Stokes number finite

In practice, decay of the spin coherence during the delay time t,i and finite optical depth flatten and broaden the anti-Stokes pulse, reducing the total number of anti-Stokes photons which can be retrieved within the coherence time of the atomic memory. For weak retrieve laser intensities, the total photon number per pulse increases with increasing laser power because the time required to read out the spin wave is longer than the characteristic decoherence time of our atomic memory ( 3gs, see Fig. 3 b). After accounting for dead-time effects, we find that once the retrieve laser power increases to 25 mW, all of the spin wave is retrieved in a time shorter than the decoherence time, resulting in a constant anti-Stokes number versus retrieve power. 

While this expression does not contain terms for momenrnm transfer between phases (and, hence, would seem to be much simpler), it is of little practical value because the spatial flux term on the left-hand side cannot be written in terms of U x. Only in the limiting case where Umx is very close to Uf (i.e. very small Stokes numbers) would it be possible to model accurately a disperse multiphase flow using a closed form of Eq. (4.106). For finite Stokes numbers, it is best to solve the separate momentum equations for the disperse and fluid phases. 

These kernels are valid for inertialess particles (i.e. Stp = 0) and can be extended to finite Stokes numbers only by employing ad hoc corrections. For example, Ammar Reeks (2009) derived for the kernel proposed by Salfman Turner (1956) a correction that is based on the local Stokes number. The relative importance of perikinetic aggregation versus orthokinetic aggregation is quantified by a Peclet number ... 

Thus, h will be finite when the Stokes number is nonzero. 

Levin (1961) has shown that inertia deposition of particles below a critical size, which corresponds to a critical Stokes number St = 1/12, is impossible. Regarding a finite size of particles, the collision is characterised by Sutherland s formula (10.11). Comparison of the results obtained from Sutherland s relation and by Levin enables to conclude that in the region of small St < St the approximation of the material point, accepted by Levin and useful at fairly big St, becomes unsuitable for Stsmall Stokes numbers were studied by Dukhin (1982 1983b) for particles of finite size. Under these conditions inertia forces retard microflotation. 

It is worth noticing that negative effects of inertia forces appear at subcritical values of Stokes numbers when a positive effect is practically absent (cf Section 10.1). The inertia-free approach of a particle and a bubble is caused by the radial particle velocity when its centre is located at a distance from the bubble surface approximately equal to a. When the particle radius tends to zero, this velocity also tends to zero and deposition depends on the finite size of the particle. 

According to this approach the consideration of a finite velocity is equivalent to the consideration of a finite size of particles in the interception effect discussed by Sutherland. Thus, it is necessary to take into consideration low radial velocity in the vicinity of the equator first because the angular dependence (cf Eq. 8.117) of the velocity on equator vanishes and, second, as a result of decreasing velocity with decreasing the particle size. Even at small centrifugal force and small Stokes numbers deposition in the neighbourhood of the equator is... 

While neglecting the finite size of a particle at subcritical Stokes numbers excludes inertia forces at all, the situation changes with the consideration of a finite particle size. Inertia forces become essential at subcritical but not too small Stokes numbers. This effect can turn out to be negative. Thus, a critical Stokes number separates the regions of positive and negative effects of inertia forces on particle deposition. 

Under potential flow Ej, can be expressed by Sutherland s equation, by Langmuir s equation for a potential flow. Note that Langmuir obtained the equation from numerical calculations of the differential equations for the particle trajectory. In this equation the particle is considered as a point mass, i.e. the particle dimension is absent in Eq. (10.10). This means there is no direct influence of the particle dimension on the trajectory. However, the particle mass, the drag coefficient, and the Stokes number depend on the particle size, only later Langmuir s equation was derived for a finite particle size. This result cannot be used in... 

A.S. Sangani, G. Mo, H.K. Tsao and D.L. Koch, Simple Shear Flows of Dense Gas-solid Suspensions at Finite Stokes Number, Journal of Fluid Mechanics 313 309-341 (1996)... 

Sangani, A. S., Mo, G., Tsao, H., and Kocb, D. L. 1996. Simple shear flows of dense gas-solid suspensions at finite Stokes numbers. /. Fluid Mech. 313, 309-341. 

