In the study of fluid dynamics and transport phenomena, key quantities of interest are often the force and torque on objects and total rate of heat/mass transfer from them. Conventionally, these integrated quantities are determined by first solving the governing equations for the detailed distribution of the field variables (i.e. velocity, pressure, temperature, concentration, etc.) and then integrating the variables or their derivatives on the surface of the objects. On the other hand, the divergence form of the conservation equations opens the door for establishing integral identities that can be used for directly calculating the integrated quantities without requiring the detailed knowledge of the distribution of the primary variables. This shortcut approach constitutes the idea of the reciprocal theorem, whose closest relative is Green’s second identity, which readers may recall from studies of partial differential equations. Despite its importance and practicality, the theorem may not be so familiar to many in the research community. Ironically, some believe that the extreme simplicity and generality of the theorem are responsible for suppressing its application! In this Perspectives piece, we provide a pedagogical introduction to the concept and application of the reciprocal theorem, with the hope of facilitating its use. Specifically, a brief history on the development of the theorem is given as a background, followed by the discussion of the main ideas in the context of elementary boundary-value problems. After that, we demonstrate how the reciprocal theorem can be utilized to solve fundamental problems in low-Reynolds-number hydrodynamics, aerodynamics, acoustics and heat/mass transfer, including convection. Throughout the article, we strive to make the materials accessible to early career researchers while keeping it interesting for more experienced scientists and engineers.

Natural and engineered surfaces are never smooth, but irregular, rough at different scales, compliant, possibly porous, liquid impregnated or superhydrophobic. The correct numerical modelling of fluid flowing through and around them is important but poses problems. For media characterized by a periodic or quasi-periodic microstructure of characteristic dimensions smaller than the relevant scales of the flow, multiscale homogenization can be used to study the effect of the surface, avoiding the numerical resolution of small details. Here, we revisit the homogenization strategy using adjoint variables to model the interaction between a fluid in motion and regularly micro-textured, permeable or impermeable walls. The approach described allows for the easy derivation of auxiliary/adjoint systems of equations which, after averaging, yield macroscopic tensorial properties, such as permeability, elasticity, slip, transpiration, etc. When the fluid in the neighbourhood of the microstructure is in the Stokes regime, classical results are recovered. Adjoint homogenization, however, permits simple extension of the analysis to the case in which the flow displays nonlinear effects. Then, the properties extracted from the auxiliary systems take the name of effective properties and do not depend only on the geometrical details of the medium, but also on the microscopic characteristics of the fluid motion. Examples are shown to demonstrate the usefulness of adjoint homogenization to extract effective tensor properties without the need for ad hoc parameters. In particular, notable results reported herein include:

1. (i) an original formulation to describe filtration in porous media in the presence of inertial effects;

2. (ii) the microscopic and macroscopic equations needed to characterize flows through poroelastic media;

3. (iii) an extended Navier’s condition to be employed at the boundary between a fluid and an impermeable rough wall, with roughness elements which can be either rigid or linearly elastic;

4. (iv) the microscopic problems needed to define the relevant parameters for a Saffman-like condition at the interface between a fluid and a porous substrate; and

5. (v) the macroscopic equations which hold at the dividing surface between a free-fluid region and a fluid-saturated poroelastic domain.

Theory and modelling remain central to improving our understanding of undulatory and oscillatory swimming. Simple models based on added mass can help to give great insight into the mechanics of undulatory swimming, as demonstrated by animals such as eels, stingrays and knifefish. To understand the swimming of oscillatory swimmers such as tuna and dolphins, models need to consider both added mass forces and circulatory forces. For all types of swimming, experiments and theory agree that the most important velocity scale is the characteristic lateral velocity of the tail motion rather than the swimming speed, which erases to a large extent the difference between results obtained in a tethered mode, compared to those obtained using a free swimming condition. There is no one-to-one connection between the integrated swimming performance and the details of the wake structure, in that similar levels of efficiency can occur with very different wake structures. Flexibility and viscous effects play crucial roles in determining the efficiency, and for isolated propulsors changing the profile shape can significantly improve both thrust and efficiency. Also, combined heave and pitch motions with an appropriate phase difference are essential to achieve high performance. Reducing the aspect ratio will always reduce thrust and efficiency, but its effects are now reasonably well understood. Planform shape can have an important mitigating influence, as do non-sinusoidal gaits and intermittent actuation.

