Technology & Engineering
Navier Stokes Cartesian
The Navier-Stokes equations in Cartesian coordinates are a set of fundamental equations that describe the motion of fluid substances. They are used in computational fluid dynamics to model and simulate fluid flow in engineering and scientific applications. These equations account for the conservation of mass, momentum, and energy, and are essential for understanding and predicting fluid behavior in various engineering systems.
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9 Key excerpts on "Navier Stokes Cartesian"
- Kansari Haldar(Author)
- 2015(Publication Date)
- Chapman and Hall/CRC(Publisher)
4 Navier–Stokes Equations in Cartesian Coordinates 4.1 Introduction Fluid mechanics deals with the physical phenomenon of fluid flow by mod-elling this phenomenon of interest, generally, in the form of Navier–Stokes equations, which are nonlinear partial differential equations. Then the model needs an effective analysis to produce a result in accordance with observation and experiment, that is, the mathematical solution must be consistent with the physical reality. Therefore, the essentially nonlinear differential equations without simplifications, a customary in research in physics and mathematics, should be solvable. Since the Navier–Stokes equations are nonlinear in character, it is very difficult to obtain analytic solutions of these equations, except in some special cases. Therefore, we take the help of linearization and restrictions, which reduce the problem to a mathematically tractable one. But the so-lution of the reduced problem is not consistent with the real world of physics and deviates much from the actual solution of the original prob-lem. As a result, we take the advantage of traditional numerical techniques in order to get a sufficiently accurate result, and the methods result in massive computations. In this chapter we have applied the decomposition method, originally developed by Adomian [1, 2], to two-dimensional Navier–Stokes equations in Cartesian coordinates and have developed the theory as far as possible. We have also considered some fluid flow problems from the boundary-layer theory of Schlichting [6] for clear illustration of the theory. 4.2 Equations of Motion Consider two-dimensional flow of viscous incompressible fluid in the planes parallel to the reference plane z = 0 . Let u ( x, y ) and v ( x, y ) be the compo-nents of velocity in the x and y directions, respectively. Then the momentum 39
eBook - ePubMathematical Economics
Mastering Mathematical Economics, Navigating the Complexities of Economic Phenomena
- Fouad Sabry(Author)
- 2023(Publication Date)
- One Billion Knowledgeable(Publisher)
Chapter 2: Navier–Stokes equations
The Navier–Stokes equations (/nævˈjeɪ stoʊks/ nav-YAY STOHKS) are partial differential equations which describe the motion of viscous fluid substances, Claude-Louis Navier and George Gabriel Stokes were the namesakes. They were both French engineers and physicists.They are the culmination of decades of research and incremental theory development, spanning from 1822 (Navier) to 1842-1850 (Stokes).Momentum conservation and balancing are quantitatively expressed by the Navier-Stokes equations for Newtonian fluids. An equation of state connecting pressure, temperature, and density is sometimes provided alongside them. Assuming that the stress in the fluid is equal to the product of a diffusing viscous component (proportional to the gradient of velocity) and a pressure term, they result from applying Isaac Newton's second law to fluid motion. The Navier-Stokes equations are similar to the Euler equations, but the Euler equations only model inviscid flow, whereas the Navier-Stokes equations also account for viscosity. With this trade-off in mathematical structure comes the improved analytic features of the Navier-Stokes, which are a parabolic equation (e.g. they are never completely integrable).The physics of many phenomena of scientific and engineering interest may be described by the Navier-Stokes equations, making them a useful tool. You may use them to simulate everything from weather to ocean currents to water flow in a conduit to air flow over a wing. Aircraft and automobile design, blood flow research, power plant construction, environmental impact assessment, and many more fields all benefit from the full and simplified Navier-Stokes equations. They can be used to model and analyze magnetohydrodynamics when coupled with Maxwell's equations.In a strictly mathematical sense, the Navier-Stokes equations are also of enormous interest. The existence of smooth solutions in three dimensions, that is, solutions that are endlessly differentiable (or even just limited) at all locations in the domain, has not been established, despite the fact that they have a broad variety of useful applications. This is known as the existence and smoothness problem for Navier-Stokes equations. For a solution or counterexample, the Clay Mathematics Institute is offering a million dollars. They consider this to be one of the seven most important open issues in mathematics.
