# Unraveling the Hidden Environmental Impacts of AI Solutions for Environment
Life Cycle Assessment of AI Solutions

Anne-Laure Ligozat  Univ. Paris-Saclay, LIMSI, CNRS, ENSIIEOrsayFrance ,
Julien Lefèvre  Univ. Aix-Marseille, CNRS, Centrale MarseilleMarseilleFrance ,
Aurélie Bugeau  Univ. Bordeaux, Bordeaux INP, CNRS, Laboratoire
LaBRITalenceFrance and  Jacques Combaz  Universite Grenoble Alpes,
VERIMAGGrenobleFrance

###### Abstract.

In the past ten years, artificial intelligence has encountered such dramatic
progress that it is now seen as a tool of choice to solve environmental issues
and in the first place greenhouse gas emissions (GHG). At the same time the
deep learning community began to realize that training models with more and
more parameters requires a lot of energy and as a consequence GHG emissions.
To our knowledge, questioning the complete net environmental impacts of AI
solutions for the environment (AI for Green), and not only GHG, has never been
addressed directly. In this article, we propose to study the possible negative
impacts of AI for Green. First, we review the different types of AI impacts,
then we present the different methodologies used to assess those impacts, and
show how to apply life cycle assessment to AI services. Finally, we discuss
how to assess the environmental usefulness of a general AI service, and point
out the limitations of existing work in AI for Green.

##  1\. Introduction

In the past few years, the AI community has begun to address the environmental
impacts of deep learning programs: (Strubell et al.,, 2019) highlighted the
impacts of training NLP models in terms of energy consumption and in terms of
carbon footprint, (Schwartz et al.,, 2020) proposed the concept of Green AI,
and the AI community created several tools to evaluate machine learning energy
consumption (Anthony et al.,, 2020; Henderson et al.,, 2020; Lacoste et al.,,
2019; Lannelongue et al.,, 2020).

These impacts are mainly expressed in terms of energy consumption and
associated greenhouse gas (GHG) emissions. Yet, as we will discuss later, this
energy consumption represents only a part of the complete environmental
impacts of such methods. (Dilhac et al.,, 2018) for example states that ”it is
in terms of their indirect effects on the global digital sector that AI
systems will have a major impact on the environment”. In the same spirit,
(Walsh et al.,, 2020) warns that ”optimising actions for a restricted set of
parameters (profit, job security, etc) without consideration of these wider
impacts can lead to consequences for others, including one’s future self as
well as future generations”.

Evaluating the impacts of an AI service is not fundamentally different from
doing it for another digital service. However, AI presents specificities, that
must be taken into account because they increase its environmental impacts.

First, AI - and in particular deep learning - methods usually require large
quantities of data. These data have to be acquired, transferred, stored and
processed. All these steps require equipment and energy, and have
environmental impacts. In the case of a surveillance satellite, the data will
probably be in large quantities, but the number of acquisition devices may be
limited; in the case of a smart building infrastructure, the data may be in
smaller quantities, but many devices will be required.

Training deep neural models also takes a lot of computation time and
resources, partly because the model itself learns a comprehensive
representation that enables it to better analyze the data. Whereas with other
models, a human will provide part of this information, often in the form of a
handcrafted solution. The computation cost can be even higher if the model
does continuous learning.

At the same time, AI’s popularity is increasing and AI is often presented as a
solution to environmental problems with AI for Green proposals (Rolnick et
al.,, 2019; Vinuesa et al.,, 2020; Gailhofer et al.,, 2021). The negative
environmental impacts can be briefly evoked - and in particular rebound
effects (Rolnick et al.,, 2019; Wu-2021) where unitary efficiency gains can
lead to global GHG increase - but no quantification of all AI’s environmental
costs is proposed to close the loop between AI for Green and Green AI. That is
why it is even more important to be able to assess the actual impacts, taking
into account both positive and negative effects.

Incidentally those works often use the term AI to actually refer to deep
learning methods, even though AI has a much wider scope with at least two
major historical trends (Cardon et al.,, 2018). In this paper, we will also
focus on deep learning methods, which pose specific environmental issues, and
as we have seen, are often presented as possible solutions to environmental
problems. We describe these impacts and discuss how to take them into account.

Our contributions are the following:

  * •

We review the existing work to assess the environmental impacts of AI and show
their limitations (Sections 2.1 and 2.2).

  * •

We present life cycle assessment (Section 2.3) and examine how it can
comprehensively evaluate the direct environmental impacts of an AI service
(Section 3).

  * •

We discuss how to assess the environmental value of an AI service designed for
environmental purposes (Section 4).

  * •

We argue that although improving the state of the art, the proposed
methodology can only show the technical potential of a service, which may not
fully realize in a real-life context (Section 5).

##  2\. Related work

This section reviews existing tools for evaluating environmental impacts of AI
as well as green applications of AI. It ends with an introduction to life
cycle assessment, a well-founded methodology for environmental impact
assessment but still not used for AI services.

###  2.1. Carbon footprint of AI

Strubell et al. (Strubell et al.,, 2019) has received much attention because
it revealed a dramatic impact of NLP algorithms in the training phase: the
authors found GHG emissions to be equivalent to 300 flights between New York
and San Francisco. Premises of such an approach were already present in (Li et
al.,, 2016) for CNN with less meaningful metrics (energy per image or power
with no indications on the global duration).

In (Schwartz et al.,, 2020) the authors observe a more general exponential
evolution in deep learning architecture parameters. Therefore they promote
”Green AI” to consider energy efficiency at the same level as accuracy in
training models, and recommend in particular to report floating-point
operations. Other authors (García-Martín et al.,, 2019) have also reviewed all
the methods to estimate energy consumption from computer architecture. They
distinguish between different levels of description, software/hardware level,
instruction/application level and they consider how those methods can be
applied to monitor training and inference phases in machine learning.

In the continuity of (Strubell et al.,, 2019) and (Schwartz et al.,, 2020),
several tools have been proposed to make the impacts of training models more
visible. They can be schematically divided into

  * •

_Integrated tools_ , such as Experiment Impact Tracker
111https://github.com/Breakend/experiment-impact-tracker (Henderson et al.,,
2020), Carbon Tracker 222https://github.com/lfwa/carbontracker (Anthony et
al.,, 2020) and CodeCarbon 333https://codecarbon.io/, which are all Python
packages reporting measured energy consumption and the associated carbon
footprint.

  * •

_Online tools_ , such as Green Algorithms 444http://www.green-algorithms.org/
(Lannelongue et al.,, 2020) and ML CO2 impact
555https://mlco2.github.io/impact/#compute (Lacoste et al.,, 2019), which
require only a few parameters, such as the training duration, the material,
the location but are less accurate.

AI literature mostly addresses a small part of direct impacts and neglects
production and end of life, thus not following recommendations such as
(ITU-T,, 2014). In (Gupta et al.,, 2020; Wu-2021) the authors point out the
methodological gaps of the previous studies focusing on the use phase. In
particular, manufacturing would account for about $75\%$ of the total
emissions of Apple or of an iPhone 5, just to give examples of various scales.
Their study is based on a life cycle methodology, relying on sustainability
reports with the GHG protocol standard. (Ligozat and Luccioni,, 2021) provides
a list of the carbon emission sources of an AI service, which gives a more
comprehensive view of the direct impacts in terms of carbon footprint only.
(kaack:hal-03368037) also advocates the need for taking indirect impacts
(e.g., behavioral or societal changes due to AI) into account when evaluating
AI services.

Some works focus on optimizing the AI processes regarding runtime, energy
consumption, or carbon footprint. For example, in (Patterson et al.,, 2021)
the authors update the results from (Strubell et al.,, 2019) and reveal a
considerable reduction of the GHG impact - by a factor of 100 - if one
considers the location of the data center used for training (low-carbon
energy) and the architecture of the deep network (sparsity). Nevertheless as
they recognize, their study evaluates the GHG emissions of operating computers
and data centers only and limits the perimeter by excluding the production and
the end-of-life phases of the life cycle. Their work also considers a highly
optimized use case, which may not be representative of real case scenarios.
The energy efficiency of machine learning has also been the subject of
dedicated workshops 666Workshop on Energy Efficient Machine Learning and
Cognitive Computing, https://www.emc2-ai.org/virtual-21.

###  2.2. AI for Green benefits

When designing an AI for Green method i.e., a method using AI to reduce energy
consumption or to benefit other environmental indicators, complete AI’s
impacts should also be considered to build meaningful costs/benefits
assessments. (Bommasani et al.,, 2021) proposes a framework for such cost-
benefit analysis of AI foundation models to evaluate environmental and
societal trade-offs. We discuss this framework in Section 4. Most AI solutions
for the environment lack a rigorous evaluation of the cost/benefit balance,
and one of our contributions is to advance this issue.

###  2.3. Life cycle assessment

LCA is a widely recognized methodology for environmental impact assessment,
with ISO standards (ISO 14040 and 14044) and a specific methodology standard
for ICT from ETSI/ITU (ITU-T,, 2014). It quantifies multiple environmental
criteria and covers the different life cycle phases of a target system.
(Hauschild et al.,, 2018) clearly states that ”to avoid the often seen problem
shifting where solutions to a problem creates several new and often ignored
problems, these decisions must take a systems perspective. They must consider
[…] the life cycle of the solution, and they need to consider all the relevant
impacts caused by the solution.” The LCA theoretical approach exposed in
(Heijungs and Suh,, 2002) describes the system of interest as a collection of
building blocks called _unit processes_ , for example ”Use phase of the
server” on which the model is trained. The set of all unit processes is called
the _technosphere_ , as opposed to the _ecosphere_. Each unit process can be
expressed in terms of _flows_ of two kinds:

  * •

_Economic flows_ are the directed links between the unit processes or said
differently exchanges inside the technosphere.

