During water entry, a projectile can entrain an air cavity that trails behind it. Most previous studies focus on the formation and pinch-off dynamics of the air cavity, but only a few have investigated the long-term cavity dynamics after pinch-off. In this study, we examine the ripple formation following the pinch-off of an air cavity generated by a cone, with different cone angles and impact velocities. The amplitude and wavelength of these ripples are measured, and the force on the cone is experimentally determined. It was observed that the ripple amplitude and wavelength increase linearly with the cone impact velocity, which is predicted by our acoustic model of the compressible air cavity. In addition, the measured force exhibits distinct amplitudes and wavelengths. By measuring the length of the cavity, the resulting pressure variation was averaged inside the air cavity leading to a theoretical force amplitude, which matched our observations. We noted that the force wavelength also follows the same acoustic model, which agrees very well with the wavelength of the ripples.

We present a theoretical and numerical study of Fourier-space triad phase dynamics in the one-dimensional stochastically forced Burgers equation at Reynolds number $Re\approx 2.7\times 10^{4}$. We demonstrate that Fourier triad phases over the inertial range display a collective behaviour characterised by intermittent periods of synchronisation and alignment, reminiscent of the Kuramoto model (Chemical Oscillations, Waves, and Turbulence, Springer, 1984) and directly related to collisions of shocks in physical space. These periods of synchronisation favour efficient energy fluxes across the inertial range towards small scales, resulting in strong bursts of dissipation and enhanced coherence of the Fourier energy spectrum. The fast time scale of the onset of synchronisation relegates energy dynamics to a passive role: this is further examined using a reduced system with the Fourier amplitudes fixed in time – a phase-only model. We show that intermittent triad phase dynamics persists without amplitude evolution and we broadly recover many of the characteristics of the full Burgers system. In addition, for both full Burgers and phase-only systems the physical-space velocity statistics reveals that triad phase alignment is directly related to the non-Gaussian statistics typically associated with structure-function intermittency in turbulent systems.

We report an experimental investigation of the wake of free falling super-hydrophobic spheres. The mutual interaction between the air layer (plastron) encapsulating the super-hydrophobic spheres and the flow is emphasised by studying the hydrodynamic performance. It is found that the air plastron adapts its shape to the flow-induced stresses which compete with the surface tension. This competition is characterised by introducing the Weber number ${\mathcal{W}}e$, whilst the plastron deformation is estimated via the aspect ratio $\unicode[STIX]{x1D712}$. While noticeable distortions are locally observed, the plastron becomes more and more spherical on average (i.e. $\unicode[STIX]{x1D712}\rightarrow 1$) as far as ${\mathcal{W}}e$ increases. The study of the falling motion reveals that the plastron compliance has a sizeable influence on the wake development. Investigating the lift force experienced by the super-hydrophobic spheres, the onset of wake instabilities is found to be triggered earlier than for smooth spheres used as reference. Surprisingly, it is also observed that the early promotion of the wake instabilities is even more pronounced beyond a critical Weber number, ${\mathcal{W}}e_{c}$, which corresponds to a critical aspect ratio $\unicode[STIX]{x1D712}_{c}$. Furthermore, the magnitude of the hydrodynamic loads is found to be dependent on the average deformation of the gas/liquid interface. Indeed, in comparison to the reference spheres, the high deformation achieved for $\unicode[STIX]{x1D712}>\unicode[STIX]{x1D712}_{c}$ (oblate shape) leads to lift and drag increase, whereas the low deformation obtained for $\unicode[STIX]{x1D712}<\unicode[STIX]{x1D712}_{c}$ (spherical shape) yields lift and drag mitigation. Accordingly, taking into account the plastron deformation provides an attractive way to explain the somehow discordant results reported in other studies at comparable Reynolds numbers. These results suggest that the amount of vorticity produced at the body surface and then released in the wake is strongly impacted by the plastron compliance. If confirmed by additional studies and extrapolated to other flow configurations, our findings would imply that plastron compliance and its feedback on the flow, which are currently neglected in most theoretical works and numerical simulations, must be accounted for to design super-hydrophobic surfaces and/or predict their performance.

