A global analysis of steady states of inviscid compressible subsonic swirling flows in a finite-length straight circular pipe is developed. A nonlinear partial differential equation for the solution of the flow stream function is derived in terms of the inlet flow specific total enthalpy, specific entropy and circulation functions. The equation reflects the complicated thermo–physical interactions in the flows. Several types of solutions of the resulting nonlinear ordinary differential equation for the columnar case together with a flow force condition describe the outlet state of the flow in the pipe. These solutions are used to form the bifurcation diagram of steady compressible flows with swirl as the inlet swirl level is increased at a fixed inlet Mach number. The approach is applied to two profiles of inlet flows, solid-body rotation and the Lamb–Oseen vortex, both with a uniform axial velocity and temperature. The computed results provide for each inlet flow profile theoretical predictions of the critical swirl levels for the appearance of vortex breakdown states as a function of the inlet Mach number, suggesting that the results are robust for a variety of inlet swirling flows. The analysis sheds light on the dynamics of compressible flows with swirl and vortex breakdown, and shows the delay in the appearance of breakdown with increase of the inlet axial flow Mach number in the subsonic range of operation. The present theory is limited to axisymmetric dynamics of swirling flows in pipes where the wall boundary layer is thin and attached and does not interact with the flow in the bulk.

The present study deals with a theoretical investigation of a close-contact melting (CCM) process involving a vertical cylinder on a horizontal isothermal surface, where the liquid phase is a non-Newtonian viscoplastic fluid that behaves according to the Bingham model. Accordingly, a new approach is formulated based on the thin layer approximation and different quasi-steady process assumptions. By analytical derivation, an algebraic equation that relates the molten layer thickness and the solid bulk height is developed. The problem is then solved numerically, coupled with another equation for the melting rate. The new model shows that as the yield stress increases the melting rate decreases and the molten layer thickness increases. It is found that under certain conditions, the model can be reduced to a form that allows an analytical solution. The approximate model predicts an exponential dependence of both the melt fraction and the molten layer thickness. Comparison between the numerical and analytical solutions shows that the analytical approximation provides an excellent estimation for sufficiently large values of the yield stress. Dimensional analysis, which is supported by the analytical model results, reveals the dimensionless groups that govern the problem. For the general case, the melt fraction is a function of two dimensionless groups. For the analytical approximation, it is shown that the melt fraction is governed by a single dimensionless group and that the molten layer thickness is governed by two dimensionless groups.

Taylor—Couette flow is ideal for studying drag-reducing polymer additives because, unlike turbulent boundary layers, the instabilities are better understood. Video records of laser-induced fluorescence experiments with and without polymers will be presented. Polyethylene-oxide (PEO) ‘oceans’ were used in concentrations of 20 and 100 p.p.m. In the Taylor number range, 3 × 104 ≤ Ta ≤ 108, Newtonian flow consisted of Taylor vortices which span the gap between cylinders and much smaller Görtler vortices at the inner cylinder wall. Measurements of core-to-core separation between counter-rotating vortices were made to estimate the Görtler instability wavenumber. These measurements show that PEO addition increases the Görtler wavenumber for a given Taylor number. At the lower Taylor numbers, Görtler vortex formation was suppressed by PEO. This implies that polymers directly affect the evolution of centrifugal instabilities.

We study the motion of a buoyant or a nearly neutrally buoyant nano-sized spheroid in a fluid filled tube without or with an imposed pressure gradient (weak Poiseuille flow). The fluctuating hydrodynamics approach and the deterministic method are both employed. We ensure that the fluctuation–dissipation relation and the principle of thermal equipartition of energy are both satisfied. The major focus is on the effect of the confining boundary. Results for the velocity and the angular velocity autocorrelations (VACF and AVACF), the diffusivities and the drag and the lift forces as functions of the shape, the aspect ratio, the inclination angle and the proximity to the wall are presented. For the parameters considered, the boundary modifies the VACF and AVACF such that three distinct regimes are discernible – an initial exponential decay followed by an algebraic decay culminating in a second exponential decay. The first is due to the thermal noise, the algebraic regime is due both to the thermal noise and the hydrodynamic correlations, while the second exponential decay shows the effect of momentum reflection from the confining wall. Our predictions display excellent comparison with published results for the algebraic regime (the only regime for which earlier results exist). We also discuss the role of the off-diagonal elements of the mobility and the diffusivity tensors that enable the quantifications of the degree of lift and margination of the nanocarrier. Our study covers a range of parameters that are of wide applicability in nanotechnology, microrheology and in targeted drug delivery.

