We examine the fluid mechanics of drinking in nature. We classify the drinking strategies of a broad range of creatures according to the principal forces involved, and present physical pictures for each style. Simple scaling arguments are developed and tested against existing data. While suction is the most common drinking strategy, various alternative styles have evolved among creatures whose morphological, physiological and environmental constraints preclude it. Particular attention is given to creatures small relative to the capillary length, whose drinking styles rely on relatively subtle interfacial effects. We also discuss attempts to rationalize various drinking strategies through consideration of constrained optimization problems. Some biomimetic applications are discussed.

Fluid mechanics plays a vital role in early vertebrate embryo development, an example being the establishment of left–right asymmetry. Following the dorsal–ventral and anterior–posterior axes, the left–right axis is the last to be established; in several species it has been shown that an important process involved with this is the production of a left–right asymmetric flow driven by ‘whirling’ cilia. It has previously been established in experimental and mathematical models of the mouse ventral node that the combination of a consistent rotational direction and posterior tilt creates left–right asymmetric flow. The zebrafish organizing structure, Kupffer’s vesicle, has a more complex internal arrangement of cilia than the mouse ventral node; experimental studies show that the flow exhibits an anticlockwise rotational motion when viewing the embryo from the dorsal roof, looking in the ventral direction. Reports of the arrangement and configuration of cilia suggest two possible mechanisms for the generation of this flow from existing axis information: (a) posterior tilt combined with increased cilia density on the dorsal roof; and (b) dorsal tilt of ‘equatorial’ cilia. We develop a mathematical model of symmetry breaking cilia-driven flow in Kupffer’s vesicle using the regularized Stokeslet boundary element method. Computations of the flow produced by tilted whirling cilia in an enclosed domain suggest that a possible mechanism capable of producing the flow field with qualitative and quantitative features closest to those observed experimentally is a combination of posteriorly tilted roof and floor cilia, and dorsally tilted equatorial cilia.

We study theoretically the hydrodynamics of a fluid drop containing oriented filaments endowed with active contractile or extensile stresses and placed on a solid surface. The active stresses alter qualitatively the wetting properties of the drop, leading to new spreading laws and novel static drop shapes. Candidate systems for testing our predictions include cytoskeletal extracts with motors and ATP, suspensions of bacteria or pulsatile cells, or fluids laden with artificial self-propelled colloids.

Many motile bacteria display wiggling trajectories, which correspond to helical swimming paths. Wiggling trajectories result from flagella pushing off-axis relative to the cell body and making the cell wobble. The spatial extent of wiggling trajectories is controlled by the swimming velocity and flagellar torque, which leads to rotation of the cell body. We employ the method of regularized stokeslets to investigate the wiggling trajectories produced by flagellar bundles, which can form at many locations and orientations relative to the cell body for peritrichously flagellated bacteria. Modelling the bundle as a rigid helix with fixed position and orientation relative to the cell body, we show that the wiggling trajectory depends on the position and orientation of the flagellar bundle relative to the cell body. We observe and quantify the helical wiggling trajectories of Bacillus subtilis, which show a wide range of trajectory pitches and radii, many with pitch larger than 4 . For this bacterium, we show that flagellar bundles with fixed orientation relative to the cell body are unlikely to produce wiggling trajectories with pitch larger than 4 . An estimate based on torque balance shows that this constraint on pitch is a result of the large torque exerted by the flagellar bundle. On the other hand, multiple rigid bundles with fixed orientation, similar to those recently observed experimentally, are able to produce wiggling trajectories with large pitches.

Since free Stokesian swimming does no work external to fluid and body, the classical thermodynamic efficiency of this activity is zero. This paper introduces a potential thermodynamic efficiency by partially tethering the body so that work is done externally and instantaneously. We compare the resulting efficiency with other definitions utilized in Stokes flow, extend the instantaneous definition to encompass a full swimming stroke, and compute it for propulsion of a spherical body by a helical flagellum.

