Internal particle velocity

Here B represents the density function of new particles which are born/created in the size range Cp, Cp + dtp De represents the density function of existing particles which disappear in the same size range. Additional terms due to external particle velocities in the y- and z-directions and internal particle velocities in the /- and z -directions should be considered along with the above terms to obtain from equation (6.2.49) the following equation ... 

Such an equation has been identified as a general dynamic equation (Friedlander, 1977)) for n r, which, to be exact, should be represented as n rp, v, tl , f) namely, it depends on particle size, fluid velocity, internal particle velocity and time. This equation does not include one term, namely a diffusion term on the right-hand side, D V n rp), which arises from random fluctuations in crystal growth rate for which the diffusion coefficient is D and the coordinate dimensions are x, y and z . 

Contact discontinuity A spatial discontinuity in one of the dependent variables other than normal stress (or pressure) and particle velocity. Examples such as density, specific internal energy, or temperature are possible. The contact discontinuity may arise because material on either side of it has experienced a different loading history. It does not give rise to further wave motion. 

Hugoniot curve A curve representing all possible final states that can be attained by a single shock wave passing into a given initial state. It may be expressed in terms of any two of the five variables shock velocity, particle velocity, density (or specific volume), normal stress (or pressure), and specific internal energy. This curve it not the loading path in thermodynamic space. 

Figure 4.1. Profile of a steady shock wave, risetime imparting a particle velocity, e.g., Uj, pressure Pi, and internal energy density E, propagating with velocity U, into material that is at rest at density pQ and internal energy density Eq.

Surface-active contaminants play an important role in damping out internal circulation in deformed bubbles and drops, as in spherical fluid particles (see Chapters 3 and 5). No systematic visualization of internal motion in ellipsoidal bubbles and drops has been reported. However, there are indications that deformations tend to decrease internal circulation velocities significantly (MI2), while shape oscillations tend to disrupt the internal circulation pattern of droplets and promote rapid mixing (R3). No secondary vortex of opposite sense to the prime internal vortex has been observed, even when the external boundary layer was found to separate (Sll). 

Sy et al (S8, S9) and Morrison and Stewart (M12) analyzed the initial motion of fluid spheres with creeping flow in both phases. For bubbles (y = 0, k = 0), the condition that internal and external Reynolds numbers remain small is sufficient to ensure a spherical shape. However, for other k and y, the Weber number must also be small to prevent significant distortion (S9). For k = 0, the equation governing the particle velocity may be transformed to an ordinary differential equation (Kl), to give a result corresponding to Eq. (11-16), i.e.,... 

Theory. If p is pressure, v - specific volume, e - specific internal energy, D detonation velocity, u - particle velocity, C - sound velocity, y - adiabatic exponent and q -specific.detonation energy, the velocity of propagation and particle velocity immediately behind any plane detonation wave in an explosive, defined by initial conditions, pD, v0, eQ, and uQ, are given by the first two Rankine-Hugoniot relations ... 

These three equations of conservation may be looked upon as defining any three of the four variables p, p, U, u in terms of the 4th, if it is assumed that the equation of the medium, f(p,p,T)=0, as well as the dependence of internal energy of any pair of these variables of state is known. Therefore, the properties of a stationary shock wave follow from the knowledge of the velocity of the piston maintaining the wave, which is also the material (particle) velocity, u... 

Evans St Ablow (Ref 2) defined the steady-flow as "a flow in which all partial derivatives with.respect to time are equal to zero . The five equations listed in their, paper (p 131), together with. appropriate initial and boundary conditions, are sufficient to solve for the dependent variables q (material or particle velocity factor), P (pressure), p (density), e (specific internal energy) and s (specific entropy) in regions which.are free of discontinuities. When dissipative irreversible effects are present, appropriate additional terms are required in the equations... 

These computed values compare well with exptl PCJ s in the previous tabulation. Ref 29 also gives computed values for density, particle velocity (U), internal energy and temps at pressures greater than PCJ- The following interpolation is proposed ... 

