Technology & Engineering
Euler's Equation Fluid
Euler's equation for fluid dynamics is a set of equations that describe the motion of an inviscid fluid. It is based on the principles of conservation of mass and momentum and is widely used in the study of fluid mechanics. The equation is named after the Swiss mathematician Leonhard Euler, who made significant contributions to the field of fluid dynamics.
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10 Key excerpts on "Euler's Equation Fluid"
eBook - PDF- P. C. Deshmukh(Author)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
Rudiments of Fluid Mechanics CHAPTER 11 Surrender to the flow of the River of Life, yet do not float down the river like a leaf or a log. While neither attempting to resist life, nor to hurry it, become the rudder and use your energy to correct your course, to avoid the whirlpools and undertow. —Jonathan Lockwood Huie 11.1 THE EULER’S EQUATION OF MOTION FOR FLUID FLOW In the previous chapter, we studied the Kelvin–Stokes theorem. When restricted to just two dimensions, it is referred to as the Green’s theorem which is therefore only a special case of the Kelvin–Stokes theorem. Using the Green’s theorem, we are automatically led to another prodigious theorem in the analysis of complex functions, known as the Cauchy’s theorem. The methods developed in Chapter 10 are of great importance in fluid dynamics, electrodynamics and also in quantum dynamics. Practical applications of these methods are abundantly found not merely in fluid dynamics but also in the study of plasma in stellar atmosphere, and the analysis of charge plasma produced by lasers. The techniques are therefore of great consequence in engineering and technology, apart of course, from basic sciences. In this chapter, we shall use these methods to develop the equation of motion for a classical fluid. The central question in mechanics is how to describe the state of a system, and how the system evolves with time. We have learned in earlier chapters that the state of a material particle system is represented by its position and momentum in the phase space. The temporal evolution of the system is provided by its equation of motion, viz., the Newton’s equation of motion, or equivalently by the Lagrange or the Hamilton’s equation of motion, which are discussed in earlier chapters. Much of the study of classical mechanics is about setting up the equation of motion, and learning to solve the same. When the medium is the continuum fluid, the system is very complex.
eBook - PDFPhysical Oceanography
A Mathematical Introduction with MATLAB
- Reza Malek-Madani(Author)
- 2012(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 6 Equations of Fluid Dynamics In this chapter we develop the equations of motion for fluid flows in an inertial coordinate system and present examples of exact solutions of this fundamental set of equations. In the subsequent chapter we develop the same equations in a rotating coordinate system, the equations of Geophysical Fluid Dynamics , and concentrate on the properties that distinguish motions of fluids in an inertial frame from the ones in a rotating frame. The equations we develop in this section are general and describe flows with temporal and spatial scales that typically arise in laboratory experiments. We consider fluid flows that are smooth, i.e., do not have sharp discontinuities and shock waves, and are often referred to as lam-inar flows . We do not develop the description of turbulent flows in this text, since the proper description of these flows requires a different set of mathematical tools, including statistical tools, which deserve their own dedicated treatment. One way to view our approach here and in the next chapter, where we consider large-scale oceanographic flows, is to note that the mathematical models we study are intended to exhibit the averaged behavior of the phenomena of interest. We refer the reader to Chapter 8 of [1] for a thorough treatment and development of the methodology that is commonly adopted when dealing with turbulent flows. We will now develop the traditional mathematical description of a fluid flow, its Eulerian description, where the attributes of a fluid ele-ment are described by the position occupied by the fluid element. That is, the velocity of the fluid element, or its temperature and salinity, or the pressure induced on it, are quantified by position and time, and not by the distinguished fluid element.
- Philip M. Gerhart, Andrew L. Gerhart, John I. Hochstein(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
The final basic restriction on use of the Bernoulli equation is that there are no mechanical devices (pumps or turbines) in the system between the two points for which the equation is applied. These devices necessarily cause unsteady flow or interrupt the streamlines (Ref. 4), if they are pres-ent. Analysis of flows with pumps and turbines is covered in Chapters 5 and 12. In this chapter we have spent considerable time investigating fluid dynamic situations gov-erned by a relatively simple analysis for steady, inviscid, incompressible flows. Many flows can be adequately analyzed by use of these ideas. However, because of the rather severe restrictions imposed, many others cannot. An understanding of these basic ideas will provide a firm foundation for the remainder of the topics in this book. 3.9 Chapter Summary and Study Guide In this chapter, several aspects of the steady flow of an inviscid, incompressible fluid are discussed. Newton’s second law, F = m a , is applied to flows for which the only important forces are those due to pressure and gravity (weight)—viscous effects are assumed negligible. The result is the often-used Bernoulli equation, which provides a simple relationship among pressure, elevation, and velocity variations along a streamline. A similar but less often used equation is also obtained to describe the variations in these parameters normal to a streamline. The concept of a stagnation point and the corresponding stagnation pressure is introduced, as are the concepts of static, dynamic, and total pressure and their related heads. Several applications of the Bernoulli equation are discussed. In some flow situations, such as the use of a Pitot-static tube to measure fluid velocity or the flow of a liquid as a free jet from a tank, a Bernoulli equation alone is sufficient for the analysis.