The large-scale structure of many turbulent flows encountered in practical situations such as aeronautics, industry, meteorology is nowadays successfully computed using the Kolmogorov–Kármán–Howarth energy cascade picture. This theory appears increasingly inaccurate when going down the energy cascade that terminates through intermittent spots of energy dissipation, at variance with the assumed homogeneity. This is problematic for the modelling of all processes that depend on small scales of turbulence, such as combustion instabilities or droplet atomization in industrial burners or cloud formation. This paper explores a paradigm shift where the homogeneity hypothesis is replaced by the assumption that turbulence contains singularities, as suggested by Onsager. This paradigm leads to a weak formulation of the Kolmogorov–Kármán–Howarth–Monin equation (WKHE) that allows taking into account explicitly the presence of singularities and their impact on the energy transfer and dissipation. It provides a local in scale, space and time description of energy transfers and dissipation, valid for any inhomogeneous, anisotropic flow, under any type of boundary conditions. The goal of this article is to discuss WKHE as a tool to get a new description of energy cascades and dissipation that goes beyond Kolmogorov and allows the description of small-scale intermittency. It puts the problem of intermittency and dissipation in turbulence into a modern framework, compatible with recent mathematical advances on the proof of Onsager’s conjecture.

Suspensions are composed of mixtures of particles and fluid and are omnipresent in natural phenomena and in industrial processes. The present paper addresses the rheology of concentrated suspensions of non-colloidal particles. While hydrodynamic interactions or lubrication forces between the particles are important in the dilute regime, they become of lesser significance when the concentration is increased, and direct particle contacts become dominant in the rheological response of concentrated suspensions, particularly those close to the maximum volume fraction where the suspension ceases to flow. The rheology of these dense suspensions can be approached via a diversity of approaches that the paper introduces successively. The mixture of particles and fluid can be seen as a fluid with effective rheological properties but also as a two-phase system wherein the fluid and particles can experience relative motion. Rheometry can be undertaken at an imposed volume fraction but also at imposed values of particle normal stress, which is particularly suited to yield examination of the rheology close to the jamming transition. The response of suspensions to unsteady or transient flows provides access to different features of the suspension rheology. Finally, beyond the problem of suspension of rigid, non-colloidal spheres in a Newtonian fluid, there are a great variety of complex mixtures of particles and fluid that remain relatively unexplored.

This article discusses the description of wall-bounded turbulence as a deterministic high-dimensional dynamical system of interacting coherent structures, defined as eddies with enough internal dynamics to behave relatively autonomously from any remaining incoherent part of the flow. The guiding principle is that randomness is not a property, but a methodological choice of what to ignore in the flow, and that a complete understanding of turbulence, including the possibility of control, requires that it be kept to a minimum. After briefly reviewing the underlying low-order statistics of flows at moderate Reynolds numbers, the article examines what two-point statistics imply for the decomposition of the flow into individual eddies. Intense eddies are examined next, including their temporal evolution, and shown to satisfy many of the properties required for coherence. In particular, it is shown that coherent structures larger than the Corrsin scale are a natural consequence of the shear. In wall-bounded turbulence, they can be classified into coherent dispersive waves and transient bursts. The former are found in the viscous layer near the wall, and as very large structures spanning the entire boundary layer. Although they are shear-driven, these waves have enough internal structure to maintain a uniform advection velocity. Conversely, bursts exist at all scales, are characteristic of the logarithmic layer, and interact almost linearly with the shear. While the waves require a wall to determine their length scale, the bursts are essentially independent from it. The article concludes with a brief review of our present theoretical understanding of turbulent structures, and with a list of open problems and future perspectives.