Microorganisms often swim upwards due to the cell’s phototaxis, chemotaxis or geotaxis, in flow fields with vertical velocity gradients. In this study, the vertical dispersion of model microorganisms was investigated under horizontal shear conditions. A microorganism was modelled as a spherical squirmer with or without bottom-heaviness. First, the three-dimensional movement of 100 identical squirmers in a homogeneous suspension was computed by the Stokesian dynamics method. The results show that the dispersion of squirmers is strongly affected by the swimming velocity and bottom-heaviness of the cells and the shear rate of the background flow. The vertical diffusion is considerably smaller than the horizontal diffusion. Interestingly, the vertical diffusion decreases as the volume fraction and the stresslet of squirmers decrease, which is opposite of the tendency in diffusion with no background flow. Next, a continuum model of a suspension of squirmers was developed using the diffusion tensor and the drift velocity to simulate the spatial distribution of squirmers in macroscopic flow fields. The results of the continuum model illustrate that the gyrotactic trapping found by Durham, Kessler & Stocker (Science, vol. 323, 2009, pp. 1067–1070) also appears in the present model considering cell–cell hydrodynamic interactions. In the case of horizontal Poiseuille flow, the volume fraction of bottom-heavy cells in the channel becomes considerably larger than that at the inlet. These fundamental findings are helpful for understanding the distribution of microorganisms in various water regimes in nature and industry.

Marangoni propulsion is a form of locomotion wherein an asymmetric release of surfactant by a body located at the surface of a liquid leads to its directed motion. We present in this paper a mathematical model for Marangoni propulsion in the viscous regime. We consider the case of a thin rigid circular disk placed at the surface of a viscous fluid and whose perimeter has a prescribed concentration of an insoluble surfactant, to which the rest of its surface is impenetrable. Assuming a linearized equation of state between surface tension and surfactant concentration, we derive analytically the surfactant, velocity and pressure fields in the asymptotic limit of low capillary, Péclet and Reynolds numbers. We then exploit these results to calculate the Marangoni propulsion speed of the disk. Neglecting the stress contribution from Marangoni flows is seen to over-predict the propulsion speed by 50 %.

Motivated by our interest in understanding collective behaviour and self-organization resulting from hydrodynamic interactions, we investigate the two-dimensional dynamics of horizontal arrays of settling cylinders at intermediate Reynolds numbers. To simulate these dynamics, we develop a direct numerical simulation based on the immersed interface method. A novel aspect of our method is its ability to efficiently and accurately couple the dynamics of the freely moving objects with the fluid. We report the falling configuration and the wake pattern of the array, and investigate their dependence on the number of particles, , as well as the initial inter-particle spacing, . We find that, in the case of odd-numbered arrays, the middle cylinder is always leading, whereas in the case of even-numbered arrays, the steady-state shape is concave-down. In large arrays , the outer pairs tend to cluster. In addition, we analyse detailed kinematics, wakes and forces of three settling cylinders. We find that the middle one experiences a higher drag force in the presence of neighbouring cylinders, compared to an isolated settling cylinder, resulting in a decrease in its settling velocity. For a small initial spacing , the middle cylinder experiences a strong sideway repulsive force, the magnitude of which increases with decreasing . During the fall, the left and right cylinders rotate outwards and shed vortices in anti-phase.

We calculate optimal driving motions for a fin ray in a two-dimensional inviscid fluid, which is a model for caudal fin locomotion. The driving is sinusoidal in time, and consists of heaving, pitching and a less-studied motion called ‘shifting’. The optimal phases of shifting relative to heaving and pitching for maximum thrust power and efficiency are calculated. The optimal phases undergo jumps at resonant combinations of fin ray bending and shear moduli, and are nearly constant in regions between resonances. In two examples, pitching- and heaving-based motions converge with the addition of optimal shifting. Shifting provides an order-one increase in output power and efficiency.

