Newton’s Law of Mechanics

Although the derivation of the continuity equation by use of a fixed control volume is perfectly satisfactory, it is less obvious how to apply Newton s laws of mechanics in this framework. The familiar use of these principles from coursework in classical mechanics is that they are applied to describe the motion of a specific body subject to various forces or torques. To apply these same laws to a fluid (i.e., a liquid or a gas), we introduce the concepts of material points and a material volume (or material control volume) that we denote as Vm(t). Now a material point is a continuum point that moves with the local continuum velocity of the fluid. A material volume Vm (t), is a macroscopic control volume whose shape at some initial instant, / = 0, is arbitrary, that contains a fixed set of material points. Because the material volume contains a fixed set of such points, it must move with the local continuum velocity of the fluid at every point. Hence, as illustrated in Fig. 2-3, it must deform and change volume in such a way that the local flux of mass through all points on its surface is identically zero for all time (though, of course, there may still be exchange of molecules due to random molecular motion). Because mass is neither created nor destroyed according to the principle of mass conservation, the total mass contained... 

Equations may express theoretical concepts or they may express empirical information. The division between these two is not very clear when closely scrutinized because most theoretical concepts and so-called fundamental principles were based in their formative stages on direct observation. As time went on, and as it became clear that there were means of expressing these observations so that they could generally predict further observations, they were then elevated to the status of principles. These principles may still be subject to modification as further experimental observations cause the principles to be re-evaluated. Such was the case for Newton s laws of mechanics when it became clear that they did not predict actions occurring at the subatomic level. 

Newton s laws in mechanics. This law of equilibrium establishing a link with the rule of zeroing the sum of concurrent forces must not be confused with Newton s laws of mechanical movement that are recalled below for insisting on the differences. 

Starting from these ideas, Albert Einstein formulated a theory that made it necessary to reconsider the theoretical treatment of all phenomena in which high velocities play a role, i.e., when vjc is not negligibly small. It turned out that Newton s laws of mechanics are asymptotically valid in the limit c oo - the nonrelativistic limit. If this limit is not attained, new phenomena become important, which are not present in the non-relativistic theory relativistic effects. 

Raoult s and Henry s laws are widely used to supply the equilibrium relations in the above equation sets. These are useful estimating approximations, not laws like Newton s laws of mechanics or the laws of thermodynamics. 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

The phenomenon of attraction of masses is one of the most amazing features of nature, and it plays a fundamental role in the gravitational method. Everything that we are going to derive is based on the fact that each body attracts other. Clearly this indicates that a body generates a force, and this attraction is observed for extremely small particles, as well as very large ones, like planets. It is a universal phenomenon. At the same time, the Newtonian theory of attraction does not attempt to explain the mechanism of transmission of a force from one body to another. In the 17th century Newton discovered this phenomenon, and, moreover, he was able to describe the role of masses and distance between them that allows us to calculate the force of interaction of two particles. To formulate this law of attraction we suppose that particles occupy elementary volumes AF( ) and AF(p), and their position is characterized by points q and p, respectively, see Fig. 1.1a. It is important to emphasize that dimensions of these volumes are much smaller than the distance Lgp between points q and p. This is the most essential feature of elementary volumes or particles, and it explains why the points q and p can be chosen anywhere inside these bodies. Then, in accordance with Newton s law of attraction the particle around point q acts on the particle around point p with the force d ip) equal to... 

In the 1920s it was found that electrons do not behave like macroscopic objects that are governed by Newton s laws of motion rather, they obey the laws of quantum mechanics. The application of these laws to atoms and molecules gave rise to orbital-based models of chemical bonding. In Chapter 3 we discuss some of the basic ideas of quantum mechanics, particularly the Pauli principle, the Heisenberg uncertainty principle, and the concept of electronic charge distribution, and we give a brief review of orbital-based models and modem ab initio calculations based on them. 

Newton s laws of motion apply to these atoms since we are treating their motion within the framework of classical mechanics. That is,... 

