DRAG ON A SPHERE

The force D) that a flowing fluid exerts upon a totally immersed body of fixed geometry, such as a sphere, will be considered next in terms of dimensional analysis. If compressibility of the fluid is ignored, the quantities of Table 6.1 suffice to define the system. 

After performing a dimensional analysis with p, d, and V as the dimensionally independent set  

If the Reynolds Number is very large, viscous forces will be negligible compared with inertia forces and p may be dropped from the dimensional analysis. In such a case, yr(R) reduces to a constant (Cj)  

A fluid for which p may be considered zero is called an ideal fluid. Air flowing at high speed approximates an ideal fluid and Eq. (6.2) may be used to find the drag on a body in such a case. The constant is approximately 0.1 for a sphere, but will have other values for bodies having other shapes. At very low values of R, inertia forces will be small compared with viscous forces and p need not be considered in the dimensional analysis. 

Stokes (1856) found C2 to be 3 r for a sphere falling slowly (Reynolds Number 1) in a large volume of viscous liquid. 

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For heterogeneous media composed of solvent and fibers, it was proposed to treat the fiber array as an effective medium, where the hydrodynamic drag is characterized by only one parameter, i.e., Darcy s permeability. This hydrodynamic parameter can be experimentally determined or estimated based upon the structural details of the network [297]. Using Brinkman s equation [49] to compute the drag on a sphere, and combining it with Einstein s equation relating the diffusion and friction coefficients, the following expression was obtained ... 

With regard to the drag on a sphere moving in a Bingham plastic medium, the drag coefficient (CD) must be a function of the Reynolds number as well as of either the Hedstrom number or the Bingham number (7V Si = /Vne//VRe = t0d/fi V). One approach is to reconsider the Reynolds number from the perspective of the ratio of inertial to viscous momentum flux. For a Newtonian fluid in a tube, this is equivalent to... 

Several expressions of varying forms and complexity have been proposed(35,36) for the prediction of the drag on a sphere moving through a power-law fluid. These are based on a combination of numerical solutions of the equations of motion and extensive experimental results. In the absence of wall effects, dimensional analysis yields the following functional relationship between the variables for the interaction between a single isolated particle and a fluid ... 

From Table 3.9 it is seen that, depending on the value of n, the drag on a sphere in a power-law fluid may be up to 46 per cent higher than that in a Newtonian fluid at the same particle Reynolds number. Practical measurements lie in the range 1 < Y < 1.8, with considerable divergences between the results of the various workers. 

The conventional correlation for the drag on a sphere in steady motion is presented as a graph, see Fig. 5.12, called the standard drag curve , where is plotted as a function of Re. Many empirical or semiempirical equations have been proposed to approximate this curve. Some of the more popular are listed in Table 5.1. None of these correlations appears to consider all available data. 

In the free-molecule flow range, the drag on a sphere is given (Sll), for diffuse molecular reflection (cr = 1), by ... 

The first term again represents drag in steady motion at the instantaneous velocity, with Cd an empirical function of Re as in Chapter 5. The other terms represent contributions from added mass and history, with empirical coefficients, Aa and Ah, to account for differences from creeping flow. From measurements of the drag on a sphere executing simple harmonic motion in a liquid, Aa and Ah appeared to depend only on the acceleration modulus according to ... 

Solve for the drag on a sphere in a flowing stream with a uniform velocity profile upstream. Solve for zero Reynolds number (Stokes flow). Compare the solution with an analytical solution in your textbook. (Hint Set the density to zero to simulate Stokes flow. The drag is obtained by integrating certain stresses over the boundary of the sphere.)... 

The drag on a sphere approaching a flat plate has been computed by solving the equations of fluid motion without inertia. The result of the calculation (which is considerably more complex than for the case of the two disks given above) can be expressed in the form... 

For h/dp 1, the drag on a sphere approaches the form (Charles and Mason, 1960)... 

The dependenee of H and IT on A. are shown in Figure 5.9. These hydrodynamic coefficients were calculated after determination of K and G, as in Equation 5-10. These coefficients can be determined by finding the drag on a sphere within a tube and the approach velocity for a sphere in parabolic flow, as described previously for spheres on the tube centerline [12] or distributed throughout the tube [11]. The sohd lines in Figure 5.9 indicate values of H and W when spheres are eonfined to the tube axis (the centerline approximation ) ... 

Equation (308) is the analog of Proudman and Pearson s (PI 1) result for the drag on a sphere at small, nonzero Reynolds numbers, quoted in Eq. (212). 

Numerical predictions of drag on a sphere moving in a power-law fluid are available for the sphere Reynolds number up to 130 [Tripathi et al, 1994 Graham and Jones, 1995] and the values of drag coefficient are best represented by the following expressions with a maximum error of 10% for shear-thinning fluids [Graham and Jones, 1995] ... 

There also exists a range of possibilities that fall between the continuum and molecular models. These are approaches that combine some features of both. For example, in the description of Brownian motion of macromolecules or latex beads in the 10-100 pm size range, one uses classical hydrodynamic results such as the Stokes law for drag on a sphere while at the same time modeling fluctuating forces from molecular collisions that arise at the level of the molecular description. In the theory of ionic... 

Returning to the Stokes-Einstein eqiratiorr, Stokes drag on a sphere is inappropriate not only at high eoneentrations when the presenee of other spheres distort Stokes flow. It is also inappropriate when the sphere is close to a wall, and for the same reason. This is important in particle collection, discussed in Chapter 3 (Section 7), where ehanges in hydrodynamics give rise to both anisotropy, (i.e., 5 Z) ) and position dependence of both on h, the perpendicular... 

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