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1. Nonlinear limiting dynamics of a shrinking interface in a Hele-Shaw cell

   https://doi.org/10.1017/jfm.2020.983  

   Zhao, Meng ; Niroobakhsh, Zahra ; Lowengrub, John ; Li, Shuwang ( March 2021 , Journal of Fluid Mechanics)

   The flow in a Hele-Shaw cell with a time-increasing gap poses a unique shrinking interface problem. When the upper plate of the cell is lifted perpendicularly at a prescribed speed, the exterior less viscous fluid penetrates the interior more viscous fluid, which generates complex, time-dependent interfacial patterns through the Saffman–Taylor instability. The pattern formation process sensitively depends on the lifting speed and is still not fully understood. For some lifting speeds, such as linear or exponential speed, the instability is transient and the interface eventually shrinks as a circle. However, linear stability analysis suggests there exist shape invariant shrinking patterns if the gap $b(t)$ is increased more rapidly: $b(t)=\left (1-({7}/{2})\tau \mathcal {C} t\right )^{-{2}/{7}}$ , where $\tau$ is the surface tension and $\mathcal {C}$ is a function of the interface perturbation mode $k$ . Here, we use a spectrally accurate boundary integral method together with an efficient time adaptive rescaling scheme, which for the first time makes it possible to explore the nonlinear limiting dynamical behaviour of a vanishing interface. When the gap is increased at a constant rate, our numerical results quantitatively agree with experimental observations (Nase et al. , Phys. Fluids , vol. 23, 2011, 123101). When wemore »use the shape invariant gap $b(t)$ , our nonlinear results reveal the existence of $k$ -fold dominant, one-dimensional, web-like networks, where the fractal dimension is reduced to almost unity at late times. We conclude by constructing a morphology diagram for pattern selection that relates the dominant mode $k$ of the vanishing interface and the control parameter $\mathcal {C}$ .« less
2. Understanding Atypical Midlevel Wind Speed Maxima in Hurricane Eyewalls

   https://doi.org/10.1175/JAS-D-19-0191.1  

   Stern, Daniel P. ; Kepert, Jeffrey D. ; Bryan, George H. ; Doyle, James D. ( May 2020 , Journal of the Atmospheric Sciences)

   In tropical cyclones (TCs), the peak wind speed is typically found near the top of the boundary layer (approximately 0.5–1 km). Recently, it was shown that in a few observed TCs, the wind speed within the eyewall can increase with height within the midtroposphere, resulting in a secondary local maximum at 4–5 km. This study presents additional evidence of such an atypical structure, using dropsonde and Doppler radar observations from Hurricane Patricia (2015). Near peak intensity, Patricia exhibited an absolute wind speed maximum at 5–6-km height, along with a weaker boundary layer maximum. Idealized simulations and a diagnostic boundary layer model are used to investigate the dynamics that result in these atypical wind profiles, which only occur in TCs that are very intense (surface wind speed > 50 m s⁻¹) and/or very small (radius of maximum winds < 20 km). The existence of multiple maxima in wind speed is a consequence of an inertial oscillation that is driven ultimately by surface friction. The vertical oscillation in the radial velocity results in a series of unbalanced tangential wind jets, whose magnitude and structure can manifest as a midlevel wind speed maximum. The wavelength of the inertial oscillation increases with vertical mixingmore »length l_∞in a turbulence parameterization, and no midlevel wind speed maximum occurs when l_∞is large. Consistent with theory, the wavelength in the simulations scales with (2 K/ I)^1/2, where K is the (vertical) turbulent diffusivity, and I²is the inertial stability. This scaling is used to explain why only small and/or strong TCs exhibit midlevel wind speed maxima.

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3. Stability of Fluid Flow through a Channel with Flexible Walls

   https://doi.org/10.1155/2021/8825677  

   Shubov, Marianna A. ; Edwards, Madeline M. ( March 2021 , International Journal of Mathematics and Mathematical Sciences)

   Solovjovs, Sergejs (Ed.)