The purpose of this note is to comment on the calculation of equilibrium gas dynamic parameters behind strong shock waves. The writer has been aroused by the appearance of a paper by Guman (1956) presenting a generalized computing scheme for ionizing shock waves in monatomic gases. In that paper the reader is not cautioned about including excited states in the Saha equation for the computation of the degree of ionization behind shock fronts at appropriate temperatures and densities. The same paper treats the adiabatic exponent γ = cp/cv as constant across strong shocks when at the same time it is implied that the computing scheme is of general validity. Hence, the unwary reader might attempt to apply the scheme in a regime where γ is not only no longer constant but is no longer a useful quantity for characterizing the shock conditions. Other authors (see, for example, Glass, Martin & Patterson (1953)) characterize flows in which a shock has excited internal degrees of freedom in terms of a variable specific heat ratio when in fact one cannot use this quantity in calculating shock front conditions.

The stability of streamwise corner flow is investigated by means of direct numerical simulation at subcritical Reynolds numbers. The flow is harmonically forced, and global modes are extracted through a spectral decomposition. Spatial amplification in the near-corner region is observed even though the flow is shown to be subcritical in terms of spatial linear theory. This apparent discrepancy is resolved by extending the local analysis to include non-modal effects. It is demonstrated that the amplification is a result of the interaction between two coexistent spatial transient growth processes that can be associated with different parts of the linear stability spectrum. A detailed investigation of the underlying mechanisms shows that the transient amplification behaviour is caused by pseudo-resonance between the inviscid corner mode, and different sets of viscous modes. By comparison with studies of other locally inflectional flows, it is found that viscid–inviscid pseudo-resonance might be a general phenomenon leading to selective noise amplification.

Amplifications of flow past a backward-facing step with respect to optimal inflow and initial perturbations are investigated at Reynolds number 500. Two mechanisms of receptivity to inflow noise are identified: the bubble-induced inflectional point instability and the misalignment effect downstream of the secondary bubble. Further development of the misalignment results in decay of perturbations from $x=28$ onwards (the step is located at $x=0$), as has been observed in previous non-normality studies (Blackburn et al., J. Fluid Mech., vol. 603, 2008, pp. 271–304), and eventually limits the receptivity. The receptivity is found to be maximized at an inflow perturbation frequency of ${\it\omega}=0.50$ and a spanwise wavenumber of ${\it\beta}=0$, where the inflow noise takes full advantage of both mechanisms and is amplified over two orders of magnitude in terms of the velocity magnitude. In direct numerical simulations (DNS) of the flow perturbed by optimal or random inflow noise, vortex shedding, flapping of bubbles, three-dimensionality and turbulence are observed in succession as the magnitude of the inflow noise increases. Similar features of linear and nonlinear receptivity are observed at higher Reynolds numbers. The Strouhal number of the bubble flapping is 0.08, at which the receptivity to inflow noise reaches a maximum. This Strouhal number is close to reported values extracted from DNS or large eddy simulations (LES) at larger Reynolds numbers (Le et al., J. Fluid Mech., vol. 330, 1997, pp. 349–374; Kaiktsis et al., J. Fluid Mech., vol. 321, 1996, pp. 157–187; Métais, New Trends in Turbulence, 2001, Springer; Wee et al., Phys. Fluids, vol. 16, 2004, pp. 3361–3373). Methods to further clarify the mechanisms of receptivity and to suppress the noise amplifications by modifying the base flow using a linearly optimal body force are proposed. It is observed that the mechanisms of optimal noise amplification are fully revealed by the distribution of the base flow modification, which weakens the bubble instabilities and misalignment effects and subsequently reduces the receptivity significantly. Comparing the base flow modifications with respect to amplifications of inflow and initial perturbations, it is found that the maximum receptivity to initial perturbations is highly correlated with the receptivity to inflow noise at the optimal frequency ${\it\omega}=0.50$, and the correlation reduces as the inflow frequency deviates from this optimal value.