  * •

_Ecnvironmental flows_ are the links from the biosphere to the technosphere or
vice versa.

The detailed description of such a system is called the life cycle inventory
(LCI) 777We include the complete LCI of a generic AI service in the
supplementary material to the paper. and it can be formulated in terms of
linear algebra. The goal of a life cycle assessment consists in computing the
sum of the environmental flows of the system associated with a _functional
unit_. To be concrete, if one considers a heating system in a smart building,
the functional unit could be ”heating 1m2 to 20∘C for one year”.

Of course, very often, the LCI does not correspond exactly to the functional
unit. The size of economic flows may not match (e.g., the functional unit may
partially use shared servers and sensors), and a process may be
_multifunctional_ , i.e., producing flows of different types at the same time
(e.g., storage capacity and computational power). Both these problems can be
solved using for instance _allocation_ methods according to a _key_. A typical
allocation key for network infrastructures would be the volume of data. For a
data center it could be the economic value of storage and computational
services when they cannot be physically isolated.

Even though LCA is widely used in many domains, it has rarely been applied to
AI services.

##  3\. Life cycle assessment of an AI solution

When it comes to quantifying the impacts of digital technologies and in
particular AI technologies, one faces several methodological choices that
deserve a specific definition of the studied system. For instance, assessing
the global impacts of the AI domain - if we could circumscribe it precisely -
is not the same as assessing the impacts of an AI algorithm or service. The
emerging field of AI’s impacts quantification still suffers from a lack of
common methodology, and in particular it very often focuses only on the Use
phase of devices involved in an AI service. To perform meaningful
quantification, we strongly suggest following the general framework of life
cycle assessment (LCA, detailed in 3.2). We will show how it can be adapted to
an AI service i.e., in this case a deep learning code used either alone or in
a larger application.

AI being part of the Information and Communication Technology (ICT) sector,
and following the taxonomies from (Hilty and Hercheui,, 2010; Horner et al.,,
2016), its impacts can be divided into first-, second- and third-order
impacts. In this section we focus only on first-order impacts while we will
discuss second and third orders in Sections 4 and 5.

We will use the term _AI service_ for all the equipment (sensors, servers…)
used by the AI, and the term _AI solution_ for the complete application using
AI. In the case of the smart building, the _AI solution_ is the smart building
itself, while the _AI service_ is the digital equipment needed for the smart
infrastructure.

###  3.1. First-order impacts of an AI service

_First-order_ \- or _direct_ \- impacts of the AI service are the impacts due
to the different life cycle phases of the equipment:

  * •

_Raw material extraction_ , which encompasses all the industrial processes
involved in the transformation from ore to metals;

  * •

_Manufacturing_ , which includes the processes that create the equipment from
the raw material;

  * •

_Transport_ , which includes all transport processes involved, including
product distribution;

  * •

_Use_ , which includes mostly the energy consumption of equipment while it is
being used;

  * •

and _End of life_ , which refers to the processes to dismantle, recycle and/or
dispose of the equipment.

For simplicity reasons, we will merge the first three items into a single
_production_ phase in the rest of the paper.

For example an AI solution in a smart building may need sensors and servers
that require resources and energy for their production, operation and end of
life.

Figure 1. LCA dimensions: the first dimension corresponds to the phases of
life cycle, the second one to the environmental impacts (see 3 for more
details on this last dimension).

A second dimension is necessary to assess the impacts, a set of environmental
criteria considered. Indeed each life cycle phase has impacts on different
environmental indicators: Greenhouse Gases emissions (usually expressed as
Global Warming Potential, GWP), water footprint, human toxicity, or abiotic
resource depletion (ADP) for instance. In general, evaluating the
environmental impact of a service requires multiple impacts criteria (ITU-T,,
2014). ISO states that ”the selection of impact categories shall reflect a
comprehensive set of environmental issues related to the product system being
studied, taking the goal and scope into consideration”. Additionally, ”the
selection of impact categories, indicators and models shall be consistent with
the goal and scope of the LCA study”. Hence, the costs must take into account
at least the criteria that are supposed to be tackled by the AI solution in
the case of AI for Green: if the AI solution is applied to reduce energy
consumption for example, the main expected gain will probably be in terms of
carbon footprint, so at least the carbon footprint of using the model should
be considered. For an application monitoring biodiversity, the most relevant
criterion may be natural biotic resources (and not carbon footprint), which
include wild animals, plants etc.

Figure 1 sums ups these two dimensions. As it has been previously stated, in
the literature, only part of the global warming potential due to the use phase
has generally been considered when evaluating AI, which corresponds to the
shaded area in the figure.

###  3.2. Life cycle assessment methodology for AI

In this section, we focus on life cycle assessment of the AI solution, and the
associated ICT equipment. We aim at proposing a methodology for applying the
general framework of LCA to AI services. For LCA of all other processes, we
refer to LCA standards and (Hauschild et al.,, 2018) for example. In order to
concretely apply the methodology presented for an AI service, we use the ITU
recommendation (ITU-T,, 2014) for environmental evaluation of ICT.

Figure 2 shows two sides of the Life cycle of an AI service. The top part of
this figure shows the different tasks involved in an AI service, from a
software point of view (data acquisition, …, inference). For each task, one or
several devices is used. The bottom part of the figure shows the life cycle
phases of each of these devices, from a hardware point of view. The
environmental impacts of the AI service will stem from the life cycle phases
of the devices. Note that all devices involved in the AI tasks should be taken
into account.

Remark on terminology: In the paper, the term ”Use phase” refers to the use
phase of the life cycle of equipment, corresponding to the devices provided
for the AI service (box ”Use of device” of the lower part in Figure 2). We
call ”Application phase” the inference phase of the AI service (green box of
the upper part in Figure 2).

AI serviceAI enhanced application(electric vehicle, smart building…)Data
acquisitionData production Learning Inference Data storage
device1(sensors…)device2(computer…)device3(supercomputer…)device4(mobile…)device5(hard
drive…) (a) Different tasks involved in an AI service

AI serviceAI enhanced application(electric vehicle, smart
building…)Productionof $device_{i}$Useof $device_{i}$End of lifeof
$device_{i}$ Production of electricity Resources (metals, etc.)Emissions
(pollution, abiotic resources depletion…) (b) Life cycle phases of each
$device_{i}$ used by the service

Figure 2. Diagram representing the Life Cycle Inventory of an AI service:
Above: an AI for green application corresponds to the inference step that
depends on other unit processes that require various devices. Below: the use
of devices is located in a more global environment, including production of
resources and impacts. In both schemes colored boxes correspond to unit
processes, black arrows correspond to economic flows (bold: material, dashed:
energy) and red arrows to environmental flows.

Concerning the system boundaries, we refer to (ADEME,, 2021) to consider the
equipment for three tiers:

  * •

_terminals_. In the case of the smart building, this can include: user
terminals used to develop, train and use the AI service; terminals in the
facility where the AI service is trained dedicated to IT support; smart
thermostats.

  * •

_network_. For the smart building case, network equipment used for training
the AI model in the facility, and network equipment in the buildings where the
thermostats are used.

  * •

_data center/server_. For the smart building case, servers on which the model
is trained and used; training and inference can be done on the same server or
not.

For each tier, all support equipment and activities may also be considered.
For example the power supply unit and HVAC of the data center should be taken
into account.

The life cycle stages to consider are the ones previously mentioned:
production, use and end of life. In particular, (ITU-T,, 2014) and (ADEME,,
2021) give classifications of unit processes according to the life cycle
stages, which can be applied to AI services, as shown in Table 1.

Table 1. Application to AI services of ITU recommendation (ITU-T,, 2014) regarding the evaluation of life cycle stages/unit processes  Life cycle id |  Life cycle stage and unit processes |  Recommendation
---|---|---
A - Raw material acquisition |  Mandatory
B - Production
|  Devices production and assembly |  Mandatory
|  Manufacturer support activities |  Recommended
|  Production of support equipment |  Mandatory
|  ICT-specific site construction |  Recommended
C - Use
|  Use of ICT equipment |  Mandatory
|  Use of support equipment |  Mandatory
|  Operator support activities |  Recommended
|  Service provider support activities |  Recommended
D - End of life
|  Preparation of ICT goods for reuse |  Mandatory
|  Storage / disassembly / dismantling / crushing |  Mandatory

If applied to our smart building use case, the unit processes that must be
taken into account would be:

  * •

For equipment that is dedicated to the application, such as the smart
thermostats: Production, Use and End of life.

  * •

For the servers on which the AI service is trained and used and their
environment (network devices, storage servers, backup servers, user terminal,
HVAC… and other potential equipment not dedicated to the application):

    * –

Production and End of life with an allocation of the impacts, with respect to
the execution time for instance.

    * –

Part of the use phase corresponding to the dynamic energy consumption i.e.,
raise of consumption due to the execution of the program.

    * –

Part of the use phase corresponding to the static consumption, with an
allocation (for example if $n$ programs are run simultaneously, $1/n$ of this
consumption) ”since equipment is switched on in part to address the computing
needs of the (Machine Learning) model” (Ligozat and Luccioni,, 2021).

