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Basic Concepts
Published in Ron Darby, Raj P. Chhabra, Chemical Engineering Fluid Mechanics, 2016
In addition to the laws for the conservation of mass, energy, momentum, etc. there are additional physical laws that govern the rate at which these conserved quantities are transported from one region to another in a continuous medium. These are called phenomenological laws because they are based on observable phenomena and logical deduction but they cannot be derived from first principles. These “rate” or “transport” models can be written for any conserved (extensive) quantity (mass, energy, momentum, electric charge, etc.) and can be expressed in a general form as Rateoftransport=DrivingforceResistance=(Conductance)×(Drivingforce)
Conservative and Dissipative Soliton Systems
Published in David S. Ricketts, Donhee Ham, Electrical Solitons, 2011
These three correspond to conservation of mass, momentum, and energy [3], which are the three commonly known conserved quantities in physics. The KdV equation, however, does not stop there. It has many, many more conserved quantities. In fact, there are an infinite number of them [51]!! The next two Ts are1 () T4=5u4−10uux2+uxx2, () T5=21u5−105u2ux2+21uuxx2−uxxx2.
Formalism
Published in Shabnam Siddiqui, Quantum Mechanics, 2018
In classical mechanics, physical variables such as momentum and energy of a system are conserved quantities when the system is closed and isolated, which means that these variables remain constant for all times. In quantum mechanics, one important quantity is the probability of locating the particle in some region of space. Thus, we must find out whether probability of a system is a conserved quantity.
Perspectives on geometric numerical integration
Published in Journal of the Royal Society of New Zealand, 2019
Success here has been due both to the importance of symplecticity and to the structure of the particular system at hand. What other properties can dynamical systems have that influence their behaviour? Some examples are conserved quantities (like energy and momentum); symmetries; time reversibility; and volume preservation (as in incompressible fluids). These properties influence the characteristic dynamics, bifurcations, and invariant sets of the systems. Thus we are asking for a natural classification of dynamical systems, ideally according to some algebraic structure. This approach is something that is a characteristic of mathematics as a whole that is distinct from other disciplines. Once an interesting example of something is known, it is natural to ask for all possible examples. This may turn out to be easy (there are just seven types of symmetry in frieze patterns Knight 1984) or extremely difficult (prime numbers are the building blocks of the integers, but understanding all their properties is difficult), but it is always important. Attempts to classify dynamical systems were made early on by Lie and by Cartan (1909), but have not led to a satisfactory general theory. Nevertheless, the known examples are important because, as most differential equations cannot be solved exactly, it is helpful to know at least the typical or characteristic features of each class.
On symmetries, conservation laws and exact solutions of the nonlinear Schrödinger–Hirota equation
Published in Waves in Random and Complex Media, 2018
Conservation laws play important roles understanding physical properties of various systems and finding PDEs exact solutions. They describe physical conserved quantities such as mass, energy, momentum, and angular momentum [6]. They are also used in the analysis of stability and global behavior of solutions. In addition, they play an essential role in the development of numerical methods and provide an essential starting point for finding nonlocally related systems and potential variables [7].
Modelling sediment transport in three-phase surface water systems
Published in Journal of Hydraulic Research, 2019
Cass T. Miller, William G. Gray, Christopher E. Kees, Iryna V. Rybak, Brittany J. Shepherd
The TCAT approach has been developed into a general procedure for formulating multiscale, multiphase models to describe transport phenomena (Gray & Miller, 2014). Microscale conservation and balance equations are developed for a target set of entities, which can include phases, interfaces that form where two phases meet, common curves that form where three phases meet, and common points that form where four phases meet. At the microscale, a point lies within a single entity, although technically the point is sufficiently large to represent averaged conditions in such a manner that the principles of continuum mechanics are applicable. The conserved quantities of interest include mass, momentum, and energy. A balance of entropy equation expresses the contributions to a rate of production of entropy. Microscale equilibrium conditions can be derived using variational methods. The TCAT approach relies on a set of averaging theorems to transform microscale conservation, balance, and thermodynamic equations to a larger scale, which for purposes of this work is the macroscale. A macroscale region is an averaging region that can include all entities, with applicable extent measures defined for each entity (e.g. volume fraction and specific interfacial area). A macroscale balance of entropy is augmented with the product of Lagrange multipliers and conservation equations to yield an expression that is order and dimensionally consistent. The entropy inequality can be manipulated, certain terms approximated, and the resultant expression placed into a flux-force form. The flux-force form provides permissibility constraints for closure relations that approximate the dissipative fluxes in terms of force functions. Kinematic evolution equations relating changes in entity extent measures can also be derived based upon only the averaging theorems, which can be used to form closed models along with the conservation equations and closure relations.