Finite-amplitude bifurcation diagrams of axisymmetric liquid bridges anchored between two plane parallel electrodes subjected to a potential difference and in the presence of an axial gravity field are found by solving simultaneously the Laplace equation for the electric potential and the Young–Laplace equation for the interface by means of the Galerkin/finite element method. Results show the strong stabilizing effect of the electric field, which plays a role somewhat similar to the inverse of the slenderness. It is also shown that the electric field may determine whether the breaking of the liquid bridge leads to two equal or unequal drops. Finally, the sensitivity of liquid bridges to an axial gravity in the presence of the electric field is studied.

This paper considers nonlinear acoustic waves propagating unidirectionally in a gas-filled tube under an axial temperature gradient, and examines whether the energy flux of the waves can be amplified by thermoacoustic effects. An array of Helmholtz resonators is connected to the tube axially to avoid shock formation which would otherwise give rise to nonlinear damping of the energy flux. The amplification is expected to be caused by action of the boundary layer doing reverse work, in the presence of the temperature gradient, on the acoustic main flow outside the boundary layer. By the linear theory, the velocity at the edge of the boundary layer is given in terms of the fractional derivatives of the axial velocity of the gas in the acoustic main flow. It is clearly seen how the temperature gradient controls the velocity at the edge. The velocity is almost in phase with the heat flux into the boundary layer from the wall. With effects of both the boundary layer and the array of resonators taken into account, nonlinear wave equations for unidirectional propagation in the tube are derived. Assuming a constant temperature gradient along the tube, the evolution of compression pulses is solved numerically by imposing the initial profiles of both an acoustic solitary wave and of a square pulse. It is revealed that when a positive gradient is imposed, the excess pressure decreases while the particle velocity increases and that the total energy flux can indeed be amplified if the gradient is suitable.

Numerical computations and order of magnitude estimates are presented for the periodic generation and coalescence of bubbles due to the injection of a constant flow rate of a gas through a circular orifice at the bottom wall of an inviscid dielectric or very polar liquid that is at rest and subject to a uniform vertical electric field far from the orifice. The problem depends on five dimensionless parameters: a Bond number based on the radius of the orifice; Weber and electric Bond numbers whose square roots are dimensionless measures of the flow rate of gas and the applied electric field; the dielectric constant of the liquid; and the contact angle of the liquid with the bottom wall. The bubbles that grow quasi-statically at the orifice for small values of the Weber number are always elongated vertically by the electric stress that acts on their surface when an electric field is applied. The volume of these bubbles at detachment may reach a maximum at a certain value of the electric Bond number, if the Bond number is sufficiently small, or decrease monotonically with the electric Bond number if the Bond number is larger. In both cases the bubbling ceases to be periodic beyond a certain value of the electric Bond number, apparently giving way to more complex bubbling regimes, which are not investigated here. Bubble interaction and eventually coalescence occur when the Weber number is increased keeping the electric Bond number in the range of periodic bubbling. Different periodic regimes are described. It is shown that a moderate electric field may increase the value of the Weber number above which coalescence occurs without changing the shape of the bubbles much. A large electric field may suppress coalescence but it also favours the development of upward and downward jets that cross the bubbles and may cause their breakdown.

