Basset history term

In Eq. (29), Vd represents the dispersed phase velocity, Fq is the drag force, Fg denotes the force of gravity, Fl is the lift force, Fs represents effects of the fluid stress gradients, Fh is the Basset history term, and F-w represents interactions with the wall. The review paper by Loth (42) presents and discusses all the forces present in Eq. (29). Flere we limit ourselves to the most important effect of drag forces. In the case of spherical solid particles of diameter d, Fd can be expressed as... 

The change of momentum for a particle in the disperse phase is typically due to body forces and fluid-particle interaction forces. Among body forces, gravity is probably the most important. However, because body forces act on each phase individually, they do not result in momentum transfer between phases. In contrast, fluid-particle forces result in momentum transfer between the continuous phase and the disperse phase. The most important of these are the buoyancy and drag forces, which, for reasons that will become clearer below, must be defined in a consistent manner. However, as detailed in the work of Maxey Riley (1983), additional forces affect the motion of a particle in the disperse phase, such as the added-mass or virtual-mass force (Auton et al., 1988), the Saffman lift force (Saffman, 1965), the Basset history term, and the Brownian and thermophoretic forces. All these forces will be discussed in the following sections, and the equations for their quantification will be presented and discussed. 

The second term is the added mass term where is the added mass coefficient. The last term in Eq. (46) is the Basset history term where C is the history coefficient, Qj is the density of the fluid, Pf the viscosity of the fluid and

dummy parameter. In most cases the added mass and the history terms are orders of magnitude less than the drag term and, consequently, may be neglected leading to the equation ... 

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. 

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... 

Terms VI are called "Basset terms . Indeed, they render an effect of the history of the particle s movement with respect to the fluid. The Basset terms that appear in... 

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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