*BY : Khawla Msheik*

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## Motivation of studying fluids

The story behind surveying fluids goes back to ancient people. Motion captured their sights and attention: the rhythmic percussion of soft shoreline waves, the entrancing swaying of leaves upon spring breezes, the dust caught in a sunlight ray, snowflakes swinging down the ground, Tsunamis crashing like huge beasts and washing down all sightseeing. They were skeptical about the air above them that filled the endless skies, and they feared the huge bulks of water, from oceans to lakes, springs and rivers, reverencing some of them, and describing others as the sacred paths to the divine self. But what is fear but the paving way to knowledge? It is at this level of recognition and conception, the level of doubt of fear, that one can start to wonder about the essence of such entities, and their fuzzy impact all around, and what astonishing results he can invest through them.

A scientific scrutiny synchronized with the appearance of human civilizations who chose to dwell near lakes and seas, and thus adaptively showed a practical knowledge of flows which was manifested in the design of flow systems (spears, sailing ships with oars) and particularly in hydraulic projects for irrigation systems, flood protection, water supplies (conduits, canals and wells), drainage, etc. But this pragmatic knowledge didn’t just depend on observatory and applicable tools, as it was noticed that Greek people started more scientific qualitative studies and postulations, and the most remarkable ones of that era were Archimedes principles (laws of buoyancy) applied on floating and submerged bodies. The majority of contact with fluids through these eras (especially at the level of patents as is the case of the invention of the barometer by Evangelista Torricelli in 1643) was almost at the level of static fluids, which are fluids in rest state, and which also have a minimized impact compared to fluids in motion as results show until today, and from here comes the out of question importance of « flows » which have made the headlines in the research field, and still. In fact, experiments and observations continued to prosper after Archimedes by many remarkable scientists: Leonardo da Vinci (1452 – 1519) who stated the equation of conservation of mass in one-dimensional steady-state flow and experimented with waves, jets, hydraulic jumps and eddy formation and with Edme Mariotte (1620 – 1684) who built the first wind tunnel and tested models in it. The first outstanding theoretical work wasn’t furnished until the 15th century when Sir Isaac Newton postulated his laws on motion and introduced the notion of linear fluids, now known as Newtonian fluids. A very little after that Daniel Bernoulli (1738) and Leonhard Euler (1757) did a pioneering, fundamental approach to write down the first set of diﬀerential equations describing the motion of fluid flow. Thanks to these equations, a colossal amount of studies and research had been conducted and a quite significant results and solutions for existing problems had been proved. Besides the distinguished approaches of both Froude and Reynold (to whom the Froude number and Reynold number correspond), it was until the 16th century that the next milestone in fluid analysis was hit by Claude-Louis Navier (1822) and George Gabriel Stokes (1842) who presented a mathematical justification of fluid flow mechanics. In particular, they wrote down the foundational axioms of fluid dynamics by exploiting Newton’s laws of motion (conservation of mass and momentum) and the first law of thermodynamics (energy conservation). These axioms are known nowadays by Navier-Stokes equations (NS) and have been the fulcrum of research in solving diﬀerent problems related to fluid flow. Various remarkable scientists after that have been able to add a scientific value and progress in the chapter of studying fluids especially studies related to understanding viscosity and turbulence such as Andrey kolmogorov and Geoﬀrey Ingram Taylor, and not to forget as well Ludwig Prandtl who remarkably introduced the Boundary-Layer theory in 1904.

It is intrinsic at this point to remark that all the above reviewed approaches to setting fluid equations do not have a » fundamental nature », but they are rather » phenomenological equations and for this reason one cannot ask too much of them « . When this concept was realized by the scientific community due to the fact that the foundational systems presented by Euler-Bernoulli and Navier-Stokes approach, in their complex absurd formulation, couldn’t answer questions using the available analysis tools, researchers started to look for condensed abbreviated models that translate the physics embodied in the preceding two systems. The main tool till today was the perturbation analysis, or asymptotic analysis, which limits the number of diﬃculties in the case of study and preserves at the same time its physical identity. Two main theories appeared as a result of such analysis: Shallow Water theory and Lubrication theory. This thesis will elaborate in the general context, the relation of such theories with NS system.

