducting
cables or textile fibers. The application of a thin layer on these solid
substrates should ensure mechanical or optical properties of the final
deposit. Hence, it is of crucial interest to control the final thickness of
the liquid film. It has been well known since [27] that a cylindrical free
surface of a fluid is unstable under the action of the surface tension. Later,
[4] described in his monograph the patterns observed in a spider web with a
sticky fluid. He reproduced the experiment with castor oil and a quartz fiber,
and detailed the spatial variations of the film thickness. For a fiber drawn
out of a bath, [18] and later [30] provided a first view of the flowing regime
as a function of the capillary number and the Goucher number defined as the
ratio between the radius of the fiber and the capillary length. Subsequently,
different configurations have been studied to identify the different
mechanisms responsible for destabilization of the film and the dependence on
various parameters such as the radius of the fiber, the viscosity, the
inertial forces or the influence of surfactant on the growth rate of the
instability [16, 17, 25, 26, 6, 10, 11, 28, 21]. More recently, we have
reported a flow regime diagram which identifies, depending on the fiber radius
and the flow rate on the fiber, the dominant physical mechanisms [11, 12, 10].
It appears that for small fiber radii compared to the capillary length and low
flow rates, the liquid film is dominated by the surface tension, and the
instability mechanism is the Rayleigh-Plateau instability whose nature is
absolute. For higher values of the parameters (fiber radius and/or flow rate)
two other regimes have been discerned respectively dominated by gravity and
drag (the drag/dravity regime) or by inertia and drag (the drag/inertia
regime). The nature of the instability is then convective for these regimes.
Extensive theoretical studies have investigated the dynamics of the film [15,
25, 20]. Among the most recent studies, we can mention the works of [21] and
[8] on film thickness of the same order as the radius fiber and with
negligible inertia contribution. Their numerical results are compared with
experimental results [11] performed in the three kinds of regime depicted
above with a predictable deviation for the Drag/Inertia regime.

In most industrial situations, the coating fluid is a polymer material or a
complex fluid and exhibits non-Newtonian properties depending on miscellaneous
parameters such as concentration, structure or flexibility of polymers. Yield
stress, shear-thinning or elastic effects are some of the non-Newtonian
behaviors which can affect the structure of the flow, the appearance of the
instability or the morphology of the patterns. [9], in the case of a “dip-
coating” configuration where the fiber is drawn out of a bath of liquid,
observed that the film swells due to the presence of polymer in the solution.
Considering the normal stress and the lubrication approximation, they found an
analytical expression of the film thickness as a function of the withdrawal
velocity and the normal stress coefficient. In general for most of the cases,
when the instability is studied for non-Newtonian fluids it reveals different
classes of patterns and the presence of polymers can drastically change the
dynamic of the system, as in Faraday or Saffman-Taylor instabilities [22]. In
the case of the instability of a liquid jet [7, 29, 13, 1], the addition of
polymers causes the formation of a “beads-on-a-string” structure where
adjacent beads are joined by a thread which grows thinner and strongly delays
the detachment of droplets. In this configuration, the flow is subject to a
strong elongation: a velocity gradient exists in the direction of the flow due
to gravity forces. This additional resistance to breakup compared to a simple
fluid is due to large extensional stresses. Contrary to the case of a liquid
jet, there is no elongational viscosity in the case of the flow down a fiber
due to the no-slip condition on the fiber as detailed further.

A complication inherent in the use of these complex fluids is that they
exhibit different non-Newtonian properties with opposite effects. Notably,
most polymer solutions are both shear-thinning and present elastic effects. In
the context of fiber coating, and to obtain independently the role of the
shear-thinning effect and normal stress on the Rayleigh-Plateau instability,
we have performed experiments with two different polymer solutions: one with a
rod-like polymer (xanthan), exhibiting a pure shear-thinning effect; the other
with a flexible polymer (polyacrylamide abbreviated as PAAm) which exhibits
non-negligible normal stress along with shear-rate dependence of viscosity
similar to xanthan solutions.

Our experiments are all performed in a regime where inertial and gravitational
forces are negligible compared with capillary, elastic and viscous forces. The
Rayleigh-Plateau instability is then absolute [12] and the flow patterns
consist of drops, where fluid is partly trapped in a recirculation zone,
sliding down a very thin (smaller than $100$ $\mu$m) and quasistatic liquid
substrate. This droplike wave train emerges from a constant film thickness,
i.e. the flat film region. We investigate the role of the non-Newtonian
properties on such a flow, i.e, the flat film and the drop-like wave train
resulting from the Rayleigh-Plateau instability.
This paper is organized as follows. In section 2.1, we proceed to a
rheological characterization of the solutions. In section 2, we present the
experimental setup and visualisation techniques. In section 3, we look at the
flat film before the appearance of the instability and the shear-thinning
effect on the film thickness. In section 4, the experimental growth rate and
the wavelength of the instability are measured experimentally for xanthan
solutions (only shear-thinning effect) and for PAAm solutions exhibited strong
normal forces and similar viscosity shear-rate dependence. These experimental
data are then compared to the results of a linear stability analysis taking
into account a non constant viscosity with the shear-rate. In the last part,
section 5, we will treat the effect of normal stress on the morphology of the
drops and provide a simplified model to explain the dependence of the drop
shape with the normal stress.

