In a variety of applications, most notably microfluidics design, slip-based boundary conditions have been sought to characterize fluid flow over patterned surfaces. We focus on laminar shear flows over surfaces with periodic height fluctuations and/or fluctuating Navier scalar slip properties. We derive a general formula for the ‘effective slip’, which describes equivalent fluid motion at the mean surface as depicted by the linear velocity profile that arises far from it. We show that the slip and the applied stress are related linearly through a tensorial mobility matrix, and the method of domain perturbation is then used to derive an approximate formula for the mobility law directly in terms of surface properties. The specific accuracy of the approximation is detailed, and the mobility relation is then utilized to address several questions, such as the determination of optimal surface shapes and the effect of random surface fluctuations on fluid slip.

The results of an experimental study of the diffraction of shock waves on plane-walled convex corners are given for a Mach number range from 1·0 to 5·0. The behaviour of the disturbances produced in the region perturbed by the corner are discussed. It is shown that the position of the slipstream and tail of the Prandtl-Meyer fan, and the velocities of the contact surface and second shock become independent of corner angle for angles greater than 75°. Comparisons with theoretical predictions of Jones, Martin & Thornhill (1951) and Parks (1952) are included. In most cases fair agreement is obtained.

The large-scale coherent motions in a realistic swirl fuel-injector geometry are analysed by direct numerical simulations (DNS), proper orthogonal decomposition (POD), and linear global modes. The aim is to identify the origin of instability in this turbulent flow in a complex internal geometry. The flow field in the nonlinear simulation is highly turbulent, but with a distinguishable coherent structure: the precessing vortex core (a spiralling mode). The most energetic POD mode pair is identified as the precessing vortex core. By analysing the fast Fourier transform (FFT) of the time coefficients of the POD modes, we conclude that the first four POD modes contain the coherent fluctuations. The remaining POD modes (incoherent fluctuations) are used to form a turbulent viscosity field, using the Newtonian eddy model. The turbulence sets in from convective shear layer instabilities even before the nonlinear flow reaches the other end of the domain, indicating that equilibrium solutions of the Navier–Stokes are never observed. Linear global modes are computed around the mean flow from DNS, applying the turbulent viscosity extracted from POD modes. A slightly stable discrete $m=1$ eigenmode is found, well separated from the continuous spectrum, in very good agreement with the POD mode shape and frequency. The structural sensitivity of the precessing vortex core is located upstream of the central recirculation zone, identifying it as a spiral vortex breakdown instability in the nozzle. Furthermore, the structural sensitivity indicates that the dominant instability mechanism is the Kelvin–Helmholtz instability at the inflection point forming near vortex breakdown. Adjoint modes are strong in the shear layer along the whole extent of the nozzle, showing that the optimal initial condition for the global mode is localized in the shear layer. We analyse the qualitative influence of turbulent dissipation in the stability problem (eddy viscosity) on the eigenmodes by comparing them to eigenmodes computed without eddy viscosity. The results show that the eddy viscosity improves the complex frequency and shape of global modes around the fuel-injector mean flow, while a qualitative wavemaker position can be obtained with or without turbulent dissipation, in agreement with previous studies. This study shows how sensitivity analysis can identify which parts of the flow in a complex geometry need to be altered in order to change its hydrodynamic stability characteristics.

Meander bends typically show certain systematic deviations from simple Cartesian sinusoidal forms. Bends tend to be round and full, or ‘fat’, often to the point of possessing double-valued plan-forms, as Langbein & Leopold (1966) have noted. Bends also tend to be characteristically skewed in such a fashion that their direction of migration can be determined directly from an aerial photograph of the planform; the water margin of the downstream accreting half of a point bar describes a convex planform, whereas the upstream eroding side has a concave shape.

