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1. Global inviscid limit of 2D, stationary Navier-Stokes and stability of Prandtl expansions

   https://doi.org/10.1017/fmp.2026.10027  

   Iyer, Sameer; Masmoudi, Nader (April 2026, Forum of Mathematics, Pi)

   Abstract In this work, we establish the convergence of 2D, stationary Navier-Stokes flows with viscosity$$\varepsilon> 0$$,$$(u^\varepsilon , v^\varepsilon )$$to the classical Prandtl boundary layer,$$(\bar {u}_p, \bar {v}_p)$$, posed on the domain$$(0, \infty ) \times (0, \infty )$$:$$ \begin{align*} \| u^\varepsilon - \bar{u}_p \|_{L^\infty_y} \lesssim \sqrt{\varepsilon} \langle x \rangle^{- \frac 1 4 + \delta}, \qquad \| v^\varepsilon - \sqrt{\varepsilon} \bar{v}_p \|_{L^\infty_y} \lesssim \sqrt{\varepsilon} \langle x \rangle^{- \frac 1 2}. \end{align*} $$ This validates Prandtl’s boundary layer theorygloballyin thex-variable for a large class of boundary layers, including the entire one parameter family of the classical Blasius profiles, with sharp decay rates. The result demonstrates asymptotic stability in two senses simultaneously: (1) asymptotic as$$\varepsilon \rightarrow 0$$and (2) asymptotic as$$x \rightarrow \infty $$. In particular, our result provides the first rigorous confirmation for the Navier-Stokes equations that the boundary layer cannot “separate” in these stable regimes, which is very important for physical and engineering applications. 

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2. Nonlinear electrophoretic velocity of a spherical colloidal particle

   https://doi.org/10.1017/jfm.2023.537  

   Cobos, Richard; Khair, Aditya S. (August 2023, Journal of Fluid Mechanics)

   Electrophoresis is the motion of a charged colloidal particle in an electrolyte under an applied electric field. The electrophoretic velocity of a spherical particle depends on the dimensionless electric field strength$$\beta =a^*e^*E_\infty ^*/k_B^*T^*$$, defined as the ratio of the product of the applied electric field magnitude$$E_\infty ^*$$and particle radius$$a^*$$, to the thermal voltage$$k_B^*T^*/e^*$$, where$$k_B^*$$is Boltzmann's constant,$$T^*$$is the absolute temperature, and$$e^*$$is the charge on a proton. In this paper, we develop a spectral element algorithm to compute the electrophoretic velocity of a spherical, rigid, dielectric particle, of fixed dimensionless surface charge density$$\sigma$$over a wide range of$$\beta$$. Here,$$\sigma =(e^*a^*/\epsilon ^*k_B^*T^*)\sigma ^*$$, where$$\sigma ^*$$is the dimensional surface charge density, and$$\epsilon ^*$$is the permittivity of the electrolyte. For moderately charged particles ($$\sigma ={O}(1)$$), the electrophoretic velocity is linear in$$\beta$$when$$\beta \ll 1$$, and its dependence on the ratio of the Debye length ($$1/\kappa ^*$$) to particle radius (denoted by$$\delta =1/(\kappa ^*a^*)$$) agrees with Henry's formula. As$$\beta$$increases, the nonlinear contribution to the electrophoretic velocity becomes prominent, and the onset of this behaviour is$$\delta$$-dependent. For$$\beta \gg 1$$, the electrophoretic velocity again becomes linear in field strength, approaching the Hückel limit of electrophoresis in a dielectric medium, for all$$\delta$$. For highly charged particles ($$\sigma \gg 1$$) in the thin-Debye-layer limit ($$\delta \ll 1$$), our computations are in good agreement with recent experimental and asymptotic results. 

