We consider here the effects of inertia on the instability of a flat liquid film under the effects of capillary and intermolecular forces (van der Waals interaction). Firstly, we perform a linear stability analysis within the long-wave approximation, which shows that the inclusion of inertia does not produce new regions of instability other than the one previously known from the usual lubrication case. The wavelength, ${\it\lambda}_{m}$, corresponding to the maximum growth, ${\it\omega}_{m}$ and the critical (marginal) wavelength do not change. The most affected feature of the instability under an increase of the Laplace number is the noticeable decrease of the growth rates of the unstable modes. In order to put in evidence the effects of the bidimensional aspects of the flow (neglected in the long-wave approximation), we also calculate the dispersion relation of the instability from the linearized version of the complete Navier–Stokes (N–S) equations. Unlike the long-wave approximation, the bidimensional model shows that ${\it\lambda}_{m}$ can vary significantly with inertia when the aspect ratio of the film is not sufficiently small. We also perform numerical simulations of the nonlinear N–S equations and analyse to which extent the linear predictions can be applied depending on both the amount of inertia involved and the aspect ratio of the film.

We derive a constitutive relation, relating the tangential stress, tangential velocity, thickness h, and viscosity μ, for a thin layer of Newtonian fluid on top of a fluid substrate. We find that the upper layer exerts a viscous tangential shear stress on the lower fluid, behaving as if it were a film with a two-dimensional shear viscosity equal to μh, and a dilatational viscosity 3μh.

The well-known Whitham theory may be applied to shocks diffracting over concave corners, provided that the diffraction results in Mach reflexion. This paper compares the theory with data obtained during experiments with diffracting shocks. If an incident shock is classified as weak or strong in the strict sense defined by von Neumann, then it is found that the Whitham theory accurately determines the Mach number of the Mach stem at the wall for both the weak and the strong cases. The theory also has some further value for strong shocks but not for weak; it is not applicable when the diffraction is regular.

A theoretical model is developed which illustrates the dynamics of layering instability, frequently realized in ocean regions with active fingering convection. Thermohaline layering is driven by the interplay between large-scale stratification and primary double-diffusive instabilities operating at the microscale – temporal and spatial scales set by molecular dissipation. This interaction is described by a combination of direct numerical simulations and an asymptotic multiscale model. The multiscale theory is used to formulate explicit and dynamically consistent flux laws, which can be readily implemented in large-scale analytical and numerical models. Most previous theoretical investigations of thermohaline layering were based on the flux-gradient model, which assumes that the vertical transport of density components is uniquely determined by their local background gradients. The key deficiency of this approach is that layering instabilities predicted by the flux-gradient model have unbounded growth rates at high wavenumbers. The resulting ultraviolet catastrophe precludes the analysis of such basic properties of layering instability as its preferred wavelength or the maximal growth rate. The multiscale model, on the other hand, incorporates hyperdiffusion terms that stabilize short layering modes. Overall, the presented theory carries the triple advantage of (i) offering an explicit description of the interaction between microstructure and layering modes, (ii) taking into account the influence of non-uniform stratification on microstructure-driven mixing, and (iii) avoiding unphysical behaviour of the flux-gradient laws at small scales. While the multiscale approach to the parametrization of time-dependent small-scale processes is illustrated here on the example of fingering convection, we expect the proposed technique to be readily adaptable to a wide range of applications.

The dynamics of a dilute, monodisperse suspension of nearly spherical particles that undergo Brownian rotations in an oscillatory simple shear flow is quantified, as a paradigm for large-amplitude oscillatory shear (LAOS) rheology of complex fluids. We focus on the ‘strongly nonlinear’ regime of LAOS, defined by ${\it\beta}\gg 1$ and ${\it\beta}/{\it\alpha}\gg 1$, where ${\it\beta}$ is a dimensionless shear rate (or Weissenberg number) and ${\it\alpha}$ is a dimensionless oscillation frequency (or Deborah number). We derive an asymptotic solution for the long-time periodic orientation probability density function of the particles. Our analysis reveals that the orientation dynamics consists of ‘core’ regions of rapid oscillation (on the time scale of the inverse of the shear-rate amplitude), separated by comparatively short ‘turning’ regions of slow evolution when the imposed flow vanishes. Uniformly valid approximations to the shear stress and normal stress differences (NSDs) of the suspension are then constructed: the non-Newtonian contribution to the shear stress, first NSD and second NSD, decays as ${\it\beta}^{-3/2}$, ${\it\beta}^{-1}$ and ${\it\beta}^{-1/2}$, respectively, at large ${\it\beta}$. These stress scalings originate from the orientation dynamics at the turning regions. Therefore, it is the occasions when the flow vanishes that dominate the rheology of this paradigmatic complex fluid under LAOS.

