Newton fluid dynamics

The result that Archimedes discovered was the first law of hydrostatics, better known as Archimedes Principle. Aixhimedes studied fluids at rest, hydrostatics, and it was nearly 2,000 years before Daniel Bernoulli took the next step when he combined Archimedes idea of pressure with Newton s laws of motion to develop the subject of fluid dynamics. 

Hydrodynamic marked the beginning of fluid dynamics—the study of the way fluids and gases behave. Each particle in a gas obeys Isaac Newton s laws of motion, but instead of simple planetary motion, a much richer variety of behavior can be observed. In the third century B.C.E., Archimedes of Syracuse studied fluids at rest, hydrostatics, but it was nearly 2,000 years before Daniel Bernoulli took the next step. Using calculus, he combined Archimedes idea of pressure with Newton s laws of motion. Fluid dynamics is a vast area of study that can be used to describe many phenomena, from the study of simple fluids such as water, to the behavior of the plasma in the interior of stars, and even interstellar gases. 

The most important method is the similarity method first proposed by Newton. In fluid dynamics, three types of similarity can be considered (Mukhlyonov el al., 1979 Treybal, 1980 Holland, 1962) ... 

We remarked at the beginning of this section that the equation of motion is the cornerstone of any discussion of fluid dynamics. When one considers the various coordinate systems in which it may be expressed, the vector identities that may transform it, or the approximations that may be used to simplify it, Equation (28a) takes on many forms, some of which are scarcely recognizable as the same relationship. The purpose of this section is to illustrate that —despite its complexity and variations —the equation of motion is really nothing more than a statement of Newton s second law of motionl... 

Rabin Y, Ottinger HCh (1990) Dilute polymer solutions internal viscosity, dynamic scaling, shear thinning, and frequency-dependent viscosity. Europhys Lett 13(5) 423—428 Rallison JM, Hinch EJ (1988) Do we understand the physics in the constitutive equation J Non-Newton Fluid Mech 29(l) 37-55... 

Mysaka, J., Lin, Z., Stepanek, P. and Zakin, J.L. (2001) Influence of salts on dynamic properties of drag reducing surfactants. J. Non-Newton. Fluid Mech, 97,251-66. 

The above form of Newton s second law of motion applies to a system of constant mass. In fluid dynamics it is not usually convenient to work with elements of mass rather, we deal with elemental control volumes such as that shown in Fig. 5-4, where mass may flow in or out of the different sides of the... 

A fluid is said to be Newtonian when it obeys Newton s law of viscosity, given by x = r y, where x is the shear stress, r is the fluid dynamic constant, and y is the shear rate. 

The mechanical response of systems of distinct particles is often adequately described by Newton s laws, which constitute the bases of classical mechanics (36) (see sect. The Discrete Element Method ). However, additional concepts are needed for deformable matter, such as stress and strain, which will be described here (37). We will focus on solid materials, but remark that the same principles are also valid for fluids (in which case the field is usually referred to as computational fluid dynamics). 

The strategy for solving fluid dynamics problems begins by putting a control volume within the fluid that matches the symmetry of the macroscopic boundaries, and balancing the forces that act on the system. The system is defined as the fluid that is contained within the control volume V, which is completely surrounded by surface S. Since a force is synonymous with the time rate of change of momentum as prescribed by Newton s laws of motion, the terms in the force balance are best viewed as momentum rate processes. The force balance for an open system is stated without proof as l = 2- -3H-4- -5, where... 

Fluid dynamics is based on Newton s axioms. It has been convenient to describe the action of a force on a body in the following forms of Newton s law (see also Chapter 4.1) ... 

Brenner [6] proposes a different description of fluid dynamics. He proposes that Vv in Newton s rheological law should not be based on the mass-based velocity of the fluid, but on its volume velocity. This is a controversial idea. The adapted Navier-Stokes equations provide a hydrodynamic description of thermophoresis and thermal creep. Bedeaux et al. [7] describe how this alternative approach of the transport equations can be validated experimentally by means of thermophoresis. 

In 1687 Sir Isaac Newton postulated his theory about fluid dynamics,... 

