We consider an internally heated fluid between parallel plates with fixed thermal fluxes. For a large class of heat sources that vary in the direction of gravity, we prove that $\smash { \smash {{\langle {\delta T} \rangle _h}} } \geq \sigma R^{-1/3} - \mu$, where $\smash { \smash {{\langle {\delta T} \rangle _h}} }$ is the average temperature difference between the bottom and top plates, $R$ is a ‘flux’ Rayleigh number and the constants $\sigma,\mu >0$ depend on the geometric properties of the internal heating. This result implies that mean downward conduction (for which $\smash { \smash {{\langle {\delta T} \rangle _h}} }< 0$) is impossible for a range of Rayleigh numbers smaller than a critical value $R_0:=(\sigma /\mu )^{3}$. The bound demonstrates that $R_0$ depends on the heating distribution and can be made arbitrarily large by concentrating the heating near the bottom plate. However, for any given fixed heating profile of the class we consider, the corresponding value of $R_0$ is always finite. This points to a fundamental difference between internally heated convection and its limiting case of Rayleigh–Bénard convection with fixed-flux boundary conditions, for which $\smash {{\langle {\delta T} \rangle _h}}$ is known to be positive for all $R$.

The Choquard equation is a partial differential equation that has gained significant interest and attention in recent decades. It is a nonlinear equation that combines elements of both the Laplace and Schrödinger operators, and it arises frequently in the study of numerous physical phenomena, from condensed matter physics to nonlinear optics.

In particular, the steady states of the Choquard equation were thoroughly investigated using a variational functional acting on the wave functions.

In this article, we introduce a dual formulation for the variational functional in terms of the potential induced by the wave function, and use it to explore the existence of steady states of a multi-state version the Choquard equation in critical and sub-critical cases.

We use the entropy method to analyse the nonlinear dynamics and stability of a continuum kinetic model of an active nematic suspension. From the time evolution of the relative entropy, an energy-like quantity in the kinetic model, we derive a variational bound on relative entropy fluctuations that can be expressed in terms of orientational order parameters. From this bound we show isotropic suspensions are nonlinearly stable for sufficiently low activity, and derive upper bounds on spatiotemporal averages in the unstable regime that are consistent with fully nonlinear simulations. This work highlights the self-organising role of activity in particle suspensions, and places limits on how organised such systems can be.

Applying a variational analysis, a minimum-enstrophy vortex in three-dimensional (3-D) fluids with continuous stratification is found, under the quasi-geostrophic hypothesis. The buoyancy frequency is held constant. This vortex is an ideal limiting state in a flow with an enstrophy decay while energy and generalized angular momentum remain fixed. The variational method used to obtain two-dimensional (2-D) minimum-enstrophy vortices is applied here to 3-D integral quantities. The solution from the first-order variation is expanded on a basis of orthogonal spherical Bessel functions. By computing second-order variations, the solution is found to be a true minimum in enstrophy. This solution is weakly unstable when inserted in a numerical code of the quasi-geostrophic equations. After a stage of linear instability, nonlinear wave interaction leads to the reorganization of this vortex into a tripolar vortex. Further work will relate our solution with maximal entropy 3-D vortices.

Successive drops of coloured ink mixed with surfactant are deposited onto a thin film of water to create marbling patterns in the Japanese art technique of Suminagashi. To understand the physics behind this and other applications where surfactant transports adsorbed passive matter at gas–liquid interfaces, we investigate the Lagrangian trajectories of material particles on the surface of a thin film of a confined viscous liquid under Marangoni-driven spreading by an insoluble surfactant. We study a model problem in which several deposits of exogenous surfactant simultaneously spread on a bounded rectangular surface containing a pre-existing endogenous surfactant. We derive Eulerian and Lagrangian formulations of the equations governing the Marangoni-driven surface flow. Both descriptions show how confinement can induce drift and flow reversal during spreading. The Lagrangian formulation captures trajectories without the need to calculate surfactant concentrations; however, concentrations can still be inferred from the Jacobian of the map from initial to current particle position. We explore a link between thin-film surfactant dynamics and optimal transport theory to find the approximate equilibrium locations of material particles for any given initial condition by solving a Monge–Ampère equation. We find that as the endogenous surfactant concentration $\delta$ vanishes, the equilibrium shapes of deposits using the Monge–Ampère approximation approach polygons with corners curving in a self-similar manner over lengths scaling as $\delta ^{1/2}$. We explore how Suminagashi patterns may be produced by using computationally efficient successive solutions of the Monge–Ampère equation.

