Conservation equations molecular derivation

Equations (2.3-11) and (2.3-12) are general equations for the conservation of momentum, thermal energy, or mass and will be used in many sections of this text. The equations consider here only molecular transport occurring and the equations do not consider other transport mechanisms such as convection, and so on, which will be considered when the specific conservation equations are derived in later sections of this text for momentum, energy, or mass. 

The conservation equations (2.218), (2.223) and (2.229) are rigorous (i.e., for mono-atomic gases) consequences of the Boltzmann equation (2.184). It is important to note that the governing conservation equations are derived without the exact form of the collision term, the only requirement is that only summation invariant properties of mono-atomic gases are considered. That is, only properties that are conserved in molecular collisions are considered. 

In addition to overall mass conservation, we are concerned with the conservation laws for individual chemical species. Beginning in a way analogous to the approach for the overall mass-conservation equation, we seek an equation for the rate of change of the mass of species k, mk. Here the extensive variable is N = mu and the intensive variable is the mass fraction, T = mk/m. Homogeneous chemical reaction can produce species within the system, and species can be transported into the system by molecular diffusion. There is convective transport as well, but it represented on the left-hand side through the substantial derivative. Thus, in the Eulerian framework, using the relationship between the system and the control volume yields... 

When appropriately applied, the equations derived in the section above can be used to predict variations in drug concentration within a tissue following administration. Por the description of molecular transport in cells or tissues, the mass conservation equations must be simplified by making appropriate... 

Analogous to the equation of change of mean molecular properties for dilute gas that was examined in Sect. 2.6, similar macroscopic conservation equations may be derived for dense gas from the Enskog equation. Multiplying the Enskog s equation (2.663) with the summation invariant property, //, and integrating over c, the result... 

The function / incorporates the screening effect of the surfactant, and is the surfactant density. The exponent x can be derived from the observation that the total interface area at late times should be proportional to p. In two dimensions, this implies R t) oc 1/ps and hence x = /n. The scaling form (20) was found to describe consistently data from Langevin simulations of systems with conserved order parameter (with n = 1/3) [217], systems which evolve according to hydrodynamic equations (with n = 1/2) [218], and also data from molecular dynamics of a microscopic off-lattice model (with n= 1/2) [155]. The data collapse has not been quite as good in Langevin simulations which include thermal noise [218]. 

There are several attractive features of such a mesoscopic description. Because the dynamics is simple, it is both easy and efficient to simulate. The equations of motion are easily written and the techniques of nonequilibriun statistical mechanics can be used to derive macroscopic laws and correlation function expressions for the transport properties. Accurate analytical expressions for the transport coefficient can be derived. The mesoscopic description can be combined with full molecular dynamics in order to describe the properties of solute species, such as polymers or colloids, in solution. Because all of the conservation laws are satisfied, hydrodynamic interactions, which play an important role in the dynamical properties of such systems, are automatically taken into account. 

Poh BT, Ong BT (1984) Dependence of viscosity of polystyrene solutions on molecular weight and concentration. Eur Polym J 20(10) 975-978 Pokrovskii VN (1970) Equations of motion of viscoelastic systems as derived from the conservation laws and the phenomenological theory of non-equilibrium processes. Polym Mech 6(5) 693—702... 

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