Deformation drag

Of obvious importance to aircraft is the smoothness of exterior surfaces. Smooth aerodynamic surfaces reduce aerodynamic drag, resulting in higher airspeeds and increased efficiency. Mechanical fasteners, even countersunk flush fasteners, introduce disruptions in the airflow over the exterior surface. Even the slight deformation of thin sheets around fasteners produces drag. Adhesively bonded structure has no fasteners to disrupt airflow and is more capable of producing the smooth continuous contours that are so common on aircraft. 

The term (v — f ) is the difference between the statistically averaged velocity of the bead (fj) and the local velocity (v) of the continuum background at the position of the bead. The drag force is zero when (v — fj) = 0, in which case the beads deform affinely with the surrounding medium. 

The Rouse and Zimm models consider only minute deformations of the molecular coil in the presence of a constant velocity field. In the presence of a velocity gradient, each bead sample has a different fluid velocity resulting in different drag forces which must be incorporated into the system of equations (21). 

Each submolecule will experience a frictional drag with the solvent represented by the frictional coefficient /0. This drag is related to the frictional coefficient of the monomer unit (0- If there are x monomer units per link then the frictional coefficient of a link is x(0- If we aPply a step strain to the polymer chain it will deform and its entropy will fall. In order to attain its equilibrium conformation and maximum entropy the chain will rearrange itself by diffusion. The instantaneous elastic response can be thought of as being due to an entropic spring . The drag on each submolecule can be treated in terms of the motion of the N+ 1 ends of the submolecules. We can think of these as beads linked... 

Fig. 9.11 Stationary relative velocity, Vp, of pentachloroethane drops in motionless water is dependent on drop diameter, dp. Very small drops behave bice rigid spheres, as shown. Larger drops have an internal circulation and are bnaUy deformed ebipticaUy. When they have reached a certain diameter, the drops in the end oschlate along and across their major axis. Their axial velocity is nearly independent of the diameter. Once the drop size is higher than a maximum value dp raax., the drop will break, owing to the drag forces. (From Ref. 2.)...

Spheroidal particles can be treated analytically, and allow study of shapes ranging from slightly deformed spheres to disks and needles. Moreover, a spheroid often provides a useful approximation for the drag on a less regular... 

Ol). Results for thin disks are obtained in the limit as 0. Approximate relationships, obtained by treating the spheroid as a slightly deformed sphere (H3, SI), are also given. The drag ratio may conveniently be expressed as the ratio of the resistance of the spheroid to that of the sphere with the same equatorial radius a ... 

All the work discussed in the preceding sections is subject to the assumptions that the fluid particles remain perfectly spherical and that surfactants play a negligible role. Deformation from a spherical shape tends to increase the drag on a bubble or drop (see Chapter 7). Likewise, any retardation at the interface leads to an increase in drag as discussed in Chapter 3. Hence the theories presented above provide lower limits for the drag and upper limits for the internal circulation of fluid particles at intermediate and high Re, just as the Hadamard-Rybzcynski solution does at low Re. 

It is an open question whether small quantities of surfactants, too small to influence the gross properties, affect the terminal velocity of liquid drops in air. This appears unlikely in view of the large values of /c, but Buzzard and Nedderman (B18) have claimed such an influence. Acceleration may have contributed to this observation. Quantities of surfactant large enough to lower a appreciably can lead to significantly increased deformation and hence to an increase in drag and a reduction in terminal velocity (R6). 

Attempts to obtain theoretical solutions for deformed bubbles and drops are limited, while no numerical solutions have been reported. A simplifying assumption adopted is that the bubble or drop is perfectly spheroidal. SalTman (SI) considered flow at the front of a spheroidal bubble in spiral or zig-zag motion. Results are in fair agreement with experiment. Harper (H4) tabulated energy dissipation values for potential flow past a true spheroid. Moore (Mil) applied a boundary layer approach to a spheroidal bubble analogous to that for spherical bubbles described in Chapter 5. The interface is again assumed to be completely free of contaminants. The drag is given by... 

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