Some general solutions for linear Bragg–Hawthorne equation

Abstract

Linear cases of Bragg-Hawthorne equation for steady axisymmetric incompressible ideal flows are systematically discussed. The equation is converted to a more convenient form in a spherical coordinate system. A new vorticity decomposition is derived. General solutions for 16 linear cases of the equation are obtained. These solutions can be specified to gain new analytical vortex flows, as examples in the paper demonstrate. A lot of well-known solutions like potential flow past a sphere, Hill's vortex with and without swirl, are included and extended in these solutions. Special relations between some vortex flows are also revealed when exploring or comparing related solutions.

Three-Dimensional Unsteady Axisymmetric Viscous Beltrami Vortex Solutions to the Navier–Stokes Equations

“…Breaking the scale invariance due to c of the dimension [length −1 ] generates a characteristic length l = c −1 and a decay time τ = l 2 /ν for a single-mode flow. It should be noted that the decay time does not depend on the wavenumber, k, appearing in (23). For air at normal pressure and room temperature, τ air ≈ 5 × 10 4 (l/m) 2 s. Similarly, for water, we have an estimation τ water ≈ 10 6 (l/m) 2 s. These periods shall be long enough for a macroscopic Beltrami vortex to persist.…”

“…Note that ψ and C are not related a priori. For recent applications of the Bragg-Hawthorne equation, see, e.g., [17,22,23]. The Bragg-Hawthorne equation is an expression of the equation of motion for inviscid flow under steadiness assumption.…”

“…Steady axisymmetric inviscid Beltramian solutions in bounded or unbounded space have been found by focusing on the so-called Bragg-Hawthorne equation [17,22,23]. How the structure of the fundamental modes admitted by the linear equation varies by depending on the coordinate system chosen for solving the equation has been revealed in [22,24].…”

A novel nanocomposite material C3F7-azo+/Ti4O92− was prepared as a sensor for the detection of ascorbic acid and uric acid

“…The 2 θ Angle of the interlaminar characteristic peak (001) is 10.38°, and the basal spacing of K 2 Ti 4 O 9 is 0.85 nm calculated by Bragg equation (2 d sin θ = λ ). 30 The sharp peaks indicate that the matrix material K 2 Ti 4 O 9 has good crystallinity. Fig.…”