Nowadays, flows are a main factor in understanding most of the natural and biological phenomena, and a resolute factor in technological and industrial applications. It is of this high criticality not just for surviving, as was the motivation of ancient civilizations, but also for development and sustainability on earth. This triggered scientists and decision makers to be better equipped with theoretical understanding and capability to use experiments and numerical tools in order to increase the eﬃciency of the result.

In order to launch an exhaustive comprehensive study on fluids, one must narrow the targets of studying. In this context, it is good to remark that describing an entity, as a fluid, in the substantial scientific concept, can adopt several perspectives. The classifications done in this dissertation line up in a selected path that identifies with the chosen subdomain of study. Any other crucially important classifications are not included for the convenience, time and space matters. The first requisite piece of information to carry on before any study is realizing the notion behind the topic of study. As a part of matter, which is mainly split into two parts: fluids-solids, fluids consist of a number of particles with specific parametric quantities, or degrees of freedom, that gives tendency to its particles, on the contrary to solids, to reposition whenever a force is applied on it, simply we often say « to flow » preserving its macroscopic properties. Out of this definition we get three eﬀective elements of study: particles, forces, and degrees of freedom which direct the perspective of research on fluids targeting the outcome element of study: the flow.

**The relative Perspective of Motion: Macroscopic and Mi-croscopic**

At the level of particles, and according to the case of application or the phenomenon aimed for study, the fluids can be viewed either from a microscopic perspective, or from a macroscopic one, a classification better know in the mathematical description as Lagrangian or Eulerian description, respectively associated to the German scientists behind setting these two frames of study. At the microscopic level, the fluid is considered to constitute of bunch of separate individual element moving freely and independently, whereas at the macroscopic one, the fluid is regarded as a bulk entity sharing constitutional properties (mass, volume..) and exhibiting others (speed, temperature,..). Though the fundamental basis applied to derive the equations of motion in both frames are similar: Newton’s laws of classical mechanics, yet there is a basic diﬀerence related to the object variable in each. A good example that illustrates this diﬀerence is the pendulum (even though it is not a fluid case), in Eulerian frame of work, the main concentration is to determine the forces acting on the pendulum in terms of the velocity field and describe the evolution by determining the history of a specific position not in the moving pendulum but with respect to a fixed reference in space, whereas a Lagrangian study would be more concerned in determining the trajectory followed by a material point on the pendulum as it moves with time, the variables of the system become functions of the path lines of this point, and one of the approaches would be writing down a Lagrangian functional defined by a minimization of the potential and kinetic energies of the point, based on the postulate that the point will follow a path of minimum energy requirement.

**Lagrangian Frame**

In case of a fluid, this classification becomes sharp and intricate depending on the length scales compared to the size of individual particle in the fluid. If the flow is occurring in con-figurations where the mean free path of the particle, defined as the » average distance traveled by the particle before undergoing collision with another particle or barrier and after which it modifies its direction or energy », is crucially valuable compared to the characteristic lengths of the domain occupied by the fluid, in such case, a Lagrangian frame of reference associated to the particle is used to determine the history and prediction of its motion; the trajectory.

Definition 1.1. In a fluid occupying initially a domain Ω0, a fluid particle (mathematical definition) is defined by the family (ϕt(x0))t, where ϕt is a bijective map mapping the initial domain Ω0 to the domain Ωt occupied by all the particles initially present in Ω0:

ϕ : Ωq0 −→ Ωt

x0 7−→ϕt(x0).

Definition 1.2. The trajectory of a fluid particle x0 is the map t 7→X(t, ti, xi), where xi is the position occupied by this particle at time ti, and X(t, ti, xi) is defined by Xi(t) = X(t, ti, xi) := ϕt(ϕ−t1(xi)). {x0} is considered to be the frame of reference.

In this context, the velocity of the fluid particle {xi} at time t is given by vLag(t, X(t, ti, xi)) = X˙(t, ti, xi) = dtdX(t, ti, xi).

And so other variables such as acceleration, pressure, temperature, etc, are being expressed. Having determined the Lagrangian frame of reference, one can derive the Lagrangian equations of motion by determining the diﬀerent forces and interactions acted upon the particle preassuming that it follows Newton’s laws of classical mechanics. This would result in a system of the very general form

dtdXi = Mi,

dtdMi(t, X) = −γMi + Fi,

where

Mi = M(t, Xi) is the momentum of the particle,

Fi = F (t, Xi) is the sum of forces on the particle due to any external potentials or interactions, γMi is a viscosity term.