The production phase is generally important for ICT equipment in terms of
global warming potential at least. Yet, when trying to assess this phase for
deep learning methods, we are faced with a lack of LCAs for Graphical
Processing Unit (GPUs) (or Tensor Processing Unit (TPUs) or equivalents).
(Berthoud et al.,, 2020) yet showed that for a CPU-only data center in France,
around 40% of the GHG emissions of the equipment were due to the production
phase.

The use phase is mostly due to the energy use, so the impacts of this part are
highly dependent on the server/facility efficiency and the carbon intensity of
the energy sources.

The end-of-life phase is difficult to assess in ICT in general because of lack
of data concerning this phase of equipment. In particular, the end of life of
many ICT equipment is poorly documented: globally, about 80% of electronic and
electrical equipment is not formally collected (Baldé et al.,, 2017).

##  4\. Assessing the usefulness of an AI for Green service

Now that we have presented how the general framework of life cycle assessment
can be adapted to AI solutions, we propose to use it for evaluating the
complete benefits of an AI for Green service.

In this section, we will consider the following setting:

  * •

A reference application $M_{1}$ which corresponds to the application without
AI. If the application is a smart building for example, $M_{1}$ will be the
building without smart capabilities.

  * •

An AI-enhanced application $M_{2}$ which corresponds to the application with
an AI service that is supposed to have a positive impact on the environment.
In the previous case, it would be the smart building.

###  4.1. Theoretical aspects

When proposing an AI for Green method, one should ensure that the overall
environmental impact is positive: the positive gain induced by using the AI
solution should be higher than the negative impacts associated to the
solution.

This requires to assess first-, second- and third-order impacts of AI (Hilty
and Hercheui,, 2010; Horner et al.,, 2016), as illustrated in Figure 3. As we
detailed in the previous section, first-order impacts come from the life cycle
phases of all the equipment necessary to develop and deploy the AI service.

_Second-order_ impacts correspond to the impacts due to the application of AI.
AI can optimize or substitute existing systems: energy consumption in a
building can be optimized using occupancy or behavior detection, energy
profiling, etc.

_Third-order_ impacts are all changes in technology or society due to the
introduction of AI solutions, possibly encompassing effects of very different
scales, from individual behavioral responses to systemic and societal
transformations, and from short-term to long-term effects. Rebounds effects
fall into this category: an increase in efficiency does not necessarily
translate into a reduction of impacts of the same magnitude, and it can even
lead to an increase in these impact (Berkhout et al.,, 2000). Rebound effects
occur because potential savings (in terms of money, time, resources, etc.) are
transformed into more consumption (Schneider et al.,, 2001). For example, due
to economical savings, smart building users may decide to increase heating
temperature for better comfort or to buy more flight tickets after an increase
in energy efficiency.

Figure 3. Overview of AI’s impacts. First-order or direct impacts result from
the equipment life cycle. Second-order impacts are the difference between the
LCAs of the reference system and the AI-enhanced system. Third-order impacts
are changes in technology or society induced by the application.

Third-order impacts are beyond the scope of the methodology proposed here, and
are briefly discussed in Section 5. According to (ITU-T,, 2014), first and
second-order impacts of the AI service should be estimated based on life cycle
assessment (LCA), the difference between the two being the scope: for first-
order impacts the scope is restricted to the equipment involved in the target
AI service (for example the AI involved in a smart building), while second-
order impacts consider the whole solution (the smart building itself).
Including second-order impacts requires to extend the scope to the whole
application AI is supposed to enhance. More specifically the net environmental
impacts considering both first and second-order effects are obtained by
computing:

(1) $\Delta(M_{2}|M_{1})=LCA(M_{2})-LCA(M_{1})\in\mathbb{R}^{d}$

with:

  * •

$M_{1}$ the reference application without using the AI service,

  * •

$M_{2}$ the application enhanced by AI,

  * •

$LCA(x)$ a quantification of $d$ types of environmental impacts (e.g., GHG
emissions, water footprint, etc.). LCA methodology is described in 3.2. Note
that $LCA(M_{2})$ includes the impacts of the AI service itself, i.e.,
$LCA_{AI}(M_{2})$.

A previous work (Bommasani et al.,, 2021) also gave a simplified scheme for
assessing the cost-benefit of deploying a foundation model, which also
includes social benefits and costs, but does not explicit the direct
environmental costs of using this model. We propose to relate our methodology
(Equation (1)) to their proposal. Adopting their equation, but focusing on the
environmental impacts only, the overall value of a model can be assessed with:

(2) $V(M)=S(M)-E(M)-O(M)$

with:

  * •

$V(M)$ the value of using the model i.e., the environmental gain induced by
its use in the practical application considered

  * •

$S(M)$ the environmental benefit that can be interpreted as the difference
between the initial impact of the application and its final impact (not taking
into account the AI solution i.e., the Learning and Inference task in the top
part of Figure 2)

  * •

$E(M)$ the energy cost of the model

  * •

$O(M)$ all other impacts including chip production, waste, risks for
biodiversity, and third-order impacts (which are not discussed here).

Regarding the well-established framework of LCA, this approach suffers from
several weaknesses. First, in the equation all the values are expressed in
dollars. This formally allows to perform addition of several kinds of impacts
but with an arbitrary consideration to the diversity of environmental issues.
By definition, LCA considers multiple criteria for the impacts, previously
described at the beginning of Section 3 (GHG emissions, water footprint…). LCA
may aggregate several impacts but with specific weights not necessarily
dependent on an economic value. As noted in (Hauschild et al.,, 2018) ”there
is no scientific basis on which to reduce the results of an LCA to a single
result or score because of the underlying ethical value-choices”.

Besides, if one considers for instance the case of an AI service dedicated to
biodiversity (see for instance 8.1 in (Rolnick et al.,, 2019)), one would
expect to precisely quantify the positive impact of this service on
biodiversity (schematically, how many species can be saved?) balanced by the
negative ones (producing chips for GPUs has an impact on the biodiversity
through several sources of pollution (Villard et al.,, 2015)). Adopting
Equation (2) will mix several impacts together and may dilute the value of
interest (e.g., biodiversity), that could be burdened by negative impacts
regarding energy to train the models, for instance.

Last, even if the equation is not wrong per se, the expression in terms of
benefit/costs is questionable, and practical means for its computation are
missing in (Bommasani et al.,, 2021).

We thus believe that Equation (1) should be used. Terms of Equation (2) can be
related to the methodology proposed in our paper as follows:

(3) $\underbrace{V(M_{2})}_{-\Delta(M_{2}|M_{1})}\approx
S(M_{2}|M_{1})\underbrace{-E(M_{2})-O(M_{2})}_{LCA_{AI}(M_{2})}$

where $\Delta(M_{2}|M_{1})$ and $LCA_{AI}(M_{2})$ are defined in Equation (1).
The negative impacts of an AI solution $M_{2}$ compared to the reference
solution $M_{1}$ are not always restricted to its AI part (i.e., to $E(M_{2})$
and $O(M_{2})$). For example, compared to a standard vehicle the negative
impacts of an autonomous vehicle are not only due to the life cycle of
(additional) ICT equipment, but also to additional aerodynamic drag due to the
presence of LIDAR on the roof (Taiebat et al.,, 2018). Hence, the nature of
the impacts in $S(M_{2}|M_{1})$ (positive or negative) cannot be stated a
priori and depends on complete LCA results for both applications $M_{2}$ and
$M_{1}$. It may also depend on the target environmental criteria.

###  4.2. Case studies

In order to review the kind of evaluation that is usually made in the AI for
Green literature, we analyzed the references for several domains of (Rolnick
et al.,, 2019), which identifies potential applications of machine learning
for climate change adaptation of mitigation 888This review was documented in a
csv file, which is given as supplementary material to the paper..

We mostly chose domains that had been flagged as having a _High Leverage_ and
noted for each paper cited in the corresponding section the kind of
environmental evaluation, with the following categories:

  1. (a)

No mention of the environmental gain.

  2. (b)

General mention of the environmental gain.

  3. (c)

A few words about the environmental gain but no quantitative evaluation or
only indirect estimation.

  4. (d)

Evaluation of the energy gain without taking the AI service into account.

  5. (e)

Evaluation of the energy gain taking the use phase of the AI service into
account.

  6. (f)

Comprehensive evaluation of the environmental gain (comparison of LCAs).

The results of the review are shown in Figure 4.

Figure 4. Sankey diagram of parts of Rolnick’s paper references in terms of
environmental evaluation _(created with the Sankey Diagram Generator by Dénes
Csala, based on the Sankey plugin for D3 by Mike Bostock;
https://sankey.csaladen.es; 2014)_

The central node is ”Rolnick et al. citations”. On its left are the domains of
the citations. For example, the Smart building section contained 15 relevant
citations.

On its right the first flows show the partition into general machine learning
applications (ML), deep learning applications (DL), and other methods (other).
For example, 20 papers corresponded to deep learning applications.

The last flows on the right show the kinds of environmental evaluation. We can
note that about half of the papers do not include any environmental
evaluation, although the focus is on applications to tackle climate change.
Many papers also give a distant proxy for evaluation, such as detailing the
possible impacts without quantification, or indicating the execution time of
the program.

A few citations evaluate the environmental gain, mostly in terms of energy
gain, but none of the papers considered took into account the AI service
impacts.

It can be noted that other papers that include an evaluation of part of these
impacts, can be found in the literature. (Bracquené et al.,, 2020) for example
present an intelligent control systems that takes into account the expected
occupancy in order to adapt the thermostat and save energy. They do not take
into account learning the occupancy model, but take into account the LCA of
the smart thermostats, and show that the energy needed for these devices
across their whole life cycle will almost always be lower that the energy
saved.

