Slender bodies

Although the derived scaling law seems to conform to the experimental results under 0-conditions where v = 0.50, the slender body hydrodynamics model is unsatisfactory in many respects. 

Figure 4.8 shows that Eq. (4-27) gives a good approximation for the drag on a cylinder with motion normal to the axis for the range in which experimental results are available. The curve obtained from the exact results for spheroids can be used to estimate for very small or large E. The slender-body result, Eq. (4-37), appears to be applicable for E > 3. 

Cox (C5) and Tchen (Tl) also obtained expressions for the drag on slender cylinders and ellipsoids which are curved to form rings or half circles. The advantages of prolate spheroidal coordinates in dealing with slender bodies have been demonstrated by Tuck (T2). Batchelor (Bl) has generalized the slender body approach to particles which are not axisymmetric and Clarke (C2) has applied it to twisted particles by considering a surface distribution rather than a line distribution. 

For needle-like bodies an electrostatic slender body theory is available (M4) which yields... 

C What is the difference between skin friction drag and pressure drag Which is usually more significant for slender bodies such as airfoils ... 

A semirigorous multiple-scattering theory of Shaqfeh and Fredrickson (1990) that accounts for multiparticle hydrodynamic interactions for slender bodies has verified Batchelor s theory and has given a slightly improved formula for str ... 

Rather rigorous analyses of hydrodynamic interactions among cylindrical among fibers using slender-body theory (Koch 1995) leads to the prediction that the effective diffusivity should be anisotropic and dependent on flow type. The diffusivity in Eq. (6-47) should then be replaced by a tensor dotted into d-tfr/dn. A reasonably accurate estimate of this diffusion tensor is... 

For semidilute suspensions, one expects n/cp to replace p in the logarithmic term in Eq, (6-55), but otherwise the expression for N] should be similar to that for dilute suspensions.] Hence, a plot of N / (prjsP ln(p)) versus y will be universal only if C scales as p and is independent of (p this is consistent with neither slender-body theory nor the simulations of Yamane et al. Hence, the effective diffusivity of rigid rods does not seem able to account for the behavior of the measured values of N in fiber suspensions. However, other possible sources may contribute to the first normal stress difference in these suspensions. For example, according to recent simulations, fiber flexibility produces a positive first normal stress difference (Yamamoto and Matsuoka 1995). Other possible sources of nonzero N include interactions of long fibers with rheometer walls, or streamline curvature. 

In the remainder of this section, we outline results for two types of problems for which internal distributions of singularities have been used to advantage. The first, originally pursued by Chwang and Wu,13 considers bodies of very simple shape - spheres, prolate ellipsoids of revolution (spheroids), and similar cases for which exact solutions can be obtained for some flows either by a point or line distribution involving only a few singularities. The second class of problems is for very slender bodies for which an approximate solution can be obtained by means of a distribution of stokeslets along the particle centerline.14... 

Approximate Solutions of the Creeping-Flow Equations by Means of Slender-Body Theory... 

Figure 8-7. A schematic sketch of the geometrical configuration for the slender-body analysis of an elongated body of arbitrary cross-sectional shape [derivation of Eq. (8-181)].

Eq. (8-124). However, for a slender body, the magnitude ofthe position vector that connects points at the body surface and a point on the centerline is very small (for the same xi), and thus the stokeslet distribution at the body surface can be accurately approximated by use of only a stokeslet distribution on the centerline. This qualitative discussion can be formalized in a straightforward manner. 

With u°° (x) specified, the algebraic equation (8-187) can thus be solved to determine a first approximation to the stokeslet density distribution in the slender-body limit, 0. 

Thus, for u°° = ei (that is, uniform flow parallel to the slender-body axis, with u00 nondi-mensionalized by the magnitude of the velocity, U),... 

---