The Lagrangian frame of work subjoined two main domains in fluid dynamics known by Kinetic Theory and Molecular Dynamics (MD). In fact, where MD is based on models similar to the above one, the models in Kinetic Theory are based on averaging techniques of the above equations in which the main state variable is a kinetic distribution F (t, x, u) which is the density of the particles with velocity u found at time t in the position x. The applications in each subdomain are vast, a good example would be in biological flows as blood, where the diameter of the red blood cell, or its mean free path, is very close to the diameter of the capillaries in which it flows, thus imposing a Lagrangian frame of work.

Nevertheless, in the whole thesis we wouldn’t be treating such kind of flows, but rather we will be considering Eulerian frame of work.

**Eulerian Frame**

Viewing the fluid at the macroscopic level induces the notion of « continuum » which is on the contrary position to the notion associated to the microscopic perspective of fluid being a scattering of individually separate particles. In the Eulerian description, a fixed geometric reference attached to the underlying physical space is adapted, for example, the Cartesian space (x, y, z ) ⊂ R3 or polar coordinates (r, θ) in a time interval [0, T ]. This allows to express the states of the continuum object viewed as one physically unified entity in this time-spatial domain. Furthermore, the continuum hypothesis assures that the state and motion variables of the fluid are continuous in the metric reference topology.

Definition 1.3. The continuum hypothesis of fluid mechanics admits the following postulates:

1. a domain Ω ⊂ R3 occupied by a fluid in an ambient space;

2. a non-negative measurable function ρ = ρ(x) defined for t ∈ (0, T ), x ∈ Ω, yielding the mass density;

3. a vector filed u = u(t, x), t ∈ (0, T ), x ∈ Ω, defining the velocity of the fluid;

4. a positive measurable function θ = θ(t, x), t ∈ (0, T ), x ∈ Ω, describing the distribution of temperature measured in the absolute Kelvin scale;

5. the thermodynamic functions: the pressure p = p(ρ, θ), the specified internal energy e = e(ρ, θ), and the specific entropy s = s(ρ, θ);

6. a tensor T = {Ti,j}3i,j=1 yielding the force per unit surface that the part of a fluid on the other side of the same surface element exerts;

7. a vector field q giving the flux of the internal energy;

8. a vector field f = f(t, x),t ∈ (0, T ), x ∈ Ω, defining the distribution of a volume force acting on a fluid;

9. a function Q = Q(t, x),t ∈ (0, T ), x ∈ Ω, yielding the rate of production of internal energy

The hypothesis guarantees the validity of the laws of mechanics and thermodynamics de-scribing the basic state variables (ρ, u, θ). The other states are then expressed in terms of these latter variables through constitutional relations that play an important role in the classification of the third eﬀective element of study (degrees of freedom), that we will discuss later. As these relations are related to forces and tensors in fluid, I will keep them to the second section where forces are discussed. The Eulerian frame is adapted in the whole remaining literature.

**Volume and Surface Forces: A Story of Tensors**

**Body and Contact forces**

Consider an element in the fluid of volume δV . We can distinguish between two kinds of forces acting on the element:

• Long-range forces (body forces): which are forces acting at a distance without having a contact between the origin of the force and the element of volume δV , such as gravity, electromagnetism, and centrifugal forces, etc. Such forces act equally on all the matter inside the element, so we can assume that the total force on the element is proportional to the size of volume of the element. If δV is centered at a position X at a certain instant t, then the total body force is given by F (X, t)δV.

• Short-range forces: these are contact forces of molecular interaction origin. Their contri-bution is more obvious in liquids than in gases due to the fact that their values decrease rapidly with the increase in the distance separating the molecules of interaction (as it the case in gases’ molecules). The forces in fact occur on the membrane of the element which is in contact with another volume element δV 0. At this common membrane, two phenom-ena can take place: 1) the transport of momentum across it when interacting molecules are in oscillatory motion, and 2) forces between molecules on the two opposite side of the common boundary. Thus, such forces act on a surface element on the boundary of δV rather than the whole volume, and furthermore-unlike long-range forces- they don’t act equally in all regions of this surface element due to the diﬀerent orientations in diﬀerent areas. This means that we cannot express the total contact force on our chosen element in a proportional way, or any relevant way, with respect to the volume δV , but instead, the total force will be determined locally on plane surface (in the whole closed surface of δV on which contact forces act) of area δA as the total force exerted on the fluid on one side of the element surface by the fluid on its other side. Thus, the total force will be proportional to δA, and at a position X centering δA at a certain time t, it will be given by Σ(n, X, t)δA, where n is the unit normal to the element. Being defined then by total force per unit area, Σ is in fact called the local stress, and a normal component of Σ in the direction of n presents a tension.

