Buoyancy force parameter

G is usually termed the buoyancy force parameter . It is a form of Richardson number. 

A similar temperature and contaminant distribution throughout the room is reached with stratification as with a piston. The driving forces of the two strategies are, however, completely different and the distribution of parameters is in practice different. Typical schemes for the vertical distribution of temperature and contaminants are presented in Fig. 8.11. While in the piston strateg) the uniform flow pattern is created by the supply air, in stratification it is caused only by the density differences inside the room, i.e., the room airflows are controlled by the buoyancy forces. As a result, the contaminant removal and temperature effectiveness are more modest than with the piston air conditioning strategy. 

Zeng et al. (1993) proposed that the dominant forces leading to bubble detachment could be the unsteady growth force and buoyancy force. In order to derive an accurate detachment criterion from a force balance, all forces should be accurately known. If a mechanism is not known precisely, then approximate expressions, one or two fitted parameters and comparison with experiments might offer a solution. Such fitting procedures have indeed been applied (Klausner et al. 1993 Mei et al. 1995a Helden et al.l995). 

It will be seen from Eq. (9.118) that die effect of the buoyancy forces on the velocity profile is characterized by the parameter ... 

This buoyancy force effect parameter can be written as ... 

It will be seen from Fig. 9.30 that the buoyancy forces increase the velocity near the hotter wall (at Y = 1). Since the total mass flow rate is fixed, the increase in velocity near the hot wall is associated with a decrease in velocity near the cooler wall (at Y = 0). As the parameter GrjtRe increases, the velocity profiles become increasingly distorted and at high values of GrjIRe flow reversal can occur adjacent to the cooler wall, i.e., a downward flow can occur near the cooler wail. The condition under whjfch such a reverse flow occurs can be deduced by considering the shear... 

Grashof number — The Grashof number (Ge) is a dimensionless parameter that relates the ratio of buoyancy forces to the viscous forces with a fluid solution. It is defined as ... 

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-... 

As the electrode temperature is larger than the bulk temperature, a modulation of the temperature of the electrochemical interface induces a modulation of the thermal gradient in the solution while the bulk temperature is kept constant. Then the temperature-dependent parameters, like buoyancy forces, are modulated inside the thermal diffusion layer adjacent to the surface and, consequently, a modulation of the velocity is induced near the electrode. Therefore, the transient material balance equation may be written as... 

The Grashof number is a dimensionless parameter used to analyse the flow patterns of a fluid. When the Gr number is much greater than 1, the viscous force is negligible compared to the buoyancy forces. When buoyant forces overcome viscous forces, the flow starts a transition to the turbulent regime [28, 34], For a flat plate in vertical orientation, this transition occurs at a Gr number of around 109. 

By definition the Monin-Obukhov length is the height at which the production of turbulence by both mechanical and buoyancy forces is equal. The parameter L, like the flux Richardson number, provides a measure of the stability of the surface layer. As we discussed, when Rf > 0 and therefore according to (16.69) L > 0 the atmosphere is stable. On the other hand, when the atmosphere is unstable, Rf < 0 and then L < 0. Because of the inverse relationship between Rf and L, an adiabatic atmosphere corresponds to very small (positive or negative) values of Rf and to very large (positive or negative) values of L. Typical values of L for different atmospheric stability conditions are given in Table 16.2. 

To illustrate the dependence of the mobility function d>y on the concentration of surfactant in the continuous phase, in Fig. 12 we present theoretical curves, calculated in Ref 138 for the nonionic surfactant Triton X-100, for the ionic surfactant SDS ( + 0.1 M NaCl) and for the protein bovine serum albumin (BSA). The parameter values, used to calculated the curves in Fig. 12, are listed in Table 4 and K are parameters of the Langmuir adsorption isotherm used to describe the dependence of surfactant adsorption, surface tension, and Gibbs elasticity on the surfactant concentration (see Tables 1 and 2). As before, we have used the approximation Dj Dj (surface diffusivity equal to the bulk dif-fusivity). The surfactant concentration in Fig. 12 is scaled with the reference concentration cq, which is also given in Table 4 for Triton X-100 and SDS + 0.1 M NaCl, cq is chosen to coincide with the cmc. The driving force, F, was taken to be the buoyancy force for dodecane drops in water. The surface force is identified with the van der Waals attraction the Hamaker function Ajj(A) was calculated by means of Eq. (86) (see below). The mean drop radius in Fig. 12 is a = 20 /pm. As seen in the figure, for such small drops 4>y = 1 for Triton X-100 and BSA, i.e., the drop sur-... 

Stability parameters Dimensionless ratio of height, Z, and the Monin-Obukhov stability length, L, which relates to the wind shear and temperature (buoyancy) effect. Thus, the term Z/L represents the relative importance of heat convection and mechanical turbulence. The Richardson number is also a dimensionless ratio of wind shear and buoyancy force. 

Results for the effect of gravity and viscous forces on trapping are given in Table 1 and are shown in Figure 3 as plots of residual saturation versus inverse Bond number with capillary number as parameter. For the type of system under study, it is estimated that no trapping will occur if the inverse Bond number is less than about 3. When the inverse Bond number is greater than about 200, buoyancy forces have no effect on the amount of trapped residual nonwetting phase saturation, and residual saturation depends only on the capillary number. Above the critical capillary number... 

Here Ms and are the mechanical moments of the gravity forces of the beam, wire, and basket of the balance on its sorbent sample site (s) and ballast or tare site (k) respectively. Likewise Bs and Bk indicate the moments of the buoyancy forces of beam, wire, and basket on the sorbent site (s) and ballast or tare site (k) respectively. Tbe other quantities and parameters in Eq. (3.1)... 

With parameter values typical for foams, a = 30 dyn/cm and Rc = 50 xm, and with the buoyancy as the driving force, from Eq. (246) one obtains hi = 14 nm, which is an uirrealistically small value. This means that the buoyancy force might be insuflicient to explain the formation of films during the hydrodynamic interaction of two bubbles. Another outer force that can be important for the emulsion and foam stability is the hydro-dynamic force in a shear or nonturbulent flows [461]. An attempt to treat the case of turbulence was performed by Kumar et al. [462,463]. For micron-sized liquid droplets. 

The capillary length X = y/gAp is defined in terms of the acceleration of gravity g and the difference Ap in mass density of the fluid phases X 3 nun for the air-water interface. For spherical colloidal particles of radius R and with a mass density of the order of Ap, the buoyancy force is given approximately by (4jt/3)/ gAp (4jt/3)y/ /A,, and the effective potential is U ff -(8ji/9) yR R/Xy ln(L/d) -10 kT (/ /pm) ln(L/d), for typical values of the parameters at room temperature. Hence, this interaction is relevant compared to the effect of the thermal agitation only for particle sizes above 10 pm roughly. Equation 2.15 holds when H = 0, but it has to be modified to account for the effect of gravity on the fluid phases, showing up in the form of a pressure field ... 

A solution that was accurate to first order in the buoyancy parameter, Gr. for near-forced convective laminar two-dimensional boundary layer flow over an isothermal vertical plate was discussed in this chapter. Derive the equations that would allow a solution that was second order accurate in Gx to be obtained. Clearly state the boundary conditions on the solution. 

---