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ME437/537 G. Ahmadi 1 HYDRODYNAMIC FORCES Drag Force and Drag Coefficient A particle suspended in a fluid is subjected to hydrodynamic forces. For low Reynolds’ number, the Stokes drag force on a spherical particle is given by FD = 3πµUd, (1) where d the particle diameter, µ is the coefficient of viscosity and U is the relative velocity of the fluid with respect to the particle. Equation (1) may be restated as Re24AU21FC2DD=ρ= (2) In Equation (3), ρ is the fluid (air) density, 4dA2π= is cross sectional area of the spherical particle, and µρ=UdRe (3) is the Reynolds number. The Stokes drag is applicable to the creeping flow regime (Stokes regime) with small Reynolds numbers (Re < 0.5). At higher Reynolds numbers, the flow the drag coefficient deviates from Equation 2. Figure 1 shows the variation of drag coefficient for a sphere for a range of Reynolds numbers. Figure 1. Variations of drag coefficient with Reynolds number for a spherical particle. DC ReEq. (2) Eq. (4)ME437/537 G. Ahmadi 2 Oseen included the inertial effect approximately and developed a correction to the Stokes drag given as Re]16Re/31[24CD+= , (4) which is shown in Figure 1. For 1 < Re < 1000, which is referred to as the transition regime, the following expressions may be used (Clift et al., 1978): Re]Re15.01[24C687.0D+=, (5) or 33.0DRe4Re24C += (6) 0 1 10 100 1000 CD0 1 10 100 1000 10000 ReStokes Eq. (5) Oseen Newton Experiment Figure 2. Predictions of various models for drag coefficient for a spherical particle.ME437/537 G. Ahmadi 3 For 53105.2Re10 ×<< , the drag coefficient is roughly constant ( 4.0CD= ). This regime is referred to as the Newton regime. At 5105.2Re ×≈ , the drag coefficient decreases sharply due to the transient from laminar to turbulent boundary layer around the sphere. That causes the separation point to shift downstream as shown in Figure 3. Laminar Boundary Layer Turbulent Boundary Layer Figure 3. Laminar and turbulent boundary layer separation. Wall Effects on Drag Coefficient For a particle moving near a wall, the drag force varies with distance of the particle from the surface. Brenner (1961) analyzed the drag acting on a particle moving toward a wall under the creeping flow condition as shown in Figure 4a. To the first order, the drag coefficient is given as )h2d1(Re24CD+= (7) (a) Motion normal to the wall (b) Motion parallel to the wall Figure 4. Particle motions near a wall. For a particle moving parallel to the wall as shown in Figure 4b, the Stokes drag force need to be modifies. For large distances from the wall, Faxon (1923) found 1543D])h2d(161)h2d(25645)h2d(81)h2d(1691[Re24C−−−+−= (8) U dh U d hME437/537 G. Ahmadi 4Cunningham Correction Factor For very small particles, when the particle size becomes comparable with the gas mean free path, slip occurs and the expression for drag must be modified accordingly. Cunningham obtained the needed correction to the Stokes drag force: cDCUd3Fπµ= , (9) where the Cunningham correction factor Cc is given by ]e4.0257.1[d21C2/d1.1cλ−+λ+= (10) Here λ denotes the molecular mean free path in the gas. Note that Cc > 1 for all values of d and λ. Figure 5 shows the variation of Cunningham correction factor with Knudsen number. It is seen that cC is about 1 for Kn <0.1 and increases sharply as Kn increases beyond 0.5. Table 4 illustrates the variation of Cunningham correction factor with particle diameter in air under normal pressure and temperature conditions with λ = 0.07 µm. Equation (10) is applicable to a wide range of 1000dKn ≤λ= that covers slip, transition and part of free molecular flows. The particle Reynolds number and Mach number (bases on relative velocity), however should be small. 1 10 100 1000 Cc0.001 0.01 0.1 1 10 100 KnFigure 5. Variation of Cunningham correction with Knudsen number.ME437/537 G. Ahmadi 5Table 4 – Variations of Cc with d for λ = 0.07 µm Diameter, µm Cc 10 µm 1.018 1 µm 1.176 0.1 µm 3.015 0.01 µm 23.775 0.001 µm 232.54 Compressibility Effect For high-speed flows with high Mach number, the compressibility could affect the drag coefficient. Many expressions were suggested in the literature to account for the effect of gas Mach number on the drag force. Henderson (1976) suggested two expressions for drag force acting on spherical particles for subsonic and supersonic flows. Accordingly, for subsonic flow ()SMMMMSSCD6.0Reexp12.01.0Re48.0Re03.01Re48.0Re03.038.05.4Re5.0expRe247.0exp567.133.4Re24821−−+++++++−+−×++=− (11) where M is Mach number based on relative velocity, pVVV −=∆ , and S=2γM is the molecular speed ratio, where γ is the specific heat ratio. For the supersonic flows with Mach numbers equal to or exceeding 1.75, the drag force is given by 2142212DReM86.11S1S058.1S22ReM86.1M34.09.0C+−++++= (12) For the flow regimes with Mach between 1 and 1.75, a linear interpolation is to be used.ME437/537 G. Ahmadi 6 Carlson and Hoglund (1964) proposed the following expression: )}MRe25.1exp(28.182.3{ReM1}Re3M427.0exp(1Re24C88.063.4D−++−−+= (13) Droplets For drag force for liquid droplets at small Reynolds numbers is given as pfpffD/13/21Ud3Fµµ+µµ+πµ= (14) where the superscripts f and p refer to the continuous fluid and discrete particles (droplets, bubbles), respectively. Non-spherical Particles For non-spherical (chains or fibers) particles, Stokes’ drag law must be modified. i.e., KUd3FeDπµ= , (15) where ed is the diameter of a sphere having the same volume as the chain or fiber. That is, 3/1e)Volume6(dπ= (16) and K is a correction factor. For a cluster of n spheres, dnde3/1= . For tightly packed clusters, k < 1.25. Some other values of K are listed in Table 5.ME437/537 G. Ahmadi 7Table 5 – Correction Coefficient Cluster Shape Correction Cluster Shape Correction Cluster Shape Correction oo K = 1.12 oooo K = 1.32 oo oo K = 1.17 ooo K =

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