Theorem 1.1. [6] Assume that the density field ρ, the velocity field u, and the body force density f are regular. Lets also assume that, the vector u being fixed, the function (t, X) → Σ(n, X, t) is continuous.

Then, there exists a tensor-valued function (t, X) → σ(X, t) such that for all (t, X) and for all unit vectors u we have Σ(n, X, t) = σ(X, t) • n. σ(X, t) is called the stress tensor of the fluid.

Since the tensor Σ across a plane surface is represented in the Cartesian reference (Eulerian description) by a vector in R3 in the same direction of the corresponding contact force, then the stress tensor is represented by a matrix in M3×3: σi,j represents the i-th component of the force per unit area exerted across a plane surface element normal to the j-th direction Moreover,σ is symmetric in the sense that σij = σji, and these latter are decomposed into two kinds of stresses:

• normal stresses represented by the diagonal entries σii which represent the amount of stretch or compression given by the normal component of the force applied on a plane surface element parallel to the i-th coordinate plane.

• shear stresses which represent the amount of distortion associated with the sliding of plane layers over each other represented by the six other entries σij.

Remark 1. Though the terminology « molecular » is associated to the surface forces described above, yet this doesn’t mean that their eﬀect is only valuable at the microscopic prospect. Nev-ertheless, their impact echos at the macroscopic level as well, and makes a significant diﬀerence in describing the motion of a fluid as a whole.

### Compressible and Incompressible Fluids

In the context of forces and tensors, it is good to remark that one of the main forces that operate on the fluid is the hydrostatic pressure. It is defined by the pressure that is exerted by a fluid at equilibrium at a given point within the fluid, due to the force of gravity. Hydrostatic pressure increases in proportion to depth measured from the surface because of the increasing weight of fluid exerting downward force from above. In fact, the range of such pressure gives rise to two types of fluids: Compressible and Incompressible fluids. In reality, all fluids are com-pressible, which means that they are vulnerable to change their density (or volume) upon acted external force. Nevertheless, as this variation is considered negligible compared to the absolute pressure in some cases, especially for liquids, then we assume that density is not changing upon exerted forces and motion, and the fluid is said to be incompressible.

**Hydrodynamics and Rheology: a story of viscosity**

The third category of classification is related to the degrees of freedom in a fluid, and by those we mean specific constitutional properties which characterize the identity of the fluid, and contributes to the variety of patterns of motion of diﬀerent fluids. These degrees of freedom serve to link motion discussed in section 1.2 with forces discussed in section 1.3 , thus completing the description of the deformation. In particular, such constitutional characteristics directly impact the surface forces discussed in the previous part which in turn impacts the motion. Two primarily properties we aim at studying here correspond in fact to the way the fluid responds upon inter molecular forces in two positions: at rest, and in motion. A fluid element at rest is vulnerable to stress eﬀect, mainly compression one. The stress tensor in this case is only formed of normal stresses (non-zero diagonal entries), whereas shear stresses are absent due to the absence of the bulk motion of the fluid. This consequently means that at rest, no resistance to motion exists by shear stresses as there is no motion supervening at the first place, and this doesn’t neglect the fact that the fluid still has the property of resistance if set to motion. It is convenient to write then the stress tensor as σ = −p Id, (1.1) where p is known as the hydrostatic (also called thermodynamic) pressure coming from com-pression eﬀects. The finding of viscosity and elasticity constitute in fact a part in a greater story of fluid mechanics: unveiling the enigma of a complete constitutional law for stresses. This latter need was definitely urgent for closing the kinetic description of the fluid’s (or solid’s) motion as will be discussed in next section (balance laws). Both viscosity and elasticity reveal the same tendency of resistance to shear motion, but it is conventional that the latter notion is used for solids whereas viscosity denotes the resistance in fluids. A rigorous scientific study of both notions synchronized starting from the 15th century, however, a primitive understanding of elasticity goes back to antiquity with the use of bows. Starting from the 15th century, especially with the work of Leonardo da Vinci on friction forces, attempts to understand and introduce these features to mechanics harmonized in a way that some milestone findings in both states of matter were delivered by the same person, and the same community. It is at this level that we can unwrap the classification done at the level of the constitutional behaviors of fluids, mainly giving rise to two classes of fluids: Newtonian fluids which are studied in Hydrodynamics and non-Newtonian fluids studied in Rheology. What is good about this category is that one paved the way to the other: the science setup started with Newtonian fluids (laws of Hydrodynamics), but when Hydrodynamics failed in interpreting various applications and phenomena, the branch of Rheology was introduced to serve the mission. Here the story continues in chronological order:

