Basset history force

Apart from the drag force, there are three other important forces acting on a dispersed phase particle, namely lift force, virtual mass force and Basset history force. When the dispersed phase particle is rising through the non-uniform flow field of the continuous phase, it will experience a lift force due to vorticity or shear in the continuous phase flow field. Auton (1983) showed that the lift force is proportional to the vector product of the slip velocity and the curl of the liquid velocity. This suggests that lift force acts in a direction perpendicular to both, the direction of slip velocity... 

There are two main approaches for the numerical simulation of the gas-solid flow 1) Eulerian framework for the gas phase and Lagrangian framework for the dispersed phase (E-L) and 2) Eulerian framework for all phases (E-E). In the E-L approach, trajectories of dispersed phase particles are calculated by solving Newton s second law of motion for each dispersed particle, and the motion of the continuous phase (gas phase) is modeled using an Eulerian framework with the coupling of the particle-gas interaction force. This approach is also referred to as the distinct element method or discrete particle method when applied to a granular system. The fluid forces acting upon particles would include the drag force, lift force, virtual mass force, and Basset history force.Moreover, particle-wall and particle-particle collision models (such as hard sphere model, soft sphere model, or Monte Carlo techniques) are commonly employed for this approach. In the E-E approach, the particle cloud is treated as a continuum. Local mean... 

Basset history force acting on a single particle (N)... 

VAN Hdjsberg, M. a. T., ten Thue Boonkkamp, J. H. M. Clercx, H. J. H. 2011 An efficient, second order method for the approximation of the Basset history force. Journal of Computational Physics 230, 1465-1478. 

Reeks, M.W. and Mckee, S. (1984). The Dispersive Effect of Basset History Forces on Particle Motion in a Turbulent Flow. Phys. Huid, Vol. 27, pp. 1573 1582. 

Olivieri S, Picano F, Sardina G, ludicone D, Brandt L The effect of the Basset history force on particle clustering in homogeneous and isotropic turbulence, Phys Fluids 26 041704, 2014. http //dx.doi.Org/10.1063/l.4871480. 

The history force was discovered independently by Boussinesq [18] and Basset [10] in their study of the oscillations of a rigid sphere in a viscous flow. 

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. 

We have so far described drag and lift forces acting on a suspended particle. There are, however, additional hydrodynamic forces, such as Basset history, Faxen correction, and virtual mass effects that act on the particles. Some of these forces could become important especially for the particles suspended in a liquid. The general equation of motion of a small spherical particle suspended in fluid as obtained by Maxey and Riley is given as... 

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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