##  5\. Discussion

In this paper, we have analyzed the environmental impacts of AI solutions, in
particular in the case of AI for Green applications, and proposed a framework
to evaluate them more completely. The proposed methodology compares, through
life cycle assessment, the impact of a reference solution with the AI one (1)
for the appropriate types of environmental impacts. The analysis of literature
on AI solutions has made salient the following issues/problems.

###  5.1. Current environmental evaluation of AI services is under-estimated

We have shown that AI for Green papers only take into account a small part of
the direct environmental impacts.

Several reasons can explain this under-estimation. The narratives about
dematerialization that would correspond to a dramatic decrease in
environmental impacts, permeate AI as a part of ICT (Bol et al.,, 2021).
However, these narratives have proven to be false until now. Attention to AI’s
GHG emissions has focused on electricity consumption (energy flows). At the
moment, material flows receive less attention in AI. However, it is beginning
to be considered (Gupta et al.,, 2020; Wu-2021).

###  5.2. AI research should use Life Cycle Assessment to assess the
usefulness of an AI service

Life cycle assessment is a solid methodology to evaluate not only global
warming potential but also other direct environmental impacts. LCA considers
all the steps from production to use and end of life. However, it has several
well-known limitations due to the complexity of processes involved in material
production. Obtaining all the information to assign reliable values to each
edge of the life cycle inventory also proves difficult, e.g., there is very
little information on manufacturing impacts of GPU either from manufacturers
or in LCA databases. To solve this problem, we could encourage the AI
community to lobby companies to open a part of their data. This approach would
be in the same spirit as what is happening for open science but would also
require taking legal issues into account.

###  5.3. AI for Green gains are only potential

Even when a properly conducted LCA concludes that an AI solution is
environmentally beneficial, such a result should be considered with caution.
Environmental benefits computed by the LCA-based methodology proposed in this
paper correspond to a technical and simplistic view of environmental problems:
it assumes that AI will enhance or replace existing applications, all other
things being equal. The ambition to solve societal problems using AI is
praiseworthy, but it should probably be accompanied by socio-technical
concerns and an evaluation of possible third-order effects. For example,
autonomous vehicles are often associated with potential efficiency gains (such
as helping car sharing, or allowing platooning) and corresponding
environmental benefits (Taiebat et al.,, 2018). However, autonomy could also
profoundly transform mobility in a non-ecological way (Coroamă and Pargman,,
2020).

###  5.4. AI services and large deployment

Evaluating third-order effects is even more critical when large-scale
deployment of the proposed solution(s) is projected, e.g., to maximize
absolute gains. This case requires special attention even in LCA, since large-
scale deployment may induce societal reorganizations for producing and
operating the solution(s). For example, the generalization of AI may lead to a
substantial increase in demand for specific materials (such as lithium or
cobalt) or energy. This increase may have non-linear environmental
consequences, e.g., opening new and less performing mines, increasing the use
of fossil fuel based power plants, etc. Hence in this case, the
_attributional_ LCA framework we suggest using in this paper needs to be
replaced by the much more complex _consequential_ one (Hauschild et al.,,
2018).

## Authors contribution

“Conceptualization, all authors; methodology, all authors; validation, all
authors; formal analysis, all authors; investigation, J.L. and A.-L. L.; data
curation, J.L. and A.-L. L.; writing—original draft preparation, all authors;
writing—review and editing, all authors; visualization, A.B. and A.-L. L.;
supervision, A.-L. L.; project administration, A.-L. L. All authors have read
and agreed to the published version of the manuscript.

## Acknowledgments

This work was partly supported by the CNRS EcoInfo group
(https://ecoinfo.cnrs.fr/).

## Abbreviations

The following abbreviations are used in this manuscript:

AI | Artificial Intelligence
---|---
CNN | Convolutional Neural Network
CPU | Central Processing Unit
DL | Deep Learning
GHG | Greenhouse Gas
GPU | Graphics Processing Unit
HVAC | Heating, Ventilation, and Air Conditioning
ICT | Information and Communications Technology
LCA | Life cycle Assessment or Analysis
LCI | Life Cycle Inventory
ML | Machine Learning
NLP | Natural Language Processing
TPU | Tensor Processing Unit

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# A note on the steady Poiseuille flow of Carreau-Yasuda fluid

N. Kutev  Nikolay Kutev, Institute of Mathematics and Informatics, Bulgarian
Academy of Sciences, Acad. G. Bontchev str., bl. 9, 1113 Sofia, Bulgaria
<EMAIL_ADDRESS>and  S. Tabakova  Sonia Tabakova, Institute of Mechanics,
Bulgarian Academy of Sciences, Acad. G. Bontchev str., bl. 4, 1113 Sofia,
Bulgaria<EMAIL_ADDRESS>
###### Abstract.

The steady Poiseuille flow of Carreau-Yasuda fluid in a pipe, caused by
constant pressure gradient, is studied theoretically. It is proved that at
some values of the viscosity model parameters, the problem has a classical
solution, while at others - generalized solution. For the latter, a necessary
and sufficient condition is found, which depends on the pressure gradient and
Carreau-Yasuda model parameters.

###### Key words and phrases:

Carreau-Yasuda fluid, Steady Poiseuille flow, classical solution, necessary
and sufficient condition

###### 2010 Mathematics Subject Classification:

76A05, 35J66, 35Q35

*corresponding author

##  1\. Introduction

The Poiseuille flow problem is one of the fundamental problems of fluid
mechanics, corresponding to a parabolic velocity profile in a pipe or channel,
when the flow is laminar and fully developed in axial direction. For Newtonian
fluid, the problem has a well known analytical solution [1]. However, for non-
Newtonian fluids, it has no analitical solution, except for some special cases
for generalized Newtonian fluid models, such as power law model of viscosity
dependence on shear rate [2]. The behavior of the so-called shear-thinning
fluids (viscosity is decreasing function of shear rate), for example polymer
solutions, polymer melts, suspensions, emulsions, and some biological fluids,
is described by different viscosity models, such as the power law model,
Carreau model, Carreau-Yasuda model and others [2]-[4].

The general unsteady and oscillatory cases of the Poiseuille flow in an
infinite channel or pipe using the Carreau or Carreau-Yasuda models are
studied in our previous papers [5] \- [8]. The most interesting case of
negative power index in these models, $n<0$, corresponds to severe
pseudoplastic behaviour [9]-[11]. It has been studied theoretically in [7],
[8], but the sufficient and necessary condition for solution existence is
still a challenge. The aim of the present work is to prove the solution
existence or non-existence for the different parameters of the steady
Poiseuille flow problem of Carreau-Yasuda fluid and to find the existence
condition at $n<0$.

The dimensionless equations of motion and continuity for the steady pipe flow
are:

(1) $\textbf{v}\cdot\nabla\textbf{v}=-\nabla p+\nabla\cdot\textbf{T},$ (2)
$\nabla\cdot\textbf{v}=0,$

where v is velocity vector, $p$ \- pressure, $\textbf{T}=\mu_{app}\dot{S}$ \-
viscous stress tensor with
$\displaystyle{\dot{S}=\frac{1}{2}\left(\nabla\textbf{v}+\nabla\textbf{v}^{T}\right)}$
as strain rate tensor and $\mu_{app}$ \- apparent viscosity (constant for
Newtonian fluid and a non-linear function of shear rate
$\displaystyle{\dot{\gamma}=\sqrt{2\dot{S}:\dot{S}}}$ for generalized
Newtonian fluid).

The flow is assumed fully developed laminar driven by a constant gradient $b$
in axial direction. Then, in the adopted cylindrical coordinate system
$(x,Y,\varphi)$ with $x$ as the axial coordinate, the flow velocity is
simplified as $\textbf{v}=(U(Y),0,0)$ and T reduces to the scalar
$\mu_{app}(\dot{\gamma})\dot{\gamma}$, where $\displaystyle{\dot{\gamma}=\mid
U_{Y}\mid}$. The Carreau-Yasuda viscosity model is assumed for the apparent
viscosity $\mu_{app}$ :

(3) $\mu_{app}=1-c+c[1+Cu^{\alpha}\dot{\gamma}^{\alpha}]^{(n-1)/\alpha},$

where $b$ is the dimensionless pressure gradient and
$\displaystyle{c=1-\frac{\mu_{\infty}}{\mu_{0}}}$, with $\mu_{0}$,
$\mu_{\infty}$ as the upper and lower limits of the viscosity corresponding to
the low and high shear rates, $Cu$ is the Carreau number (Weissenberg number),
$\alpha$ and $n$ are empirically determined for the considered fluid.

Applying the upper assumptions, the system (1) - (3) is reduced to one
elliptic equation for the axial velocity:

(4)
$L(U(Y))=\frac{1}{Y}\frac{d}{dY}\Bigg{\\{}\left[1-c+c\left(1+Cu^{\alpha}\mid
U_{Y}\mid^{\alpha}\right)^{\frac{n-1}{\alpha}}\right]YU_{Y}\Bigg{\\}}=b$

with the axisymmetrical and no-slip boundary conditions:

(5) $U_{Y}(0)=U(R)=0.$

where $R$ is the dimensionless pipe radius (the pipe radius or diameter are
usually taken as characteristic length). Here $Cu>0$, $b\in\mathbb{R}$,
$\alpha>0$, $n\in\mathbb{R}$, $c\in[0,1]$ are empirically determined
parameters.

