Hydrodynamic time

One would prefer to be able to calculate aU of them by molecular dynamics simulations, exclusively. This is unfortunately not possible at present. In fact, some indices p, v of Eq. (6) refer to electronically excited molecules, which decay through population relaxation on the pico- and nanosecond time scales. The other indices p, v denote molecules that remain in their electronic ground state, and hydrodynamic time scales beyond microseconds intervene. The presence of these long times precludes the exclusive use of molecular dynamics, and a recourse to hydrodynamics of continuous media is inevitable. This concession has a high price. Macroscopic hydrodynamics assume a local thermodynamic equilibrium, which does not exist at times prior to 100 ps. These times are thus excluded from these studies. 

In the final, hydrodynamic stage, the system is described by the density, the average velocity, and the local temperature and evolves towards equilibrium by means of the effect of transport phenomena (conductivity, diffusion, viscosity,. . . ). This takes place in times of the order of the hydrodynamic time rh,... 

Characteristic diffusion time Characteristic hydrodynamic time... 

Zel dovich noted (Ref 6) that in order for a stationary burning rate of propellant to be recovered after a pressure fluctuation, enough time must elapse to allow a layer of the propellant to heat up. When the pressure varies rapidly, and there is insufficient time for the solid to heat up, the derivative du/dp (rate of burning velocity increase for a given pressure rise) and the pressure exponent become larger than during steady burning. Therefore, in a small chamber at low pressure, when the characteristic hydrodynamic time of pressure variation is shorter than the burn-up time of the heated layer, stability can be lost... 

This form is adopted from the molecular dynamics computer results of Levesque and Verlet. - The first term in Eq. (8.4) is the short-time colli-sional contribution characterized by the collisional time while the second term originates from long-range many-body interactions and is characterized by the hydrodynamic time t. From the discussion of Section VI it follows that in highly non-Markovian situations (such as large cuj, in barrier dominated processes) the collisional term in Eq. (8.4) makes the dominant contribution. (This observation is very significant for the analysis of the viscosity dependence of the rate. ) Within this model and using available informa-... 

Analytical models of pore collapse follow two approaches, termed hydrodynamic [162] or viscoplastic. In Mader s hydrodynamic model [162] (Fig. 17a), a steep planar shock front hits the upstream surface of an empty spherical pore of diameter d (for nanopores, gas inside the pore can be neglected [52,162]), it accelerating the free surface to velocity 2Up [5]. The pore s free surface undergoes hydrodynamic focusing [61,162]. A material spike strikes the downstream surface, causing impact heating. In molecular dynamics simulations, individual molecules from the upstream surface are observed to break off and strike the downstream surface [61,163]. The hydrodynamic time constant for pore collapse is approximately the material transit time across the pore. 

As widely described in the literature (Janssen et al, 2000 Perner-Nochta and Posten, 2007 Pruvost et al, 2008 Richmond, 2004a RoseUo Sastre et al, 2007), this dynamic fluctuating regime can influence photosynthetic growth and thereby process efficiency. Note, however, that hydrodynamic time-scales are several orders of magnitude greater than photosynthesis timescales, so the effects of L/D cycles due to hydrodynamics can in most cases be considered neghgible (Pruvost et al, 2008), which is not the case for the presence of dark zones, as shown later (Section 3). 

The system evolves on two time scales, a collision time tc and a hydrodynamic time with corresponding length scales Ic and /h, respectively. 

Increase injection (hydrodynamic) time until analyte resolution for a given separation is impaired. If sensitivity is still inadequate the following may be tried ... 

The shortest hydrodynamical time scale for the system is the time for a sound wave to cross the system, that is the time needed for mechanical equilibrium. This is approximately 2000 time steps of integration where, as usual for such models, the time step is one hundredth of a picosecond. The largest time scale, on the other hand, is the time needed for diffusion over a distance L, (D g/L ) , and this is approximately 500.000 time steps. This last time is out of range for our computer studies as in one hour of CPU time on a... 

The Damkohler number Da = tx/ chem - ratio of the characteristic hydrodynamic time scale (for example, turbulent mixing time x) to the chemical reaction characteristic time tchem- Large Damkohler numbers (Da >> 1) correspond to a very fast chemical reaction in comparison with other processes. Small Damkohler numbers (Da << 1) correspond to a slow chemical reaction in relation to other processes. 

Two methods dissipative particle dynamics (DPD) was initially devised by Hoogerbrugge and Koelman as a particle-based off-lattice simulation method for the flow of complex fluids and to tackle hydrodynamic time and space scales beyond those available with MD. Since DPD is a coarse-grained model and individual atoms or molecules are not represented directly by the particles but they are grouped together into beads, these beads represent local fluid packages able to move independently. 

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