Resistance-drag coefficient

If we consider very slow motion of a macromolecular coil with constant velocity, the force of internal resistance can be neglected and the resistance-drag coefficient for the external force can be written down as... 

If the influence function r/(r) is known, the resistance-drag coefficient of a Brownian particle can be calculated (see Appendix E) as... 

The resistance (drag) coefficient Cx is a function of the Reynolds number, i.e., Cx = /(Re). In order to compare values of Cx for spherical and irregular particles, we introduce the sphericity factor k to account for the particle shape (see p. 168). For particles of various shapes with diameters from 1 to 100 fxm in water, with Reynolds numbers below 0.2, the sphericity factor has the following values ... 

To estimate the dependence of coefficient B in (45) on the length of the macromolecule, we consider, following Pokrovskii and Pyshnograi [61], very slow motion, for which the resistance-drag coefficient according to (A.5) can be written down as < B. So, the dimensionless quantity B is a measure of the increase in the friction coefficient, due to the fact that the particle has behind it a wake of ambient macromolecules. Let us imagine the shear motion of the system as a motion of overlapping macromolecular coils each of which is... 

The situation is complicated, however, because some of the drag on a skidding tire is due to the elastic hysteresis effect discussed in Section XII-2E. That is, asperities in the road surface produce a traveling depression in the tire with energy loss due to imperfect elasticity of the tire material. In fact, tires made of high-elastic hysteresis material will tend to show superior skid resistance and coefficient of friction. 

The term essentially a drag coefficient for the dust cake particles, should be a function of the median particle size and particle size distribution, the particle shape, and the packing density. Experimental data are the only reflable source for predicting cake resistance to flow. Bag filters are often selected for some desired maximum pressure drop (500—1750 Pa = 3.75-13 mm Hg) and the cleaning interval is then set to limit pressure drop to a chosen maximum value. 

We now have all the information necessary to develop some working expressions for particle settling. Look back at equation 3 (the resistance force exerted by the water), and the expressions for the drag coefficient (sidebar discussion on page 261). The important factor for us to realize is that the settling velocity of a particle is that velocity when accelerating and resisting forces are equal ... 

The mean velocity of migration Vj depends on the external driving force = ZjFE and on the resistance to motion set up by the medium s viscosity. This retarding force as a rule is proportional to the velocity. Under the influence of the external force, the velocity will increase until it attains the value Vj where the retarding force VjQ (9 is the drag coefficient) becomes equal to the external driving force. Hence,... 

Intrinsic resistance to dislocation motion can be measured in either of two ways direct measurements of individual dislocation velocities (Vreeland and Jassby, 1973) or by measurements of internal friction (Granato, 1968). In both cases, for pure simple metals there is little or no static barrier to motion. As a result of viscosity there is dynamic resistance, but the viscous drag coefficient is very small (10" to 10" Poise). This is only 0.1 to 1 percent of the viscosity of water (at STP) and about 1 percent of the viscosity of liquid metals at their... 

Air resistance (or drag) is quantified by a dimensionless drag coefficient which is related to the external configuration of the rocket. Other factors that influence drag being the air density, the diameter of the rocket and the square of the rocket velocity. 

FIG. 6-57 Drag coefficients for spheres, disks, and cylinders A = area of particle projected on a plane normal to direction of motion C = overall drag coefficient, dimensionless Dp - diameter of particle Fd = drag or resistance to motion of body in fluid Re = Reynolds number, dimensionless u = relative velocity between particle and main body of fluid (I = fluid viscosity and p = fluid density. (From Lapple and Shepherd, Ind. Eng. Chem., 32, 60S [1940].)... 

If the above values for typical velocity are substituted into the above equation, a maximum range of 5 X 105 ft is possible. Therefore, it is clearly necessary to include air resistance. To include air resistance, a value of CD, the drag coefficient, must be estimated. The drag coefficient ranges from 0.48 for a sphere to 2 for flow perpendicular to a flat strip, and for most fragments ranges from 1.5 to 2.0. 

Similarity of flow will occur around similarly shaped bodies in those cases where the ratio of forces over the bodies surfaces is the same this is equivalent to saying that there will be similar resisting forces when the Reynolds numbers of the two bodies are the same. But then the drag coefficients for the two cases are also the same, i.e.,... 

The origin of the spherical polar coordinate system (r, 9, cp) is held fixed at the center of one particle and the polar axis (9 = 0) is set parallel to E. Let the electrolyte be composed of M ionic mobile species of valence zt and drag coefficient A,-(/ = 1, 2,. . . , M), and let nf be the concentration (number density) of the ith ionic species in the electroneutral solution. We also assume that fixed charges are distributed with a density of pflx. We adopt the model of Debye-Bueche where the polymer segments are regarded as resistance centers distributed in the polyelectrolyte... 

Figure 6.5 displays the relationship between drag coefficient and Reynolds number for particles of different shapes. As the velocity increases the particles tend to rotate to give maximum resistance to drag. 

The equations describe that stage of the particle movement when the liquid interlayer is thin and the distance between the centres of a particle and the bubble is equal to a. The rates of interlayer thinning and particle movement are identical and are controlled by the action of the pressing force F and the resistance force (product of Stokes drag coefficient and the... 

Use of symbolic drag coefficients (Section I1,C,2) and symbolic heat-and mass-transfer coefficients (Section IV, A) furnishes a unique method for describing the intrinsic, interphase transport properties of particles for a wide variety of boundary conditions. Here, the particle resistance is characterized by a partial differential operator that represents its intrinsic resistance to vector or scalar transfer, independently of the physical properties of the fluid, the state of motion of the particle, or of the unperturbed velocity or temperature fields at infinity. Though restricted as yet in applicability, the general ideas underlying the existence of these operators appear capable of extension in a variety of ways. 

For Re 300000 the resistance crisis occurs. The drag force depends on wake dimensions and these dimensions also depend upon the position of the separation line between the wake and the boundary layer. As Re increases, the turbulent separation point moves back as a consequence the wake dimensions diminish and the drag coefficient drops sharply from 0.4 to 0.1 (Fig. 20). All of these considerations are reported in detail by Torobin and Gauvin Boulos, Bhattacharyya and... 

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