Convection involves the transfer of heat by means of a fluid, including gases and liquids. Typically, convection describes heat transfer from a solid surface to an adjacent fluid, but it can also describe the bulk movement of fluid and the associate transport of heat energy, as in the case of a hot, rising gas. Recall that there are two general types of convection forced convection and natural (free) convection. In the former, fluid is forced past an object by mechanical means, such as a pump or a fan, whereas the latter describes the free motion of fluid elements due primarily to density differences. It is common for both types of convection to occur simultaneously in what is termed mixed convection. In such instance, a modified form of Fourier s Law is applied, called Newton s Law of Cooling, where the thermal conductivity is replaced with what is called the heat transfer coefficient, h ... 

In classical mechanics, Newton s laws of motion determine the path or time evolution of a particle of mass, m. In quantum mechanics what is the corresponding equation that governs the time evolution of the wave function, F(r, t) Obviously this equation cannot be obtained from classical physics. However, it can be derived using a plausibility argument that is centred on the principle of wave-particle duality. Consider first the case of a free particle travelling in one dimension on which no forces act, that is, it moves in a region of constant potential, V. Then by the conservation of energy... 

Calculation of molecular structure at successive small time intervals using a molecular mechanics force field with the shifts determined using Newton s laws of motion. 

If the end-points are fixed, the integrated term vanishes, and A is stationary if and only if the final integral vanishes. Since Sxa is arbitrary, the integrand must vanish, which is Newton s law of motion. Hence Lagrange s derivation proves that the principle of least action is equivalent to Newtonian mechanics if energy is conserved and end-point coordinates are specified. 

It is also clear that Newton s law of cooling is a special case of Fourier s law. The foregoing provides the reason for only two commonly recognized basic heat transfer mechanisms. But owing to the complexity of fluid motion, convection is often treated as a separate heat transfer mode. 

In the nineteenth century the universe was thought of as a collection of particles, the atoms, which obeyed Newton s laws of motion. The future position of every particle was therefore determined by the positions and motions of the particles at any given time. The course of events was therefore fixed by natural law. Free will was impossible. A man seemed to decide what he would do, but the process by which he decided was controlled by natural laws, and the result was determined beforehand. According to this idea it was difficult to believe in any supernatural powers controlling the evolution of the universe, and in particular that of life on the earth. It was, however, extremely difficult not to believe in free will. The conclusion that there is no free will seems to be contrary to the facts and so requires the theory to be modified. Moreover no one really believed that art, literature, religion, and all the other human activities of a more or less spiritual character could be regarded as the results of a purely mechanical process based on Newton s laws of motion. The theory was obviously quite inadequate to explain these facts. 

Molecular-dynamic simulations are characterized by a solution of Newton s laws of motion for the molecules travelling through the zeolite pore system under control of the force field given by the properties of the host lattice, by interactions between the host and the molecules, and by interactions between the molecules. To date this has been possible only for the diffusion of simple molecules (e.g. methane or benzene) inside a zeolite lattice of limited dimensions [29, 37, 54], To take into account the effects of a chemical reaction as well would require quantum-mechanical considerations however, such simulations are in their infancy. 

For what follows one can accept this equation as a fundamental natural law of the mechanics of electrons, the correctness of which has been tested many times by experiment, just as Newton s law of attraction is a fundamental law of macroscopic mechanics. De Broglie s relation is then a special result of this equation. 

Figure 11.1 A schematic that illustrates the analogy between the theories for mechanical motions and for chemical dynamics. Newton s law of motion, governing a collection of particles with positions x (t), X2(t), , Xj/(t), arises from Schrodinger s equation for the wave function f in the limit h - 0. Similarly, the chemical master equation for p(n, n2, , ftat, t) yields the law of mass action in the limit V -> oo.

B Understand Ihe basic mechanisms of heat transfer, v/hich are conduction, convection, and radiation, and Fourier s law of heat conduction, Newton s law of cooling, and the Stefaa-Boltzmann law of radiation,... 

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