We study the linear stability of bubbles in a capillary tube under external flow. Yu et al. (J. Fluid Mech., vol. 911, 2021, pp. 1–19) showed that a rich variety of bubble dynamics occurs when a downward external flow is applied, opposing the buoyancy-driven ascent of the bubble. They found experimentally and numerically the existence of two branches of solutions that overlap over a finite range of the capillary number of the downward external flow in cases where the Reynolds number is small and the Bond number is larger than the critical value for which the bubble can rise spontaneously (Bretherton, J. Fluid Mech., vol. 10, issue 2, 1961, pp. 166–188). Furthermore, inertialess, symmetry-breaking steady-state shapes were found as the bubble transits near the tipping points of the solution branches. In this work, using steady axisymmetric simulations, we show that the reported multiplicity of solutions can be described using bifurcation diagrams with three branches of steady axisymmetric solutions and two limit points. The linear global stability analysis of the different branches of the stationary axisymmetric solutions demonstrates that the symmetry breaking is due to the development of three-dimensional instabilities with azimuthal wavenumber $|m|=1$.

A neural network framework is used to design a new Ni-based superalloy that surpasses the performance of IN718 for laser-blown-powder directed-energy-deposition repair applications. The framework utilized a large database comprising physical and thermodynamic properties for different alloy compositions to learn both composition to property and also property to property relationships. The alloy composition space was based on IN718, although, W was additionally included and the limiting Al and Co content were allowed to increase compared standard IN718, thereby allowing the alloy to approach the composition of ATI 718Plus® (718Plus). The composition with the highest probability of satisfying target properties including phase stability, solidification strain, and tensile strength was identified. The alloy was fabricated, and the properties were experimentally investigated. The testing confirms that this alloy offers advantages for additive repair applications over standard IN718.

Diffusiophoresis refers to the movement of colloidal particles in the presence of a concentration gradient of a solute and enables directed motion of colloidal particles in geometries that are inaccessible, such as dead-end pores, without imposing an external field. Previous experimental reports on dead-end pore geometries show that, even in the absence of mean flow, colloidal particles moving through diffusiophoresis exhibit significant dispersion. Existing models of diffusiophoresis are not able to predict the dispersion and thus the comparison between the experiments and the models is largely qualitative. To address these quantitative differences between the experiments and models, we derive an effective one-dimensional equation, similar to a Taylor dispersion analysis, that accounts for the dispersion created by diffusioosmotic flow from the channel sidewalls. We derive the effective dispersion coefficient and validate our results by comparing them with direct numerical simulations. We also compare our model with experiments and obtain quantitative agreement for a wide range of colloidal particle sizes. Our analysis reveals two important conclusions. First, in the absence of mean flow, dispersion is driven by the flow created by diffusioosmotic wall slip such that spreading can be reduced by decreasing the channel wall diffusioosmotic mobility. Second, the model can explain the spreading of colloids in a dead-end pore for a wide range of particle sizes. We note that, while the analysis presented here focuses on a dead-end pore geometry with no mean flow, our theoretical framework is general and can be adapted to other geometries and other background flows.

At low Reynolds numbers, axisymmetric ellipsoidal particles immersed in a shear flow undergo periodic tumbling motions known as Jeffery orbits, with the orbit determined by the initial orientation. Understanding this motion is important for predicting the overall dynamics of a suspension. While slender fibres may follow Jeffery orbits, many such particles in nature are neither straight nor rigid. Recent work exploring the dynamics of curved or elastic fibres have found Jeffery-like behaviour along with chaotic orbits, decaying orbital constants and cross-streamline drift. Most work focuses on particles with reflectional symmetry; we instead consider the behaviour of a composite asymmetric slender body made of two straight rods, suspended in a two-dimensional shear flow, to understand the effects of the shape on the dynamics. We find that for certain geometries the particle does not rotate and undergoes persistent drift across streamlines, the magnitude of which is consistent with other previously identified forms of cross-streamline drift. For this class of particles, such geometry-driven cross-streamline motion may be important in giving rise to dispersion in channel flows, thereby potentially enhancing mixing.

The motion of finite-length cylindrical rods moving near a planar rigid surface is a scenario common across many engineering and natural settings. We study the low-Reynolds-number flow around finite rods that are allowed to rotate or translate in directions perpendicular or parallel to the plane. We develop a three-dimensional lubrication theory to characterize the pressure and hydrodynamic resistances of the cylinders through a special consideration of the cylinder's end effects. In addition, we use three-dimensional numerical simulations to solve these Stokes flows for cylinders of varying lengths and with varying gap sizes between the cylinder and plane, and the numerical results support the developed analytical descriptions. We also use visualizations of the flow to provide qualitative insights and rationalize the effect of the ends on the dynamics of the cylinders. The numerical simulations and theoretical predictions show good agreement in the long (isolated ends) and short (disk-like) limits.

