Faxen force

Saltation of solids occurs in the turbulent boundary layer where the wall effects on the particle motion must be accounted for. Such effects include the lift due to the imposed mean shear (Saffman lift, see 3.2.3) and particle rotation (Magnus effect, see 3.2.4), as well as an increase in drag force (Faxen effect). In pneumatic conveying, the motion of a particle in the boundary layer is primarily affected by the shear-induced lift. In addition, the added mass effect and Basset force can be neglected for most cases where the particle... 

This result should be checked against Faxen s law, which will be proven in Chap. 8. Faxen s law for a solid sphere holds that the force and torque that are due to an arbitrary undisturbed flow, Uoo = u(r), that satisfies the creeping-motion equations can be calculated from the following formula ... 

This important result is known as Faxen s law.22 According to this law, if we specify the undisturbed velocity u°°(x), then the force on a sphere can be calculated directly from the formula (8-220), without any need to actually solve the flow problem corresponding to the free-stream velocity u°°(x). 

A number of interesting results can be obtained from the Faxen formula, (8-220). The first is to note that the force on a stationary sphere in an arbitrary linear flow,... 

The symbolic operators (87)-(88) for the sphere possess a greater degree of generality than do Faxen s laws. In particular, if we consider any Stokes flow v(r, r/r) vanishing at infinity and satisfying the arbitrary boundary condition V = f(r/r) s f(0, < ) at r = a, then the force on the sphere may be obtained directly from the prescribed velocity boundary condition via the expression... 

The expressions for e and were obtained by using the generalized Faxen theorem giving the force experienced by a particle of arbitrary shape into a fluid imder non-stationary flow conditions (Mazur Bedeaux, 1974 Santamaria Holek, 2005). For spherical particles at low Reynolds numbers, e = el, e is given by... 

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... 

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