If $b=0$, then (4), (5) has only the trivial solution $U(Y)\equiv 0$. It is
clear that $U(Y,-b)=-U(Y,b)$, where $U(Y,b)$, $U(Y,-b)$ are solutions of (4),
(5) with right hand side $b$ and $-b$, correspondingly. That is why, further
on we consider only the case $b>0$.

The outline of the paper is as follows. In section 2 the main results are
formulated in four theorems to be proved in section 3. Two applications,
presented by other three theorems, together with some illustrations of the
theorems’ results are found in section 4. Finally, comparison between the
criteria proved for the unsteady and steady case are given as an appendix.

##  2\. Main results

In order to formulate the main results, we introduce the function

(6)
$F(\zeta)=\left[1-c+c\left(1+Cu^{\alpha}\zeta^{\alpha}\right)^{\frac{n-1}{\alpha}}\right]\zeta\quad\textrm{for}\quad\zeta\geq
0,$

where $\displaystyle{\zeta=\mid U_{Y}\mid}$.

######  Theorem 2.1.

Suppose $b>0$, $\alpha>0$ and either $c=0$, $Cu=0$ or $n=1$. Then problem (4),
(5) is Newtonian one (classical Poiseuille problem in a pipe) and has a unique
classical solution:

(7) $U(Y)=\frac{b}{4}(Y^{2}-R^{2})\quad\textrm{for}\quad every\quad
Y\in[0,R].$

######  Theorem 2.2.

Suppose $b>0$, $\alpha>0$, $c\in(0,1)$ and either one of the following
conditions holds:

(8) $n\geq 0;$ (9) $n<0\quad
and\quad\left(1-\frac{\alpha+1}{n}\right)^{\frac{n-1-\alpha}{\alpha}}<\frac{1-c}{\alpha
c}.$

Then problem (4), (5) has a unique classical solution $U(Y)\in C^{2}([0,R])$
satisfying

(10) $U(Y)=-\int^{R}_{Y}F^{-1}\left(\frac{bs}{2}\right)ds\quad\textrm{and}$
(11) $0\leq U_{Y}(Y)\leq\
F^{-1}\left(\frac{bR}{2}\right)\quad\textrm{for}\quad Y\in[0,R].$

######  Theorem 2.3.

Suppose $b>0$, $\alpha>0$, $n<0$, $c\in(0,1)$ and

(12)
$\left(1-\frac{\alpha+1}{n}\right)^{\frac{n-1-\alpha}{\alpha}}=\frac{1-c}{\alpha
c}$

If

(13)
$\left[1-c+c\left(1-\frac{\alpha+1}{n}\right)^{\frac{n-1}{\alpha}}\right]Cu^{-1}\left(-\frac{\alpha+1}{n}\right)^{\frac{1}{\alpha}}\geq\frac{bR}{2},$

then problem (4), (5) has a unique classical solution $U(Y)$ satisfying (10),
(11);
If

(14)
$\left[1-c+c\left(1-\frac{\alpha+1}{n}\right)^{\frac{n-1}{\alpha}}\right]Cu^{-1}\left(-\frac{\alpha+1}{n}\right)^{\frac{1}{\alpha}}<\frac{bR}{2},$

then problem (4), (5) has a unique generalized solution $U(Y)\in
C^{2}([0,R]\setminus{Y_{0}})\bigcap C^{1}([0,R])$ satisfying (10), (11),
$U_{YY}(Y_{0})=\infty$, where

(15)
$Y_{0}=\frac{2}{b}\left[1-c+c\left(1-\frac{\alpha+1}{n}\right)^{\frac{n-1}{\alpha}}\right]Cu^{-1}\left(-\frac{\alpha+1}{n}\right)^{\frac{1}{\alpha}}\quad\in(0,R).$

######  Remark 2.1.

If (13) is strict inequality then $U(Y)\in C^{2}([0,R])$, while when (13) is
an equality, then $U(Y)\in C^{2}([0,R))\bigcap C^{1}([0,R])$,
$U_{YY}(R)=\infty$.

######  Theorem 2.4.

Suppose $b>0$, $\alpha>0$, $n<0$, $c\in(0,1)$ and

(16)
$\left(1-\frac{\alpha+1}{n}\right)^{\frac{n-1-\alpha}{\alpha}}>\frac{1-c}{\alpha
c}.$

Then problem (4), (5) has a unique classical solution iff

(17)
$\frac{bR}{2}\leq\left[1-c+c\left(1+Cu^{\alpha}\zeta_{1}^{\alpha}\right)^{\frac{n-1}{\alpha}}\right]\zeta_{1}=F(\zeta_{1}),$

where $\zeta_{1}$ is the first positive zero of $F^{\prime}(\zeta)=0$.
Moreover, $U(Y)$ satisfies (10), (11) and

(18) $0\leq F^{-1}\left(\frac{bR}{2}\right)\leq\zeta_{1}\quad\textrm{for}\quad
Y\in[0,R].$

######  Remark 2.2.

If (17) is a strict inequality, then $U(Y)\in C^{2}([0,R])$. When (17) is an
equality, then $U(Y)\in C^{2}([0,R))\bigcap C^{1}([0,R])$, $U_{YY}(R)=\infty$.

##  3\. Proofs of the main results

Integrating $Y.L(U(Y))$ from $0$ to $Y\in(0,R]$, we get the identity

(19) $\left[1-c+c\left(1+Cu^{\alpha}\mid
U_{Y}\mid^{\alpha}\right)^{\frac{n-1}{\alpha}}\right]U_{Y}=\frac{bY}{2}\quad\textrm{for}\quad
Y\in[0,R].$

Hence $U_{Y}(Y)>0$ for $Y\in(0,R]$ and (19) is equivalent to

(20)
$F(U_{Y}(Y))=\left[1-c+c\left(1+Cu^{\alpha}U_{Y}^{\alpha}\right)^{\frac{n-1}{\alpha}}\right]U_{Y}=\frac{bY}{2}\quad\textrm{for}\quad
Y\in[0,R].$

###### Proof.

of Theorem 2.1. For $c=0$ or $n=1$ we have $F(\zeta)=\zeta$ and (20) becomes

(21) $U_{Y}(Y)=\frac{bY}{2}\quad\textrm{for}\quad Y\in[0,R].$

Integrating (21) from $0$ to $Y\in(0,R]$, we obtain from (5) that
$\displaystyle{U(Y)=\frac{b}{4}(Y^{2}-R^{2})}$, which proves Theorem 2.1.

∎

###### Proof.

of Theorem 2.2. Simple computations give us for $\zeta\geq 0$

(22)
$F^{\prime}(\zeta)=1-c+c\left(1+Cu^{\alpha}\zeta^{\alpha}\right)^{\frac{n-1-\alpha}{\alpha}}\left(1+nCu^{\alpha}\zeta^{\alpha}\right),$
(23)
$F^{\prime\prime}(\zeta)=c(n-1)Cu^{\alpha}\zeta^{\alpha-1}\left(1+Cu^{\alpha}\zeta^{\alpha}\right)^{\frac{n-1-2\alpha}{\alpha}}\left(\alpha+1+nCu^{\alpha}\zeta^{\alpha}\right).$

Thus from (22) we get

(24) $F^{\prime}(\zeta)\geq
1-c+c\left(1+Cu^{\alpha}\zeta^{\alpha}\right)^{\frac{n-1-\alpha}{\alpha}}>0\quad\textrm{for}\quad\zeta\geq
0,\quad n\geq 0\quad\textrm{and}\quad c\in(0,1].$

If (9) holds then from (23) we obtain

(25) $F^{\prime\prime}(\zeta_{0})=0\quad\textrm{for}\quad
Cu^{\alpha}\zeta_{0}^{\alpha}=-\frac{\alpha+1}{n}>0,$ (26)
$F^{\prime\prime}(\zeta)<0\quad\textrm{for}\quad\zeta<\zeta_{0}\quad\textrm{i.e.
for}\quad Cu^{\alpha}\zeta^{\alpha}<-\frac{\alpha+1}{n}\quad\textrm{and}$ (27)
$F^{\prime\prime}(\zeta)>0\quad\textrm{for}\quad\zeta>\zeta_{0}\quad\textrm{i.e.
for}\quad Cu^{\alpha}\zeta^{\alpha}>-\frac{\alpha+1}{n}.$

From (25)-(27) it follows that the function $F^{\prime}(\zeta)$ has a global
minimum in the interval $[0,\infty)$ at the point $\zeta=\zeta_{0}$. Since
from (9), we have

(28) $F^{\prime}(\zeta_{0})=1-c-\alpha
c\left(1-\frac{\alpha+1}{n}\right)^{\frac{n-1-\alpha}{\alpha}},$ (29)
$F^{\prime}(\zeta)\geq F^{\prime}(\zeta_{0})>0\quad\textrm{for}\quad\zeta\geq
0.$

Under the conditions (8), (9) and from (24), (19), it is seen that
$F^{\prime}(\zeta)>0$ and $F(\zeta)$ is strictly monotone increasing.
Moreover,

(30) $F(0)=0,\quad\lim_{\zeta\rightarrow\infty}F(\zeta)=\infty,$

so that there exits the inverse function $F^{-1}(\zeta)$. From

(31) $(F^{-1}(\zeta))^{\prime}=\frac{1}{F^{\prime}(F^{-1}(\zeta))}>0$

it follows that $F^{-1}(\zeta):[0,\infty)\rightarrow[0,\infty)$ is strictly
monotone increasing function and (20) is equivalent to

(32) $U_{Y}(Y)=F^{-1}\left(\frac{bY}{2}\right)\quad\textrm{for}\quad
Y\in[0,R],$

where

(33)
$F^{-1}\left(\frac{bY}{2}\right):[0,R]\rightarrow\left[0,F^{-1}\left(\frac{bR}{2}\right)\right].$

Integrating (32) from $Y\in[0,R)$ to $R$, we get from (5) that the function

(34)
$U(Y)=-\int^{R}_{Y}F^{-1}\left(\frac{bs}{2}\right)ds\quad\textrm{for}\quad
Y\in[0,R]$

is the unique classical solution of (4), (5), when one of the conditions (8),
(9) is satisfied. Moreover, from the monotonicity of $F^{-1}(\zeta)$ and (32),
estimate (11) holds and Theorem 2.2 is completed.