In the present paper a generalized nonlinear equation of bend migration is treated based on the analysis of Part 1 (Ikeda, Parker & Sawai 1981). An expansion technique reminiscent of the Stokes expansion for water waves is developed to perform a nonlinear stability analysis. This analysis provides an explanation of skewing and fattening, and also indicates that lateral and downstream migration rates should increase as bend amplitude develops. These results agree qualitatively with field observations.

We have studied the transition between oscillatory and steady convection in a simplified model of two-dimensional thermosolutal convection. This model is exact to second order in the amplitude of the motion and is qualitatively accurate for larger amplitudes. If the ratio of the solutal diffusivity to the thermal diffusivity is sufficiently small and the solutal Rayleigh number, RS, sufficiently large, convection sets in as overstable oscillations, and these oscillations grow in amplitude as the thermal Rayleigh number, RT, is increased. In addition to this oscillatory branch, there is a branch of steady solutions that bifurcates from the static equilibrium towards lower values of RT; this subcritical branch is initially unstable but acquires stability as it turns round towards increasing values of RT. For moderate values of RS the oscillatory branch ends on the unstable (subcritical) portion of the steady branch, where the period of the oscillations becomes infinite. For larger values of RS a birfurcation from symmetrical to asymmetrical oscillations is followed by a succession of bifurcations, at each of which the period doubles, until the motion becomes aperiodic at some finite value of RT. The chaotic solutions persist as RT is further increased but eventually they lose stability and there is a transition to the stable steady branch. These results are consistent with the behaviour of solutions of the full two-dimensional problem and suggest that period-doubling, followed by the appearance of a strange attractor, is a characteristic feature of double-diffusive convection.

A fluid droplet located on a superhydrophobic surface makes contact with the surface only at small isolated regions, and is mostly in contact with the surrounding air. As a result, a fluid in motion near such a surface experiences very low friction, and superhydrophobic surfaces display strong drag reduction in the laminar regime. Here we consider theoretically a superhydrophobic surface composed of circular posts (so-called fakir geometry) located on a planar rectangular lattice. Using a superposition of point forces with suitably spatially dependent strength, we derive the effective surface-slip length for a planar shear flow on such a fakir-like surface as the solution to an infinite series of linear equations. In the asymptotic limit of small surface coverage by the posts, the series can be interpreted as Riemann sums, and the slip length can be obtained analytically. For posts on a square lattice, our analytical prediction of the dimensionless slip length, in the low surface coverage limit, is in excellent quantitative agreement with previous numerical computations.

The collisional flow of a slightly inelastic granular material down a rough inclined plane is usually described by kinetic theories. We present an experimental study aimed at analysing the assumptions and the quantitative predictions of such theories. A two-dimensional channel coupled to a model granular material and image analysis allow detailed and complete measurement of the kinematics and structure of the flows. We determine the range of inclination and particle flux for which the flow is stationary and uniform. The characteristic profiles of solid fraction, mean velocity and granular temperature are systematically measured. Both the true collisional and the dilute kinetic regimes are examined. We show that a quasi-hydrodynamic description of these regimes seems relevant, and that the pressure and the viscosity terms are in good qualitative agreement with the prediction of the kinetic theory. The profiles are well described by the kinetic theory near the top of the flow, at low solid fraction. Conversely there are large discrepancies near the rough plane, where the material is structured in layers.

The three-dimensional incompressible magnetohydrodynamic (MHD) equations for rectangular geometry and periodic boundary conditions are solved numerically using the spectral method of Orszag & Patterson (1972). The calculations are restricted to a magnetic Prandtl number of one and to Gaussian random initial conditions with zero mean magnetic and momentum fields. We permit non-mirror-symmetric (helical) flows. In all cases, there is a continuous transfer of energy from the momentum field to the magnetic field. A proposed mechanism for this transfer involves the cascading of energy from the large scales of the momentum field to the small scales, thence a redistribution of energy between the momentum and magnetic fields by Alfvén waves, and, finally, an inverse cascade of energy from the small scales of the magnetic field to the large scales. This inverse cascade is found when magnetic helicity (〈a. b〉, where b = curl a is the magnetic induction) is present in the flow.

