Spheres accelerated motion

Torobin, L. B. and Gauvin, W. H. Can. J. Chem. Eng. 38 (1959) 129, 167, 224. Fundamental aspects of solids-gas flow. Part I Introductory concepts and idealized sphere-motion in viscous regime. Part II The sphere wake in steady laminar fluids. Part III Accelerated motion of a particle in a fluid. 

The only rigid particle for which accelerated motion beyond the creeping flow range has been considered in detail is the sphere. Odar and Hamilton (06) suggested that Eq. (11-11) be extended to higher Re as ... 

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... 

Problem 7-7. Accelerating Motion of a Sphere in a Quiescent Fluid. Consider a rigid sphere of radius a, which executes a rectilinear oscillatory motion with velocity... 

In fact, the diffusion constant in solutions has the form of an Einstein diffusion of hard spheres with radius Re. For a diffusing chain the solvent within the coil is apparently also set in motion and does not contribute to the friction. Thus, the long-range hydrodynamic interactions lead, in comparison to the Rouse model, to qualitatively different results for both the center-of-mass diffusion—which is not proportional to the number of monomers exerting friction - as well as for the segment diffusion - which is considerably accelerated and follows a modified time law t2/3 instead of t1/2. 

Consider a spherical particle of diameter dp and density pp falling from rest in a stationary fluid of density p and dynamic viscosity p.. The particle will accelerate until it reaches its terminal velocity a,. At any time t, let a be the particle s velocity. Recalling that the drag force acting on a sphere in the Stokes regime is of magnitude iirdppu, application of Newton s second law of motion can be written as... 

The first term again represents drag in steady motion at the instantaneous velocity, with Cd an empirical function of Re as in Chapter 5. The other terms represent contributions from added mass and history, with empirical coefficients, Aa and Ah, to account for differences from creeping flow. From measurements of the drag on a sphere executing simple harmonic motion in a liquid, Aa and Ah appeared to depend only on the acceleration modulus according to ... 

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. 

In evaluating the particle resistance due to relative motion to the fluid, it is convenient to begin with the constant velocity case and work to the time-dependent case. Consider the case of a sphere moving in a gravitational field and assume that the gravitational acceleration is in the same direction as the sphere motion. Denote Fz as the resistance. Since the total resistance is due to the surface shear and surface pressure, and the flow is axisymmetric, Fz can be postulated as... 

Photographic methods. The camera is one of the most valuable tools in a fluid mechanics research laboratory. In studying the motion of water, for example, a series of small spheres consisting of a mixture of benzene and carbon tetrachloride adjusted to the same specific gravity as the water can be introduced into the flow through suitable nozzles. When illuminated from the direction of the camera, these spheres will stand out in a picture. If successive exposures are taken on the same film, the velocities and the accelerations of the particles can be determined. 

The virtual mass coefficient for a sphere in an invicid fluid is thus Cy = The basic model (5.111) is often slightly extended to take into account the self-motion of the fluid. In general the added mass force is expressed in terms of the relative acceleration of the fluid with respect to the particle acceleration. 

Stokes law can be used in conjunction with a force balance on the sphere to obtain an estimate for the sphere velocity for gravity-driven motion at steady-state, where acceleration effects are not present. For this purpose, we need to know that the net buoyancy force on the sphere is... 

Figure 8.4 (a) Rotational motion of a sphere induced by a shear field this motion is resisted by viscous effects over the surface of the sphere, (b) Resultant distortion of the flow field of the liquid dotted line - flow field in the absence of the sphere full line - effect of the presence of the sphere, which slows down the liquid around the upper hemisphere and accelerates it in the lower. 

Each element has a tendency to move towards its own sphere. If, by some mmatural action, an earthly object is moved beyond the sphere of earth, it strives to return where it belongs. The bigger the object the faster it falls back. On approaching its own sphere more closely the more friendly environment stimulates the natural motion as the object accelerates towards its target. By this theory an object which is launched in a horizontal direction... 

The same argument imphed that celestial objects should accelerate towards higher spheres. Beyond the ring of fire the realm of Ptolemaic planetary spheres commenced. The celestial bodies carried by these spheres consisted of more subtle matter than terrestrial objects. The god-like spirits which controlled the seven planetary spheres were now identified with the seven archangels. The fundamental characteristic of these crystal (transparent) spheres was that they moved in the perfect mode of permanent, uniform, circular motion. 

At infinite Reynolds number, potential flow theory may be used to investigate bubble motion. As a consequence of D Alembert s paradox (see, e.g., Ref. 10), potential flow theory predicts no drag on a steadily rising bubble. However, it provides a result for the force on an accelerating sphere ... 

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