A New Two-Parameter Heteromorphic Elliptic Equation: Properties and Applications

The ellipse and the superellipse are both planar closed curves with a double axis of symmetry. Here we show the isoconcentration contour of the simplified two-dimensional advection-diffusion equation from a stable line source in the center of a wide river. A new two-parameter heteromorphic elliptic equation with a single axis of symmetry is defined. The values of heights, at the point of the maximum width and that of the centroid of the heteromorphic ellipse, are derived through mathematical analysis. Taking the compression coefficient θ = b/a = 1 as the criterion, the shape classification of H-type, Standard-type and W-type for heteromorphic ellipse have been given. The area formula, the perimeter theorem, and the radius of curvature of heteromorphic ellipses, and the geometric properties of the rotating body are subsequently proposed. An illustrative analysis shows that the inner contour curve of a heteromorphic elliptic tunnel has obvious advantages over the multiple- arc splicing cross section. This work demonstrates that the heteromorphic ellipses have extensive prospects of application in all categories of tunnels, liquid transport tanks, aircraft and submarines, bridges, buildings, furniture, and crafts.

Experimental Study of Vertical Two-dimensional Diffusion Concentration Distribution in Sloped Wave Bank Angular Field

[Objectives] This study was conducted to detect the two-dimensional diffusion concentration distribution from sloped wave bank. [Methods] Diffusion experiments of instantaneous line source discharge were carried out using two sloped wave banks with different inclination angles based on the developed two-dimensional diffusion tank device for sloped wave banks by the apex discharge method under grid oscillation frequencies n=15, 20, 40 and 60 r/min. The image acquisition and digital image processing techniques were applied to measure the two-dimensional concentration field distribution and to analyze the distribution laws of the pollutant in the angular field. [Results] The diffusion of the pollutant in the sloped wave bank area became faster with the increase of the grid oscillation frequency, and the pollution range became wider with the diffusion time. The point concentration of the pollutant at the water surface monotonically decreased with the increase of the abscissa, and the vertical concentration distribution decreased with the increase of water depth. The transverse diffusion rate of the pollutant in water was greater than the vertical diffusion rate, and its concentration distribution exhibited a distribution characteristic of farther diffusion in the adjacent area on the water surface. The diffusion experiment area of the sloped wave bank at θ=30° had a higher concentration of the pollutant at each point compared with the diffusion experiment of the sloped bank at θ=45°, under the same experimental conditions. A large-scale vortex appeared near the sloped wave bank at θ=45° during the experiment, and the presence of the vortex made the concentration distribution of the pollutant in the direction along the bank slope first decrease and then increase, while no obvious vortex was observed near the sloped wave bank at θ=30°, and the concentration of the pollutant decreased monotonously along the bank slope direction. [Conclusions] This study is of great significance for the concentration distribution laws and the lateral and vertical diffusion coefficients of side discharge at complex bank slopes and river banks.