We analyse the pressure-driven flow of the Oldroyd-B fluid in slowly varying arbitrarily shaped, narrow channels and present a theoretical framework for calculating the relationship between the flow rate $q$ and pressure drop $\Delta p$. We first identify the characteristic scales and dimensionless parameters governing the flow in the lubrication limit. Employing a perturbation expansion in powers of the Deborah number ($De$), we provide analytical expressions for the velocity, stress and the $q$–$\Delta p$ relation in the weakly viscoelastic limit up to $O(De^2)$. Furthermore, we exploit the reciprocal theorem derived by Boyko $\&$ Stone (Phys. Rev. Fluids, vol. 6, 2021, L081301) to obtain the $q$–$\Delta p$ relation at the next order, $O(De^3)$, using only the velocity and stress fields at the previous orders. We validate our analytical results with two-dimensional numerical simulations in the case of a hyperbolic, symmetric contracting channel and find excellent agreement. While the velocity remains approximately Newtonian in the weakly viscoelastic limit (i.e. the theorem of Tanner and Pipkin), we reveal that the pressure drop strongly depends on the viscoelastic effects and decreases with $De$. We elucidate the relative importance of different terms in the momentum equation contributing to the pressure drop along the symmetry line and identify that a pressure drop reduction for narrow contracting geometries is primarily due to gradients in the viscoelastic shear stresses. We further show that, although for narrow geometries the viscoelastic axial stresses are negligible along the symmetry line, they are comparable or larger than shear stresses in the rest of the domain.

Air exchange between people has emerged in the COVID-19 pandemic as the important vector for transmission of the SARS-CoV-2 virus. We study the airflow and exchange between two unmasked individuals conversing face-to-face at short range, which can potentially transfer a high dose of a pathogen, because the dilution is small when compared to long-range airborne transmission. We conduct flow visualization experiments and direct numerical simulations of colliding respiratory jets mimicking the initial phase of a conversation. The evolution and dynamics of the jets are affected by the vertical offset between the mouths of the speakers. At low offsets the head-on collision of jets results in a `blocking effect', temporarily shielding the susceptible speaker from the pathogen carrying jet, although, the lateral spread of the jets is enhanced. Sufficiently large offsets prevent the interaction of the jets. At intermediate offsets (8-10 cm for 1 m separation), jet entrainment and the inhaled breath assist the transport of the pathogen-loaded saliva droplets towards the susceptible speaker's mouth. Air exchange is expected, in spite of the blocking effect arising from the interaction of the respiratory jets from the two speakers.

We study the motion of large bubbles in curved channels both semi-analytically using the lubrication approximation and computationally using a finite-volume/front-tracking method. The steady film thickness is governed by the classical Landau–Levich–Derjaguin–Bretherton (LLDB) equation in the low-capillary-number limit but with the boundary conditions modified to account for the channel curvature. The lubrication results show that the film is thinner on the inside of a bend than on the outside of a bend. They also indicate that the bubble velocity relative to the average liquid velocity is always larger in a curved channel than that in a corresponding straight channel and increases monotonically with increasing channel curvature. Numerical computations are performed for two-dimensional cases and the computational results are found to be in a good agreement with the lubrication theory for small capillary numbers and small or moderate channel curvatures. For moderate capillary numbers the numerical results for the film thickness, when rescaled to account for channel curvature as suggested in the lubrication calculation, essentially collapse onto the corresponding results for a bubble in a straight tube. The lubrication theory is also extended to the motion of large bubbles in a curved channel of circular cross-section.

We derive an accurate estimate for the diffusive evaporation rates of multiple droplets of different sizes and arbitrary contact angles placed on a horizontal substrate. The derivation, which is based on a combination of Green's second identity and the method of reflections, simply makes use of the solution for the evaporation of a single droplet. The theoretical results can serve as a guide for future computational and experimental studies on the collective evaporation of arrays of droplets, as well as similar multi-body, diffusion-dominated transport problems.

