Timescales dissipation

This response time should be compared to the turbulent eddy lifetime to estimate whether the drops will follow the turbulent flow. The timescale for the large turbulent eddies can be estimated from the turbulent kinetic energy k and the rate of dissipation e, Xc = 30-50 ms, for most chemical reactors. The Stokes number is an estimation of the effect of external flow on the particle movement, St = r /tc. If the Stokes number is above 1, the particles will have some random movement that increases the probability for coalescence. If St 1, the drops move with the turbulent eddies, and the rates of collisions and coalescence are very small. Coalescence will mainly be seen in shear layers at a high volume fraction of the dispersed phase. 

The phenomenological approach does not preclude a consideration of the molecular origins of the characteristic timescales within the material. It is these timescales that determine whether the observation you make is one which sees the material as elastic, viscous or viscoelastic. There are great differences between timescales and length scales for atomic, molecular and macromolecular materials. When an instantaneous deformation is applied to a body the particles forming the body are displaced from their normal positions. They diffuse from these positions with time and gradually dissipate the stress. The diffusion coefficient relates the distance diffused to the timescale characteristic of this motion. The form of the diffusion coefficient depends on the extent of ordering within the material. 

One feature of the Maxwell model is that it allows the complete relaxation of any applied strain, i.e. we do not observe any energy stored in the sample, and all the energy stored in the springs is dissipated in flow. Such a material is termed a viscoelastic fluid or viscoelastic liquid. However, it is feasible for a material to show an apparent yield stress at low shear rates or stresses (Section 6.2). We can think of this as an elastic response at low stresses or strains regardless of the application time (over all practical timescales). We can only obtain such a response by removing one of the dashpots from the viscoelastic model in Figure 4.8. When a... 

Typical timescales for the process are of the order of l(T13-l(T9s in condensed phases, and the excess vibrational energy is dissipated as heat. 

Using preformed pores in a DMPC bilayer, Gurtovenko and Vattulainen [82] investigated the translocation of DMPC across a pore. It was shown that multiple lipids diffused across the pore before it dissipated, providing support for pore-mediated flip-flop as mechanism for passive flip-flop. The timescale for pore dissipation was found to be 35-200 ns, at the limits of current computational capability for equilibrium simulations. 

To establish timescales, one needs to study the generator of the dynamics, providing the foundation for Hamiltonian and Liouvillian isometric and contractive evolution, see Appendix F and Refs. [28, 102, 122] for technical discussions involving ensuing organization of appropriate levels of description. As will be seen, the dimension n is controlled by the physicochemical conditions of the dissipative system. As has been shown in Appendix E, the theoretical formulation is founded on the transformation B... 

The conductivity of the buffer solution is, however, also of great relevance in this context (Eq. 10) and capillaries of very small diameter with very efficient heat dissipation are commercially available. Using well designed buffer solutions and conventional 50 pm capillaries, fast protein separations at 2000 V/cm have been described by Hjerten et al. [47]. Using a laser photolysis based gating technique, the group of Jorgenson has reported extremely fast CE separations on a subsecond timescale at up to 2500 V/cm in a 6 pm and at 3300 V/cm in a 10 pm inner diameter capillary respectively [48,49]. 

Although the disk mass is dominated by hydrogen, much less is known about its dispersal. Tracers of hot gas in the innermost disk regions show a one-to-one correspondence to the presence of hot dust (Harfigan et al. 1995) and gas accretion to the stars declines at the same rate as hot dust disperses. Spitzer studies of mid-infrared ro-vibrational lines probe warm gas on orbits similar to Jupiter s and demonstrate the loss of gas in few tens of millions of years (Pascucci et al. 2007). Gas in the coldest disk regions can be traced through CO rotational lines such studies also suggest a gas depletion by 10 Myr. The combined astronomical evidence shows that (1) dust disks dissipate in 3—8 Myr via rapid inside-out dispersal (2) gas dissipates in a similar, or perhaps even shorter timescale. 

Using the dissipation rate (e) as the only relevant flow parameter in the v —> 0 limit, from dimensional considerations the typical fluctuations of the velocity field over a distance l and the corresponding characteristic timescales r(Z), usually interpreted as eddy turnover time , can be estimated as... 

When the Schmidt number (or Prandtl number in case of temperature) is larger than unity the Obukhov-Corrsin scaling (2.120) remains valid for scales above the Kolmogorov scale. However, the scalar fluctuations are not yet dissipated at the viscous cut-off, rj, and are transferred further to smaller scales by chaotic advection in a spatially smooth unsteady flow. This is the so called Batchelor regime, where the advection is dominated by vortices of a single characteristic size l rj, with characteristic timescale tv e l/2v1/2. 

This is the Batchelor spectrum, that is valid in the viscous-convective range that extends from the Kolmogorov scale down to the diffusive scale, where the scalar variance is finally dissipated by molecular diffusion. The diffusive scale in this case is the length scale at which the diffusion time l2/D is comparable to the timescale of advection corresponding to the Kolmogorov scale eddies, that gives... 

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