The filament model

Equation (2.87) represents the competition between the convergent component of the advecting flow, that compresses the concentration towards the origin, and diffusive dispersion. Since this simple model equation aims only to describe the dominant processes, we neglect secondary effects like the fluctuations of the stretching rate and curvature effects due to the folding of the filament. We 

This is because it represents an open one-dimensional flow in which fluid continuously escapes along the unstable foliation, that is not included in the model (2.87). The loss of fluid along the unstable direction, however, is balanced by the apparent compressibility of the flow along the x direction, dvx/dx = —A, which is consistent with the loss rate in (2.88). 

The asymptotic solution of (2.87) for large times is a Gaussian concentration profile with exponentially decaying amplitude 

This also implies that the concentration within the filament should decrease as the initial patch is diluted by the fluid entrained into the filament along the contracting direction. 

The general time-dependent solution which is normalizable (i.e. with finite Ct) of Eq. (2.87) can be found in terms of Hermite polynomials as 

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The question of what controls the asymptotic decay rate and how is it related to characteristic properties of the velocity field has been an area of active research recently, and uncovered the existence of two possible mechanisms leading to different estimates of the decay rate. Each of these can be dominant depending on the particular system. One theoretical approach focuses on the small scale structure of the concentration field, and relates it to the Lagrangian stretching histories encountered along the trajectories of the fluid parcels. This leads to an estimate of the decay rate based on the distribution of finite-time Lyapunov exponents of the chaotic advection. Details of this type of description can be found in Antonsen et al. (1996) Balkovsky and Fouxon (1999) Thiffeault (2008). Here we give a simplified version of this approach in term of the filament model based... 

This is exactly the filament model (2.87) except for the linear growth term. There are two differences with respect to the KiSS model. One is the possible time-dependence of the growth rate //(t), which is a simplified linear representation of nonlinear interactions and predation on phytoplankton following the initial stage of growth. The other is the advective term —AxdxP that models a local strain. 

In summary, the bistable system shows a scenario quite similar to the autocatalytic case. The main difference is the discontinuous character of the transition between the two regimes, associated with the discontinuous behavior of the observed filament at Dac, and the existence of a threshold perturbation needed to initiate it. All these features can also be recovered from the filament model (Eqs. (7.2)-... 

These observations can be interpreted again in terms of the filament model of Sect. 2.7.1. The interesting point is that there exists a stable steady state filament solution with the excited state in the center, even though in the homogeneous system the excited state is not steady. This can be explained qualitatively by the different timescales corresponding to the dynamics of the two reaction com-... 

Figure 7.13 Different stable filament solutions of the excitable system (7.33), with the flow of the filament model of Sect. 2.7.1.

Examples of stable filament solutions of the excitable model (7.33), obtained numerically for the flow of the filament model of Sect. 2.7.1, are shown in Fig. 7.13 (Neufeld et al., 2002c Hernandez-Garcfa et ah, 2003). At Da = Dac 12.5 the stable filament solution collide with the unstable pulse (7.29) in a saddle-node bifurcation, so that no... 

Since the limit cycle is invariant with respect to a phase shift, one of the Floquet exponents is zero and the asymptotic stability of the limit cycle implies that the other Floquet exponents have all negative real parts. Thus the amplitude of all modes decay in time and the uniform oscillatory state is stable to small localized perturbations (within the filament model and in the linear regime to which Eq. (8.6) applies). The slowest decaying mode (i.e. for p = 0 and k = 0) decays on average as exp(—At), that is the same as the decay rate of a passive scalar inhomogeneity in the filament model. 

Figure 14.11 The sliding filament model of muscle contraction. The actin (red) and myosin (green) filaments slide past each other without shortening.

Studies on muscle contraction carried out between 1930 and 1960 heralded the modem era of research on cytoskeletal stmctures. Actin and myosin were identified as the major contractile proteins of muscle, and detailed electron microscopic studies on sarcomeres by H.E. Huxley and associates in the 1950s produced the concept of the sliding filament model, which remains the keystone to an understanding of the molecular mechanisms responsible for cytoskeletal motility. 

Figure 4. The Brownian ratchet model of lamellar protrusion (Peskin et al., 1993). According to this hypothesis, the distance between the plasma membrane (PM) and the filament end fluctuates randomly. At a point in time when the PM is most distant from the filament end, a new monomer is able to add on. Consequently, the PM is no longer able to return to its former position since the filament is now longer. The filament cannot be pushed backwards by the returning PM as it is locked into the mass of the cell cortex by actin binding proteins. In this way, the PM is permitted to diffuse only in an outward direction. The maximum force which a single filament can exert (the stalling force) is related to the thermal energy of the actin monomer by kinetic theory according to the following equation ...

In order to understand the effect of each process variable, a fundamental understanding of the heat transfer and polymer curing kinetics is needed. A systematic experimental approach to optimize the process would be expensive and time consuming. This motivated the authors to use a mathematical model of the filament winding process to optimize processing conditions. 

This paper will discuss the formulation of the simulator for the filament winding process which describes the temperature and extent of cure in a cross-section of a composite part. The model consists of two parts the kinetic model to predict the curing kinetics of the polymeric system and the heat transfer model which incorporates the kinetic model. A Galerkin finite element code was written to solve the specially and time dependent system. The program was implemented on a microcomputer to minimize computer costs. 

A series of kinetic studies on the carbon filament formation by methane decomposition over Ni catalysts was reported by Snoeck et al. [116]. The authors derived a rigorous kinetic model for the formation of the filamentous carbon and hydrogen by methane cracking. The model includes the following steps ... 

The rigorous kinetic modeling with the incorporation of the diffusion step allows explaining the deactivation of the carbon filament growth and the influence of the affinity for carbon formation on the nucleation of the filamentous carbon. 

At this early stage we present a modification of Yoon s model by introducing the crossing length distribution. This probability function h u) of the lengths of chains which pass through a cross section of the filament is defined by... 

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