Dissipative flux

The firsttwo terms on the right-hand side of this expression are responsible for spatial transport of scalar dissipation. In high-Reynolds-number turbulent flows, the scalar-dissipation flux (iijC ) is the dominant term. The other terms on the right-hand side are similar to the corresponding terms in the dissipation transport equation ((2.125), p. 52), and are defined as follows. 

The joint-dissipation-flux term ujcap and the molecular-transport term T f, defined by... 

The second term on the right-hand side of (A.41) represents the flux of scalar dissipation into the scalar dissipation range, and can be rewritten in terms of known quantities. From (A.39), it can be seen that Taa(icD, t) = TD(t). Likewise, using the definition of kd, it follows that 2raK = CD(e/v)1/2. The scalar-dissipation flux term can thus be expressed as... 

The energy dissipation is the highest at the discharge of the device where the pressure is the highest as measured by the increase in zone temperature. The energy dissipation flux is described by Eq. 5.29 below ... 

Extended nonequilibrium thermodynamics is not based on the local equilibrium hypothesis, and uses the conserved variables and nonconserved dissipative fluxes as the independent variables to establish evolution equations for the dissipative fluxes satisfying the second law of thermodynamics. For conservation laws in hydrodynamic systems, the independent variables are the mass density, p, velocity, v, and specific internal energy, u, while the nonconserved variables are the heat flux, shear and bulk viscous pressure, diffusion flux, and electrical flux. For the generalized entropy with the properties of additivity and convex function considered, extended nonequilibrium thermodynamics formulations provide a more complete formulation of transport and rate processes beyond local equilibrium. The formulations can relate microscopic phenomena to a macroscopic thermodynamic interpretation by deriving the generalized transport laws expressed in terms of the generalized frequency and wave-vector-dependent transport coefficients. 

As mentioned above, the time-integral of the dissipation function takes on the value of the extensive generalised entropy production, 2, over a period, t under suitable circumstances. The main requirement is that the dynamics satisfies the condition know as the adiabatic incompressibility of phase space . In this case, 2, = —JtFefiV, where is the dissipative flux caused by the field, F, p = IKk T) where T is the temperature of the corresponding initial system and V is the volume of the system. An example where such a relation can be applied is if a molten salt at equilibrium was exposed to a constant electric field. In that case the entropy production would be directly proportional to the current induced, and the FR would describe the probability that it would be observed to flow in the + ve or — ve... 

The notation A = iLA, A = (iL)M,... recalls that classical and quantum Liouvillians are time derivatives. The dissipative fluxes refer to the dynamically accessible microstates. They also provide the basis of the so-called extended thermodynamics [66]. For example, in the linear response theory [67], the dynamical susceptibilities are proportional to the time derivatives of the correlation functions [63]... 

In addition to the dissipative fluxes, other short-lived observables directly implied in the dynamics may be required. An important example is the transition state on a reaction path of a chemical reaction (see Section 4.3). 

Complex systems such as solutions of macromolecules, magnetic hysteresis bodies, visco-elastic fluids, polarizable media require some extra variables in the fundamental equation of Gibbs. Dissipative fluxes (heat, diffusion, viscous pressure tensor and viscous pressure) are included in the Gibbs function in new formalism. In the formalism of extended irreversible thermodynamics (EIT), the dissipative fluxes are the independent variables in addition to classical variables of thermostatics [1]. 

The popular viewpoint up to about the mid-1960s is summarized as follows. Find all the conserved variables and order parameters for a particular system, use these (for a particular k) as a set of variables in Eq. (10) or one of its many equivalent forms, expand the equations to at least order and linear laws, valid for the long times over which the variables of interest fluctuate, and for k all lengths but ic, will be obtained. The Onsager coefficients in the linear laws will be given by integrals over dissipative flux correlation functions that contain no slowly varying character and no order parameter character, i.e., no critical anomaly. ... 

It is now desirable to deal with the nonclassical behavior of the kernel in the linear laws in a precise, formal way. Of course, one could simply try to improve the crude method just discussed such an approach is perfectly valid. However, we feel that an alternate procedure, which has almost always been used in the literature, is preferable. Mori s method allows the writing of equations with well-behaved kernels if the proper set of variables is chosen. The kernel in the linear laws is badly behaved due to the influence of the nonlinear variable. If we include the linear and nonlinear variables in the set of variables to which Mori s method is applied, the random forces and the dissipative fluxes (/ will be defined precisely in this section) will be projected orthogonal to all of these variables. The kernels in the resulting equations, the nonlinear Langevin equations, should behave classically. Thus, convolutions involving K will be converted into scalar multiplication by the classical relation. 

Now, suppose we choose nZ and nZ+kg-k, fc [Pg.279]

Calculate the current-flow on a given resistivity on an arbitrary DC circuit. Prove that this task could be solved by the minimization of the sum of dissipation fluxes or by the principle of minimal entropy production, like we did in the case of parallel connection of the Figure 6., when the circuit regarding the contacts of the resistor is replaced with the Norton s current source equivalent circuit [53]. 

The general NEMD algorithm (Evans Morriss 1990) introduces a known but (usually) fictitious applied field, X, into the equations of motion and hence generates a dissipative thermodynamic flux, J. The dissipative flux is defined as the work performed on the system per unit time, by the applied field, X, where... 

If we design the coupling of the external field to the system in such a way that the dissipative flux is equal to one of the Navier-Stokes fluxes (such as the shear stress in planar Couette flow or the heat flux in thermal conductivity), it can be shown - provided the system satisfies a number of fairly simple conditions (Evans Morriss 1990) - that the response is proportional to the Green-Kubo time integral for the corresponding Navier-Stokes transport coefficient. This means that the linear response of the system to the fictitious external field is exactly related to linear response of a real system to a real Navier-Stokes force, thereby enabling the calculation of the relevant transport coefficient. 

It is straightforward to show that these equations of motion have the heat-flux vector as their dissipative flux (that is, = — Jg Fg), and therefore the thermal conductivity... 

The flux in the x direction is associated with the particles velocity, as in the deterministic case. The flux in the v direction consists of two parts. The deterministic Newtonian part, results from the acceleration associated with the potential V, and the dissipative part, results from the coupling to the thermal environment. Note that this dissipative flux does not depend on the potential V. 

As current density increases (corresponding also to reduced cell voltage), the thermal energy dissipation flux will increase. The first term on the right-hand side of Eq. (5.110) represents Peltier heating. The second term on the right-hand side represents the sum of the activation (kinetic), concentration, ohmic, and crossover contributions to the heat generation. 

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