Author: Dr. Jonathan Kenigson, FRSA

Author Affiliation: Athanasian Hall, Cambridge LTD (Cambridge, UK) *

The Navier-Stokes equation is a fundamental equation of Fluid Mechanics which describes the motion of an “Incompressible” fluid. More general and robust compression conditions have been adapted over the past 150 years. Gauss-Green Theory, Divergence Theory, Tensor Calculus, and Differential Geometry have permitted novel mathematical and physical insights into the Navier-Stokes equations [1]. The original equation was derived by Claude-Louis Navier and George Gabriel Stokes in 1845: In summary, it states that the rate of change of a fluid’s momentum is equal to the sum of the forces acting upon it [1]. This includes forces like gravity, viscosity, and surface tension. The equation is important in the study of a variety of fields, from aerodynamics to hydrodynamics. It’s also used to model various physical phenomena, from the flow of air around an airplane to the tides of the ocean [2]. Parabolic Partial Differential Equations (PDE) are so intimately to the theory of Diffusion via the Heat Equation that one often forgets – albeit momentarily – that Heat Kernels appear in Fluid Mechanics and wave mechanics as well. The Classical Heat Equation is a linear, second-order partial differential equation that describes the evolution of a diffusion function over time [3].

With both Energy and Classical pointwise techniques, Heat-Kernel analysis can be used to describe wave motion, fluid flow, and other phenomena [4]. If physical contexts are considered, it is evident how Heat and Fluid Transfer are twin processes and share critical commonalities. Heat Equations and Fluid Mechanics are useful for solving problems related to the propagation and time-dependent distribution of energy, such as finding the temperature distribution within a body of material [3]. Heat Kernel techniques can thus be used to model a wide variety of physical systems, from heat transfer in a building to the motion of particles in plasmas. Even where there is no patently obvious connection, the Heat Kernel acts as an *ersatz* for fluid-transfer and wave-transfer processes [3]. Heat Kernel approximation is a key aspect of Navier-Stokes Theory in both the Compressible and Incompressible regimes except, per the typical physical demands of Continuum Mechanics, Tensor formalisms must be introduced and associated notational conventions established [4][5].

Clearly, there is no single Navier-Stokes equation. There are different versions of the equations depending upon the initial and boundary conditions supposed for the fluid in question. The main distinction in Navier-Stokes Theory is between Incompressible and Compressible Fluids. However, a rich and varied literature addresses a plethora of peculiarities either derived from physical experiments on individual fluids (as in Applied Engineering) or more ponderous theoretical excursions into Well-Posedness, Existence and Uniqueness, and Stability Analysis, among other fields of inquiry. Our discussion herein will address the most fundamental commonalities assumed by most (but by no means all) of such approaches. Navier-Stokes in such forms typically assumes:

- That the fluid is Newtonian [6]. A Newtonian fluid is a type of fluid with a constant viscosity that is independent of the force applied to it. That is, its viscosity does not change due to a change in pressure, shear rate, or temperature. This type of fluid is named after Sir Isaac Newton, who first theorized its properties. Water is the most common example of a Newtonian fluid, as it has a constant viscosity regardless of the pressure, shear rate, or temperature. Other examples include air, carbon dioxide, and many gases. Non-Newtonian fluids, on the other hand, have a viscosity that changes with pressure, shear rate, or temperature. Common examples include honey, ketchup, and shaving cream.
- That we are dealing with Viscous Flow [6]. In fluid dynamics, Viscous Flow and Inviscid Flow are two different kinds of flow models used to describe the flow of fluids. Viscous Flow is the flow that is caused by viscosity, or the resistance of a fluid to flow due to its internal structure. The equations of motion that govern Viscous Flow are the Navier-Stokes equations, which are based on Newton’s law of motion and the laws of thermodynamics. Inviscid Flow, on the other hand, is flow that is not affected by viscosity. It is usually used to describe laminar flow, which is flow in which the fluid particles move in parallel layers. Inviscid Flow is governed by the Euler equations, which are derived from the conservation of momentum and energy.
- One assumes that the mass of the fluid is conserved throughout the entire lifecycle of the process under consideration [6]. This means that if a certain amount of mass is present in a system, the same amount will remain in the system. This law applies not only to fluids, but also to solids and gases. In Fluid Mechanics, it is applied to calculate the flow rate, or the amount of fluid flowing through a pipe per unit time. By understanding the conservation of mass, engineers can create more efficient systems for the transportation of fluids. Conservation of mass is also critical for understanding the behavior of fluids in a wide variety of applications, such as in water distribution systems, automotive cooling systems, and aircraft engines.
- One assumes that the momenta of particles in the fluid are conserved [6]. Conservation of Momentum is an important concept in Fluid Mechanics. This principle states that the total momentum of a system is conserved, meaning that it cannot be changed by external forces. This means that the momentum of a fluid flowing through a pipe is the same at the start and end of the pipe. To understand this concept better, let’s look at an example. If a fluid is flowing through a pipe and there are no changes or obstructions, then the momentum of the fluid will remain constant. However, if something like an obstacle is introduced, then the momentum of the fluid will either increase or decrease, depending on the direction and magnitude of the force.