A series of direct numerical simulations of mass transfer across the air–water interface driven by buoyancy-induced convection have been carried out to elucidate the physical mechanisms that play a role in the transfer of heat and atmospheric gases. The buoyant instability is caused by the presence of a thin layer of cold water situated on top of a body of warm water. In time, heat and atmospheric gases diffuse into the uppermost part of the thermal boundary layer and are subsequently transported down into the bulk by falling sheets and plumes of cold water. Using a specifically designed numerical code for the discretization of scalar convection and diffusion, it was possible to accurately resolve this buoyant-instability-induced transport of atmospheric gases into the bulk at a realistic Prandtl number ($\mathit{Pr}=6$) and Schmidt numbers ranging from $\mathit{Sc}=20$ to $\mathit{Sc}=500$. The simulations presented here provided a detailed insight into instantaneous gas transfer processes. The falling plumes with highly gas-saturated fluid in their core were found to penetrate deep inside the bulk. With an initial temperature difference between the water surface and the bulk of slightly above $2$  K, peaks in the instantaneous heat flux in excess of $1600~\text{W}~\text{m}^{-2}$ were observed, proving the potential effectiveness of buoyant-convective heat and gas transfer. Furthermore, the validity of the scaling law for the ratio of gas and heat transfer velocities $K_{L}/H_{L}\propto (\mathit{Pr}/\mathit{Sc})^{0.5}$ for the entire range of Schmidt numbers considered was confirmed. A good time-accurate approximation of $K_{L}$ was found using surface information such as velocity fluctuations and convection cell size or surface divergence. A reasonable time accuracy for the $K_{L}$ estimation was obtained using the horizontal integral length scale and the root mean square of the horizontal velocity fluctuations in the upper part of the bulk.

The dynamics of spreading of a macroscopic liquid droplet over a wetting surface is often described by a power-law relaxation, namely, the droplet radius increases as $t^{m}$ for time $t$, which is known as Tanner’s law. Here we show, by both experiments and theory, that when the liquid spreading takes place between a thin soap film and a glass fibre penetrating the film, the spreading is significantly slowed down. When the film thickness $\ell$ becomes smaller than the fibre diameter $d$, the strong hydrodynamic confinement effect of the soap film gives rise to a logarithmic relaxation with fibre creeping time $t$. Such a slow dynamics of spreading is observed for hours both in the measured time-dependent height of capillary rise $h(t)$ on the fibre surface and viscous friction coefficient $\unicode[STIX]{x1D709}_{s}(t)$ felt by the glass fibre in contact with the soap film. A new theoretical approach based on the Onsager variational principle is developed to describe the dynamics of thin film spreading along a fibre. The newly derived equations of motion provide the analytical solutions of $h(t)$ and contact angle $\unicode[STIX]{x1D703}(t)$, which are found to be in good agreement with the experimental results. Our work thus provides a common framework for understanding the confinement effect of thin soap films on the dynamics of spreading along a fibre.

We study the linear stability of two-dimensional high-Reynolds-number flow in a rigid parallel-sided channel, of which part of one wall has been replaced by a flexible membrane under longitudinal tension . Far upstream the flow is parallel Poiseuille flow at Reynolds number ; the width of the channel is and the length of the membrane is , where . Steady flow was studied using interactive boundary-layer theory by Guneratne & Pedley (J. Fluid Mech., vol. 569, 2006, pp. 151–184) for various values of the pressure difference across the membrane at its upstream end. Here unsteady interactive boundary-layer theory is used to investigate the stability of the trivial steady solution for . An unexpected finding is that the flow is always unstable, with a growth rate that increases with . In other words, the stability problem is ill-posed. However, when the pressure difference is held fixed () at the downstream end of the membrane, or a little further downstream, the problem is well-posed and all solutions are stable. The physical mechanisms underlying these findings are explored using a simple inviscid model; the crucial factor in the fluid dynamics is the vorticity gradient across the incoming Poiseuille flow.

