An approach is presented that uses velocimetry data to estimate accurately the spatial distribution of viscosity in steady laminar parallel flows of incompressible linearly viscous fluids. The approach is generally applicable to Newtonian fluids with spatially varying viscosity or to particle-suspension flows where a non-uniform distribution of the particles contributes to spatial variations in the local effective viscosity of the suspension. Emphasis is placed on the application of these methods to steady axisymmetric blood flow in cylindrical glass capillary tubes and microvessels. In this context, the spatial variations in viscosity over the vessel cross-section are predicted where it is assumed that the rheological properties associated with a heterogeneous red blood cell suspension can be well approximated by a continuous generalized linearly viscous fluid having a spatially non-uniform viscosity. For such a fluid, an expression for the viscosity profile over the vessel cross-section is derived that satisfies the conservation principles of mass and momentum and depends upon the a priori determined velocity distribution, which is extracted from fluorescent micro-particle image velocimetry data obtained from microvessels in vivo. These profiles provide useful information about dynamic, kinematic and rheological properties of the flow that include expressions for the axial pressure-gradient component, the local shear stress distribution, and the relative apparent viscosity. In microvessels, the effect of the glycocalyx surface layer on the vessel wall is also accounted for in the analysis by modelling the layer as a uniformly thick porous medium. Velocimetry data are presented from in vivo measurements made in venules after the application of a light-dye treatment to degrade the glycocalyx. Results reveal that these methods are sufficiently sensitive to detect a reduction in glycocalyx thickness of ${\sim}\,0.3\,\umu{\rm m}$, which represents a fractional decrease in thickness of ${\sim}\,60$–70% when compared with results from a separately published data set obtained from venules having an intact glycocalyx.

The linear stability of Bingham-plastic fluid flow between two concentric cylinders rotating independently and with axial sliding of the inner cylinder (spiral Couette flow) is studied. Bingham fluid exhibits a yield stress in addition to the plastic viscosity, which has some inhibiting effects on the competition between the centrifugal and shear instability mechanisms owing to the inter-relationship of the azimuthal and axial velocities. Islands of instability, which are found in the spiral Couette flow of Newtonian fluids, may not exist owing to the effect of yield stress. The possibility of the yield surface falling between the cylinders is analysed. Although small perturbation waves appearing on the yield surface are considered, the yield surface, which has been treated as a free surface, has little effect on the flow stability. The effects of the axisymmetric and non-axisymmetric perturbation on flow stability are both presented. Both the rotation of the outer cylinder and a decrease of the gap between the cylinders have stabilizing effects.

A low-Reynolds-number zero-pressure-gradient incompressible turbulent boundary layer was investigated using a volumetric imaging technique. The Reynolds number based on momentum thickness was 700. The flow was tagged with a passive scalar from two spanwise dye slots to distinguish between fluid motions originating in the inner and outer portions of the boundary layer. The resulting volumetric scalar field was interrogated using a laser sheet scanner developed for this study. Two- and three-dimensional time-dependent visualizations of a 50 volume time series are presented (equivalent to 17$\delta$ in length). In the outer portion of the boundary layer, scalar structures were observed to lie along lines in the ($x$, $z$)-plane, inclined to the streamwise ($x$-)direction in the range $\pm 50^\circ$. The ejection of brightly dyed fluid packets from the near-wall region was observed to be spatially organized, and related to the passage of the large-scale scalar structures.

The spatial growth of small perturbations developing on a fully developed base flow in a duct with two inhomogeneous cross-flow directions is examined. For a laminar mean flow it is shown that optimally configured vortices at an upstream cross-section induce large transient amplification of a disturbance energy norm downstream. Such a linear growth is a likely initial stage of transition in ducts for which the cross-section is square or of moderate aspect ratio, asymptotically stable according to conventional eigen-analyis. With the increase of the cross-sectional aspect ratio the transient growth results of the plane channel flow case are approached. The optimization methodology is then employed to study the transient amplification of large-scale secondary structures developing in a square duct for which the unidirectional base motion has a turbulent-like profile. Under the hypothesis that fine-scale turbulence does not affect the growth of large-scale secondary flows, it is shown that cross-stream vortices are sustained by the mean flow and grow maximally over a streamwise distance of several thousand viscous units of length.

We consider the steady sedimentation under gravity of a viscous drop, suspended in a viscous liquid, along a plane tilted at a small angle $\alpha$ to the horizontal. The drop does not wet the wall but is supported by a thin lubricating film of liquid. In the Stokes-flow limit, the problem is parameterized by $\alpha$, the ratio $B$ of buoyancy to capillary forces (a Bond number) and a viscosity ratio $\lambda$. Provided $B$ is not too large ($B\,{\ll}\,\alpha^{-1}$ in two dimensions, $B\,{\ll}\,\alpha^{-4/3}$ in three dimensions), the drop's motion can be described asymptotically by combining a capillary-statics approximation for the drop shape away from the wall, lubrication theory for the thin film and a combination of lubrication theory and a half-plane boundary-integral method for the drop interior.