From this point onward, it is necessary to determine whether one is modeling a fluid that is either Newtonian and Compressible or Newtonian and Incompressible. Incompressible fluids are fluids whose density remains constant, even when subjected to pressure. This means that the volume of the fluid does not change when pressure is applied. Examples of incompressible fluids include water and air at low pressure. Compressible fluids, on the other hand, are fluids whose density will change when subjected to pressure. This means that the volume of the fluid will decrease when pressure is applied. Examples of compressible fluids include air at high pressure, steam, and gases. The simplest case structure assumes Galilean Invariance of the Stress Tensor and Isotropy of the fluid.

- The Cauchy Momentum Equation applies in either case [7]. The Cauchy Momentum Equation is a fundamental expression used in the analysis of motion in physics. It states that the total momentum of a system is conserved and is equal to the sum of the momentum of all the individual particles in the system. This equation is used to describe the motion of objects in one, two, or three spatial dimensions and is derived from Newton’s second law of motion. Succinctly, the Cauchy Momentum Equation states that the rate of change of momentum of a system is equal to the net external force applied to it [7][8]. Theoretically, this equation can be used to calculate the acceleration, velocity, and position of an object over time. In practice, uncertainty in the initial conditions, measurement errors, and state changes in the system render pointwise solution either impossible or inordinately computationally expensive [7].
- In either compressible or incompressible flow, when one speaks of Stress in the context of mechanics, one typically implies a Tensorial quantity [9]. Navier-Stokes is no different, and methods that render one problem tractable in Continuum Mechanics typically afford insights into many ostensibly unrelated phenomena. Stress plays a major role in Continuum Mechanics, a branch of physics that deals with the behavior of continuous materials like fluid, solids, and gases. In Continuum Mechanics, stress is a measure of how much force is applied to a material per unit area. It is usually measured in Pascals, and it is one of the most important concepts in Fluid Mechanics. Stress can be further broken down into two types: normal stress and shear stress [10]. Normal stress is the force that is applied perpendicular to the surface of a material, while shear stress is the force applied parallel to the surface. Both types of stress are important for understanding the behavior of fluids and other materials in various situations. Stress is used to calculate the strength and elasticity of materials, as well as to determine flow patterns [10].
- In either compressible or incompressible flow, one typically assumes upon the isotropy of the fluid [11]. An isotropic fluid is a type of fluid that has the same physical properties in all directions. The most common example of an isotropic fluid is water, which is the same in any direction. Other examples include air, liquids, and gases, if they are in equilibrium. Isotropic fluids have several properties that make them desirable for engineering applications. For example, they can flow from one point to another without changing their behavior, making them ideal for fluid systems. Additionally, the fluid density is equal in all directions, so the forces caused by the fluid don’t have to be considered when designing a system. Finally, the viscosity of an isotropic fluid is constant in all directions, making it easier to predict its behavior [11].
- In either compressible or incompressible flow, one assumes the Galilean Invariance of the fluid in addition to the isotropy of the fluid. In its broadest form, Galilean Invariance states that the laws of fluid motion should be independent of the reference frame used to describe them [12]. In other words, the physical properties of a fluid should remain the same regardless of where you are observing it from or how it is moving. This invariance allows us to calculate the motion of a fluid in any reference frame, making it easier to understand and predict fluid motion. It also has important implications for the conservation of energy, since energy must remain constant when the reference frame is changed [12].
- In Compressible Flow, we typically ask that the Stress Tensor be linear [13]. The Linearity of the Stress Tensor is no different from Linearity defined on any arbitrary Tensor Field and is essentially identical to the paradigm employed in General Relativity for the Metric Tensor for WKB and related approximations. Roughly, a linear Stress Tensor is defined as a second-order tensor that maps a vector of forces onto a vector of displacements or speeds. This tensor is linear in the sense that it is unaffected by the coordinates in which it is expressed. It is also symmetric, meaning that the stresses in different directions are related in a predictable way. As is typical in Relativity and Fluid Mechanics, the Stress Tensor is represented by a three-by-three matrix, and it has nine components, which each describe different types of stress. These components are broken down into two categories: normal stresses, which describe compression and tension along a given axis, and shear stresses, which describe the internal friction between particles. Note that our first excursion into Stress did not employ Tensor formalism and was characteristically imprecise – a notion that the language of Tensor Calculus obviates almost entirely. Deviations from Stress Linearity can be explored via Perturbation Theory or Approximation Theory used to afford Real Linearization of locally or globally nonlinear fields [14].
- In cases of incompressible flow, it is no longer sensible to ask for linearity of the Stress Tensor. One instead obtains an Equation of State (EoS) for the “Deviatoric Stress Tensor” of a flow in 3-space [15]. The Stokes Stress Constitutive Equation describes the relationship between the viscous stress and the strain rate in the fluid [1][2]. More specifically, it states that the rate of strain in the fluid is proportional to the viscous stress. This equation is used to describe the motion of fluids, and it is especially useful in studying the behavior of fluids such as air and water. It can also be used to calculate the force required to move a fluid under certain conditions [15].