We consider the translation of a rigid, axisymmetric, tightly fitting object through a cylindrical elastic tube filled with viscous fluid, using a combination of theory and direct numerical simulations. The intruding object is assumed to be wider than the undeformed tube radius, forcing solid–solid contact in the absence of relative motion. The motion of the object establishes a thin fluid film that lubricates this contact. Our theory couples lubrication theory to a geometrically nonlinear membrane description of the tube's elasticity, and applies to a slender intruding object and a thin tube with negligible bending rigidity. We show using asymptotic and numerical solutions of the theory, that the thickness of the thin fluid film scales with the square root of the relative speed for small speeds, set by a balance of hoop stresses, membrane tension and fluid pressure. While membrane tension is relatively small at the entrance of the film, it dominates near the exit and produces undulations of the film thickness, even in the limit of vanishing speeds and slender objects. We find that the drag force on the intruding object depends on the slope of its surface at the entrance to the thin fluid film, and scales as the square root of the relative speed. The predictions of the lubricated membrane theory for the shape of the film and the force on the intruder are in quantitative agreement with three-dimensional direct numerical simulations of the coupled fluid–elastic problem.

Non-Newtonian fluids are characterized by complex rheological behaviour that affects the hydrodynamic features, such as the flow rate–pressure drop relation. While flow rate–pressure drop measurements of such fluids are common in the literature, a comparison of experimental data with theory is rare, even for shear-thinning fluids at low Reynolds number, presumably due to the lack of analytical expressions for the flow rate–pressure drop relation covering the entire range of pressures and flow rates. Such a comparison, however, is of fundamental importance as it may provide insight into the adequacy of the constitutive model that was used and the values of the rheological parameters. In this work, we present a theoretical approach to calculating the flow rate–pressure drop relation of shear-thinning fluids in long, narrow channels that can be used for comparison with experimental measurements. We utilize the Carreau constitutive model and provide a semi-analytical expression for the flow rate–pressure drop relation. In particular, we derive three asymptotic solutions for small, intermediate and large values of the dimensionless pressures or flow rates, which agree with distinct limits previously known and allow us to approximate analytically the entire flow rate–pressure drop curve. We compare our semi-analytical and asymptotic results with the experimental measurements of Pipe et al. (Rheol. Acta, vol. 47, 2008, pp. 621–642) and find excellent agreement. Our results rationalize the change in the slope of the flow rate–pressure drop data, when reported in log–log coordinates, at high flow rates, which cannot be explained using a simple power-law model.

We present a numerical study of the dynamics of an elastic fibre in a shear flow at low Reynolds number, and seek to understand several aspects of the fibre's motion using the equations for slender-body theory coupled to the elastica. The numerical simulations are performed in the bead-spring framework including hydrodynamic interactions in two theoretical schemes: the generalized Rotne–Prager–Yamakawa model and a multipole expansion corrected for lubrication forces. In general, the two schemes yield similar results, including for the dominant scaling features of the shape that we identify. In particular, we focus on the evolution of an initially straight fibre oriented in the flow direction and show that the time scales of fibre bending, curling and rotation, which depend on its length and stiffness, determine the overall motion and evolution of the shapes. We document several characteristic time scales and curvatures representative of the shape that vary as power laws of the bending stiffness and fibre length. The numerical results are further supported by an interpretation using an elastica model.

We study bubble motion in a vertical capillary tube under an external flow. Bretherton (J. Fluid Mech., vol. 10, issue 2, 1961, pp. 166–188) has shown that, without external flow, a bubble can spontaneously rise when the Bond number (${Bo} \equiv \rho g R^2 / \gamma$) is above the critical value ${Bo}_{cr}=0.842$, where $\rho$ is the liquid density, $g$ the gravitational acceleration, $R$ the tube radius and $\gamma$ the surface tension. It was then shown by Magnini et al. (Phys. Rev. Fluids, vol. 4, issue 2, 2019, 023601) that the presence of an imposed liquid flow, in the same (upward) direction as buoyancy, accelerates the bubble and thickens the liquid film around it. In this work we carry out a systematic study of the bubble motion under a wide range of upward and downward external flows, focusing on the inertialess regime with Bond numbers above the critical value. We show that a rich variety of bubble dynamics occurs when an external downward flow is applied, opposing the buoyancy-driven rise of the bubble. We reveal the existence of a critical capillary number of the external downward flow (${Ca}_l \equiv \mu U_l/\gamma$, where $\mu$ is the fluid viscosity and $U_l$ is the mean liquid speed) at which the bubble arrests and changes its translational direction. Depending on the relative direction of gravity and the external flow, the thickness of the film separating the bubble surface and the tube inner wall follows two distinct solution branches. The results from theory, experiments and numerical simulations confirm the existence of the two solution branches and reveal that the two branches overlap over a finite range of ${Ca}_l$, thus suggesting non-unique, history-dependent solutions for the steady-state film thickness under the same external flow conditions. Furthermore, inertialess symmetry-breaking shape profiles at steady state are found as the bubble transits near the tipping points of the solution branches, which are shown in both experiments and three-dimensional numerical simulations.

