Time-dependent Boundary Regions

In certain problems, the nature of the boundary conditions [Christensen (1982)] allows one to factor the solution as follows  

This will only be possible under the proportionality assumption discussed in Sect. 1.8. Applying this assumption, it is easy to show that f(t), g t) are connected by a simple formula. One can show furthermore that W/(r), Oij(r) are given by their static elasticity form while the time functions are easily determined from the boundary conditions. 

General Theorems and Methods of Solution of Boundary Value Problems 

Consider relations of the form (2.1.6, 7) for the viscoelastic displacements and stresses at any point in the medium, where co, 0) 

The next four sections will be devoted to the application of this strategy, in fairly general terms, to certain problem categories. We will make the assumption that the elastic problem, or class of problems, corresponding to the viscoelastic configuration of interest, can be solved, in the sense that the Green s functions for the problem are known. 

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Ting, T.C.T. (1969) A Mixed Boundary Value Problem in Viscoelasticity with Time-Dependent Boundary Regions , in Proceedings of the Eleventh Midwestern Mechanics Conference, ed. by H.J. Weiss, D.F. Young, W.F. Riley, T.R. Rogge (Iowa University Press) pp. 591- 598 Titchmarsh, E.C. (1937) Introduction to the Theory of Fourier Integrals (Oxford University Press, Oxford)... 

Viscoelastic contact problems have drawn the attention of researchers for some time [2,3,104,105]. The mathematical peculiarity of these problems is their time-dependent boundaries. This has limited the ability to quantify the boundary value contact problems by the tools used in elasticity. The normal displacement (u) and pressure (p) fields in the contact region for non-adhesive contact of viscoelastic materials are obtained by a self-consistent solution to the governing singular integral equation given by [106] ... 

In Fig. 5.5, the flow configuration and velocity and temperature distributions at the time instant of 12.5 s are depicted. Even though the flow is subsonic, due to the high Reynolds number, the flow structure in the region upstream of the solid propellant is minimally affected by the time-dependent boundary shape due to phase change. However, the thermal characteristics near the propellant interface show clear signs of time dependency, indicating that the mass flux of... 

Sihcate solutions of equivalent composition may exhibit different physical properties and chemical reactivities because of differences in the distributions of polymer sihcate species. This effect is keenly observed in commercial alkah sihcate solutions with compositions that he in the metastable region near the solubihty limit of amorphous sihca. Experimental studies have shown that the precipitation boundaries of sodium sihcate solutions expand as a function of time, depending on the concentration of metal salts (29,58). Apparently, the high viscosity of concentrated alkah sihcate solutions contributes to the slow approach to equihbrium. 

A major shortage of the method is that the border positions and boundary pressure distributions between the hydrodynamic and contact regions have to be calculated at every step of computation. It is a difficult and laborious procedure because the asperity contacts may produce many contact regions with irregular and time-dependent contours, which complicates the algorithm implementation, increases the computational work, and perhaps spoils the convergence of the solutions. 

A typical trajectory has nonzero values of both P and Q. It is part of neither the NHIM itself nor the NHIM s stable or unstable manifolds. As illustrated in Fig. la, these typical trajectories fall into four distinct classes. Some trajectories cross the barrier from the reactant side q < 0 to the product side q > 0 (reactive) or from the product side to the reactant side (backward reactive). Other trajectories approach the barrier from either the reactant or the product side but do not cross it. They return on the side from which they approached (nonreactive trajectories). The boundaries or separatrices between regions of reactive and nonreactive trajectories in phase space are formed by the stable and unstable manifolds of the NHIM. Thus once these manifolds are known, one can predict the fate of a trajectory that approaches the barrier with certainty, without having to follow the trajectory until it leaves the barrier region again. This predictive value of the invariant manifolds constitutes the power of the geometric approach to TST, and when we are discussing driven systems, we mainly strive to construct time-dependent analogues of these manifolds. 

Fig. 2. Schematic configuration space for the reaction AB + CD — A + BCD. R is the radial coordinate between the center-of-mass of the two diatoms, and r is the vibrational coordinate of the reactive AB diatom. I denotes the interaction region and II denotes the asymptotic region. The shaded regions are the absorption zones for the time-dependent wavefunction to avoid boundary reflections. The reactive flux is evaluated at the r = rB surface.

A more phenomenological approach 25) allows to overcome this limitation. The space around each particle is divided between two regions the boundary is echo-time dependent. In the first one, near the particle, the gradients are too large for the refocusing pulses to be effective, so that the moments situated in this region will be rapidly dephased. These protons contribute a fast signal decay that is unobservable with MRI techniques. 

The time-dependent development of the initial absorption scans of the PVP transport in dextran, monitored at 237 nm, is shown in Fig. 8. The anomalous feature of these scans is that material which absorbs at 237 nm rapidly accumulates on the left-hand side of the boundary. This material appears to be evenly distributed in this region and would therefore not be detected by Schlieren optics. We have shown that accumulation of absorbing material on the LHS of the boundary is exactly balanced by the depletion of absorbing material on the RHS of the boundary. 

An alternative way of portraying the pattern formation behaviour in systems of the sort under consideration here is to delineate the regions in chemical parameter space (the h k plane) over which the uniform state is unstable to non-uniform perturbations. We have already seen in chapter 4, and in Fig. 10.3, that we can locate the boundary of Hopf instability (where the uniform state is unstable to a uniform perturbation and at which spatially uniform time-dependent oscillations set in). We can use the equations derived in 10.3.2 to draw similar loci for instability to spatial pattern formation. For this, we can choose a value for the ratio of the diffusivities / and then find the conditions where eqn (10.48), regarded as a quadratic in either y or n, has two real positive solutions. The latter requires that... 

Thus, in one lifetime (t = 75 pis) the distance which would be traveled is tvp — 2.5 cm. It is thus obvious that serious errors in the measurement of collision-free decay constants could be made if fluorescence cells with dimensions on the order of tdp are used. The limiting dimensions also apply to the observation region since, once the excited molecule moves out of the detection area, the effect is the same as if it had been quenched. Equations for the time dependence of fluorescence which is influenced by migration of long-lived excited molecules to the boundaries of a cylindrical observation region have been developed by Sackett and Yardley. Also included in these equations is the effect of the variation in detection efficiency over the volume of the fluorescence cell. 

This linear growth also holds for the length of material lines, e.g. representing a boundary between two fluid regions with different properties. As we will see later, time-dependent flows allow for much faster, accelerating growth of the length of material lines. 

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