Flow Structure and Deformation of Two Bubbles Rising Side by Side in a Quiescent Liquid

In the motion of two spherical bubbles rising side by side, the bubbles are known to attract each other at a high Reynolds number (Re = ρUd/μ). Furthermore, spherical bubbles kiss and bounce under certain conditions; however, deformable bubbles repel each other without kissing. This paper experimentally and numerically presents the flow structures and shape of the nonkissing repulsion of deformable bubbles. For the experimental analysis, we organized bubble behaviors by Galilei number (Ga = ρg1/2d3/2/μ) and Bond number (Bo = ρgd2/σ). The bubbles repelled each other without kissing near the unstable critical curve of a single bubble. The curvature inside the gap, which is similar to the shape of a zigzag behavior bubble, was large. For the numerical analysis, the velocity of the equatorial plane inside the gap was larger due to the potential interaction, although the velocity behind was the opposite due to the strengthened vorticity generated at the surface. Furthermore, the double-threaded wake emerged behind the interacting bubbles, and it showed that the rotation direction was repulsion regardless of whether the bubbles attracted or repelled each other. The streamline behind the bubbles in the 2D plane was from the outside to the inside.

Geoethics education and climate literacy: Bridging the gap – interactively

<p>In previous years, the authors have addressed questions related to <strong>geoethics education,</strong> or what we have called <strong>geo-edu-ethics</strong> (<strong>GEE</strong>), in relation to geo-problems in general (such as global warming, pollution, sea-level rise, deforestation, ocean acidification, biodiversity).</p><p>In this session we wish to focus in on the greatest of all geo-problems, that of <strong>climate change</strong> (<strong>CC</strong>), which necessarily entails the urgent need for massive, widespread <strong>climate literacy</strong> (<strong>CL</strong>) – both education and learning.  We wish to examine the relationships between GEE and CL, their overlaps and differences, and how they may mutually reinforce each other.  In so doing, we will also touch on the ethics of educational and learning methods that are used to help people learn about geoethics and CC.</p><p>Currently, it seems that the two areas work in parallel, maybe even separated by a mindset of splendid isolation, and yet the apparent overlap, not least in their visons and missions, beckons us to bring the two closer together.  This is what we will attempt in our presentation.  The questions that we plan to address include the following:</p><ul><li>Is it true, or a misconception, that GEE and CL tend to work separately, often ignorant of each other?</li> <li>What do GEE and CL have in common?</li> <li>Their ethos, their content, their methods, their audience?</li> <li>Is it possible to unify the GEE and CL into an overarching rational and thereby form a coherent community of practice?</li> <li>What can practitioners in each bubble learn from each other? What will it take for the two bubbles to merge?</li> <li>How can each group maintain its own professional identity (if that is deemed important) and yet work hand in hand with the other, to their mutual benefit?</li> <li>What are the most effective ways forward, given the geoethical urgency of acting to slow CC?</li> </ul><p>The presentation will be interactive, as we will invite the audience to contribute their own ideas and experience.  If we are permitted to have breakout rooms, we will divide into small groups for a short time, and then bring everyone together for a plenary sharing.</p>

Blowing-up solutions for a supercritical elliptic equation

<p style='text-indent:20px;'>This paper concerns the existence of solutions of the following supercritical PDE: <inline-formula><tex-math id="M1">\begin{document}$ (P_\varepsilon) $\end{document}</tex-math></inline-formula>: <inline-formula><tex-math id="M2">\begin{document}$ -\Delta u = K|u|^{\frac{4}{n-2}+\varepsilon}u\; \mbox{ in }\Omega, \; u = 0 \mbox{ on }\partial\Omega, $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a smooth bounded domain in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ n\geq 3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ K $\end{document}</tex-math></inline-formula> is a <inline-formula><tex-math id="M7">\begin{document}$ C^3 $\end{document}</tex-math></inline-formula> positive function and <inline-formula><tex-math id="M8">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> is a small positive real. Our method is inspired from the work of Bahri-Li-Rey. It consists to reduce the existence of a critical point to a finite dimensional system. Using a fixed-point theorem, we are able to construct positive solutions of <inline-formula><tex-math id="M9">\begin{document}$ (P_\varepsilon) $\end{document}</tex-math></inline-formula> having the form of two bubbles with non comparable speeds and which have only one blow-up point in <inline-formula><tex-math id="M10">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>. That means that this blow-up point is non simple. This represents a new phenomenon compared with the subcritical case.</p>