Tomotika time

Moreover, the interface relaxation time, characterized by the Tomotika time, is very small. Let us recall that the Tomotika time—or capillary time— noted is the time taken by a distorted liquid-air interface to... 

At the microscale, using our typical numerical values, we obtain T 10 — 10 seconds. The capillary time is much smaller than the time taken by the flow to fill even a small distance of the channel. In summary, even if the quasi-static approach does not account for the dynamics, it produces plausible results because the capillary number is much smaller than unity and the Tomotika time much smaller than the flow time scale in the channel. Hence Evolver produces a realistic succession of steady-state location of the interface, but does not accoimt for the flow velocity. 

Although linear stability theory does not predict the correct number and size of drops, the time for breakup is reasonably estimated by the time for the amplitude of the fastest growing disturbance to become equal to the average radius (Tomotika, 1935) ... 

For the case of a thread breaking during flow, the analysis is complicated because the wavelength of each disturbance is stretched along with the thread. This causes the dominant disturbance to change over time, which results in a delay of actual breakup. Tomotika (1936) and Mikami et al. (1975) analyzed breakup of threads during flow for 3D extensional flow, and Khakhar and Ottino (1987) extended the analysis to general linear flows. Each of these works uses a perturbation analysis to describe an equation for the evolution of a disturbance. 

Lord Rayleigh (31) was the first to investigate the stability of an infinitely long, liquid cylinder embedded in an immiscible liquid matrix driven by surface tension, taking into account inertia. Weber (32) considered stresses in the thread, and Tomotika (33) included the viscosity of the matrix as well. The analysis follows the evolution in time of small Rayleigh sinusoidal disturbance in diameter (Fig. 7.19) ... 

The values of 0(x, A.) can be calculated from Tomotika s equations. The linear dependence of ln(a) on the fiber disintegration time enables estimation of q and, consequently, v -... 

Thus, tb is an important parameter describing the breakup process for fibers subjected to lower stresses than those required for fibrillation, i.e., k < 2. The above indicates that breakup is less likely at low interfacial tension. Since the matrix viscosity appears in the left side of the equation (in the capUlary number), one may expect shorter breakup times with lower matrix viscosity, but it is noteworthy that this term also changes the Tomotika function on the right side of the equation. Figure 7.15 shows the distortion growth rate at the dominant wavelength as a function of viscosity ratio. To obtain a low value of 0(1, X), thread viscosity should be high and the matrix viscosity has to be low (Potschke and Paul 2003). 

The determination of the interfacial tension is performed by means of a recently developed rheo-optical technique [22] based on Tomotika s theory of fibril break-up [33] When a long fluid filament is present in a quiescent fluid matrix, interfacial instabilities due to thermal fluctuations will occur. These so-called Rayleigh instabilities will start to grow and will eventually disintegrate the fibril. Tomotika derived the following formula for the break-up time iB of a Newtonian fibril immersed in a quiescent Newtonian matrix [33]... 

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