# KU-DMIS-MSRA at RadSum23: Pre-trained Vision-Language Model for Radiology
Report Summarization

Gangwoo Kim1⋄ Hajung Kim1 Lei Ji2 Seongsu Bae3 Chanhwi Kim1 Mujeen Sung1
Hyunjae Kim1 Kun Yan4 Eric Chang5 Jaewoo Kang1,6†
1Korea University∗ 2Microsoft Research Asia 3KAIST AI 4Beihang University
5Centre for Perceptual and Interactive Intelligence, Hong Kong 6AIGEN Sciences
Inc.
{gangwoo_kim<EMAIL_ADDRESS>
###### Abstract

In this paper, we introduce CheXOFA, a new pre-trained vision-language model
(VLM) for the chest X-ray domain. Our model is initially pre-trained on
various multimodal datasets within the general domain before being transferred
to the chest X-ray domain. Following a prominent VLM, we unify various domain-
specific tasks into a simple sequence-to-sequence schema. It enables the model
to effectively learn the required knowledge and skills from limited resources
in the domain. Demonstrating superior performance on the benchmark datasets
provided by the BioNLP shared task Delbrouck et al. (2023), our model benefits
from its training across multiple tasks and domains. With subtle techniques
including ensemble and factual calibration, our system achieves first place on
the RadSum23 leaderboard for the hidden test set.

††$\diamond$ Work done while interning at Microsoft Research Asia ††$\dagger$
Corresponding author ††$*$ Department of Computer Science and Engineering

##  1 Introduction

Chest radiography is a widely used imaging modality for assessing the thorax
and diagnosing cardiopulmonary conditions. However, there is a significant
shortage of clinical doctors in several under-resourced regions, delaying
diagnosis and treatment and reducing the quality of care. Developing an
automated system for analyzing radiographs can improve radiologist workflow
efficiency and expand healthcare services to these regions.

To promote research in this direction, the BioNLP 2023 workshop opens a new
shared task called RadSum23 Delbrouck et al. (2023) for the radiology report
summarization.111vilmedic.app/misc/bionlp23/sharedtask In this task,
participants are asked to build a model that takes a findings section as
input, which is a generic radiology report of a given X-ray image, and then
outputs an impression section, which is a summary of key observations in the
given report. High-resolution X-ray images can be used as input along with
findings sections. The radiology report summarization task aims to effectively
distill complex clinical observations from chest X-ray images into concise and
coherent summaries.

In this paper, we introduce CheXOFA (One For All tasks with Chest X-ray), a
novel vision-language model designed for the chest X-ray domain. Specifically,
we initialize our model using OFA Wang et al. (2022), a Transformer model pre-
trained with a unified sequence-to-sequence schema on diverse uni- or cross-
modal tasks such as image classification, language modeling, and image
captioning. We further train the model to generate full-text reports from the
chest X-ray image using the MIMIC-CXR dataset Johnson et al. (2019). Then, we
fine-tune the model on the radiology report summarization task. When
summarizing the report, our model jointly encodes visual information from the
chest X-ray image with the full-text report, taking advantage of its
multimodal nature. Additionally, we employ subtle techniques such as task-
specific ensemble Dai et al. (2021) and factual calibration to further improve
the model performance.

Our experiments demonstrate that our proposed methods largely enhance model
performances on two test sets of the shared task. On the official leaderboard
222vilmedic.app/misc/bionlp23/leaderboard/ for MIMIC-CXR hidden test set, our
system ranked first place, achieving the state-of-the-art performances in most
evaluation metrics. Especially, it surpasses the second-best model by 2.3 and
2.9 in BLEU and F1-CheXbert score. In the ablation study, we demonstrate how
much each method contributes to the improvement.

##  2 Related Works

###  2.1 Automated Radiology Report Generation

Utilizing radiographic datasets such as MIMIC-CXR Johnson et al. (2019) and
CheXpert Irvin et al. (2019), various methodologies for automated report
generation have recently attracted attention. Notably, these datasets often
include both chest X-ray images and free-text reports, enabling the use of
automated rule-based labelers Irvin et al. (2019) or neural models Smit et al.
(2020) to extract disease labels from the reports. It can be categorized into
two tasks: 1) radiology report generation, which is similar to medical image
captioning and aims to describe radiology images in detail (findings section),
having seen significant progress in recent years Chen et al. (2020); Zhang et
al. (2020b); Liu et al. (2021a, b); Miura et al. (2021); Chen et al. (2022);
2) radiology report summarization Zhang et al. (2018), which focuses on
summarizing findings section into impressions section in radiology reports.
Most existing research Zhang et al. (2020c); Hu et al. (2021, 2022b); Karn et
al. (2022) tends to focus on text-based summarization while some image-
incorporating studies Delbrouck et al. (2021); Hu et al. (2022a) use
suboptimal methods lacking appropriate multi-modal pre-training objectives for
the generative task.

###  2.2 Multimodal Foundation Models

Vision-language pretraining models are becoming the foundation models
effective for multimodal tasks including Vilbert Lu et al. (2019), OFA Wang et
al. (2022), Flamingo Alayrac et al. (2022) for open-domain, and MedViLL Moon
et al. (2022), Clinical-BERT Yan and Pei (2022), BioViL Boecking et al. (2022)
and M3AE Chen et al. (2022) for biomedical domain. However, most of the
existing works for clinical domain mainly focus on pretraining the encoder for
understanding tasks like medical VQA, classification and so on, and rarely
take this radiology report summarization task as downstream task. In this
paper, we propose CheXOFA, a pre-trained generative VLM that learns the
required capabilities for the radiology report summarization task.

Figure 1: Overview of CheXOFA. It is pre-trained in the report generation
task. We then fine-tune the pre-trained model in the report summarization
task. We use task-specific instructions.

##  3 CheXOFA

We propose a novel vision-language model, CheXOFA, specifically designed for
the radiology report summarization. We first initialize our model parameters
with OFA Wang et al. (2022), which has been shown to be effective in the
general domain. Then, we pre-train and fine-tune the model with the various
tasks in the medical imaging domain. In addition, we newly introduce a factual
calibration technique to further improve the model performance.

###  3.1 Multimodal Architecture

The backbone of the CheXOFA model is Transformer Vaswani et al. (2017)
architecture with a sequence-to-sequence framework specialized for generative
tasks. Following BART Lewis et al. (2020) and GPT Radford et al. , we utilize
byte-pair encoding (BPE) Sennrich et al. (2016) to transform text sequences
into linguistic features of subword sequences. On the vision side, we use a
visual extractor to encode an image into a sequence of hidden representations.
Specifically, we divide an input image into fixed size of patches. Then,
ResNet He et al. (2016) modules are used to convolve the visual information
into visual features $x^{v}\in\mathcal{R}^{|P|\times d}$, where $|P|$ is the
number of patches and $d$ is the dimension of hidden representation. Overall,
the linguistic features and the visual features are concatenated into one
sequence for feeding into the encoder-decoder Transformer for modality fusion
and sequence generation.

###  3.2 Training and Inference

Our model is optimized with the cross-entropy loss to the ground-truth
sequences. Given an input $x$ and an output $y$, we train model parameters
$\theta$ by minimizing $\mathcal{L}=-\sum_{t=1}^{|y|}\log
P_{\theta}(y_{t}|y_{<t},x)$, where $x$ can be composed of instruction $x^{i}$,
linguistic and visual features $x^{l},x^{v}$. In the inference phase, we
choose the beam search as the decoding strategy to obtain better text
sequences. Additionally, we adjust the length penalty that assigns weights
according to the length of each beam. We control the output sequence length
and explore the optimal value for it.

###  3.3 Pre-training and Downstream Tasks

Following Wang et al. (2022), we unify cross-modal tasks into a simple
sequence-to-sequence format. We design a unified learning framework for pre-
training and downstream tasks, which require multimodal reasoning ability.
CheXOFA’s versatile design allows it to tackle a wide range of tasks using a
single model. Furthermore, the model is designed with multitasking
capabilities, allowing it to simultaneously handle multiple tasks across
various modalities. To this end, the model shares its parameters and schema
across all tasks. Meanwhile, we employ task-specific instructions manually
crafted for each task.

We pre-train CheXOFA with the report generation (RGen) task and then fine-tune
it with report summarization (RSum) on MIMIC-CXR dataset Johnson et al.
(2019). In the RGen task, the model learns to generate findings section of the
report, based on the chest X-ray image. We use the same instruction $x^{i}$
with that of the image captioning task, “What does the image describe?”. For
our target task, RSum, the model is trained to generate impression section,
given findings section of the report. We also exploit the corresponding chest
X-ray image to jointly leverage the visual information. Hence, an input $x$ is
composed of visual features, subword tokens of findings section, and
instruction, “what is the summary of the following article?”. Furthermore, we
newly design the classification-supported RSum task (cls-RSum) to enhance the
factual correctness of the summary. In the task, the model additionally
performs a classification task for observed disease from the X-ray image or
report. Then, it generates summaries based on the identified category,
ensuring relevance and coherence.

###  3.4 Ensemble with Factual Calibration

To improve the overall performance, we utilize an ensemble method that
combines various predictions from multiple models. Following Dai et al.
(2021), we select the best prediction based on the mutual similarity score. In
particular, we calculate similarity scores for every possible pair of
predictions, creating a mutual similarity matrix. Subsequently, we aggregate
the matrix in a row-wise fashion, averaging value for each row. The prediction
with the highest score is selected as the final output. If multiple outputs
have the same highest score, we randomly select the final output, ensuring
that it is chosen in an unbiased manner. To compute the similarity, we use
F1-RadGraph as a scoring function. Through the combination of diverse
predictions, we are able to obtain an optimal summary and mitigate the failure
of individual models.

