Trapped modes in an infinite tube with inhomogeneity

Symmetric and anti-symmetric trapped modes in a cylindrical tube with a segment of higher density are studied. The problem is reduced to an eigenvalue problem of the spatial Helmholtz equation subject to vanishing Dirichlet boundary condition in the cylindrical coordinate system. Through the domain decomposition method and matching technique, multiple frequency parameters are determined by solving the characteristic equation, and the corresponding n-fold periodic trapped modes can be constructed. It is found that in addition to the fundamental mode, the second- and higher-order trapped modes exist, which depend on the density ratio and length of the inhomogeneity. The local inhomogeneity leads to a decrease of the cutoff frequencies of the homogeneous tube and the corresponding vibration mode is localized near the inhomogeneous segment.

Motion of a helical vortex

This paper deals with the motion of a single helical vortex in an unbounded inviscid incompressible fluid. The vortex is an infinite tube whose centreline is a helix and whose cross-section is a small circle where the vorticity is uniform and parallel to the centreline. Ever since Joukowsky (Trudy Otd. Fiz. Nauk Mosk. Obshch. Lyub. Estest., vol. 16, 1912, pp. 1–31) deduced that this vortex translates and rotates steadily without change of form, numerous attempts have been made to compute the velocities. Here, Hardin’s (Phys. Fluids, vol. 25, 1982, pp. 1949–1952) solution for the velocity field is used to find new expressions for the linear and angular velocities of the vortex. The theoretical results are verified by numerically computing the velocity at a single point using the Helmholtz integral and the Rosenhead–Moore approximation to the Biot–Savart law, and by numerically simulating the vortex evolution, under the Euler equations, in a triple-periodic cube. The new formulae are also shown to be more accurate than previous results over the whole range of values of the vortex pitch and cross-section.

A non-interpenetrating lead-organic framework with large channels based on 1D tube-shaped SBUs

A 3D open MOF based on 1D infinite tube-shaped SBUs shows efficient adsorption of I2 and selective adsorption of some dyes containing the SO3− group.