eBook - PDF- Yulii D. Shikhmurzaev(Author)
- 2007(Publication Date)
- Chapman and Hall/CRC(Publisher)
Then, using the definition of material derivative (2.7) we arrive at (2.8). If the Eulerian description of motion is known, the corresponding La-grangian one can be obtained in the following way. By integrating a set of ordinary differential equations dx u ( x, y, z, t ) = dy v ( x, y, z, t ) = dz w ( x, y, z, t ) = dt, (2.9) one can find trajectories of fluid particles, where the three constants of inte-gration can be interpreted as the Lagrangian coordinates individualizing the trajectories. The result will have the form of (2.5). Given that all parameters of the fluid are known as functions of the Eulerian variables, one can use (2.5) to express them in terms of the Lagrangian ones. It should be noted that in a general case trajectories of fluid particles differ from the streamlines, which are a set of lines tangential to the velocity field at a given moment. The latter are determined by the set of equations dx u ( x, y, z, t ) = dy v ( x, y, z, t ) = dz w ( x, y, z, t ) = dλ, (2.10) where λ is the independent variable, and t is a parameter. For steady flows, streamlines and trajectories obviously coincide. If the flow is specified in the Lagrangian variables, one can also obtain its description in the Eulerian ones. Adding the velocity field found from (2.6) to other parameters of fluid known as functions of ξ 1 , ξ 2 , ξ 3 and t we can then find ξ 1 , ξ 2 , ξ 3 as functions of x 1 , x 2 , x 3 and t from the set of algebraic equations (2.5). 2.2 Governing equations The set of equations of fluid mechanics comprise two subsets: the equations expressing fundamental conservation laws (conservation of mass, momentum, Governing equations 25 angular momentum, energy) and supplementary equations, which include con-stitutive equations and equations of state. Conservation laws are essentially the same for all media, and their mathematical representation defines and quantifies the basic characteristics to be used to describe the fluid.
eBook - PDF- E. Bobok(Author)
- 1993(Publication Date)
- Elsevier Science(Publisher)
Typical cases for which this assumption is valid are flow through orifices, the discharge from large tanks, the flow around aerofoils, etc. In such problems the flow of the main fluid mass has the greater importance. Viscous effects are signifi- cant only in the flow immediately adjacent to the solid boundaries; moving away from the walls the shear stresses decrease rapidly. That region of the flow where fluid friction is important is known as the boundary layer. The main flow, outside of the boundary layer can be described by the equations derived for perfect fluids. 60 4 PERFECT FLUID FLOW 4.2 Euler’s equation The equation of motion for a perfect fluid is obtained by substitution the stress T= -PI, (4.2) tensor into Cauchy’s equation of motion: aa 1 . -=G+ -Dlv(-pI) at e (4.3) After expanding the terms representing the local and convective components of the acceleration, and taking into account the identity Div ( -PI) = -grad p , the result is: (4.4) aa 1 - +(fiV) a=G- -gradp. This equation of motion is the so-called Euler equation. Its physical meaning is obvious: the acceleration equals the sum of body force and the pressure force acting on a unit mass of flowing fluid. Its components in Cartesian coordinates are: at e av, av, avx ax y ay aZ e ax at at ax ay aZ y ay 9 av, av av av at -++v,-++v -+ o au.= 9 x -- -9 aP - avY +v,- avy +o,- avy + v , L av = g - - 1 - aP -+ + o , L + u ---“+v I= aP 92- - - ax ay aZ e Euler’s equation can be also written in cylindrical coordinates: av, av av v2 av l a -+ , L + X L -2 +v2- =g,- - -(rp), at ‘ a r r acp r ar er ar av av av av 1 aP -+,,A + 2 A + 2.2 + v 2 3 = g - - -, at ar r acp r aZ ’ er aq av, av, av, av, 1 aP +v,- + 2 - +v,- =g2- - -. at ar r acp a Z e - (4.5) Following Lamb Euler’s equation can be further modified. The convective component of the acceleration may be expanded by applying the identity: V L (aV)C=grad- +curlaxi, 2
- Andrew L. Gerhart, John I. Hochstein, Philip M. Gerhart(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
82 CHAPTER 3 Elementary Fluid Dynamics—The Bernoulli Equation In general it is not possible to integrate the pressure term because the density may not be constant and, therefore, cannot be removed from under the integral sign. To carry out this integration we must know specifically how the density varies with pressure. This is not always easily determined. For example, for an ideal gas the density, pressure, and temperature are related according to ρ = p/RT, where R is the gas constant. To know how the density varies with pressure, we must also know the temperature variation. For now we will assume that the density and specific weight are constant (incompressible flow). The justification for this assumption and the consequences of compressibility will be considered further in Section 