##  2 Materials and characterizations

Before considering the non-Newtonian properties, certain conditions are
required: first, to avoid inertial forces, and second to ensure a perfect
wetting on the fiber. In a previous paper [11], we have presented a diagram of
the expected flow regime which details the dominant physical mechanisms in the
plane of the dimensionless numbers $R/l_{c}$ versus $h_{0}/R$ with $R$, the
radius of the fiber, $l_{c}$ the capillary number and $h_{0}$ the flat film
thickness. To ensure that the flow is dominated by capillary forces, with no
inertia, some conditions on the fiber radius, the surface tension and the
viscosity of the fluid must be fulfilled in agreement with the flow regime
diagram mentioned above. In that capillary region, the flow on the fiber
consists of drops sliding on a quasistatic thin film. To ensure a good wetting
on the fiber and consequently an axisymmetric pattern on the fiber, the
surface tension has to be lower than $40$ mN/m. Finally, these two conditions
are satisfied by using a fiber radius equal to $0.28$ mm and solutions
composed of a mixture of water, glycerol (to increase viscosity) and
surfactant to reach a surface tension close to $30$ mN/m.

Some previous experiments carried out with Newtonian fluids on the same
experiment [11], have shown that film thicknesses between $0.1$ and $1$ mm are
possible with velocities ranging from $1$ to $10$ cm/s and viscosities between
$50$ and $500$ mPa.s. This implies that the shear-rate range encountered in
our experiments is from $10$ to $1000$ s-1.

The liquids used are then semi-dilute solutions of polymers: xanthan and
Polyacrylamide (PAAm) purchased from Sigma-Aldrich. Both present shear-
thinning behavior but only the second one exhibits large elastic effects in
the shear-rate range considered in the experiments. To discern the effect of
elasticity, Boger fluids would have been theoretically more appropriate.
Nevertheless, in practice, the large quantities of liquid required for our
experiments and the possible degradation of the nylon fiber by the solvents
used in Boger fluids, are two reasons prohibiting their use.

###  2.1 Samples preparation

####  2.1.1 Polyacrylamide

PAAm is a high molecular weight chain resulting in long flexible chains
($M\simeq 1.5\times 10^{7}$ g/mol). The samples are prepared very carefully to
get homogeneous solutions. First, a solvent is prepared by mixing $50$% of
purified water and $50$% of pure glycerol (all percentages presented in this
article are weight percentages). Six polymer concentrations were studied in
the range from $0.1\%$ to $0.6\%$. The surfactant selected to reduce the
surface tension is Triton X-100 (TX-100). This choice was motivated by its
mixing properties at high polymer concentrations and for the resulting low
surface tension [32]. The TX-100 concentration is $4.5$% (about $300$ times
the Critical Micelle Concentration, CMC, in water). For this high
concentration we assume that the surfactant mobility timescale is higher than
the timescale for the instability growth rate. Indeed, the time variation of
the surface tension due to diffusion of surfactant is $10^{-2}$ s for a TX-100
concentration $50$ times the CMC in water [14]. This time scale is of the same
order of magnitude as the characteristic growth rate of the instability:
$\frac{\eta R^{4}}{\gamma h^{3}}\sim\frac{0.1\times(0.6\times
10^{-3})^{4}}{30\times 10^{-3}(0.2\times 10^{-3})^{3}}=0.05$ s (using typical
values for the viscosity $\eta$, the fiber radius $R$, the surface tension
$\gamma$, and the film thickness $h$). Consequently, in the range of
concentrations considered in our experiments we assume that the interface is
rapidly saturated with surfactant molecules before the instability occurs. The
surface tension of the final solution is then $\gamma=32.3\pm 0.5$ mN/m. We
should note that no apparent rheological modifications are observed by varying
the TX-100 concentration.

####  2.1.2 xanthan

xanthan is a rigid rod-like polymer ($M\simeq 5\times 10^{6}$ g/mol). A
preparation protocol similar to the PAAm solutions was used for xanthan
solutions. The solvent is slightly different: $60$% of glycerol and $40$% of
water. Since xanthan is a polyelectrolyte polymer, the resulting rheological
properties of this polymer are known to be modified by the addition of salt
[31]. Thus, different concentrations of NaCl allow adjustment of the
rheological properties. After the addition of TX-100, the surface tension of
the final solution is $\gamma=32.7\pm 0.8$ mN/m independent of the salt
concentration.

###  2.2 Rheological characterization

The rheology of polymer solutions was performed using an “Anton Paar”
rheometer with a cone and plate geometry. We have chosen a large cone (radius:
$49.988$ mm) with small angle (angle: $0.484^{\circ}$) in order to measure
precisely normal stress in a large range of shear-rates. The sample
temperature was fixed at $20.00\pm 0.05^{\circ}$C.