We perform a calibration to improve the factual correctness of the ensembled
prediction. We first extract medical observations from the prediction by using
cheXbert labeler Smit et al. (2020). Then, we check whether they are matched
with identified labels by the cls-RSum model. If not, the cls-RSum result is
chosen as the final prediction. We find the factual calibration performs
effectively when the ensemble process yields a low aggregated similarity
score, which is in proportion to the model’s confidence.

##  4 Experiment

Track | Rank | Team Name | BLEU | ROUGE | Bertscore | CheXbert | Radgraph
---|---|---|---|---|---|---|---
F1 | F1
Hidden Test | 1 | ku-dmis-msra (ours) | 18.62 | 34.57 | 55.90 | 72.36 | 43.20
2 | utsa-nlp | 16.33 | 34.97 | 55.54 | 69.41 | 42.86
3 | knowlab | 14.41 | 33.63 | 54.72 | 67.20 | 39.98
4 | shs-te-dti-mai | 14.59 | 32.43 | 53.99 | 6 8.99 | 38.40
5 | aimi | 5.15 | 31.84 | 47.83 | 64.18 | 32.05
Public Test | 1 | utsa-nlp | 25.87 | 47.86 | 64.74 | 77.93 | 51.84
2 | ku-dmis-msra (ours) | 25.58 | 47.75 | 64.80 | 76.29 | 50.96
3 | knowlab | 22.97 | 46.15 | 63.43 | 75.14 | 48.04
4 | e-health csiro | 17.97 | 44.14 | 61.47 | 71.67 | 44.95
5 | iuteam1 | 10.10 | 40.44 | 56.44 | 58.01 | 39.48

Table 1: Official results of the leaderboards on MIMIC-CXR hidden test set and
MIMIC-CXR test set. The Models are ranked based on F1-Radgraph score. The best
score is displayed in bold typeface and the score of the second best model is
underlined.

###  4.1 Experimental Setup

#### Evaluation Metric

For evaluation, we used five metrics: BLEU4 (Papineni et al., 2002), ROUGE-L
(Lin, 2004), Bertscore (Zhang et al., 2020a), F1-cheXbert (Zhang et al.,
2020d), and F1-RadGraph (Delbrouck et al., 2022a). BLUE4 and ROUGE-L measure
syntactical similarity based on n-gram overlap between reference and generated
summaries, while Bertscore measures semantic similarity to handle synonyms and
paraphrasing. To evaluate the factual correctness of generated summaries,
F1-cheXbert and F1-RadGraph are used. Out of these five metrics, F1-RadGraph
is selected as the primary metric for ranking participating systems, as it
uses RadGraph (Jain et al., 2021) annotations to better consider the
consistency and correctness of extracted entities and relations.

#### Benchmark Datasets

Our model was trained on MIMIC-CXR (Johnson et al., 2019) and evaluated on
both the test set of MIMIC-CXR and a newly-collected hidden test set. MIMIC-
CXR is a publicly available large dataset consisting of 128,032 report-image
pairs with 227,835 multi-view images333The mentioned statistics are derived
from reports containing both the findings and impression sections
simultaneously.. The dataset is split into training, validation, and test
sets, which comprise 125,417, 991, and 1,624 image-report pairs, respectively.
The MIMIC-CXR hidden test set was newly introduced in the RadSum23 challenge
served by vilmedic Delbrouck et al. (2022b), and it comprises 1,000 out-of-
domain image-report pairs collected from CheXpert images Irvin et al.
(2019)444Although it comes from CheXpert dataset, we name it as MIMIC-CXR
hidden test set, following the challenge description.

###  4.2 Results

Table 1 shows the official results of the leaderboard on the hidden test set.
Our model ranked first place among other systems on the leaderboard and
achieved the state-of-the-art performance in four out of five evaluation
metrics. Especially, our model significantly outperformed the second-best
model by 2.29 BLEU4 and 2.95 F1-cheXbert.

Table 1 also presents the official results on the public test set of MIMIC-
CXR. Our model achieved competitive performance with the best model in four
out of five metrics and the best performance based on Bertscore. Overall
results indicate that our method could generalize to diverse datasets,
achieving outstanding performances on both hidden and visible test sets.
Conclusively, our method has remarkable capabilities to summarize radiology
reports by capturing essential medical observations.

Model | ROUGE | CheXbert | Radgraph
---|---|---|---
F1 | F1
CheXOFA (Ensem.) | 47.75 | 76.29 | 50.96
w/o. Fact Calib. | 47.04 | 75.98 | 50.14
CheXOFA (Single) | 46.03 | 75.67 | 48.16
Text-only | 46.30 | 73.59 | 47.78
w/o. pre-training | 43.79 | 73.67 | 45.73

Table 2: Ablation study on the public test set. We provide the performance of
ensembled (Ensem.) and single results. Every component such as factual
calibration (Fact Calib.), encoding multimodal inputs (Text-only), and pre-
training contributes to the improvement.

###  4.3 Ablation Study

We performed an ablation study to analyze how each method contributes to the
overall performance. Table 2 shows the evaluation scores when removing each
method from our best model on the test set of MIMIC-CXR. Factual calibration
improves the factual correctness scores, 0.3 of the F1-CheXbert score and 0.8
of F1-Radgraph score. Using a single CheXOFA model shows a performance drop
compared to the ensemble model by approximately 1.8 in F1-Radgraph.
Nevertheless, it achieves competitive performances with other participating
systems. Allowing the model to focus on the report (Text-only) achieves
similar performance in ROUGE-L and F1-Radgraph scores relevant to the lexical
overlap. However, F1-cheXbert score significantly degrades, which indicates
that models benefit from using multimodal information. Fine-tuning a vanilla
OFA model performs poorly in most scores, which shows the importance of the
pre-training task.

##  5 Conclusion

We proposed a pre-trained VLM, CheXOFA, for the chest X-ray domain. We showed
that pre-traing with the report generation task improves the downstream task,
report summarization. Taking advantage of its multimodal nature, we improved
the performance by jointly encoding visual and linguistic features.
Furthermore, we explored subtle techniques such as ensemble and factual
calibration to improve the model performance. Our experimental results
demonstrated that proposed methods benefit the summarization performance. Our
model ranked first on the hidden test set in RadSum23 shared task. We showed
promising results about the domain-specific VLM in the chest X-ray tasks. We
hope that our method can shed light on automating radiology report generation.

## Acknowledgement

This research was supported by the MSIT(Ministry of Science, ICT), Korea,
under the High-Potential Individuals Global Training Program)
(RS-2022-00155958) and the ICT Creative Consilience
program(IITP-2023-2020-0-01819) supervised by the IITP(Institute for
Information & Communications Technology Planning & Evaluation). It was also
supported by the National Research Foundation of Korea, South Korea
(NRF-2023R1A2C3004176) and a grant of the Korea Health Technology R&D Project
through the Korea Health Industry Development Institute (KHIDI), funded by the
Ministry of Health & Welfare, Republic of Korea (grant number: HR20C0021).

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BIASING PHENOMENON111Paper presented at the Fourth Zeldovich meeting, an
international conference in honor of Ya. B. Zeldovich held in Minsk, Belarus
on September 7–11, 2020\. Published by the recommendation of the special
editors: S. Ya. Kilin, R. Ruffini and G. V. Vereshchagin.

Jaan Einasto

Tartu Observatory, 61602 Tõravere, Estonia

###### Abstract

We study biasing as a physical phenomenon by analysing power spectra (PS) and
correlation functions (CF) of simulated galaxy samples and dark matter (DM)
samples. We apply an algorithm based on the local densities of particles,
$\rho$, to form populations of simulated galaxies, using particles with
$\rho\geq\rho_{0}$. We calculate two-point CF of projected (2D) and spatial
(3D) density fields of simulated galaxies for various particle-density limits
$\rho_{0}$. We compare 3D and 2D CFs; in 2D case we use samples of various
thickness to find the dependence of 2D CFs on thickness of samples. Dominant
elements of the cosmic web are clusters and filaments, separated by voids
filling most of the volume. In individual 2D sheets positions of clusters and
filaments do not coincide. As a result, in projection clusters and filaments
fill in 2D voids. This leads to the decrease of amplitudes of CFs in
projection. For this reason amplitudes of 2D CFs are lower than amplitudes of
3D CFs, the difference is the larger, the thicker are 2D samples. Using PS and
CFs of simulated galaxies and DM we estimate the bias factor for $L^{\ast}$
galaxies, $b^{\ast}=1.85\pm 0.15$.

Keywords: Cosmology: large-scale structure of Universe; Cosmology: dark
matter; Cosmology: theory; Galaxies: clusters; Methods: numerical

##  1 Introduction

The biasing problem is as old as our understanding on the presence of the
cosmic web. Early data on the distribution of galaxies were obtained from
counts of galaxies on the sky. The most complete counts were made in Lick
Observatory, and reduced by Soneira and Peebles [1], see Fig. 1. This survey
shows the presence of numerous clusters of galaxies and a few superclusters,
most of galaxies belong to an essentially random field population, which fills
most the whole sky in projection.

Figure 1: Left: Distribution of galaxies according to Lick survey, reduced by
Soneira and Peebles [1]. Right: distribution of particles in 2D numerical
simulation by Doroshkevich and Shandarin [6].

Figure 2: Left: The distribution of Zwicky clusters of the distance class near
in the Perseus area of the sky in redshift interval 3500 to 6500 km/s [2].
Right: distribution of galaxies in the Perseus-Pisces region [3].