eBook - PDF- Jos Stam(Author)
- 2015(Publication Date)
- A K Peters/CRC Press(Publisher)
This time it stands for nu basically the Greek version of n. Again do not ask me why they chose this symbol. † At this point, some experts carrying their fluid mechanics luggage will ask: what happened to our dear pressure? If you want the pressure for an incompressible fluid, I can give it to you at any time. See below. Euler–Newton Equations or Navier–Stokes Equations ◾ 65 pretty complicated right? Still they are simple compared to the Lagrangian of the Standard Model in particle physics. Remember Figure 2.12. The Navier–Stokes equations are even scarier looking when the velocity of the fluid is described in spherical coordinates : each point is now described by two angles and a radius. The two angles determine a point on a sphere of radius one and the radius models by how much the point is removed from the center of the sphere. These equations are shown in Figure 3.24. There is a more compact version however. I was able to write down the Navier–Stokes equations in the sand while on vacation on one of my many trips to the beach of Puerto Vallarta in Mexico as shown in Figure 3.25. I used part of a wooden stick I found to write a more elegant version of the Holy Scriptures. This is of course not my invention. Just a standard FIGURE 3.23 The usual form of the Navier–Stokes equations in Cartesian coordinates. FIGURE 3.24 The Navier–Stokes equations in spherical coordinates. 66 ◾ The Art of Fluid Animation condensed version. But it was fun to write these equations down on the beach of Puerto Vallarta. Why did Navier and Stokes get all the credit and not Euler and Newton? Claude-Louis Navier (1785–1836) was a French engineer who studied at the École des Ponts et Chaussées .* He went onto become a mathematician, however, and he was the first person to combine Euler and Newton’s work to formulate the equations shown in Figure 3.23.
eBook - PDF- P.G. Ciarlet, CIARLET(Authors)
- 2003(Publication Date)
- Elsevier Science(Publisher)
C HAPTER I The Navier–Stokes Equations for Incompressible Viscous Fluids Introduction: Synopsis The Navier–Stokes equations have been known for more than a century and they still provide the most commonly used mathematical model to describe and study the motion of viscous fluids, including phenomena as complicated as turbulent flow . One can only marvel at the fact that these equations accurately describe phenomena whose length scales (resp. time scale) range from fractions of a millimeter (resp. of a second) to thousands of kilometers (resp. several years). Indeed, the Navier–Stokes equations have been validated by numerous comparisons between analytical or computational results and experimental measurements; some of these comparisons are reported in Chapter IX (see also C ANUTO , H USSAINI , Q UARTERONI and Z ANG [1988, p. 29], L ESIEUR [1990], G UYON , H ULIN and P ETIT [1991]). The content of this chapter is as follows: • In Section 1 we shall briefly discuss the derivation of the Navier–Stokes equations . • In Section 2 we shall address the important issue of the boundary conditions . • In Section 3 we shall discuss the stream function-vorticity formulation of the Navier–Stokes equations. • In Section 4 we shall introduce functional spaces (of the Sobolev type) and use them to derive variational formulations of the Navier–Stokes equations in Section 5. • Finally, in Section 6, we shall mention some mathematical results concerning the existence and/or the uniqueness of the solutions to the Navier–Stokes equations. 1. Derivation of the Navier–Stokes equations for viscous fluids Let Ω be an open and connected region (i.e. a domain ) of R 3 filled with a fluid . The generic point of R 3 will be denoted by x = { x i } 3 i = 1 while d x will denote the elementary volume d x 1 d x 2 d x 3 .