The majority of polymer flow processes are characterized as low Reynolds number Stokes (i.e. creeping) flow regimes. Therefore in the formulation of finite element models for polymeric flow systems the inertia terms in the equation of motion are usually neglected. In addition, highly viscous polymer flow systems are, in general, dominated by stress and pressure variations and in comparison the body forces acting upon them are small and can be safely ignored. 

The simplest case of fluid modeling is the technique known as computational fluid dynamics. These calculations model the fluid as a continuum that has various properties of viscosity, Reynolds number, and so on. The flow of that fluid is then modeled by using numerical techniques, such as a finite element calculation, to determine the properties of the system as predicted by the Navier-Stokes equation. These techniques are generally the realm of the engineering community and will not be discussed further here. 

Of much greater relevance in micro reactors are rectangular channels, which were the subject of a study by Cheng et al. [110], among others. They solved the Navier-Stokes equation for channel cross-sections with an aspect ratio between 0.5 and 5 and Dean numbers between 5 and 715 using a finite-difference method. The vortex patterns obtained as a result of their computations are depicted in Figure 2.20 for two different Dean numbers. 

CFD may be loosely thought of as computational methods applied to the study of quantities that flow. This would include both methods that solve differential equations and finite automata methods that simulate the motion of fluid particles. We shall include both of these in our discussions of the applications of CFD to packed-tube simulation in Sections III and IV. For our purposes in the present section, we consider CFD to imply the numerical solution of the Navier-Stokes momentum equations and the energy and species balances. The differential forms of these balances are solved over a large number of control volumes. These small control volumes when properly combined form the entire flow geometry. The size and number of control volumes (mesh density) are user determined and together with the chosen discretization will influence the accuracy of the solutions. After boundary conditions have been implemented, the flow and energy balances are solved numerically an iteration process decreases the error in the solution until a satisfactory result has been reached. 

It is well known that, for the Navier-Stokes equations, the prescription of the velocity field or of the traction on the boundary leads to a well-posed problem. On the other hand, viscoeleistic fluids have memory the flow inside the domain depends on the deformations that the fluid has experienced before it entered the domain, and one needs to specify conditions at the inflow boundary. For integral models, an infinite number of such conditions are required. For differential models only a finite number of conditions are necessary (more and more as the number of relaxation times increases,. ..). The number... 

It is important to point out that even at 77 = 0 the observed value V = 0.942 0.006 is far from its ideal value of V = 0. One important source of error is the finite retrieval efficiency, which is limited by two factors. Due to the atomic memory decoherence rate 7C, the finite retrieval time Tr always results in a finite loss probability p 7c Tr. For the correlation measurements we use a relatively weak retrieve laser ( 2 mW) to reduce the number of background photons and to avoid APD dead-time effects. The resulting anti-Stokes pulse width is on the order of the measured decoherence time, so the atomic excitation decays before it is fully retrieved. Moreover, even as 7C —> 0 the retrieval efficiency is limited by the finite optical depth q of the ensemble, which yields an error scaling as p 1/ y/rj. The measured maximum retrieval efficiency at 77 = 0 corresponds to about 0.3. In addition to finite retrieval efficiency, many other factors reduce correlations, including losses in the detection system, background photons, APD afterpulsing effects, and imperfect spatial mode-matching. 

For 2D and axisymmetric flows, however, the general representation results, (7-31) and (7 32), do lead to a very significant simplification, both for creeping flows and for flows at finite Reynolds numbers where we must retain the ftdl Navier Stokes equations. The reason for this simplification is that the vector potential A can be represented in terms of a single... 

An alternative point of view is that vorticity accumulates at the rear of the body, which leads to a large recirculating eddy structure, and as a consequence, the flow in the vicinity of the body surface is forced to detach from the surface. This is quite a different mechanism from the first one because it assumes that the primary process leading to separation is the production and accumulation of vorticity rather than the local dynamics within the boundary layer.25 However, viscosity still plays a critical role for a solid body in the production of vorticity. In fact, for any finite Reynolds number, there is probably some element of truth in both explanations. Furthermore, it is unlikely that experimental evidence (or evidence based on numerical solutions of the complete Navier Stokes equations) will be able to distinguish between these ideas, because such evidence for steady flows will inevitably be limited to moderate Reynolds numbers. 

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