∎

###### Proof.

of Theorem 2.3. From (12) inequalities (28), (29) become

(35) $F^{\prime}(\zeta_{0})=0,\quad
F^{\prime}(\zeta)>0\quad\textrm{for}\quad\zeta\geq 0,\quad\zeta\neq\zeta_{0}.$

Thus $F(\zeta)$ is monotone increasing function and the inverse function
$F^{-1}(\zeta)$ exists and is monotone increasing, which is evident from (31).
Moreover, (30) holds and $F(\zeta):[0,\infty)\rightarrow[0,\infty)$,
$F^{-1}(\zeta):[0,\infty)\rightarrow[0,\infty)$. Hence, equation (20) is
equivalent to (32) and after integration, as in the proof of Theorem 2.2, the
unique solution of (4), (5) is given by (34).

Let us analyze the regularity of $U(Y)$. It is clear that $U(Y)\in
C^{1}([0,R])$ and from the monotonicity of $F^{-1}$ the estimate (11) holds.
Since (13) is equivalent to

(36) $F(\zeta_{0})\geq\frac{bR}{2},\quad\textrm{i.e.,}\quad\zeta_{0}\geq
F^{-1}\left(\frac{bR}{2}\right)$

and after differentiating (32), we get from (31), (26) the estimates

(37) $U_{YY}(Y)=\frac{\partial}{\partial
Y}\left(F^{-1}\left(\frac{bY}{2}\right)\right)=\frac{b}{2F^{\prime}(F^{-1}\left(\frac{bY}{2}\right))}\leq\frac{b}{2F^{\prime}(F^{-1}\left(\frac{bR}{2}\right))}$
$\leq\frac{b}{2F^{\prime}(F^{-1}\left(\zeta_{0}\right))}=\infty\quad\textrm{for}\quad
Y\in[0,R].$

Thus

(38) $0<U_{YY}(Y)<\infty\quad\textrm{for}\quad Y\in[0,R)\quad\textrm{and}\quad
U_{YY}(R)=\infty,$

when (13), equivalently (36), is a strict inequality, because

$\displaystyle{\frac{b}{2F^{\prime}(F^{-1}\left(\frac{bR}{2}\right))}<\frac{b}{2F^{\prime}(F^{-1}\left(\zeta_{0}\right))}=\infty}.$

However, when (13), equivalently (36), is equality, then

$\displaystyle{U_{YY}(R)=\frac{b}{2F^{\prime}(F^{-1}\left(\frac{bR}{2}\right))}=\frac{b}{2F^{\prime}(F^{-1}\left(\zeta_{0}\right))}=\infty}.$

Remark 2.1. and the proof of (13) is completed.

Suppose that (14) holds. As in the proof of (13) $U(Y)\in C^{1}([0,R])$ is
unique solution of (4), (5) satisfying (10), (11). Conditions (14), (15) are
equivalent to

(39) $F(\zeta_{0})<\frac{bR}{2},\quad
F(\zeta_{0})=\frac{bY_{0}}{2},\quad\textrm{i.e.,}\quad\zeta_{0}<F^{-1}\left(\frac{bR}{2}\right),\quad\zeta_{0}=F^{-1}\left(\frac{bY_{0}}{2}\right)$

and therefore $Y_{0}\in(0,R)$.
From the monotonicity of $F^{-1}$ we get from (39)

$F^{-1}\left(\frac{bY}{2}\right)<F^{-1}\left(\frac{bY_{0}}{2}\right)=\zeta_{0}\quad\textrm{for}\quad
Y\in[0,Y_{0}),$ (40)
$F^{-1}\left(\frac{bY}{2}\right)>F^{-1}\left(\frac{bY_{0}}{2}\right)=\zeta_{0}\quad\textrm{for}\quad
Y\in(Y_{0},R].$

Repeating the estimate (37), from (40) and the monotonicity of
$F^{\prime}(\zeta)$, see (26), (27), we have

$U_{YY}(Y_{0})=\infty,\quad
0<U_{YY}(Y)<U_{YY}(Y_{0})=\infty\quad\textrm{for}\quad
Y\in[0,R]\setminus{Y_{0}},$

i.e., $U(Y)\in C^{2}([0,R]\setminus{Y_{0}})\bigcap C^{1}([0,R])$, which proves
(14) and Theorem 2.3.

∎

###### Proof.

of Theorem 2.4. Under conditions (16), (28) it follows that
$F^{\prime}(\zeta)$ has a strictly negative minimum in the interval
$[0,\infty)$ at the point $\zeta_{0}$. Since

(41)
$F^{\prime}(0)=0,\quad\lim_{\zeta\rightarrow\infty}F^{\prime}(\zeta)=1-c>0$

the function $F^{\prime}(\zeta)$ has two positive roots
$0<\zeta_{1}<\zeta_{2}<\infty$, $F^{\prime}(\zeta_{i})=0,\quad i=1,2$,
$\zeta_{0}\in(\zeta_{1},\zeta_{2})$. Moreover, from (41) and (26) it follows
that

(42) $F^{\prime}(\zeta)>0,\quad
F^{\prime\prime}(\zeta)<0\quad\textrm{for}\quad\zeta\in[0,\zeta_{1}),\quad
F^{\prime}(\zeta_{1})=0,$

i.e., $F^{\prime}(\zeta)$ is strictly monotone decreasing, while $F(\zeta)$ is
strictly monotone increasing for $\zeta\in[0,\zeta_{1})$. Hence, the inverse
function

(43) $F^{-1}(\zeta):[0,\zeta_{1})\rightarrow[0,F^{-1}(\zeta_{1})]$

is well defined and from (31), (42) is strictly monotone increasing.

Sufficiency: From (17) we get

(44) $\zeta_{1}\geq F^{-1}\left(\frac{bR}{2}\right)\quad\textrm{and}\quad
F^{-1}\left(\frac{bY}{2}\right):[0,R]\rightarrow\left[0,F^{-1}\left(\frac{bR}{2}\right)\right].$

Since
$\displaystyle{\left[0,F^{-1}\left(\frac{bR}{2}\right)\right]\subset[0,\zeta_{1})}$,
from (43) equation (20) is equivalent to (32) and after integration $U(Y)\in
C^{1}([0,R)$ satisfies (34). Moreover, from the monotonicity of
$F^{-1}(\zeta)$ and (44), the estimate (18) is satisfied because

$0\leq U_{Y}(Y)\leq U_{Y}(R)=F^{-1}\left(\frac{bR}{2}\right)\leq\zeta_{1}.$

Repeating the estimate (37) we get

$0<U_{YY}(Y)<\infty\quad\textrm{for}\quad Y\in[0,R],$

when (17), equivalently (44), is a strict inequality, i.e., $U(Y)\in
C^{2}([0,R])$.

If (17), equivalently (44), is an equality, then

$U_{YY}(R)=\frac{b}{2F^{\prime}(F^{-1}\left(\frac{bR}{2}\right))}=\frac{b}{2F^{\prime}(\zeta_{1})}=\infty,\quad
U(Y)\in C^{2}([0,R))\cap C^{1}([0,R]),$

which proves the sufficiency in Theorem 2.4 and Remark 2.2.

Necessity: Suppose $U(Y)\in C^{2}((0,R))\cap C^{1}([0,R])$ is a classical
solution of (4), (5). Then $U(Y)$ satisfies (20). We assume by contradiction
that (17) fails, i.e.,

(45) $\frac{bR}{2}>F(\zeta_{1}).$

If $\displaystyle{Y_{0}=\frac{2F(\zeta_{1})}{b}}$, then from (45), we have
$Y_{0}\in(0,R)$ and from (42), (43)

(46) $\zeta_{1}=F^{-1}\left(\frac{bY_{0}}{2}\right).$

Thus (46) and (37) for $Y=Y_{0}$ give us

(47)
$U_{YY}(Y_{0})=\frac{b}{2F^{\prime}(F^{-1}\left(\frac{bY_{0}}{2}\right))}=\frac{b}{2F^{\prime}(\zeta_{1})}=\infty.$

Since $Y_{0}$ is an interior point of the interval $(0,R)$, it follows from
(47) that $U(Y)$ is not a classical $C^{2}(0,R)$ smooth solution of (4), (5)
and Theorem 2.4 is completed.

∎

##  4\. Applications

###  4.1. Case c=1

In the applications of polymer pipe flows, the special case of $c=1$ is
important ($\mu_{\infty}\rightarrow 0$ [2]).

