Liouville-type theorems for the steady incompressible Navier–Stokes system are investigated for solutions in a three-dimensional (3-D) slab with either no-slip boundary conditions or periodic boundary conditions. When the no-slip boundary conditions are prescribed, we prove that any bounded solution is trivial if it is axisymmetric or $ru^r$ is bounded, and that general 3-D solutions must be Poiseuille flows when the velocity is not big in $L^\infty$ space. When the periodic boundary conditions are imposed on the slab boundaries, we prove that the bounded solutions must be constant vectors if either the swirl or radial velocity is independent of the angular variable, or $ru^r$ decays to zero as $r$ tends to infinity. The proofs are based on the fundamental structure of the equations and energy estimates. The key technique is to establish a Saint-Venant type estimate that characterizes the growth of the Dirichlet integral of non-trivial solutions.

We identify forcing mechanisms that separately amplify subsonic and supersonic features obtained from a linearised Navier–Stokes based model for compressible parallel boundary layers. Resolvent analysis is used to analyse the linear model, where the nonlinear terms of the linearised equations act as a forcing to the linear terms. Considering subsonic modes, only the solenoidal component of the forcing to the momentum equations amplify these modes. When considering supersonic modes, we find that these are pressure fluctuations that radiate into the free stream. Within the free stream, these modes closely follow the trends of inviscid Mach waves. There are two distinct forcing mechanisms that amplify the supersonic modes: (i) the ‘direct route’, where the forcing to the continuity and energy equations and the dilatational component of the forcing to the momentum equations directly force the mode; and (ii) the ‘indirect route’, where the solenoidal component of the forcing to the momentum equations force a response in wall-normal velocity, and this wall-normal velocity in turn forces the supersonic mode. A majority of the supersonic modes considered are dominantly forced by the direct route. However, when considering Mach waves that are, like in direct numerical simulations, forced from the buffer layer of the flow, the indirect route of forcing becomes significant. We find that these observations are also valid for a streamwise developing boundary layer. These results are consistent with, and extend, the observations in the literature regarding the solenoidal and dilatational components of velocity in compressible turbulent wall-bounded flows.

Predicting and perhaps mitigating against rare, extreme events in fluid flows is an important challenge. Due to the time-localised nature of these events, Fourier-based methods prove inefficient in capturing them. Instead, this paper uses wavelet-based methods to understand the underlying patterns in a forced flow over a 2-torus which has intermittent high-energy burst events interrupting an ambient low-energy ‘quiet’ flow. Two wavelet-based methods are examined to predict burst events: (i) a wavelet proper orthogonal decomposition (WPOD) based method which uncovers and utilises the key flow patterns seen in the quiet regions and the bursting episodes; and (ii) a wavelet resolvent analysis (WRA) based method that relies on the forcing structures which amplify the underlying flow patterns. These methods are compared with a straightforward energy tracking approach which acts as a benchmark. Both the wavelet-based approaches succeed in producing better predictions than a simple energy criterion, i.e. earlier prediction times and/or fewer false positives and the WRA-based technique always performs better than WPOD. However, the improvement of WRA over WPOD is not as substantial as anticipated. We conjecture that this is because the mechanism for the bursts in the flow studied is found to be largely modal, associated with the unstable eigenfunction of the Navier–Stokes operator linearised around the mean flow. The WRA approach should deliver much better improvement over the WPOD approach for generically non-modal bursting mechanisms where there is a lag between the imposed forcing and the final response pattern.

Most of the existing theories on electrophoresis are based on the consideration of a weak applied electric field and ions as point charges, which create a mean electric potential and neglect ion–solvent interactions. These theories cannot demonstrate the dependence of electrophoretic mobility on the applied electric field (nonlinear electrophoresis), reversal in mobility with increasing ion concentration and/or surface charge density or counterion saturation in the electric double layer. In this study we consider a modified electrokinetic model to analyse nonlinear electrophoresis by taking into account the finite ion size effects and ion–ion electrostatic correlations. In this approach, the mean-field-based model is extended to capture the many-body phenomena by considering the non-local electrostatic contribution in the ion free energy functional and the ion–ion hydrodynamic steric interactions are incorporated through the volume exclusion effect in the electrochemical potential. The viscosity of the medium is considered to vary with the local ionic volume fraction. Stronger correlations for multivalent counterions create ion layering, charge density oscillation and mobility reversal. Such phenomena are captured by the present continuum model. The ion crowding attenuates the growth of the electrophoretic mobility with the electric field. At a higher range of the imposed electric field, the ion concentration in the electric double layer enhances, which modifies both the overscreening and ion crowding processes.