**Hydrodynamics: Study of Newtonian Fluids**

The name’s origin is merely related to Sir Isaac Newton who was the first to shed light on the viscous notion. In 1676, Robert Hook published the first quantitative concept of elasticity which showed a linear relation between forces and the extension of springs. In 1675, Edme Mariotte rediscovered the same law for fluids in France, and connected what was NOT known at that time by « stress » to the state of the fluid. But the real notion of viscosity wasn’t established until 1687 by Sir Isaac Newton. The story of viscosity goes back to the experiment that he did in 1687, know as the Couette flow. In his work “Philosophie Principia Mathematica”, Newton defines viscosity as The resistance which arises from the lack of slipperiness of the parts of the liquid, other things being equal, is proportional to the velocity with which the parts of the liquid are separated from one another.

Newton’s new interest in viscosity was due to his attempt to disprove Descart’s theory of cosmic vortices. He was wondering how spinning planets could entrain fluid bulks in a vortex motion. In the Couette flow experiment, Newton placed water between two parallel plates being apart by a distance d, one fixed on a plane surface, and the upper one moving with a speed u. He noticed that diﬀerent layers of the fluid possess diﬀerent speeds ranging from 0 at the first layer at the bottom to u at the top layer adjacent to the upper plate. His observation pointed out to the presence of a force that is resisting the relative motion of the layers. If layer A is moving with a constant speed uA, then the layer just below it, say layer B with constant velocity uB, will apply a force on A in an opposite direction to the motion of A, and thus an external force-acted by A-should be exerted on B to keep it moving with velocity uB. Newton signified that the total of such force between layers and particles, is inversely proportional to d, but directly proportional to u. Under the above hypotheses, Newtonian fluids satisfy the following properties:

1. The viscous stress tensor S in a flow depends only on the strain rate tensor D(u).

2. The dependence of S on D(u) is linear, i.e a Newtonian fluid is characterized by a constant viscosity.

3. The relation linking S to D(u) is isotropic, i.e it is invariant under a change of the orthog-onal frame of reference.

Newton’s fundamental finding gives rise to the notion of viscosity established through this rela-tion

F = µAud, (1.2) where F is the total resistance force acted upon the fluid, µ is the viscosity coeﬃcient and A is the total surface area on which F acts. Technically, the force due to viscous eﬀects per unit area F represents the viscous tensor S, and u is congruent to the rate of change of velocity, i.e the spatialA gradient of velocity, thus we can haved the following relation known as Newton’s law of viscosity S ∼ µru.

Later on, Euler, along with Bernoulli’s pioneering work on kinematics of deformation, set down a rigorous system for the motion of fluid using partial diﬀerential equations. Nevertheless, Euler neglected completely Newton’s laws of viscosity and these stresses, and his set of equations-till this day- govern the motion of a perfect fluid, known also for d’Alembert. The d’Alambert-Euler constitutional equation represents simply an abstract fluid at rest σ = −p Id .

Many mathematicians and physicist after that tried to further study elasticity and viscosity as dynamical quantities such as Coulomb and Young. At the level of elasticity, Young could elaborate a constitutional relation for solids involving shear stresses where he recognized a new concept: shear strain. As for the fluids, the constitutional relation wasn’t clear by this time, but scientists could only approve comprising a viscous eﬀect in it. In accordance with the property of the fluid of not being able to withstand any deformation tendency by applied forces without changing its volume, the pressure force acts equally in all directions, which is technically exhibited by being represented by a diagonal matrix of equal diagonal entries. Once the fluid is set to motion, shear stresses contribute again to a new resisting eﬀect to motion, the viscous eﬀect, which along with the hydrostatic pressure furnishes a general formulation of the stress tensor perfectly displayed by the following law σ = S − p Id, (1.3) with S being the viscous tensor. The form and dependency of this tensor on the state variables of the system remained a debatable question till the work of Navier. On the basis of Coulomb’s work, who elaborated strongly on shear stresses in fluids, Navier derived the fundamental equa-tion of elasticity [7]. He considered a system of spherical particles among which central forces operate, and the equation developed for small displacement was given by µ(r2u + 2rΘ) + f = 0, where Θ denoted the volumetric strain of the element, µ the shear modulus (which is related to Young’s modulus of elasticity) and f the external body forces.

