Drag force on a sphere

Fig. 8. The ratio of the drag force to the weight of an a-pinene droplet with initial diameter 29.8 /tm evaporating in nitrogen at 293 K. The solid line is the prediction based on Stokes law for the drag force on a sphere, assuming a quasi-steady process.

Dimensional analysis (see drag force on a sphere example) yields CD =/ (Re). If we ensure that the value of Re is the same from prototype to model, then we will have dynamic similarity, and therefore Cd must have the same value from model to prototype. 

This can best be described by an example Suppose that we are interested in the drag force on a sphere that is submerged in a moving stream of fluid. The velocity of the fluid stream some distance ahead of the sphere is V. The diameter of the sphere is D. The density of the fluid is p and the dynamic coefficient of viscosity is p.. We know that these are the only pertinent variables that will affect the drag force F. However, we do not know the relationship of these variables. We can write an almost equation. 

Regardless of how we vary the values of the individual original variables, for each value of (VDpl/x) there is only one unique value of (F/pV D ). So our problem is greatly simplified. We have reduced a five-variable problem to a two-variable problem. We still do not know the analytical or physical relationship of all the variables, but we now know enough to be able to determine the drag force on a sphere for any given combination of the other four independent variables. 

At very low Reynolds numbers no flow separation occurs. In Chapter 5 the Reynolds number was shown to represent a ratio of inertial and viscous forces. At very low Reynolds numbers the inertial forces are small, and the inertial terms of the momentum equation become negligibly small compared to the viscous terms. Under these conditions the drag force on a sphere of diameter D is found to be... 

The earliest treatment of such behavior was that of Newton, who proposed that the drag force on a sphere for flowing air was... 

N. Lecoq, R. Anthore, B. Cichocki, P. Szymczak, and F. Feuillebois, Drag force on a sphere moving towards a corrugated wall,/ Fluid Mech., 513, 247-264 [2004). 

E. S. Asmolov, A. V. Belyaev, and 0. I. Vinogradova, Drag force on a sphere moving towards an anisotropic super-hydrophobic plane, Phys. Rev. E. 84, 026330 (2011). 

The drag force on a single particle falling through a fluid is a function of a dimensionless drag coefficient, CD, the projected area of the particle and the inertia of the fluid.6 For a sphere ... 

This is identical to the drag force on a solid sphere at whose surface perfect slip occurs. 

Drag force on a single rigid sphere in laminar flow... 

The aerodynamic diameter dj, is the diameter of spheres of unit density po, which reach the same velocity as nonspherical particles of density p in the air stream Cd Re) is calculated for calibration particles of diameter dp, and Cd(i e, cp) is calculated for particles with diameter dv and sphericity 9. Sphericity is defined as the ratio of the surface area of a sphere with equivalent volume to the actual surface area of the particle determined, for example, by means of specific surface area measurements (24). The aerodynamic shape factor X is defined as the ratio of the drag force on a particle to the drag force on the particle volume-equivalent sphere at the same velocity. For the Stokesian flow regime and spherical particles (9 = 1, X drag... 

In a flow with quadratic dependence on spatial position, on the other hand, the hydrodynamic force on a sphere will not simply equal Stokes drag. For example, let us suppose that we have a sphere in a 2D, Poiseuille flow ... 

The 0(Re) term in (9-119) also can be associated with the existence of a defect in the flux of momentum downstream of the sphere, at i] = 1, relative to that approaching the sphere from upstream. This momentum defect is proportional in magnitude to the drag force on the sphere. 

By virtue of its yield stress, a viscoplastic material in an unsheared state will support an immersed particle for an indefinite period of time. In recent years, this property has been successfiilly exploited in the design of slurry pipelines, as briefly discussed in section 4.3. Before undertaking an examination of the drag force on a spherical particle in a viscoplastic medium, the question of static equilibrium will be discussed and a criterion will be developed to delineate the conditions under which a sphere will either settle or be held stationary in a liquid exhibiting a yield stress. 

Bird et al. (2002) discuss the biharmonic operator E and show how Equation 7.20 can be solved for flow around a solid sphere and how quantities like the drag force on the sphere can be evaluated. The solution for the case of a fluid sphere is given in Levich (1%2) using a somewhat different method of attack. Following the lines of development pursued here, Equations 7.20 through 7.22 apply in both the iimer and outer fluids, but the boimdary conditions are different. We first list the boundary conditions... 

Another method that has been applied to a number of problems of gas flows is based on a variational principle that can be derived from the linearized Boltzmann equation/ One particularly useful feature of the variational method is that quantities of physical interest such as the drag on an object or the flow rate can be directly related to the stationary point in the variational procedure. One particularly striking application of this method is the computa-tion of the force on a sphere in a gas stream at low Mach numbers for all values of the ratio of the mean free path to the diameter of the sphere. The results are in very good agreement with the experimental results from the Millikan oil drop experiment. ... 

More recently, Proudman and Pearson (1957) provided the key to a more straightforward calculation of further correction terms. The results for the drag force on the sphere in accordance with Stokes , Oseen s, and Proudman and Pearson s calculations are given in Table 7. A very large and important collection of solutions of such problems can be found in the book by Happel and Brenner (1965). 

Drag Force on a Single Rigid Sphere in Laminar Flow... 

Equation 3.1d is the theoretical equation for drag force on a spherical droplet however, this equation does not consider the impact of viscosity or the dynamics of the flow arotmd the sphere. The impact of viscosity and flow dynamics around the sphere is introduced through another coefficient, popularly known as drag coefficient (C ), and Equation 3.1d becomes... 

Therefore, the inertia forces have an insignificant influence on the sedimentation process in this regime. Theoretically, their influence is equal to zero. In contrast, the forces of viscous friction are at a maximum. Evaluating the coefficient B in equation 55 for a = 1 results in a value of 24. Hence, we have derived the expression for the drag coefficient of a sphere, = 24/Re. 

As a matter of fact, one may think of a multiscale approach coupling a macroscale simulation (preferably, a LES) of the whole vessel to meso or microscale simulations (DNS) of local processes. A rather simple, off-line way of doing this is to incorporate the effect of microscale phenomena into the full-scale simulation of the vessel by means of phenomenological coefficients derived from microscale simulations. Kandhai et al. (2003) demonstrated the power of this approach by deriving the functional dependence of the singleparticle drag force in a swarm of particles on volume fraction by means of DNS of the fluid flow through disordered arrays of spheres in a periodic box this functional dependence now can be used in full-scale simulations of any flow device. 

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