Variational equation density-only form

Incompressible Limit In order to obtain the more familiar form of the Navier-Stokes equations (9.16), we take the low-velocity (i,e. low Mach number M = u I /cs) limit of equation 9,104, We also take a cue from the continuous case, where, if the incompressible Navier-Stokes equations are derived via a Mach-number expansion of the full compressible equations, density variations become negligible everywhere except in the pressure term [frisch87]. Thus setting p = peq + p and allowing density fluctuations only in the pressure term, the low-velocity limit of equation 9,104 becomes... 

Hohenberg and Kohn have proved generally that the total ground state energy E of a collection of electrons in the presence of an externally applied potential (e.g. the valence electrons in the presence of the periodic potential due to the cores in a lattice), when no net magnetic moment is present, depends only on the average density of electrons n(R). By this proof, n(R) becomes the fundamental variable of the system (as it is in the Thomas-Fermi theory ). Variational minimization of the most general form of E, with respect to n lends to the Hartree-Fock equations formalism. 

Also, a third method exists. This involves calculating the concentrations of both the protonated and deprotonated forms at different pHs. The pK will correspond to the pH where the concentrations of both forms are equal. The derivation of the mathematical equation used to determine the pK can be obtained as follows. The variation of the optical density of the dye with the pH is sigmoidal. At low pHs in the presence of the protonated form only, the optical density (OD(ah)) is... 

Clearly, equation (49) reduces to equation (24) if tr is replaced by the approximation (18) formally it now takes full account of the (rapid) variation of electron density in the atom, in contrast to the semiclassical TF Euler equation. Unfortunately, tr is only presently known in two special cases (i) to low order in gradient expansion corrections to equation (18) as in equation (76) below and (ii) in a perturbative development about the uniform electron assembly.13 Form (i) will be referred to again below. However, as Scott14 was first to argue for the neutral atom, the origin of the Z2 term in equation (48) resides in the inhomogeneity correction to the TF theory, which is formally contained in equation (49). Fortunately, an approximation based on the Coulomb field treatment of Section 4 suffices to gain a useful estimate of the order of the Z2 term in the neutral atom. 

To simplify the problems represented by Eqs. (2 88) and (2 93), one of two possible assumptions is normally introduced. If the bounding surfaces are all at the same temperature, so that the only source of heat is viscous dissipation, it is often a good approximation to assume that the fluid is isothermal (i.e., the temperature is a constant, independent of spatial position or time). In this case, we do not need to consider (2 93) at all because it is an equation that is to be used to determine the temperature as a function of position and time. In addition, as noted in Chap. 2, the equation of motion also simplifies when the fluid is isothermal to either the form (2-89) or to (2 91) if the density in the system is also a constant, independent of position. When the temperature at the bounding surfaces is not constant, a different approximation known as the Boussinesq approximation is often used to simplify the problem. A detailed discussion of this approximation is postponed to a later point in the book. Here, we simply note the basic idea, which is that the material properties may be approximated as constants, provided that the temperature changes are not too large. In this case, the values of these properties can be evaluated at a representative temperature of the system, such as its mean value. An exception, as we shall see later, is that we sometimes cannot ignore the spatial variations of the density p in the body-force term of (2-88) even when the temperature changes are modest. Such density variations can produce motion... 

With X not explicitly dependent on the fields, there are only two potential sources of dependence on p or T of the tension in (3.18) first, - W, as given in (3.21), depends expliddy on the fields second, — W and X depend on the densities, and the p(z) and i (z) that minimize a- in (3.18) depend, in turn, on the fields. But by the variational prindple, equilibrium forms so, in the end, the field-dependence of field-dependence of -W. By the Gibbs-Duhem equation p( i, T) is such that... 

This is exactly the same form as equation 3.2, the overvoltage for one electrode. So, whether the activation overvoltage arises mainly at one electrode only or at both electrodes, the equation that models the voltage is of the same form. Furthermore, in aU cases, the item in the equation that shows the most variation is the exchange current density io rather than A. 

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