Initial motion fluid spheres

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.,... 

As for steady motion, shape changes and oscillations may complicate the accelerated motion of bubbles and drops. Here we consider only acceleration of drops and bubbles which have already been formed formation processes are considered in Chapter 12. As for solid spheres, initial motion of fluid spheres is controlled by added mass, and the initial acceleration under gravity is g y - l)/ y + ) (El, H15, W2). Quantitative measurements beyond the initial stages are scant, and limited to falling drops with intermediate Re, and rising... 

Free-fall experiments with Re >10 show that a sphere released from rest initially accelerates vertically, and then moves horizontally while its vertical velocity falls sharply (R3, S2, S3, V2). As for steady motion discussed in Chapter 5, secondary motion results from asymmetric shedding of fluid from the wake (S3, V2). Wake-shedding limits applicability of the equations given above. Data on the point at which wake-shedding occurs are scant, but lateral motion has been detected for in the range 4-5 (C7). Deceleration occurs for Re > 0.9 Re. The first asymmetric shedding occurs at much higher Re than in steady motion (Re = 200 see Chapter 5), due to the relatively slow downstream development, as shown in Fig. 11.12. 

MEISs and macroscopic kinetics. Formalization of constraints on chemical kinetics and transfer processes. Reduction of initial equations determining the limiting rates of processes. Development of the formalization methods of kinetic constraints direct application of kinetics equations, transition from the kinetic to the thermodynamic space, and direct setting of thermodynamic constraints on individual stages of the studied process. Specific features of description of constraints on motion of the ideal and nonideal fluids, heat and mass exchange, transfer of electric charges, radiation, and cross effects. Physicochemical and computational analysis of MEISs with kinetic constraints and the spheres of their effective application. 

To illustrate this fact, we may adopt the simplest model fluid - the billiard-ball gas -and refer to the simple situation shown in Fig. 2 1. Here we consider a fluid made up of two species namely, black billiard balls and white billiard balls, which are identical apart from their color. By billiard-ball gas we mean that the molecules are modeled as hard spheres that interact only when they colhde. The motion of each billiard ball (or molecule) is stochastic and thus time dependent, but we assume that there is a nonzero, steady macroscopic velocity field u. At an initial moment in time, we imagine a configuration in which the two species are separated by a surface in the fluid that is defined to be locally... 

Solve Newton s second law for a sphere initially moving upward in a fluid with velocity component given by u lO) and initially at z = 0. Draw a sketch of its trajectory as a function of time. The equation of motion is an inhomogeneous equation. ... 

Here, the filament is modeled by a bead-spring configuration that additionally resists bending like a worm-like chain (81). TTius, each bead in the filament experiences a force caused by stretching and bending as described in Eq. (9.12). This offers an approach to treat hydrodynamic friction of the filament with the surrounding fluid beyond resistive force theory. Each bead moving under the influence of a force initiates a flow field that influences the motion of other beads and vice versa, so a complicated many-body problem arises. At low Reynolds number the flow field ( , t) around the spheres is described by the Stokes equations and the incompressibiUty condition ... 

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