Euler's Equation Fluid

6 Key excerpts on "Euler's Equation Fluid"

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  eBook - ePub

  Fluid Mechanics, Hydraulics, Hydrology and Water Resources for Civil Engineers

  • Amithirigala Widhanelage Jayawardena(Author)
  • 2021(Publication Date)
  • CRC Press

    (Publisher)

  Since then, there has not been much significant developments until Leonardo da Vinci (1459–1519) carried out several experiments in fluid mechanics as well as introducing the concept of the hydrological cycle as it is understood today. The basic laws of fluid motion were introduced by Isaac Newton (1649–1727) together with the linear law of viscosity, which identifies a fluid as either Newtonian or non-Newtonian. Newton’s laws paved the way to describe fluid motion in differential equations for inviscid fluids. These include the Euler equation (Leonhard Euler, 1707–1783) and the Bernoulli equation (Daniel Bernoulli, 1700–1782) for incompressible fluids, which have now become household names in fluid mechanics. Dimensional analysis, which is a powerful technique for model testing, was developed by Lord Rayleigh (1849–1919). The familiar dimensionless number in fluid mechanics, the Reynolds Number, was introduced by Osborn Reynolds (1849–1912) based on his extensive experimental studies with pipe flow. The general equations of fluid flow that includes fluid friction were developed independently by Claude Louis Navier (1785–1836) and George Gabriel Stokes (1819–1903), although at that time the equations had no simple solutions. At present, these equations form the basis of almost all fluid flow problems. Modern-day Computational Fluid Dynamics (CFD) software solve these equations numerically using various types of numerical techniques. The next important contribution came from Ludwig Prandtl (1875–1953) through the introduction of the boundary layer theory that enabled a fluid flow to be divided into two regions: a layer of flow near the wall known as the boundary layer where fluid friction is taken into account and a layer outside the boundary layer where the fluid friction is negligible. Outside the boundary layer, the Euler equation and the Bernoulli equation are applicable.

  In the 20th century, notable contributions to the development of fluid mechanics have been made by Theodore von Karman (1875–1963) and Sir Geoffrey Ingram Taylor (1886–1975). An area of fluid mechanics still not well understood is turbulence where important contributions have been made by Joseph Valentin Boussinesq (1877), who hypothesized that turbulent stresses are linearly proportional to the large scale mean strain rates, Ludwig Prandtl (1925) who introduced the mixing length theory and the logarithmic velocity profile near a solid wall, G. I. Taylor (1921) who introduced the idea of presenting turbulence in statistical terms as well as the concept of mixing length and the statistical theory of turbulence, Lewis Fry Richardson (1922) who introduced the concept of energy cascade which was followed by Andrey Kolmogorov (1941) who postulated that the statistics of small scales are isotropic and uniquely determined by the length scale, l , the kinematic viscosity, ν, and an average rate of kinetic energy dissipation per unit mass, ε

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  eBook - ePub

  Introduction to Engineering Mechanics

  A Continuum Approach, Second Edition

  • Jenn Stroud Rossmann, Clive L. Dym, Lori Bassman(Authors)
  • 2015(Publication Date)
  • CRC Press

    (Publisher)

  a along a streamline, may be integrated to yield the following equation:

  p ρ + g z + 1 2 V 2 = constant   along   a   streamline ,

  (18.34)

  where we have assumed that gravity acts in the negative z-direction, and where V is the velocity in the s direction, simply the magnitude of the velocity vector since V is in the s direction. This equation is known as the Bernoulli equation, and it is true for steady flow of an incompressible fluid under inviscid conditions. For convenience, we write the equation together with its restrictions:

  p ρ

  + g z +

  1 2

  V 2

  = constant

  • On a streamline

  • For steady flow

  • For incompressible fluid

  • If viscous effects neglected

  Many problems can be solved using the Bernoulli equation, allowing us to dodge having to solve the full Euler or Navier–Stokes equations. It should not escape our notice that the Bernoulli equation, derived from ∑F = ma, looks like an energy conservation equation. This is even easier to see if we multiply through by the (assumed constant) density: Equation 18.34 becomes

  p + ρ g z + 1 2 ρ V 2 = constant ,

  (18.35)

  and we can think of pressure p as a measure of flow work, ρgz as a gravitational potential energy, and

  1 2

  ρ

  V 2

  as a kinetic energy, all per unit volume of fluid. Daniel Bernoulli actually first arrived at Equation 18.34 by performing an energy balance, even though the concept of energy was still a bit fuzzy in 1738.

  One of the most useful applications of the Bernoulli equation is a device known as a Pitot* tube, and its cousin the Pitot-static tube, used to measure flow velocities. The tube (Figure 18.9a ) contains a column of air. When an oncoming fluid flow impinges on the nose of the Pitot tube, it displaces this air. As we know from hydrostatics, the displacement will be proportional to the pressure at the stagnation point on the Pitot tube nose. The stagnation point is at the divide between the flow that goes up-and-over and that which goes down-and-under (imagining a flow in the plane of the page for simplicity) and there the velocity must be zero. The difference between this “stagnation pressure” (where the fluid has speed V = 0) and the “static pressure” elsewhere in the flow (where the fluid has average speed V∞

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  From Vehicles to Grid to Electric Vehicles to Green Grid

  Many a Little Makes a Miracle

  • Fuhuo Li, Shigeru Kanemitsu;Jianjie Zhang(Authors)
  • 2019(Publication Date)
  • WSPC

    (Publisher)

  Chapter 3

  Fluid dynamics: Navier-Stokes equation

  Abstract

  This chapter paves the foundations of fluid dynamics as a precursor of the study on the theory of lubrication, which is indispensable in reducing the friction—the most important challenge for human beings in the 21st century.

