Basset force

For simplicity, we start with the derivation in cylindrical coordinates. The resultant equations are then transformed to polar coordinates in which the final form of the Basset force is obtained. 

Consider a sphere of radius a moving along a straight line in an infinite stagnant fluid medium at a velocity V. The cylindrical coordinates are selected such that the z-axis coincides with the path of the sphere, and the origin is an arbitrarily fixed point on the path. Assuming that the fluid is incompressible and its motion is axisymmetric, the stream function if may be defined such that 

Eliminating the pressure term from the preceding equations leads to 

The preceding equations are obtained with the coordinate defined relative to a fixed origin. Now, consider a coordinate defined with the center of the sphere as the origin and denote as the distance from a fixed point. We may express the axial component of velocity uz as 

The second term on the right-side of Eq. (3.14) is of the same order as the square of the velocity and so represents an inertia. Hence, this term can be omitted in Stokes flows. Therefore, it can be stated that both Eq. (3.5) and Eq. (3.6) are valid whether the origin is fixed or in motion. 

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The Basset force can be substantial when the particle is accelerated at a high rate. The total force on a particle in acceleration can be many times that in a steady state [Hughes and Gilliland, 1952]. In a simple model with constant acceleration, the ratio of the Basset force to the Stokes drag, / gs> was derived [Wallis, 1969] and rearranged to [Rudinger, 1980]... 

The Basset force may be negligible when the fluid-particle density ratio is small, e.g., in most gas-solid suspensions, and the time change is much longer than the Stokes relaxation time or the acceleration rate is low. 

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... 

In gas-solid flows well beyond the Stokes regime, the effect of convective acceleration of the gas surrounding the particle is important. To incorporate this effect into the preceding formulation, modifications of the expressions for the Stokes drag, carried mass, and Basset force in the BBO equation are necessary [Odar and Hamilton, 1964]. The modified BBO equation takes the form [Hansell et al., 1992]... 

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... 

Assuming the effect of Basset force is negligible and noting that in gas-solid flows Pp/p 5>> 1, Eq. (5.183) reduces to... 

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. 

For pneumatic transport of solids in a dilute suspension, the effects of apparent mass, Basset force, diffusion, and electric charge of the particles may be ignored. Thus, the dynamic equation of a small particle in a gas medium is given by... 

A is the friction force coefficient B is the inertia correction coefficient and C is the resistance coefficient due to the Basset force. Equation (6.65) gives... 

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... 

Without losing generality, in this section we only consider case (3), where the pipe bend is located in the vertical plane with a vertical gas-solid suspension flow at the inlet, as shown in Fig. 11.10. It is assumed that the carried mass and the Basset force are neglected. In addition, the particles slide along the outer surface of the bend by centrifugal force and by the inertia effect of particles. The rebounding effect due to particle collisions with the wall is neglected. 

The terms on the right-hand side of Eq. (11.4) correspond to interphase drag force, virtual mass force. Basset force and lift force, respectively, /l is a transversal lift... 

Basset force coefficient Transversal lift coefficient Eulerian quantity at node n ... 

In particular applications alternative relations for the slip velocity (3.428) can be derived introducing suitable simplifying assumptions about the dispersed phase momentum equations comparing the relative importance of the pressure gradient, the drag force, the added mass force, the Basset force, the Magnus force and the Saffman lift force [125, 119, 58]. For gas-liquid flows it is frequently assumed that the last four effects are negligible [201, 19[. 

In summary, the Boussinesq-Basset, Brownian, and thermophoretic forces are rarely used in disperse multiphase flow simulations for different reasons. The Boussinesq-Basset force is neglected because it is needed only for rapidly accelerating particles and because its form makes its simulation difficult to implement. The Brownian and thermophoretic forces are important for very small particles, which usually implies that the particle Stokes number is near zero. For such particles, it is not necessary to solve transport equations for the disperse-phase momentum density. Instead, the Brownian and thermophoretic forces generate real-space diffusion terms in the particle-concentration transport equation (which is coupled to the fluid-phase momentum equation). 

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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