‘Chance is the name we give to what we choose to ignore (Voltaire)’

Binary fluid mixtures are examples of complex fluids whose microstructure and flow are strongly coupled. For pairs of simple fluids, the microstructure consists of droplets or bicontinuous demixed domains and the physics is controlled by the interfaces between these domains. At continuum level, the structure is defined by a composition field whose gradients – which are steep near interfaces – drive its diffusive current. These gradients also cause thermodynamic stresses which can drive fluid flow. Fluid flow in turn advects the composition field, while thermal noise creates additional random fluxes that allow the system to explore its configuration space and move towards the Boltzmann distribution. This article introduces continuum models of binary fluids, first covering some well-studied areas such as the thermodynamics and kinetics of phase separation, and emulsion stability. We then address cases where one of the fluid components has anisotropic structure at mesoscopic scales creating nematic (or polar) liquid-crystalline order; this can be described through an additional tensor (or vector) order parameter field. We conclude by outlining a thriving area of current research, namely active emulsions, in which one of the binary components consists of living or synthetic material that is continuously converting chemical energy into mechanical work. Such activity can be modelled with judicious additional terms in the equations of motion for simple or liquid-crystalline binary fluids. Throughout, the emphasis of the article is on presenting the theoretical tools needed to address a wide range of physical phenomena. Examples include the kinetics of fluid–fluid demixing from an initially uniform state; the result of imposing a steady macroscopic shear flow on this demixing process; and the diffusive coarsening, Brownian motion and coalescence of emulsion droplets. We discuss strategies to create long-lived emulsions by adding trapped species, solid particles, or surfactants; to address the latter, we outline the theory of bending energy for interfacial films. In emulsions where one of the components is liquid-crystalline, ‘anchoring’ terms can create preferential orientation tangential or normal to the fluid–fluid interface. These allow droplets of an isotropic fluid in a liquid crystal (or vice versa) to support a variety of topological defects, which we describe, altering their interactions and stability. Addition of active terms to the equations of motion for binary simple fluids creates a model of ‘motility-induced’ phase separation, where demixing stems from self-propulsion of particles rather than their interaction forces, altering the relation between interfacial structure and fluid stress. Coupling activity to binary liquid crystal dynamics creates models of active liquid-crystalline emulsion droplets. Such droplets show various modes of locomotion, some of which strikingly resemble the swimming or crawling motions of biological cells.

Known structures and self-sustaining mechanisms of wall turbulence are reviewed and explored in the context of the scale interactions implied by the nonlinear advective term in the Navier–Stokes equations. The viewpoint is shaped by the systems approach provided by the resolvent framework for wall turbulence proposed by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), in which the nonlinearity is interpreted as providing the forcing to the linear Navier–Stokes operator (the resolvent). Elements of the structure of wall turbulence that can be uncovered as the treatment of the nonlinearity ranges from data-informed approximation to analysis of exact solutions of the Navier–Stokes equations (so-called exact coherent states) are discussed. The article concludes with an outline of the feasibility of extending this kind of approach to high-Reynolds-number wall turbulence in canonical flows and beyond.

The route to turbulence in pipe flow is a complex, nonlinear, spatiotemporal process for which an increasingly clear understanding has emerged in recent years. This paper presents a theoretical perspective on the problem, focusing on what can be understood from relatively few physical features and models that encompass these features. The paper proceeds step-by-step with increasing detail about the transition process, first discussing the relationship to phase transitions and then exploiting an even deeper connection between pipe flow and excitable and bistable media. In the end a picture emerges for all stages of the transition process, from transient turbulence, to the onset of sustained turbulence in a percolation transition, to the modest and then rapid expansion of turbulence, ultimately leading to fully turbulent pipe flow.