In mid 1970s the number of galaxies with known redshifts was large enough to
find the spatial distribution of galaxies. New three-dimensional (3D) data
showed that in the distribution of galaxies there exists a certain pattern,
called presently the cosmic web. Galaxies and clusters of galaxies are
concentrated to filaments and chains, the space between filaments is almost
empty, as shown by Jõeveer and Einasto [2], [3] in the IAU Symposium on Large
Scale Structure of the Universe, see Fig. 2. A theory of galaxy formation was
suggested by Zeldovich [4]. First two-dimensional (2D) numerical simulations,
based on Zeldovich model, were made by Doroshkevich and Shandarin [5], [6],
see Fig. 1. Simulations show the formation of the cosmic web with rich
superclusters, connected by filaments, and large voids, populated by a
rarefied field of test particles. This picture is similar to the observed
distribution of galaxies. However, there is an important difference. In the
simulation model there exist test particles in low density regions, in the
real universe voids are almost empty. On the other hand, as noted by Jõeveer
and Einasto [2], it is very difficult to imagine a process of galaxy and
supercluster formation which is effective enough to evacuate completely such
large volumes as cell interiors are. There must be invisible dark matter (DM)
in voids. This was the first hint to differences in distributions of mass (DM)
and galaxies. Differences between distributions of galaxies of different
luminosity and clusters of galaxies were studied by Kaiser [7], who used the
term “biasing” for this phenomenon.

We consider biasing as a physical phenomenon, concerning the relation between
distributions of matter and galaxies on large scales. In discussing the global
biasing phenomenon we assume that gravity is the dominating force, determining
the formation and evolution of the cosmic web on large scales. We consider the
bias function $b$ as a function, which quantitatively relates differences
between distributions of matter and galaxies. The numerical value of the bias
function can be found by the power spectrum (PS) and the correlation function
(CF) analyses. We shall compare PS and CFs of simulated galaxy samples and
full samples of particles in models of the evolution of the cosmic web. The
use of simulated samples of galaxies has the advantage that all data are
available for galaxies and matter.

##  2 Biasing model

Already first data on the distribution of galaxies and matter, discussed
above, suggested that biasing is a threshold phenomenon — in low-density
regions there are no galaxies. For this reason we cannot use a simple biasing
model, $b=\rho_{g}(X)/\rho_{m}(X)$, where $\rho_{g}(X)$ and $\rho_{m}(X)$ are
densities of galaxies and matter on location $X$, respectively. Instead we use
a biasing model, where galaxies do not form in low-density regions at all, or
are too faint to be included into flux-limited galaxy surveys. We label all
particles of $\Lambda$CDM simulation with local density value, $\rho$, and
apply a sharp particle-density limit, $\rho_{0}$. Simulated galaxy samples
consist of particles with density labels, $\rho\geq\rho_{0}$. Full DM samples
consist of all particles, i.e. particles with density limit $\rho_{0}=0$.

We shall describe the biasing phenomenon quantitatively by the bias function
$b$, which can be calculated from the CF,

$b(r,\rho_{0})=\sqrt{\xi_{C}(r,\rho_{0})/\xi_{m}(r)},$ (1)

where $\xi_{C}(r,\rho_{0})$ is the CF of clustered matter — galaxies, and
$\xi_{m}(r)$ is the CF of all matter, $r$ is the separation of
particles/galaxies, and $\rho_{0}$ is the matter density limit, which selects
particles to be included. Another possibility is to use the PS,
$b(k,\rho_{0})=\sqrt{P_{C}(k,\rho_{0})/P_{m}(k)}$, where $P_{C}(k,\rho_{0})$
is the PS of true or simulated galaxies, $P_{m}(k)$ is the PS of mass, and $k$
is wavenumber.

Both the CF and the PS methods have been used to study the clustering of
galaxies. Norberg [8] investigated the clustering of galaxies of the 2dF
Galaxy Redshift Survey. Tegmark et al. [9] and Zehavi et al. [10] determined
power spectra of galaxies of the SDSS survey, see Fig. 4. To avoid redshift
distortions authors calculated as first step 2D CF of galaxies. These studies
allowed to determine with good accuracy relative bias functions of galaxies of
various magnitude. Authors found that galaxies approximately follow matter. In
our view this result is difficult to accept, since already a visual comparison
of distributions of real and simulated galaxies with the distribution of
matter shows large differences. To understand the reason for this difference
in global bias parameter it is needed to study in more detail properties of 2D
CF.

The main difference between distributions of matter and galaxies is the
presence of DM in low-density regions, with no corresponding population of
galaxies. Thus a straightforward application of the equation (1) is difficult,
since there is no observable population of DM. We shall overcome this
difficulty by using a model distribution both for simulated galaxies and
matter (DM). We assume that modern models of $\Lambda$ cold dark matter
($\Lambda$CDM) represent the real universe well. Further we apply a threshold
clipping biasing model, described above. Also we apply a new approach to
measure CF and PS — the density field method by Szapudi et al. [13]. In this
approach the input information is identical for CF and PS analysis – high-
resolution density field. This approach allows to have a new look to CF,
similar to the PS analysis. Both start from density fields, use FFT, PS
analysis finds the dependence of bias function on wavelength scale $k$, CF on
relative spatial coordinates, $r$. In both analyses we used a $\Lambda$CDM
simulation with cube size $L_{0}=512$ $h^{-1}$ Mpc, and number of particles
$N_{\mathrm{part}}=512^{3}$, assuming cosmological parameters
$\Omega_{\mathrm{m}}=0.28$, $\Omega_{\Lambda}=0.72$, $\sigma_{8}=0.84$, and
the dimensionless Hubble constant $h=0.73$.

##  3 Correlation analysis of $\Lambda$CDM universe

Here we analyse properties of 3D and 2D CFs of the $\Lambda$CDM model, based
on Einasto et al. [11] and [12]. The analysis of PS of the $\Lambda$CDM model
by Einasto et al. [14] yields similar results for the biasing problem. In both
analyses the same $\Lambda$CDM simulation in a cube of size $L_{0}=512$
$h^{-1}$ Mpc was used. Simulated galaxy samples with particle density limits
$\rho_{0}$ were selected with $\rho_{0}=1,~{}2,5,~{}\dots,100$, and with
$\rho_{0}=0$, i.e. full DM model sample. Einasto et al. [14] compared spatial
density fields of simulated galaxy samples and galaxy samples from Sloan
Digital Sky Survey (SDSS). This comparison shows that density fields of
simulated galaxies with particle density limits $\rho_{0}\approx 5$ are
similar to density fields of SDSS galaxies of absolute magnitude
$M_{r}\approx-18$. $L^{\ast}$ galaxies of magnitude $M_{r}=-20.5$ have spatial
distribution properties, similar to spatial distribution of simulated
$\Lambda$CDM samples with density threshold $\rho_{0}=10$.

Figure 3: CFs of simulated galaxies in $\Lambda$CDM universe: left in 3D,
right in 2D [11], [12].

In Fig 3 we show in the left panel 3D CFs of simulated galaxies in our
$\Lambda$CDM universe. Solid bold line corresponds to the CF of all matter
with particle density threshold $\rho_{0}=0$, blue dashed line is for the
sample $\rho_{0}=5$, red solid line for the sample $\rho_{0}=10$, and violet
dashed line for the sample $\rho_{0}=30$. These coloured lines correspond to
real galaxies of absolute red magnitudes $M_{r}=-18,~{}-20.5,~{}-22.0$. We see
that all 3D CFs are approximately parallel, see Fig. 4. As shown by Einasto et
al. [14], the percolation analysis suggests for galaxies of absolute magnitude
$M_{r}=-19.0$ the bias parameter $b=1.68\pm 0.09$, and for galaxies of
absolute magnitude $M_{r}=-20.5$ the bias parameter $b=1.85\pm 0.15$. This
magnitude corresponds to $L^{\ast}$ galaxies.

In the distant observer approximation we can calculate projected 2D CFs from
2D density fields, using model data in rectangular spatial coordinates. First
we calculated 2D density fields by integrating 3D field by ignoring
$z$-coordinates in calculation of density fields in the selected $z$ range. We
made the integration of cubical samples to get $n$ sequentially located 2D
sheets of thickness $T=L_{0}/n$, with $n=1,~{}2,~{}4,\dots 2048$. $n=1$
corresponds to the whole sample in $z$-direction of thickness, $T=L_{0}=512$
$h^{-1}$ Mpc, $n=2$ corresponds to thickness $512/2=256$ $h^{-1}$ Mpc. For
each $n$ we calculated 2D CFs for all $n$ sheets, and then found the mean CF
and its error for a given $n$. We show 2D CFs in the right panel of Fig. 3 for
a sample of simulated galaxies with $\rho_{0}=10$, corresponding to $L^{\ast}$
galaxies. Bold dashed black line shows 3D CF of this sample. Solid coloured
lines correspond to 2D CFs of various $n$ and thickness.

Fig. 3 shows that the integration of density fields in the radial direction
decreases the amplitude of 2D CFs. 2D CF of the thickest slice with thickness
$T=L_{0}/n=512$ $h^{-1}$ Mpc has at separation $r=6$ $h^{-1}$ Mpc an
amplitude, about 25 times lower than the 3D CF of the same sample. With
increasing $n$ and decreasing thickness $T$ the difference in amplitudes of 2D
and 3D CFs decreases. Only the mean 2D CF for $n=2048$ and thickness $T=0.25$
$h^{-1}$ Mpc 2D CF is identical to 3D CF of the same sample.

Figure 4: Left: Bias function found by Einasto et al. [14] is marked by red
circles with error bars. Black, blue, and red lines show bias functions by
Norberg et al. [8], Tegmark et al. [9], and Zehavi et al. [10]. Dashed line
shows the fit by Norberg et al. [8], applying bias normalising factor
$b_{\circ}=~{}1.85$. Right: Cross section of density fields: blue line is for
early universe at redshift $z=30$, black line is for DM at the present epoch,
red line corresponds to matter in galaxies at the present epoch [14].