An idealized plane shock is characterized by its propagation velocity U, particle velocity u, density p (or specific volume which is 1/p), peak pressure P, and the internal energy E and temp T immediately behind the shock front. 

Assumptions Same as microdistributed balance. Only internal coordinate particle velocities are considered. 

We found in the previous session that five basic parameters were involved to describe fully, and therefore calculate, a shock wave P (pressure), U (shock velocity), u (particle velocity), p (density), and e (specific internal energy). We had derived the three equations based upon mass, momentum, and energy conservation ... 

Figure 27.6b shows the trajectory of an individual synthetic virus during such an internalization process [29] (Movie, see supplementary material of [29]). Three different phases can be identified In phase I, binding to the plasma membrane is followed by a slow movement with drift, which can be deduced from the quadratic dependence of the mean square displacement as a function of time. Furthermore, a strong correlation between neighboring particles is seen and subsequent internalization is observed, and can be proven by quenching experiments. During this phase, the particles are subjected to actin-driven processes mediated by transmembrane proteins. Phase II is characterized by a sudden increase in particle velocity and random movement, often followed by confined movement. 

A special case of considerable interest occurs when the internal-coordinate vector is the particle-velocity vector, which we will denote by the phase-space variable v. In fact, particle velocity is a special internal coordinate since it is related to particle position (i.e. external coordinates) through Newton s law, and therefore a special treatment is necessary. We will come back to this aspect later, but for the time being let us imagine that otherwise identical particles are moving with velocities that may be different from particle to particle (and different from the surrounding fluid velocity). It is therefore possible to define a velocity-based NDF nv(t, x, v) that is parameterized by the velocity components V = (vi, V2, V3). In order to obtain the total number concentration (i.e. number of particles per unit volume) it is sufficient to integrate over all possible values of particle velocity Oy ... 

Equation (2.14) must be coupled with initial conditions given for the starting time and with boundaries conditions in physical space O and in phase space O. Analytical solutions to Eq. (2.14) are available for a few special cases and only under conditions specified by some very simple hypotheses. However, numerical methods can be used to solve this equation and will be presented in Chapters 7 and 8. The numerical solution of Eq. (2.14) provides knowledge of the NDE for each time instant and at every physical point in the computational domain, as well as at every point in phase space. As has already been mentioned, sometimes the population of particles is described by just one internal coordinate, for example particle length (i.e. f = L), and the PBE is said to be univariate. When two internal coordinates are needed, for example particle volume and surface area (i.e. = (v, a)), the PBE is said to be bivariate. More generally, higher-dimensional cases are referred to as multivariate PBEs. Another important case occurs when part of the internal-coordinate vector is equal to the particle-velocity vector (i.e. when the particles are characterized not by a unique velocity field but by their own velocity distribution). In that case, the PBE becomes the GPBE, as described next. 

By using a very similar approach to the one outlined above for the PBE, it is possible to derive a GPBE for an NDF that includes particle velocity as an internal variable. We will denote this general NDF as n(t,x,, ) (i.e. without subscripts on n). The simplest GPBE (i.e. velocity without other internal coordinates) is known as the Boltzmann kinetic equation and was first derived in the context of gas theory (Chapman Cowling, 1961). The final form of the GPBE is... 

When developing models for polydisperse multiphase flows, it is often useful to resort to conditioning on particle size. For example, in gas-solid flows the momentum-exchange terms between the gas phase and a solid particle will depend on the particle size. Thus, the conditional particle velocity given that the particle has internal-coordinate vector will... 

But mi is usually not zero when the internal coordinate represents particle mass, surface area, size, etc. In these cases the PD algorithm can be safely used. The case of null mi occurs more often when the internal coordinate is a particle velocity that, ranging from negative to positive real values, can result in distributions with zero mean velocity. Another frequent case in which the mean is null is when central moments (moments translated with respect to the mean of the distribution) are used to build the quadrature approximation. These cases will be discussed later on, when describing the algorithms for building multivariate quadratures. 

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