The order of the dispersion relation for the propagation of hydromagnetic waves along a magnetized cylindrical plasma falls by unity when the plasma resistivity, σ−1, tends to zero. A consequence of this is that the two boundary conditions necessary on an insulating wall are reduced to a single condition, a reduction brought about by the development of a current sheet. If the ratio, $\Omega \equiv \omega|\omega_{ci}$ of the wave frequency to the ion cyclotron frequency is also assumed to be vanishingly small, then the nature of the single boundary condition to be adopted in the limit σ−1 → 0 depends, for the slow hydromagnetic wave, on the limiting value of ρ½Ω2 . Similarly, if Ω [Gt ] 1, and the fast hydromagnetic wave is being considered, then the relevant boundary condition is found to depend on the limiting value of Ωσ−½.

The ‘resistive’ waves that are found to accompany the fast and slow waves, in order to satisfy the boundary conditions for small but finite values of σ−1, are studied in some detail and their contribution to the wave damping is determined.

A self-induced free-surface oscillation termed ‘self-induced sloshing’ was observed in a rectangular tank with a submerged and horizontally injected water jet. Self-induced sloshing is excited by the flow itself without any external force. Its behaviour was examined by experiment. The dominant frequency was found to be close to the first or second eigenvalue of fluid in a tank. The conditions of sloshing excitation were obtained for four tank geometries. They were called the ‘sloshing condition’, and defined in terms of inlet velocity and water level. Sloshing conditions were found to be strongly dependent on inlet velocity and tank geometry. A two-dimensional numerical simulation code was developed to simulate self-induced sloshing. The code was based on the boundary-fitted coordinate (BFC) method with height function. The numerical results were qualitatively verified by the experimental results, and were found to correlate well in terms of flow pattern, free-surface shape and sloshing conditions. In this study, sloshing growth was evaluated quantitatively using the simulation results. Oscillation energy supplied for the sloshing motion during a sloshing period (Econ) was calculated from simulation results. Sloshing growth was found to be strongly related to the sign and magnitude of Econ. The distribution of Econ showed that jet flow had a strong correlation with the sloshing growth. It was clarified that sloshing growth was primarily dependent on the spatial phase state of jet fluctuation. A governing parameter of self-induced sloshing, the modified Strouhal number Sts, was proposed on the basis of numerical evaluations of oscillation energy. The value of Sts suggests that one or two large vortices generated by jet fluctuations exist between the inlet and outlet during a sloshing period. When Sts is approximately either 1 (first stage) or 2 (second stage), self-induced sloshing occurs consistently in all experimental cases. The dependence of sloshing on inlet velocity, water level and tank geometry was revealed using Sts. For several tank geometries, a sloshing mode shift or jet mode (stage) transition was found to occur due to changes in inlet jet velocity. The combination of sloshing mode and jet stage can determine the state of the self-induced sloshing. As a result of this study, we propose a new excitation mechanism of self-induced sloshing, represented by a simple feedback loop closed by sloshing motion and jet fluctuation. The overall physical oscillation mechanism of self-induced sloshing was clarified using this feedback loop.

The dynamics of free and forced inertial waves inside cylinders of different aspect ratios ($\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}A=H_0/2R_0$) were investigated experimentally in this study. The liquid metal GaInSn was chosen as the fluid in order to enable a contactless stimulation of the flow by means of alternating electromagnetic fields. A rotating magnetic field generates the rotating motion of the liquid, whereas periodic modulations of the field strength and short pulses excite specific wave modes. Ultrasound Doppler velocimetry was used to record the flow structure and to identify inertial waves in the set-up. Our experiments demonstrate selective excitation of different inertial wave modes by deliberate variation of the magnetic field parameters. Furthermore, it was found that turbulent perturbations in the boundary layers of the swirling flow are able to induce an inertial wave mode that survives over a long time. Experiments at the fundamental resonance have shown that multiple harmonic wave modes appeared simultaneously. The measured inertial wave frequencies were compared to the predictions of the linear inviscid theory.