A spherical blast wave generated by a sudden release of a sphere of compressed gas is an important model problem to understand blast phenomena such as volcanic eruptions and explosive detonations. The resulting explosion flow physics, such as the instability at the gas contact discontinuity and the interaction between the shock wave and the gas contact, are dictated by the initial pressure and sound-speed ratios between the compressed gas and the ambience. Since the initial pressure and sound-speed ratios vary over a wide range in practical applications, it is of interest to investigate the scaling laws and similarity solutions for the spherical symmetric explosion flow. In the present study, numerical simulation of a spherical blast wave is performed. A long-term length scale that incorporates the initial charge radius and the initial pressure ratio is introduced. The trajectories of the main shock normalized by the long-term length scale for a wide range of parameters collapse after a short transition time, indicating an asymptotic similarity solution exists for the far field in the long term. With the assistance of this similarity solution, the full evolution of the main shock can be obtained semi-analytically. For near-field features, i.e. the gas contact and the secondary shock wave, only semi-similarity solutions are observed, which depend on the initial sound-speed ratio but not the initial pressure ratio. The gas contact and the secondary shock share the same scaling relations. Asymptotic analysis is performed to obtain the short-term dynamics of the gas contact, including the gas contact acceleration and the Atwood number, which are the key parameters determining the Rayleigh–Taylor instability development at the gas contact. The asymptotic contact radius as $t\rightarrow \infty$ is also obtained, which is found to be well represented by the long-term length scale and thus only depends on the initial pressure ratio. A simple model of an oscillating bubble is employed to explain the scaling relation of the asymptotic gas contact radius.

Understanding the dynamics of a droplet pushed by an external fluid in a confined geometry calls for the identification of all the dissipation mechanisms at play in the lubrication film between droplet and cell wall. Experimentally, reflection interference contrast microscopy has proven an efficient tool to measure the thickness of such lubrication films for microfluidic droplets, with a precision of a few nanometres (Huerre et al., Lab on a Chip, vol. 16 (5), 2016, pp. 911–916). The present work takes advantage of the high accuracy of this technique to chart quantitatively the lubrication film between oil droplets and the glass wall of a microfluidic chamber. We find that the lubrication films exhibit a complex three-dimensional shape, which we are able to rationalize using a hydrodynamical model in the lubrication approximation. We show that the complete topography cannot be recovered using a single model boundary condition along the whole interface. Rather, surface tension gradients are negligible at the front of the droplet, whereas they significantly modify the film profile at the rear, where surfactant accumulation induces local thickening of the lubrication film. The presence of ravines on the sides of the droplet is due to three-dimensional effects which can be qualitatively reproduced numerically. To our knowledge, this is the first experimental investigation of such local effects on travelling droplets.

Direct numerical simulations are used to examine large-scale motions with a streamwise length $2\sim 4h$ ($h$ denotes the channel half-width) in the logarithmic and outer regions of a turbulent channel flow. We test a minimal ‘streamwise’ flow unit (Toh & Itano, J. Fluid Mech., vol. 524, 2005, pp. 249–262) (or MSU) for larger Kármán numbers ($h^{+}=395$ and 1020) than in the original work. This flow unit consists of a sufficiently long (${L_{x}}^{+}\approx 400$) streamwise domain to maintain near-wall turbulence (Jiménez & Moin, J. Fluid Mech., vol. 225, 1991, pp. 213–240) and a spanwise domain which is large enough to represent the spanwise behaviour of inner and outer structures correctly; as $h^{+}$ increases, the streamwise extent of the MSU domain decreases with respect to $h$. Particular attention is given to whether the spanwise organization of the large-scale structures may be represented properly in this simplified system at sufficiently large $h^{+}$ and how these structures are associated with the mean streamwise velocity $\overline{U}$. It is shown that, in the MSU, the large-scale structures become approximately two-dimensional at $h^{+}=1020$. In this case, the streamwise velocity fluctuation $u$ is energized, whereas the spanwise velocity fluctuation $w$ is weakened significantly. Indeed, there is a reduced energy redistribution arising from the impaired global nature of the pressure, which is linked to the reduced linear–nonlinear interaction in the Poisson equation (i.e. the rapid pressure). The logarithmic dependence of $\overline{ww}$ is also more evident due to the reduced large-scale spanwise meandering. On the other hand, the spanwise organization of the large-scale $u$ structures is essentially identical for the MSU and large streamwise domain (LSD). One discernible difference, relative to the LSD, is that the large-scale structures in the MSU are more energized in the outer region due to a reduced turbulent diffusion. In this region, there is a tight coupling between neighbouring structures, which yields antisymmetric pairs (with respect to centreline) of large-scale structures with a spanwise spacing of approximately $3h$; this is intrinsically identical with the outer energetic mode in the optimal transient growth of perturbations (del Álamo & Jiménez, J. Fluid Mech., vol. 561, 2006, pp. 329–358).