######  Theorem 4.1.

Suppose $b>0$, $\alpha>0$, $n\in\mathbb{R}$ and $c=1$. Then

  * •

(a) for

(48) $\quad n>0$

problem (4), (5) has a unique classical solution $U(Y)\in C^{2}([0,R])$
satisfying (10), (11);

  * •

(b) for

(49) $\quad n=0$

problem (4), (5) has a unique classical solution $U(Y)\in C^{2}((0,R))\cap
C^{1}([0,R])$ iff

(50) $\frac{bR}{2}\leq Cu^{-1}$

and $U(Y)$ satisfies (10), (11);

  * •

(c) for

(51) $\quad n<0$

problem (4), (5) has a unique classical solution $U(Y)\in C^{2}((0,R))\cap
C^{1}([0,R])$ iff

(52) $\frac{bR}{2}\leq
Cu^{-1}\left(\frac{n-1}{n}\right)^{\frac{n-1}{\alpha}}\left(-\frac{1}{n}\right)^{\frac{1}{\alpha}}=F\left(Cu^{-1}\left(-\frac{1}{n}\right)^{\frac{1}{\alpha}}\right)$

and $U(Y)$ satisfies (10), (11).

######  Remark 4.1.

If (50) or (52) is strict inequality then $U(Y)\in C^{2}([0,R])$. When (50) or
(52) is equality then $U(Y)\in C^{2}([0,R))\cap C^{1}([0,R])$ and
$U_{YY}(R)=\infty$.

###### Proof.

  * •

(a) Suppose (48) holds. Then from (24)

(53)
$F^{\prime}(\zeta)\geq\left(1+Cu^{\alpha}\zeta^{\alpha}\right)^{\frac{n-1-\alpha}{\alpha}}>0\quad\textrm{for}\quad\zeta\geq
0$

and $F(\zeta)$ is strictly monotone increasing function.

Since $F(0)=0$, $\lim_{\zeta\rightarrow\infty}F(\zeta)=\infty$, there exists
the inverse function $F^{-1}(\zeta):[0,\infty)\rightarrow[0,\infty)$, which is
strictly monotone increasing function from (31) and (53). Thus problem (4),
(5) is equivalent to (20) and after integration, $U(Y)$ satisfies (10), (11).

  * •

(b) Suppose (49) holds. Here, again (53) is valid, from which $F(\zeta)$ is
strictly monotone increasing function for $\zeta\geq 0$. Since $F(0)=0$,
$\lim_{\zeta\rightarrow\infty}F(\zeta)=Cu^{-1}=\sup_{\zeta\geq 0}F(\zeta)$,
there exists the inverse function

(54) $F^{-1}(\zeta):[0,Cu^{-1})\rightarrow[0,\infty),$

which is strictly monotone increasing function from (31) and (53).

Sufficiency: From (50)
$\displaystyle{F^{-1}\left(\frac{bY}{2}\right):\left[0,\frac{bR}{2}\right]\rightarrow\left[0,F^{-1}\left(\frac{bR}{2}\right)\right]}$
is well defined for $Y\in[0,R]$ because
$\displaystyle{\left[0,\frac{bR}{2}\right]\subset[0,Cu^{-1}]}$. The rest of
the proof is identical with the proof of (a) of Theorem 4.1.

Necessity: If $U(Y)\in C^{2}((0,R))\cap C^{1}([0,R])$ is a classical solution
of (4), (5), we suppose by contradiction that
$\displaystyle{\frac{bR}{2}>Cu^{-1}}$.

Then $\displaystyle{Y_{0}=\frac{2}{bCu}\in(0,R)}$ and from (20) and the
monotonicity of $F(\zeta)$ we have for $Y\in(Y_{0},R)$ the following
impossible chain of inequalities

$Cu^{-1}=\frac{bY_{0}}{2}<\frac{bY}{2}=F(U_{Y}(Y))\leq\sup_{\zeta\geq
0}F(\zeta)=Cu^{-1}.$

  * •

(c) Suppose (51) holds. From (22), (23) we get

(55) $\displaystyle
F^{\prime}(\zeta)=\left(1+Cu^{\alpha}\zeta^{\alpha}\right)^{\frac{n-1-\alpha}{\alpha}}\left(1+nCu^{\alpha}\zeta^{\alpha}\right),$
(56) $\displaystyle
F^{\prime\prime}(\zeta)=(n-1)Cu^{\alpha}\zeta^{\alpha-1}\left(1+Cu^{\alpha}\zeta^{\alpha}\right)^{\frac{n-1-\alpha}{\alpha}}\left(\alpha+1+nCu^{\alpha}\zeta^{\alpha}\right).$

and $F^{\prime}(\zeta)>0$ for
$\displaystyle{Cu^{\alpha}\zeta^{\alpha}<-\frac{1}{n}}$,
$F^{\prime}(\zeta_{1})=0$ for
$\displaystyle{Cu^{\alpha}\zeta^{\alpha}=-\frac{1}{n}}$ and
$F^{\prime}(\zeta)<0$ for
$\displaystyle{Cu^{\alpha}\zeta^{\alpha}>-\frac{1}{n}}$. Thus the function
$F(\zeta)$ has a global maximum in the interval $[0,\infty)$ at the point
$\displaystyle{\zeta_{1}=Cu^{-1}\left(-\frac{1}{n}\right)^{\frac{1}{\alpha}}}$
and

$F(\zeta_{1})=Cu^{-1}\left(\frac{n-1}{n}\right)^{\frac{n-1}{\alpha}}\left(-\frac{1}{n}\right)^{\frac{1}{\alpha}}>0.$

The rest of the proof is identical with the proof of Theorem 2.4 and we omit
it. The proof of remark 4.1 is identical with the proof of Remarks 2.1 and
2.2.

∎

In practice the necessary and sufficient condition (52) corresponds to three
different scenarios for solution existence of (4), (5), which depend on the
values of the parameters $n$, $\alpha$, $Cu$, $b$ and $R$. Here, we present
some examples to illustrate each of these three cases.

  * •

(i) if (52) is strict inequality, then $\zeta_{1}$ is not reached for any
$Y\in[0,R]$, i.e., $\zeta(R)<\zeta_{1}$ and $\zeta_{1}=\zeta(Y_{1})$, where
$Y_{1}>R$. For example, at $n=-10$, $\alpha=10$, $Cu=b=R=1$, we have
$\displaystyle{Cu^{-1}\left(\frac{n-1}{n}\right)^{\frac{n-1}{\alpha}}\left(-\frac{1}{n}\right)^{\frac{1}{\alpha}}=0.7153}$,
$\displaystyle{\frac{bR}{2}=0.5}$ and (52) is fulfilled. Moreover,
$\zeta_{1}=0.7943$ and $Y_{1}=1.4305>R=1$. This means that if
$\zeta\leq\zeta_{1}$, then the classical solution exists everywhere inside the
pipe. From the equation $\displaystyle{F(\zeta)=\frac{bY}{2}}$, the inverse
function $\displaystyle{F^{-1}\left(\frac{bY}{2}\right)}$ is constructed
numerically using the computational algebra package MAPLE. Two different
functions $\displaystyle{F^{-1}\left(\frac{bY}{2}\right)}$ are found:

$F^{-1}_{1}\approx
0.02232Y^{9}-0.07589Y^{7}+0.08834Y^{5}-0.03763Y^{3}+0.5044Y,$
$F^{-1}_{2}\approx\frac{16.03Y}{(Y^{2})^{2.064}+16.73(Y^{2})^{0.5535}}$

The last function is not bounded at $Y=0$, i.e.,
$F^{-1}_{2}(0)=U_{2Y}(0)\rightarrow\infty$, which means that $U_{2Y}\neq 0$
does not fulfill the condition (5). Then, only the function $F^{-1}_{1}$
corresponds to the velocity $U(Y)$:

$U_{1}(Y)=0.00223Y^{10}-0.00949Y^{8}+0.01472Y^{6}-0.0094Y^{4}+0.2522Y^{2}-0.25026,$

which is solution of (4), (5).

  * •

(ii) if (52) is equality, then $\displaystyle{F(\zeta_{1})=\frac{bR}{2}}$ and
$\displaystyle{\zeta_{1}=F^{-1}\left(\frac{bR}{2}\right)}$ is reached at
$Y_{1}=R$. For example, at $n=-5$, $\alpha=3.9$, $Cu=b=R=1$, we have
$\zeta_{1}=0.662$, $Y_{1}=R=1$ and
$\displaystyle{F(\zeta_{1})=\frac{bR}{2}}=0.5$. The inverse function

$F^{-1}\left(\frac{bY}{2}\right)\approx
0.7761Y^{9}-1.301Y^{7}+0.798Y^{5}-0.1547Y^{3}+0.5085Y$

is again constructed numerically and

$U(Y)=0.07761Y^{10}-0.16262Y^{8}+0.133Y^{6}-0.03868Y^{4}+0.25425Y^{2}-0.26356$

which is solution of (4) and (5).

  * •

(iii) if (52) is not fulfilled, then $\zeta_{1}$ is reached at some inner
point $Y_{1}\in[0,R)$, such that $\zeta_{1}=\zeta(Y_{1})<\zeta(R)$. This means
that the equation (4) together with the first condition in (5) has solution,
which is not a solution of the problem (4), (5). For example, at $n=-3$,
$\alpha=2$, $Cu=b=R=1$, it is obtained that $\zeta_{1}=0.325$,
$Y_{1}=0.65<R=1$ and $\displaystyle{F(\zeta_{1})=0.325<\frac{bR}{2}}=0.5$.
Thus if $\zeta\leq\zeta_{1}$, the classical solution exists only in
$[0,Y_{1}]$. There is no solution in the rest of the pipe cross section, i.e.,
in $(Y_{1},R]$. In $[0,Y_{1}]$ the inverse function is

$F^{-1}\left(\frac{bY}{2}\right)\approx
48.1Y^{9}-33.83Y^{7}+8.55Y^{5}-0.4777Y^{3}+0.5168Y.$

This function corresponds to a single velocity profile

$U(Y)=4.81Y^{10}-4.2287Y^{8}+1.425Y^{6}-0.11942Y^{4}+0.2584Y^{2}+const.,$

with an unknown constant depending on an eventual additional condition at
$Y=Y_{1}$ or at $Y=0$.

