Maxey-Riley equation

This new approach is described by Lee and Wiesler [90] and developed into a model that explains the transversal movement of particles as a result of the turbulent diffusion. The basis of their modeling is a particular form of the Maxey-Riley equation [100] ... 

The Maxey-Riley equation takes the following form ... 

Let us consider the specific case of a bubble. In most cases, one can ignore the mass of the bubble, trip, in comparison with the mass of the displaced liquid, mi. If one also assumes that the disturbed flow varies slowly enough that the terms involving the Lapladan may be neglected, one can simplify the Maxey-Riley equation as follows ... 

The Maxey-Riley equation is valid provided that the flow around the particle can be approximated, to lowest order, by unsteady Stokes flow. The Reynolds number. Re, of the flow field in which the particle moves may be very large as long as the particle-scale Reynolds numbers are smaller than unity. In practice, this may be very restrictive. For example, let us consider air bubbles in water. If one estimates the Reynolds number of the bubble from the buoyancy-driven rise velocity of the bubble, one obtains... 

The Reynolds number is equal to unity for a bubble with a diameter equal to 0.12 mm. In more viscous liquids, the Maxey-Riley equation should be valid for larger bubbles provided that the interface may be assumed to be immobilized. For example, in an aqueous sugar solution having a viscosity that is 10 times larger than the viscosity of pure water, one might expect the equation to be valid for 0.55-mm bubbles. 

The Maxey-Riley equation does not include inertial effects such as the lift force. Saffman [48] derived an expression for the lift force on a small rigid sphere in a linear shear flow. The leading order lift force on the sphere is caused by an interaction between the slip velocity between the particle and the flow and the shear. Fig. 4 shows a schematic illustration of a particle in a... 

Small bubbles in polar liquids can be treated as rigid spheres because of the effects of surfactants. Provided that the bubble Reynolds number is 0(1) or smaller, one can use the Maxey-Riley equation, which is given in Eq. (14). However, this may be a very restrictive assumption. For example, based on its rise velocity, a 120-/im bubble in water has a Reynolds number roughly equal to unity. It reasonable to assume that bubbles as large as 1 mm are spherical in water. Using Ryskin and Leal s [94,95,96] finite difference methods, McLaughlin [68] found that the axis ratio of a freely rising 1-mm bubble in pure water (i.e., a mobile interface) was 1.12. However, the axis ratio of a fully contaminated 1 -mm bubble was 1.01. In the latter case, the Reynolds number based on the rise velocity and the equivalent spherical diameter was 110. This... 

As a second example, we consider the kinetic equation (KE) for monodisperse, isothermal solid particles suspended in a constant-density gas phase. For clarity, we assume that the particle material density is significantly larger than that of the gas so that only the fluid drag and buoyancy terms are needed to account for momentum exchange between the two phases (Maxey Riley, 1983). In this example, the particles are large enough to have finite inertia and thus they evolve with a velocity that can be quite different than that of the gas phase. 

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. 

M. Maxey and J. Riley. Equation of motion of a small rigid sphere in a nonuniform flow. Phys. Fluids, 26 883-889, 1983. 

Maxey MR, Riley JJ (1983) Equation of motion for a a mall rigid sphere in a non-uniform flow. Phys Fluids 26 (4) 883-889. 

Magnaudet JJM, Takagi S, Legendre D (2003) Drag, deformation and lateral migration of a buoyant drop moving near a wall. J Fluid Mech 476 115-157 Maxey MR, Riley JJ (1983) Equation of motion for a a mall rigid sphere in a non-uniform flow. Phys Fluids 26 (4) 883-889. 

Maxey and Riley [47] derived an equation of motion for a small rigid sphere of radius R in a nonuniform flow. If one considers small bubbles moving in a polar liquid, this equation might be appropriate because surfactants would tend to immobilize the surface of a bubble and make it behave like a rigid sphere. Maxey and Riley assumed that the Reynolds number based on the difference between the sphere velocity and the undisturbed fluid velocity was small compared to unity. In addition, they assumed that the spatial nonuni-formity of the undisturbed flow was sufficiently small that the modified drag due to particle rotation and the Saffman [48] lift force could be neglected. Finally, they ignored interactions between particles. 

The equation of motion given by Maxey and Riley is valid provided that two Reynolds numbers based on the radius of the sphere are small compared to unity. The two Reynolds numbers are uqRIv and R uol(Lv), where uq is a velocity that is characteristic of the undisturbed fluid, wq is a velocity that is characteristic of the relative motion between the particle and the undisturbed fluid, and T is a characteristic length of the undisturbed flow. These conditions imply that the time required for a significant change in the relative velocity is large compared to the timescale for viscous diffusion, and that viscous diffusion remains the dominant mechanism for the transfer of vorticity away from the sphere. 

Extensive reviews of turbulent diffusion were provided by Levich andHinze. Tchen ° was the first investigator who modified the Basset-Boussinesq-Oseen (BBO) equation and applied it to study motions of small particles in a turbulent flow. Corrsin and Lumley pointed out some inconsistencies of Tchen s modifications. Csanady showed that the inertia effect on particle dispersion in the atmosphere is negligible, but the crossing trajectory effect is appreciable. Ahmadi and Ahmadi and Goldschmidt smdied the effect of the Basset term on the particle diffusivity. Maxey and Riley obtained a corrected version of the BBO equation, which includes the Faxen correction for unsteady spatially varying Stokes flows. 

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... 

In the following we resume the notations used by Maxey and Riley (1983) and write the Basset, Boussinesq, Oseen, and Tchen (BBOT) equations in a form that is quasi-identical to that of Maxey and Riley. This formulation is interesting insofar as it highlights the relative movement of the particle with respect to the fluid. The three components of the particle s relative velocity with respect to the fluid are obtained by solving the following differential equations ... 

The differences with the equations obtained by Maxey and Riley are the corrections brought in by Anton (1987, 1988). An in-depth discussion of these equations can be found in the book by Michaehdes. ... 

With a view to applying the concept of the BBO equation to turbulent flows, Tchen (1947) extended the BBO equation for cases where both the particle and the fluid undergo acceleration. Corrsin and Lumley (1956) and Maxey and Riley (1982) identified several inconsistencies in Tchen s work. Their resulting equations comprise many more small terms which seem to be important for applications such as the dispersion of very fine particles in a turbulent atmosphere. For a more detailed discussion on these terms and their relevance, the reader is referred to Maxey and Riley (1982) and Olivieri et al (2014). When the density of the particle is sufficiently larger than that of the ambient fluid, the equation of motion may be simplified again (Yamamoto et al, 2001). Armenio and Fiorotto (2001) reported on numerical simulations assessing systematically the importance of added mass and Basset force in turbulent flows. 

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