System mechanical: conservative

Thermal NO mechanisms For premixed combustion systems a conservative estimate of the thermal contribution to NO formation can be made by consideration of the equilibrium system given by reaction (8.48) ... 

Once the continuum hypothesis has been adopted, the usual macroscopic laws of classical continuum physics are invoked to provide a mathematical description of fluid motion and/or heat transfer in nonisothermal systems - namely, conservation of mass, conservation of linear and angular momentum (the basic principles of Newtonian mechanics), and conservation of energy (the first law of thermodynamics). Although the second law of thermodynamics does not contribute directly to the derivation of the governing equations, we shall see that it does provide constraints on the allowable forms for the so-called constitutive models that relate the velocity gradients in the fluid to the short-range forces that act across surfaces within the fluid. 

Benson M, Bennett CR, Harry JE, Patel MK and Cross M (2000) The recovery mechanism of platinum group metals from catalytic converters in spent automotive exhaust systems. Resources, conservation and recycling 31 1-7. 

If electroneutrality may be assumed, the fluid mechanical conservation equations of mass, momentum, and energy remain unchanged from those discussed in the last section for multicomponent systems. If electroneutrality is not assumed, the mass conservation equation remains unchanged but the Lorentz body force must be added to the right-hand side of the Navier-Stokes equation. In addition, to the right-hand side of the energy equation is added the corresponding work term p E u. 

In the various quenching mechanisms which we are considering, the total energy of the system is conserved by converting the proper amount of electronic excitation energy into vibrational excitation of M... 

Sir William Rowan Hamilton (1805-1865) devised an alternative form of Newton s equations of motion involving a function H, the Hamiltonian function for the system. For a system where the potential energy is a function of the coordinates only, the total energy remains constant with time that is, E is conserved. We shall restrict ourselves to such conservative systems. For conservative systems, the classical-mechanical Hamiltonian function turns out to be simply the total energy expressed in terms of coordinates and conjugate momenta. For Cartesian coordinates x, y, z, the conjugate momenta are the components of linear momentum in the x, y, and z directions p, Py, and p. ... 

Shearer, M. Nonuniqueness of admissible solutions of Riemann initial value problems for a system of conservation laws of mixed type. Archive for Rational Mechanics and Analysis 93 (1986) 45-59. 

The above system of conservation equations is usually called the Navier-Stokes equations in fluid mechanics. 

Inertia forces are the uncommon forces that disobey the laws of classical Newton mechanics. Indeed, in a noninertia reference system we are unable to indicate a body whose action can explain the appearance of inertia forces. This signifies that Newtonian laws are not executed in noninertial reference systems. Figuratively speaking, there exists a force of actions (the force of inertia), but no force of counteraction. In noninertial reference systems, these particularities of inertia forces do not allow the selection of a closed system of bodies (refer to 1.3.7), since for any body in a noninertial system the inertia forces are the internal ones. Thus, in the noninertial reference system the conservation laws of energy and momentum, which will be considered below (see Section 1.5), are not valid. 

Conservation laws at a microscopic level of molecular interactions play an important role. In particular, energy as a conserved variable plays a central role in statistical mechanics. Another important concept for equilibrium systems is the law of detailed balance. Molecular motion can be viewed as a sequence of collisions, each of which is akin to a reaction. Most often it is the momentum, energy and angrilar momentum of each of the constituents that is changed during a collision if the molecular structure is altered, one has a chemical reaction. The law of detailed balance implies that, in equilibrium, the number of each reaction in the forward direction is the same as that in the reverse direction i.e. each microscopic reaction is in equilibrium. This is a consequence of the time reversal syimnetry of mechanics. 

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 tliree conservation laws of mass, momentum and energy play a central role in the hydrodynamic description. For a one-component system, these are the only hydrodynamic variables. The mass density has an interesting feature in the associated continuity equation the mass current (flux) is the momentum density and thus itself is conserved, in the absence of external forces. The mass density p(r,0 satisfies a continuity equation which can be expressed in the fonn (see, for example, the book on fluid mechanics by Landau and Lifshitz, cited in the Furtlier Reading)... 

Bornemann, F. A. Homogenization in Time of Singularly Perturbed Conservative Mechanical Systems. Manuscript (1997) 146pp... 

We have used a common notation from mechanics in Eq. (5-4) by denoting velocity, the first time derivative of a , x, and acceleration, the second time derivative, x. In a conservative system (one having no frictional loss), potential energy is dependent only on the location and the force on a particle = —f, hence, by differentiating Eq. (5-3),... 

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