####  2.2.1 Polyacrylamide

The evolution of the apparent shear viscosity, $\eta$, is plotted versus the
shear-rate, $\dot{\gamma}$, for two concentrations in figure 1. As usual,
$\eta$ decreases with $\dot{\gamma}$ and, at a given shear-rate, increases
with the polymer concentration. Such curves are typical for shear-thinning
fluids where a constant Newtonian low-shear viscosity is followed by a power-
law dependence before reaching the viscosity of the solvent at high shear-
rates. These measurements can be reasonably fitted by a four-parameter Carreau
model [24]:

$\eta=\eta_{\infty}+\frac{\eta_{0}-\eta_{\infty}}{\left(1+(\tau\dot{\gamma})^{2}\right)^{\frac{1-n}{2}}}$
(1)

where $\eta_{0}$ and $\eta_{\infty}$ are respectively the viscosity for the
zero-shear limit and infinite shear-rate, and $\tau$ denotes a characteristic
time scale that measures the scale at which the shear-thinning effect becomes
important. The exponent $n$ is the power of the following Ostwald power-law
equation:

$\eta=\beta\dot{\gamma}^{n-1}$ (2)

The zero-shear limit increases rapidly with the polymer concentration, that is
typical of entangled polymer solutions (see the inset of figure 1). Recently,
it has been shown [23] that in a good solvent the entangled concentration
$c_{e}$ for PAAm is about nine times the crossover concentration $c^{*}\simeq
0.2$ g/L. The temperature effect on the samples indicates that the viscosity
decreases by $10$% for a $5^{\circ}$C increase.

Figure 1: Variations of the shear viscosity $\eta$, of PAAm solutions vs.
shear-rate $\dot{\gamma}$, in a log-log scale. Each symbol refers to a
different polymer concentration. Plain curves correspond to data fits with a
Carreau model. The inset shows the zero shear-rate viscosity $\eta_{0}$, as a
function of the concentration in PAAm.

Figure 2: (a) Log-log plot of the normal stress, $N_{1}$, as a function of the
shear-rate, $\dot{\gamma}$, for PAAm solutions in water-glycerol ($50$% :
$50$%) solvent; $4.5$% TX-100 surfactant was added to the solutions. Plain
curves correspond to data fits using equation 3. (b)
$\frac{N_{1}}{\eta(\dot{\gamma})\dot{\gamma}}$ versus shear-rate,
$\dot{\gamma}$ , for PAAm solutions

The normal stress measurements are presented in figure 2 as a function of the
shear-rate for different PAAm concentrations over a wide range of shear-rates.
A significant increase of the normal stress with the shear-rate is observed in
accordance with:

$N_{1}=\psi_{1}\dot{\gamma}^{2}$ (3)

where $\psi_{1}$ is the first normal stress coefficient characterizing the
fluid [2]. Values are given in figure 2 for several Polyacrylamide
concentrations. In the inset graph, we have presented the data using a double
log-plot. It appears more clearly that, except for PAAm solutions at $0.6\%$,
there is a discrepancy between the data and the curve fit for low shear-rates
(lower than $200$ s-1), indicating dependence of the first normal stress
difference with the shear-rate. Thus the $0.6\%$ solution would be the best
candidate for studying the normal force effect in section 5. Normal stress
magnitude can be compared to viscous stress by estimating the ratio
$\frac{N_{1}}{\eta(\dot{\gamma})\dot{\gamma}}$ as a function of the shear-rate
2. This ratio increases with the shear-rate, highlighting the importance of
the normal stress, which starts to be dominant compared to the shear-thinning
effect for shear-rates larger than $100$ s-1. One should then expect a large
amount of normal stress in the drops, for which the shear-rate is always
larger than $100$ s-1.

####  2.2.2 xanthan

Rheological measurements are typical of shear-thinning fluids where a constant
Newtonian low-shear viscosity is followed by a power-law dependence before
reaching the viscosity of the solvent at high shear-rates (figure 3).
Nevertheless, as shown in figure 3, the power-law behavior failed to fit the
experimental results in the whole range of shear-rates. Since xanthan is a
polyelectrolyte, the solution rheology and molecular configuration are greatly
affected by the solution’s ionic strength. Thus by adding $0.8$% NaCl to the
solution, the shear-thinning effect can be adjusted to $n=0.73$ in a
reasonable range of shear-rates from $5$ to $2000$ s-1. No significant normal
stress has been detected for xanthan below $4000$ s-1, a shear-rate which is
not expected to be reached in the experiment. Subsequently, we will
exclusively use salted xanthan solutions as pure shear-thinning solutions.

Figure 3: Viscosity $\eta$, as a function of shear-rate $\dot{\gamma}$, for
two solutions constituted by water-glycerol ($50$% : $50$%), $4.5$% TX-100 and
$0.4$% xanthan. Red cross correspond to free-salt solution and blue one are
for a salt concentration up to $0.8$%. Numbers indicate the slopes of the
Ostwald power-law model from fits over a $\dot{\gamma}$ range: $[10;2000]$
s-1.

###  2.3 Experimental setup

As depicted in figure 4, the fluid flows from an upper reservoir (diameter:
$14$ cm) down a nylon fiber (diameter: $0.56$ mm). The relative pressure
variation is about $0.001$% during one minute for the highest measured flow
rates. The flow rate is controlled by a valve composed of two axisymmetric
cones. The mass flow rate $Q$ is measured from the weight variation of a
collecting tank recorded by a computer-controlled scale. A transparent nozzle
guides the fluid on the fiber. Its verticality is crucial to obtaining an
axisymmetric flow and it is ensured by a mechanical device which enables very
accurate fiber displacements with a sensibility of $2.4$ arc sec. Two
perpendicular cameras with zoom lens help to control the axisymmetry of the
film flowing down the fiber.

