We study the thermocapillary migration of two-dimensional droplets of partially wetting liquids on a non-uniform heated substrate. An equation for the thickness profile of the droplet is derived by employing lubrication approximations. The model includes the effect of a non-zero contact angle introduced through a disjoining–conjoining pressure term. Instead of assuming a fixed shape for the droplet, as in previous works, here we allow the droplet to change its profile with time. We identify and describe three different regimes of behaviour. For small contact angles, the droplet spreads into a long film profile with a capillary ridge near the leading edge, a behaviour that resembles the experiments on Marangoni films reported by Ludviksson & Lightfoot (Am. Inst. Chem. Eng. J., vol. 17, 1971, pp. 1166). For large contact angles, the droplet moves as a single entity, weakly distorted from its static shape. This regime is the usual one reported in experiments on thermocapillary migration of droplets. We also show some intriguing morphologies that appear in the transition between these two regimes. The occurrence of these three regimes and their dependence on various parameters is analysed.

Consideration is given to the flow of an idealized elastico-viscous liquid in a curved pipe under a pressure gradient. By using the series expansion method of Dean (1927, 1928) in powers of a/R where a is the radius of the pipe and R the radius of curvature of its ‘central line’, it is shown that the general nature of the motion is similar to that of the motion of a Newtonian viscous liquid, the liquid elements moving along the pipe in two sets of spirals separated by the central plane. However elasticity of the type considered could strongly affect the pitch of these spirals. To the approximation considered, the flow pattern of the elastico-viscous liquid depends only on the limiting (zero-shear-rate) viscosity η0 and the first moment, K0, of the distribution of relaxation times. The corresponding stress components involve also the second moment of this distribution.

It is also shown that the presence of elasticity in the liquid increases the rate of discharge of the liquid.

Experiments have been performed to examine the turbulence structure and development of a pair of counter-rotating wing-tip vortices. The vortices were generated by two rectangular NACA 0012 half wings placed tip to tip, separated by 0.25 chordlengths. Preliminary studies showed the vortices to be insensitive to the introduction of a probe and subject only to small wandering motions. Meaningful measurements could therefore be made using hot-wire probes. Three-component velocity measurements were made 10 and 30 chordlengths downstream of the wing leading edges for a chord Reynolds number of 260000.

At 10 chordlengths the vortex cores are laminar. True turbulence levels within them are low and vary little with radius. The turbulence that surrounds the cores is formed by the roll-up of and interaction of the wing wakes that spiral around them. This turbulence is stretched and organized but apparently not produced by the circulating mean velocity fields of the vortices.

At 30 chordlengths the vortex cores have become turbulent. True turbulence levels within them are larger and increase rapidly with radius. The turbulent region surrounding the cores has doubled in size and turbulence levels have not diminished, apparently being sustained by outward diffusion from the core regions. The distribution of the turbulence has also changed, the wake spirals having been replaced by a much more core-centred turbulence field.

This change in flow structure contrasts sharply with what is seen in the equivalent isolated tip vortex, produced when one of the wings is removed. Here the vortex core remains laminar and the turbulence surrounding it decays rapidly with downstream distance. This implies that the transition to turbulence in the cores of the vortex pair is stimulated by interaction between the vortices. Spectral measurements at 10 chordlengths suggest that short-wave instability may be the cause.

An investigation of the stability of plane Couette flow with viscous heating of a Navier–Stokes–Pourier fluid with an exponential dependence of viscosity upon temperature is presented. Using classical small perturbation theory, the stability of the flow can be described by a sixth-order set of coupled ordinary differential equations. Using Galerkin's method, these equations are reduced to an algebraic eigenvalue problem. An eigenvalue with a negative real part means that the flow is unstable.

Neutral stability curves are determined at Brinkman numbers of 15, 19, 25, 30,40,80 and 600 for Prandtl numbers of 1, s and 50. A Brinkman number of 19 corresponds approximately to the maximum shear stress which can be applied to the system.

The results indicate that four different modes of instability occur: one termed an inviscid mode, arising from an inflexion point in the primary flow; a viscous mode, due to the stratification of viscosity in the flow field and an associated diffusive mechanism; a coupling mode, resulting from the convective and viscous dissipation terms in the energy equation; and finally a purely thermal mode.

