We see that in the case of a liquid bridge rupturing in air close to the singularity, the slender-body approximation works excellently. Vakhtang Putkaradze, in Interface Science and Technology, 2004 3 PHYSICAL EXPERIMENTS SHOWING CONICAL INTERFACES 3.1 Lava lamps: break-up of a thread of one fluid in another fluid Therefore, for sufficiently small α, one may write The lateral velocity of the fluid relative to the moving body is made up of (i) the component of U normal to the inclined body, equal to – U sin α, where α = tan −1( ∂w/∂x), and (ii) the lateral velocity of the body, ∂w/∂t, reversed, if at that instant the body is moving upwards as in the inset of Figure 2.5(b).
Hence, only the velocity component related to (2.81b) remains, namely Then, according to slender-body theory, the flow can be regarded as compounded of (a) the steady flow around the stretched-straight body, which we shall ignore here since d R/d x is nearly or exactly zero over most of the length of the body, and (b) the flow due to displacements w( x, t) ( Lighthill 1960). The body is subjected to an oscillatory lateral displacement w( x, t) in the θ = 1/2π plane. 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.Ĭonsider next a very slender cylindrical body for which d R/d x ⋍ 0, or exactly 0, except near the extremities. 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 to 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 2ℓ and radius a ≪ ℓ moving with uniform velocity U in a viscous fluid. SBT is concerned with the second (and by extension the third) 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. 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 rod-like bodies whose length greatly exceeds their other two dimensions. Ribe, in Treatise on Geophysics, 2007 7.04.5.7 Slender-Body Theory
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