Viscous fingering is observed experimentally when a bidisperse suspension displaces air inside a Hele-Shaw cell, despite the stabilising viscosity ratio between the invading (suspension) and defending (air) phases. Careful experiments are carried out to characterise this instability by either systematically varying the large-particle concentrations $\unicode[STIX]{x1D719}_{l0}$ at constant total concentrations $\unicode[STIX]{x1D719}_{0}$, or changing $\unicode[STIX]{x1D719}_{0}$ with fixed $\unicode[STIX]{x1D719}_{l0}$. Leading to the instability, we observe that larger particles consistently enrich the fluid–fluid interface at a faster rate than small particles. This size-dependent enrichment of the interface leads to an earlier onset of the fingering instability for bidisperse suspensions, compared to their monodisperse counterpart of all small particles. In particular, even the small presence of large particles is shown to effectively lower the total particle concentration needed for fingering, compared to the all-small-particle case. We hypothesise that the key mechanism behind this enhanced viscous fingering is the size-dependent nature of shear-induced migration of particles far upstream from the interface. A reduced equilibrium model is derived based on the modified suspension balance model to verify this hypothesis, in reasonable agreement with experiments.

The Stewartson E½- and E¼-layers in a rapidly rotating compressible fluid are considered within the framework of linearized equations and the boundary-layer method. The fluid is contained in a cylinder made of a thermally insulated side wall and conducting top and bottom end plates. The end plates and the side wall rotate with slightly different angular velocities. The case of an incompressible fluid was discussed by Stewartson, who found that the flow is restricted to the side-wall boundary layers. In the case of a compressible fluid, however, the solutions are strongly dependent upon the thermal boundary conditions assumed on the side wall. In particular, if the wall is insulated the fluid in the inner core is dragged along too since it is coupled strongly to the flow in the side-wall Stewartson layers. The critical parameter governing the solutions is found to be $(\gamma -1) PrG_0 E^{-\frac{1}{3}}/4\gamma$, where γ is the ratio of specific heats, Pr the Prandtl number, G0 the square of the rotational Mach number and E the Ekman number.

Analytical solutions for the quasi-one-dimensional flow of a gas not in thermodynamic equilibrium are presented for two distinct types of rate equation, namely, the linear rate equation which governs vibrational relaxation, and the nonlinear rate equation which governs dissociation. The solutions are derived for the case when, to a first approximation, the rate equation is uncoupled from the remaining flow equations.

There are, in general, three distinct regions in non-equilibrium nozzle flow. First, a so-called near equilibrium region where a perturbation solution is expected to hold. This region is followed by a narrow transition-layer in which there is a rapid departure from equilibrium. Finally, downstream of this layer, the energy in the lagging mode tends asymptotically to some constant ‘frozenout’ value. The solutions applicable to each of these three regions are derived for both rate equations, the boundary conditions for the transition-layer solution and the asymptotic solution are obtained from appropriate matching procedures.

In particular the structures of the asymptotic solutions are discussed. Several approximate methods for determining the asymptotic frozen level of theenergy in the lagging mode have been proposed in the literature. For the present case, when there is only a small amount of energy in the lagging mode, it is shown that none of these approximate methods is mathematically correct.

The characteristics of the laminar boundary layer on a continuous moving surface are described and an experiment is performed to demonstrate that such a flow is physically realizable. The hydrodynamic stability of the flow is analysed within the framework of small-perturbation stability theory. A complete stability diagram is mapped out. The critical Reynolds number is found to be substantially higher than that for the Blasius flow and, correspondingly, the critical layer lies closer to the wall. The disturbance amplitude function and its derivative are numerically evaluated, from which are derived the vector flow field of the disturbance, the resultant flow field (main flow plus disturbances), the root-mean-square distributions of the disturbance velocity components, and the distributions of the kinetic energy and the Reynolds stress. The energy criterion for stability is also investigated and is found to be consistent with the solutions of the eigenvalue problem.

The work that follows considers the velocity profiles within the boundary layer at the wall of an arbitrarily converging funnel. The occurrence of super-velocities, i.e. components of velocity within the boundary layer exceeding their corresponding free stream component, is investigated and the relevance of such a phenomenon to the efficiency of discharge discussed.