   In the present paper, we summarize the results of the research devoted to the problem of stability of the fluid flow moving in a channel with flexible walls and interacting with the walls. The walls of the vessel are subject to traveling waves. Experimental data show that the energy of the flowing fluid can be transferred and consumed by the structure (the walls), inducing “traveling wave flutter.” The problem of stability of fluid-structure interaction splits into two parts: (a) stability of fluid flow in the channel with harmonically moving walls and (b) stability of solid structure participating in the energy exchange with the flow. Stability of fluid flow, the main focus of the research, is obtained by solving the initial boundary value problem for the stream function. The main findings of the paper are the following: (i) rigorous formulation of the initial boundary problem for the stream function, ψ x , y , t , induced by the fluid-structure interaction model, which takes into account the axisymmetric pattern of the flow and “no-slip” condition near the channel walls; (ii) application of a double integral transformation (the Fourier transformation and Laplace transformation) to both the equation and boundary and initial conditions,more »which reduces the original partial differential equation to a parameter-dependent ordinary differential equation; (iii) derivation of the explicit formula for the Fourier transform of the stream function, ψ ˜ k , y , t ; (iv) evaluation of the inverse Fourier transform of ψ ˜ k , y , t and proving that reconstruction of ψ x , y , t can be obtained through a limiting process in the complex k -plane, which allows us to use the Residue theorem and represent the solution in the form of an infinite series of residues. The result of this research is an analytical solution describing blood flowing through a channel with flexible walls that are being perturbed in the form of a traveling wave.« less
4. A New Vibrational Control System in Nature: Flapping Flight

   Taha, H ; Kiani, M ( January 2019 , AIAA SciTech)

   Vibrational control is an open loop stabilization technique via the application of highamplitude, high-frequency oscillatory inputs. The averaging theory has been the standard technique for designing vibrational control systems. However, it stipulates too high oscillation frequency that may not be practically feasible. Therefore, although vibrational control is very robust and elegant (stabilization without feedback), it is rarely used in practical applications. The only well-known example is the Kapitza pendulum; an inverted pendulum shose pivot is subject to vertical oscillation. the unstable equilibrium of the inverted pendulum gains asymptotic stability due to the high-frequency oscillation of the pivot. In this paper, we provide a new vibrational control system from Nature; flapping flight dynamics. Flapping flight is a rich dynamical system as a representative model will typically be nonlinear, time-varying, multi-body, multi-time-scale dynamical system. Over the last two decades, using direct averaging, there has been consensus in the flapping flight dynamics community that insects are unstable at the hovering equilibrium due to the lack of pitch stiffness. In this work, we perform higher-order averaging of the time-periodic dynamics of flapping flight to show a vibrational control mechanism due to the oscillation of the driving aerodynamic forces. We also experimentally demonstrate such amore »phenomenon on a flapping apparatus that has two degrees of freedom: forward translation and pitching motion. It is found that the time-periodic dynamics of the flapping micro-air-vehicle is naturally (without feedback) stabilized beyond a certain threshold. Moreover, if the averaged aerodynamic thrust force is produced by a propeller revolving at a constant speed while maintaining the wings stationary at their mean positions, no stabilization is observed. Hence, it is concluded that the observed stabilization in the flapping system at high frequencies is due to the oscillation of the driving aerodynamic force and, as such, flapping flight indeed enjoys vibrational stabilization.« less
5. Stability of asymptotic behaviour within polarized T2-symmetric vacuum solutions with cosmological constant

   https://doi.org/10.1098/rsta.2021.0173  

   Ames, Ellery ; Beyer, Florian ; Isenberg, James ; Oliynyk, Todd A. ( May 2022 , Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   We prove the nonlinear stability of the asymptotic behaviour of perturbations of subfamilies of Kasner solutions in the contracting time direction within the class of polarized T 2 -symmetric solutions of the vacuum Einstein equations with arbitrary cosmological constant Λ . This stability result generalizes the results proven in Ames E et al. (2022 Stability of AVTD Behavior within the Polarized T 2 -symmetric vacuum spacetimes. Ann. Henri Poincaré . ( doi:10.1007/s00023-021-01142-0 )), which focus on the Λ = 0 case, and as in that article, the proof relies on an areal time foliation and Fuchsian techniques. Even for Λ = 0 , the results established here apply to a wider class of perturbations of Kasner solutions within the family of polarized T 2 -symmetric vacuum solutions than those considered in Ames E et al. (2022 Stability of AVTD Behavior within the Polarized T 2 -symmetric vacuum spacetimes. Ann. Henri Poincaré . ( doi:10.1007/s00023-021-01142-0 )) and Fournodavlos G et al. (2020 Stable Big Bang formation for Einstein’s equations: the complete sub-critical regime . Preprint. ( http://arxiv.org/abs/2012.05888 )). Our results establish that the areal time coordinate takes all values in ( 0 , T 0 ] for some T 0 >more »0 , for certain families of polarized T 2 -symmetric solutions with cosmological constant. This article is part of the theme issue ‘The future of mathematical cosmology, Volume 1’.« less