Self-similar breakup of near-inviscid liquids.

The final stages of pinchoff and breakup of dripping droplets of near-inviscid Newtonian fluids are studied experimentally for pure water and ethanol. High-speed imaging and image analysis are used to determine the angle and the minimum neck size of the cone-shaped extrema of the ligaments attached to dripping droplets in the final microseconds before pinchoff. The angle is shown to steadily approach the value of 18.0 ± 0.4°, independently of the initial flow conditions or the type of breakup. The filament thins and necks following a τ(2/3) law in terms of the time remaining until pinchoff, regardless of the initial conditions. The observed behavior confirms theoretical predictions.this work was sponsored by EPSRC grant number RG5560

Motion of a non-axisymmetric particle in viscous shear flow

We examine the motion in a shear flow at zero Reynolds number of particles with two planes of symmetry. We show that in most cases the rotational motion is qualitatively similar to that of a non-axisymmetric ellipsoid, and characterised by a combination of chaotic and quasiperiodic orbits. We use Kolmogorov–Arnold–Moser (KAM) theory and related ideas in dynamical systems to elucidate the underlying mathematical structure of the motion and thence to explain why such a large class of particles all rotate in essentially the same manner. Numerical simulations are presented for curved spheroids of varying centreline curvature, which are found to drift persistently across the streamlines of the flow for certain initial orientations. We explain the origin of this migration as the result of a lack of symmetries of the particle’s orientation orbit.</jats:p

Viscous control of shallow elastic fracture: Peeling without precursors

We consider peeling of an elastic sheet away from an elastic substrate through propagation of a fluid-filled crack along the interface between the two. The peeling is driven by a bending moment applied to the sheet and is resisted by viscous flow towards the crack tip and by the toughness of any bonding between the sheet and the substrate. Travelling-wave solutions are determined using lubrication theory coupled to the full equations of elasticity and fracture. The propagation speed $v$ scales like M^{3}/\unicode[STIX]{x1D707}\bar{E}^{2}d^{5}=Bd\unicode[STIX]{x1D705}^{3}/144\unicode[STIX]{x1D707}, where $d$ is the sheet’s thickness, $B=\bar{E}d^{3}/12$ its stiffness, \bar{E}=E/(1-\unicode[STIX]{x1D708}^{2}) its plane-strain modulus, \unicode[STIX]{x1D707} the fluid viscosity, $M$ the applied bending moment and \unicode[STIX]{x1D705}=M/B the sheet’s curvature due to bending; and the prefactor depends on the dimensionless toughness. If the toughness is small then there is a region of dry shear failure ahead of the fluid-filled region. The expressions for the propagation speed have been used to derive new similarity solutions for the spread of an axisymmetric fluid-filled blister in a variety of regimes: constant-flux injection resisted by elastohydrodynamics in the tip leads to spread proportional to $t^{4/13}$, $t^{4/17}$ and $t^{7/19}$ for peeling-by-bending, gravitational spreading and peeling-by-pulling, respectively.EPSR

Buoyancy-driven plumes in a layered porous medium

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Quantification and propagation of errors when converting vertebrate biomineral oxygen isotope data to temperature for palaeoclimate reconstruction

Oxygen isotope analysis of bioapatite in vertebrate remains (bones and teeth) is commonly used to address questions on palaeoclimate from the Eocene to the recent past. Researchers currently use a range of methods to calibrate their data, enabling the isotopic composition of precipitation and the air temperature to be estimated. In some situations the regression method used can significantly affect the resulting palaeoclimatic interpretations. Furthermore, to understand the uncertainties in the results, it is necessary to quantify the errors involved in calibration. Studies in which isotopic data are converted rarely address these points, and a better understanding of the calibration process is needed. This paper compares regression methods employed in recent publications to calibrate isotopic data for palaeoclimatic interpretation and determines that least-squares regression inverted to x=(y-b)/a is the most appropriate method to use for calibrating causal isotopic relationships. We also identify the main sources of error introduced at each conversion stage, and investigate ways to minimise this error. We demonstrate that larger sample sizes substantially reduce the uncertainties inherent within the calibration process: typical uncertainty in temperature inferred from a single sample is at least ±4°C, which multiple samples can reduce to ±1-2°C. Moreover, the gain even from one to four samples is greater than the gain from any further increases. We also show that when converting δ18Oprecipitation to temperature, use of annually averaged data can give significantly less uncertainty in inferred temperatures than use of monthly rainfall data. Equations and an online spreadsheet for the quantification of errors are provided for general use, and could be extended to contexts beyond the specific application of this paper.Palaeotemperature estimation from isotopic data can be highly informative for our understanding of past climates and their impact on humans and animals. However, for such estimates to be useful, there must be confidence in their accuracy, and this includes an assessment of calibration error. We give a series of recommendations for assessing uncertainty when making calibrations of δ18Obioapatite-δ18Oprecipitation-Temperature. Use of these guidelines will provide a more solid foundation for palaeoclimate inferences made from vertebrate isotopic data.We are grateful to the University of Cambridge (AJEP) and the Royal Society (RES) for financial support

