Phase diffusion equation

Continuum models encompass both micro and macro scales and in li-ion models the microscale is governed by the solid phase diffusion equation. The coupling of the microscale and the macroscale variables pose computational limitations. 

Derivation of the nonlinear phase diffusion equation. The longwave nature of the two basic instabilities of roll patterns described above shows that longwave distortions of rolls are of major interest. Let us consider longwave solutions of the NWS equation (73)... 

At the next order, a system of coupled equations for R2 and Oq is obtained. After eliminating R2, one can obtain the following nonlinear phase diffusion equation [49] (below we drop the subscript 0) ... 

Equation (3.3.5) represents a nonlinear phase diffusion equation. It is equivalent to the Burgers equation in the case of one space dimension (Chap. 6). It is known that the Burgers equation can be reduced to a linear diffusion equation through a transformation called the Hopf-Cole transformation (Burgers, 1974), and essentially the same is true for (3.3.5) in an arbitrary dimension. We shall take advantage of this fact in Chap. 6 when analytically discussing a certain form of chemical waves. 

The nonlinear phase diffusion equation (3.3.5) now takes the explicit form dy/... 

Generalization of the Nonlinear Phase Diffusion Equation dQtit)... 

We are now ready to take ep(X) as representing a diffusion term, and to generalize the nonlinear phase diffusion equation (3.3.5). To say that 6 is equivalent to saying that the operator V carries the smallness factor j/e, that is, whenever a spatial derivative appears, it generates a small quantity of order /s. In fact, V always appears as the combination j/e V in the theory below. 

As a preliminary to investigating two-dimensional wave patterns, we first make a brief inspection of some particular solutions of the nonlinear phase diffusion equation in one dimension. 

Note that (6.6.3) reduces to the nonlinear phase diffusion equation if we neglect the space dependence of R, The previous numerical simulation suggests that R... 

This shows the anticipated fact that the instability of the uniform oscillation to long wavelength fluctuations corresponds precisely to the negative sign of the phase diffusion constant. The equality = —y may also be confirmed, where y is the quantity which appeared in (4.2.36) and is the abbreviation of —a> defined in (4.2.35). More generally, it is possible to prove that the dispersion curve of the phaselike branch has an exact correspondence to the linearized form of the phase diffusion equation (4.2.36), or one may possibly have... 

Although some physical implications of the nonlinear phase diffusion equation with positive a have been discussed in Sect. 6.2, we have not yet discussed the same equation in relation to the wavefront dynamics this should be done before going into the phase turbulence equation. Let the wavefront form a straight line which is slightly non-parallel to the y direction (Fig. 7.9). Then the nonlinear phase diffusion equation becomes... 

When a is positive, (7.3.15) is consistent with the ordinary picture that locally convex fronts tend to be flattened. If a given front is concave, the flattening effect will ultimately be balanced with the sharpening effect (coming from the very fact that the front has a finite propagation velocity), so that formation of a shocklike structure is expected (Fig. 7.10). We already know, in fact, that the nonlinear phase diffusion equation admits a family of shock solutions (though in a different physical context see Sect. 6.2). In the present notation, the shock solutions (6.2.6) are expressed as... 

For polygonal structures the phase is a two dimensional vector. For instance in the case of a hexagonal structure, the phase diffusion equation takes the form... 

Linear stability analysis then yields a phase diffusion equation [2,28,53] for the slow phase modulations Y, T) that evolve on time and length scales larger than those of the modulus of A ... 

The linear phase diffusion equations have also been derived for patterns of hexagonal symmetry (M — 3) [58, 59] for a variational model (otherwise... 

Although the derivation of the phase equations has been sketched in the weakly nonlinear regime, i.e. starting from the amplitude equations, they are far more general and may be derived far from the onset of the structures. The starting point is then the finite amplitude planform the wavevectors of which are allowed to vary slowly over the extent of the reactor. In this context, that follows the work of Witham on nonlinear wavetrains, the phase diffusion equations appear as solvability conditions [28,61]. 

In this approach, originally developed by Fuller et al. [18, 53] based on the porous electrode theory [42], the active material is assumed to consist of spherical particles with a specific size, and solid phase diffusion in the radial direction is assumed to be the predominant mode of transport. The electrolyte phase concentration (Cg) and the potentials (4>s,4>e) are assumed to vary along the principal (i.e., thickness) direction only, and are henceforth referred to as the x direction. In other words, this model implicitly considers two length scales (1D + 1D), that is, the r direction inside the spherical particle and the x direction along the thickness. All other equations described earlier continue to remain valid except the solid phase diffusion. Equation 25.19 and the corresponding boundary/initial conditions. The solid phase diffusion equation now takes the following form ... 

The bioheat (equation 12.47) is used together with the fabric energy, solid phase continuity and gas phase diffusivity equations (equations 12.38-40) to obtain the temperature profile in multiple layers of skin. These are used in Henriques and Moritz [50] bum integral equation to calculate the maximum durations of the flash fire exposure before the human skin can get second and third-degree bums ... 

---