We investigate experimentally and theoretically how a droplet of a binary mixture evaporates when placed on a solid substrate. Our focus is the limit at which the two liquid components have different vapour pressures. Using physicochemical effects, we directly visualize the selective evaporation of the more volatile component and so document the space and time dependence of the chemical distribution in the droplet. In particular, we observe that a mixture consisting of an organic solvent and deionized water dissolves suspended fluorescent polystyrene particles if the lower volatility organic solvent reaches a critical concentration. Consequently, we show that for a small contact angle ($\unicode[STIX]{x1D703}<\unicode[STIX]{x03C0}/2$) the suspended polystyrene particles begin to disappear from near the contact line, which indicates that the volatile component, here water, evaporates rapidly compared to the other component(s). Finally, we show that a diffusion-dominated model for evaporation of a binary mixture can predict well the experimental results where convective and diffusive mixing effects are negligible, in which case there is significant chemical segregation in the drop.

In this paper we study the essential role of Darrieus–Landau (DL), hydrodynamic, cellular flame-front instability in the propagation of expanding turbulent flames. First, we analyse and compare the characteristic time scales of flame wrinkling under the simultaneous actions of DL instability and turbulent eddies, based on which three turbulent flame propagation regimes are identified, namely, instability dominated, instability–turbulence interaction and turbulence dominated regimes. We then perform experiments over an extensive range of conditions, including high pressures, to promote and manipulate the DL instability. The results clearly demonstrate the increase in the acceleration exponent of the turbulent flame propagation as these three regimes are traversed from the weakest to the strongest, which are respectively similar to those of the laminar cellularly unstable flame and the turbulent flame without flame-front instability, and thus validating the scaling analysis. Finally, based on the scaling analysis and the experimental results, we propose a modification of the conventional turbulent flame regime diagram to account for the effects of DL instability.

We study experimentally the dynamics and statistics of capillary waves forced by random steep gravity waves mechanically generated in the laboratory. Capillary waves are produced here by gravity waves from nonlinear wave interactions. Using a spatio-temporal measurement of the free surface, we characterize statistically the random regimes of capillary waves in the spatial and temporal Fourier spaces. For a significant wave steepness (0.2–0.3), power-law spectra are observed both in space and time, defining a turbulent regime of capillary waves transferring energy from the large scale to the small scale. Analysis of temporal fluctuations of the spatial spectrum demonstrates that the capillary power-law spectra result from the temporal averaging over intermittent and strong nonlinear events transferring energy to the small scale in a fast time scale, when capillary wave trains are generated in a way similar to the parasitic capillary wave generation mechanism. The frequency and wavenumber power-law exponents of the wave spectra are found to be in agreement with those of the weakly nonlinear wave turbulence theory. However, the energy flux is not constant through the scales and the wave spectrum scaling with this flux is not in good agreement with wave turbulence theory. These results suggest that theoretical developments beyond the classic wave turbulence theory are necessary to describe the dynamics and statistics of capillary waves in a natural environment. In particular, in the presence of broad-scale viscous dissipation and strong nonlinearity, the role of non-local and non-resonant interactions should be reconsidered.