Recent experiments have shown that when a near-hemispherical lipid vesicle attached to a solid surface is subjected to a simple shear flow it exhibits a pattern of membrane circulation much like a dipole vortex. This is in marked contrast to the toroidal circulation that would occur in the related problem of a drop of immiscible fluid attached to a surface and subjected to shear. This profound difference in flow patterns arises from the lateral incompressibility of the membrane, which restricts the observable flows to those in which the velocity field in the membrane is two-dimensionally divergence free. Here we study these circulation patterns within the simplest model of membrane fluid dynamics. A systematic expansion of the flow field based on Papkovich–Neuber potentials is developed for general viscosity ratios between the membrane and the surrounding fluids. Comparison with experimental results (Vézy, Massiera & Viallat, Soft Matt., vol. 3, 2007, pp. 844–851) is made, and it is shown how such studies could allow measurements of the membrane viscosity. Issues of symmetry-breaking and pattern selection are discussed.

The motion and deformation of a spherical elastic capsule freely flowing in a pore of comparable dimension is studied. The thin capsule membrane has a neo-Hookean shear softening constitutive law. The three-dimensional fluid–structure interactions are modelled by coupling a boundary integral method (for the internal and external fluid motion) with a finite element method (for the membrane deformation). In a cylindrical tube with a circular cross-section, the confinement effect of the channel walls leads to compression of the capsule in the hoop direction. The membrane then tends to buckle and to fold as observed experimentally. The capsule deformation is three-dimensional but can be fairly well approximated by an axisymmetric model that ignores the folds. In a microfluidic pore with a square cross-section, the capsule deformation is fully three-dimensional. For the same size ratio and flow rate, a capsule is more deformed in a circular than in a square cross-section pore. We provide new graphs of the deformation parameters and capsule velocity as a function of flow strength and size ratio in a square section pore. We show how these graphs can be used to analyse experimental data on the deformation of artificial capsules in such channels.

A two-dimensional model is used to simulate the motion and deformation of a single mammalian red blood cell (RBC) flowing close to the wall of a microvessel, taking into account the effects of a porous endothelial surface layer (ESL) lining the vessel wall. Migration of RBCs away from the wall leads to the formation of a cell-depleted layer near the wall, which has a large effect on the resistance to blood flow in microvessels. The objective is to examine the mechanical factors causing this migration, including the effects of the ESL. The vessel is represented as a straight parallel-sided channel. The RBC is represented as a set of interconnected viscoelastic elements, suspended in plasma, a Newtonian fluid. The ESL is represented as a porous medium, and plasma flow in the layer is computed using the Brinkman approximation. It is shown that an initially circular cell positioned close to the ESL in a shear flow is deformed into an asymmetric shape. This breaking of symmetry leads to migration away from the wall. With increasing hydraulic resistivity of the layer, the rate of lateral migration increases. It is concluded that mechanical interactions of RBCs flowing in microvessels with a porous wall layer may reduce the rate of lateral migration and hence reduce the width of the cell-depleted zone external to the ESL, relative to the cell-depleted zone that would be formed if the interface between the ESL and free-flowing plasma were replaced by an impermeable boundary.

We consider axially uniform, two-phase flow through a rigid curved tube in which a fluid (air) core is surrounded by a film of a second, immiscible fluid (water): a simplified model for flow in a conducting airway of the lung. Jensen (1997) showed that, in the absence of a core flow, surface tension drives the system towards a configuration in which the film thickness tends to zero on the inner wall of the bend. In the present work, we demonstrate that the presence of a core flow, driven by a steady axial pressure gradient, allows the existence of steady states in which the film thickness remains finite, a consequence of the fact that the tangential stresses at the interface, imposed by secondary flows in the core, can oppose the surface-tension-driven flow. For sufficiently strong surface tension, the steady configurations are symmetric about the plane containing the tube’s centreline, but as the surface tension decreases the symmetry is lost through a pitchfork bifurcation, which is closely followed by a limit point on the symmetric solution branch. This solution structure is found both in simulations of the Navier–Stokes equations and a thin-film model appropriate for weakly curved tubes. Analysis of the thin-film model reveals that the bifurcation structure arises from a perturbation of the translational degeneracy of the interface location in a straight tube.

