Particle accelerators history

The first term of Eq. (11-11) is the Stokes drag for steady motion at the instantaneous velocity. The second term is the added mass or virtual mass contribution which arises because acceleration of the particle requires acceleration of the fluid. The volume of the added mass of fluid is 0.5 F, the same as obtained from potential flow theory. In general, the instantaneous drag depends not only on the instantaneous velocities and accelerations, but also on conditions which prevailed during development of the flow. The final term in Eq. (11-11) includes the Basset history integral, in which past acceleration is included, weighted as t — 5) , where (t — s) is the time elapsed since the past acceleration. The form of the history integral results from diffusion of vorticity from the particle. 

Neglect of added mass and history simplifies calculation of unsteady motion considerably. However, for y characteristic of particles in liquids, this introduces substantial errors as illustrated by curve 4 in Fig. 11.7. The accuracy of the simplification improves as y and Re increase, but even for y as high as 10 trajectories calculated neglecting history and added mass substantially underpredict the duration of accelerated motion. Neglect of added mass causes the predicted trajectory to be in error from the start of the motion. Since it is the... 

While the virtual mass force accounts for the form drag on the particle due to relative acceleration between the particle and the surrounding fluid, the history term accounts for the corresponding viscous effects. Moreover, the history force originates from the unsteady diffusion of the vorticity around the particle so there is a delay in the boundary layer development as the relative velocity changes with time [96, 97, 22]. This means that when the relative velocity between the particle and the fluid varies, the vorticity present at the particle surface changes and the surrounding flow needs a flnite time to readapt to the new conditions. 

Term Sf is the sum of forces caused by various inertial effects and by effects of flow nonhomogeneity. When there are concentrated suspensions, an analytical expression for this term has been so far obtained only for fine spherical particles whose Reynolds number is smaller than unity [24]. In the case of fine suspensions, the inertial part of Sf includes 1) an inertial force due to acceleration of the virtual fluid mass by the moving particle, 2) a contribution to the buoyancy which is caused by the field of inertial body forces in the same way as buoyancy is usually caused by the field of external body forces, 3) a hereditary force whose strength and direction depend on the flow history (Basset force), and 4) a new force due to frequency dispersion of the suspension effective viscosity. As the suspension concentration comes to zero, the first three force constituents of the inertial part of Sf tend to manifest themselves as forces similar to those experienced by a single... 

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