3.8.1 and more fully in Chapter 11. With the additional assumption that the density remains constant (a very good assump- tion for liquids and also for gases if the speed is “not too high”), Eq. 3.6 assumes the following simple representation for steady, inviscid, incompressible flow. p + 1 _ 2 ρV 2 + γ z = constant along streamline (3.7) This is the celebrated Bernoulli equation—a very powerful tool in fluid mechanics. In 1738 Daniel Bernoulli (1700–1782) published his Hydrodynamics in which an equivalent of this famous equation first appeared. To use it correctly we must constantly remember the basic assumptions used in its derivation: (1) viscous effects are assumed negligible, (2) the flow is assumed to be steady, (3) the flow is assumed to be incompressible, and (4) the equation is applicable along a streamline. In the derivation of Eq. 3.7, we assume that the flow takes place in a plane (the x–z plane). In general, this equation is valid for both planar and nonplanar (three-dimensional) flows, provided it is applied along the streamline.
- John I. Hochstein, Andrew L. Gerhart(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
72 LEARNING OBJECTIVES After completing this chapter, you should be able to: CHAPTER 3 Elementary Fluid Dynamics— The Bernoulli Equation • discuss the application of Newton’s second law to fluid flows. • explain the development, uses, and limitations of the Bernoulli equation. • use the Bernoulli equation (stand-alone or in combination with the simple continuity equation) to solve flow problems. In this chapter we investigate some typical fluid motions (fluid dynamics) in an elementary way. We will discuss in some detail the use of Newton’s second law (F = ma) as it is applied to fluid particle motion that is “ideal” in some sense. We will obtain the celebrated Bernoulli equation and apply it to various flows. Although this equation is one of the oldest in fluid mechanics and the assumptions involved in its derivation are numerous, it can be used effectively to predict, understand, and analyze a variety of flow situations. However, if the equation is applied with- out proper respect for its restrictions, serious errors can arise. Indeed, the Bernoulli equation is appropriately called the most used and the most abused equation in fluid mechanics. A thorough understanding of the elementary approach to fluid dynamics involved in this chapter will be useful on its own. It also provides a good foundation for the material in the fol- lowing chapters where some of the present restrictions are removed and “more nearly exact” results are presented. 3.1 Newton’s Second Law Consider a really tiny volume of fluid, which is still large enough to contain a significant num- ber of molecules. This volume is called a fluid particle (with further description in Section 4.1). As the fluid particle moves from one location to another, it usually experiences an acceleration or deceleration.
- Andrew L. Gerhart, John I. Hochstein, Philip M. Gerhart(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
139 • apply the concepts of static, stagnation, dynamic, and total pressures. • calculate various flow properties using the energy and hydraulic grade lines. • discuss the application of Newton’s second law to fluid flows. • explain the development, uses, and limitations of the Bernoulli equation. • use the Bernoulli equation (stand-alone or in combination with the simple continuity equation) to solve flow problems. LEARNING OBJECTIVES After completing this chapter, you should be able to: In this chapter we investigate some typical fluid motions (fluid dynamics) in an elementary way. We will discuss in some detail the use of Newton’s second law (F = ma) as it is applied to fluid particle motion that is “ideal” in some sense. We will obtain the celebrated Bernoulli equation and apply it to various flows. Although this equation is one of the oldest in fluid me- chanics and the assumptions involved in its derivation are numerous, it can be used effectively to predict, understand, and analyze a variety of flow situations. However, if the equation is applied without proper respect for its restrictions, serious errors can arise. Indeed, the Ber- noulli equation is appropriately called the most used and the most abused equation in fluid mechanics. A thorough understanding of the elementary approach to fluid dynamics involved in this chapter will be useful on its own. It also provides a good foundation for the material in the fol- lowing chapters where some of the present restrictions are removed and “more nearly exact” results are presented. 4.1 NEWTON’S SECOND LAW Consider a really tiny volume of fluid, which is still large enough to contain a significant num- ber of molecules. This volume is called a fluid particle (with further description in Section 4.1). As the fluid particle moves from one location to another, it usually experiences an acceleration or deceleration.
