Ribe, in Treatise on Geophysics, 2007 7.04.5.7 Slender-Body Theory Whittaker and Lister (2008b) used SBT to calculate the trajectories of plumes rising and widening diffusively in a background shear flow. Whittaker and Lister (2008a) used SBT to determine the structure of the self-similar wake trailing a rising buoyant thermal in Stokes flow. Whittaker and Lister (2006a) presented a model for a creeping plume above a planar boundary from a point source of buoyancy, in which they modeled the flow outside the plume as that due to a line distribution of Stokeslets.
Slender body type free#
Koch and Koch (1995) used an expression analogous to eqn for an expanding ring to model the buoyant spreading of a viscous drop beneath the free surface of a much more viscous fluid. Geophysically relevant applications of SBT include Olson and Singer (1985), who used eqn to predict the rise velocity of buoyant quasi-cylindrical (diapiric) plumes. For further details and extensions of SBT, see Batchelor (1970), Cox (1970), Keller and Rubinow (1976), and Johnson (1980). The lift vanishes only when U is perpendicular to or parallel to e, and the drag is twice as large in the former case as in the latter. The force F includes both drag (parallel to U) and lift (perpendicular to U) components in general. The basic idea of the MMAE is to obtain two different asymptotic expansions for the velocity field that are valid in the inner and outer regions, respectively, and then to match them together in an intermediate or overlap region where both expansions must coincide.
Here, the fluid is unaffected by the rod's finite radius and sees it as a line distribution of point forces with effectively zero thickness.
The second, outer region ρ ≫ a is at distances from the rod that are large compared to its radius. In this region, the fluid is not affected by the ends of the rod and sees it as an infinite cylinder with radius a. The first or inner region includes points whose radial distance ρ from the rod is small compared with their distance from the rod's nearer end. The solution can be found using the MMAE, which exploits the fact that the flow field comprises two distinct regions characterized by very different length scales. The canonical problem of SBT is to determine the force F on a rod of length 2 ℓ and radius a ≪ ℓ moving with uniform velocity U in a viscous fluid. SBT is concerned with the second of these possibilities. However, the problem can be regularized in one of three ways: by including inertia, by making the length of the cylinder finite, or by making the domain bounded ( Section 7.04.5.6.3). The approach takes its departure from Stokes's paradox: the fact that a solution of the equations for Stokes flow around an infinitely long circular cylinder moving steadily in an unbounded viscous fluid does not exist, due to a logarithmic singularity that makes it impossible to satisfy all the boundary conditions ( Batchelor, 1967).
Slender-body theory (SBT) is concerned with Stokes flow around thin rodlike bodies whose length greatly exceeds their other two dimensions. Ribe, in Treatise on Geophysics (Second Edition), 2015 7.04.5.7 Slender-Body Theory
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