Force Saffman

Saffman Force and Other Gradient-Related Forces... 

Besides the drag force, Basset force, and Saffman force, another force may act on the particle as a result of the existence of a pressure gradient in the fluid. Using the axisymmetric condition, the force on a differential element of a sphere in a pressure gradient field shown in Fig. 3.2 can be expressed by... 

For general particle motion formulation, additional forces such as the Saffman force, Magnus force, and electrostatic force should be included. Assuming that all forces applied on the moving particle are additive, the equation of motion of a particle in an arbitrary flow can be expressed by... 

It should be noted that the forces in Eq. (3.104) are not generally linearly additive. The drag force, Basset force, Saffman force, and Magnus force all depend on the same flow... 

Fe m F M Fs Lorentz force vector Magnus force vector Saffman force vector Pi Pi Stokes expansion Pressure of fluid Dipole moment vector... 

For a fixed spherical particle in a fully developed laminar pipe flow, determine the Saffinan force on the particle at various radial positions. Identify the location of the maximum Saffman force. Discuss the case if the flow is turbulent (using the 1/7 power law for the velocity profile). 

To simplify the following analysis, we assume that (1) the particles are spherical and of identical size (2) for the momentum interaction between the gas and solid phases, only the drag force in a locally uniform flow field is considered, i.e., all other forces such as Magnus force, Saffman force, Basset force, and electrostatic force are negligible and (3) the solids concentration is low so that particle-particle interactions are excluded. 

Fig. 5.5. The Saffman force on a particle in a shear flow. The sketch illustrates that this lift force is caused by the pressure distribution developed around the sphere due to particle rotation induced by the shear flow velocity gradient.

Lawler and Lu [85] reviewed the classical experimental observations on transversal migration of spherical particles and concluded that neither the original Magnus nor the Saffman force models are capable of explaining all these observations. They thus propose that the lift forces might be expressed in terms of the relative particle-fluid angular velocity rather than the absolute angular velocity of the particle as used in all the classical models. Crowe et al [26] also made similar extensions of the classical lift force models. 

Based on the model of Lee and Wiesler [90] and the measurements of Lee and Borner [88] Lee [87] formulated a set of governing equations for the mean motion of particles in a suspension turbulent flow. The model contained both pseudo Stokes drag and pseudo Saffman forces which were expressed in terms of a modified viscosity, and supposedly valid for larger particle Reynolds numbers. 

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