In France 1827, Cauchy introduced the symmetric stress tensor D(u) = ru+ruT , and he set down the first complete realistic constitutional equation for solids that governed2 the basis of classical elasticity σ = ΛΘ Id +2η D(u).

Λ and η where called Lame’s coeﬃcients, Θ the volume strain given by div u. Simultaneously, a new investigation for motion was done at the level of viscous fluids by Navier. Similar to his first attempt of considering system of spherical bodies, but in fluid, he derived the first equation of motion to account for viscosity f − rp = ρDDut − µr2u, where µ being the viscosity, ρ the mass density,u the velocity and p the hydrostatic pressure. In 1845, Stokes added the last chapter in this story [8] joining the work of Cauchy and Navier, by fulfilling the notion of resistance to attempted or actual volumetric changes in a general fluid via a complete constitutional law that is still considered till today σ = −p Id +λ div u Id +2µ D u.

The term in front of λ expresses volumetric dilatation; the rate of change in volume which is in essence a change of the density, while the term in front of µ corresponds to the rate of linear dilatation—a change in shape with fixed volume. We notice here the presence of two viscosity coeﬃcients µ and λ. Those latter are also known by Lame’s coeﬃcients as called after Cauchy’s work. The first viscosity is also known as the dynamic or shear viscosity. It is associated to shear deformation and it is the same one spotted in Newton’s experiment. In his attempt to close the expressions of stresses, and upon the lack of measurements of λ, Stokes postulated a relation between the two viscosities so that the flow will depend on just one viscosity determined from the basis of empirical assumptions postulated in the continuum hypothesis. However this relation is not valid for all fluids, but it has been tremendously exploited in most of fluids as λ = −d2µ which means that the pure volumetric changes without shearing do not dissipate energy (see notion of bulk viscosity in the next paragraph). Stokes work was later supported by many experimental results. It is good to mention that the agreement between Stokes theory and experiment is related to boundary conditions on the wall of the capillary: no-slip boundary condition. Such law now is only guaranteed for fluids with small molecules, i,e Newtonian fluids, and under low shear stresses. In fluid dynamics, this strain tensor is decomposed into two parts in order to identify the diﬀerent deformations involved and in a way to split aside the diagonal involving div u which expresses isotropic deformations associated with volumetric dilatation from the diagonal-free part called the deviatoric part of the deformation rate tensor and associated to the rate of linear dilatation. In Md×d, we have D(u) = div u Id +(D(u) − div u Id). (1.4)

Bulk viscosity. In fact, Stokes relation gives rise to a third viscosity type known as bulk viscosity. Bulk viscosity, denoted µB = λ + 2 µ, is associated to normal stresses and related to change of volume in the fluid parcel. The notiond of this viscosity can be understood from this phenomenon: Supposing that we have an element fluid as a sphere, normally subjected to normal stresses from all directions being equal to the hydrostatic pressure p inside the fluid. Upon sudden work acted on the sphere, causing the normal stresses to increase in a quasi-static reversible manner, an increase in the internal energy will start to occur due to the first law of thermodynamics. This energy will be depicted in three forms of molecular motion: translation, rotation or vibration. Upon very rapid compression, a need of time-lag is noticed in order for this internal energy to re-partition into the diﬀerent modes. This time is due to a resistance property in the sphere which makes it exert pressure to oppose the volumetric constraint due the exerted work of com-pression. This eﬀect is treated as viscosity eﬀect and it is what we call bulk viscosity. Due to Stokes law of viscosity, which defines the bulk viscosity as a zero quantity, most of the studies on fluids neglect this kind of resistance. It is proven to be eﬃcient in supersonic flows (high Mach number) and multidimensional reaction flows.