A variational principle is proposed to derive the governing equations for the problem of ocean wave interactions with a floating ice shelf, where the ice shelf is modelled by the full linear equations of elasticity and has an Archimedean draught. The variational principle is used to form a thin-plate approximation for the ice shelf, which includes water–ice coupling at the shelf front and extensional waves in the shelf, in contrast to the benchmark thin-plate approximation for ocean wave interactions with an ice shelf. The thin-plate approximation is combined with a single-mode approximation in the water, where the vertical motion is constrained to the eigenfunction that supports propagating waves. The new terms in the approximation are shown to have a major impact on predictions of ice shelf strains for wave periods in the swell regime.

This paper presents a novel Hamiltonian formulation of the isotropic Navier–Stokes problem based on a minimum-action principle derived from the principle of least squares. This formulation uses the velocities $u_{i}(x_{j},t)$ and pressure $p(x_{j},t)$ as the field quantities to be varied, along with canonically conjugate momenta deduced from the analysis. From these, a conserved Hamiltonian functional $H^{*}$ satisfying Hamilton's canonical equations is constructed, and the associated Hamilton–Jacobi equation is formulated for both compressible and incompressible flows. This Hamilton–Jacobi equation reduces the problem of finding four separate field quantities ($u_{i}$,$p$) to that of finding a single scalar functional in those fields – Hamilton's principal functional ${S}^{*}[u_{i},p,t]$. Moreover, the transformation theory of Hamilton and Jacobi now provides a prescribed recipe for solving the Navier–Stokes problem: find ${S}^{*}$. If an analytical expression for ${S}^{*}$ can be obtained, it will lead via canonical transformation to a new set of fields which are simply equal to their initial values, giving analytical expressions for the original velocity and pressure fields. Failing that, if one can only show that a complete solution to this Hamilton–Jacobi equation does or does not exist, that will also resolve the question of existence of solutions. The method employed here is not specific to the Navier–Stokes problem or even to classical mechanics, and can be applied to any traditionally non-Hamiltonian problem.

Cavitation inception originates from nuclei in a liquid. This paper proposes a Gibbs free energy approach that provides a smooth transition from homogeneous to heterogeneous nucleation when gas is present. The impact of gas content on nucleation is explored. It is found that the gas content stabilises nuclei, a phenomenon not present in pure liquid–vapour systems. This reduces the energy barrier over that required to nucleate a vapour bubble. Different gas saturation levels are studied. Gas content can significantly reduce the energy barrier required for nucleation, and under certain circumstances eliminate it. An analytic solution for the critical radius and activation energy is obtained that accounts for gas content. The classical Blake radius is recovered as a limiting case. The hysteresis between incipience and desinence is explained using the asymmetry observed in the critical radii. The solution is used to obtain the initial bubble radius, given a critical pressure condition in cavitation susceptibility meter experiments. The relationship between initial bubble diameter and critical pressure is described by an analytic solution that accounts for gas content. A model for the derivative of the cumulative nuclei histogram with respect to bubble diameter is proposed. An analytic expression is obtained that shows good agreement with decades worth of experimental data compiled by Khoo et al. (Exp. Fluids, vol. 61, issue 2, 2020, pp. 1–20) from ocean to water tunnels. The expression recovers the $-4$ power law that is observed experimentally.