We study how the properties of forcing and dissipation affect the scaling behaviour of the vortex population in the two-dimensional turbulent inverse energy cascade. When the flow is forced at scales intermediate between the domain and dissipation scales, the growth rates of the largest vortex area and the spectral peak length scale are robust to all simulation parameters. For white-in-time forcing the number density distribution of vortex areas follows the scaling theory predictions of Burgess & Scott (J. Fluid Mech., vol. 811, 2017, pp. 742–756) and shows little sensitivity either to the forcing bandwidth or to the nature of the small-scale dissipation: both narrowband and broadband forcing generate nearly identical vortex populations, as do Laplacian diffusion and hyperdiffusion. The greatest differences arise in comparing simulations with correlated forcing to those with white-in-time forcing: in flows with correlated forcing the intermediate range in the vortex number density steepens significantly past the predicted scale-invariant $A^{-1}$ scaling. We also study the impact of the forcing Reynolds number $Re_{f}$, a measure of the relative importance of nonlinear terms and dissipation at the forcing scale, on vortex formation and the scaling of the number density. As $Re_{f}$ decreases, the flow changes from one dominated by intense circular vortices surrounded by filaments to a less structured flow in which vortex formation becomes progressively more suppressed and the filamentary nature of the surrounding vorticity field is lost. However, even at very small $Re_{f}$, and in the absence of intense coherent vortex formation, regions of anomalously high vorticity merge and grow in area as predicted by the scaling theory, generating a three-part number density similar to that found at higher $Re_{f}$. At late enough stages the aggregation process results in the formation of long-lived circular vortices, demonstrating a strong tendency to vortex formation, and via a route distinct from the axisymmetrization of forcing extrema seen at higher $Re_{f}$. Our results establish coherent vortices as a robust feature of the two-dimensional inverse energy cascade, and provide clues as to the dynamical mechanisms shaping their statistics.

We consider a passive zero-mean scalar field organised into two layers of different concentrations in a three-dimensional plane channel flow subjected to a constant along-stream pressure gradient. We employ a nonlinear direct-adjoint-looping method to identify the optimal initial perturbation of the velocity field with given initial energy which yields ‘maximal’ mixing by a target time horizon, where maximal mixing is defined here as the minimisation of the spatially integrated variance of the concentration field. We verify in three-dimensional flows the conjecture by Foures et al. (J. Fluid Mech., vol. 748, 2014, pp. 241–277) that the initial perturbation which maximises the time-averaged energy gain of the flow leads to relatively weak mixing, and is qualitatively different from the optimal initial ‘mixing’ perturbation which exploits classical Taylor dispersion. We carry out the analysis for two different Reynolds numbers ($Re=U_{m}h/\unicode[STIX]{x1D708}=500$ and $Re=3000$, where $U_{m}$ is the maximum flow speed of the unperturbed flow, $h$ is the channel half-depth and $\unicode[STIX]{x1D708}$ is the kinematic viscosity of the fluid) demonstrating that this key finding is robust with respect to the transition to turbulence. We also identify the initial perturbations that minimise, at chosen target times, the ‘mix-norm’ of the concentration field, i.e. a Sobolev norm of negative index in the class introduced by Mathew et al. (Physica D, vol. 211, 2005, pp. 23–46). We show that the ‘true’ variance-based mixing strategy can be successfully and practicably approximated by the mix-norm minimisation since we find that the mix-norm-optimal initial perturbations are far less sensitive to changes in the target time horizon than their optimal variance-minimising counterparts.

We consider the asymptotic structure of a steady developed viscous thin film passing the sharp trailing edge of a horizontally aligned flat plate under the weak action of gravity acting vertically and surface tension. The surprisingly rich details of the flow in the immediate vicinity of the trailing edge are elucidated both analytically and numerically. As a central innovation, we demonstrate how streamline curvature serves to regularise the edge singularity apparent on larger scales via generic viscous–inviscid interaction. This is shown to be provoked by weak disturbances of accordingly strong exponential downstream growth, which we trace from the virtual origin of the flow towards the trailing edge. They represent a prototype of the precursor to free interaction in the most general sense, which, interestingly, has not attracted due attention previously. Moreover, we delineate how an increased effect of gravity involves marginally choked flow at the edge.