#### Rheology: An Introduction To Non-Newtonian Fluids

Newton’s hypothesis on the behavior of a fluid in response to deformation were accurate in plenty of study cases. However, the rise of polymers in the industrial world had spot the light on a diﬀerent kind of behavior among these fluids that would refute Newton’s postulates. Newton’s linear constitutional law was no more valid to define the viscosity of such types of fluids-called by analogy non Newtonian- and as a result a need to a generalized constitutional law appeared. Non Newtonian fluids are defined as those fluids whose viscous stress tensor is not linearly depending on the rate of deformation, and thus the resisting behavior they exhibit depends on the state variables of the system, for example this resistance may increase with the increase in applied stress, or decrease (i.e to say viscosity depends on rate of deformation), it may be also time dependent, or have a yield stress, and so on. Non-Newtonian fluids come in a number of diﬀerent types: viscoelastic, Thixotropic, Rheopectic, Pseudoplastic, and dilatant. Our main concern in the thesis would be that of shear thinning fluids as Pseudoplastic fluids, and with viscoelastic fluids. As for the development of this branch of science, it would be surprising to know that curiosity and usage of non-Newtonian fluids predates its formal framing into a branch of science called Rheology, which was enunciated circa the twentieth century. The concept of « thinness » and « thickness » was illustrated in ancient knowledge by many applications as that of the water clock: Clepsydra, which originated in Babylon and migrated to Egypt and China, where water had been heated by those people in winter in order to keep time accurate, i.e they manipulated using heat the viscosity or « stickness » of water. Inspired by what the the Greek philosopher Heraklitus said: »Everything flows », Marcus Reiner, one of the eminent founders of Rheology, once said in reference to variable viscosity: « Everything flows if you wait long enough, even the mountains. »

This non linear behavior in matter started to be studied in the beginning of 19-th century. Weber (1841) was the first to highlight new observations in silk fibers and visco-elastic eﬀects. His experiment of a silk fiber attached to a load and then released freely to contract showed that the silk reattains its initial length, which means that the elasticity in silk fibers is not perfect, and he called this behavior visco-elastic eﬀect in metals. He presented a law on the observed extension by experiments x˙ = bx6.82.

Between 1847 and 1866, Kohlrauch continued the pioneering work of his father, and estab-lished the linearity of torsional phenomenon by separating time and magnitude eﬀects of the response through experiments. The first theoretical law towards Rheology was noted in 1867 by Maxwell who stated that « viscosity in all bodies may be described independent of hypothesis » by the equation

dσdt = E d D(dtu) − στ where E is Young’s modulus, and τ is a time constant. One firmly can say that Maxwell’s equation is the basic formula that all constitutional models and equations for solids and non Newtonian fluids were based on. However, at the time of Maxwell, and lacking the real inter-pretation of this formula, he used it experimentally to calculate the viscosity Eτ in gases.

Later, Oskar E Meyer in 1874 proposed that the shear stress σ and strain D(u) could be written in the form of σ = G D(u) + η dtd D(u) which describes what is known now by Kelvin-Voigt body. Kelvin then did damping experiments on metals using the law introduced by Meyer, and Voigt later in 1889 generalized Meyer’s ideas to anisotropic media. The next cornestone in Rheology of solids was set by Boltzman who is best known for contributions in Kinetic theory and entropy concepts. His additions were based on correcting Meyer’s work:

• he presented the concept of « fading memory »; he assumed that the stress at time t depends not only on the strain at that time, but also on those in previous times, and that the longer the time interval from present to past, the smaller is the contribution of strain to stress. His new notion of behavior is known as « fading memory ».

where wji = σxx = λΘ(t) + 2G(D(u))xx(t) − Z0 ∞ dwϕ1(w)Θ(t − w) − 2 Z0 ∞ dwϕ2(w)exx(t − w) where ϕi are memory functions and w = t − τ, with τ being a past time. Shear stresses where given in their usual form.