Figure 4: (a) Experimental setup. Scheme showing a fluid flowing down on a
fiber from a upper tank. The flow rate is controlled by a valve and guided
with a nozzle. (b) Picture snapped by a high-speed camera with a telecentric
lens ($1\times$). The white bar length is $5$ mm. (c) Spatiotemporal evolution
of the film obtained by a vertical linear camera passing through top drops
(black lines). Letters A, B and C denote respectively the flat film, the
ordered and the disordered pattern regions.

As depicted in figure 4, the flow presents three regions along the fiber. A
meniscus is followed by a flat film with a constant thickness on a distance
called the healing length, which increases slightly with the flow rate [12].
Then, the Rayleigh-Plateau instability leads to the formation of a regular
pattern of beads flowing on a very thin and flat film. In this paper, we will
deal only with regime dominated by capillary forces, so we exclude low flow
rates (the dripping regime) and high flow rates where inertial flow dominates.
The flow regime is then absolute. The film thickness and the shape of the
drops are captured by a high-speed digital camera with a telecentric lens. The
interface position is detected in both space and time, so we are able to
measure the film thickness $h(z,t)$ with an accuracy of $0.02$ mm. A linear
camera provides spatiotemporal diagrams which deliver information on the
dynamic of the flow. A vertical pixel line passing through the peaks of the
drops is recorded and stored at constant time intervals. The resulting
spatiotemporal diagram produces the $(z,t)$ trajectories of the drops along
the fiber. A typical spatiotemporal diagram is shown on figure 4. The uniform
grey region, located at the upper part of the fiber is the place where the
film is flat (region A) and gives rise to a zone of regular stripes with a
constant wavelength and velocity for the drops (region B). Finally,
downstream, some coalescences between drops lead to the formation of a
disordered pattern (region C).

##  3 Flat film

In this section we focus on the region close to the inlet where the film
thickness is constant (flat film, grey uniform region, see figure 4). In the
case of Newtonian fluids, the thickness of the film, $h$, is given by the
classical Nusselt solution [12]. In the case of very thin films ($h\ll R$),
i.e. the planar case, there is a cubic relation between the flow rate on the
fiber and $h$. We define the cylindrical coordinates system $(r,\theta,z)$,
where $r$ is the radial coordinate (the fiber center is the origin), $\theta$
the azimuthal coordinate and $z$ the axial coordinate oriented downward in the
flow direction. In the case of a shear-thinning solution exhibiting normal
stress effects, for a steady axisymmetric flow, the stress balance in the
axial direction $z$ is written as:

$\frac{\partial\sigma_{zz}}{\partial
z}+\frac{1}{r}\frac{\partial(r\sigma_{rz})}{\partial r}=\frac{\partial
p}{\partial z}-\rho g$ (4)

for $R<r<R+h(z)$, where $\sigma$ denotes the stress tensor, $p$ is the
pressure field in the film, $\rho$ and $g$ are respectively the fluid density
and the gravitational acceleration. Since the first normal stress difference,
$N_{1}$, can be expressed as
$\sigma_{zz}-\sigma_{rr}=\psi_{1}\left(\frac{\partial v}{\partial
r}\right)^{2}$, with $v(r,z)$ the axial velocity, which varies along the film
thickness, equation (4) becomes

$\frac{\partial N_{1}}{\partial
z}+\frac{1}{r}\frac{\partial(r\sigma_{rz})}{\partial r}=\frac{\partial
p}{\partial z}-\frac{\partial\sigma_{rr}}{\partial z}-\rho g$ (5)

As no spatial variations are detected in the flat film, $z$-invariance of the
velocity field implies that $\frac{\partial N_{1}}{\partial z}=0$. Thus,
normal stress has no effect on the flat film.

To calculate the velocity profile in the flowing film, we model the shear-
thinning effect by a power-law in accordance with expression 2. Thus (5)
becomes

$1+\frac{1}{1+\tilde{r}}\frac{d}{d\tilde{r}}\left((1+\tilde{r})\left(\frac{d\tilde{v}(\tilde{r})}{d\tilde{r}}\right)^{n}\right)=0$
(6)

where the following dimensionless variables are introduced:
$\tilde{r}=\frac{r-R}{R}$, $\tilde{h}=\frac{h}{R}$, $\tilde{v}=\frac{v}{V}$
and $V\equiv R\left(\frac{\rho gR}{\beta}\right)^{1/n}$.

The fluid velocity satisfies two boundary conditions: no-slip on the fiber
($\tilde{v}(\tilde{r}=0)=0$) and zero tangential stress at the liquid-air
interface ($\partial_{\tilde{r}}\tilde{v}(\tilde{r}=\tilde{h})=0$).
Considering this last boundary condition, equation (6) becomes:

$\frac{d\tilde{v}}{d\tilde{r}}=\left(\frac{1}{1+\tilde{r}}\left(\tilde{h}(1+\tilde{h}/2)-\tilde{r}(1+\tilde{r}/2)\right)\right)^{\frac{1}{n}}$
(7)

The flow rate per unit length $q=\frac{Q}{2\pi\rho R}$ is written in
dimensionless form as

$\tilde{q}=q\frac{1}{RV}=\int_{0}^{\tilde{h}}\tilde{v}(\tilde{r}+1)d\tilde{r}$
(8)

We note that for the Newtonian case $n=1$ and $\beta=\eta$, we recover the
analytical Nusselt solution $v=\frac{\rho
g}{4\eta}\left(2(R+h)^{2}\ln\left(\frac{r}{R}\right)-(r^{2}-R^{2})\right)$.