eBook - PDF- William Graebel(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
Combining these with the momentum equation, we have the NavierStokes equations. Its uses are discussed along with some simple applications. 1. The Local Continuity Equation In order to derive our local equations, which hold true at any point in our fluid, we locate a general point in the fluid and draw an infinitesimal box around it. This box is our control volume. For derivation purposes, it is easiest to work in Cartesian coordinates; the box will then have dimensions dx by dy by dz, as shown in Figure 4.1. Positive sign convention for the velocity components are given. We represent the density and velocity components at the center of the box by ρ, v x , v y , and v z , respectfully. Since these quantities vary throughout the fluid, they will vary also throughout our box. We represent this variation by considering 194 Differential Analysis the quantities at any point as the zero- and first-order terms in a Taylor series. Thus density for instance would vary within the box according to + higher order terms. Here the subscript 0 is the center of the box, and (x, y, z) is any point in the box. Derivatives are evaluated at the box center. In order to obtain our local continuity equation, we will sum mass flow rates and find that the lowest order terms are all multiplied by the box volume. Dividing by the volume and then letting the volume shrink to zero eliminates all higher order terms, so we need not consider them. A figure showing the variation of all components needed in three dimensions to our analysis soon becomes very complicated. In Figure 4.2 we show the simplified case for two dimensions. Table 4.1 summarizes the mass rates of flow in three dimensions for the six faces of our box. Recall from the previous chapter that the mass rate of flow is given by m=ρv normal A, where A is the surface area and v normal is the velocity component normal to that area.
eBook - PDF- P. A. Lagerstrom(Author)
- 2022(Publication Date)
- Princeton University Press(Publisher)
CHAPTER ONE The Navier-Stokes Equations for a Viscous Heat-Conducting Compressible Fluid B,l. Introduction. Physical foundations. In this chapter a general system of equations for fluid mechanics is developed. It includes the classical Navier-Stokes equations suitably modified to take into account heat conduction and changes in density and temperature, as well as an energy law and one or more equations of state, taken from equilibrium thermodynamics. This system is referred to simply as the Navier-Stokes equations, and the subsequent theoretical treatment of the present section is based on these equations or on approximations to these equations. It is generally believed that the Navier-Stokes equations are capable of describing most phe- nomena observed in fluid mechanics. However, it should be remembered that any useful system of equations for fluid mechanics may always be regarded as an approximation to a more accurate system. There are indications that certain special flow problems arising, for example, in the dynamics of strong shock waves or highly rarefied gases, may require more accurate equations for an adequate treatment. For a further dis- cussion of such problems the reader is referred to I,D, 1,1, and III,H. In principle, the range of validity of the Navier-Stokes equations should be apparent from a derivation of these equations. There are two different schemes for deriving equations for fluid mechanics. One 1 This Section was completed in 1956 and it was not possible to revise it to include newer results and appropriate references to more recent publications. Various people, in particular GALCIT staff members, have rendered invaluable help to the author in preparing this section. H. W. Liepmann worked as co-author during the early stages of writing. Anatol Roshko and Donald Coles are in essence the authors of the articles on experimental results.
Available until 4 Dec |Learn moreFluid Mechanics
An Intermediate Approach
- Bijay Sultanian(Author)
- 2015(Publication Date)
- CRC Press(Publisher)
313 7 Navier–Stokes Equations: Exact Solutions 7.1 Introduction In Chapter 6, we discussed potential flows, which are ideal fluid flows with constant density and zero viscosity. Although a fluid flow with constant density is physically pos-sible, one with zero viscosity is not. Nevertheless, one of the major contributions of the potential flow theory, which forms most of classical fluid mechanics, has been to provide a good insight into how a lift force is generated on a body. Euler’s equations governing potential flows provide a system of partial differential equations whose numerical solu-tions (using a CFD method) yield three-dimensional results in complex geometries. These solutions, without the wall boundary layers found in real flows, provide a good insight into these flows with their reasonably accurate static pressure distributions. A logical next step in our journey to understand real fluid flows is to introduce constant fluid viscosity in the mathematical modeling of these flows. The resulting governing partial differential equations are the Navier–Stokes equations, which are discussed at some length in this chapter along with their few exact solutions under special situations of fully developed two-dimensional laminar flows. In a number of practical applications such as electronics cooling, bio-fluid mechan-ics, and lubrication, the flow may indeed be modeled as a laminar flow. To start with, turbulent flows were numerically analyzed like a laminar flow with the fluid viscosity modified by the so-called turbulent viscosity, a practice continued to date in most CFD methods overviewed in Chapter 10. The material presented in this chapter provides a good framework and building blocks for developing better intuitive understanding of internal shear flows. 7.2 Forces on a Fluid Element 7.2.1 Surface Forces due to Stresses Figure 7.1 shows all nine components of the 3 × 3 stress tensor at a point on a differential fluid element.