**Introduction **

**1 Introduction to Continuum Mechanics **

1.1 Motivation of studying fluids

1.2 The relative Perspective of Motion: Macroscopic and Microscopic

1.2.1 Lagrangian Frame

1.2.2 Eulerian Frame

1.3 Volume and Surface Forces: A Story of Tensors

1.3.1 Body and Contact forces

1.3.2 Compressible and Incompressible Fluids

1.4 Hydrodynamics and Rheology: a story of viscosity

1.4.1 Hydrodynamics: Study of Newtonian Fluids

1.4.2 Rheology: An Introduction To Non-Newtonian Fluids

1.5 Balance Laws

1.5.1 Mass Conservation

1.5.2 Momentum Conservation

1.5.3 Energy Conservation

1.6 Constitutional relations

1.7 Boundary conditions

1.8 Modelling : Scaling and Important Parameters

**2 A Scope on the Thesis **

2.1 Brief Historical Review on Modeling Gravity Driven Films

2.2 Shallow Water Theory

2.2.1 Shallow Water Theory From the Modeling Point of View

2.2.2 A Word On The Well Posedness Of Shallow Water Models

2.2.3 Contribution to the Thesis

2.2.3.1 Refined 2 and 3 Equation Models Using WRM

2.2.3.2 Bi-viscous Shallow Water model

2.3 Lubrication Theory

2.3.1 Lubrication Theory From the Modeling Point of View

2.3.2 Contribution to the Thesis

2.3.2.1 Bi-viscous Lubrication Equations

2.3.2.2 Existence Results for Some Lubrication-type Equations

2.3.2.3 Bingham Lubrication Equation Revisited

2.4 Existence Result for Degenerate Lake model for Bingham fluids

2.5 Dissipative Solutions for Oldroyd-B Fluids

**II Newtonian Flows with Free Surfaces **

**3 Derivation Of Viscous Newtonian Shallow Water Models **

3.1 Introduction

3.2 Preliminary System, Scaling and Main Results

3.3 Overview on the Momentum Integral Method (MIM) Justified in [1]

3.4 Overview on the Weighted Residual Method (WRM) Explored in [2]

3.5 Overview On The Three Equation Model Derived in [3]

3.6 A Revisit of the Three-Equation Approach

3.7 A 3-Equation Model for 2D and 3D flows Using the Weighted Residual Method

3.7.1 A 3-Equation Model for the 2D flow

3.7.2 A 3-Equation Model for the 3D flow

3.8 Numerical validation

**4 BD Entropy and BF Dissipative Entropy **

4.1 Introduction

4.2 The limit of a viscous shallow water model formally derived in [4] and justified in [5]

4.2.1 Formal limit

4.2.2 Mathematical justification

4.2.3 BF-entropy information include in the limit part of the BD-entropy

4.3 The limit of a viscous compressible system with a general drag term

4.3.1 Ansatz between n and m on the physical basis

4.3.2 Mathematical justification

4.3.3 Convergence of the BD-entropy

4.4 A more general framework

4.4.1 Limit problem

4.5 Appendix

**III Bi-viscous Rheology: Lubrication and Shallow Water Equations **

**5 A Lubrication Equation for a Simplified Model of Shear-Thinning Fluid **

5.1 Introduction

5.2 Mathematical model

5.2.1 Rheology

5.2.2 Scalings

5.3 Lubrication equation

5.4 Numerical illustrations

**6 Bi-viscous Shallow Water Model **

6.1 Introduction

6.2 Starting model, scaling choices and main result

6.2.1 Adimensionalized System and Boundary Conditions

6.2.2 Scaling and expansion

6.2.3 Main result – Shallow Water type systems

6.2.3.1 Comparison with Bingham model (6.1)

6.2.3.2 Comparison with Newtonian Model

6.3 Shallow-water equation derivation.

6.3.1 Main order profile and hydrostatic pressure constraint.

6.3.2 Calculating xz(0)

6.4 Conclusion

**IV Degenerate Lake System for Bingham Fluids **

**7 On The Rigid-Lid Approximation of Shallow Water Bingham Model **

7.1 Introduction

7.2 Functional spaces

7.3 Variational Inequality

7.4 Main Results

7.5 Newtonian fluids as a limit of Non-Newtonian fluids

7.6 Numerical Scheme

7.6.1 Semi-discrete scheme

7.6.2 Discrete scheme

7.6.3 When b ! 0

7.6.4 Boundary conditions

7.6.5 Simulation results

**V Dissipative Solutions for Oldroyd Systems **

**8 Dissipative Solution for Oldroyd Systems **

8.1 Introduction

8.2 Main Results

8.3 Dissipative solution for the Incompressible Oldroyd system

8.3.1 Modulated free energy

8.4 Appendix

8.4.1 System Setup

8.4.2 Free Energy of the Regularized System: A Priori Estimates

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Keywords : Free surface, Shallow water, lubrication, pseudo plastic, Bingham, BD entropy, BF entropy, Oldroyd.