Our previous research has shown that an active flow control strategy using two-dimensional (2-D) harmonic blowing and suction with properly chosen frequency and amplitude can significantly reduce the separation region, delay transition to turbulence and can even relaminarize the flow. How such effective flow control for transition delay and relaminarization is affected by free-stream turbulence (FST) remains an open question. In order to answer this question, highly resolved direct numerical simulations (DNS) are carried out where very low-amplitude isotropic FST fluctuations are introduced at the inflow boundary of the computational domain. With FST the effectiveness of the flow control is not diminished, and the extent of the separated flow region is reduced by the same amount as for the zero FST case. However, a striking difference observed in the DNS is the fact that in the presence of even very low levels of FST, the flow transitions shortly downstream of the reattachment location of the bubble, contrary to the case without FST. It appears that this different behaviour for even very small levels of FST is caused by an interaction between the high-amplitude 2-D disturbances introduced by the flow control forcing and 3-D Klebanoff modes (K-modes) that are generated by the FST. The streamwise elongated streaks due to the K-modes cause a spanwise-periodic modulation of the basic flow and subsequently of the primary 2-D wave. The disturbances associated with this modulation exhibit strong growth and initiate the breakdown process to turbulence. Linear secondary instability investigations with respect to low-frequency 3-D disturbances are carried out based on the linearized Navier–Stokes equations. The response of the forced flow to the low-frequency 3-D disturbances reveals that the time-periodic base flow is unstable with respect to a wide range of 3-D modes. In particular, the wavelength associated with the spanwise spacing of the K-mode falls into the range of, and is in fact very close to, the most unstable 3-D disturbances. Results from the secondary instability analysis and the comparison with DNS results, support the conjecture that the forcing amplitude has a major impact on the onset and amplification rate of the K-modes: lowering the forcing amplitude postpones the onset of the growth of the K-modes and reduces the growth rate of the K-modes for a given FST intensity. The net effect of these two events is a delay of the transition onset. Nevertheless, the instability mechanism that governs the transition process is the same as previously identified, i.e. interaction of the K-mode and 2-D primary wave. Furthermore, for low levels of FST, the amplitude of the low-frequency K-modes scales linearly with the FST intensity in the approach boundary layer up to the secondary instability regime.

Any tilt of a Rayleigh–Bénard convection cell against gravity changes the global flow structure inside the cell, which leads to a change of the heat and momentum transport. Especially sensitive to the inclination angle is the heat transport in low-Prandtl-number fluids and confined geometries. The purpose of the present work is to investigate the global flow structure and its influence on the global heat transport in inclined convection in a cylindrical container of diameter-to-height aspect ratio $\unicode[STIX]{x1D6E4}=1/5$. The study is based on direct numerical simulations where two different Prandtl numbers $Pr=0.1$ and 1.0 are considered, while the Rayleigh number, $Ra$, ranges from $10^{6}$ to $10^{9}$. For each combination of $Ra$ and $Pr$, the inclination angle is varied between 0 and $\unicode[STIX]{x03C0}/2$. An optimal inclination angle of the convection cell, which provides the maximal global heat transport, is determined. For inclined convection we observe the formation of two system-sized plume columns, a hot and a cold one, that impinge on the opposite boundary layers. These are related to a strong increase in the heat transport.

This study examines the transition of stable falling to tumbling flight for freely falling heavy plates in a two-dimensional viscous fluid, solved via direct numerical simulation with the immersed boundary method. The simulations are performed at a range of Reynolds number ($Re$) of up to 500 and a dimensionless moment of inertia ($I^{\ast }$) up to 10. It is found that a plate may settle to stable falling or develop into tumbling descent depending on the initial angle of release $\unicode[STIX]{x1D703}_{0}$. The characteristics and performance that distinguish two flight states are investigated. This bistability is analysed with phase portraits and the region mapped across the regime of $I^{\ast }$ and $Re$ at a specific thickness ratio. In determining the flight state, the respective critical $\unicode[STIX]{x1D703}_{0}$ is found to follow a power law through $I^{\ast }$ and $Re$. It is suggested that the changing slope of the lift curve that the plate undergoes sets the two flight states apart. Flow fields also reveal that the recirculation behind the plate is confined by the vortex structures and provides an additional rotation to the plate. An experiment is performed suggesting that bistability also occurs at Re ${\sim}O(10^{4})$. Other shapes are also simulated and the different bistable effects are discussed.

