Boussinesq number

Pr Raz is sometimes called the Boussinesq number, Box. In terms of this number, Eq. (5.89) becomes Nux BolJ. ... 

The Boussinesq number (Bo) measures the ratio of interfacial and bulk viscous effects in the draining film ... 

Figure 9. Dimensionless drainage time for the film to drain from a dimensionless thickness, h(, to the thickness, by versus Boussinesq number, at various values of the dimensionless interfacial elasticity number. (Reproduced with permission from reference 21. Copyright 1991 Butterworth— Heinemann.)...

Boussinesq number concentration diffusivity surface diffusivity entering coefficient Gibbs elasticity... 

Eddy Viscosity Models. A large number of closure models are based on the Boussinesq concept of eddy viscosity ... 

In some way, introducing an increased particle drag by means of Eq. (17) resembles the earlier proposal raised by Bakker and Van den Akker (1994b) to increase viscosity in the particle Reynolds number due to turbulence (in agreement with the very old conclusion due to Boussinesq, see Frisch, 1995) with the view of increasing the particle drag coefficient and eventually the bubble holdup in the vessel. Lane et al. (2000) compared the two approaches for an aerated stirred vessel and found neither proposal to yield a correct spatial gas distribution. 

A variety of statistical models are available for predictions of multiphase turbulent flows [85]. A large number of the application oriented investigations are based on the Eulerian description utilizing turbulence closures for both the dispersed and the carrier phases. The closure schemes for the carrier phase are mostly limited to Boussinesq type approximations in conjunction with modified forms of the conventional k-e model [87]. The models for the dispersed phase are typically via the Hinze-Tchen algebraic relation [88] which relates the eddy viscosity of the dispersed phase to that of the carrier phase. While the simplicity of this model has promoted its use, its nonuniversality has been widely recognized [88]. 

To dress up in gingham the number of Bingham, or regard Boussinesq s as a gimmick,... 

In the first case (Figure 8a), the side walls are adiabatic, and the reactor height (2 cm) is low enough to make natural convection unimportant. The fluid-particle trajectories are not perturbed, except for the gas expansion at the beginning of the reactor that is caused by the thermal expansion of the cold gas upon approaching the hot susceptor. On the basis of the mean temperature, the effective Rayleigh number, Rat, is 596, which is less than the Rayleigh number of 1844 necessary for the existence of a two-dimensional, stable, steady-state solution with flow in the transverse direction that was computed for equivalent Boussinesq conditions (188). 

The BBO equation delineates a sphere in linear motion at low Reynolds numbers [Basset, 1888 Boussinesq, 1903 Oseen, 1927], For a spherical particle, the BBO equation, based upon Eq. (3.37) with the replacement of the buoyancy force with the pressure gradient force, can be expressed as [Soo, 1990]... 

The boundary layer equations can, as previously discussed, only be applied to flows in which the Reynolds number is relatively large and in which there is no significant areas of reversed flow. This, in particular, severely limits the applicability of these equations in situations involving opposing flow. When these conditions are not satisfied, the solution must be obtained using the full governing equations. For example, if the flow can be assumed to be two-dimensional and if the Boussinesq approximations are applicable, the equations governing the flow are Eqs. (9.5) to (9.7). If the x-axis is vertical, these equations become ... 

When Boussinesq approximation is adopted in full conservation equations, it is noted that the effect of buoyancy force appears in terms of GrjRe where Gr is the Grashof number and Re is the Reynolds number defined in terms of appropriate length, velocity and temperature scales. However, Leal et al. (1973) and Sparrow Minkowycz (1962) have shown that the equivalent buoyancy parameter with the boundary layer assump-... 

The numerical convection model that is used to illustrate the visualization and quantification of mixing (Figures 1 -10) is based on the solution of the equations governing convection in the Earth s mantle, assuming that the mantle can be described as an anelastic and weakly compressible fluid at infinite Prandtl number. Under the extended Boussinesq approximation, we can write the equation of motion as... 

The turbulent Schmidt numbers (a ) were included for all variables that are modeled via the gradient and Boussinesq hypotheses. These Schmidt numbers were set to 1.0. The only exception was [Pg.1149]

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). 

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