Monotonic, travelling

In this chapter we shall treat some particular instances of the system (3.1.15) and the related phenomena. Thus in 3.2, we shall concentrate upon binary ion-exchange and discuss the relevant single nonlinear diffusion equation. It will be seen that in a certain range of parameters this equation reduces to the porous medium equation with diffusivity proportional to concentration. Furthermore, it turns out that in another parameter range the binary ion-exchange is described by the fast diffusion equation with diffusivity inversely proportional to concentration. It will be shown that in the latter case some monotonic travelling concentration waves may arise. 

In this section we shall study still another peculiarity of (3.2.3)—the occurrence of uniformly bounded, monotonic travelling waves. These waves, very common in reaction-diffusion (see, e.g., [25]—[27]), seem fairly unexpected in the reactionless diffusion under discussion. Their occurrence here is directly related to the singularity of diffusivity in (3.2.3) and thus can be viewed as the fast diffusional counterpart of the aforementioned peculiarities of the porous medium equation. 

Observe that a monotonic travelling wave solution to (3.2.2a) with boundary conditions... 

Equation (3.2.24b) (or more generally for m < 0, the appropriate integral of (3.2.21b), satisfying the boundary condition (3.2.20b)), represents a monotonic wave, travelling from left to right with speed c. In order to specify c the boundary condition (3.2.20b) has to be modified to... 

A traveling wave for which X is bounded and X does not vanish as C varies over (-cd, ao ) describes a steady, continuous drawing process of the type used in the synthetic fiber industry to cold-draw yarn to several times its length, for improvement of such physical properties as stiffness and tenacity. In such a traveling wave is monotone with the numbers... 

In order that there be values of C for which (5.41) has solutions that describe traveling necks and also periodic solutions, it suffices that g have a single turning point X (V) with dg(X)/dX > 0 for X < X (V) and with dg(X)/dX < 0 for X > X (V) clearly, there are values of V which give g this property whenever T is monotone increasing and strictly concave. Similarly, when T is a strictly convex increasing function, one can find values of V... 

As p increases from its limiting value of 0 for unequal resistances to 1/4 for equal resistances, the right-hand side increases by a factor of 4 from 2 to 8. This is nicely explained by the fact that for unequal resistances, the change starts at one interface and has to spread monotonically across the whole layer to the other interface. On the other hand for the symmetrical case, changes start equally at both interfaces and meet eventually in the middle. The distance over which perturbations have to travel is therefore halved from L to L/2, and hence D/L is four times larger. 

In general the two velocity functions need not necessarily coincide. For the level set formulation, the velocity function is adequately defined without constraints, whereas the velocity function applied in the stationary formulation is supposed to be monotonic. The travel-time value, which a node acquires as the front sweeps by, is retained because the monotonicity prohibits a second sweep and and so an update of the already computed value. 

Coupons were tested in two loading conditions monotonic to failure and unload/reload to failure. Strain was measured in an average sense using a 25.4 mm extensometer with 0.5 mm travel. 

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