We examine the possibility that the Earth's outer core, as a tidally distorted fluid-filled rotating spheroid, may be the seat of an elliptical instability. The instability mechanism is described within the framework of a simple Earth-like model. The preferred forms of wave disturbance are explored and a likely growth rate supremum deduced. Estimates are made of the Ohmic and viscous decay rates of such hydromagnetic waves in the outer core. Rather than a conclusive disparity of scales, we find that typical elliptical growth rates, Ohmic decay rates and viscous decay rates all have the same order for plausible core fields and core-to-mantle conductivities. This study is all the more timely considering the recent realization that the Earth's precession may also drive similar instabilities at comparable strengths in the outer core.

Kolmogorov flow in two dimensions – the two-dimensional (2D) Navier–Stokes equations with a sinusoidal body force – is considered over extended periodic domains to reveal localised spatiotemporal complexity. The flow response mimics the forcing at small forcing amplitudes but beyond a critical value develops a long wavelength instability. The ensuing state is described by a Cahn–Hilliard-type equation and as a result coarsening dynamics is observed for random initial data. After further bifurcations, this regime gives way to multiple attractors, some of which possess spatially localised time dependence. Co-existence of such attractors in a large domain gives rise to interesting collisional dynamics which is captured by a system of 5 (1-space and 1-time) partial differential equations (PDEs) based on a long wavelength limit. The coarsening regime reinstates itself at yet higher forcing amplitudes in the sense that only longest-wavelength solutions remain attractors. Eventually, there is one global longest-wavelength attractor which possesses two localised chaotic regions – a kink and antikink – which connect two steady one-dimensional (1D) flow regions of essentially half the domain width each. The wealth of spatiotemporal complexity uncovered presents a bountiful arena in which to study the existence of simple invariant localised solutions which presumably underpin all of the observed behaviour.

The effects of incident shock strength on the mixing transition in the Richtmyer–Meshkov instability (RMI) are experimentally investigated using simultaneous density–velocity measurements. This effort uses a shock with an incident Mach number of 1.9, in concert with previous work at Mach 1.55 (Mohaghar et al., J. Fluid Mech., vol. 831, 2017 pp. 779–825) where each case is followed by a reshock wave. Single- and multi-mode interfaces are used to quantify the effect of initial conditions on the evolution of the RMI. The interface between light and heavy gases ($\text{N}_{2}/\text{CO}_{2}$, Atwood number, $A\approx 0.22$; amplitude to wavelength ratio of 0.088) is created in an inclined shock tube at $80^{\circ }$ relative to the horizontal, resulting in a predominantly single-mode perturbation. To investigate the effects of initial perturbations on the mixing transition, a multi-mode inclined interface is also created via shear and buoyancy superposed on the dominant inclined perturbation. The evolution of mixing is investigated via the density fields by computing mixed mass and mixed-mass thickness, along with mixing width, mixedness and the density self-correlation (DSC). It is shown that the amount of mixing is dependent on both initial conditions and incident shock Mach number. Evolution of the density self-correlation is discussed and the relative importance of different DSC terms is shown through fields and spanwise-averaged profiles. The localized distribution of vorticity and the development of roll-up features in the flow are studied through the evolution of interface wrinkling and length of the interface edge, which indicate that the vorticity concentration shows a strong dependence on the Mach number. The contribution of different terms in the Favre-averaged Reynolds stress is shown, and while the mean density-velocity fluctuation correlation term, $\langle \unicode[STIX]{x1D70C}\rangle \langle u_{i}^{\prime }u_{j}^{\prime }\rangle$, is dominant, a high dependency on the initial condition and reshock is observed for the turbulent mass-flux term. Mixing transition is analysed through two criteria: the Reynolds number (Dimotakis, J. Fluid Mech., vol. 409, 2000, pp. 69–98) for mixing transition and Zhou (Phys. Plasmas, vol. 14 (8), 2007, 082701 for minimum state) and the time-dependent length scales (Robey et al., Phys. Plasmas, vol. 10 (3), 2003, 614622; Zhou et al., Phys. Rev. E, vol. 67 (5), 2003, 056305). The Reynolds number threshold is surpassed in all cases after reshock. In addition, the Reynolds number is around the threshold range for the multi-mode, high Mach number case ($M\sim 1.9$) before reshock. However, the time-dependent length-scale threshold is surpassed by all cases only at the latest time after reshock, while all cases at early times after reshock and the high Mach number case at the latest time before reshock fall around the threshold. The scaling analysis of the turbulent kinetic energy spectra after reshock at the latest time, at which mixing transition analysis suggests that an inertial range has formed, indicates power scaling of $-1.8\pm 0.05$ for the low Mach number case and $-2.1\pm 0.1$ for the higher Mach number case. This could possibly be related to the high anisotropy observed in this flow resulting from strong, large-scale streamwise fluctuations produced by large-scale shear.