##  4 Discussion

Right panel of Figure 4 shows a cross section of the density field. We plot
here the density contrast $\delta(x)=\rho(x)-1$. The blue line shows the
density contrast at the early epoch, corresponding to a redshift of $z=30$. At
this early epoch, the amplitude of density fluctuations is small, and density
fluctuations are approximately equal everywhere. The black line gives the
density contrast for the same cross section at the present epoch for the
sample with all particles. This density field is dominated by fluctuations of
the density contrasts in the range $-1<\delta(x)<2$, with a mean around
$\delta\approx-0.5$. The red line shows the same cross section for the
simulated galaxy sample for $L^{\ast}$. Here high-density peaks of the density
field are the same as in the full model, but weak DM knots of medium density,
seen in the full DM sample, are gone. In contrast to the full sample, over
most of the density field the density of the simulated galaxy sample has zero
density and density contrast $\delta=-1$.

CFs of our models were calculated using density fields applying the method by
Szapudi et al. [13]. As input for both correlation and PS analysis is the
density contrast field, $\delta$, used for PS analysis as follows:
$P(k)=\langle|\delta_{k}|^{2}\rangle$. In DM model there are no cells with
zero density, low-density cells have a mean density contrast $|\delta|\approx
0.5$. In $L^{\ast}$ model sample most of these cells are empty with
$|\delta|=1$. From the above equation it follows that this increases the
amplitude of PS. The increase of the amplitude is the greater the larger is
the fraction of zero density cells. The fraction of zero density cells is 95%
of all cells both in the SDSS sample and the $\Lambda$CDM model sample for
$L^{\ast}$. The PS and CF are proportional to the sum over all density
contrasts. The greater the fraction of cells with zero density and density
contrast $|\delta|=1$, the greater the sum. For this reason, the PS and CF of
all simulated galaxy samples have higher amplitudes than the full DM model;
the larger the fraction of zero-density regions, the higher the amplitude.

Figure 5: 2D density fields of various thickness. In left panel a view of
thickness 512 $h^{-1}$ Mpc, in right panel of thickness 32 $h^{-1}$ Mpc [12].

We show in Fig. 5 2D density fields of the model $\Lambda$CDM using particle
density limit $\rho_{0}=10$, and having various thickness. In left panel we
present 2D density field of thickness $T=512$ $h^{-1}$ Mpc, i.e. the whole
cube of our $\Lambda$CDM model. In the right panel the thickness is $T=32$
$h^{-1}$ Mpc. Morphological properties of 2D thin slices are close to
properties of the 3D density field of the same model: the fraction of high-
and low-density regions, mutual distances between high-density regions.
Essential elements of the cosmic web are high-density regions (clusters and
filaments) and voids, which together form the pattern of the web. The detailed
structure of the web is seen in right panel of Figure. High-density regions
are surrounded by zero density voids, occupying most of the volume of the
density field. Clusters and filaments in sheets at different $z$ locations are
located in various $x,y$-positions. In stacked thick 2D sheets clusters and
filaments at various $z$ are projected to the 2D $x,y$ plane at different
positions, and in this way fill in voids in the 2D density field. This is
clearly seen in the left panel of Fig. 5. In earlier studies this effect was
ignored. The lowering of the amplitude of 2D CFs can be the main reason why
amplitudes of bias functions in earlier studies were so low.

##  5 Conclusions

  1. 1.

The presence in galaxy density field of large number of cells with zero
density increases the amplitude of the galaxy PS and CF over the amplitude of
matter PS and CF. Thus the amplitude of the galaxy PS and CF is always higher
than the amplitude of PS and CF of matter, i.e. bias parameter is $b>1$. The
bias parameter of $L^{\ast}$ galaxies, as found from $\Lambda$CDM power
spectra is $b=1.85\pm 0.15$.

  2. 2.

Dominant elements of the cosmic web are clusters and filaments, separated by
voids filling most of the volume. In 2D sheets clusters and filaments are
located at different positions fill in 2D voids, which leads to the decrease
of amplitudes of CFs. For this reason amplitudes of 2D CFs are lower than
amplitudes of 3D CFs, the difference is the larger, the thicker are 2D
samples.

  3. 3.

Biasing problem is a surprisingly complex phenomenon, where theories of galaxy
formation meet with studies of the structure of the cosmic web. The use of the
density field in CF and PS analyses allows to see these functions from a
different point of view.

I thank my coauthors in the study of the cosmic web Maret Einasto, Gert Hütsi,
Juhan Liivamägi, Ivan Suhhonenko, Elmo Tempel for fruitful and enjoyable
collaboration, and Enn Saar for discussion.

## Funding

This work was supported by institutional research funding IUT40-2 of the
Estonian Ministry of Education and Research, by the Estonian Research Council
grant PRG803, and by Mobilitas Plus grant MOBTT5. We acknowledge the support
by the Centre of Excellence“Dark side of the Universe” (TK133) financed by the
European Union through the European Regional Development Fund.

## References

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††thanks<EMAIL_ADDRESS>
# Electron Magneto-Hydrodynamics in Graphene

Jack N. Engdahl  School of Physics, University of New South Wales, Sydney
2052, Australia  Aydın Cem Keser  CSIRO, Bradfield Road, West Lindfield NSW
2070, Australia  Thomas Schmidt  Department of Physics and Materials Science,
University of Luxembourg, Luxembourg  School of Chemical and Physical
Sciences, Victoria University of Wellington,
P.O. Box 600, Wellington 6140, New Zealand  Oleg P. Sushkov  School of
Physics, University of New South Wales, Sydney 2052, Australia

###### Abstract

We consider the hydrodynamic flow of an electron fluid in a channel formed in
a two-dimensional electron gas (2DEG) with no-slip boundary conditions. To
generate vorticity in the fluid the flow is influenced by an array of
micromagnets on the top of 2DEG. We analyse the viscous boundary layer and
demonstrate anti-Poiseuille behaviour in this region. Furthermore we predict a
longitudinal Hall effect, where a periodic magnetic field generates a voltage
modulation in the direction of transport. From the experimental point of view
we propose a method for a precise measurements of the properties of different
electron fluids. The results are applicable to graphene away from the charge
neutrality point and to semiconductors.

##  I Introduction

It has been over half a century since the theory of viscous electron
hydrodynamics was first developed [1] and in recent years viscous effects have
been observed in the electron fluid experimentally [2, 3]. Electron transport
transitions to the hydrodynamic regime when the momentum-conserving electron-
electron scattering length, $l_{ee}$, is the shortest length scale in the
system and becomes much shorter than the momentum-nonconserving diffusive
scattering length, $l_{\rm mfp}$, and the width of the sample, $W$. In this
regime electron-electron interactions dominate and a local thermodynamic
equilibrium is established among the electrons, enabling them to flow as a
fluid. Since this condition requires samples of very high purity, the
theoretical prediction of the hydrodynamic regime in the 1960s [1, 4] could
only be realized experimentally after the development of high-mobility
semiconductors [5] and graphene [6, 2].

Hydrodynamic electron flow manifests itself in a range of effects that have
been observed. While hydrodynamic electron transport was suggested to have
been achieved already in 1995 in Ref. [5], this result was immediately
challenged by Gurzhi in Ref. [7]: the conditions for hydrodynamics in the bulk
material were satisfied, but they were not satisfied for hydrodynamics in a
2DEG. Reliable evidence of the hydrodynamic regime was first presented two
decades later in Refs. [8, 9, 10]. Notable consequences of hydrodynamic
behavior are the decrease of resistance with increasing temperature known as
the Gurzhi effect [5, 11, 12, 13, 14, 15], Poiseuille flow profiles [16, 17,
18], negative non-local resistance and formation of whirlpools [19, 20, 8,
21], the formation of local viscous regions [22], Hall viscosity [14, 23],
negative magnetoresistance [24, 25, 13, 26, 10, 27, 28], the violation of the
Wiedemann-Franz law111Although recent work in Ref. [29] suggests that the
result of a well-known experimental work [30] is not hydrodynamic in nature.
[30, 31, 32, 33, 34], anomalous scaling of the resistance with the channel
width [31, 9], resonant photoresistance in magnetic fields [35, 36, 37, 38,
27], and a quantum-critical dynamic conductivity [39]. Furthermore, many novel
phenomena have been proposed, including the elimination of the Landauer-
Sharvin resistance [40], anisotropic fluids [41, 42], dynamo effect in the
electron-hole plasma [43] and the prospect of hydrodynamic spin transport [44,
45, 46] inspired by experiments on liquid mercury [47, 48]. In addition to the
Poiseuille-like flow expected from electron hydrodynamics, anomalous flow
profiles have been predicted in the boundary layer in charge neutral graphene
in a uniform perpendicular magnetic field, with this “anti-Poiseuille” flow
attributed to recombination effects [49, 50, 51].

Hydrodynamic flow depends qualitatively on boundary conditions, with no-slip
boundaries giving rise to the well known Poiseuille flow profile. On the other
hand, it is also possible to achieve perfect-slip boundaries by using gates to
create a channel in a clean semiconductor. This unique property was used in
Ref. [13] for precise measurements of the viscosity in clean GaAs. The
measurements indicate deviations from the predictions of the conventional
theory of electron-electron scattering and this exciting observation requires
more careful studies.