Spatial growth of mechanically generated water waves under the action of wind has been measured in a laboratory wind-wave flume both for pure water and for water containing a surfactant (sodium lauryl sulphate, concentration 2.6 × 10−2%). I n the latter case, no wind waves develop on the surface of the mechanically generated waves as well as on the still water surface for wind speeds up to U10≈ 15 m/s, where U10 is the wind velocity at the height Z = 10 m. Therefore we can study the wind-induced growth of monochromatic waves without the effects of co-existing short wind waves. The mechanically generated waves grew exponentially under the action of the wind, with fetch in both cases. The measured growth rate β for the pure water can be fitted by β/f = 0.34(U*/C)2 0.1 [lsime ] U*/C [lsime ] 1.0, where f is the frequency of the waves, C is the corresponding phase velocity, and U, is the friction velocity obtained from vertical wind profiles. The effect of the wave steepness H/L on the dimensionless growth rate β/f is not clear, but seems to be small. For water containing the surfactant, the measured growth rate is smaller than that for pure water, but the friction velocity of the wind is also small, and the above relation between β/f and U*/C holds approximately if the measured friction velocity U* is used for the relation.

The stability of a heavy top, containing a cylindrical cavity partly full of liquid, for small displacements from the sleeping position is studied. It is shown theoretically that instability can occur when any one of the periods of free oscillation of the liquid, which are doubly infinite in number, is sufficiently near to the period of nutation of the empty top. In experiments carried out by Prof. Ward, only the two principal instabilities could be distinguished.

The Kelvin-Helmholtz model for the formation of surface waves at the interface between two fluids in relative motion is generalized for parallel shear flows. It is assumed that phase changes across the flow are negligible and hence that the aerodynamic pressure on the wave is in phase with its displacement (rather than its slope). A variational formulation is established and leads to the determination of appropriately weighted means for the velocity profiles. The principal application is to flow of a light inviscid fluid over a viscous liquid; it is shown that the principle of exchange of stabilities is applicable to such a configuration, and a critical wind speed in satisfactory agreement with observation is predicted for an air-oil interface. The results also are applied to an air-water interface and lead to the conclusion that Kelvin-Helmholtz instability of such an interface is unlikely at commonly observed wind speeds. A more general formulation of the Kelvin-Helmholtz boundary-value problem and variational principle, allowing for variations in both velocity and density, is given in two appendices.

For the systematic development of feedback flow controllers, a numerical model that captures the dynamic behaviour of the flow field to be controlled is required. This poses a particular challenge for flow fields where the dynamic behaviour is nonlinear, and the governing equations cannot easily be solved in closed form. This has led to many versions of low-dimensional modelling techniques, which we extend in this work to represent better the impact of actuation on the flow. For the benchmark problem of a circular cylinder wake in the laminar regime, we introduce a novel extension to the proper orthogonal decomposition (POD) procedure that facilitates mode construction from transient data sets. We demonstrate the performance of this new decomposition by applying it to a data set from the development of the limit cycle oscillation of a circular cylinder wake simulation as well as an ensemble of transient forced simulation results. The modes obtained from this decomposition, which we refer to as the double POD (DPOD) method, correctly track the changes of the spatial modes both during the evolution of the limit cycle and when forcing is applied by transverse translation of the cylinder. The mode amplitudes, which are obtained by projecting the original data sets onto the truncated DPOD modes, can be used to construct a dynamic mathematical model of the wake that accurately predicts the wake flow dynamics within the lock-in region at low forcing amplitudes. This low-dimensional model, derived using nonlinear artificial neural network based system identification methods, is robust and accurate and can be used to simulate the dynamic behaviour of the wake flow. We demonstrate this ability not just for unforced and open-loop forced data, but also for a feedback-controlled simulation that leads to a 90% reduction in lift fluctuations. This indicates the possibility of constructing accurate dynamic low-dimensional models for feedback control by using unforced and transient forced data only.