We consider the two-dimensional Rayleigh–Bénard convection problem between Navier-slip fixed-temperature boundary conditions, and present a new upper bound for the Nusselt number (${\textit {Nu}}$). The result, based on a localization principle for the Nusselt number and an interpolation bound, exploits the regularity of the flow. On one hand our method yields a shorter proof of the celebrated result of Whitehead & Doering (Phys. Rev. Lett., vol. 106, 2011, 244501) in the case of free-slip boundary conditions. On the other hand, its combination with a new, refined estimate for the pressure gives a substantial improvement of the interpolation bounds in Drivas et al. (Phil. Trans. R. Soc. A, vol. 380, issue 2225, 2022, 20210025) for slippery boundaries. A rich description of the scaling behaviour arises from our result: depending on the magnitude of the Prandtl number (${\textit {Pr}}$) and slip length, our upper bounds indicate five possible scaling laws (where ${\textit {Ra}}$ is the Rayleigh number): ${\textit {Nu}}\sim (L_s^{-1}\,{\textit {Ra}})^{{1}/{3}}$, ${\textit {Nu}}\sim (L_s^{-2/5}\,{\textit {Ra}})^{{5}/{13}}$, ${\textit {Nu}}\sim {\textit {Ra}}^{{5}/{12}}$, ${\textit {Nu}}\sim {\textit {Pr}}^{-1/6}(L_s^{-4/3}\,{\textit {Ra}})^{{1}/{2}}$ and ${\textit {Nu}}\sim {\textit {Pr}}^{-1/6}(L_s^{-1/3}\,{\textit {Ra}})^{{1}/{2}}$.

Drop collision with a solid particle is a ubiquitous phenomenon in a wide range of applications, including rain, spray coating, cooling or cleaning, particle encapsulation, inkjet printing, and additive manufacturing. Understanding the dynamics of drop collision is essential for optimizing these processes. In this study, we present a comprehensive experimental and analytical investigation of non-axisymmetric as well as axisymmetric drop impact on a solid particle. We use a high-speed video system to visualize the drop profile during the impact, and measure the drop height and spreading diameter for different liquid viscosities, ratios of the target to drop diameters, offsets, and various other impact parameters. We then develop a theoretical model for drop spreading on a solid spherical particle that relies on the formulation of a remote asymptotic solution for the inviscid flows, generated by non-axisymmetric drop impact. Next, the viscous effects in a thin viscous boundary layer are considered, which allows the formulation of an expression for the residual lamella thickness and maximum spreading. The theoretically predicted evolution of the lamella thickness, the residual film thickness, and the maximum spreading angle agree well with the experimental data presented in this work and the literature. Finally, we present a novel approach for in situ measurement of liquid viscosity, drop impact viscometry, at high shear rates via a single drop impact experiment, with potential application in industries where non-Newtonian drops play a major role, such as pesticide spraying, paint droplet spreading, blood drop impact and fuel injectors.

Two-dimensional turbulence transfers its energy towards the lowest mode in the domain, but domain geometry exerts a powerful control. On the sphere, with its three axes of rotational symmetry, angular momentum conservation prevents energy from entering the three lowest modes – those corresponding to the spherical harmonics $Y_1^0$ and $Y_1^{\pm 1}$ – because the amplitudes of these three modes are proportional to the three conserved components of the angular momentum vector. Non-spherical ellipsoids partly or completely break the rotational symmetry corresponding to angular momentum conservation. The flow on spheroids, which have only one axis of rotational symmetry, conserves only a single component of angular momentum. If the axis of symmetry is taken to be the $z$-axis, then only the $z$-component of angular momentum is conserved. Energy can flow into the other two lowest modes. The general triaxial ellipsoid breaks all rotational symmetries, thus angular momentum is not conserved, and energy can flow into any mode. We describe numerical experiments that confirm these predictions.