There exists a generic problem in electron hydrodynamics, namely disentangling
Ohmic effects, boundary effects, and hydrodynamic effects in experimental
data. Viscous dissipation is proportional to the vorticity of the fluid, while
the Ohmic dissipation is independent of vorticity. To enhance hydrodynamics
one thus needs to enhance the vorticity. This might be achieved by generating
turbulent flow of the electron fluid, but the electron fluid is
extraordinarily viscous and therefore it is practically impossible to approach
turbulence.

In the laminar regime in a uniform magnetic field, vorticity is generated only
near the boundaries of the channel in a layer of width $\sim\sqrt{l_{ee}l_{\rm
mfp}}$. The flow in this “Knudsen layer” is influenced not only by the viscous
fluid effects, but also by the boundary conditions that are device specific
and typically not well known. Therefore it is hard to disentangle the unknown
boundary effects and unknown viscous effects.

In Ref. [13] the problem was partially resolved by using a gated channel in a
GaAs 2DEG that supports perfect-slip boundary conditions. This allowed for
relatively accurate measurements of the viscosity of the electron fluid.
However, the accuracy of the measurements was limited by Ohmic dissipation
which was more than an order of magnitude larger than the viscous one.

In Ref. [52] we suggested a method that involved the use of micromagnets to
create wiggling hydrodynamic flow and hence create vorticity throughout the
bulk of the sample, not just in the boundary region. This allows for a
dramatic enhancement of viscous hydrodynamic dissipation. Hydrodynamics in a
finite channel is inherently a boundary problem and in Ref. [52] the problem
was solved only for a straight channel in the unique case of perfect-slip
boundaries. In the present work we extend the method to a straight channel
with arbitrary boundary conditions, either no-slip or finite-slip boundaries.
Of course the bulk solution is the same, but the structure of the Knudsen
layer depends on the boundary condition and we determine the structure in the
present work. Hence, the present work extends the magneto-hydrodynamic method
to graphene and also to $\delta$-doped semiconductors.

The structure of this paper is as follows. In Sec. II we briefly describe the
solution for the bulk flow through a sinusoidal magnetic field found in Ref.
[52]. The difference from Ref. [52] is that we present numerical estimates for
graphene instead of GaAs. In Sec. III we study the boundary layer in a
straight channel using no-slip boundary condition and characterize the flow
near the boundary. We first solve the boundary conditions using perturbation
theory and then expand this idea to develop a numerically exact solution. The
key results of Sec. III are the width of the boundary layer and the emergence
of pseudo anti-Poiseuille flow at high magnetic field strengths. Understanding
the width of the boundary layer allows a lower limit to be placed on the
sample size in order to use the bulk solution for the “zigzag” flow. In Sec.
VI we calculate the magnetic field of periodic ferromagnetic stripes and hence
determine parameters of micromagnets necessary for the experimental set up
assuming monolayer graphene with experimentally realistic density and
mobility. Sec. VII presents our conclusions.

##  II Bulk Solution for Electron Hydrodynamics in a sinusoidal magnetic field

Consider a two-dimensional channel where $x$ is the longitudinal direction,
$y$ the transverse direction and $z$ is perpendicular to the plane. The
electron fluid may be controlled through the Lorentz force and thus obeys the
modified stationary Navier-Stokes equation at constant density [53],

$\displaystyle\mathbf{v}/\tau-\nu\nabla^{2}\mathbf{v}+\mathbf{v}\cdot{\nabla}\mathbf{v}$
$\displaystyle=-\nabla\Phi/m^{*}+q/m^{*}\mathbf{v}\times\mathbf{B},$
$\displaystyle\nabla\cdot\mathbf{v}$ $\displaystyle=0.$ (1)

Here, $\mathbf{v}$ is the fluid velocity field, $\nu$ is the kinematic
viscosity, $\tau$ is the relaxation time related to Ohmic processes
(scattering from impurities and phonons), $\Phi$ is the electro-chemical
potential, $q$ is the charge of the fluid particle ($q=-e<0$ for electrons),
$m^{*}$ is the effective hydrodynamic mass and $\mathbf{B}$ is the magnetic
field. For a uniform field, it is known that this equation predicts the usual
Hall effect, whereas recently it has been realized that non-uniform fields
have remarkable consequences such as a local suppression of flow [54] and
vortex layers in the bulk [52]. The electron-electron scattering length,
$l_{ee}$, and the momentum diffusive scattering length, $l_{\rm mfp}$, are
defined by the following relations

$\displaystyle\nu=\frac{l_{ee}v_{F}}{4}$ $\displaystyle\tau=\frac{l_{\rm
mfp}}{v_{F}},$ (2)

where $v_{F}$ is the Fermi velocity at zero temperature.

A superlattice of micromagnets modulated in the $x$ direction generates a
sinusoidal magnetic field along the $z$ direction in the plane of the 2DEG,

$\displaystyle B_{z}=B_{0}\sin(gx)\ ,$ (3)

where $g=2\pi/a$ and $a$ is the separation between adjacent micromagnets. As
previously mentioned, the electron fluid is very viscous, so the “Bernoulli
term”, $\mathbf{v}\cdot{\nabla}\mathbf{v}$, in the left hand side of Eq. (II)
can be neglected. Hence, we obtain a linear Stokes equation that can be solved
using the stream function $\psi$ defined by $v_{x}=\partial_{y}\psi$ and
$v_{y}=-\partial_{x}\psi$. For an infinitely long and infinitely wide channel
with fixed averaged current density $\mathbf{J}=qnv_{0}\mathbf{\hat{x}}$ the
solution is

$\psi_{\infty}=v_{0}\left(y+\frac{\omega\tau}{g(1+D^{2}g^{2})}\cos(gx)\right),$
(4)

where $D=\sqrt{\nu\tau}=\frac{1}{2}\sqrt{l_{ee}l_{\rm mfp}}$ is the
characteristic viscous length and $\omega=|e|B_{0}/m^{*}$ is the cyclotron
frequency. The viscous and Ohmic dissipation rates follow directly from Eq.
(4), were $L$ and $W$ are the length and the width of the channel,
respectively,

$\displaystyle\dot{E}_{\Omega}$ $\displaystyle=$
$\displaystyle\frac{nm^{*}}{\tau}\int d^{2}r\>v^{2},$
$\displaystyle\dot{E}_{\nu}$ $\displaystyle=$ $\displaystyle nm^{*}\nu\int
d^{2}r\>\mathbf{v}\cdot\nabla^{2}\mathbf{v}$ (5)

For the bulk solution (4) this gives

$\displaystyle\dot{E}_{\Omega}$ $\displaystyle=$
$\displaystyle\frac{nm^{*}v_{0}^{2}}{\tau}\
\left(1+\frac{\omega^{2}\tau^{2}}{2(1+D^{2}g^{2})^{2}}\right)LW,$
$\displaystyle\dot{E}_{\nu}$ $\displaystyle=$
$\displaystyle\frac{nm^{*}v_{0}^{2}}{\tau}\
\frac{g^{2}D^{2}\omega^{2}\tau^{2}}{2(1+D^{2}g^{2})^{2}}LW.$ (6)

The coefficient $nm^{*}v_{0}^{2}LW/\tau$ can be identified as the dissipation
due to the Drude resistance in a conventional conductor with diffusive
transport. The ratio of viscous to Ohmic dissipation is dependent on the
dimensionless parameters $\omega\tau$ and $gD$ which, for fixed $\omega\tau$,
may be shown to be maximized at $gD=[1+(\omega\tau)^{2}/2]^{1/4}$.

So far we have not explicitly solved for the electro-chemical potential in the
Stokes equation (II). Substituting $\mathbf{v}$ calculated from the stream
function (4), it is easy to derive an expression for the electro-chemical
potential, representing the external force on each fluid particle,

$\displaystyle\partial_{x}\Phi=-\frac{m^{*}v_{0}}{\tau}\left(1+\frac{\omega^{2}\tau^{2}\sin^{2}(gx)}{1+D^{2}g^{2}}\right),$
$\displaystyle\partial_{y}\Phi=0.$ (7)

The electro-chemical potential does not vary in $y$ in an infinite sample as
there is translational symmetry in the $y$ direction, whereas the periodic
structure of the magnetic field in the $x$ direction gives rise to a modulated
electro-chemical potential as a function of $x$. Furthermore, it is clear that
the rate of work done by the external force $\dot{E}=-\int
d^{2}r\>\mathbf{v}\cdot\nabla\Phi$ matches the total dissipation
$\dot{E}_{\Omega}+\dot{E}_{\nu}$ obtained in Eq. (II).

As the electron number density is approximately constant throughout the
sample, the thermodynamic pressure gradients vanish. Therefore, the
electrochemical potential gradients are predominantly electric in nature, and
we can write the voltage as $\varphi=-\Phi/|e|$,

$\displaystyle\varphi=\frac{m^{*}v_{0}}{|e|\tau}\left(x+\frac{\omega^{2}\tau^{2}}{2(1+D^{2}g^{2})}\left(x-\frac{\sin(2gx)}{2g}\right)\right).$
(8)

From Eq. (8) it is clear that the electric field varies with $x$. As this
effect arises from the presence of a periodic magnetic field and the
modulation in voltage is in the same direction as transport it is appropriate
to label this a “longitudinal Hall effect”.

