Slender-body theory

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

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. 

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

G. K. Batchelor, Slender-body theory for particles of arbitrary cross-section in Stokes flow, J. Fluid Mech. 44, 419-40 (1970) R. G. Cox, The motion of long slender bodies in a viscous fluid, Part 1, General theory, J. Fluid Mech. 44, 791-810, (1970) Part 2, Shear Flow, J. Fluid Mech. 45, 625-657 (1971) J. B. Keller and S. T. Rubinow, Slender-body theory for slow viscous flow, J. Fluid Mech. 75, 705-14 (1976) R. E. Johnson, An improved slender-body theory for Stokes flow, J. Fluid Mech. 99, 411-31 (1980) A. Sellier, Stokes flow past a slender particle, Proc. R. Soc. London Ser. A 455, 2975-3002 (1999). 

Batchelor, G. K., Slender-body theory for particles of arbitrary cross-section in Stokes flow, J. Fluid Mech., 44,419-440 (1970). 

Many different numerical methods can be used to calculate the ship squat. Their only common point is that they calculate the velocity components and the pressure of the flow surrounding the ship. Depending on whether the fluid is modeled as viscous, a potential velocity function can be used or a more sophisticated flow model has to be applied. Some models are based on slender body theory, whereas others use the boundary elements method (BEM) or the finite element method (FEM). 

Tuck established a mathematical expression for squat with a slender body theory. The slender body theory assumes that the beam, draft, and water depth are very small relative to ship length. This theory uses potential flow where the continuity equation becomes Laplace s equation. The flow is taken to be inviscid and incompressible and is steady and irrotational. In restricted water, the problem is divided into the inner and the outer problems, following a technique of matched asymptotic expansions to construct an approximate solution. The inner problem deals with flow very close to the ship. The potential is only a function of y and 2 in the Cartesian coordinate system. In the cross-flow sections, the potential function... 

Dand and Ferguson and Beck used slender body theory and found good agreement with squat measurements for ratios of water depth to ship draft h/T >2. Dand used the cross-sectional strip theory of Korvin-Kroukovsky. The slender... 

Gourlay extended the slender body theory of Tuck with the unsteady slender body theory. This improvement allows one to consider a ship moving in a non-uniform depth since the coordinate system is now earth-flxed, whereas it is ship-fixed for classic numerical methods. The ID system still uses vertical cross-sections and decomposition into an inner and outer expansion. The pressure integration is only made on the ship length based on the ship section B(x) at each x along the hull. Resolution of the ID equation is made with the finite difference method. Comparison with experimental results for soft squat situations h/T > 4) showed good agreement with numerical results. No tests were made for hard squat conditions (i.e., shallow depths) where flow around the ship is affected. 

The analysis of electrospinning process is based on the slender-body theory. It is widely used in fiber spinning of viscoelastic liquid. To simplify the mathematical description, a few idealizing assumptions are made. The jet radius R decreases slowly along the axial direction dR Z) dZ fluid velocity x> is uniform in the cross section of the jet. 

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