Diffuse interface models are an important class of models used to describe multi-phase flows. In the case of incompressible viscous fluids there are a number of different diffuse interface models which have been known for many years. Nevertheless, a model based on mixture theory with a full set of conservation laws for the conservation of linear momentum of each constituent was not yet available. This gap was filled by ten Eikelder et al. (J. Fluid Mech., in press) recently and a first comparison with known models of Navier–Stokes/Cahn–Hilliard type is given. A detailed understanding of the relations between these models remains an important question.

A theory of incompressible turbulent plane jets (TPJs) is proposed by advancing an improved boundary layer approximation over the limiting classical – retaining more terms in the momentum balance equations. A pressure deficit inside the jet (with respect to the ambient) must exist due to transverse turbulence (Miller & Comings, J. Fluid Mech., vol. 3, 1957, pp. 1–16; Hussain & Clarke, Phys. Fluids, vol. 20, 1977, pp. 1416–1426). Contrary to the universally accepted invariance of the total momentum flux $J_T(x)$ (non-dimensionalized by its inlet value) as a function of the streamwise distance $x$, we prove that $J_T(x) >1$ – a condition that all TPJs must satisfy; surprisingly, prior theories and most experiments do not satisfy this condition. This motivated us to apply Lie symmetry analysis with translational and dilatational transformations of the modified equations (incorporating $J_T>1$), which yields scaling laws for key jet measures: the mean streamwise and transverse velocities $U(x,y)$ and $V(x,y)$, the turbulence intensities, the Reynolds shear stress $-\rho \,\overline {u'v'}(x,y)$, the mean pressure $P(x,y)$, etc. Experiments satisfying $J_T(x)>1$ validate our predictions for all jet measures, including, among others, the profiles of $U$, $V$ and $-\rho \,\overline {u'v'}$. We further predict $U \sim x^{-0.24}$, $V \sim x^{-0.45}$, $-\rho \,\overline {u'v'}\sim x^{-0.69}$, the mass flux $Q_m \sim x^{0.55}$, and $J_T$ increases to approximately 1.5. Contrary to the classical linear jet spread, we find sublinear spread, with the jet half-width growing like $b(x)\sim x^{0.79}$, indicating a narrower jet. Our predictions differ notably from most results reported in the literature. These contradictions demand revisiting jet studies involving carefully designed facilities and boundary conditions, and highly resolved simulations.

The prototypical diffuse-interface model for incompressible fluid mixtures is the Navier–Stokes Cahn–Hilliard (Allen–Cahn) model. Despite its foundation in continuum mixture theory, it is not fully compatible with this theory due to the diffusive flux approximation. This paper introduces a class of thermodynamically consistent diffuse-interface incompressible fluid mixture models that is fully compatible with the continuum theory of mixtures. The proposed models can be formulated in either constituent or mixture quantities, enabling a direct comparison with the Navier–Stokes Cahn–Hilliard (Allen–Cahn) model with non-matching densities. This comparison reveals the key modelling simplifications employed in the latter.

We study the stationary Navier–Stokes equations in the region between two rotating concentric cylinders. We first prove that, for a small Reynolds number, if the fluid flow is axisymmetric and if its velocity is sufficiently small in the $L^\infty$-norm, then it is necessarily the Taylor–Couette–Poiseuille flow. If, in addition, the associated pressure is bounded or periodic in the $z$ axis, then it coincides with the well-known canonical Taylor–Couette flow. We discuss the relation between uniqueness and stability of such a flow in terms of the Taylor number in the case of narrow gap of two cylinders. The investigation in comparison with two Reynolds numbers based on inner and outer cylinder rotational velocities is also conducted. Next, we give a certain bound of the Reynolds number and the $L^\infty$-norm of the velocity such that the fluid is, indeed, necessarily axisymmetric. As a result, it is clarified that smallness of Reynolds number of the fluid in the two rotating concentric cylinders governs both axisymmetry and the Taylor–Couette–Poiseuille flow with the exact form of the pressure.