Works Consulted.

[1]. Gilles, Pierre, and Lemarié Rieusset. *The Navier∼ Stokes Problem in the 21st Century*. Chapman and Hall/CRC, 2018.

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[4]. Vassilevich, Dmitri V. “Heat kernel expansion: user’s manual.” *Physics reports* 388.5-6 (2003): 279-360.

[5]. Neuenschwander, Dwight E. *Tensor calculus for physics*. John Hopkins University Press, 2014.

[6]. Raymond, J-P. “Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions.” *Annales de l’IHP Analyse non linéaire*. Vol. 24. No. 6. 2007.

[7]. Goraj, Robert. “Transformation of the Navier-Stokes equation to the Cauchy Momentum equation using a novel mathematical notation.” *Applied Mathematics* 7.10 (2016): 1068-1073.

[8]. Green, Albert E., and Ronald S. Rivlin. “On Cauchy’s equations of motion.” *Zeitschrift für angewandte Mathematik und Physik ZAMP* 15.3 (1964): 290-292.

[9]. Brenner, Howard. “Navier–Stokes revisited.” *Physica A: Statistical Mechanics and its Applications* 349.1-2 (2005): 60-132.

[10]. Lazar, Markus, and Helmut OK Kirchner. “The Eshelby stress tensor, angular momentum tensor and dilatation flux in gradient elasticity.” *International Journal of Solids and Structures* 44.7-8 (2007): 2477-2486.

[11]. Condiff, Duane W., and John S. Dahler. “Fluid mechanical aspects of antisymmetric stress.” *The Physics of Fluids* 7.6 (1964): 842-854.

[12]. Ruggeri, T. “Galilean invariance and entropy principle for systems of balance laws.” *Continuum Mechanics and Thermodynamics* 1.1 (1989): 3-20.

[13]. Iosifidis, Damianos. “Solving Linear Tensor Equations.” *Universe* 7.10 (2021): 383.

[14]. O’Malley Jr, Robert E. “Singular perturbation theory: a viscous flow out of Göttingen.” *Annual review of fluid mechanics* 42 (2010): 1-17.

[15]. Dumbser, Michael, and Vincenzo Casulli. “A conservative, weakly nonlinear semi-implicit finite volume scheme for the compressible Navier− Stokes equations with general equation of state.” *Applied Mathematics and Computation* 272 (2016): 479-497.

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