To quantitatively understand the correlation between the molecular structure of an erythrocyte (red blood cell, RBC) and its mechanical response, and to predict mechanically induced structural remodelling in physiological conditions, we developed a computational model by coupling a multiscale approach of RBC membranes with a boundary element method (BEM) for surrounding Stokes flows. The membrane is depicted at three levels: in the whole cell level, a finite element method (FEM) is employed to model the lipid bilayer and the cytoskeleton as two distinct layers of continuum shells. The mechanical properties of the cytoskeleton are obtained from a molecular-detailed model of the junctional complex. The spectrin, a major protein of the cytoskeleton, is simulated using a molecular-based constitutive model. The BEM model is coupled with the FEM model through a staggered coupling algorithm. Using this technique, we first simulated RBC dynamics in capillary flow and found that the protein density variation and bilayer–skeleton interaction forces are much lower than those in micropipette aspiration, and the maximum interaction force occurs at the trailing edge. Then we investigated mechanical responses of RBCs in shear flow during tumbling, tank-treading and swinging motions. The dependencies of tank-treading frequency on the blood plasma viscosity and the membrane viscosity we found match well with benchmark data. The simulation results show that during tank-treading the protein density variation is insignificant for healthy erythrocytes, but significant for cells with a smaller bilayer–skeleton friction coefficient, which may be the case in hereditary spherocytosis.

A primarily experimental investigation was undertaken to determine the internal structure of steady, quasi-uniform, non-separated, axisymmetric flows in circular pipes with sinusoidal wall profiles. The quantities measured include radial and longitudinal distributions of mean velocity, pressure, and total head; the Reynolds shear stress and all three components of turbulence velocity; and boundary shear stress and pressure. Two different wall-wave steepnesses were investigated, and a constant Reynolds number of 1·13 × 105 (based on the average pipe diameter) was maintained in most experiments. The boundary shear stress was found to be shifted upstream relative to the boundary wave, whereas the wall pressure is shifted slightly downstream. The turbulence measurements revealed that there is a central core extending over some 60% of the pipe radius in which the turbulence quantities are constant along the pipe. Near the boundary, however, the turbulence velocities and stress vary periodically along the boundary waves. The longitudinal component of mean velocity was found to be distributed radially according to the power law, but with an exponent that varies along each wave; a simple analytical model is constructed to predict the variation of the exponent. It was not found possible to relate the local boundary shear stress to just the local flow characteristics, since the convective or ‘history’ effects play a significant role in its determination. An empirical formula is derived relating the local boundary shear stress to the local velocity distribution and the first two derivatives of the boundary profile.

Flexible structures placed within an oncoming flow exhibit far more complex vortex-induced dynamics than flexibly mounted rigid cylinders, because they involve the distributed interaction between the structural and wake dynamics along the entire span. Hence, mapping the well-understood properties of rigid cylinder vibrations to those of strings and beams has been elusive. We show here with a combination of experiments, conducted at Reynolds number, $Re$ from 250 to 2300, and computational fluid dynamics that such a mapping is possible for flexible structures in uniform flow undergoing combined cross-flow and in-line oscillations, but only when additional concepts are introduced to model the extended coupling of the flow and the structure. The in-line response consists of largely standing waves that define cells, each cell spanning the distance between adjacent nodes, over which stable vortical patterns form, whose features (‘2S’ versus ‘P$+$S’) depend strongly on the true reduced velocity, $V_{r}=U/f_{y}d$, where $U$ is the inflow velocity, $f_{y}$ is the cross-flow vibration frequency and $d$ is the cylinder diameter, and the phase angle between in-line and cross-flow response; while the cross-flow response may contain travelling waves, breaking the symmetry of the problem. The axial distribution of the highly variable effective added masses in the cross-flow and in-line directions, and the local phase angle between in-line and cross-flow motion determine the single frequency of cross-flow response, while the in-line response vibrates at twice the cross-flow frequency. The cross-flow and in-line lift coefficients in phase with velocity depend strongly on the true reduced velocity but also on the local phase angle between in-line and cross-flow motions. Modal shapes can be defined for in-line and cross-flow, based on the resemblance of the response to conventional modes, which can be in the ratio of either ‘$2n/n$’ or ‘$(2n-1)/n$’, where $n$ is the order of the cross-flow response mode. We use an underwater optical tracking system to reconstruct the sectional fluid forces in a flexible structure and show that, once the cross-flow and in-line motion features are known, employing strip theory and the hydrodynamic coefficients obtained from forced rigid cylinder experiments allows us to predict the distributed forces accurately.