The leakage from a fracture network to a surrounding medium during drainage, or backflow, driven by elastic relaxation, is considered. A network model is extended to include the effects of permeable boundaries, with the permeation through the wall assumed to be proportional to the local pressure. The regimes in which leakage is dominant relative to the parallel flow along the channel are evaluated at different times. Results show that, when the aperture of the channel is large enough, the parallel flow is greater than the permeation through the wall, and the channel thickness decreases in time, $t$, with a $t^{-1/3}$ behaviour, as reported previously. However, when the aperture is small, the channel thickness decreases exponentially in time. An asymptotic investigation of the solution for a single fracture is performed and extended to network systems. The study provides insight into the influence leakage may have on squeezing-induced flows, which is relevant to natural and engineering systems.

Understanding the physics of viscoelastic liquid jets is relevant to jet-based printing and deposition techniques. In this paper we study the behaviour of jets induced from viscoelastic liquid films, using the mechanical impulse provided by a laser pulse to actuate jet formation. We present direct numerical simulations of viscoelastic liquid jets solving the two-phase flow problem, accounting for the Oldroyd-B rheology. We describe how the jet extension time and length are controlled by the Deborah number (ratio of the elastic and inertia-capillary time scales), the viscous dissipation described by the Ohnesorge number (ratio of the viscous-capillary and inertia-capillary time scales), as well as the ratio of laser impulse energy to the energy required to create free surface during jet formation and propagation. Using the droplet ejection laser threshold energy of a Newtonian liquid, we investigate the influence of increasing viscoelastic effects. We show that viscoelastic effects can modify the effective drop size at the tip of the jet, while the maximum jet length increases with increasing Deborah number. Using the simulations, we identify a high-Deborah-number regime, where the time of maximum jet extension can be described as $t_{max} = c_1 De^{1/4}$, with $c_1$ depending on the Ohnesorge number and blister geometry, while the length of maximum extension reaches an asymptotic value $L_{max}^{\infty }$ for $De>100$, $L_{max}^{\infty }$ depending on the Ohnesorge number and laser energy. The observed asymptotic relationships are in good agreement with experiments performed at much higher Deborah numbers.

Under a steady DC electric field of sufficient strength, a weakly conducting dielectric sphere in a dielectric solvent with higher conductivity can undergo spontaneous spinning (Quincke rotation) through a pitchfork bifurcation. We design an object composed of a dielectric sphere and an elastic filament. By solving an elasto-electro-hydrodynamic (EEH) problem numerically, we uncover an EEH instability exhibiting diverse dynamic responses. Varying the bending stiffness of the filament, the composite object displays three behaviours: a stationary state, undulatory swimming and steady spinning, where the swimming results from a self-oscillatory instability through a Hopf bifurcation. By conducting a linear stability analysis incorporating an elastohydrodynamic model, we theoretically predict the growth rates and critical conditions, which agree well with the numerical counterparts. We also propose a reduced model system consisting of a minimal elastic structure which reproduces the EEH instability. The elasto-viscous response of the composite structure is able to transform the pitchfork bifurcation into a Hopf bifurcation, leading to self-oscillation. Our results imply a new way of harnessing elastic media to engineer self-oscillations, and more generally, to manipulate and diversify the bifurcations and the corresponding instabilities. These ideas will be useful in designing soft, environmentally adaptive machines.

Marangoni flow is the motion induced by a surface tension gradient along a fluid–fluid interface. In this study, we report a Marangoni flow generated when a bath of surfactant contacts a pre-wetted film of deionized water on a vertical substrate. The thickness profile of the pre-wetted film is set by gravitational drainage and so varies with the drainage time. The surface tension is lower in the bath due to the surfactant, and thus a liquid film climbs upwards along the vertical substrate due to the surface tension difference. Particle tracking velocimetry is performed to measure the dynamics in the film, where the mean fluid velocity reverses direction as the draining film encounters the front of the climbing film. The effect of the surfactant concentration and the pre-wetted film thickness on the film climbing is then studied. High-speed interferometry is used to measure the front position of the climbing film and the film thickness profile. As a result, higher surfactant concentration induces a faster and thicker climbing film. Also, for high surfactant concentrations, where Marangoni driving dominates, increasing the film thickness increases the rise speed of the climbing front, since viscous resistance is less important. In contrast, for low surfactant concentrations, where Marangoni driving balances gravitational drainage, increasing the film thickness decreases the rise speed of the climbing front while enhancing gravitational drainage. We rationalize these observations by utilizing a dimensionless parameter that compares the magnitudes of the Marangoni stress and gravitational drainage. A model is established to analyse the climbing front, either in the Marangoni-driving-dominated region or in the Marangoni-balanced drainage region. Our work highlights the effects of the gravitational drainage on the Marangoni flow, both by setting the thickness of a pre-wetted film and by resisting the film climbing.