Capillary folding is the process of folding planar objects into three-dimensional (3-D) structures using capillary force. It has many important industrial applications, e.g. the fabrication of microelectromechanical systems. In this work, we propose a 3-D model for the capillary folding of thin elastic sheets with pinned contact lines. The energy of the system consists of interfacial energies between the different phases and the elastic energy of the sheet. The latter is described by the nonlinear Koiter's model, which can accommodate large deformations of the sheet. From the energy, we derive the governing equations for the static system using a variational approach. We then develop a numerical method to find equilibrium solutions via a relaxation dynamics. At each time step, we evolve the sheet by using the subdivision element method, and update the droplet surface by minimizing a squared area functional using linear finite elements. Qualitatively, numerical solutions for the equilibrium configurations of the sheet–droplet system agree well with those observed in experiments. Furthermore, we identify the critical bendabilities and present bifurcation diagrams for the folding of a triangular sheet. The results exhibit rich and fully 3-D behaviours that were not captured by previous two-dimensional models. Our results provide new insights into the nonlinear process of capillary folding, and may contribute to the advancement of microfabrication techniques.

The super-temporal-resolution (STR) reconstruction of turbulent flows is an important data augmentation application for increasing the data reach in measurement techniques and understanding turbulence dynamics. This paper proposes a data assimilation (DA) strategy based on weak-constraint four-dimensional variation to conduct an STR reconstruction in a turbulent jet beyond the Nyquist limit from given low-sampling-rate observations. Highly resolved large-eddy simulation (LES) data are used to produce synthetic measurements, which are used as observations and for validation. A segregated assimilation procedure is realised to assimilate the initial condition, inflow boundary condition and model error separately. Different types of observational data are tested. The first type is down-sampled LES data containing many small-scale turbulence structures with or without synthetic noise. The DA results show that the temporal variation of the small-scale structures is well recovered even with noise in the observations. The spectra are resolved to a frequency approximately one order of magnitude higher than what can be captured within the Nyquist limit. The second type of observation is low-sampling-rate tomographic particle image velocimetry (tomo-PIV) data with or without the injection of small-scale structures. The modulation between the large-scale structures contained in the tomo-PIV fields and the small scales injected from the observations is improved. The resultant small scales in the STR reconstruction have the characteristics of authentic turbulence to a considerable extent. Additionally, DA yields much smaller errors in the prediction of particle positions when compared with the Wiener filter, demonstrating the great potential for Lagrangian particle tracking in measurement techniques.

Invariant solutions of the Navier–Stokes equations play an important role in the spatiotemporally chaotic dynamics of turbulent shear flows. Despite the significance of these solutions, their identification remains a computational challenge, rendering many solutions inaccessible and thus hindering progress towards a dynamical description of turbulence in terms of invariant solutions. We compute equilibria of three-dimensional wall-bounded shear flows using an adjoint-based matrix-free variational approach. To address the challenge of computing pressure in the presence of solid walls, we develop a formulation that circumvents the explicit construction of pressure and instead employs the influence matrix method. Together with a data-driven convergence acceleration technique based on dynamic mode decomposition, this yields a practically feasible alternative to state-of-the-art Newton methods for converging equilibrium solutions. We compute multiple equilibria of plane Couette flow starting from inaccurate guesses extracted from a turbulent time series. The variational method outperforms Newton(-hookstep) iterations in converging successfully from poor initial guesses, suggesting a larger convergence radius.

This paper first studies the multiplicity of normalized solutions to the non-autonomous Schrödinger equation with mixed nonlinearities

\begin{equation*}\begin{cases}-\Delta u=\lambda u+h(\epsilon x)|u|^{q-2}u+\eta |u|^{p-2}u,\quad x\in \mathbb{R}^N, \\\int_{\mathbb{R}^N}|u|^2\,\textrm{d}x=a^2,\end{cases}\end{equation*}

$a, \epsilon, \eta \gt 0$

$\lambda\in \mathbb{R}$

$p=2N/(N-2)$

The Schrödinger–Poisson system describes standing waves for the nonlinear Schrödinger equation interacting with the electrostatic field. In this paper, we are concerned with the existence of positive ground states to the planar Schrödinger–Poisson system with a nonlinearity having either a subcritical or a critical exponential growth in the sense of Trudinger–Moser. A feature of this paper is that neither the finite steep potential nor the reaction satisfies any symmetry or periodicity hypotheses. The analysis developed in this paper seems to be the first attempt in the study of planar Schrödinger–Poisson systems with lack of symmetry.

