Contracting bubbles in Hele-Shaw cells with a power-law fluid

The problem of bubble contraction in a Hele-Shaw cell is studied for the case in which the surrounding fluid is of power-law type. A small perturbation of the radially symmetric problem is first considered, focussing on the behaviour just before the bubble vanishes, it being found that for shear-thinning fluids the radially symmetric solution is stable, while for shear-thickening fluids the aspect ratio of the bubble boundary increases. The borderline (Newtonian) case considered previously is neutrally stable, the bubble boundary becoming elliptic in shape with the eccentricity of the ellipse depending on the initial data. Further light is shed on the bubble contraction problem by considering a long thin Hele-Shaw cell: for early times the leading-order behaviour is one-dimensional in this limit; however, as the bubble contracts its evolution is ultimately determined by the solution of a Wiener-Hopf problem, the transition between the long-thin limit and the extinction limit in which the bubble vanishes being described by what is in effect a similarity solution of the second kind. This same solution describes the generic (slit-like) extinction behaviour for shear-thickening fluids, the interface profiles that generalise the ellipses that characterise the Newtonian case being constructed by the Wiener-Hopf calculation.

Corner and finger formation in Hele-Shaw flow with kinetic undercooling regularisation

We examine the effect of a kinetic undercooling condition on the evolution of a free boundary in Hele--Shaw flow, in both bubble and channel geometries. We present analytical and numerical evidence that the bubble boundary is unstable and may develop one or more corners in finite time, for both expansion and contraction cases. This loss of regularity is interesting because it occurs regardless of whether the less viscous fluid is displacing the more viscous fluid, or vice versa. We show that small contracting bubbles are described to leading order by a well-studied geometric flow rule. Exact solutions to this asymptotic problem continue past the corner formation until the bubble contracts to a point as a slit in the limit. Lastly, we consider the evolving boundary with kinetic undercooling in a Saffman--Taylor channel geometry. The boundary may either form corners in finite time, or evolve to a single long finger travelling at constant speed, depending on the strength of kinetic undercooling. We demonstrate these two different behaviours numerically. For the travelling finger, we present results of a numerical solution method similar to that used to demonstrate the selection of discrete fingers by surface tension. With kinetic undercooling, a continuum of corner-free travelling fingers exists for any finger width above a critical value, which goes to zero as the kinetic undercooling vanishes. We have not been able to compute the discrete family of analytic solutions, predicted by previous asymptotic analysis, because the numerical scheme cannot distinguish between solutions characterised by analytic fingers and those which are corner-free but non-analytic.

An accurate numerical scheme for the contraction of a bubble in a Hele-Shaw cell

We report on an accurate numerical scheme for the evolution of an inviscid bubble in radial Hele-Shaw flow, where the nonlinear boundary effects of surface tension and kinetic undercooling are included on the bubble-fluid interface. As well as demonstrating the onset of the Saffman-Taylor instability for growing bubbles, the numerical method is used to show the effect of the boundary conditions on the separation (pinch-off) of a contracting bubble into multiple bubbles, and the existence of multiple possible asymptotic bubble shapes in the extinction limit. The numerical scheme also allows for the accurate computation of bubbles which pinch off very close to the theoretical extinction time, raising the possibility of computing solutions for the evolution of bubbles with non-generic extinction behaviour.

Influence of surfactant solubility on the deformation and breakup of a bubble or capillary jet in a viscous fluid

In a previous study [M. Hameed, J. Fluid Mech. 594, 307 (2008)] the authors investigated the influence of insoluble surfactant on the evolution of a stretched, inviscid bubble surrounded by a viscous fluid via direct numerical simulation of the Navier-Stokes equations, and showed that the presence of surfactant can cause the bubble to contract and form a quasisteady slender thread connecting parent bubbles, instead of proceeding directly toward pinch-off as occurs for a surfactant-free bubble. Insoluble surfactant significantly retards pinch-off and the thread is stabilized by a balance between internal pressure and reduced capillary pressure due to a high concentration of surfactant that develops during the initial stage of contraction. In the present study we investigate the influence of surfactant solubility on thread formation. The adsorption-desorption kinetics for solubility is in the diffusion controlled regime. A long-wave model for the evolution of a capillary jet is also studied in the Stokes flow limit, and shows dynamics that are similar to those of the evolving bubble. With soluble surfactant, depending on parameter values, a slender thread forms but can pinch-off later due to exchange of surfactant between the interface and exterior bulk flow. © 2009 American Institute of Physics.