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3. Bridging the Rossby number gap in rapidly rotating thermal convection

   https://doi.org/10.1017/jfm.2025.290  

   van_Kan, Adrian; Julien, Keith; Miquel, Benjamin; Knobloch, Edgar (May 2025, Journal of Fluid Mechanics)

   Geophysical and astrophysical fluid flows are typically driven by buoyancy and strongly constrained at large scales by planetary rotation. Rapidly rotating Rayleigh–Bénard convection (RRRBC) provides a paradigm for experiments and direct numerical simulations (DNS) of such flows, but the accessible parameter space remains restricted to moderately fast rotation rates (Ekman numbers$${ {Ek}} \gtrsim 10^{-8}$$), while realistic$${Ek}$$for geo- and astrophysical applications are orders of magnitude smaller. On the other hand, previously derived reduced equations of motion describing the leading-order behaviour in the limit of very rapid rotation ($$ {Ek}\to 0$$) cannot capture finite rotation effects, and the physically most relevant part of parameter space with small but finite$${Ek}$$has remained elusive. Here, we employ the rescaled rapidly rotating incompressible Navier–Stokes equations (RRRiNSE) – a reformulation of the Navier–Stokes–Boussinesq equations informed by the scalings valid for$${Ek}\to 0$$, recently introduced by Julienet al.(2024) – to provide full DNS of RRRBC at unprecedented rotation strengths down to$$ {Ek}=10^{-15}$$and below, revealing the disappearance of cyclone–anticyclone asymmetry at previously unattainable Ekman numbers ($${Ek}\approx 10^{-9}$$). We also identify an overshoot in the heat transport as$${Ek}$$is varied at fixed$$\widetilde { {Ra}} \equiv {Ra}{Ek}^{4/3}$$, where$$Ra$$is the Rayleigh number, associated with dissipation due to ageostrophic motions in the boundary layers. The simulations validate theoretical predictions based on thermal boundary layer theory for RRRBC and show that the solutions of RRRiNSE agree with the reduced equations at very small$${Ek}$$. These results represent a first foray into the vast, largely unexplored parameter space of very rapidly rotating convection rendered accessible by RRRiNSE. 

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4. Hydrodynamics and scaling laws for intermittent S-start swimming

   https://doi.org/10.1017/jfm.2024.103  

   Yang, Dewu; Wu, Jie; Khedkar, Kaustubh; Chao, Li-Ming; Bhalla, Amneet Pal (April 2024, Journal of Fluid Mechanics)

   The hydrodynamics of a self-propelling swimmer undergoing intermittent S-start swimming are investigated extensively with varying duty cycle$$DC$$, swimming period$$T$$, and tailbeat amplitude$$A$$. We find that the steady time-averaged swimming speed$$\bar {U}_x$$increases directly with$$A$$, but varies inversely with$$DC$$and$$T$$, where there is a maximal improvement of$$541.29\,\%$$over continuous cruising swimming. Our results reveal two scaling laws, in the form of input versus output relations, that relate the swimmer's kinematics to its hydrodynamic performance: swimming speed and efficiency. A smaller$$DC$$causes increased fluctuations in the swimmer's velocity generation. A larger$$A$$, on the other hand, allows the swimmer to reach steady swimming more quickly. Although we set out to determine scaling laws for intermittent S-start swimming, these scaling laws extend naturally to burst-and-coast and continuous modes of swimming. Additionally, we have identified, categorized and linked the wake structures produced by intermittent S-start swimmers with their velocity generation. 

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5. On higher dimensional point sets in general position

   https://doi.org/10.1017/S0963548325100254  

   Suk, Andrew; Zeng, Ji (January 2026, Combinatorics, Probability and Computing)

   Abstract A finite point set in$$\mathbb{R}^d$$is in general position if no$$d + 1$$points lie on a common hyperplane. Let$$\alpha _d(N)$$be the largest integer such that any set of$$N$$points in$$\mathbb{R}^d$$, with no$$d + 2$$members on a common hyperplane, contains a subset of size$$\alpha _d(N)$$in general position. Using the method of hypergraph containers, Balogh and Solymosi showed that$$\alpha _2(N) \lt N^{5/6 + o(1)}$$. In this paper, we also use the container method to obtain new upper bounds for$$\alpha _d(N)$$when$$d \geq 3$$. More precisely, we show that if$$d$$is odd, then$$\alpha _d(N) \lt N^{\frac {1}{2} + \frac {1}{2d} + o(1)}$$, and if$$d$$is even, we have$$\alpha _d(N) \lt N^{\frac {1}{2} + \frac {1}{d-1} + o(1)}$$. We also study the classical problem of determining$$a(d,k,n)$$, the maximum number of points selected from the grid$$[n]^d$$such that no$$k + 2$$members lie on a$$k$$-flat, and improve the previously best known bound for$$a(d,k,n)$$, due to Lefmann in 2008, by a polynomial factor when$$k$$= 2 or 3 (mod 4). 

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