In 1945–1949, Lars Onsager made an exact analysis of the high-Reynolds-number limit for individual turbulent flow realisations modelled by incompressible Navier–Stokes equations, motivated by experimental observations that dissipation of kinetic energy does not vanish. I review here developments spurred by his key idea that such flows are well described by distributional or ‘weak’ solutions of ideal Euler equations. 1/3 Hölder singularities of the velocity field were predicted by Onsager and since observed. His theory describes turbulent energy cascade without probabilistic assumptions and yields a local, deterministic version of the Kolmogorov $4/5$th law. The approach is closely related to renormalisation group methods in physics and envisages ‘conservation-law anomalies’, as discovered later in quantum field theory. There are also deep connections with large-eddy simulation modelling. More recently, dissipative Euler solutions of the type conjectured by Onsager have been constructed and his $1/3$ Hölder singularity proved to be the sharp threshold for anomalous dissipation. This progress has been achieved by an unexpected connection with work of John Nash on isometric embeddings of low regularity or ‘convex integration’ techniques. The dissipative Euler solutions yielded by this method are wildly non-unique for fixed initial data, suggesting ‘spontaneously stochastic’ behaviour of high-Reynolds-number solutions. I focus in particular on applications to wall-bounded turbulence, leading to novel concepts of spatial cascades of momentum, energy and vorticity to or from the wall as deterministic, space–time local phenomena. This theory thus makes testable predictions and offers new perspectives on large-eddy simulation in the presence of solid walls.

The force and torque on a solid body in a viscous potential flow are often taken to be independent of viscosity. Joseph et al. (Eur. J. Mech. B/Fluids, vol. 12, 1993, pp. 97–106; J. Fluid Mech., vol. 265, 1994, pp. 1–23) proved that this holds for (i) the force (not the torque) in any two-dimensional flow, and (ii) the drag force experienced by a purely translating three-dimensional body. The remaining components of the force and torque, along with general three-dimensional flows, were not considered. Importantly, the flow was assumed to be unbounded and irrotational everywhere. We eliminate this rarely satisfied assumption and consider the viscous force and torque experienced by any closed surface where the flow is irrotational locally; this can include a body's surface. Any vorticity distribution is permitted away from the closed irrotational surface. In so doing, we complete the analysis of Joseph et al. for all components of the viscous force and torque in two and three dimensions and enable application to real flows that inevitably contain regions of vorticity.

First-order sensitivities and adjoint analysis are used widely to control the linear stability of unstable flows. Second-order sensitivities have recently helped to increase accuracy. In this paper, a method is presented to calculate arbitrary high-order sensitivities based on Taylor expansions of the incompressible base flow and its eigenproblem around a scalar parameter. For the incompressible Navier–Stokes equations, general expressions for the sensitivities are derived, into which parameter-specific information can be inserted. The computational costs are low since, for all orders, a linear equation system has to be solved, of which the left-hand-side matrix stays constant and thus its preconditioning can be exploited. Two flow scenarios are examined. First, the cylinder flow equations are expanded around the inverse of the Reynolds number, enabling the prediction of the two-dimensional cylinder base flow and its leading eigenvalue as a function of the Reynolds number. This approach computes accurately the base flow and eigenvalue even in the unstable regime, providing, when executed subsequently, a mean to calculate unstable base flows. This case gives a clear introduction into the method and allows us to discuss its constraints regarding convergence behaviour. Second, a small control cylinder is introduced into the domain of the cylinder flow for stabilization. Higher-order sensitivity maps are calculated by modelling the small cylinder with a steady forcing. These maps help to identify stabilizing areas of the flow field for Reynolds numbers within the laminar vortex shedding regime, with the required number of orders increasing as the Reynolds number rises. The results obtained through the proposed method align well with numerically calculated eigenvalues that incorporate the cylinder directly into the grid.