We study the heat or mass transfer from a neutrally buoyant spherical drop embedded in an ambient Newtonian medium, undergoing a general shearing flow, in the strong convection limit. The latter limit corresponds to the drop Péclet number being large ($Pe\gg 1$). We consider two families of ambient linear flows: (i) planar linear flows with open streamlines (parametrized by $\unicode[STIX]{x1D6FC}$ with $0\leqslant \unicode[STIX]{x1D6FC}\leqslant 1$, the extremal members being simple shear flow ($\unicode[STIX]{x1D6FC}=0$) and planar extension ($\unicode[STIX]{x1D6FC}=1$)) and (ii) three-dimensional extensional flows (parameterized by $\unicode[STIX]{x1D716}$, with $0\leqslant \unicode[STIX]{x1D716}\leqslant 1$, the extremal members being planar ($\unicode[STIX]{x1D716}=0$) and axisymmetric extension ($\unicode[STIX]{x1D716}=1$)). For the first family, an analysis of the exterior flow field in the inertialess limit (the drop Reynolds number, $Re$, being vanishingly small) shows that there exist two distinct streamline topologies separated by a critical drop-to-medium viscosity ratio ($\unicode[STIX]{x1D706}$) given by $\unicode[STIX]{x1D706}_{c}=2\unicode[STIX]{x1D6FC}/(1-\unicode[STIX]{x1D6FC})$. For $\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{c}$ all streamlines are open, while the near-field streamlines are closed for $\unicode[STIX]{x1D706}>\unicode[STIX]{x1D706}_{c}$. For the second family, the exterior streamlines remain open regardless of $\unicode[STIX]{x1D706}$. The two streamline topologies lead to qualitatively different mechanisms of transport for large $Pe$. The transport in the open streamline regime is enhanced in the usual manner via the formation of a boundary layer. In sharp contrast, the closed-streamline regime displays diffusion-limited transport, so there is only a finite enhancement even as $Pe\rightarrow \infty$. For $Re=0$, the drop surface streamlines in a planar linear flow may be regarded as generalized Jeffery orbits with a flow and viscosity dependent aspect ratio Jeffery orbits denote the aspect-ratio-dependent inertialess trajectories of a rigid axisymmetric particle in a simple shear flow; see Jeffery (Proc. R. Soc. Lond. A, vol. 102 (715), 1922, pp. 161–179). A Jeffery-orbit-based non-orthogonal coordinate system thus serves as a natural candidate to tackle the transport problem from a drop, in a planar linear flow, in the limit $Pe\gg 1$. Use of this system allows one to derive a closed-form expression for the dimensionless rate of transport (the Nusselt number $Nu$) from a drop in the open-streamline regime ($\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{c}$). Symmetry arguments point to a Jeffery-orbit-based coordinate system for any linear flow, and a variant of this coordinate system is therefore used to derive the Nusselt number for the family of three-dimensional extensional flows. For both classes of flows considered, the boundary-layer-enhanced transport implies that the Nusselt number takes the form $Nu={\mathcal{F}}(P,\unicode[STIX]{x1D706})Pe^{1/2}$, with the parameter $P$ being $\unicode[STIX]{x1D6FC}$ or $\unicode[STIX]{x1D716}$, and ${\mathcal{F}}(P,\unicode[STIX]{x1D706})$ given as a one and two-dimensional integral, respectively, which is readily evaluated numerically.

In this work, the scaling statistics of the dissipation along Lagrangian trajectories are investigated by using fluid tracer particles obtained from a high-resolution direct numerical simulation with $\mathit{Re}_{\lambda }=400$. Both the energy dissipation rate $\epsilon $ and the local time-averaged $\epsilon _{\tau }$ agree rather well with the lognormal distribution hypothesis. Several statistics are then examined. It is found that the autocorrelation function $\rho (\tau )$ of $\ln (\epsilon (t))$ and variance $\sigma ^2(\tau )$ of $\ln (\epsilon _{\tau }(t))$ obey a log-law with scaling exponent $\beta '=\beta =0.30$ compatible with the intermittency parameter $\mu =0.30$. The $q{\rm th}$-order moment of $\epsilon _{\tau }$ has a clear power law on the inertial range $10<\tau /\tau _{\eta }<100$. The measured scaling exponent $K_L(q)$ agrees remarkably with $q-\zeta _L(2q)$ where $\zeta _L(2q)$ is the scaling exponent estimated using the Hilbert methodology. All of these results suggest that the dissipation along Lagrangian trajectories could be modelled by a multiplicative cascade.

We numerically investigated the unsteady dynamics of a two-dimensional airfoil undergoing a continuous, prescribed pitch-up motion and freely translating as a response to aerodynamic forces and the gravity field. The pitch-up motion was applied about an axis located $1/6$ chord away from the leading edge and was parameterized using the shape change number, with a Reynolds number set to 2000. It was shown that the minimum kinetic energy reached by the airfoil depends stochastically and asymptotically on shape change numbers for values below and above 1, respectively. Very low kinetic energy levels (close to zero) can be reached in both stochastic and asymptotic regions but high shape change numbers are accompanied by significant gain in altitude which may be undesirable from a practical perspective. Rather, shape change numbers in the range [0.1–0.3] allow us to reach relatively low levels of kinetic energy for close perching locations. We showed that highly nonlinear fluid–structure interactions induced by massive flow separations and strong vortices are conducive to low kinetic energy, but responsible for the stochastic dependence of kinetic energy to shape change number, which can make perching manoeuvres hardly controllable for flying vehicles.