Early-time free-surface flow driven by a deforming boundary

AbstractWhen a solid boundary deforms rapidly into a quiescent liquid layer, a flow is induced that can lead to jet formation. An asymptotic analytical solution is presented for this flow, driven by a solid boundary deforming with dimensionless vertical velocity $V_{b}(x,t)={\it\epsilon}(1+\cos x)\,f(t)$, where the amplitude ${\it\epsilon}$ is small relative to the wavelength and the time dependence $f(t)$ approaches 0 for large $t$. Initially, the flow is directed outwards from the crest of the deformation and slows with the slowing of the boundary motion. A domain-perturbation method is used to reveal that, when the boundary stops moving, nonlinear interactions with the free surface leave a remnant momentum directed back towards the crest, and this momentum can be a precursor to jet formation. This scenario arises in a laser-induced printing technique in which an expanding blister imparts momentum into a liquid film to form a jet. The analysis provides insight into the physics underlying the interaction between the deforming boundary and free surface, in particular, the dependence of the remnant flow on the thickness of the liquid layer and the deformation amplitude and wavelength. Numerical simulations are used to show the range of validity of the analytical results, and the domain-perturbation solution is extended to an axisymmetric domain with a Gaussian boundary deformation to compare with previous numerical simulations of blister-actuated laser-induced forward transfer.The authors gratefully acknowledge financial support for this research from the Na- tional Science Foundation MRSEC program through the Princeton Center for Complex Materials (grant DMR-0819860).This is the accepted manuscript. The final published version is available from CUP at http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=9571518&fileId=S0022112015000749

Nondecaying hydrodynamic interactions along narrow channels

This is the author accepted manuscript. The final version is available from the publisher via the DOI in this record.Particle-particle interactions are of paramount importance in every multibody system as they determine the collective behavior and coupling strength. Many well-known interactions such as electrostatic, van der Waals, or screened Coulomb interactions, decay exponentially or with negative powers of the particle spacing r. Similarly, hydrodynamic interactions between particles undergoing Brownian motion decay as 1/r in bulk, and are assumed to decay in small channels. Such interactions are ubiquitous in biological and technological systems. Here we confine two particles undergoing Brownian motion in narrow, microfluidic channels and study their coupling through hydrodynamic interactions. Our experiments show that the hydrodynamic particle-particle interactions are distance independent in these channels. This finding is of fundamental importance for the interpretation of experiments where dense mixtures of particles or molecules diffuse through finite length, water-filled channels or pore networks.U. F. K. was supported by an ERC starting Grant No. (PassMembrane 261101). S. P. acknowledges funding from a Leverhulme Early Career Fellowship. K. M. was supported by a grant from the EPSRC. E. L. was supported by a Marie Curie CIG grant from the EU

Development of a single droplet freezing apparatus for studying crystallisation in cocoa butter droplets

The single droplet freezing apparatus described by Pore et al. (J. Am. Oil. Chem. Soc., 86, 215-225), which allows crystallisation to be monitored in situ by X-ray diffraction, was modified to allow rapid switching of coolant gas and monitoring by video microscopy. The apparatus was used to study drops of cocoa butter undergoing simulated spray freezing at high cooling rates, e.g. 130 K/min. The transformation of an Ivory Coast cocoa butter to the Form V polymorph was significantly faster in drops (~40 h) than in static bulk samples (10 days) crystallised under isothermal conditions. Phase transformation was observed from Forms I/II → III → IV → melt → V, with Form V crystallising directly from the melt at 28.6°C. Numerical simulations of the temperature evolution within the droplet established that the drops are not isothermal, explaining why nucleation was initially observed in the lower (upstream) part of the droplet.The provision of an EPSRC studentship for AMT and project support from Nestlé PTC York is gratefully acknowledged. The apparatus was constructed by Lee Pratt, Gary Chapman, Kevin Swan and Wei-Yao Ma. Assistance with the DSC testing from Zlatko Saraçevic, video microscopy from Dr Simon Butler, and general X-ray analysis from Dr Joanna Stasiak are all gratefully acknowledged.This is the final version of the article. It first appeared from Elsevier via http://dx.doi.org/10.1016/j.jfoodeng.2015.02.01

Stability of three-dimensional columnar convection in a porous medium

The stability of steady convective exchange flow with a rectangular planform in an unbounded three-dimensional porous medium is explored. The base flow comprises a balance between vertical advection with amplitude $A$ in interleaving rectangular columns with aspect ratio \unicode[STIX]{x1D709}\leqslant 1 and horizontal diffusion between the columns. Columnar flow with a square planform (\unicode[STIX]{x1D709}=1) is found to be weakly unstable to a large-scale perturbation of the background temperature gradient, irrespective of $A$, but to have no stronger instability on the scale of the columns. This result provides a stark contrast to two-dimensional columnar flow (Hewitt et al., J. Fluid Mech., vol. 737, 2013, pp. 205–231), which, as $A$ is increased, is increasingly unstable to a perturbation on the scale of the columnar wavelength. For rectangular planforms with \unicode[STIX]{x1D709}<1, a critical aspect ratio is identified, below which a perturbation on the scale of the columns is the fastest growing mode, as in two dimensions. Scalings for the growth rate and the structure of this mode are identified, and are explained by means of an asymptotic expansion in the limit \unicode[STIX]{x1D709}\rightarrow 0. The difference between the stabilities of two-dimensional and three-dimensional exchange flow provides a potential explanation for the apparent difference in dominant horizontal scale observed in direct numerical simulations of two-dimensional and three-dimensional statistically steady ‘Rayleigh–Darcy’ convection at high Rayleigh numbers.</jats:p