Plumes generated by vertically distributed sources of buoyancy have been observed to have substantially lower entrainment coefficients than their equivalent-geometry constant buoyancy flux plumes. Two differences between distributed and localized sources of buoyancy are the presence of a wall shear stress at the source and that non-ideal source conditions are distributed over the whole height of the enclosure for a vertically distributed source. Herein the impact of non-ideal source and boundary conditions on vertically distributed plumes is analysed. It is shown that, at small heights, the plume volume flow rate is significantly influenced by the wall-source volume flux. At larger heights the wall-source buoyancy is greater than the mean plume buoyancy, creating a non-self-similar horizontal buoyancy distribution within the plume. Recent experiments into the behaviour of a vertically distributed source of buoyancy in a confined region have also shown that the plume partially detrains in the stratified region of the enclosure. This detrainment has not been observed for constant buoyancy flux plumes in a confined region. Although models have been proposed to quantify the detrainment process, it is still unclear why vertically distributed buoyancy sources detrain while constant buoyancy flux plumes do not in the same physical geometry. The impact of source and boundary effects on previously published experiments on vertically distributed plumes are reviewed and the possible implications for plume entrainment and detrainment are discussed.

While the classical problem of a flat plate impact on a water surface at zero dead-rise angle has been studied for a long time both theoretically and experimentally, it still presents a number of challenges and unsolved questions. Hitherto, the details of the flow field – especially at early times and close to the plate edge, where the classical inviscid theory predicts a singularity in the velocity field and thus in the free surface deflection, so-called ejecta – have not been studied experimentally, which led to mutually contradicting suppositions in the literature. On one hand, it motivated Yakimov’s self-similar scaling near the plate edge. On the other hand, a removal of the singularity was previously suggested with the help of the Kutta–Joukowsky condition at the plate edge, i.e. enforcing the free surface to depart tangentially to the plate. In the present experimental study we were able to overcome challenges with optical access and investigate, for moderate Reynolds ($0.5<Re<25\,000$) and Weber ($1<We<800$) numbers, both the flow fields and the free surface dynamics at the early stage of the water impact, when the penetration depth is small compared to the plate size, thus allowing us to compare to the classical water impact theory valid in the short time limit. This, in particular, enabled us to uncover the effects of viscosity and surface tension on the velocity field and ejecta evolution usually neglected in theoretical studies. While we were able to confirm the far-field inviscid and the near-edge Stokes theoretical scalings of the free surface profiles, Yakimov’s scaling of the velocity field proved to be inapplicable and the Kutta–Joukowsky condition not satisfied universally in the studied range of Reynolds and Weber numbers. Since the local near-edge phenomena cannot be considered independently of the complete water impact event, the experiments were also set up to study the entirety of the water impact phenomena under realistic conditions – presence of air phase and finite depth of penetration. This allowed us to obtain insights also into other key aspects of the water impact phenomena such as air entrapment and pocketing at the later stage when the impactor bottoms out. In our experiments the volume of trapped air proved not to decrease necessarily with the impact speed, an effect that has not been reported before. The observed fast initial retraction of the trapped air film along the plate bottom turned out to be a consequence of a negative pressure impulse generated upon the abrupt deceleration of the plate. This abrupt deceleration is also the cause of the subsequent air pocketing. Quantitative measurements are complemented with basic scaling models explaining the nature of both retraction of the trapped air and air pocket formation.

