Energy diffusion equation, generalization

It should be noted that Eq. (5.63) may be derived from a microscopic model ° only for the special case where the friction kernel does not depend on the particle s velocity. This is not generally the case, and a rigorous derivation of reduced stochastic equations describing the motion of a subsystem coupled nonlinearly to its thermal environment leads to more complicated equations. (See the References for further discussions of this issue . ) Equation (5.63) may still be derived for special cases. An analysis very similar to that presented above leads to the energy diffusion equation (5.48) where now D(E) is given by... 

The general heat-conduction equation, along with the familiar diffusion equation, are both consequences of energy conservation and, like we have just seen for the Navier-Stokes equation, require a first-order approximation to the solution of Boltz-man s equation. 

Over the last four decades or so, transport phenomena research has benefited from the substantial efforts made to replace empiricism by fundamental knowledge based on computer simulations and theoretical modeling of transport phenomena. These efforts were spurred on by the publication in 1960 by Bird et al. (6) of the first edition of their quintessential monograph on the interrelationships among the three fundamental types of transport phenomena mass transport, energy transport, and momentum transport. All transport phenomena follow the same pattern in accordance with the generalized diffusion equation (GDE). The unidimensional flux, or overall transport rate per unit area in one direction, is expressed as a system property multiplied by a gradient (5)... 

Generalized Langevin equation, 3,70, 94 energy diffusion regime of GLE, 18, 20-21... 

We propose the balance principles for an immiscible mixture of continua with microstructure in presence of phenomena of chemical reactions, adsorption and diffusion by generalizing previous multiphase mixture [9] and use a new formulation for the balance of rotational momentum. New terms are also included in the energy equations corresponding to work done by respective terms in the micromomentum balances. 

In the operator L, the first term represents convection and the second diffusion. Equation (44) therefore describes a balance of convective, diffusive, and reactive effects. Such balances are very common in combustion and often are employed as points of departure in theories that do not begin with derivations of conservation equations. If the steady-flow approximation is relaxed, then an additional term, d(p(x)/dt, appears in L this term represents accumulation of thermal energy or chemical species. For species conservation, equations (48) and (49) may be derived with this generalized definition of L, in the absence of the assumptions of low-speed flow and of a Lewis... 

We shall not examine the equations of change in their most general form but will instead limit our attention to dilute or binary solutions in incompressible flow, without species production. For electrolyte solutions we will restrict ourselves to electrically neutral binary solutions or highly conducting solutions in which the electric field is small so that the convective diffusion equation for neutral species is applicable and the momentum and energy equations ate unaltered. For simplicity the transport and physical properties are also taken to be constant. 

Addition of a potential energy results in a generalized diffusion equation. 

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