In this paper we present results from a numerical investigation of turbulent channel flow in the presence of a compliant wall. The compliant wall is modelled as a homogeneous spring-supported plate. The simulation code is validated both by comparison with an alternative code and by reproducing results of linear stability theory. Our results demonstrate that with the wall compliance we used in the simulation there is little change in the very long-time behaviour of the turbulent skin friction drag and little modification to the near-wall turbulent coherent structures. The values of pertinent statistical quantities of the turbulence near the compliant walls converge to those near a rigid wall and the statistical effect of the wall compliance on the turbulent channel flow is small.

Sources of vorticity are examined for a laboratory-generated spilling breaking wave. Two cases are studied. For the first case, based on the breaker height, the Reynolds and Froude numbers are 7370 and 2.04, respectively. The breaker is preceded by 1 mm wavelength capillary waves, with the largest amplitude-to-wavelength ratio equal to 0.18. For this case, it is found that the dominant source of vorticity flux is a viscous process, due to the deceleration of a thin layer of the surface fluid. For the second case, the Reynolds and Froude numbers based on the wave height are 1050 and 1.62, respectively. No breaking is observed for this case; rather a capillary–gravity wave is observed with 4 mm wavelength capillaries preceding the gravity wave. The largest amplitude-to-wavelength ratio of these capillaries is 0.28. This case shows that capillary waves do not contribute to the vorticity flux; rather the only dominant source of the vorticity flux into the flow is the free-surface fluid deceleration.

Lastly, a thin free-surface jet that is relatively vorticity-free is found to precede the spilling breaker. Analyses suggest that our wave-breaking phenomena can be modelled by a hydraulic jump phenomenon where the Froude number is based on the thickness of the free-surface jet, and on the velocity of the free-surface jet just prior to breaking. We believe this to be a more physically descriptive value of the Froude number. For the high-speed case, the Froude number based on the thickness of the free-surface jet is 4.78, while for the lower-speed case it is 2.14.

Nominally zero-pressure-gradient fully developed flat-plate turbulent boundary layers with heated and unheated isothermal walls at supercritical pressures are studied by solving the full compressible Navier–Stokes equations using direct numerical simulation. With a heated isothermal wall, the wall temperature sets such that the flow temperature varies through the pseudo-critical temperature, and thus pseudo-boiling phenomena occur within the boundary layers. The pseudo-boiling process induces strongly nonlinear real-fluid effects in the flow and interacts with near-wall turbulence. The peculiar abrupt density variations through the pseudo-boiling process induce significant near-wall density fluctuations $\sqrt{\overline{\unicode[STIX]{x1D70C}^{\prime }\unicode[STIX]{x1D70C}^{\prime }}}/\overline{\unicode[STIX]{x1D70C}}\approx 0.4{-}1.0$ within the heated transcritical turbulent boundary layers. The large near-wall density fluctuations induce a turbulent mass flux $\unicode[STIX]{x1D70C}^{\prime }u_{i}^{\prime }$, and the turbulent mass flux amplifies the Favre-averaged velocity fluctuations $u_{i}^{\prime \prime }$ in the near-wall predominant structures of streamwise low-speed streaks that are associated with the ejection (where $u^{\prime \prime }<0$ and $v^{\prime \prime }>0$), while reducing the velocity fluctuations in the high-speed streaks associated with the sweep ($u^{\prime \prime }>0$ and $v^{\prime \prime }<0$). Although the near-wall low-speed and high-speed streak structures dominate the Reynolds-shear-stress generation, the energized Favre-averaged velocity fluctuations in the low-speed streaks enhance both the mean-density- and density-fluctuation-related Reynolds shear stresses ($-\overline{\unicode[STIX]{x1D70C}}\overline{u^{\prime \prime }v^{\prime \prime }}$ and $-\overline{\unicode[STIX]{x1D70C}^{\prime }u^{\prime \prime }v^{\prime \prime }}$) in the ejection event and, as a result, alter the Reynolds-shear-stress profile. The large density fluctuations also alter the near-wall viscous-stress profile and induce a near-wall convective flux $-\overline{\unicode[STIX]{x1D70C}}\widetilde{u}\widetilde{v}$ (due to non-zero $\widetilde{v}$). The changes in the contributions in the stress-balance equation result in a failure of existing velocity transformations to collapse to the universal law of the wall. The large density fluctuations also greatly contribute to the turbulent kinetic energy budget, and especially the mass flux contribution term becomes noticeable as one of the main positive terms. The unheated non-transcritical turbulent boundary layers show a negligible contribution of the real-fluid effects, and the turbulence statistics agree well with the statistics of an incompressible constant-property turbulent boundary layer with a perfect-gas law.