eBook - PDF- Richard W. Johnson(Author)
- 2016(Publication Date)
- CRC Press(Publisher)
mechanics and some-times in acoustics where the displacements from the initial positions are small. Fluid displacements continually increase, making the method less useful than the Eulerian. While the expressions for velocity and acceleration are simplified in the Lagrangian description, the equations of motion are not. All properties (pressure, density, etc.) are functions of both the fluid particle and of time. Identify each fluid particle by its position at t = t o , r = r o . Then, in the Lagrangian description, the continuity equation is (Serrin, 1959) ρ ρ J = o (4.167) where J x y z x y z o o o = ∂ ∂ ( , , ) ( , , ) (4.168) is the Jacobian of the transformation of the particle volume during its motion from r o to r while the density changes from ρ o to ρ . In Cartesian coordinates the components of the Lagrangian momentum equation for incompressible, inviscid flow are ∂ ∂ - ∂ ∂ + ∂ ∂ - ∂ ∂ + ∂ ∂ - 2 2 2 2 2 2 x t f x x y t f y x z t f B o B o B x y z ∂ ∂ = -∂ ∂ z x p x o o 1 ρ (4.169) ∂ ∂ - ∂ ∂ + ∂ ∂ - ∂ ∂ + ∂ ∂ - 2 2 2 2 2 2 x t f x y y t f y y z t f B o B o B x y z ∂ ∂ = -∂ ∂ z y p y o o 1 ρ (4.170) ∂ ∂ - ∂ ∂ + ∂ ∂ - ∂ ∂ + ∂ ∂ - 2 2 2 2 2 2 x t f x z y t f y z z t f B o B o B x y z ∂ ∂ = -∂ ∂ z z p z o o 1 ρ (4.171) These equations are very difficult to solve in practice. In addi-tion, the Lagrangian method yields more information than is normally required and the Eulerian method yields satisfactory information. 4.2 Thermodynamic State Equations Ronald H. Aungier 4.2.1 Introduction Thermodynamic state equations supplement the basic equa-tions of fluid dynamics and thermodynamics by characteriz-ing the specific fluid of interest. These equations are developed 4 -16 Handbook of Fluid Dynamics from experimental data or derived from kinetic theory or sta-tistical mechanics.
- Paul G. Tucker(Author)
- 2016(Publication Date)
- Cambridge University Press(Publisher)
Equation (2.131) in its polar coordinate form is ∂ ∂ z L 3 μ ∂ p ∂ z + 1 r ∂ ∂θ L 3 μr ∂ p ∂θ = 6U ∂ L r∂θ + 12 ∂ L ∂ t (2.132) The last term comes into play when the shaft orbits in the bearing housing. This is quite common in reciprocating combustion engines. Due to accuracy and cost reas- ons, the Reynolds equations are frequently used in computer packages for bearing design. The Reynolds equation assumptions are summarised as follows: 1) Since the mass of the fluid is negligible, so are body forces; 2) Since the y or r direction extent is small relative to the other directions, the pressure and lubricant viscosity variation in these directions is negligible; 3) Fluid inertia forces are negligible and 4) The fluid is Newtonian. It is not until high speeds are reached or if the frequency of shaft oscillation/orbiting becomes large that it is necessary to include inertial terms and solve the full Navier- Stokes equations (this is an unusual requirement). Normally, for bearings the equa- tions are solved in a curvilinear coordinate form. 2.10 Equations in Solids 51 2.9.6 Other Forms of the Flow-Governing Equations Many further forms and simplifications of the flow-governing equations can be written. For example, there is Crocco’s equation, which hybridizes thermodynamic and fluid dynamic equations. It is essentially a vorticity transport equation. However, the flow driving pressure gradient is expressed in terms of the entropy gradient. This is achieved using Equation (2.161), discussed later. Also, all the equations so far can be written in linearized forms where they are decomposed into a mean, φ, and a component fluctuation about the mean, φ . This is achieved through the substitution φ = φ + φ (2.133) The solution of linearized equations can be useful in gaining an estimate of unsteady temporal information such as the magnitude of acoustic waves or unsteady surface pressure loads.
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