Oscillatory flows have become an indispensable tool in microfluidics, inducing inertial effects for displacing and manipulating fluid-borne objects in a reliable, controllable and label-free fashion. However, the quantitative description of such effects has been confined to limit cases and specialized scenarios. Here we develop an analytical formalism yielding the equation of motion of density-mismatched spherical particles in oscillatory background flows, generalizing previous work. Inertial force terms are systematically derived from the geometry of the flow field together with analytically known Stokes number dependences. Supported by independent, first-principles direct numerical simulations, we find that these forces are important even for nearly density-matched objects such as cells or bacteria, enabling their fast displacement and separation. Our formalism thus consistently incorporates particle inertia into the Maxey–Riley equation, and in doing so provides a generalization of Auton's modification to added mass, as well as recovering the description of acoustic radiation forces on particles as a limiting case.

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.

This article is concerned with the problem of determining an unknown source of non-potential, external time-dependent perturbations of an incompressible fluid from large-scale observations on the flow field. A relaxation-based approach is proposed for accomplishing this, which makes use of a nonlinear property of the equations of motions to asymptotically enslave small scales to large scales. In particular, an algorithm is introduced that systematically produces approximations of the flow field on the unobserved scales in order to generate an approximation to the unknown force; the process is then repeated to generate an improved approximation of the unobserved scales, and so on. A mathematical proof of convergence of this algorithm is established in the context of the two-dimensional Navier–Stokes equations with periodic boundary conditions under the assumption that the force belongs to the observational subspace of phase space; at each stage in the algorithm, it is shown that the model error, represented as the difference between the approximating and true force, asymptotically decreases to zero in a geometric fashion provided that sufficiently many scales are observed and certain parameters of the algorithm are appropriately tuned.

In this work, we analyse wall-bounded flows in the continuum to transition regime with the help of higher-order transport equations (super-set of the Navier–Stokes equation). Towards this, we incorporate second-order in Knudsen number accurate terms in the single-particle distribution function, and with this complete representation, we first derive the second-order accurate extended-OBurnett (EOBurnett) and third-order accurate super-OBurnett (SOBurnett) equations. We then demonstrate that these newly derived equations exhibit unconditional linear stability. We finally validate the equations by solving for plane Poiseuille flow and derive closed-form analytical solutions for the pressure and velocity fields. The pressure and velocity results thus obtained have been compared with direct simulation Monte Carlo (DSMC) data in the transition regime. The results from both the EOBurnett and SOBurnett equations are found to yield better agreement with DSMC data than that obtained from the Navier–Stokes equations. This improved agreement is attributed to the presence of additional terms in the proposed equations, which effectively capture the effect of the Knudsen layer near the wall. The obtained higher-order transport equations and the closed-form solution presented in this work are novel. The ability of the equations to describe the flow in the transition regime should form the basis for conducting further realistic analytical studies of wall-bounded flows in the future.

This study introduces an analytical solution for the laminar entrance flow in circular pipes, aiming to confirm the occurrence of velocity overshoot. Velocity overshoot is characterised by the maximum axial velocity appearing near the pipe wall instead of the central axis. Similar to the previous studies, the analytical solution is derived from the parabolised Navier–Stokes equation; however, the specific approach used in linearising the momentum equation has not been attempted before. The accuracy of this analytical solution has been verified through a comprehensive comparison with various published experimental data. The existence of velocity overshoot at a short distance from the inlet, which is evident in numerous numerical calculations based on the full Navier–Stokes equations and corroborated by recent magnetic resonance (MR) velocimetry experiments, is identified analytically for the first time. The parabolised Navier–Stokes equation has inherent self-similarity with respect to the Reynolds number, implying that $Re$ is incorporated into the dimensionless variables rather than serving as an independent flow parameter. According to both MR velocimetry measurements and the present analytical solution, the self-similarity is not valid immediately following the pipe inlet, and this becomes more evident as $Re$ decreases; hence, the analytical solution derived from the parabolised Navier–Stokes equation cannot accurately predict the evolution of the velocity profile within this region near the pipe inlet.