Disease states characterized by airway fluid occlusion and pulmonary surfactant insufficiency, such as respiratory distress syndrome, have a high mortality rate. Understanding the mechanics of airway reopening, particularly involving surfactant transport, may provide an avenue to increase patient survival via optimized mechanical ventilation waveforms. We model the occluded airway as a liquid-filled rigid tube with the fluid phase displaced by a finger of air that propagates with both mean and sinusoidal velocity components. Finite-time Lyapunov exponent (FTLE) fields are employed to analyse the convective transport characteristics, taking note of Lagrangian coherent structures (LCSs) and their effects on transport. The Lagrangian perspective of these techniques reveals flow characteristics that are not readily apparent by observing Eulerian measures. These analysis techniques are applied to surfactant-free velocity fields determined computationally, with the boundary element method, and measured experimentally with micro particle image velocimetry (-PIV). We find that the LCS divides the fluid into two regimes, one advected upstream (into the thin residual film) and the other downstream ahead of the advancing bubble. At higher oscillatory frequencies particles originating immediately inside the LCS experience long residence times at the air–liquid interface, which may be conducive to surfactant transport. At high frequencies a well-mixed attractor region is identified; this volume of fluid cyclically travels along the interface and into the bulk fluid. The Lagrangian analysis is applied to velocity data measured with 0.01 mg ml−1 of the clinical pulmonary surfactant Infasurf in the bulk fluid, demonstrating flow field modifications with respect to the surfactant-free system that were not visible in the Eulerian frame.

We study numerically the steady creeping motion of Bingham liquid plugs in two-dimensional channels as a model of mucus behaviour during airway reopening in pulmonary airways. In addition to flow analysis related to propagation of the plug, the stress distribution on the wall is studied for better understanding of potential airway epithelial cell injury mechanisms. The yield stress behaviour of the fluid was implemented through a regularized constitutive equation. The capillary number, , and the Bingham number, , which is the ratio of the yield stress to a characteristic viscous stress, varied over the ranges 0.025–0.1 and 0–1.5, respectively. For the range of parameters studied, it was found that, while the yield stress reduces the magnitude of the shearing along the wall, it can magnify the amplitude of the wall shear stress gradient significantly, and also it can elevate the magnitude of the wall shear stress and wall pressure gradient up to 30 % and 15 %, respectively. Therefore, the motion of mucus plugs can be more damaging to the airway epithelial cells due to the yield stress properties of mucus. The yield stress also modifies the profile of the plug where the amplitude of the capillary waves at the leading meniscus decreases with increase in . Other findings are that: the thickness of the static film increases with increasing ; the driving pressure difference increases linearly with ; and increasing extends any wall stagnation point beneath the leading meniscus to an unyielded line segment beneath the leading meniscus. With an increase in , the unyielded areas appear and grow in the adjacent wall film as well as the core region of the plug between the two menisci. The plug length, , mostly modifies the topology of the yield surfaces. It was found that the unyielded area in the core region between the two menisci grows as the plug length decreases. The very short Bingham plug behaves like a solid lamella. In all computed liquid plugs moving steadily, the von Mises stress attains its maximum value near the interface of the leading meniscus in the transition region. For Bingham plugs moving very slowly, , the driving pressure is non-zero.

