Time periodic flow

The explanation for the approach to a log-normal distribution is as follows. Consider, for simplicity, flows as those considered by Muzzio, Swanson, and Ottino (1991a) two-dimensional time periodic flows. Let An,k denote the length stretch experienced by a fluid element between periods n and k. The total stretching after m periods of the flow, A0,m, can be written as the product of the stretchings from each individual period ... 

D time periodic flows and spatially periodic duct flows can produce chaos. A necessary condition for chaos is crossing of streamlines. 

Niederkom, T. C, and Ottino, J. M., Mixing of viscoelastic fluids in time-periodic flows. J. Fluid Mech. 256, 243-268 (1993). 

Horizontal Reactors. Horizontal reactor flow may involve both transverse and longitudinal rolls, as well as time-periodic flows. Insights into these phenomena may be gained from previous analysis of idealized, analogous systems, as well as from recent experiments and computations. Analytical studies of flow between two plates of infinite size differentially heated from below (180) and horizontal channel flow (181) indicate that the development of transverse and longitudinal rolls depends on the relative and absolute magnitudes of the dimensionless Rayleigh and Reynolds numbers, Ra and Re, as well as the aspect ratio. 

In soil pollution studies, the use of composite samples produces an average over a definite area or space. In air or water pollution studies, composite sampling is usually a means of integrating over a longer time period. Flow-proportional composite sampling is often applied, especially in river water analysis. 

Long-term noise can be erroneously considered as late-eluting peaks, while drift affects the obtained data when the instrument is running over a long time period. Flow sensitivity affects flow rate and gradient programs using constant pressure pumps. 

Figure 1. Schematics of the continuous chaotic stirrer developed by Kim and Beskok [25], The stirrer consists of periodically repeating mixing bocks with zeta potential patterned surfaces (a) and an electric field parallel to the x-axis is externally applied resulting in an electroosmotic flow (b). Combining a unidirectional (x-direction) pressure-driven flow (c) with electroosmotic flow under time-periodic external electric field (in the form of a Cosine wave with a frequency ra), a 2-D time-periodic flow is induced to achieve chaotic stirring in the mixer. Two fluid streams colored with red and blue are pumped into the mixer from the left and are almost well mixed after eight repeating mixing blocks for Re = 0.01, St = I2%, Pe= 1,000, and T = 0.8(d).

Figure 2.5 Sketch of the typical structures associated with the breakup of a resonant torus on the Poincare section in a weakly time-periodic flow producing a set of new elliptic and hyperbolic points.

The situation is similar in the case of time-periodic flows (Liu and Haller, 2004), where the eigenmodes are also time-dependent. The same advection and diffusion process is applied to the concentration field during each period T, that is given by the linear propagator... 

In the so called geometric optics limit when the reaction front is very thin and can be approximated by a sharp discontinuity, the front speed was found to be vf U with a logarithmic correction (Cencini et ah, 2003b). In this limit the case of time-periodic flow was also studied. As a result of the interplay between the spatial periodicity... 

In concluding this discussion of Taylor dispersion, we mention Brenner s generalization to nonrectilinear spatially and time-periodic flows (Brenner ... 

Wheeler EK, Eischer P, Puller GG (1998) Time-periodic flow induced structures and instabilities in a viscoelastic surfactant solution. J Non-Newt Fluid Mech 75(2-3) 193-208... 

In the remaining sections of this chapter we discuss the characterization of mixing and the implementation of these statistical and compntational techniques using a simple time-periodic flow as an example. Since the implementation of the algorithms used here is mostly sequential, it is natural to describe them in... 

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