Density Stokesian

As a reference to something more familiar, consider the case of a fluid where incompressibility is enforced via a Lagrange multiplier. For a Stokesian fluid, it is assumed that the constitutive variables (stress, energy, heat flux) are a function of density, p, temperature, T, rate of deformation tensor, d, and possibly other variables (such as the gradients of density and temperature). Exploiting the entropy inequality in this framework produces the following constitutive restriction for the Cauchy stress tensor [10]... 

The particle size distribution for the humic acid fraction is depicted in Figure 4. No material sedimented out until the most extreme conditions were applied (40,000 rpm for 24 hr), when some lightening of color at the top of the solution was observed. The sedimented particles had a Stokesian diameter of around 2 nm, which means that a particle size gap of three orders of magnitude exists between these and the next largest particles detected (5 xm). From the experimentally determined coal particle density of 1.43 g/cm, it was calculated that a solid sphere of diameter 2 nm would have a molecular mass of 4000. If the molecules were rod-shaped, even smaller molecular masses would be predicted. Literature values of the molecular mass of regenerated humic acids range between 800 and 20,000, with the values clustering around 1,000 and 10,000 (i5, 16, 17). 

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... 

Here u is the particle velocity, U/ i.s the local fluid velocity, and / is the Stokes friction coefficient. We call particles that obey this equation of motion Stokesian particles. The use of (4.2S) is equivalent to employing (4.19), neglecting the acceleration terms containing the gas density. Because (4.19) was derived for rectilinear motion, the extension to flows with velocity gradients and curved streamlines adds further uncertainty to this approximate method. 

The particle sphericity mainly enters the analysis because it influences the particle terminal velocity. We can account for its effect if we use the Stokesian diameter as a measure of particle size x rather than, for instance, a volume or mass equivalent diameter. We recall from Chap. 2 that the Stokesian (or dynamically equivalent ) diameter is the diameter of a sphere having the same terminal settling velocity and density as the particle under consideration. 

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