In this note we provide an analytical proof of the zero helicity condition for systems governed by the Gross–Pitaevskii equation (GPE). The proof is based on the hydrodynamic interpretation of the GPE, and the direct use of Noether's theorem by applying Kleinert's multi-valued gauge theory. As a by-product we also demonstrate the conservation and quantization of the circulation for the GPE.

We investigate the equilibrium of an axisymmetric system consisting of sessile and pendent drops on pre-stretched nonlinear elastic membranes. The membrane experiences large deformations due to the drop's weight and interfacial interactions. We first show that force balance alone leads to non-unique equilibrium solutions. Identifying the system's equilibrium with the minimum of its free energy, we then demonstrate that the equilibrium solution is made unique by requiring the continuity of meridional stretches across the three-phase contact circle. For a special class of nonlinear elastic materials – $I_2$ materials – we then compute the equilibrium configurations of the drop–membrane system for a range of drop volumes and membrane pre-tensions. Finally, the present work facilitates two important applications: (a) the membrane's pre-tension and current tension are related exactly to help in utilizing the system as an elastocapillary probe for membrane pre-tension and (b) we suggest an experimental protocol for measuring the membrane's surface properties.

This paper investigates heat transport in penetrative convection with a marginally stable temporal-horizontal-averaged field or background field. Assuming that the background field is steady and is stabilised by the nonlinear perturbation terms, we obtain an eigenvalue problem with an unknown background temperature $\tau$ by truncating the nonlinear terms. Using a piecewise profile for $\tau$, we derived an analytical scaling law for heat transport in penetrative convection as $Ra\rightarrow \infty$: $Nu=(1/8)(1-T_M)^{5/3}Ra^{1/3}$ ($Nu$ is the Nusselt number; $Ra$ is the Rayleigh number and $T_M$ corresponds to the temperature at which the density is maximal). A conditional lower bound on $Nu$, under the marginal stability assumption, is then derived from a variational problem. All the solutions to the full system should deliver a higher heat flux than the lower bound if they satisfy the marginal stability assumption. However, data from the present direct numerical simulations and previous optimal steady solutions by Ding & Wu (J. Fluid Mech., vol. 920, 2021, A48) exhibit smaller $Nu$ than the lower bound at large $Ra$, indicating that these averaged fields are over-stabilised by the nonlinear terms. To incorporate a more physically plausible constraint to bound heat transport, an alternative approach, i.e. the quasilinear approach is invoked which delivers the highest heat transport and agrees well with Veronis's assumption, i.e. $Nu\sim Ra^{1/3}$ (Astrophys. J., vol. 137, 1963, p. 641). Interestingly, the background temperature $\tau$ yielded by the quasilinear approach can be non-unique when instability is subcritical.

In the present study, the shape of a two-dimensional cylinder is optimised to minimise the mean drag in laminar unsteady flow under a noisy environment. A small inline stochastic oscillation in the free-stream velocity, which follows the Ornstein–Uhlenbeck process, is considered for the noise. The small noise is found to yield a large random fluctuation in instantaneous drag of the cylinder due to the effect of added mass. Subject to the strong random fluctuation of drag, the shape optimisation is performed using an ensemble-variation-based method (EnVar), as the conventional adjoint-based optimisation is not applicable to such a flow environment with unknown free-stream noise. The optimised cylinder geometry is found to be a nearly-symmetric slender oval at a low Reynolds number. As the Reynolds number is increased, two optimal shapes emerge: one is identical to the oval obtained at the low Reynolds number, and the other is an asymmetric oval, the rear side of which is more slender than the front side, reminiscent of an aerofoil. Despite the large random fluctuation in the instantaneous drag, the optimal cylinder shapes obtained for different levels of the upstream noise are found to be almost identical. It is shown that the robust nature of the optimal cylinder shape originates from the limited influence of the small upstream noise on the mean flow properties of the cylinder wake. Finally, the optimised cylinder primarily reduces the pressure component of the drag, associated mainly with vortex shedding in the wake, and this is achieved by marginally increasing the viscous drag through the shape change.

