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Abstract When$$k\geqslant 4$$and$$0\leqslant d\leqslant (k-2)/4$$, we consider the system of Diophantine equations\begin{align*}x_1^j+\ldots +x_k^j=y_1^j+\ldots +y_k^j\quad (1\leqslant j\leqslant k,\, j\ne k-d).\end{align*}We show that in this cousin of a Vinogradov system, there is a paucity of non-diagonal positive integral solutions. Our quantitative estimates are particularly sharp when$$d=o\!\left(k^{1/4}\right)$$.
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This content will become publicly available on May 10, 2026
Free surface proximity effects on flow dynamics around a flat plate
Flow dynamics around a stationary flat plate near a free surface is investigated using time-resolved two-dimensional particle image velocimetry. The study examines variations in angle of attack ($$\theta =0^\circ {-}35^\circ {}$$), Reynolds number ($$Re$$$$\approx$$$$10^3$$$$-$$3$$\times$$$$10^4$$) and plate proximity to the free surface ($$H^*$$). Under symmetric boundary conditions ($$H^*\geqslant {15}$$), increasing$$\theta$$intensifies fluid–plate interaction, resulting in the shedding of leading-edge and trailing-edge vortices (LEV and TEV), each characterised by distinct strengths and sizes. In both symmetric ($$H^*\geqslant {15}$$) and asymmetric ($$H^*=5$$) boundary conditions at$$\theta \lt 5^\circ {}$$, fluid flow follows the contour of the plate, unaffected by Reynolds number. However, at$$H^*=5$$, three flow regimes emerge: the first Coanda effect (CI), regular shedding (RS) and the second Coanda effect (CII), each influenced by$$\theta$$and$$Re$$. The CI regime dominates at lower angles ($$5^\circ {}\leqslant \theta \leqslant 25^\circ {}$$) and$$Re \leqslant 12\,500$$, featuring a Coanda-induced jet-like flow pattern. As the Reynolds number increases, the flow transitions into the RS regime, leading to detachment from the upper surface of the plate. This detachment results in the formation of LEV and TEV in the wake, along with surface deformation, secondary vortices and wavy shear layers beneath the free surface. At$$22\,360\lt Re \leqslant 32\,200$$and$$5^\circ {} \leqslant \theta \leqslant 25^\circ {}$$, in the CII regime, significant surface deformation causes the Coanda effect to reattach the flow to the plate, forming a unique jet-like flow.
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- Award ID(s):
- 2143263
- PAR ID:
- 10650373
- Publisher / Repository:
- Cambridge University
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 1010
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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