Electron transport in high-mobility graphene may be hydrodynamic in the
temperature range $150\>\mathrm{K}\leq T<300\>\mathrm{K}$ [3]. To be specific
we will present numerical estimates for a monolayer graphene device with
mobility $\mu\approx 2\times 10^{5}$ cm2$/$Vs and density $n\approx 10^{12}$
cm-2 at temperature $T=220$ K with parameters similar to the device used in
Ref. [8],

$\displaystyle n=10^{12}\>\text{cm}^{-2}$ $\displaystyle T=220\>\text{K}$
$\displaystyle l_{\rm mfp}=1.9\>\mu\text{m}$ $\displaystyle
l_{ee}=0.24\>\mu\text{m}$ $\displaystyle\tau=1.9\>\text{ps}$ (9)

For a magnetic field amplitude $B_{0}=75$ mT this gives $\omega\tau=1.2$. Here
we keep in mind that the hydrodynamic mass for monolayer graphene is
$m^{*}=p_{F}/v_{F}$, with Fermi momentum $p_{F}=\hbar\sqrt{n\pi}$, accounting
for spin and valley degeneracy in graphene. With these parameters the
characteristic viscous length is $D\approx 0.34\>\mu\mathrm{m}$ and the
optimal wavevector of the micromagnet superlattice is $g=3.4$ $\mu$m-1, with
corresponding lattice spacing $a=1.9$ $\mu$m.

With these parameters and assuming $v_{0}=100$ m/s, corresponding to $I=0.3$
$\mu$A in a narrow $W=2$ $\mu$m channel from Ref. [8], the oscillation
amplitude of the longitudinal Hall effect is approximately $0.3$ $\mu$V.

##  III Boundary layer for a straight semi-infinite channel with no-slip
boundary condition

###  III.1 Boundary layer flow in the perturbative limit $\omega\tau\ll 1$

Consider a semi-infinite straight channel with a no-slip boundary at $y=0$ and
fluid flow in the region $y>0$. In conjunction with the Stokes equation (II),
the stream function must satisfy the no-penetration and no-slip boundary
conditions,

$v_{y}|_{y=0}=-\partial_{x}\psi|_{y=0}=0,\;v_{x}|_{y=0}=\partial_{y}\psi|_{y=0}=0.$
(10)

The no-penetration condition ensures that no fluid can cross the boundary
whereas the no-slip condition ensures that the fluid has zero tangential
velocity at the boundary surface. Furthermore, the stream function must decay
to the bulk solution (4) far from the boundary. We look for a solution as an
expansion in powers of $\omega\tau\ll 1$, i.e.,
$\psi=\sum_{n=0}^{\infty}(\omega\tau)^{n}\psi^{(n)}$. First we solve for the
zero-order solution, corresponding to flow in the absence of a magnetic field.
Solving the Stokes equation for $\omega\tau=0$ using the boundary conditions
leads to the Poiseuille solution

$\psi^{(0)}=v_{0}\left(y+De^{-y/D}\right).$ (11)

Next, introducing the magnetic field as a perturbation we derive the first-
order correction in the solution, $\psi=\psi^{(0)}+\omega\tau\psi^{(1)}$,

$\displaystyle\psi^{(1)}=v_{0}\cos(gx)\left(\frac{1}{g(1+g^{2}D^{2})}+Ae^{-y/D}\right.$
$\displaystyle+\left.C_{1}e^{-gy}+C_{2}e^{-\sqrt{g^{2}+\frac{1}{D^{2}}}y}\right),$
(12)

where the coefficients $A$, $C_{1}$ and $C_{2}$ are defined as

$\displaystyle A$ $\displaystyle=\frac{1}{g(1-g^{2}D^{2})},$ $\displaystyle
C_{1}$
$\displaystyle=\frac{2(g^{2}D^{2}+1)^{-\frac{1}{2}}-1}{g(1-g^{2}D^{2})(gD-\sqrt{g^{2}D^{2}+1})},$
$\displaystyle C_{2}$
$\displaystyle=\frac{1-gD}{g(1+gD)(1+g^{2}D^{2})(gD-\sqrt{g^{2}D^{2}+1})}.$

Note that in the case of $gD\to 1$ the coefficients $A,C_{1}\to\infty$.
However, it is easy to check that the divergences compensate each other such
that the solution remains finite.

To maximize the viscous dissipation we are interested in the case $gD>1$.
Since the slowest decaying boundary term in Eq. (III.1) decays as $e^{-y/D}$,
the characteristic viscous length $D$ sets the width of the boundary layer.

###  III.2 Exact numerical solution for the boundary layer flow

While the perturbative solution for the boundary layer presented in the
previous subsection is illuminating, for practical purposes we must find the
solution at $\omega\tau>1$. This can only be done numerically. To do so we
reduce the Stokes equation (II) to an infinite series of linear algebraic
equations. We then numerically solve these equations by truncating them at a
suitable order.

Figure 1: Streamlines in a semi-infinite channel with a no-slip boundary at
$y=0$. The parameters are chosen as $gD=1.16$, $\omega\tau=0.3,1.2,4.5$. Solid
lines correspond to the numerically exact solution described in Sec. III.2 and
dashed lines correspond to the solution given by Eq. (4) in the infinite
device.

We consider a semi-infinite channel $y\geq 0$ and use length units where
$D=\sqrt{\nu\tau}=1$ and velocity units where $v_{0}=1$. As the magnetic field
is periodic in the $x$ direction, so is the stream function $\psi$, which can
thus be presented as the following Fourier series,

$\displaystyle\psi=y+e^{-y}+\frac{\omega\tau}{g(g^{2}+1)}\cos(gx)$
$\displaystyle+\sum_{m=-\infty}^{\infty}e^{igmx}G_{m}(y)\ .$ (13)

Here $m$ is an integer and $G_{m}(y)$ are unknown functions that must decay
with increasing $y$ as the solution tends to the bulk solution far from the
boundary. Thus, using the perturbative solution as a starting point we
consider solutions of the form

$G_{m}(y)=f_{m}e^{-y}+\sum_{a}A_{m,a}e^{-\lambda_{a}y},$ (14)

where $f_{m}$ and $A_{m,a}$ are complex coefficients. The index $a=1,2,3...$
enumerates “eigenvalues” $\lambda_{a}$, which can be real or complex. The real
parts of $\lambda_{a}$ are positive by design of Eq. (14). In principle the
index $a$ can be arbitrarily large, but due to the truncation it is never
larger than $\sim 10$. Since the “Bernoulli term” is neglected, the Stokes
equation (II) is a linear equation. Let us consider the Poiseuille solution
without magnetic field,

$\displaystyle\psi_{0}=y+e^{-y}\ ,$ (15)

as a base solution. Now, in a magnetic field the stream function has a driven
part correction, $\delta\psi_{d}$, driven by $\psi_{0}$, and a free part
correction, $\delta\psi_{f}$,

$\displaystyle\psi$ $\displaystyle=\psi_{0}+\delta\psi_{d}+\delta\psi_{f},$
$\displaystyle\delta\psi_{d}$
$\displaystyle=\frac{\omega\tau}{g(g^{2}+1)}\cos(gx)+\sum_{m=-\infty}^{\infty}e^{igmx}f_{m}e^{-y},$
$\displaystyle\delta\psi_{f}$
$\displaystyle=\sum_{m=-\infty}^{\infty}e^{igmx}\sum_{a}A_{m,a}e^{-\lambda_{a}y}$
(16)

The first term in $\delta\psi_{d}$ is driven by the first term in Eq. (15) and
the second term in $\delta\psi_{d}$ is driven by the second term in Eq. (15).
Hence, the function $G_{m}(y)$ in Eq. (14) is a combination of a driven part
and a free part. For a linear differential equation the driven part is fully
determined by the equation, but the coefficients in the free part are
determined by the boundary conditions.

It is convenient to define the following differential operators

$\displaystyle{\hat{D}}$ $\displaystyle\equiv-\nabla^{2}+\nabla^{4},$
$\displaystyle{\hat{d}}$ $\displaystyle\equiv\omega\tau
g\cos(gx)\partial_{y}.$ (17)

Using these notations the Stokes equation can be rewritten as

$\displaystyle(\hat{D}-{\hat{d}})\delta\psi_{d}$
$\displaystyle=\hat{d}\psi_{0},$
$\displaystyle(\hat{D}-\hat{d})\delta\psi_{f}$ $\displaystyle=0$ (18)

Using the representation (III.2) the driven equation is transformed to an
infinite set of linear algebraic equations for the coefficients $f_{m}$

$(g^{2}m^{2})(g^{2}m^{2}-1)f_{m}+\frac{\omega\tau g}{2}(f_{m+1}+f_{m-1})\\\
=-\frac{\omega\tau g}{2}(\delta_{m,1}+\delta_{m,-1}).$ (19)

The coefficients in this equation are real and even in $m$. Therefore, the
coefficients $f_{m}$ are also real and even in $m$. For this reason only
equations for non-negative $m$ are independent. A numerical solution of Eq.
(19) is straightforward, with truncation beyond $m>N_{m}$, resulting in
$N_{m}+1$ independent equations. The second equation in Eq.(III.2), i.e., the
free equation, is transformed to an infinite set of linear algebraic equations
for the coefficients $A_{m}$,

$\displaystyle(g^{2}m^{2}-\lambda_{a}^{2})(1+g^{2}m^{2}-\lambda_{a}^{2})A_{m,a}$
$\displaystyle+\frac{\omega\tau g}{2}\lambda_{a}(A_{m+1,a}+A_{m-1,a})=0,$ (20)

noting that $\lambda_{a}$ is independent of $m$. Again, only equations with
non-negative m are independent and we truncate $m$ such that $m\leq N_{m}$.
Hence the number of independent equations is again $N_{m}+1$. The right hand
side of Eq. (III.2) is zero, so for a nontrivial solution the determinant of
the matrix must be equal to zero. Hence, this is an eigenvalue problem but it
is not a standard linear eigenvalue problem. To solve the eigenvalue problem
we calculate the determinant as a polynomial in $\lambda$ and numerically
determine the roots of the polynomial. The polynomial order is $4(N_{m}+1)$.
After we discard roots with non-positive real part we are left with $2N_{m}+1$
roots. Interestingly, at a sufficiently large $\omega\tau$ there are complex
roots. The coefficients in the polynomial are real, so complex roots always
come in complex conjugate pairs. We enumerate $2N_{m}+1$ roots by the index
$a$. For any found $\lambda_{a}$ we solve Eqs. (III.2) and hence find the
coefficients $A_{m,a}$, numerically determined to arbitrary normalization. We
now write $A_{m,a}=\alpha_{a}A_{m,a}^{(n)}$, where $A_{m,a}^{(n)}$ is
normalized such that $\sum_{m}|A_{m,a}^{(n)}|^{2}=1$, and $\alpha_{a}$ is an
arbitrary coefficient.