Equation (7) is solved using a Runge-Kutta algorithm (starting at
$\tilde{v}(\tilde{r}=0)=0$ in order to satisfy the boundary condition on the
fiber). First, the influence of the parameter $n$ is studied for a constant
flow rate $\tilde{q}=1$, and the integral of the equation 8 is estimated by
the trapezium rule. We choose two values for the $\tilde{h}$ parameter
($h_{\textrm{min}}$ and $h_{\textrm{max}}$) satisfying
$\tilde{q}(\tilde{h}_{\textrm{min}})\leq
1\leq\tilde{q}(\tilde{h}_{\textrm{max}})$ and we find the film thickness by a
bisection method for which the condition $\tilde{q}=1$ is satisfied at $0.1$%.
The results are shown in figure 5. The numerical solution for the Newtonian
case, $n=1$ is identical to the analytical Nusselt solution. For a constant
flow rate, an increase in the shear-thinning effect modifies the velocity
profile shape: the parabolic profile tends to be replaced by a plug-like
profile. This results in higher velocity gradient close to the fiber, whereas
close to the interface, the velocity gradient is almost zero. The film
thickness is always smaller than for Newtonian fluids. The film thickness is
plotted as a function of the flow rate on the film for a PAAm solution (0.4%,
n=0.71), in figure (6). We choose to present experimental data only for PAAm
solutions, since for xanthan solutions, the healing length is two or three
times smaller than for PAAm solutions. The good agreement between the
numerical solutions and our experimental data validates the choice of the
Oswald power-law model for the viscosity and also our assumption on the
negligible effect of surfactant on the zero-shear stress boundary condition at
the liquid-air interface.

Figure 5: Velocity profiles of the flowing film at a constant flow rate
($\tilde{q}=1$) for several $n$ values from a Newtonian fluid ($n=1$) to a
high shear-thinning effect ($n=0.4$). The inset is a close-up of the liquid-
air interface region. Figure 6:  Numerical solution (solid line) and
experimental results for a concentration in PAAm equal to $0.4$%. The dashed
curve is the analytical solution for $h\ll R$ given by the equation 16 for
$n=0.73$ .

##  4 Instability growth rate

The impact of the shear-thinning and elastic effects on the growth rate of the
Rayleigh-Plateau instability is investigated through experiments with xanthan
(0.8% NaCl) and PAAm solutions (0.4%) in order to distinguish the role of each
non-Newtonian property. After the flat film region, some variations on the
film thickness are detected and a regular pattern of drops emerges due to the
Rayleigh-Plateau instability as shown in figure 4. The wavelength of the drop-
like pattern is plotted in figure 7 as a function of the distance to the
entrance nozzle. The wavelength increases (regime A in figure 7) until it
reaches a well-defined value (regime B in figure 7). Then, lower down, some
coalescence events can disrupt the regular pattern (regime C particularly in
figure 7a). The wavelengths of the regular pattern for non-Newtonian fluids
are somewhat higher than those expected with Newtonian fluids. Nevertheless,
in both cases, the classical Rayleigh Plateau wavelength fails to fit the
experimental data and is always smaller. The length of the regular pattern
depends on the flow rate (at high flow rates, coalescence events occur
earlier) but it is typically is of the order of seven centimeters for PAAm
solutions (figure 7a) and shorter, about four centimeters, for xanthan
solutions (figure 7b). In the former case, we can note that the axisymmetric
conformation is not the only case observed on the fiber: some non-axially
symmetric conformations can be observed with asymmetric drops. Such
conformations have been described by [5] as a roll-up transition and must be
avoided in our case.

Figure 7: Wavelength at different flow rates for (a) xanthan and (b) PAAm
solution. Letters A, B and C denote respectively the growth of the
instability, the ordered and the disordered pattern regions.

In order to characterize the instability growth rate, we record a stack of
images at a typical frame rate of 1000 images per second. Then, the position
of the film interface is detected over space and time: $h(z,t)$. Figure 8
shows the average film thickness over the time $<h(z,t)>_{t}$ and the extremal
film positions for a PAAm solution. It shows successively the meniscus, the
flat film and the onset of the instability which is marked by a strong
variation of the film thickness. The velocity of the interface is calculated
for each stack. Then, a point on the interface (chosen to become a point of
maximum height) is followed at this velocity using the set of data $h(z,t)$.
The resulting values of the normalized profile $(h-h_{0})/h_{0}$ as a function
of time for a typical experiment are plotted in the inset of figure 9 and
fitted by an exponential law $\frac{h-h_{0}}{h_{0}}=Ae^{\Omega t}$ in the
early linear stages. From this fit, the growth rate, $\Omega$, is extracted
and averaged over several other experiments; the results for xanthan and PAAm
solutions were reported in figure 9.

Figure 8: Average film thickness in time $<h>_{t}$ along the $z$ fiber axis
($Q=0.032$ g/s). Bars indicate the extreme values of the film thickness. The
flat film thickness is denoted by $h_{0}$.  Figure 9: The growth rate,
$\Omega$, is plotted versus the flat film thickness, $h_{0}$. The experimental
data are represented by the dots and the equation 22 by curves. The inset
shows the growth of the film using the method described in section 4.