The behavioural characteristics of thermal boundary layer inception dictate the efficiency of heat exchangers and the operational limits of fluid machinery. The specific time required by the thermal boundary layer to be established is vital to optimize flow control strategies, as well as the thermal management of systems exposed to ephemeral phenomena, typically on the millisecond scale. This paper presents the time characterization of the momentum and thermal boundary layer development in transient turbulent compressible air flows. We present a new framework to perform such estimations based on detailed unsteady Reynolds averaged Navier–Stokes simulations that may be extended to higher fidelity simulations. First of all, the aerodynamic boundary layer initiation is described using adiabatic simulations. Additional numerical calculations were then performed by setting the isothermal wall condition to evaluate the additional time required by the thermal boundary layer to establish after the aerodynamic boundary layer reaches its steady state. Finally, full conjugate simulations were executed to compute the warm up effect of the solid during the blowdown of a hot fluid over a colder metallic test model. The transient performance of the turbulent thermal and momentum boundary layers is quantified through numerical simulations of air blowdown over a flat plate for different mainstream flow conditions. The effects of Reynolds number, free stream velocity, transient duration, test article length and free stream temperature were independently assessed, to then define a mathematical expression of the momentum boundary layer settlement. This paper presents a novel numerical correlation of the additional time required by the thermal boundary layer to be stablished after the settlement of the momentum boundary layer. The time scales of the aerodynamic and thermal boundary layers are presented as a function of relevant non-dimensional numbers, as well as the description of the response of the near wall flow to sudden free stream changes. The characterization of the boundary layer mechanisms discussed in this paper contribute to the establishment of an evidence-based foundation for advances in the field of flow control.

Intricate manipulation of droplets in fluidic confinements may turn out to be critically important for achieving their controlled transverse distributions. Here, we study the migration characteristics of a suspended deformable droplet in a parallel plate channel under the combined influence of a constant temperature gradient in the transverse direction and an imposed pressure driven flow. An outstanding question concerning the resultant non-trivial dynamical features that we address here pertains to the nonlinearity that results as a consequence of the shape deformation, which does not permit us to analyse the combined transport as a mere linear superposition of the results for the thermocapillary and imposed flow driven droplet migration in an effort to obtain the final solution. For the analytical solution, an asymptotic approach is used, where we neglect any effect of inertia or thermal convection of the fluid in either of the phases. To obtain a numerical solution, we use the conservative level set method. We perform numerical simulations over a wide range of governing parameters and obtain the dependence of the transverse steady position of the droplet on different parameters. In order to address practical microfluidic set-ups, the influence of a bounding wall as well as the effect of thermal convection and finite shape deformation on the cross-stream migration of the droplet is investigated through numerical simulations. Increase in the thermal Marangoni stress shifts the steady-state transverse position of the droplet further away from the channel centreline, for any particular value of the capillary number (which signifies the ratio of the viscous force to the surface tension force). The confinement ratio, which is the ratio of the droplet radius to the channel height, plays an important role in predicting the transverse position of the droplet and thus has immense consequences for the design of droplet-based microfluidic devices with enhanced functionalities. A large confinement ratio drives the droplet towards the channel centre, whereas a smaller confinement ratio causes the droplet to move towards the wall. Moreover, for a fixed droplet radius and constant imposed temperature gradient, an increase in the channel height results in an increase in the time required for the droplet to reach the steady-state position. However, the final steady-state position of the droplet is independent of its initial position but at the same time dependent on the droplet phase thermal conductivity. A larger droplet thermal conductivity compared with the carrier phase results in a steady-state droplet position closer to the channel centreline. A higher fluid inertia, on the other hand, shifts the steady-state position towards the channel wall.

The incompressible Stokes equations can classically be recast in a boundary integral (BI) representation, which provides a general method to solve low-Reynolds-number problems analytically and computationally. Alternatively, one can solve the Stokes equations by using an appropriate distribution of flow singularities of the right strength within the boundary, a method that is particularly useful to describe the dynamics of long slender objects for which the numerical implementation of the BI representation becomes cumbersome. While the BI approach is a mathematical consequence of the Stokes equations, the singularity method involves making judicious guesses that can only be justified a posteriori. In this paper, we use matched asymptotic expansions to derive an algebraically accurate slender-body theory directly from the BI representation able to handle arbitrary surface velocities and surface tractions. This expansion procedure leads to sets of uncoupled linear equations and to a single one-dimensional integral equation identical to that derived by Keller & Rubinow (J. Fluid Mech., vol. 75, 1976, p. 705) and Johnson (J. Fluid Mech., vol. 99, 1979, p. 411) using the singularity method. Hence, we show that it is a mathematical consequence of the BI approach that the leading-order flow around a slender body can be represented using a distribution of singularities along its centreline. Furthermore, when derived from either the single-layer or the double-layer modified BI representation, general slender solutions are only possible in certain types of flow, in accordance with the limitations of these representations.