  In §3.3 which is mostly based on [Kanemitsu et al. (2016)], [Li et al. (2018a)] we elucidate basic facts in fluid dynamics in [Khono (1989)], [Sena and Ohta (2007)], [Toyokura and Kamemoto (1976)] etc. We show that by the use of chain rule, Euler’s form of velocity vectors is made clear. The main novelty of our method is a very effective use of differential forms coupled with the general form of Stokes’ theorem to make many results in fluid mechanics simpler and clearer. In particular, we shall elucidate the notion of divergence and circulation in the 3-dimensional flow case in Theorem 3.6 . Further in the case of 2-dimensional flow by the use of complex analysis, we reestablish the results in that theorem.

  The main purpose of this chapter is to prove Theorem 3.12 to the effect that the Cauchy equation of motion implies the Navier-Stokes equation, thereby correcting the coefficients in [Grodins (1963)]. For this purpose we develop the vectorial version of the general Stokes theorem. From it we deduce the most important unicity theorem from which the equation of continuity follows immediately. In the same vein that the Cauchy equation of motion is a consequence of Newton’s second law of motion follows immediately. We shall give a starter for the project of deriving the deepest results in vector analysis by the theory of generalized functions, which are known mainly as distributions. Our standpoint is, however, that of [Imai (1963)] which incorporates the Sato hyper-function as a main tool for interpreting the whirl flow.

  For the sake of completeness we assemble some basic results from the theory of complex functions, cf. e.g. [Chakraborty et al

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  eBook - ePub

  Introduction to Fluid Mechanics, Sixth Edition

  • 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

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  Principles of Polymer Processing

  • Zehev Tadmor, Costas G. Gogos(Authors)
  • 2013(Publication Date)
  • Wiley-Interscience

    (Publisher)

  Equation 2.8-1 holds only for simple shearing flow, namely, when there is one velocity component changing in one (normal) spatial direction. The most general Newtonian constitutive equation that we can write for an arbitrary flow field takes the form:

  (2.8-2)

  where κ is the dilatational viscosity. For an incompressible fluid (and polymers are generally treated as such), ∇ · v = 0 and Eq. 2.8-2 reduces to:

  (2.8-3)

  Equations 2.8-2 and 2.8-3 are coordinate-independent compact tensorial forms of the Newtonian constitutive equation. In any particular coordinate system these equations break up into nine (six independent) scalar equations. Table 2.3 lists these equations in rectangular, cylindrical and spherical coordinate systems.

  TABLE 2.3 The Components of in Several Coordinate Systems

  Inserting Eq. 2.8-3 into the equation of motion, 2.5-18 , we get10 the celebrated Navier–Stokes11 equation:

  (2.8-4)

  The symbol defined as ∇2 is called the Laplacian. Table 2.4 lists the components of the Navier–Stokes equation in the various coordinate systems.

  TABLE 2.4 The Navier–Stokes Equation in Several Coordinate Systems

  We should note that the Navier–Stokes equation holds only for Newtonian fluids and incompressible flows. Yet this equation, together with the equation of continuity and with proper initial and boundary conditions, provides all the equations needed to solve (analytically or numerically) any laminar, isothermal flow problem. Solution of these equations yields the pressure and velocity fields that, in turn, give the stress and rate of strain fields and the flow rate. If the flow is nonisothermal, then simultaneously with the foregoing equations, we must solve the thermal energy equation, which is discussed later in this chapter. In this case, if the temperature differences are significant, we must also account for the temperature dependence of the viscosity, density, and thermal conductivity.

  Polymer processing flows are always laminar and generally creeping type flows. A creeping flow is one in which viscous forces predominate over forces of inertia and acceleration. Classic examples of such flows include those treated by the hydrodynamic theory of lubrication. For these types of flows, the second term on the left-hand side of Eq. 2.5-18

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  Elements of Gas Dynamics

  • H. W. Liepmann, A. Roshko(Authors)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  Chapter 7 , Here we shall give the set of equations that describe the motion of a viscous, heat-conducting, compressible fluid. These equations are usually called the Navier-Stokes equations. In a viscous fluid the surface forces acting on a particular mass of fluid are not necessarily normal to the surface element. Thus the forces in the momentum equation are different from the corresponding terms in the Euler equations. Furthermore, there may now be heat flow in the fluid, as well as irreversible transformation of kinetic energy into heat due to the action of the viscous stresses. The energy equation has to be rewritten to take account of the exchange ofi energy between kinetic energy, internal energy, and heat. The continuity equation remains unchanged, siiice it does not involve forces or energy.

  FIG . 13.9 Temperature profile through a normal shock wave. M 1 = 1.82 in helium. Temperatures measured with a resistance wire thermometer. [From F. S. Sherman, “A Low-Density Wind Tunnel Study of Shock Wave Structure and Relaxation Phenomena in Gases” NACA Tech. Note 3298 (1955).]

  As in Chapter 7 , consider a volume V enclosed by a surface A . The surface force acting on dA can be written

  where P denotes a stress vector. For a frictionless fluid, P is parallel to n, the vector of dA , and the factor of proportionality is – p . For a viscous fluid P is not necessarily parallel or proportional to n, but is a linear 12 function of n. Hence

  The

  Tik

  are the components of a stress tensor. It is more convenient to separate the viscous and nonviscous terms in

  Tik

  and write

  where the

  Tik

  vanish for a frictionless fluid and the

  Tik

  thus reduce to the proper stress for the Euler equations.

  Using Eq. 13.62 we can easily generalize the momentum equations (7.12) . We have for the equilibrium of forces on the volume V the equation:

  Only the last term differs from the corresponding one in the derivation of the Euler equations (7.13 to 7.19 ). This term is

  If Eq. 13.64

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