A first simplified attempt to obtain an expression for the growth rate
consists in a linear stability analysis. The fluid is assumed to exhibit pure
shear-thinning effects with $\eta(\dot{\gamma})=\beta\dot{\gamma}^{n-1}$, and
we assume very thin films such that $h \ll R$ (planar approximation). Thus, in
cartesian coordinates, the following momentum equation holds, in the
lubrication approximation:

$0=\Pi+\frac{\partial\eta(\dot{\gamma})\dot{\gamma}}{\partial r}$ (9)

where $\Pi$ is the pressure gradient given by:

$\Pi=\rho
g+\gamma\left(\frac{\partial_{z}h}{(R+h)^{2}}+\frac{\partial^{3}h}{\partial
z^{3}}\right)\simeq\rho
g+\gamma\left(\frac{\partial_{z}h}{R^{2}}+\frac{\partial^{3}h}{\partial
z^{3}}\right)$ (10)

GIven that there is no fluid slippage on the fiber and no stress on the
liquid-air interface, the velocity is calculated from the momentum equation,
so that

$v(r)=\frac{1}{1+1/n}\left(\frac{\Pi}{\beta}\right)^{1/n}\left[h^{1+1/n}-(h-r)^{1+1/n}\right]$
(11)

The flow rate per unit length, defined as $q_{p}=\int_{0}^{h}vdr$, is given
by:

$q_{p}=\left(\frac{\Pi}{\beta}\right)^{1/n}\frac{h^{2+1/n}}{2+1/n}$ (12)

and also satisfies the mass conservation equation:

$\frac{\partial h}{\partial t}+\frac{\partial q_{p}}{\partial z}=0$ (13)

We assume infinitesimal perturbations around the uniform film thickness,
$h_{0}$, and for the corresponding flow rate, $q_{p0}$, so that

$\displaystyle h(z,t)=h_{0}+h_{1}(z,t),\qquad$ (14) $\displaystyle
q_{p}=q_{p0}+q_{p1},\qquad$ (15)

This results in the following expressions

$\displaystyle q_{p0}$ $\displaystyle=$ $\displaystyle\left(\frac{\rho
g}{\beta}\right)^{1/n}\frac{h_{0}^{2+1/n}}{2+1/n}$ (16) $\displaystyle q_{p1}$
$\displaystyle=$ $\displaystyle q_{p0}\left[\frac{\gamma}{n\rho
g}\left(\frac{\partial_{z}h_{1}}{R^{2}}+\frac{\partial^{3}h_{1}}{\partial
z^{3}}\right)+\left(2+\frac{1}{n}\right)\frac{h_{1}}{h_{0}}\right]$ (17)

and in the linearized equation:

$-\frac{\partial h_{1}}{\partial t}=\frac{q_{p0}\gamma}{n\rho
g}\left(\frac{\partial_{z}^{2}h_{1}}{R^{2}}+\frac{\partial^{4}h_{1}}{\partial
z^{4}}\right)+\left(2+\frac{1}{n}\right)\frac{q_{p0}}{h_{0}}\frac{\partial
h_{1}}{\partial z}$ (19)

Developing the thickness perturbation as $h_{1}=Ae^{i(kz-\omega t)}$ leads to
the dispersion relation:

$\omega(k)=k\frac{q_{p0}}{h_{0}}\left(2+\frac{1}{n}\right)+i\frac{q_{p0}\gamma}{n\rho
g}\left(\frac{k^{2}}{R^{2}}-k^{4}\right)$ (20)

The maximum of $\mbox{Im}(\omega(k))$ gives:

$\Omega=\frac{q_{p0}\gamma}{4n\rho gR^{4}},\qquad h\ll R\\\ $ (21)

As shown in figure 6, the flow rate given by 16 does not reproduce the
experiment where the planar approximation ($h\ll R$) is not valid. In previous
works on Newtonian fluids, it has been shown [8] that for $h\sim R$, the
expression for the growth rate is similar except that $R$ should be replaced
by $R+h$. Moreover, we made the choice to use our numerical calculation
described in section 3 which provides $q_{\textrm{num}}(h)$ without assumption
on the film thickness. Finally, the addition of a large amount of surfactant
modifies the growth rate by a factor $4$, as described by [6] since
surfactants change the surface elasticity.

Taking into account these corrections, we obtain the expression

$\Omega=\frac{1}{4}\frac{q_{\textrm{num}}(h)\gamma}{4n\rho g(R+h)^{4}}\\\ $
(22)

which is plotted in figure 9 for both chemical systems.

Data for the xanthan solution are well described by 22. To validate our method
and to compare with a Newtonian fluid of similar surface tension
($\gamma=20.9$ mN/m) and viscosity ($\eta=0.965$ Pa.s), we have performed an
experiment with a silicon oil (with $\rho=96.5$ kg.m-3 and $h_{0}=0.55$ mm)
and measured a growth rate $\Omega$ equal to $10.9\pm 0.7$ s-1. Equation 22
for $n=1$ gives $\Omega$ equal to $11.1$ s-1. The small variations in growth
rate between the xanthan solution and the silicon oil are reasonable as the
value of the viscosity is of the same order. Concerning PAAm solutions, there
is significant deviation from the theory due to the normal stress of this
solution. A qualitative explanation of the role of normal stress can be
provided by the “hoop stress” effect [19]. A short description of this effect
can be made by considering the liquid surface as an infinite cylindrical shell
of thickness $e$ (figure 10). For a cylinder of radius $R+h$, the balance
between the internal pressure $P$ and the stretching stress
$\sigma_{\theta\theta}$, leads to $2(R+h)LP=2L\sigma_{\theta\theta}e$. The
internal pressure $P$ is generated by the normal stress $N_{1}$ in the bulk.
Interpreting the stretching force per unit length, $2\sigma_{\theta\theta}e$,
in term of surface tension $\gamma_{\psi_{1}}$, we obtain