Certain two-dimensional, laminar bounday layers are considered whose streamlines are closed. The speed at the solid boundary is supposed uniform, the bounday outline being stationary, and the speed in the boundary layer is supposed to differ only slightly from that of the boundary. A formal solution is then obtained for the motion in the boundary layer. The analysis confirms that a closed boundary layer may exist and yields a condition needed to determine the inviscid motion. The condition is extracted in a simplified but approximate form and two examples of its use are given. A further class of closed boundary layers, namely those for which the pressure is uniform, is also considered. For this class the condition needed to determine the inviscid motion may be derived in a form both simple and exact.

Finite-amplitude, forced and free oscillations of capillary bridges are studied. They are characterized by a resonant frequency and a damping rate which, in turn, depend on fluid properties, dimensions of the bridge, gravitational force relative to surface tension and amplitude of the external disturbance. The Navier–Stokes equations are solved numerically using the Galerkin/finite-element methodology for discretization in space and implicit finite differences with adaptive time stepping for discretization in time. It is found that the resonant frequency decreases and the damping rate increases almost linearly with the oscillation amplitude. Their relative changes from their corresponding values at infinitesimal amplitude depend on fluid properties and dimensions of the bridge. Moreover, careful measurement of the resonant frequency and damping rate in a well-controlled experiment may provide quite accurate values for properties of the liquid over a wide range of modified Reynolds numbers.

Our 1975 paper reported the results of experiments on shock reflexion in a wind tunnel and a shock tube; further results are presented here. For strong shocks it is shown that transition to Mach reflexion takes place continuously at the shock wave incidence angle ω0 corresponding to the normal shock point ω0 = ωN, unless the downstream boundaries form a throat. In this event transition can be promoted anywhere within the range ω0 [les ] ωN, and it is even possible to suppress regular reflexion altogether! However when ω0 < ωN the transition is discontinuous and accompanied by hysteresis. Again for strong shocks evidence is presented which suggests that the famous persistence of regular reflexion beyond the ωN point ω0 > ωN is spurious. For weak shocks the transition condition is not known but it is found that even for regular reflexion a marked discrepancy between theory and experiment develops as the shocks become progressively weaker. Also when weak shocks diffract over single concave corners there is a somewhat surprising discontinuity in the regular reflexion range. It seems that none of these phenomena can be adequately explained by real gas effects such as viscosity and variation of specific heats.

A theory of finite-amplitude secondary flow between concentric rotating cylinders has been published by Davey (1962). A necessary feature of the theory is the generation of harmonics of the spatial periodicity in the axial direction of the velocity field. A method has been devised to measure the amplitude of each harmonic separately and experimental results for the fundamental and first three harmonics are presented here for Taylor numbers up to 100 times the critical value. The agreement with Davey's theory is excellent, and the agreement extends far beyond the range where the theory is expected to be valid. It is shown that all the harmonics are in phase with the fundamental. This result requires that jets and shock-like structure must be present in the velocity field.

The stability of a rotating doubly diffusive fluid is considered. It is shown that (1) a non-rotating layer can be destabilized by rotation, (2) a rotating layer can be destabilized by the addition of a bottom-heavy solute gradient, and (3) under some conditions, three thermal Rayleigh numbers are required to specify linear stability criteria. Numerical results are presented on the basis of which the explanation by Acheson (1979) of the second of these three anomalies can be assessed, and Acheson's explanation is adapted to the two other anomalies.