Systematic scaling arguments for both two- and three-dimensional drops, supported by detailed calculations, are used to survey $(B,\lambda)$-parameter space for fixed $\alpha\,{\ll}\,1$. We find a strong coupling between drop shape (ranging from nearly round to a flattened pancake), kinematics (including slipping, sliding, rolling and tank-treading motions) and the site of dominant viscous dissipation (the edges of the thin film, the bulk of the thin film or the drop interior). Predictions of drop speed and shape are compared with available experimental and computational data.

We describe here liquid drops in rotation, and first classify the different shapes possible. Then, we study the behaviour of liquid marbles, which are droplets coated with hydrophobic grains, running down inclines. Because these marbles roll as they move, this device allows us to achieve revolving drops. We report the different forms generated during the descent, and quantify their size by way of scaling arguments. We finally focus on the transformations which may occur between different drop shapes.

The aero-optical distortions caused by compressible flows have been used by researchers for flow diagnostics and accepted by designers of airborne optical systems as a performance penalty. In order to estimate these distortions, an understanding of the optical distortion mechanism is required. This article examines the mechanisms which produce a variable-density field (and accompanying index-of-refraction field) in a nearly incompressible shear-layer flow. The two-dimensional-shear-layer velocity field was approximated using a discrete vortex model. From this ‘known’ velocity field, the pressure and density fields were determined by iteratively solving the unsteady Euler equations. The resulting index-of-refraction field produced simulated schlieren images which closely resemble experimental schlierens. Optical wavefronts computed from the simulation reasonably match the behaviour of large-scale aberrations measured in a transonic wind tunnel. Small-scale distortions in the experimental data may have been caused by boundary layers on the splitter plate and tunnel walls or by three-dimensional effects that were not simulated.

In the second-moment-closure (SMC) method of turbulence modelling, measures to ensure a realizable turbulence model are currently limited to constraining the Reynolds stress to physically plausible values. These constraints address neither the realizability of the other statistical moments (e.g. pressure–strain correlation) nor the underlying causes of unrealizable Reynolds stress. For achieving increased consistency with flow physics in SMC, we propose the additional requirement that the closure model for each of the unclosed statistical moments in the Reynolds stress equation be individually realizable. We then proceed to derive two realizability constraints on the rapid-pressure statistics: (i) the rapid pressure-gradient variance must be positive which leads to the requirement that the $M_{ijkl}$ tensor must be positive semi-definite, and (ii) the rapid pressure–strain correlation closure must satisfy the Schwarz inequality. Calculations with currently popular models show that unrealizable rapid-pressure–strain correlation precedes unrealizable Reynolds stress. It is also demonstrated that when the Launder, Reece and Rodi (LRR) rapid-pressure–strain correlation model is modified (truncated) to satisfy the new constraints, Reynolds stress realizability is always preserved. These findings clearly indicate that an unrealizable closure model is the cause of Reynolds stress realizability violation and highlight the importance of the new constraints.

The linear stability of finite-amplitude surface solitary waves with respect to long-wavelength transverse perturbations is examined by asymptotic analysis for small wavenumbers of perturbations. The sufficient condition for the transverse instability is explicitly derived, and it is found that there exist transversely unstable surface solitary waves whose amplitude-to-depth ratio is greater than 0.713. This critical ratio is well below that for the one-dimensional instability (${=}\,0.781$) obtained by Tanaka (Phys. Fluids, 1986, vol. 29, p. 650).

Contraction of a filament of an incompressible Newtonian liquid in a passive ambient fluid is studied computationally to provide insights into the dynamics of satellite drops created during drop formation. This free boundary problem, which is composed of the Navier–Stokes system and the associated initial and boundary conditions that govern the evolution in time of the filament shape and the velocity and pressure fields within it, is solved by the method of lines incorporating the finite element method for spatial discretization. The finite element algorithm developed here utilizes an adaptive elliptic mesh generation technique that is capable of tracking the dynamics of the filament up to the incipience of pinch-off without the use of remeshing. The correctness of the algorithm is verified by demonstrating that its predictions accord with (a) previously published results of Basaran (1992) on the analysis of finite-amplitude oscillations of viscous drops, (b) simulations of the dynamics of contracting filaments carried out with the well-benchmarked algorithm of Wilkes et al. (1999), and (c) scaling laws governing interface rupture and transitions that can occur from one scaling law to another as pinch-off is approached. In dimensionless form, just two parameters govern the problem: the dimensionless half-length $L_o$ and the Ohnesorge number $Oh$ which measures the relative importance of viscous force to capillary force. Regions of the parameter space are identified where filaments (a) contract to a sphere without breaking into multiple droplets, (b) break via the so-called endpinching mechanism where daughter drops pinch-off from the ends of the main filament, and (c) break after undergoing a series of complex oscillations. Predictions made with the new algorithm are also compared to those made with a model based on the slender-jet approximation. A region of the parameter space is found where the slender-jet approximation fares poorly, and its cause is elucidated by examination of the vorticity dynamics and flow fields within contracting filaments.

