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The theory of slow viscous flow around a slender body is generalized to the situation where the ambient fluid has a yield stress. The local flow around a cylinder that is moving along or perpendicular to its axis, and rotating, provides a first step in this theory. Unlike for a Newtonian fluid, the nonlinearity associated with the viscoplastic constitutive law precludes one from linearly superposing solutions corresponding to each independent component of motion, and instead demands a full numerical approach to the problem. This is accomplished for the case of a Bingham fluid, along with a consideration of some asymptotic limits in which analytical progress is possible. Since the yield stress of the fluid strongly localizes the flow around the body, the leading-order slender-body approximation is rendered significantly more accurate than the equivalent Newtonian problem. The theory is applied to the sedimentation of inclined cylinders, bent rods and helices, and compared with some experimental data. Finally, the theory is applied to the locomotion of a cylindrical filament driven by helical waves through a viscoplastic fluid.

A yield stress is added to Taylor’s (Proc. R. Soc. Lond. A, vol. 209, 1951, pp. 447–461) model of a two-dimensional flexible sheet swimming through a viscous fluid. Both transverse waves along the sheet, as in Taylor’s original model, and longitudinal waves are considered as means of locomotion. In each case, numerical solutions are provided over a range of the two key parameters of the problem: the wave amplitude relative to the wavelength and a Bingham number which describes the strength of the yield stress. The numerical solutions are supplemented with discussions of various limits of the problem in which analytical progress is possible. When the yield stress is large, the swimming speed for low wave amplitude is exactly double that for a Newtonian fluid, for either type of wave.

An experimental investigation is conducted into the collapse of granular columns inside rectangular channels. The final shape is documented for slumps inside relatively wide channels, and for collapses inside much narrower slots. In both cases, the collapse is initiated by withdrawing a swinging gate or sliding door, and the flow remains fairly two-dimensional. Four different granular media are used; the properties of the materials vary significantly, notably in their angles of friction for basal sliding and internal deformation. If $H$ is the initial height of the column, $h_{\\infty}$ the maximum final height of the column and $a$ the initial aspect ratio, then the data suggest that $H/h_{\\infty} \\,{\\sim} a^{0.6}$ in wide channels and $H/h_{\\infty} \\,{\\sim}\\, a^{0.5}$ for narrow slots. For the runout, we find that $(l_{\\infty}\\,{-}\\,L)/L \\,{\\sim}\\, a^{0.9\\pm 0.1}$ for wide channels, and $(l_{\\infty}\\,{-}\\,L)/L \\,{\\sim}\\, a^{0.65\\pm0.05}$ or $l_\\infty/L \\,{\\sim}\\, a^{0.55\\pm0.05}$ for narrow slots, where $l_{\\infty}$ is the maximum runout of the material and $L$ the initial length of the column along the channel ($a\\,{:=}\\,H/L$). In all cases, the numerical constant of proportionality in these scaling relations shows clear material dependence. In wide slots, there is no obvious universal scaling behaviour of the final profile, but such a behaviour is evident in narrow slots. The experimental results are compared with theoretical results based on a shallow granular-flow model. The qualitative behaviour of the slump in the wide slot is reproduced by the theoretical model. However, there is qualitative disagreement between theory and the experiments in the narrow slot because of the occurrence of secondary surface avalanching.

A model is derived for long peristaltic waves propagating steadily down a fluid-filled, axisymmetric tube. The waves are driven by imposing a radial force of prescribed form on the tube. The resulting deformation of the tube wall is modelled using linear elasticity and the internal flow using the lubrication approximation. Numerical solutions for periodic wave trains and solitary waves are presented, along with asymptotic solutions at both small and large forcing amplitudes. Large-amplitude periodic waves are characterized by narrow blisters adjoining long occluded sections of the tube, whereas a solitary wave of strong contraction produces a long inflated bow wave that propels a large quantity of fluid. A measure of pumping efficacy is given by the ratio of the net fluid displacement to the power input, and is highest for a large-amplitude solitary wave.

Spreading and stationary droplets of a thermally responsive fluid on a heated surface are studied. The fluid undergoes a reversible gel formation at elevated temperature. The spatio-temporal pattern of gel formation within the droplet is examined using an experimental method based on spectral domain optical coherence tomography and time varying speckle patterns. Two stages of gel formation can be distinguished: first, a thin crust appears starting at the contact line. Second, a gel layer appears above the heated plate and then expands upward. We attribute the first stage of gel formation to solvent evaporation and heating through the air and the second to thermal conduction through the fluid from the base. Gel formation at the contact line is likely responsible for the arrest of spreading droplets, but was not detectable with our experimental protocol at the time of contact line arrest, suggesting that this arose over a microscopic length scale. Overall, substrate heating provides an effective way to control the final shape of droplets of thermo-responsive fluids.