With the increasing popularity of combustion driving in strong shock tube experiments, the possibility of detonation must be considered seriously. In addition to the extreme and relatively well-known conditions associated with a detonation, there exists a phase during the transition from a flame to a detonation during which pressures and velocities of propagation can exist which are higher than those associated with the fully formed detonation. This has been discussed theoretically by Oppenheim (1952) and has been shown experimentally by several workers. A luminous front having a higher than detonation velocity has been observed by Bone, Fraser & Wheeler (1936) and others. J. B. Smith (1949), using calibrated burst diaphragms at the end of a pipe, has observed reflected pressures in fuel gas-air, acetylene-air, and hydrogen-air mixtures which were approximately four times greater than those associated with a reflected normal detonation. Turin & Huebler (1951), using quartz pressure transducers in the side of a tube, observed pressures during this transition process about three and a half times higher than those at a later time when detonation was fully developed. They worked with Toledo natural gas. Mooradian & Gordon (1951) also have clear indications of this phenomenon with hydrogen-air mixtures.

The onset of convection of a Boussinesq fluid in a horizontal plane layer is studied. The system rotates with constant angular velocity $\Omega$, which is inclined at an angle $\vartheta$ to the vertical. The layer is sheared by keeping the upper boundary fixed, while the lower boundary moves parallel to itself with constant velocity ${\bm U}_0$ normal to the plane containing the rotation vector and gravity ${\bm g}$ (i.e. ${\bm U}_0\,\parallel\,{\bm g}\,{\times}\,\Omega)$. The system is characterized by five dimensionless parameters: the Rayleigh number $Ra$, the Taylor number $\tau^2$, the Reynolds number $Re$ (based on ${\bm U}_0$), the Prandtl number $Pr$ and the angle $\vartheta$. The basic equilibrium state consists of a linear temperature profile and an Ekman–Couette flow, both dependent only on the vertical coordinate $z$. Our linear stability study involves determining the critical Rayleigh number $Ra_c$ as a function of $\tau$ and $Re$ for representative values of $\vartheta$ and $Pr$.

Our main results relate to the case of large Reynolds number, for which there is the possibility of hydrodynamic instability. When the rotation is vertical $\vartheta=0$ and $\tau\gg 1$, so-called type I and type II Ekman layer instabilities are possible. With the inclusion of buoyancy $Ra\not=0$ mode competition occurs. On increasing $\tau$ from zero, with fixed large $Re$, we identify four types of mode: a convective mode stabilized by the strong shear for moderate $\tau$, hydrodynamic type I and II modes either assisted $(Ra>0)$ or suppressed $(Ra<0)$ by buoyancy forces at numerically large $\tau$, and a convective mode for very large $\tau$ that is largely uninfluenced by the thin Ekman shear layer, except in that it provides a selection mechanism for roll orientation which would otherwise be arbitrary. Significantly, in the case of oblique rotation $\vartheta\not=0$, the symmetry associated with ${\bm U}_0\leftrightarrow -{\bm U}_0$ for the vertical rotation is broken and so the cases of positive and negative $Re$ exhibit distinct stability characteristics, which we consider separately. Detailed numerical results were obtained for the representative case $\vartheta=\pi/4$. Though the overall features of the stability results are broadly similar to the case of vertical rotation, their detailed structure possesses a surprising variety of subtle differences.