This paper describes a new computational model developed based on the filtered mass density function (FMDF) for large-eddy simulation (LES) of two-phase turbulent reacting flows. The model is implemented with a unique Lagrangian–Eulerian–Lagrangian computational methodology. In this methodology, the resolved carrier gas velocity field is obtained by solving the filtered form of the compressible Navier–Stokes equations with high-order finite difference (FD) schemes. The gas scalar (temperature and species mass fractions) field and the liquid (droplet) phase are both obtained by Lagrangian methods. The two-way interactions between the phases and all the Eulerian and Lagrangian fields are included in the new two-phase LES/FMDF methodology. The results generated by LES/FMDF are compared with direct numerical simulation (DNS) data for a spatially developing non-reacting and reacting evaporating mixing layer. Results for two more complex and practical flows (a dump combustor and a double-swirl burner) are also considered. For all flows, it is shown that the two-phase LES/FMDF results are consistent and accurate.

A general method for studying two-dimensional problems in hydrodynamic stability is presented and applied to the classical problem of predicting instability in plane Poiseuille flow. The disturbance stream function is expanded in a Fourier series in the axial space dimension which, on substitution into the Navier-Stokes equation, leads to a system of parabolic partial differential equations in the coefficient functions. An efficient, stable and accurate numerical method is presented for solving these equations. It is demonstrated that the numerical process is capable of accurate reproduction of known results from the linear theory of hydrodynamic stability.

Disturbances that are stable according to linear theory are shown to become unstable with the addition of finite amplitude effects. This seems to be the first work of quantitative value for disturbances of moderate and larger amplitudes. A relationship between critical amplitude and Reynolds number is reported, the form of which indicates the existence of an absolute critical Reynolds number below which an arbitrary disturbance cannot be made unstable, no matter how large its initial amplitude. The critical curve shows significantly less effect of amplitude than do those obtained by earlier workers.

Wicking flow inside capillary tubes can attain considerable momentum so as to produce a liquid jet at the end of the tube. Auto-ejection refers to the formation of droplets at the tip of such a jet. Experimental observations suggest that a tapering nozzle at the end of the capillary tube is necessary for auto-ejection; it has never been reported for a straight tube. Besides, most experimental realizations require microgravity, although it is possible under normal gravity if the nozzle has a sufficiently sharp contraction. This computational study focuses on two related issues: the critical condition for auto-ejection, and the hydrodynamics of the liquid meniscus as affected by geometric parameters. We adopt a diffuse-interface Cahn–Hilliard model for the moving contact line, and allow the dynamic contact angle to deviate from the static one through wall energy relaxation. From analyzing the dynamics of the meniscus in the straight tube and the nozzle, we establish a critical condition for the onset of auto-ejection based on a Weber number defined at the exit of the nozzle and an effective length that encompasses the geometric features of the tube–nozzle combination. In particular, this shows that capillary ejection is not possible in straight tubes. With steeper contraction in the nozzle, we predict two additional regimes of interfacial rupture: rapid ejection of multiple droplets and air bubble entrapment. The numerical results are in general agreement with available experiments.

We investigate the evolution of spherical clouds of charged particles that migrate under the action of a uniform external electrostatic field. Hydrodynamic interactions are modelled by Oseen equations and the Coulomb repulsion is calculated through pairwise summation. It is shown that strong long-range Coulomb repulsion can prevent the breakup of the clouds covering a wide range of particle Reynolds number $Re_{p}$ and cloud-to-particle size ratio $R_{0}/r_{p}$. A dimensionless charge parameter $\unicode[STIX]{x1D705}_{q}$ is constructed to quantify the effect of the repulsion, and a critical value $\unicode[STIX]{x1D705}_{q,t}$ is deduced, which successfully captures the transition of a cloud from hydrodynamically controlled regime to repulsion-controlled regime. Our results also reveal that, with sufficiently strong repulsion, the cloud undergoes a universal self-similar expansion. Scaling laws of cloud radius $R_{cl}$ and particle number density $n$ are obtained by solving a continuum convection equation.