We establish multiplicity results for the following class of quasilinear problems P

\begin{equation*} \left\{ \begin{array}{@{}l} -\Delta_{\Phi}u=f(x,u) \quad \mbox{in} \quad \Omega, \\ u=0 \quad \mbox{on} \quad \partial \Omega, \end{array} \right. \end{equation*}

$\Delta _{\Phi }u=\text {div}(\varphi (x,|\nabla u|)\nabla u)$

$\Phi (x,t)=\int _{0}^{|t|}\varphi (x,s)s\,ds$

$\Omega \subset \mathbb {R}^{N}$

$\Omega _N$

$\Omega _p$

$\overline {\Omega _N}\cap \overline {\Omega _p}=\emptyset$

$(P)$

$-\Delta _{\Phi }$

$-\Delta _N$

$\Omega _N$

$-\Delta _p$

$\Omega _p$

$f:\Omega \times \mathbb {R}\to \mathbb {R}$

$e^{\alpha |t|^{\frac {N}{N-1}}}$

$\Omega _N$

$|t|^{p^{*}-2}t$

$\Omega _p$

$|t|$

$\alpha >0$

$p^{*}=\frac {Np}{N-p}$

$1< p< N$

$f$

$f$

$\Omega$

$(P)$

This paper is concerned with the optimal upper bound on mean quantities (torque, dissipation and the Nusselt number) obtained in the framework of the background method for the Taylor–Couette flow with a stationary outer cylinder. Along the way, we perform the energy stability analysis of the laminar flow, and demonstrate that below radius ratio $0.0556$, the marginally stable perturbations are not the axisymmetric Taylor vortices but rather a fully three-dimensional flow. The main result of the paper is an analytical expression of the optimal bound as a function of the radius ratio. To obtain this bound, we begin by deriving a suboptimal analytical bound using analysis techniques. We use a definition of the background flow with two boundary layers, whose relative thicknesses are optimized to obtain the bound. In the limit of high Reynolds number, the dependence of this suboptimal bound on the radius ratio (the geometrical scaling) turns out to be the same as that of numerically computed optimal bounds in three different cases: (1) the perturbed flow only satisfies the homogeneous boundary conditions but need not be incompressible; (2) the perturbed flow is three-dimensional and incompressible; (3) the perturbed flow is two-dimensional and incompressible. We compare the geometrical scaling with the observations from the turbulent Taylor–Couette flow, and find that the analytical result indeed agrees well with the available direct numerical simulations data. In this paper, we also dismiss the applicability of the background method to certain flow problems and therefore establish the limitation of this method.

In this paper, we consider the problem $-\Delta u =-u^{-\beta }\chi _{\{u>0\}} + f(u)$ in $\Omega$ with $u=0$ on $\partial \Omega$, where $0<\beta <1$ and $\Omega$ is a smooth bounded domain in $\mathbb {R}^{N}$, $N\geq 2$. We are able to solve this problem provided $f$ has subcritical growth and satisfy certain hypothesis. We also consider this problem with $f(s)=\lambda s+s^{\frac {N+2}{N-2}}$ and $N\geq 3$. In this case, we are able to obtain a solution for large values of $\lambda$. We replace the singular function $u^{-\beta }$ by a function $g_\epsilon (u)$ which pointwisely converges to $u^{-\beta }$ as $\epsilon \rightarrow 0$. The corresponding energy functional to the perturbed equation $-\Delta u + g_\epsilon (u) = f(u)$ has a critical point $u_\epsilon$ in $H_0^{1}(\Omega )$, which converges to a non-trivial non-negative solution of the original problem as $\epsilon \rightarrow 0$.