The effects of vascular rarefaction (the loss of small arteries) on the circulation of blood are studied using a multiscale mathematical model that can predict blood flow and pressure in the systemic and pulmonary arteries. We augmented a model originally developed for the systemic arteries by Olufsen and coworkers and Ottesen et al. (2004) to (a) predict flow and pressure in the pulmonary arteries, and (b) predict pressure propagation along the small arteries in the vascular beds. The systemic and pulmonary arteries are modelled as separate bifurcating trees of compliant and tapering vessels. Each tree is divided into two parts representing the ‘large’ and ‘small’ arteries. Blood flow and pressure in the large arteries are predicted using a nonlinear cross-sectional-area-averaged model for a Newtonian fluid in an elastic tube with inflow obtained from magnetic resonance measurements. Each terminal vessel within the network of the large arteries is coupled to a vascular bed of small ‘resistance’ arteries, which are modelled as asymmetric structured trees with specified area and asymmetry ratios between the parent and daughter arteries. For the systemic circulation, each structured tree represents a specific vascular bed corresponding to major organs and limbs. For the pulmonary circulation, there are four vascular beds supplied by the interlobar arteries. This paper presents the first theoretical calculations of the propagation of the pressure and flow waves along systemic and pulmonary large and small arteries. Results for all networks are in agreement with published observations. Two studies were done with this model. First, we showed how rarefaction can be modelled by pruning the tree of arteries in the microvascular system. This was done by modulating parameters used for designing the structured trees. Results showed that rarefaction leads to increased mean and decreased pulse pressure in the large arteries. Second, we investigated the impact of decreasing vessel compliance in both large and small arteries. Results showed that the effects of decreased compliance in the large arteries far outweigh the effects observed when decreasing the compliance of the small arteries. We further showed that a decrease of compliance in the large arteries results in pressure increases consistent with observations of isolated systolic hypertension, as occurs in ageing.

We determine optimal inflow boundary perturbations to steady flow through a straight inflexible tube with a smooth axisymmetric stenosis at a bulk-flow Reynolds number , for which the flow is asymptotically stable. The perturbations computed produce an optimal gain, i.e. kinetic energy in the domain at a given time horizon normalized by a measure of time-integrated energy on the inflow boundary segment. We demonstrate that similarly to the optimal initial condition problem, the gain can be interpreted as the leading singular value of the forward linearized operator that evolves the boundary conditions to the final state at a fixed time. In this investigation we restrict our attention to problems where the temporal profile of the perturbations examined is a product of a Gaussian bell and a sinusoid, whose frequency is selected to excite axial wavelengths similar to those of the optimal initial perturbations in the same geometry. Comparison of the final state induced by the optimal boundary perturbation with that induced by the optimal initial condition demonstrates a close agreement for the selected problem. Previous works dealing with optimal boundary perturbation considered a prescribed spatial structure and computed an optimal temporal variation of a wall-normal velocity component, whereas in this paper we consider the problem of a prescribed temporal structure and compute the optimal spatial variation of velocity boundary conditions over a one-dimensional inflow boundary segment. The methodology is capable of optimizing boundary perturbations in general non-parallel two- and three-dimensional flows.

A flow-induced instability in a tube with flexible walls is studied experimentally. Tubes of diameter 0.8 and 1.2 mm are cast in polydimethylsiloxane (PDMS) polymer gels, and the catalyst concentration in these gels is varied to obtain shear modulus in the range 17–550 kPa. A pressure drop between the inlet and outlet of the tube is used to drive fluid flow, and the friction factor is measured as a function of the Reynolds number . From these measurements, it is found that the laminar flow becomes unstable, and there is a transition to a more complicated flow profile, for Reynolds numbers as low as 500 for the softest gels used here. The nature of the – curves is also qualitatively different from that in the flow past rigid tubes; in contrast to the discontinuous increase in the friction factor at transition in a rigid tube, it is found that there is a continuous increase in the friction factor from the laminar value of in a flexible tube. The onset of transition is also detected by a dye-stream method, where a stream of dye is injected into the centre of the tube. It is found that there is a continuous increase of the amplitude of perturbations at the onset of transition in a flexible tube, in contrast to the abrupt disruption of the dye stream at transition in a rigid tube. There are oscillations in the wall of the tube at the onset of transition, which is detected from the laser scattering off the walls of the tube. This indicates that the coupling between the fluid stresses and the elastic stresses in the wall results in an instability of the laminar flow.