Bubble geometry and chaotic pricing behaviour

This paper is a deductive theoretical enquiry into the flow of effects from the geometry of price bubbles/busts, to price indices, to pricing behaviours of sellers and buyers, and back to price bubbles/busts. The intent of the analysis is to suggest analytical approaches to identify the presence, maturity, and/or sustainability of a price bubble. We present a pricing model to emulate market behaviour, including numeric examples and charts of the interaction of supply and demand. The model extends into dynamic market solutions myopic (single- and multi-period) backward looking rational expectations to demonstrate how buyers and sellers interact to affect supply and demand and to show how capital gain expectations can be a destabilising influence – i.e. the lagged effects of past price gains can drive the market price away from long-run market-worth. Investing based on the outputs of past price-based valuation models appear to be more of a game-of-chance than a sound investment strategy.

The organic fraction of bubble-generated, accumulation mode Sea Spray Aerosol (SSA)

Recent studies have detected a dominant accumulation mode (~100 nm) in the Sea Spray Aerosol (SSA) number distribution. There is evidence to suggest that particles in this mode are composed primarily of organics. To investigate this hypothesis we conducted experiments on NaCl, artificial SSA and natural SSA particles with a Volatility-Hygroscopicity-Tandem-Differential-Mobility-Analyser (VH-TDMA). NaCl particles were atomiser generated and a bubble generator was constructed to produce artificial and natural SSA particles. Natural seawater samples for use in the bubble generator were collected from biologically active, terrestrially-affected coastal water in Moreton Bay, Australia. Differences in the VH-TDMA-measured volatility curves of artificial and natural SSA particles were used to investigate and quantify the organic fraction of natural SSA particles. Hygroscopic Growth Factor (HGF) data, also obtained by the VH-TDMA, were used to confirm the conclusions drawn from the volatility data. Both datasets indicated that the organic fraction of our natural SSA particles evaporated in the VH-TDMA over the temperature range 170–200°C. The organic volume fraction for 71–77 nm natural SSA particles was 8±6%. Organic volume fraction did not vary significantly with varying water residence time (40 secs to 24 hrs) in the bubble generator or SSA particle diameter in the range 38–173 nm. At room temperature we measured shape- and Kelvin-corrected HGF at 90% RH of 2.46±0.02 for NaCl, 2.35±0.02 for artifical SSA and 2.26±0.02 for natural SSA particles. Overall, these results suggest that the natural accumulation mode SSA particles produced in these experiments contained only a minor organic fraction, which had little effect on hygroscopic growth. Our measurement of 8±6% is an order of magnitude below two previous measurements of the organic fraction in SSA particles of comparable sizes. We stress that our results were obtained using coastal seawater and they can’t necessarily be applied on a regional or global ocean scale. Nevertheless, considering the order of magnitude discrepancy between this and previous studies, further research with independent measurement techniques and a variety of different seawaters is required to better quantify how much organic material is present in accumulation mode SSA.

Bubble rise velocity and trajectory in xanthan gum crystal suspension

An experimental set-up was used to visually observe the characteristics of bubbles as they moved up a column holding xanthan gum crystal suspensions. The bubble rise characteristics in xanthan gum solutions with crystal suspension are presented in this paper. The suspensions were made by using different concentrations of xanthan gum solutions with 0.23 mm mean diameter polystyrene crystal particles. The influence of the dimensionless quantities; namely the Reynolds number, Re, the Weber number, We, and the drag co-efficient, cd, are identified for the determination of the bubble rise velocity. The effect of these dimensionless groups together with the Eötvös number, Eo, the Froude number, Fr, and the bubble deformation parameter, D, on the bubble rise velocity and bubble trajectory are analysed. The experimental results show that the average bubble velocity increases with the increase in bubble volume for xanthan gum crystal suspensions. At high We, Eo and Re, bubbles are spherical-capped and their velocities are found to be very high. At low We and Eo, the surface tension force is significant compared to the inertia force. The viscous forces were shown to have no substantial effect on the bubble rise velocity for 45 < Re < 299. The results show that the drag co-efficient decreases with the increase in bubble velocity and Re. The trajectory analysis showed that small bubbles followed a zigzag motion while larger bubbles followed a spiral motion. The smaller bubbles experienced less horizontal motion in crystal suspended xanthan gum solutions while larger bubbles exhibited a greater degree of spiral motion than those seen in the previous studies on the bubble rise in xanthan gum solutions without crystal.

New exact solutions for Hele-Shaw flow in doubly connected regions

Radial Hele-Shaw flows are treated analytically using conformal mapping techniques. The geometry of interest has a doubly-connected annular region of viscous fluid surrounding an inviscid bubble that is either expanding or contracting due to a pressure difference caused by injection or suction of the inviscid fluid. The zero-surface-tension problem is ill-posed for both bubble expansion and contraction, as both scenarios involve viscous fluid displacing inviscid fluid. Exact solutions are derived by tracking the location of singularities and critical points in the analytic continuation of the mapping function. We show that by treating the critical points, it is easy to observe finite-time blow-up, and the evolution equations may be written in exact form using complex residues. We present solutions that start with cusps on one interface and end with cusps on the other, as well as solutions that have the bubble contracting to a point. For the latter solutions, the bubble approaches an ellipse in shape at extinction.