The boundary conditions (10) provide a set of equations to determine the
$2N_{m}+1$ coefficients $\alpha_{a}$

$\displaystyle 0=v_{y}|_{y=0}\Longrightarrow$ $\displaystyle
m(f_{m}+\sum_{a}\alpha_{a}A_{m,a}^{(n)})+\frac{\omega\tau}{2g(1+g^{2})}(\delta_{m,1}-\delta_{m,-1})=0,$
$\displaystyle 0=v_{x}|_{y=0}\Longrightarrow$ $\displaystyle
f_{m}+\sum_{a}\lambda_{a}\alpha_{a}A_{m,a}^{(n)}=0.$ (21)

The first of these equations is nontrivial at $m=1,2...N_{m}$ and the second
equation is nontrivial at $m=0,1...N_{m}$, so altogether we have $2N_{m}+1$
equations to determine $2N_{m}+1$ unknown coefficients $\alpha_{a}$. Thus the
boundary layer problem is solved.

The boundary layer solution considered in this Section is valid for the no-
slip boundary condition. However, the method can be easily extended to the
case of an arbitrary slip length, considering that only Eqs. (III.2) depend on
the slip length.

The boundary layer solution is extended to finite width channels in App. A.
After the method is developed it is instructive to present plots for specific
cases. For the numerical solution we set $gD=1.16$, which corresponds to the
device described at the end of Sec. II. We truncate beyond $N_{m}=5$, which
provides more than sufficient accuracy for the values of $\omega\tau$ we
present. Plots of the streamlines obtained for $\omega\tau=0.3$, $1.2$ and
$4.5$ are presented in Fig. 1. These values of $\omega\tau$ were selected as
$\omega\tau=0.3$ has only real eigenvalues $\lambda_{a}$ and corresponds to
the perturbative regime, $\omega\tau=1.2$ has a complex conjugate pair of
eigenvalues $\lambda_{a}$ and corresponds to the device described in Eq. (II)
and $\omega\tau=4.5$ has two complex conjugate pairs of eigenvalues
$\lambda_{a}$. As the magnetic field increases ($\omega\tau$ increases) both
the magnitude of the boundary effects and the width of the boundary layer
increase, but in the cases considered the effective width of the boundary
layer does not extend beyond $y/D\approx 4$. Thus, considering symmetry, it is
appropriate to use the bulk solution in devices with width $W\gg 8D$.

##  IV Numerical estimates, anti-Poiseuille flow

Following the numerical calculation of the stream function, we may calculate
the the viscous dissipation rate and the Ohmic dissipation rate given by Eqs.
(II). The most important parameter is the ratio
$\dot{E}_{\nu}/\dot{E}_{\Omega}$, which is dependent on the device considered,
so to provide calculated values we take the device with parameters presented
in Eq. (II). This corresponds to $D=0.34\>\mu\text{m}$. We also take the
lattice spacing $a=1.9\>\mu\text{m}$ as discussed in the end of Section II,
resulting in $gD=1.16$. The ratio $\dot{E}_{\nu}/\dot{E}_{\Omega}$ is plotted
in Fig. 2 versus the channel width $W$, for two values of the magnetic field
amplitude, $B_{0}=75$ mT,$150$ mT, corresponding to $\omega\tau=1.2$, $2.4$
respectively. In these plots we assume that the viscosity is independent of
the magnetic field, $\nu(B)=\nu(0)$. Asymptotic values of the ratio
$\dot{E}_{\nu}/\dot{E}_{\Omega}$ at $W\to\infty$ are $0.16$ ($B_{0}=75$ mT)
and $0.48$ ($B_{0}=150$ mT).

Figure 2: Ohmic (blue line) and viscous (red line) dissipation as a function
of the width of the channel normalized to the $\dot{E}_{0}$ which is due to
Drude resistance only, i.e., what is expected in the absence of B-field and
hydrodynamics ($\dot{E}_{0}/LW=nm^{*}v_{0}^{2}/\tau$). The value
$\omega\tau=1.2$ corresponds to $B_{0}=75$ mT and $\omega\tau=2.4$ corresponds
to $B_{0}=150$ mT. In these plots we assume that the viscosity is independent
of the magnetic field, $\nu(B)=\nu(0)$. The plots correspond to the parameter
set (II), where $gD=1.16$.

However, the viscosity depends on magnetic field, with possible dependence
[10]

$\displaystyle\nu\propto\frac{1}{1+(B/B^{*})^{2}}\;,\;B^{*}=\frac{p_{F}}{2|e|l_{ee}}$
(22)

For the device parameters (II) the characteristic magnetic field is
$B^{*}\approx 240\>\text{mT}$. Therefore, for $B_{0}=150$ mT the approximation
$\nu(B)=\nu(0)$ is questionable, but the possible correction does not exceed
10-20%. The viscous dissipation comes from the regions of maximum vorticity
and in this setup the vorticity is maximum at the regions where $B\to 0$, so
the true correction for $B_{0}=150$ mT is lower than that given simply by Eq.
(22). It is not a problem to solve the Navier-Stokes equation with position
dependent viscosity, but it is not necessary at this stage.

Interestingly, the fluid flow in the Knudsen layer shows a sort of anti-
Poiseuille behaviour. To illustrate this, we plot in Fig. 3 the longitudinal
velocity $v_{x}$ versus $y$ (i.e., across the channel). The velocity $v_{x}$
is a function of both $x$ and $y$ and here we present plots at $x=0$. At a
sufficiently large magnetic field, $\omega\tau\geq 2.4$, there is a clear
anti-Poiseuille profile near the left boundary of the channel. In this regime
the peak velocity is not $v_{0}$ in the bulk, but instead the longitudinal
velocity has a maximum in the boundary layer before decaying towards $v_{0}$
in the bulk. There is a conventional Poiseuille behaviour at the right
boundary of the channel, although at high magnetic field a small amount of
counterflow develops due to the development of vortices in the boundary layer.
If we shift $x=0\to x=a/2$ the plots are effectively reflected about the
centre of the channel: the anti-Poiseuille profile is located at the right
boundary and the Poiseuille profile is at the left boundary. Mathematically,
the anti-Poiseuille profile is related to the complex eigenvalues
$\lambda_{a}$. This behaviour is qualitatively similar to the anti-Poiseuille
flow predicted for narrow graphene channels near charge neutrality in the
presence of a uniform magnetic field perpendicular to the plane [49, 50, 51].
Near charge neutrality this effect is largely attributed to electron-hole
recombination. In our case the physical origin of the anti-Poiseuille profile
is different as we do not have any holes. While we consider no-slip
boundaries, a similar anti-Poiseuille profile exists also for boundary
conditions with a finite slip length. The location of the anti-Poiseuille
maximum may be used to extract the slip length in a sample with finite-slip
boundaries.

Figure 3: Longitudinal velocity profile at $x/D=0$ with $gD=1.16$ for various
values of $\omega\tau$. At large magnetic fields anti-Poiseuille flow is
observed and a small counterflow is observed at the opposite boundary due to
the formation of a viscous vortex.

##  V Finite-elements simulations

To complement our numerical results and to extend them towards finite systems,
we also performed numerical simulations using the finite-element method (FEM).
The first step is to bring Eq. (II) to the form of a weak differential
equation by multiplying each of the equation with a test function and
integrating over all space. We again neglect the Bernoulli term because we
assume small flow velocity and assume the fluid to be incompressible. For the
finite-element analysis, we use the space of Lagrange polynomials of second
order as the function space for the velocity field and the space of Lagrange
polynomials of first order as the function space for the potential. Together
these form a Taylor-Hood element appropriate for the numerical solution of
Navier-Stokes equations. We have numerically implemented the problem using the
FEniCS package [55].

Using FEM, we consider a finite system with a rectangular geometry with width
$W$ in $y$ direction and length $L$ in $x$ direction. To simulate a finite
array of micromagnets, we consider a flow driven by an applied electric
potential difference, so we assume Dirichlet boundary conditions for the
voltage $\varphi(x=-L/2,y)=\varphi_{L}$ and $\varphi(x=L/2,y)=\varphi_{R}$ at
inlet and outlet. In addition to this, we assume that $v_{y}(x=\pm L/2,y)=0$
at both inlet and outlet. Finally, we impose no-slip boundary conditions at
the walls, i.e., $\mathbf{v}(x,y=\pm W/2)=0$.

Figure 4: FEM results for the absolute value $|\mathbf{v}|$ of the velocity in
the channel for $gD=0.5$. Note that this color plot should not be confused
with lines of flow, which are shown in Fig. 1 The velocity is measured in
units convenient for numerical solution and the system contains six magnetic
strips. The region of high velocity, anti-Poiseuille flow, is shifted by half
a period at opposite boundaries.

Figure 4 shows that the results of these simulations qualitatively reproduce
all features found for the infinite system. At strong enough magnetic fields,
an anti-Poiseuille flow emerges near the boundary. Moreover, at strong fields,
a small counterflow becomes visible at the opposite boundary.