Microfluidic systems are usually fabricated with soft materials that deform due to the fluid stresses. Recent experimental and theoretical studies on the steady flow in shallow deformable microchannels have shown that the flow rate is a nonlinear function of the pressure drop due to the deformation of the upper soft wall. Here, we extend the steady theory of Christov et al. (J. Fluid Mech., vol. 841, 2018, pp. 267–286) by considering the start-up flow from rest, both in pressure-controlled and in flow-rate-controlled configurations. The characteristic scales and relevant parameters governing the transient flow are first identified, followed by the development of an unsteady lubrication theory assuming that the inertia of the fluid is negligible, and that the upper wall can be modelled as an elastic plate under pure bending satisfying the Kirchhoff–Love equation. The model is governed by two non-geometrical dimensionless numbers: a compliance parameter $\unicode[STIX]{x1D6FD}$, which compares the characteristic displacement of the upper wall with the undeformed channel height, and a parameter $\unicode[STIX]{x1D6FE}$ that compares the inertia of the solid with its flexural rigidity. In the limit of negligible solid inertia, $\unicode[STIX]{x1D6FE}\rightarrow 0$, a quasi-steady model is developed, whereby the fluid pressure satisfies a nonlinear diffusion equation, with $\unicode[STIX]{x1D6FD}$ as the only parameter, which admits a self-similar solution under pressure-controlled conditions. This simplified lubrication description is validated with coupled three-dimensional numerical simulations of the Navier equations for the elastic solid and the Navier–Stokes equations for the fluid. The agreement is very good when the hypotheses behind the model are satisfied. Unexpectedly, we find fair agreement even in cases where the solid and liquid inertia cannot be neglected.

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.

We consider the time-dependent flow of a fluid of density $\unicode[STIX]{x1D70C}_{1}$ in a vertical cylindrical container embedded in a fluid of density $\unicode[STIX]{x1D70C}_{2}~({<}\unicode[STIX]{x1D70C}_{1})$ whose side boundary is suddenly removed and the fluid drains freely from the edge. We show that in the inertial–buoyancy regime (large initial Reynolds number) the flow is modelled by the shallow-water equations and bears similarities to a gravity current released from a lock (the dam-break problem) driven by the reduced gravity $g^{\prime }=(1-\unicode[STIX]{x1D70C}_{2}/\unicode[STIX]{x1D70C}_{1})g$. This formulation is amenable to an efficient finite-difference solution. Moreover, we demonstrate that similarity solutions exist, and show that the flow created by the dam break approaches the predicted self-similar behaviour when the volume ratio ${\mathcal{V}}(t)/{\mathcal{V}}(0)\approx 1/2$ where $t$ is time elapsed from the dam break. We considered two cases of drainage: (i) outward from the outer boundary in a full-radius reservoir; and (ii) inward from the inner radius in an annular-shaped reservoir. For the first case the similarity solution is expressed analytically, while the second case is more complicated and requires a numerical solution. In both cases ${\mathcal{V}}(t)/{\mathcal{V}}(0)$ decays like $t^{-2}$, but the details are different. The similarity solutions admit an adjustable virtual-origin constant, which we determine by matching with the finite-difference solution. The analysis is valid for both Boussinesq and non-Boussinesq systems, and a wide range of geometric parameters (inner and outer radii, and height). The importance of the neglected viscous terms increases with time, and eventually the inertial–buoyancy model becomes invalid. An estimate for this occurrence is also provided. The predictions of the model are compared to results of direct numerical simulations; there is good agreement for the position of the interface and for the averaged radial velocity, and excellent agreement for ${\mathcal{V}}(t)/{\mathcal{V}}(0)$. A box model is used for estimating the effect of a partial (over a sector) dam break. This study is an extension of the work for a rectangular reservoir of Momen et al. (J. Fluid Mech., vol. 827, 2017, pp. 640–663). We demonstrate that there are some similarities, but also significant differences, between the rectangular and the cylindrical reservoirs concerning the velocity, shape of the interface and rate of drainage, which are of interest in applications. The overall conclusion is that this simple model captures very well the flow field under consideration.