A generalized model of flow in meandering subaqueous and subaerial channels is developed. The conservation equations of mass and momentum are depth/layer integrated, normalized, and represented as deviations from a straight base state. This allows the determination of integrable forms which can be solved at both linear and nonlinear levels. The effects of various flow and geometric parameters on the flow dynamics are studied. Although the model is not limited to any specific planform, this study focuses on sine-generated curves. In analysing the flow patterns, the turbidity current of the subaqueous case is simplified to a conservative density flow with water entrainment from above neglected. The subaqueous model thus formally corresponds to a subcritical or only mildly supercritical mud-rich turbidity current. By extension, however the analysis can be applied to a depositional or erosional current carrying sand that is changing only slowly in the streamwise direction. By bringing the subaqueous and subaerial cases into a common form, flow behaviour in the two environments can be compared under similar geometric and boundary conditions. A major difference between the two cases is the degree of superelevation of channel flow around bends, which is modest in the subaerial case but substantial in the subaqueous case. Another difference concerns Coriolis effects: some of the largest subaqueous meandering systems are so large that Coriolis effects can become important. The model is applied to meander bends on the youngest channel in the mid-fan region of the Amazon Fan and a mildly sinuous bend of the North-West Atlantic Mid-Ocean Channel. In the absence of specific data on the turbid flows that created the channel, the model can be used to make inferences about the flow, and in particular the range of values of flow velocity and sediment concentration that would allow the growth and downfan migration of meander bends.

Equations governing the current system in the upper layers of oceans and lakes were derived by Craik & Leibovich (1976). These incorporate the dominant effects of both wind and waves. Solutions comprising the mean wind-driven current and a system of ‘Langmuir’ cells aligned parallel to the wind were found for cases in which the wave field consisted of just a pair of plane waves. However, it was not clear that such cellular motions would persist for the more realistic case of a continuous wave spectrum.

The present paper shows that, in the latter case, infinitesimal spanwise periodic perturbations will grow on account of an instability mechanism. Mathematically, the instability is closely similar to the onset of thermal convection in horizontal fluid layers. Physically, the mechanism is governed by kinematical processes involving the mean (Eulerian) wind-driven current and the (Lagrangian) Stokes drift associated with the waves. The relationship of this mechanism to instability models of Garrett and Gammelsrød is clarified.

We present an experimental approach used to measure both normal stress differences and the particle phase contribution to the normal stresses in suspensions of non-Brownian hard spheres. The methodology consists of measuring the radial profile of the normal stress along the velocity gradient direction in a torsional flow between two parallel discs. The values of the first and the second normal stress differences, ${N}_{1} $ and ${N}_{2} $, are deduced from the measurement of the slope and of the origin ordinate. The measurements are carried out for a wide range of particle volume fractions (between 0.2 and 0.5). As expected, ${N}_{2} $ is measured to be negative but ${N}_{1} $ is found to be positive. We discuss the validity of the method and present numerous tests that have been carried out in order to validate our results. The experimental setup also allows the pore pressure to be measured. Then, subtracting the pore pressure from the total stress, ${\mbrm{\Sigma} }_{\mathbf{22} } $, the contribution of the particles to the normal stress ${ \mbrm{\Sigma} }_{\mathbf{22} }^{\mathbi{p}} $ is obtained. Most of our results compare well with the different experimental and numerical data present in the literature. In particular, our results show that the magnitude of the particle stress tensor component and their dependence on the particle volume fraction used in the suspension model balance proposed by Morris & Boulay (J. Rheol., vol. 43, 1999, p. 1213) are suitable.