$\gamma_{\psi_{1}}=-\psi_{1}\dot{\gamma}^{2}(R+h)$ (23)

An estimation for $\dot{\gamma}=100$ s-1 gives a $\gamma_{\psi_{1}}$ of about
$-10$ mN/m significantly lowering the effective surface tension. The growth
rate should therefore be estimated with the effective surface tension lower
than the fluid surface tension and the resulting curve for the growth rate
would be shifted and enable us to recover the experimental data.

Figure 10: Stretching stress, $\sigma_{\theta\theta}$, in a thin cylindrical
shell of radius $R+h$ and thickness $e\ll R+h$.

##  5 Drop morphologies: normal stress effect

This section is devoted to a comparison between the patterns of flowing films
of PAAm and xanthan, focusing particularly on the normal stress effect on the
shape of the drops. Such a comparison requires polymeric solutions having
similar shear-thinning properties. Further, $0.8$% NaCl was added to xanthan
($0.4$%) solutions to decrease the high shear-thinning effect (figure 3).
Optimal adjustment of the shear-thinning of PAAm solutions ($0.6$%) is
achieved as shown in the inset of figure 11. Thus, the difference between the
two solutions concerns only the presence or absence of normal stress.

The typical pattern observed on the fiber consists of an axisymmetric film of
constant thickness. Then the film breaks up spontaneously into a drop-like
wave train as described in the previous section.

For axisymmetric patterns, the superposition of drop profiles in a PAAm and
xanthan films is shown in figure 11. Clear differences can be noticed in the
profiles, notably the steepening of the drop front for the PAAm solution
compared to the xanthan solution. In both profiles, there is a clear asymmetry
between the front and back of the drops which is more accentuated for the
xanthan drop. This remark suggests that the shape of the drops is affected by
gravity. The apex heights of both drops are identical as well as the film
substrate between drops (the trailing edge for PAAm is longer than for
xanthan). These observations confirm the fact that the normal stress plays a
significant role in the thin regions, close to the tail and the front of the
drops, but exhibits no effect in the center of the drop (the thick region). To
quantify experimentally the swelling effect observed with PAAm solution, we
define the slope of the front $H/L$ as shown in figure 11. This parameter is
plotted in PAAm and xanthan solutions for different flowing rates in figure
12.

Figure 11: Drop shapes for two polymer solutions. The drop front of the
viscoelastic solution (dashed green line) is swollen compared with the pure
shear-thinning liquid (solid red line). Figure 12: Slopes of drop fronts for
xanthan and PAAm as a function of the capillary number Ca.

To highlight the swelling process, we consider a scaling law analysis,
starting from the stress balance 5. Since the film is not flat the
$z$-invariance is no longer valid. Exhibiting the contribution of the normal
stress difference $\sigma_{zz}-\sigma_{rr}$, and shear stress $\sigma_{rz}$,
equation 5 becomes

$\frac{\partial(\psi_{1}\left(\frac{\partial v}{\partial
r}\right)^{2})}{\partial
z}+\frac{1}{r}\frac{\partial(r\eta(\dot{\gamma})\dot{\gamma})}{\partial
r}=\frac{\partial p}{\partial z}-\frac{\partial\sigma_{rr}}{\partial z}-\rho
g$ (24)

The axial velocity $v(r,z)$, is determined using a series of the form [3]:

$v(r,z)=a_{0}(z)+a_{1}(z)r+a_{2}(z)r^{2}$ (25)

Functions $a_{0}(z)$, $a_{1}(z)$ et $a_{3}(z)$ are calculated using the
following three equations, two for the boundary conditions and the last one
for the mass conservation:

  * •

Boundary condition at the interface with the fiber $v(r=R,z)=0$;

  * •

Boundary condition at the liquid/air interface $\frac{\partial
v(r=R+h(z))}{\partial r}=0$;

  * •

The mass conservation equation,

$\frac{\partial h}{\partial t}+\frac{\partial}{\partial
z}\int_{R}^{R+h(z,t)}v(r,z)dr=0$ (26)

Considering equation 26 in the reference frame of a drop moving at a velocity
$U$ and using the condition that the mean flow rate satisfies
$\overline{q}\sim Uh\underset{h\rightarrow 0}{\longrightarrow}0$, the axial
velocity is

$v(r,z)=\frac{3}{2}U\frac{(R-r)(r-2h(z)-R)}{2h^{2}(z)}$ (27)

Assuming a constant viscosity, $\eta$, the stress balance equation 24 to zero-
order in $r$ is

$3\eta
U\frac{h(z)-R}{2Rh^{2}(z)}-U^{2}\psi_{1}\frac{9h^{\prime}}{2h^{3}(z)}=\frac{\partial
p}{\partial z}-\frac{\partial\sigma_{rr}}{\partial z}-\rho g$ (28)