A new class of analytic solutions to the problem of two-dimensional potential fnow is presented here. The method of solution has features of both direct and indirect solutions. The bodies about which flow is computed are semi-infinite and have forward regions that either are flat or consist of a circular arc, which may be convex or concave to the flow. Closed-form solutions are obtained for the surface velocity. Afterbody shapes are defined by implicit equations containing a quadrature. Certain analytic properties of the solutions are investigated. An interesting feature of the bodies is the presence of a ‘pseudo corner’ where the slope angle is continuous but the curvature is infinite. The surface velocity becomes logarithmically infinite at these points in contrast to the power-law behaviour at a true corner. One case of the convex circular arc has finite velocity everywhere, and in some sense represents flow over a circular cylinder with a ‘natural’ separation point. This point occurs at 77·45° from the front stagnation point, which is close to the separation point for incompressible laminar boundarylayer flow.

Flow in a closed-loop thermosyphon heated from below exhibits a sequence of bifurcations with increasing Grashof number. Using the Navier–Stokes equations in the Boussinesq approximation we have derived a model where, in the case of a slender circular loop, the first Fourier modes exactly decouple from all other Fourier modes, leaving a system of three coupled nonlinear partial differential equations that completely describes the flow in the thermosyphon. We have characterized the flow through two bifurcations, identifying stable periodic solutions for flows of Prandtl number greater than 18.5, a much lower value than predicted previously. Because of the quadratic nonlinearity in this system of equations, it is possible to find the global stability limit, and we have proved that it is identical to the first bifurcation point.

The numerical study of the model equations is based on a highly accurate Fourier–Chebyshev spectral method, combined with asymptotic analysis at the various bifurcation points. Three-dimensional computations with a finite element method computational fluid dynamics code (MPSalsa), are also pursued. All three approaches are in close agreement.

A variety of numerical methods exist for the study of deformable particles in dense suspensions. None of the standard tools, however, currently include volume-changing objects such as oscillating microbubbles in three-dimensional periodic domains. In the first part of this work, we develop a novel method to include such entities based on the boundary integral method. We show that the well-known boundary integral equation must be amended with two additional terms containing the volume flux through the bubble surface. We rigorously prove the existence and uniqueness of the solution. Our proof contains as a subset the simpler boundary integral equation without volume-changing objects (such as red blood cell or capsule suspensions) which is widely used but for which a formal proof in periodic domains has not been published to date. In the second part, we apply our method to study microbubbles for targeted drug delivery. The ideal drug delivery agent should stay away from the biochemically active vessel walls during circulation. However, upon reaching its target it should attain a near-wall position for efficient drug uptake. Though seemingly contradictory, we show that lipid-coated microbubbles in conjunction with a localized ultrasound pulse possess precisely these two properties. This ultrasound-triggered margination is due to hydrodynamic interactions between the red blood cells and the oscillating lipid-coated microbubbles which alternate between a soft and a stiff state. We find that the effect is very robust, existing even if the duration in the stiff state is more than three times lower than the opposing time in the soft state.

Natural convection in a cube of fluid-saturated porous medium heated from below and cooled from the top is studied numerically using a non-Darcy flow model. All vertical sidewalls are considered to be impermeable and adiabatic. The evolution of the various flow patterns is investigated from onset up to a Rayleigh number of 1000 where irregularly fluctuating convection prevails. New flow patterns have been found to exist in addition to those mentioned in the previous studies. In the present study, a total of ten steady flow patterns have been identified, of which five show oscillatory behaviour in some Rayleigh-number range. The results are presented in terms of average Nusselt number curves consisting of the solution branches of the convective patterns. The convective patterns are classified in terms of their symmetry properties, and the symmetries broken or gained during bifurcations from one flow structure to the other are identified.

The incompressible, inviscid, parallel flow of a layer of fluid over a slightly denser fluid, in the presence of gravity, is investigated for stability. Two idealized piece-wise-linear steady velocity profiles are examined analytically, and a comparison with related experimental results is made.

The dimensionless parameter U/(gh)½, where U is flow velocity, g the acceleration of gravity, and h the thickness of the upper layer, appears to have a critical value between 0.2 and 0.7, below which stable flows can persist. At the onset of instability disturbances of wavelength about h or 2h are predicted, with more violent disturbances of longer wavelength occurring at higher values of U/(gh)½ For continuous steady velocity profiles this instability phenomenon is found to be relatively insensitive to the ratio of the densities of the two fluids.