Consider the gravity-driven flow of a thin liquid film down a vertical fibre. A model of two coupled evolution equations for the local film thickness h and the local flow rate q is formulated within the framework of the long-wave and boundary-layer approximations. The model accounts for inertia and streamwise viscous diffusion. Evolution equations obtained by previous authors are recovered in the appropriate limit. Comparisons to experimental results show good agreement in both linear and nonlinear regimes. Viscous diffusion effects are found to have a stabilizing dispersive effect on the linear waves. Time-dependent computations of the spatial evolution of the film reveal a strong influence of streamwise viscous diffusion on the dynamics of the flow and the wave selection process.

Most previous numerical studies on capsule dynamics in shear flow have ignored the role of membrane viscosity. Here we present a numerical method for large deformation of capsules using a Kelvin–Voigt viscoelastic model for the membrane. After introducing the model and the related numerical implementation, we present a comprehensive analysis of the influence of the membrane viscosity on buckling, deformation and dynamics. We observe that the membrane viscosity leads to buckling in the range of shear rate in which no buckling is observed for capsules with purely elastic membrane. For moderate to large shear rates, the wrinkles on the capsule surface appear in the same range of the membrane viscosity that was reported earlier for artificial capsules and red blood cells based on experimental measurements. In order to obtain stable shapes, it is necessary to introduce the bending stiffness. It is observed that the range of the bending stiffness required is also in the same range as that reported for the red blood cells, but considerably higher than that estimated for artificial capsules. Using the stable shapes obtained in the presence of bending stiffness, we analyse the influence of membrane viscosity on deformation, inclination and tank-treading frequency of initially spherical capsules. Membrane viscosity is observed to reduce the capsule deformation, and introduce a damped oscillation in time-dependent deformation and inclination. The time-averaged inclination angle shows a non-monotonic trend with an initial decrease reaching a minimum and a subsequent increase with increasing membrane viscosity. A similar non-monotonic trend is also observed in the tank-treading frequency. We then consider the influence of the membrane viscosity on the unsteady dynamics of an initially oblate capsule. The dynamics is observed to change from a swinging motion to a tumbling motion with increasing membrane viscosity. Further, a transient dynamics is also observed in which a capsule starts with one type of dynamics, but settles with a different dynamics over a long time.

Individual travelling cavitation bubbles generated on two axisymmetric headforms were detected using a surface electrode probe. The growth and collapse of the bubbles were studied photographically, and these observations are related to the pressure fields and viscous flow patterns associated with each headform. Measurements of the acoustic impulse generated by the bubble collapse are analysed and found to correlate with the maximum volume of the bubble for each headform. These results are compared to the observed bubble dynamics and numerical solutions of the Rayleigh–Plesset equation. Finally, the cavitation nuclei flux was measured and predicted cavitation event rates and bubble maximum size distributions are compared with the measurements of these quantities.

Experimental results on free convection in a vertical annulus filled with a saturated porous medium are reported for height-to-gap ratios of 1.46, 1 and 0.545, and radius ratio of 5.338. In these experiments, the inner and outer walls are maintained at constant temperatures. The use of several fluid–solid combinations indicates a divergence in the Nusselt-number–Rayleigh-number relation, as also reported by previous investigators for horizontal layers and vertical cavities. The reason for this divergence is the use of the stagnant thermal conductivity of the fluid-filled solid matrix. A simple model is presented to obtain an effective thermal conductivity as a function of the convective state, and thereby eliminate the aforementioned divergence. A reasonable agreement between experimentally and theoretically determined Nusselt numbers is then achieved for the present and previous experimental results. It is thus concluded that a unique relationship exists between the Nusselt and Rayleigh numbers unless Darcy's law is inapplicable. The factors that influence the breakdown of Darcian behaviour are characterized and their effects on heat-transfer rates are explained. It is observed that, once the relation between the Nusselt and Rayleigh numbers branches out from that obtained via the mathematical formulation based on Darcy's law, its slope approaches that for a fluid-filled enclosure of the same geometry when the Rayleigh number is large enough. An iterative scheme is also presented for estimation of effective thermal conductivity of a saturated porous medium by using the existing results for overall heat transfer.