Stokes drag

To estimate the available interaction time in a free-fall droplet experiment the steady-state balance between gravity and Stokes drag can be analyzed. A free-falling droplet will be subject to both gravitational and drag forces. Assuming that... 

Fig. 17.4 Steady state velocity of freely falling microdroplets as function of droplet radius calculated from the balance between gravitation and Stokes drag...

Substitution of Equation (3.62) into Equation (3.60) gives the relative zero shear viscosity. When the shear rate makes a significant contribution to the interparticle interactions, the mean minimum separation can be estimated from balancing the radial hydrodynamic force, Fhr, with the electrostatic repulsive force, Fe. The maximum radial forces occur along the principle axes of shear, i.e. at an orientation of the line joining the particle centres to the streamlines of 6 = 45°. This is the orientation shown in Figure 3.19. The hydrodynamic force is calculated from the Stokes drag, 6nr 0au, where u is the particle velocity, which is simply... 

Authors efforts in this part of the work have been concentrated on developing turbulence closures for the statistical description of two-phase turbulent flows. The primary emphasis is on development of models which are more rigorous, but can be more easily employed. The main subjects of the modeling are the Reynolds stresses (in both phases), the cross-correlation between the velocities of the two phases, and the turbulent fluxes of the void fraction. Transport of an incompressible fluid (the carrier gas) laden with monosize particles (the dispersed phase) is considered. The Stokes drag relation is used for phase interactions and there is no mass transfer between the two phases. The particle-particle interactions are neglected the dispersed phase viscosity and pressure do not appear in the particle momentum equation. 

The difference is of smaller order than the error in either solution and Eq. (3-35) is exact to 0(Re) (P3). In fact, the Re term in (Cq/Cdsi — 1) can be deduced from the Stokes drag alone for any three-dimensional body symmetrical about a plane normal to the direction of motion (C6). 

Equation (3-45) is analogous to the Oseen correction to the Stokes drag, and is accurate to 0[Pe]." It applies for any rigid or fluid sphere at any Re, provided that Pe - 0 and the velocity remote from the particle is uniform. Figure 3.10 shows that Eq. (3-45) is accurate for Pe < 0.5. Acrivos and Taylor (A2) extended the solution to higher terms, but, as for drag, the additional terms only yield slight improvement at Pe < 1. 

The first term of Eq. (11-11) is the Stokes drag for steady motion at the instantaneous velocity. The second term is the added mass or virtual mass contribution which arises because acceleration of the particle requires acceleration of the fluid. The volume of the added mass of fluid is 0.5 F, the same as obtained from potential flow theory. In general, the instantaneous drag depends not only on the instantaneous velocities and accelerations, but also on conditions which prevailed during development of the flow. The final term in Eq. (11-11) includes the Basset history integral, in which past acceleration is included, weighted as t — 5) , where (t — s) is the time elapsed since the past acceleration. The form of the history integral results from diffusion of vorticity from the particle. 

Figure 2. (Left) Experimental setup in force flow measurements. Optical tweezers are used to trap beads but forces are applied on the RNApol-DNA molecular complex using the Stokes drag force acting on the left bead immersed in the flow. In this setup, force assists RNA transcription as the DNA tether between beads increases in length as a function of time, (a) The contour length of the DNA tether as a function of time and (b) the transcription rate as a function of the contour length. Pauses (temporary arrests of transcription) are shown as vertical arrows. (From Ref. 25.) (See color insert.)...

Electrophoresis causes the migration of charged molecules in solution along the %-axis in response to an electric field. The migration rate or velocity v can be obtained from a balance of electrical force and Stokes drag and expressed by Eq. (9.15). 

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 first term on the right-hand side of Eq. (3.80) is the total drag force in the opposite direction of Up, including both Stokes drag and Oseen drag. The second term represents a lift force in the direction perpendicular to Up. Thus, the lift force or Magnus force for a spinning sphere in a uniform flow at low Reynolds numbers is obtained as... 

The ratio of the Magnus force to the Stokes drag 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]... 

Example 3.3 Consider a rotating gas flow in a cylindrical chamber with a small particle injected into the flow. Assume that the gas rotates as a rigid body with a constant angular velocity co and the only driving force is the Stokes drag [Kriebel, 1961]. Initially, the relative particle velocity is normal to the flow. Develop the equations for the particle trajectory in this rotating flow and discuss the effect of particle sizes on the trajectory. 

Assume that the only force imposed on the particle in the oscillating flow field is the Stokes drag. Derive expressions for the particle velocity and the phase lag between the particle velocity and the gas velocity for the condition when the particle is placed in the flow field for sufficiently long time (i.e., t rs). 

To estimate the particle migration velocity, it is assumed that (1) particles are spherical and have the same size (2) all particles are charged to the same extent (3) the particle motion is governed by the Coulomb force and the Stokes drag only and (4) the direction of the applied electric field is perpendicular to the direction of the suspension flow. 

Example 11.2 Use Eq. (11.17) to derive a general expression for the acceleration length for dilute gas-solid pipe flows. Assume that the Stokes drag coefficient can be used. The friction coefficient of particles at the wall can be estimated by [Konno and Saito, 1969]... 

If Kn is not assumed to be zero, then the Stokes drag force on the particle also must be corrected for a slippage of gas at the particle surface. Experiments of Robert Millikan and others showed that the Stokes drag force could be corrected in a straightforward way. Using the theory of motion in a rarified gas, the mobility takes the form ... 

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