- Heinz-Otto Kreiss, Jens Lorenz(Authors)
- 1989(Publication Date)
- Academic Press(Publisher)
1 The Navier-Stokes Equations In this preliminary chapter we first outline some questions which will be treated in this book. Then we derive the Navier-Stokes equations. Though the deriva- tion will not be used later, it is of interest to understand the underlying logical and physical assumptions, because the mathematical theory of the equations is not complete. There is no existence proof except for small time intervals. Thus it has been questioned whether the N-S equations really describe general flows. If one changes the stress tensor such that diffusion increases when the velocities become large, then existence can be shown. This change of the equa- tions does not seem to be justified physically, however. For example, certain similarity laws - valid for the Navier-Stokes equations - are well-established experimentally, but the modified equations do not allow the corresponding sim- ilarity transformations. Possibly a lack of mathematical ingenuity is the reason for the missing existence proof, and the N-S equations are physically correct. The N-S equations form a quasilinear differential system, and much of our un- derstanding of such systems is gained through the study of linearized equations. These will, in general, have variable coefficients. By fr-eezing the coefficients in such a problem, one obtains systems with constant coefficients. It is much easier to analyse the latter, as will be later shown in Chapter 2. However, the relation between variable-coefficient and constant-coefficient equations is not trivial. The fundamental ideas of linearization and localization are discussed in Section 1.3. 2 Initial-Boundary Value Problems and the Navier-Stokes Equations 1.1.
eBook - ePub- William S. Janna(Author)
- 2020(Publication Date)
- CRC Press(Publisher)
If the continuity and momentum equations are written for all three principal directions, and the fluid is Newtonian with constant properties of density and viscosity, a set of differential equations results. The momentum equation written for each principal direction gives what are called the Navier–Stokes equations. Their derivation is lengthy, involved, and beyond the scope of this text. The equations will be given without derivation here, but interested readers may refer to the specific references cited at the end of the book. FIGURE 11.1 A differential fluid element. For the general problem in fluid mechanics, assuming that we have a Newtonian fluid with constant properties, the governing equations in Cartesian coordinates are the following. Continuity equation: ∂ ρ ∂ t + ∂ (ρ V x) ∂ x + ∂ (ρ V y) ∂ y + ∂ (ρ V z) ∂ z = 0 (11.1) Navier–Stokes. equations: x -component: ρ ∂ V x ∂ t + V x ∂ V x ∂ x + V y ∂ V x ∂ y + V z ∂ V x ∂ z = − ∂ p ∂ x + μ ∂ 2 V x ∂ x 2 + ∂ 2 V x ∂ y 2 + ∂ 2 V x ∂ z 2 + ρ g x (11.2a) y -component: ρ ∂ V y ∂ t + V x ∂ V[. --=PLGO-SEPARATOR=--]y ∂ x + V y ∂ V y ∂ y + V z ∂ V y ∂ z = − ∂ p ∂ y + μ ∂ 2 V y ∂ x 2 + ∂ 2 V y ∂ y 2 + ∂ 2 V y ∂ z 2 + ρ g y (11.2b) z -component: ρ ∂ V z ∂ t + V x ∂ V z ∂ x + V y ∂ V z ∂ y + V. z ∂ V z ∂ z = − ∂ p ∂ z + μ ∂ 2 V z ∂ x 2 + ∂ 2 V z ∂ y 2 + ∂ 2 V z ∂ z 2 + ρ g z (11.2c) The left-hand sides of Equations 11.2 are acceleration terms. These terms are nonlinear and present difficulties in trying to solve the equations. Even though a variety of exact solutions for specific flows have been found, the equations have not been solved in general—owing primarily to the presence of the nonlinear terms. The right-hand side of the equations includes pressure, gravitational or body, and viscous forces. In polar cylindrical coordinates, these equations are Continuity equation: ∂ p ∂ t + 1 r ∂ (ρ r V r) ∂ r + 1 r ∂ (ρ V θ) ∂ θ + ∂ (ρ V z) ∂ z = 0 (11.3) Navier–Stokes
