Fluid particles, definition

Either (1) all intermediate states involve more than one excited fluid particle (this is just the old definition ),... 

As shown above in (6.162), the Lagrangian fluid-particle PDF can be related to the Eulerian velocity, composition PDF by integrating over all initial conditions. As shown below in (6.168), for the Lagrangian notional-particle PDF, the same transformation introduces a weighting factor which involves the PDF of the initial positions y) and the PDF of the current position /x.(x t). If we let V denote a closed volume containing a fixed mass of fluid, then, by definition, x, y e V. The first condition needed to reproduce the Eulerian PDF is that the initial locations be uniform ... 

In Chapter 2 considerable effort is devoted to establishing the relationship between the stress tensor and the strain-rate tensor. The normal and shear stresses that act on the surfaces of a fluid particle are found to depend on the velocity field in a definite, but relatively complex, manner (Eqs. 2.140 and 2.180). Therefore, when these expressions for the forces are substituted into the momentum equation, Eq. 3.53, an equation emerges that has velocities (and pressure) as the dependent variables. This is a very important result. If the forces were not explicit functions of the velocity field, then more dependent variables would likely be needed and a larger, more complex system of equations would emerge. In terms of the velocity field, the Navier-Stokes equations are stated as... 

Continuous Mixers In continuous mixers, exiting fluid particles experience both different shear rate histories and residence times therefore they have acquired different strains. Following the considerations outlined previously and parallel to the definition of residence-time distribution function, the SDF for a continuous mixer/(y) dy is defined as the fraction of exiting flow rate that experienced a strain between y and y I dy, or it is the probability of an entering fluid particle to acquire strain y. The cumulative SDF, F(y), defined by... 

The term incompressible flow is applied to any situation where changes in the density of a fluid particle are negligible [119]. A mathematical definition is... 

The main challenge in formulating these equations is related to the definition of the collision operator. So far this approach has been restricted to the formulation of the population balance equation. That is, in most cases a general transport equation which is complemented with postulated source term formulations for the particle behavior is used. Randolph [80] and Randolph and Larson [81] used this approach deriving a microscopic population balance equation for the purpose of describing the behavior of particulate systems. Ramkrishna [79] provides further details on this approach considering also fluid particle systems. 

The expression after the first equal sign in Eq. (7.18) provides the statistical definition of the singlet density in the disordered system where 5 q — qt) — S r — Vi) for a simple fluid without internal degrees of freedom, whereas q — qi) — 6 r - Vi) d u - (jJi) for anisotropic fluid particles. The second... 

For each set of initial conditions, Eqs. (4.1)-(4.3) can be solved to And X ", U ", and The initial conditions are randomly selected from known distribution functions, and we can assume that there is an infinite number of possible combinations. Each combination is called a realization of the granular flow, and the set of all possible realizations forms an ensemble. Note that, because the particles have finite size, they cannot be located at the same point thus X " 4 X for n 4 m. Also, the collision operator will generate chaotic trajectories and thus the particle positions will become uncorrelated after a relatively small number of collisions. In contrast, for particles suspended in a fluid the collisions are suppressed and correlations can be long-lived and of long range. We will make these concepts more precise when we introduce fluid-particle systems later. While the exact nature of the particle correlations is not a factor in the definition of the multi-particle joint PDF introduced below, it is important to keep in mind that they will have... 

Finally, using the definition of the fluid-particle NDF given in Fq. (4.29), we will derive the GPBF. 

The other two collision source vectors, and can be evaluated using the definitions in Eqs. (6.104) and (6.106). As mentioned earlier, will be closed in terms of the moments of order two and lower, and their gradients. In contrast, C will not be closed in terms of any finite set of moments. Nevertheless, it can be approximated using quadrature-based moment methods as described in Section 6.5. In the fluid-particle limit d d2), neither CI2 i or C will contribute terms involving spatial gradients of the fluid properties (i.e. buoyancy, lift, etc.) to the fluid-phase momentum equation. As mentioned earlier, such terms result from the model for gapi i-n) and would appear, for example, on using the expression in Eq. (6.81). With the latter, Eq. (6.161) becomes... 

Let /a and be the fractions of the total COj exchange due to assimilation and respiration, respectively, so that for a moving fluid particle, /a = 4>Adt)/( [Pg.54]

On the other hand, we can also calculate the Stokes drift velocity from the solution of Eq. 8. We can follow a certain fluid particle s path x(xq, t), where xq is a reference point independent of the time. From the definition of the flow velocity, we can write... 

For a laminar flow through a straight microchannel, the fluid particles move in definite paths called streamlines and there are no components of fluid velocity normal to the duct axis. The projection of Eq. 6 along the axial direction (z) gives... 

At finite deformations, equation 59 can be shown to be incorrect because it is not objective i.e., it predicts results which erroneously depend on the orientation of the sample with respect to laboratory coordinates. This error can be eliminated by replacing j/j in equation 59 by the components of a corotational rate-of-strain tensor or the components of one of several possible codeformational rate-of-strain tensors either of these replacements ensures that the unwanted dependence of cy on the instantaneous orientation of a fluid particle in space is removed. If the stress-strain relations are linear within the changing coordinate frame, equation 59 is modified only be replacing y,-j with a different strain rate tensor whose definition is complicated and beyond the scope of this discussion. The corresponding corotational model is that of Goddard and Miller and the codeformational models correspond to those of Lodge or Oldroyd, Walters, and Fredrickson. ... 

Figure 2.1. Definition of the intrinsic coordinate system associated with the path of a fluid particle for a steady plane flow...

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