##  VI Magnetic field of periodic ferromagnetic stripes

The setup considered in this paper is similar to the setup previously proposed
for GaAs heterostructure in Ref. [52], but adapted to suit monolayer graphene
with parameters described in Eq.(II). A superlattice of rectangular magnetic
bars oriented perpendicular to the longitudinal axis of the device is placed
above the 2DEG as seen in the left panel of Fig. 5.

Figure 5: Left: top view of a periodic array of magnetic bars perpendicular to
the flow direction shown by the red arrow. The period of the array is $a$ and
the width of the bar is $d$. The bars are magnetized in parallel to the
longitudinal axis of the channel. The direction of the magnetization is shown
by the blue arrows. Right: Side view of the device. The vertical distance from
the bar magnets to the 2DEG is $z$ and the thickness of the rectangular bar is
$\delta z$.

The bars are magnetized along the longitudinal axis of the channel. The bar
thickness is $\delta z$ and the distance from the base of the bar to the 2DEG
is $z$, shown in the right panel of Fig. 5. The period of modulation is
$a=1.9$ $\mu$m as found in Sec. II. We tune the remaining free parameters
$d,z,\delta z$ such that the $z$ component of the magnetic field in the 2DEG
is approximately $B_{0}\sin(2\pi x/a)$ with $B_{0}\approx 75$ mT. A stronger
magnetic field may be also achieved.

Assuming the magnetization is uniform throughout the volume of each bar, and
the bars’ extent in the transverse ($y$) direction is wider than the width $W$
of the sample, we calculate the magnetic field in the plane of the 2DEG as a
function of longitudinal coordinate $x$ from the Biot-Savart law as

$\displaystyle\frac{B_{z}(x)}{B_{m}}\bigg{|}_{\rm 2DEG}$
$\displaystyle=-\sum_{n>1}\frac{\sin\left(gnd/2\right)}{\pi n}\sin(gnx)Z_{n},$
$\displaystyle Z_{n}(z,\delta z)$ $\displaystyle=e^{-gnz}(1-e^{-gn\delta z}),$
(23)

where $B_{m}$ is the saturation field, and $x$ is measured from the center of
one of the magnets. For example, at the edge of the magnet $x=d/2$, the
harmonics of the field in the 2DEG plane is $\propto\sin^{2}(gnd/2)$.
Therefore, $d=a/2=\pi/g$ maximizes the first harmonic and suppresses the
second harmonic. We numerically calculate $B$ from Eq. (VI) and further
confirm that $d=a/2=0.93\>\mu$m is optimal.

In order to achieve a field with magnitude $B_{0}=75$ mT we consider a NiFe
alloy that has saturation magnetic field $1500$ mT. [56] Through concurrently
tuning $\delta z$ and $z$ we find that we may generate the desired magnetic
field profile with $\delta z=0.15$ $\mu$m and $z=0.30$ $\mu$m. Plots of
$B_{z}$ versus $x$ and $B_{x}$ versus $x$ are presented in Fig. 6.

Figure 6:  Magnetic field in the 2DEG plane for NiFe bars. Left panel: $B_{z}$
versus $x$. Right panel: $B_{x}$ versus $x$. The parameters are $\delta
z=0.15\>\mu\mathrm{m},\ z=0.30\>\mu\mathrm{m}$.

The $x$ component of the magnetic field is not relevant, but we present it for
completeness. The $x$ dependence of $B_{z}$ is not quite sinusoidal, however
this is not a big issue as using the finite element method one can also solve
the Navier-Stokes equation for this magnetic field profile.

##  VII Conclusions

We develop a theory for the no-slip boundary layer in the magnetohydrodynamic
flow of an electron fluid in a straight channel. The fluid flow is modified by
an array of micromagnets on the top of two-dimensional electron gas. The
micromagnets create a wiggling flow of the fluid, which dramatically enhances
hydrodynamic effects in the fluid. To be specific, the analysis was performed
for no-slip boundary conditions, but the method can be extended to boundary
conditions with arbitrary slip length. We demonstrate that the fluid velocity
profile in the boundary layer manifests an anti-Poiseuille behaviour. In
addition, we show that the micromagnet array creates a longitudinal Hall
effect in the sample. From the experimental point of view we propose a method
for a precise measurements of the properties of different electron fluids. The
results are applicable to graphene away from the charge neutrality point and
also to semiconductors.

##  Acknowledgements

We acknowledge important discussions with Alexander Hamilton, Yonatan Ashlea
Alava, Oleh Klochan, Daisy Wang and Zeb Krix. This work was supported by the
Australian Research Council Centre of Excellence in Future Low- Energy
Electronics Technologies (CE170100039).

##  Appendix A The method for exact solution for a finite width channel

Here we expand the method of Sec.III to a finite channel with no slip
boundaries at $y=\pm W/2$. Here we use units $D=v_{0}=1$. As before we split
the stream function into the zero field term which corresponds to Poiseuille
flow, $\psi_{0}$, the driven magnetic field correction term, $\delta\psi_{f}$
and the free magnetic field correction term $\delta\psi_{d}$.

$\displaystyle\psi$ $\displaystyle=\psi_{0}+\delta\psi_{d}+\delta\psi_{f},$
$\displaystyle\psi_{0}$ $\displaystyle=y-\frac{\sinh{y}}{\cosh{W/2}}$
$\displaystyle\delta\psi_{d}$
$\displaystyle=\frac{\omega\tau}{g(g^{2}+1)}\cos(gx)+\sum_{m=-\infty}^{\infty}e^{igmx}\left(f_{m}e^{-y}+h_{m}e^{y}\right),$
$\displaystyle\delta\psi_{f}$
$\displaystyle=\sum_{m=-\infty}^{\infty}e^{igmx}\sum_{a}\left(A_{m,a}e^{-\lambda_{a}y}+C_{m,a}e^{\lambda_{a}y}\right)$
(24)

Note that we absorb the constant exponential terms into the unknown
coefficients. The stream function solves the Stokes’ equation as in
Eq.(III.2). From the driven equation we obtain a set of equations for
coefficients $f_{m}$ and $h_{m}$.

$\displaystyle(g^{2}m^{2})(g^{2}m^{2}-1)f_{m}+\frac{\omega\tau
g}{2}(f_{m+1}+f_{m-1})$ (25) $\displaystyle=-\frac{\omega\tau
g}{4\cosh{W/2}}(\delta_{m,1}+\delta_{m,-1})$
$\displaystyle(g^{2}m^{2})(g^{2}m^{2}-1)h_{m}-\frac{\omega\tau
g}{2}(h_{m+1}+h_{m-1})$ $\displaystyle=-\frac{\omega\tau
g}{4\cosh{W/2}}(\delta_{m,1}+\delta_{m,-1})$

As the coefficients are real and even in $m$, $f_{m}$ and $h_{m}$ are also
real and even in $m$. The free equation gives eigenequations for eigenvalues
$\lambda_{a}$ and eigenvectors $A_{m,a}$ and $C_{m,a}$.

$\displaystyle(g^{2}m^{2}-\lambda_{a}^{2})(1+g^{2}m^{2}-\lambda_{a}^{2}))A_{m,a}$
$\displaystyle+\frac{\omega\tau g}{2}\lambda_{a}(A_{m+1}+A_{m-1})=0$
$\displaystyle(g^{2}m^{2}-\lambda_{a}^{2})(1+g^{2}m^{2}-\lambda_{a}^{2}))C_{m,a}$
$\displaystyle-\frac{\omega\tau g}{2}\lambda_{a}(C_{m+1}+C_{m-1})=0$ (26)

Note that the determinant of the $A$-equation is identical to that of the
$C$-equation. Hence, there is only one set of eigenvalues $\lambda_{a}$ which
we find following the method described in the main text. The eigenvectors
$A_{m,a}$ and $C_{m,a}$ are different and we find them solving Eqs. (A). The
eigenvectors are determined up to some arbitrary normalization. Similarly to
the approach in Sec.III, we define coefficients $\alpha_{a}$ and $\gamma_{a}$
and write $A_{m,a}=\alpha_{a}A_{m,a}^{(n)}$ and
$C_{m,a}=\gamma_{a}C_{m,a}^{(n)}$, where
$\sum_{m}\left|A_{m,a}^{(n)}\right|^{2}=1$ and
$\sum_{m}\left|C_{m,a}^{(n)}\right|^{2}=1$, and $\alpha_{a}$ and $\gamma_{a}$
are arbitrary coefficients. These coefficients are determined by considering
the boundary conditions at $y=\pm\frac{W}{2}$.

$\displaystyle 0=v_{y}|_{y=\pm W/2}\to$ $\displaystyle m(f_{m}e^{\mp
W/2}+h_{m}e^{\pm W/2}+\sum_{a}(\alpha_{a}A_{m,a}^{(n)})e^{\mp\lambda_{a}W/2}$
$\displaystyle+\gamma_{a}C_{m,a}^{(n)}e^{\pm\lambda_{a}W/2}))+\frac{\omega\tau}{2g(1+g^{2})}(\delta_{m,1}-\delta_{m,-1})=0$
$\displaystyle 0=v_{x}|_{y=\pm W/2}\to$
$\displaystyle\sum_{a}\lambda_{a}(\alpha_{a}A_{m,a}^{(n)}e^{\mp\lambda_{a}W/2}-\gamma_{a}C_{m,a}^{(n)})$
$\displaystyle+f_{m}e^{\mp W/2}-h_{m}e^{\pm W/2}+e^{\pm\lambda_{a}W/2}=0$ (27)

Upon solving Eq. (A) and obtaining $\alpha_{a}$ and $\gamma_{a}$ the stream
function is entirely characterized and velocity and dissipation profiles may
be calculated.

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