The pattern of motion on the surface of a model shaped in plaster of paris and immersed in a water stream, can be made visible by reason of the marks caused when small discontinuities in the surface excite local fluctuations of velocity which lead to differential solution of the plaster and to small features of relief. This technique, which has so far been applied in geological studies, is illustrated by reference to motions about a cylinder on a flat plate and over symmetrical skewed steps. Current crescents and sand waves are briefly discussed in the light of these motions.

The effect of free-surface drift layers on the maximum height that a steady wave can attain without breaking is explored through experiments and numerical simulations. In the experiments, the waves are generated by towing a two-dimensional fully submerged hydrofoil at constant depth, speed and angle of attack. The drift layer is generated by towing a plastic sheet on the water surface ahead of the hydrofoil. It is found that the presence of this drift layer (free-surface wake) dramatically reduces the maximum non-breaking wave height and that this wave height correlates well with the surface drift velocity. In the simulations, the inviscid two-dimensional fully nonlinear Euler equations are solved numerically. Initially symmetric wave profiles are superimposed on a parallel drift layer whose mean flow characteristics match those in the experiments. It is found that for large enough initial wave amplitudes a bulge forms at the crest on the forward face of the wave and the vorticity fluctuations just under the surface in this region grow dramatically in time. This behaviour is taken as a criterion to indicate impending wave breaking. The maximum non-breaking wave elevations obtained in this way are in good agreement with the experimental findings.

We examine the collective dynamics of polarizable, Brownian, sedimenting rods of high aspect ratio. Previous work of Koch and Shaqfeh (J. Fluids Mech., vol. 209, 1989 pp. 521–542) has shown that in the absence of Brownian motion, sedimenting suspensions of rods are unstable to concentration fluctuations and form dense streamers via interparticle hydrodynamic interactions. Recently, Saintillan, Shaqfeh & Darve (Phys. Fluids, vol. 18 (121701), 2006b p. 1) demonstrated that electric fields can act to stabilize these non-Brownian suspensions of polarizable rods through induced-charge electrokinetic rotation, which forces particle alignment. In this paper, we employ a mean-field linear stability analysis as well as Brownian dynamics simulations to study the effect of thermal motion on the onset of instability. We find that in the absence of electric fields, Brownian motion consistently suppresses instability formation through randomization of particle orientation. However, when electric fields are applied, thermal motion can act to induce instability by counteracting the stabilizing effect of induced-charge orientation.

A particle mesh Ewald (PME) Stokesian dynamics algorithm has been developed to model hydrodynamic interactions in suspensions of non-spherical dicolloidal particles. Dicolloids, which have recently been synthesized by a number of independent research groups (Johnson, van Kats & van Blaaderen (Langmuir, vol. 21, 2005, p. 11510), Mock et al. (Langmuir, vol. 22, 2006, p. 4037), Kim, Larsen & Weitz (J. Am. Chem. Soc., vol. 128, 2006, p. 14374)), consist of two intersecting spheres of varying radii and centre-to-centre separation. One-body resistance tensors and disturbance velocity fields are computed for general linear flows using a superposition of Stokes singularities along the symmetry axis of the dicolloid particles. The coefficients and the locations of the singularities are optimized to minimize the norm of the velocity error on the particle surface. The one-body solution provides all coefficients required for the far-field many-body interactions in the Stokesian dynamics algorithm. These generalize the analytical results for spheres employed in the classic algorithm. Modified lubrication interaction tensors are developed for dicolloids for the singular near-field lubrication interactions. Accuracy of the one-body solutions and two-body generalized Stokesian dynamics solutions are validated by comparison with high-precision numerical solutions computed with the spectral boundary element method of Muldowney & Higdon (J. Fluid Mech., vol. 298, 1995, p. 167). The newly developed PME Stokesian dynamics algorithm was used to study transport properties in dicolloidal suspensions over a range of volume fractions (φ ≤ 0.5). The effects of the degree of anisotropy on the properties of the suspension are discussed. For these mildly anisotropic particles, the transport properties remain close to those of spheres, however certain interesting trends emerge, with non-monotonic viscosity dependence as a function of increasing aspect ratio. The minimum viscosity in concentrated suspensions is lower than that for spheres with equal volume fraction over a range of volume fractions.