THE DRAG FORCE ON A SPHERICAL PARTICLE

The most satisfactory way of representing the relation between drag force and velocity involves the use of two dimensionless groups, similar to those used for correlating information on the pressure drop for flow of fluids in pipes. 

The second is the group R /pu2, in which R is the force per unit projected area of particle in a plane perpendicular to the direction of motion. For a sphere, the projected area is that of a circle of the same diameter as the sphere. 

R /pu2 is a form of drag coefficient, often denoted by the symbol C D. Frequently, a drag coefficient CD is defined as the ratio of R to pu2. 

It is seen that C D is analogous to the friction factor / (= R/pu2) for pipe flow, and f/) is analogous to the Fanning friction factor /. 

When the force F is given by Stokes law (equation 3.1), then  

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The drag force on a spherical particle moving in water at low speeds (low Reynolds number) is F = —6TTfiaU, where (x, is the viscosity of the water and U is the particle speed. 

By virtue of its yield stress, a viscoplastic material in an unsheared state will support an immersed particle for an indefinite period of time. In recent years, this property has been successfiilly exploited in the design of slurry pipelines, as briefly discussed in section 4.3. Before undertaking an examination of the drag force on a spherical particle in a viscoplastic medium, the question of static equilibrium will be discussed and a criterion will be developed to delineate the conditions under which a sphere will either settle or be held stationary in a liquid exhibiting a yield stress. 

According to Stokes (1891) the drag force on a spherical particle is entirely due to viscous resistance and is described by following the equation ... 

These are the isothermal drag force and the thermal force on a spherical particle. Various difficulties in current theory for these phenomena are cited. 

The standard of comparison for the isothermal drag force on a spherical particle for Ma - 1 has for many years been the empirical relation of MILLIKAN and coworkers [2.96] based on extensive experimental measurements ... 

When the velocity is further increased, the bed becomes less dense and finally the particles are blown out at the terminal or free-fall velocity. The velocity can be calculated by comparing the drag exerted on a spherical particle by the upflowing gas, Fd, and the gravity force, %dl, /6) (p - Pa)g, so that... 

A particle suspended in a fluid is subjected to hydrodynamic forces. For low Reynolds number, the Stokes drag force on a spherical particle is given by... 

The expression for the DEP force was presented above, using Stokes derivation when only the drag and the DEP forces on a spherical particle are considered ... 

The aerodynamic diameter dj, is the diameter of spheres of unit density po, which reach the same velocity as nonspherical particles of density p in the air stream Cd Re) is calculated for calibration particles of diameter dp, and Cd(i e, cp) is calculated for particles with diameter dv and sphericity 9. Sphericity is defined as the ratio of the surface area of a sphere with equivalent volume to the actual surface area of the particle determined, for example, by means of specific surface area measurements (24). The aerodynamic shape factor X is defined as the ratio of the drag force on a particle to the drag force on the particle volume-equivalent sphere at the same velocity. For the Stokesian flow regime and spherical particles (9 = 1, X drag... 

Problem. Calculate the drag force exerted on a spherical particle, diameter 25 mm, as it moves through water at 2 m s-1. 

This is equivalent to stating that the drag on a spherical particle falling in a fluid of infinite extent is due entirely to viscous forces within the fluid. Combining equation (6.3), (6.5) and (6.6) gives, for low Reynolds numbers ... 

When the distance h between a spherical particle of radius a and a solid boundary becomes sufficiently small (hla 1), hydrodynamic interactions between the particle and wall hinder the Brownian motion of the particle. Such effects are critical to near-wall measurements and the accuracy of velocimetry techniques, which rely on an accurate accounting of particle displacements to infer fluid velocity. By applying the evanescent wave-based 3D PTV techniques to freely suspended fluorescent particles, anisotropic hindered Brownian motion has been quantified for particle gap sizes hla 1 with 200 nm diameter tracers [8] and hla 1 with 3 pm diameter tracers [9]. These results confirm the increase of hydrodynamic drag when a particle approaches a solid boundary, and such correction shall be applied to not only Brownian motion but also other translational motion of particles where the drag force is of concern. 

The friction drag force over a spherical charge particle moving through a liquid electrolyte is given based on Stake s law as... 

The Stokes-Einstein equation that follows provides us with the diffusion coefficient for a spherical particle suspended in a liquid by taking into account the drag force on the particle ... 

For validating the flow-induced forces acting on a particle, the drag coefficient of single spherical particles fixed in space was calculated for a wide range of particle Reynolds numbers (i.e. 0.3-480) and compared to experimental data. The simulations captured the main features of the flow structure around the sphere. Furthermore, the drag coefficient was predicted with reasonable accuracy [10,21]. 

The ability separation depends on hydrodynamic drag and DEP force stown as Fig2. The DEP depends on radius particles and polarizable particles in the solution under non-uniform electric fields. The time-averaged DEP force, Fdep, acting on a spherical particle of radius r suspended in a medium of relative permittivity c , is given by ... 

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... 

Rowe and Henwood(26) made similar studies by supporting a spherical particle 12.7 mm diameter, in water, at the end of a 100 mm length of fine nichrome wire. The force exerted by the water when flowing in a 150 mm square duct was calculated from the measured deflection of the wire. The experiments were carried out at low Reynolds numbers with respect to the duct (< 1200), corresponding to between 32 and 96 relative to the particle. The experimental values of the drag force were about 10 per cent higher than those calculated from the Schiller and Naumann equation. The work was then extended to cover the measurement of the force on a particle surrounded by an assemblage of particles, as described in Chapter 5. 

Because most shear-thinning fluids, particularly polymer solutions and flocculated suspensions, have high apparent viscosities, even relatively coarse particles may have velocities in the creeping-flow of Stokes law regime. Chhabra(35,36) has proposed that both theoretical and experimental results for the drag force F on an isolated spherical particle of diameter d moving at a velocity u may be expressed as a modified form of Stokes law ... 

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