We use a dynamical systems approach to extend Prandtl's steady separation criterion to two-dimensional unsteady flows with no-slip boundaries. Viewing separation profiles as non-hyperbolic unstable manifolds in the Lagrangian frame, we obtain explicit Eulerian formulae for the location of flow separation and reattachment on fixed and moving boundaries. We also derive high-order approximations for the unsteady separation profile in the vicinity of the boundary. Our criteria and formulae only use the derivatives of the velocity field along the boundary, and hence are of use in monitoring and controlling separation. In particular, we predict unsteady flow separation points and separation angles from distributed pressure and skin-friction measurements along the wall. As an example, we predict and verify separation points and separation profiles in variants of a two-dimensional oscillating separation-bubble flow.

The characteristics of the unsteady boundary layer and stall events occurring on an oscillating NACA 0012 airfoil were investigated by using closely spaced multiple hot-film sensor arrays at $Re\,{=}\,1.35\,{\times}\,10^{5}$. Aerodynamic forces and pitching moments, integrated from surface pressure measurements, and smoke-flow visualizations were also obtained to supplement the hot-film measurements. Special attention was focused on the behaviour of the spatial-temporal progression of the locations of the boundary-layer transition and separation, and reattachment and relaminarization points, compared to the static values, for a range of oscillation frequency and amplitude both prior to, during and after the stall. The initiation, growth and rearward convection of a leading-edge vortex, and the role of the laminar separation bubble leading to the dynamic stall, as well as the mechanisms responsible for the stall events observed at different test conditions were also characterized. The hot-film measurements were also correlated with the aerodynamic load and pitching moment results to quantify the values of lift increment and stall angle delay as a result of the observed boundary layer and stall events. The results reported here provide an insight into the detailed nature of the unsteady boundary-layer events as well as the stalling mechanisms at work at different stages in the dynamic-stall process.

This paper investigates the control of self-excited oscillations in spatially developing flow systems such as jets and wakes using ${\mathcal H}_{\infty}$ control theory on a complex Ginzburg–Landau (CGL) model. The coefficients used in this one-dimensional equation, which serves as a simple model of the evolution of hydrodynamic instability waves, are those selected by Roussopoulos & Monkewitz (Physica D 1996, vol. 97, p. 264) to model the behaviour of the near-wake of a circular cylinder. Based on noisy measurements at a point sensor typically located inside the cylinder wake, the compensator uses a linear ${\mathcal H}_{\infty}$ filter based on the CGL model to construct a state estimate. This estimate is then used to compute linear ${\mathcal H}_{\infty}$ control feedback at a point actuator location, which is typically located upstream of the sensor. The goal of the control scheme is to stabilize the system by minimizing a weighted average of the ‘system response’ and the ‘control effort’ while rigorously bounding the response of the controlled linear system to external disturbances. The application of such modern control and estimation rules stabilizes the linear CGL system at Reynolds numbers far above the critical Reynolds number $Re_c \,{\approx}\, 47$ at which linear global instability appears in the uncontrolled system. In so doing, many unstable modes of the uncontrolled CGL system are linearly stabilized by the single actuator/sensor pair and the model-based feedback control strategy. Further, the linear performance of the closed-loop system, in terms of the relevant transfer function norms quantifying the linear response of the controlled system to external disturbances, is substantially improved beyond that possible with the simple proportional measurement feedback proposed in previous studies. Above $Re\,{\approx}\, 84$, the ${\mathcal H}_{\infty}$ control designs significantly outperform the corresponding ${\mathcal H}_2$ control designs in terms of their ability to stabilize the CGL system in the presence of worst-case disturbances. The extension of these control and estimation rules to the nonlinear CGL system on its attractor (a simple limit cycle) stabilizes the full nonlinear system back to the stationary state at Reynolds numbers up to $Re\,{\approx}\, 97$ using a single actuator/sensor pair, fixed-gain linear feedback and an extended Kalman filter incorporating the system nolinearity.