The planar flow arising in an initially quiescent viscous fluid under the action of a localized dipolar-type forcing has been studied analytically and experimentally. The force dipole, with non-dimensional forcing amplitude Re, brings net zero momentum into the fluid and gives rise to the formation of a quadrupolar vortex: a system of two dipolar vortices moving apart. Experimentally, the action of a force dipole was modelled by a vertical cylinder oscillating horizontally in the shallow upper layer of a two-layer fluid. Two cases were studied: single quadrupoles and an array of quadrupoles. It is found that single quadrupoles develop in a self-similar manner: the length L and the translation velocity Ū of the quadrupolar vortex change with time as L ∼ t1/2 and Ū ∼ t-1/2. These quantities are characterized by non-dimensional functions α(Re) and β(Re), respectively, which have been determined theoretically for small Re-values and experimentally for Re-values in the range 160–2200.

To produce an array of quadrupoles an array of oscillating vertical rods was used. Two stages in the flow evolution were studied experimentally: the initial stage, when the interactions between the quadrupoles are weak, and the intermediate stage when the interactions play an essential role and the flow is (two-dimensionally) turbulent. It is found that at both stages the width H of the region with intense vortical motions increases with time as H ∼ t1/2. A theoretical explanation of the experimental results is given.

A differencing scheme known as the ‘Fluid-in-Cell’ method has been used in the numerical simulation of a choked jet of air impinging on a flat plate. Before sonic conditions are applied at the nozzle exit, the field of interest is at rest at the ambient pressure and temperature. The instantaneous application of sonic conditions at the nozzle exit is tantamount to the sudden appearance of a normal shock wave whose strength is determined by the experimental conditions.

The results of the simulation describe the decay of the initial shock wave and its reflection at the plate; the formation of a second shock wave and its merging with the reflected shock giving rise to a detached shock wave which oscillates; and the growth and subsequent motion of a toroidal vortex that is generated between this shock wave and the plate. The results show clearly how the flow field which has been observed in physical experiments under stable operating conditions is developed.

A possible mechanism for the occurrence of the phenomenon of erratic drift of bubbles in liquids subjected to acoustic waves was proposed by Benjamin & Ellis (1990) who showed that nonlinear interactions between adjacent perturbation modes expressed in terms of spherical harmonics of any order may lead to the excitation of mode 1 which is equivalent to a displacement of the bubble centroid. We show that indeed such a mechanism can give rise to a chaotic process at least under the conditions experimentally investigated by Benjamin & Ellis (1990). In fact we examine the case in which the angular frequency ω of the incident wave is sufficiently close to both the natural frequency of mode n + 1 (ωn + 1) and twice the natural frequency of mode n (2ωn) thus exciting simultaneously a subharmonic mode n and a synchronous mode n + 1. The value of n is set equal to 3 in accordance with Benjamin & Ellis' (1990) observation. A classical multiple scale analysis allows us to follow the development of these perturbations in the weakly nonlinear regime to find an autonomous system of quadratically coupled nonlinear differential equations governing the evolution of the amplitudes of the perturbations on a slow time scale. As obtained by Gu & Sethna (1987) for the Faraday resonance problem, we find both regular and chaotic solutions of the above system. Chaos is found to develop for large enough values of the amplitude of the acoustic excitation within some region in the parameter space and is reached through a period-doubling sequence displaying the typical characteristics of Feigenbaum scenario.

Pressure-wave propagation through a separated gas-liquid layer at rest in a duct of constant rectangular cross-section and infinite length is considered. Such a system is dispersive, possessing an infinite number of modes which depend on the ratios of the densities, thicknesses and sound speeds of the two phases. The transitional variation of an infinitesimal disturbance initially having a step profile is investigated analytically and numerically. In addition, it is shown that a weak but finite disturbance is described asymptotically by the solution of the Korteweg-de Vries equation.

The heat transfer across an inclined air layer in the longitudinal roll regime often falls below expected values. A companion paper (Ruth et al. 1980) presents the heat transfer measurement and flow visualization observations; the present paper discusses the mechanisms which are thought to be responsible for this anomalous behaviour and formulates a simple model which correlates the results of our previous study and that of Hart (1971). The observed suppression of heat transfer below values expected for pure longitudinal roll motion is explained in terms of a shear instability. A stabilizing mechansim, which would restore the longitudinal rolls at higher Rayleigh numbers, is described; the return of the heat transfer to expected levels at higher Rayleigh numbers is ascribed to this mechanism. Comparisons are made between the results of the present model and the analyses of Clever & Busse (1977).