Small parameters fast time scale

Owing to the presence of the small parameter , the model in Equation (7.1) is stiff and can potentially exhibit a dynamic behavior with multiple time scales. Proceeding in a manner similar to the one adopted in Chapter 6, we define the fast time scale r = t/e and rewrite (7.1) as... 

A parameter indicating whether viscoelastic effects are important is the Deborah number, which is the ratio of the characteristic relaxation time of the fluid to the characteristic time scale of the flow. For small Deborah numbers, the relaxation is fast compared to the characteristic time of the flow, and the fluid behavior is purely viscous. For veiy large Deborah numbers, the behavior closely resembles that of an elastic solid. 

Here, w = m, n, and S. V represents the membrane potential, n is the opening probability of the potassium channels, and S accounts for the presence of a slow dynamics in the system. Ic and Ik are the calcium and potassium currents, gca = 3.6 and gx = 10.0 are the associated conductances, and Vca = 25 mV and Vk = -75 mV are the respective Nernst (or reversal) potentials. The ratio r/r s defines the relation between the fast (V and n) and the slow (S) time scales. The time constant for the membrane potential is determined by the capacitance and typical conductance of the cell membrane. With r = 0.02 s and ts = 35 s, the ratio ks = r/r s is quite small, and the cell model is numerically stiff. The calcium current Ica is assumed to adjust immediately to variations in V. For fixed values of the membrane potential, the gating variables n and S relax exponentially towards the voltage-dependent steady-state values noo (V) and S00 (V). Together with the ratio ks of the fast to the slow time constant, Vs is used as the main bifurcation parameter. This parameter determines the membrane potential at which the steady-state value for the gating variable S attains one-half of its maximum value. The other parameters are assumed to take the following values gs = 4.0, Vm = -20 mV, Vn = -16 mV, 9m = 12 mV, 9n = 5.6 mV, 9s = 10 mV, and a = 0.85. These values are all adjusted to fit experimentally observed relationships. In accordance with the formulation used by Sherman et al. [53], there is no capacitance in Eq. (6), and all the conductances are dimensionless. To eliminate any dependence on the cell size, all conductances are scaled with the typical conductance. Hence, we may consider the model to represent a cluster of closely coupled / -cells that share the combined capacity and conductance of the entire membrane area. 

Strictly, the application of the algebraic method in non-linear perturbation theory requires the existence of a small parameter e in the equations, and this can be revealed by a non-dimensionalization procedure. However, even for such a simple set of equations, this is a complicated process which can be avoided by carrying out a numerical investigation of the time-scales present in the problem. By examining the eigenvalues of a linear approximation to the system as described in Section 4.7, it becomes clear that there are two negative fast modes in the above equations over all conditions tested. These are indicated by the presence of two large... 

By looking at the location of the small parameters in the equations, one can guess three time scales r, el, and eV. The original equations (I) were written for the very fast scale t... 

Suppose that the process time scale (or the time window of interest) is bounded between and tmax (tmm reaction time scale spectrum. Then the species reacting on time scales longer than /niax remain dormant their concentrations are hardly different from their initial values. The state of these species may be treated as system parameters. On the other hand, species reacting with time scales shorter than tnim niay be considered relaxed. The relaxed state of these fast-reacting species may be treated as system initial conditions. These considerations naturally help reduce the system dimensionality. 

While in some cases considering the environment is sufficient to reproduce experimental values of the g and hyperfine tensors, there are molecules presenting fast motions in the neighborhood of the unpaired electron. Dependence of the magnetic parameters on these small geometric variations can be very significant [57, 74—76]. These motions are usually too fast with respect to the ESR time scale window so the effective contribution is a correction that can be calculated as an average over short-time dynamics calculated at a QM level [77, 78]. 

The constitutive relation is plotted on a log-log scale in Figure 12.16. There are three fitting parameters Uy, and the exponent 3. The yield stress strain rates, that is, a Uy. It represents the minimum stress needed to sustain steady flow a nonzero value reflects the fact that the material deforms elastically, instead of flowing, for small stresses. We anticipate that the yield stress will depend on packing fraction ( ) and will vanish at where the material no longer displays elastic response. The parameter Tx is a crossover time scale, reminiscent of the crossover frequency to in oscillatory response. For strain rates that are fast compared to 1/Tx, the yield stress no longer dominates the stress. Instead, the stress grows nonlinearly with the strain rate, a yP. The exponent (3 is called the Herschel-Bulkley exponent and is less than one in many materials. 

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