Sir Isaac Newton put forth his three laws in the l700 s. These laws play a fundamental part in many branches of science, including fluid dynamics. In addition to the term hydrodynamics, Bernoulli s contribution to fluid dynamics was the realization that pressure decreases as velocity increases. This understanding is essential to the understanding of lift. Leonhard Euler, the father of fluid dynamics, is considered by many to be the preeminent mathematician of the eighteenth century. He is the one who derived what is today known as the Bernoulli equation from the work of Daniel Bernoulli. Euler also developed equations for inviscid flows. These equations were based on his own work and are still used for compressible and incompressible fluids. 

In recent years, DEM has been used in combination with computational fluid dynamics (CFD) aiming at investigating particulate behavior in fluid phase. For a two-phase particle-fluid system, the solid motion and fluid mechanics are solved through the application of Newton s equations of motion for the discrete particles and Navier-Stokes equations for the continuum fluid [2]. 

We have considered a one-dimensional flow case for a Newtonian fluid (Newton s Law of Viscosity) as well as a phenomenological consideration of fluid dynamics (the Reynolds experiment, the Reynolds number, velocity profiles). Now, let us direct our attention to the concepts of the multidimensional cases. 

On log r vs. log y coordinates, a power-law fluid is represented by a straight line with slope n. Thus, for n = 1, it reduces to Newton s law, for n< 1, the fluid is pseudoplastic, and for n > 1, the fluid is diiatant. The power law can reasonably approximate only portions of actual flow curves over one or two decades of shear rate (see Fig. 1S.6), but it does so with ir mathematical simplicity and has been adequate for many engineering purposes. Many useful relations have been obtained simply by replacing Newton s law with the power law in the usual fluid-dynamic equations. 

Because of their complex structure the mechanical behavior of polymeric materials is not well described by the classical constitutive equations Hooke s law (for elastic solids) or Newton s law (for viscous liquids). Polymeric materials are said to be viscoelastic inasmuch as they exhibit both viscous and elastic responses. This viscoelastic behavior has played a key role in the development of the understanding of polymer structure. Viscoelasticity is also important in the understanding of various measuring devices needed for rheometric measurements. In the fluid dynamics of polymeric liquids, viscoelasticity also plays a crucial role. " Also in the polymer-processing industry it is necessary to include the role of viscoelastic behavior in careful analysis and design. Finally there are important connections between viscoelasticity and flow birefringence. ... 

The equations required to describe the motions of a planetary atmosphere include Newton s second law, the mass continuity equation, the first law of thermodynamics, and an equation of state for the atmospheric gas. These relations are briefly reviewed from a general point of view. More detailed discussions of the governing equations and their applications can be found in texts on dynamical meteorology and geophysical fluid dynamics, such as Pedloskey (1979), Haltiner Williams (1980), Holton (1992), and Salby (1996). 

The dynamic viscosity, or coefficient of viscosity, 77 of a Newtonian fluid is defined as the force per unit area necessary to maintain a unit velocity gradient at right angles to the direction of flow between two parallel planes a unit distance apart. The SI unit is pascal-second or newton-second per meter squared [N s m ]. The c.g.s. unit of viscosity is the poise [P] 1 cP = 1 mN s m . The dynamic viscosity decreases with the temperature approximately according to the equation log rj = A + BIT. Values of A and B for a large number of liquids are given by Barrer, Trans. Faraday Soc. 39 48 (1943). 

Book II investigates the dynamical conditions of fluid motion. Book III displays the law of gi avitatioii at work in the solar system. It is demonstrated from the revolutions of the six known planets, including Earth, and their satellites, though Newton could never quite perfect the difficult theory of the Moon s motion. It is also demonstrated from the motions of comets. The gravitational forces of the heavenly bodies are used to calculate their relative masses. The tidal ebb and flow and the precession of the equinoxes is explained m terms of the forces exerted by the Sun and Moon. These demonstrations are carried out with precise calculations. 

In 1687, Newton summarized his discoveries in terrestrial and celestial mechanics in his Philosophiae naturalis principia mathematica (Mathematical Principles of Natural Philosophy), one of the greatest milestones in the history of science. In this work he showed how his (45) principle of universal gravitation provided an explanation both of falling bodies on the earth and of the motions of planets, comets, and other bodies in the heavens. The first part of the Principia, devoted to dynamics, includes Newton s three laws of motion the second part to fluid motion and other topics and the third part to the system of the (50) world, in which, among other things, he provides an explanation of Kepler s laws of planetary motion. 

---