The given examples are arbitrary chosen with the only aim to illustrate the
importance of the condition (52).

###  4.2. Revision of unsteady case

Another application of Theorem 2.4 and Theorem 4.1. is connected with the
unsteady Carreau-Yasuda flow in a pipe. We revise the results in [8], where
the following problem is studied

(57) $8\beta^{2}W_{T}-\frac{1}{Y}\frac{\partial}{\partial
Y}\left\\{\left[1-c+c\left(1+Cu^{\alpha}\mid
W_{Y}\mid^{\alpha}\right)^{\frac{n-1}{\alpha}}\right]YW_{Y}\right\\}=f(T),$
$\textrm{for}\quad T>0,\quad Y\in(0,R),\qquad$ (58)
$W_{Y}(T,0)=W(T,R)=0\quad\textrm{for}\quad T\geq 0,\quad
W(0,Y)=\Psi(Y)\quad\textrm{for}\quad Y\in[0,R],$

with $\Psi(Y)\in C^{4}([0,R])$ satisfying the compatibility conditions:

(59) $\Psi^{\prime}(0)=\Psi^{\prime\prime\prime}(0)=\Psi(R)=0.$

and $f\in C^{1}([0,\infty))$ bounded as

(60) $\sup_{T\geq 0}\mid f(T)\mid<\infty.$

The problem parameters are: $c\in[0,1)$, $\alpha>0$, $n\in\mathbb{R}$,
$\beta>0$.

As it is proved in [8], under the condition

(61)
$\alpha\left(1-\frac{\alpha+1}{n}\right)^{\frac{n-1-\alpha}{\alpha}}>\frac{1-c}{c},\quad
n<0$

equation (57) becomes forward-backward parabolic equation. If (61) holds, then
the function $h(\eta)$

(62) $h(\eta)=1-c+c\Phi(\eta),\quad\textrm{where}$ (63)
$\eta=\left|\frac{\partial W}{\partial
Y}\right|,\quad\Phi(\eta)=(1-n)\left(1+Cu^{\alpha}\eta^{\alpha}\right)^{\frac{n-1-\alpha}{\alpha}}+n\left(1+Cu^{\alpha}\eta^{\alpha}\right)^{\frac{n-1}{\alpha}}$

has two positive roots, $0<\eta_{1}<\eta_{2}$, such that (57) is forward
parabolic equation for $\eta\in[0,\infty)\setminus(\eta_{1},\eta_{2})$ and
backward parabolic one for $\eta\in(\eta_{1},\eta_{2})$.

We recall Theorem 3 in [8], where sufficient conditions for existence of
global solutions of (57)-(60) are given.

######  Theorem 4.2.

(Theorem 3 in [8]) Suppose $W(T,Y)\in C^{3}\left([0,T_{m}]\times[0,R]\right)$
is a solution of (57) - (60), $0\leq T_{m}\leq\infty$, $\alpha>0$, $\beta>0$,
$n<0$, $c\in(0,1)$ and (61) hold. If $K_{1}<\eta_{1}$, where $K_{1}$ is given
in (64) and $\eta_{1}$ is the first positive root of $h(\eta)$, then
$T_{m}=\infty$ and the estimates

$\left|W(T,Y)\right|\leq K_{1}\left(R^{2}-Y^{2}\right),$
$\left|W_{Y}(T,Y)\right|\leq K_{1}$

are satisfied for $T\geq 0,Y\in[0,R]$. The constant $K_{1}$ is defined in the
following way

(64)
$K_{1}=RM\left[1-c+c\left(1+Cu^{\alpha}\lim_{m\rightarrow\infty}(C_{m})^{\alpha}\right)^{\frac{n-1}{\alpha}}\right]^{-1},\quad\textrm{where}$
$M=\max\Bigg{\\{}\sup_{T\geq 0}\mid
f(T)\mid,\frac{1}{R}\sup_{Y\in[0,R]}\mid\Psi^{\prime}(Y)\mid,2\sup_{Y\in[0,R]}\frac{\mid\Psi(Y)\mid}{R^{2}-Y^{2}}\Bigg{\\}},$
$C_{m}=RM\left[1-c+c\left(1+Cu^{\alpha}(C_{m-1})^{\alpha}\right)^{\frac{n-1}{\alpha}}\right]^{-1},\quad
m=1,2,\cdots$ $C_{0}=\frac{RM}{1-c}$

By means of Theorem 2.4 and Theorem 4.1, we have the following precise variant
of the upper Theorem 4.2. For this purpose we take the right-hand side of (4)
as $b=\sup_{T\geq 0}\left|f(T)\right|$, $\alpha>0$, $\beta>0$, $n<0$,
$c\in(0,1)$ and when (16), (17) hold.

######  Theorem 4.3.

Suppose $\alpha>0$, $\beta>0$, $n<0$, $c\in(0,1]$ and (16), (17) hold. If

(65) $-U(Y)\leq\Psi(Y)\leq U(Y)\quad\textrm{for}\quad Y\in[0,R],$ (66)
$\max\Bigg{\\{}F^{-1}\left(\frac{R}{2}\sup_{T\geq 0}\mid
f(T)\mid\right),\sup_{Y\in[0,R]}\mid\Psi^{\prime}(Y)\mid\Bigg{\\}}<\zeta_{1}$

then the problem (57) - (60) has a unique global classical solution $W(T,Y)\in
C^{3}\left([0,\infty)\times[0,R]\right)$ and the estimate

(67) $\left|W_{Y}(T,Y)\right|\leq\zeta_{1}\quad\textrm{for}\quad T\geq 0,\quad
Y\in[0,R]$

is satisfied, where $\zeta_{1}$ is well defined in Theorem 2.4.

###### Proof.

From (53) and the comparison principle, Theorem 1 in [8], we get the estimates

(68) $-U(Y)\leq W(T,Y)\leq U(Y)\quad\textrm{for}\quad T\geq 0,\quad
Y\in[0,R],$ (69)
$\left|W_{Y}(T,Y)\right|\leq\left|U_{Y}(Y)\right|=\left|F^{-1}\left(\frac{R}{2}\sup_{T\geq
0}\mid f(T)\mid\right)\right|\quad\textrm{for}\quad T\geq 0$

The rest of the proof follows from (66), (69) and the global gradient
estimates, Theorem 2 in [8].

∎

### A Appendix

We will prove that $\zeta_{1}\equiv\eta_{1}$, where $\zeta_{1}$ is the first
positive zero of $F^{\prime}(\zeta)=0$ and $\eta_{1}$ is the first positive
zero of $h(\eta)=0$.

From (22) we define $Z=1+Cu^{\alpha}\zeta_{1}^{\alpha}$ as a solution of the
equation

$1+nCu^{\alpha}\zeta_{1}^{\alpha}=-\left(\frac{1-c}{c}\right)\left(1+Cu^{\alpha}\zeta_{1}^{\alpha}\right)^{\frac{1+\alpha-n}{\alpha}},\quad\textrm{i.e.,}$
$\left(\frac{1-c}{c}\right)Z^{\frac{1+\alpha-n}{\alpha}}+nZ-n+1=0.$

In the same way, $\eta_{1}$ is a solution of the equation

$1+nCu^{\alpha}\eta_{1}^{\alpha}=-\left(\frac{1-c}{c}\right)\left(1+Cu^{\alpha}\eta_{1}^{\alpha}\right)^{\frac{1+\alpha-n}{\alpha}},\quad\textrm{i.e.,}$

and for $X=1+Cu^{\alpha}\eta_{1}^{\alpha}$, we get

$\left(\frac{1-c}{c}\right)X^{\frac{1+\alpha-n}{\alpha}}+nX-n+1=0.$

Since $\zeta_{1}<\zeta_{0}$, where
$\displaystyle{Cu^{\alpha}\zeta_{0}^{\alpha}=-\frac{\alpha+1}{n}}$ and
$\eta_{1}<\eta_{0}$, where
$\displaystyle{Cu^{\alpha}\eta_{0}^{\alpha}=-\frac{\alpha+1}{n}}$, we conclude
that $\zeta_{1}=\eta_{1}$. Here $\eta_{0}$ is the maximum point for the
function $h(\eta)$, i.e.,

$h^{\prime}(\eta_{0})=0,h^{\prime}(\eta)=c(n-1)Cu^{\alpha}\left(1+Cu^{\alpha}\eta^{\alpha}\right)^{\frac{n-1-2\alpha}{\alpha}}\left(1+\alpha+nCu^{\alpha}\eta^{\alpha}\right)\eta^{\alpha-1}.$

### Acknowledgment

N.K. has been partially supported by the National Scientific Program
”Information and Communication Technologies for a Single Digital Market in
Science, Education and Security (ICTinSES)”, contract No D01-–205/23.11.2018,
financed by the Ministry of Education and Science in Bulgaria and by the Grant
No BG05M2OP001-1.001-0003-C01, financed by the Science and Education for Smart
Growth Operational Program (2018-2023).

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