The normal stress balance at the free surface of the film assumes that
$-P+\sigma_{rr}=\gamma\kappa$ with $\kappa$ the curvature of the interface. If
$L$ is the characteristic length in the axial direction and if $H$ is the
characteristic apex height of the drop, then the normal stress balance at the
free surface of the film, accounting for the curvature, is given by

$\frac{\partial p}{\partial z}-\frac{\partial\sigma_{rr}}{\partial
z}\sim-\gamma\left(\frac{\partial_{z}h}{R^{2}}+\frac{\partial^{3}h}{\partial
z^{3}}\right)$ (29)

Thus the right-hand side of equation 28 becomes:

$\frac{\partial p}{\partial z}-\frac{\partial\sigma_{rr}}{\partial z}-\rho
g\sim-\gamma\left(\frac{H}{L^{3}}+\frac{H}{LR^{2}}+l_{c}^{-2}\right)$ (30)

where $l_{c}=\sqrt{\frac{\gamma}{\rho g}}$ is the capillary length.

Experimental observations suggest that $\frac{H}{L^{3}}\ll\frac{H}{LR^{2}}$
and $\kappa^{2}\ll\frac{H}{LR^{2}}$. So, the scaling law analysis leads to the
following equation:

$\frac{H^{3}}{L^{3}}\sim\frac{R^{2}}{L^{2}}\textrm{Ca}\left(1-6\frac{\mathscr{L}}{L}\right)$
(31)

where $\textrm{Ca}=\frac{\eta U}{\gamma}$ is the capillary number and
$\mathscr{L}=\frac{\psi_{1}U}{\eta}$ is the normal stress characteristic
length.

This scaling law gives the slope of the front of a drop, $H/L$, as a function
of the viscoelastic properties of the polymeric solution. In particular, in
the case of polymeric solutions exhibiting normal stress, $\psi_{1}\neq 0$,
the expression 31 clearly shows that $H/L$ decreases. A comparison between the
experimental data and the results of the scaling analysis is presented in
figure 12 for different flow rates. There is a good agreement between the
experiment and the model which succeeds in highlighting the swelling effect on
the drop shape induced by the normal stress effect.

##  6 Conclusion

The effects of non-Newtonian properties of fluids have been investigated in
the case of a film flowing down a vertical fiber. The flow on the fiber can be
divided into three regions: (A) at the inlet, the film exhibits a uniform
thickness, i.e the flat film region; (B) the uniform film is progressively
replaced by a well-defined pattern of drops separated by a thin film, i.e the
Rayleigh-Plateau region; (C) the coalescence of drops disrupts the flow and
give rise to a disordered pattern. In order to disentangle the role of the
shear-thinning effect and of the normal stress, we have considered two kinds
of polymer solutions. The first consists of rigid rod-like polymers (xanthan),
exhibiting a strong shear-thinning behaviour but negligible elastic effects.
For the second solution, we used flexible polymers (PAAm) exhibiting strong
elastic effects and shear-thinning effects similar to those of xanthan under
certain physico-chemical conditions. Some adjustments have been made by
modifying both the polymer concentration and the physico-chemical properties
of the solutions to enhance or reduce one of the non-Newtonian properties :
shear-thinning or elastic effect.

Consequences of both effects have been investigated in the first two regions
of the flowing film. In the flat film region, due to the invariance of the
film thickness in the axial direction, only the shear-thinning effect is
effective. At a constant flow rate, our experiments demonstrate that, as a
consequence of the shear-thinning effect, the thickness of the film is always
smaller than in the case of Newtonian fluid. Our results clearly show the
influence of the shear-thinning effect on the velocity profile: a parabolic
profile in the Newtonian case tends to become a plug-like profile. Thus, an
increase of the shear-thinning effect yields a thinner, unperturbed film.
Further downstream on the fiber, the film undergoes the Rayleigh-Plateau
instability. The growth rate of the instability has been investigated
experimentally and theoretically using a linear stability analysis. Good
agreement is found between the experimental data for xanthan and the model.
For PAAm solutions, and to take into account the normal forces, we consider an
effective surface tension (lower than the fluid surface tension) which tends
to decrease the growth rate and to recover the experimental data. The
morphology of the patterns resulting from the instability depends on the non-
Newtonian properties. In particular, the drops formed with PAAm solutions
exhibit a swelling effect compared to drops observed with xanthan solution,
for a similar shear-thinning effect. We observe that the drop of fluid with
normal forces is less rounded compared with the case of a pure shear-thinning
drop. This swelling effect has been quantified by a scaling law analysis where
the slope of the drop front is expressed as a function of the normal stress.

As a conclusion, by considering two kinds of polymeric solutions with the same
shear-thinning effect, which differ from each other in the presence of normal
forces, we have succeeded in understanding the relationship between the
rheological properties and the destabilization of the flowing film on a fiber
as well as the morphology of the observed patterns. This should be helpful in
understanding what happens with more complex fluids, in particular fluids
which exhibit more elastic effects where the elasticity could prevent the
growth of the instability.

##### Acknowledgments:

We thank Fédération Paris VI (high-speed camera) and Triangle de la Physique
(rheometer apparatus). The authors thank Liyan Yu and Prof. John Hinch for
fruitful discussions. Also we thank Lionel Auffray, Rafael Pidoux and Alban
Aubertin for the experiment engineering and technical improvements.

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