In the limit of a large yield stress, or equivalently at the initiation of motion, viscoplastic flows can develop narrow boundary layers that provide either surfaces of failure between rigid plugs, the lubrication between a plugged flow and a wall or buffers for regions of predominantly plastic deformation. Oldroyd (Proc. Camb. Phil. Soc., vol. 43, 1947, pp. 383–395) presented the first theoretical discussion of these viscoplastic boundary layers, offering an asymptotic reduction of the governing equations and a discussion of some model flow problems. However, the complicated nonlinear form of Oldroyd’s boundary-layer equations has evidently precluded further discussion of them. In the current paper, we revisit Oldroyd’s viscoplastic boundary-layer analysis and his canonical examples of a jet-like intrusion and flow past a thin plate. We also consider flow down channels with either sudden expansions or wavy walls. In all these examples, we verify that viscoplastic boundary layers form as envisioned by Oldroyd. For each example, we extract the dependence of the boundary-layer thickness and flow profiles on the dimensionless yield-stress parameter (Bingham number). We find that, while Oldroyd’s boundary-layer theory applies to free viscoplastic shear layers, it does not apply when the boundary layer is adjacent to a wall, as has been observed previously for two-dimensional flow around circular obstructions. Instead, the boundary-layer thickness scales in a different fashion with the Bingham number, as suggested by classical solutions for plane-parallel flows, lubrication theory and, for flow around a plate, by Piau (J. Non-Newtonian Fluid Mech., vol. 102, 2002, pp. 193–218); we rationalize this second scaling and provide an alternative boundary-layer theory.

Experiments are conducted to study the transition from episodic avalanching (slumping) to continuous flow (rolling) in drums half full of granular material. The width and radius of the drum is varied and different granular materials are used, ranging from glass spheres with different radii to irregularly shaped sand. Image processing is performed in real time to extract relatively long time series of the surface slope derived from a linear fit to the granular surface. For the drums with glass spheres, the transition mostly takes the form of a blend of the characteristics of episodic avalanching and continuous flow, that gradually switches from slumping to rolling as the rotation rate increases. For sand, a hysteretic transition can be observed in which one observes prolonged episodic avalanching or continuous flow at the same rotation rate, spanning a window of rotation speeds. For drums with the smallest spheres (1 mm diameter), the transition takes the form of noise-driven intermittent switching between clearly identifiable phases of episodic avalanching or continuous flow. This style of transition is also found for the sand in either the largest or smallest drum (by volume). We formulate dimensionless groupings of the experimental parameters to locate the transition and characterize the mean surface slope and its fluctuations. We extract statistics for episodic avalanching, including angle distributions for avalanche initiation and cessation, the correlations between successive collapses, mean avalanche profiles and durations, and characteristic frequencies and spectra.

Shallow-water equations with bottom drag and viscosity are used to study the dynamics of roll waves. First, we explore the effect of bottom topography on linear stability of turbulent flow over uneven surfaces. Low-amplitude topography is found to destabilize turbulent roll waves and lower the critical Froude number required for instability. At higher amplitude, the trend reverses and topography stabilizes roll waves. At intermediate topographic amplitude, instability can be created at much lower Froude numbers due to the development of hydraulic jumps in the equilibrium flow. Second, the nonlinear dynamics of the roll waves is explored, with numerical solutions of the shallow-water equations complementing an asymptotic theory relevant near onset. We find that trains of roll waves undergo coarsening due to waves overtaking one another and merging, lengthening the scale of the pattern. Unlike previous investigations, we find that coarsening does not always continue to its ultimate conclusion (a single roll wave with the largest spatial scale). Instead, coarsening becomes interrupted at intermediate scales, creating patterns with preferred wavelengths. We quantify the coarsening dynamics in terms of linear stability of steady roll-wave trains.

A mathematical model is developed for long peristaltic waves propelling a suspended rigid object down a fluid-filled axisymmetric tube. The fluid flow is described using lubrication theory and the deformation of the tube using linear elasticity. The object is taken to be either an infinitely long rod of constant radius or a parabolic-shaped lozenge of finite length. The system is driven by a radial force imposed on the tube wall that translates at constant speed down the tube axis and with a form chosen to generate a periodic wave train or a solitary wave. These waves exert a traction on the enclosed object, forcing it into motion. Periodic waves drive the infinite rod at a speed that attains a maximum at a moderate forcing amplitude and approaches approximately one quarter of the wave speed in the large-amplitude limit. The finite lozenge can be entrained and driven at the same speed as a solitary wave or periodic wave train if the forcing is sufficiently strong. For weaker forcing, the lozenge is either left behind the solitary wave or interacts repeatedly with the waves in the periodic train to generate stuttering forward progress. The threshold forcing amplitude for entrainment increases weakly with the radial span of the enclosed object, but strongly with the axial length, with entrainment becoming impossible if the object is too long.

Slender-body theory is used to study axisymmetric swimmers propelled by motions of their surfaces. To leading order, the locomotion speed is given by an integral involving the fluid velocity at the surface of the slender body. Locomotion speeds are calculated for fixed-shape swimmers with prescribed fluid surface velocities and for impermeable swimmers driven by propagating surface waves. Next, the internal mechanics is considered, modelling the swimmer as a viscous fluid bounded by an elastic shell. Prescribed forces are exerted on the shell to drive both the internal and external fluid flow and the surface waves. The internal fluid mechanics is determined using lubrication theory. Locomotion speeds are calculated for transverse and longitudinal waves of surface deformation, and the efficiency of the motions is determined. Transverse surface waves are both weaker and less efficient at driving locomotion than longitudinal waves. The results indicate how estimates of swimming speed based on nearly spherical swimmers with low-amplitude surface waves can be adapted for slender swimmers with nonlinear surface deformations.