Basset integral

There large imcertainty in the calculated results for the super-critical Stokes numbers is due to three reasons. The Basset integral (Basset 1988, Thomas 1992) is not incorporated in the calculation for the trajectory, and the particle tangential velocity according to Eq. (11.76) is used to calculate the centrifugal forces, 0 and 0,. 

The two last terms on the right-hand side of (2.2.1) relate to fast, unsteady motion. The added mass term accounts for the fact that when accelerating a particle from rest, the surrounding fluid must also be accelerated. This appears to add mass to the particle. The Basset integral says that the drag will, by rapidly changing motion, depend not only on its instantaneous velocity relative to the fluid, but also on the previous motion since the fluid flow pattern may not have had time to adjust, due to the fluid inertia. These two terms are zero in steady movement. 

Clift et al. (2005) showed that ignoring these two unsteady terms (in particular the Basset integral) can lead to errors for a rapidly changing motion in liquid. Fortunately, in the case of gas cyclones we can safely ignore them, even when calculating the rapid, small-scale turbulent motion, since the gas inertia is relatively small. In fact, it turns out that this is true even for the case... 

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. 

Here, the first term is the added mass contribution, the second is Stokes law, and the third is known as the Basset memory integral contribution. Evaluate this expression for... 

The Basset term incorporates, via the time integral, the recent history of the particle s movement. The quantity H sjn t-T) inside the integral allows for a... 

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