Rate of torque and electromyographic development during anticipated eccentric contraction is lower in previously strained hamstrings

Background: Hamstring strain injuries are prevalent in sport and re-injury rates have been high for many years. Whilst much focus has centred on the impact of previous hamstring strain injury on maximal eccentric strength, high rates of torque development is also of interest, given the important role of the hamstrings during the terminal swing phase of running. The impact of prior strain injury on myoelectrical activity of the hamstrings during tasks requiring high rates of torque development has received little attention. Purpose: To determine if recreational athletes with a history of unilateral hamstring strain injury, who have returned to training and competition, will exhibit lower levels of myoelectrical activity during eccentric contraction, rate of torque development and impulse 30, 50 and 100ms after the onset of myoelectrical activity or torque development in the previously injured limb compared to the uninjured limb. Study design: Case-control study Methods: Twenty-six recreational athletes were recruited. Of these, 13 athletes had a history of unilateral hamstring strain injury (all confined to biceps femoris long head) and 13 had no history of hamstring strain injury. Following familiarisation, all athletes undertook isokinetic dynamometry testing and surface electromyography assessment of the biceps femoris long head and medial hamstrings during eccentric contractions at -60 and -1800.s-1. Results: In the injured limb of the injured group, compared to the contralateral uninjured limb rate of torque development and impulse was lower during -600.s-1 eccentric contractions at 50 (RTD, injured limb = 312.27 ± 191.78Nm.s-1 vs. uninjured limb = 518.54 ± 172.81Nm.s-1, p=0.008; IMP, injured limb = 0.73 ± 0.30 Nm.s vs. uninjured limb = 0.97 ± 0.23 Nm.s, p=0.005) and 100ms (RTD, injured limb = 280.03 ± 131.42Nm.s-1 vs. uninjured limb = 460.54.54 ± 152.94Nm.s-1,p=0.001; IMP, injured limb = 2.15 ± 0.89 Nm.s vs. uninjured limb = 3.07 ± 0.63 Nm.s, p<0.001) after the onset of contraction. Biceps femoris long head muscle activation was lower at 100ms at both contraction speeds (-600.s-1, normalised iEMG activity (x1000), injured limb = 26.25 ± 10.11 vs. uninjured limb 33.57 ± 8.29, p=0.009; -1800.s-1, normalised iEMG activity (x1000), injured limb = 31.16 ± 10.01 vs. uninjured limb 39.64 ± 8.36, p=0.009). Medial hamstring activation did not differ between limbs in the injured group. Comparisons in the uninjured group showed no significant between limbs difference for any variables. Conclusion: Previously injured hamstrings displayed lower rate of torque development and impulse during slow maximal eccentric contraction compared to the contralateral uninjured limb. Lower myoelectrical activity was confined to the biceps femoris long head. Regardless of whether these deficits are the cause of or the result of injury, these findings could have important implications for hamstring strain injury and re-injury. Particularly, given the importance of high levels of muscle activity to bring about specific muscular adaptations, lower levels of myoelectrical activity may limit the adaptive response to rehabilitation interventions and suggest greater attention be given to neural function of the knee flexors following hamstring strain injury.

Impact of previous hamstring strain injury on rate of torque and EMG development during eccentric contraction.

Background: Hamstring strain injuries (HSIs) are prevalent in sport and re-injury rates have been high for many years. Whilst much focus has centred on the impact of previous hamstring strain injury on maximal eccentric strength, high rates of torque development is also of interest, given the important role of the hamstrings during the terminal swing phase of gait. The impact of prior strain injury on neuromuscular function of the hamstrings during tasks requiring high rates of torque development has received little attention. The purpose of this study is to determine if recreational athletes with a history of unilateral hamstring strain injury, who have returned to training and competition, will exhibit lower levels of eccentric muscle activation, rate of torque development and impulse 30, 50 and 100ms after the onset of electromyographical or torque development in the previously injured limb compared to the uninjured limb. Methods: Twenty-six recreational athletes were recruited. Of these, 13 athletes had a history of unilateral hamstring strain injury (all confined to biceps femoris long head) and 13 had no history of hamstring strain injury. Following familiarisation, all athletes undertook isokinetic dynamometry testing and surface electromyography assessment of the biceps femoris long head and medial hamstrings during eccentric contractions at -60 and -1800.s-1. Results: In the injured limb of the injured group, compared to the contralateral uninjured limb rate of torque development and impulse was lower during -600.s-1 eccentric contractions at 50 (RTD, p=0.008; IMP, p=0.005) and 100ms (RTD, p=0.001; IMP p<0.001) after the onset of contraction. There was also a non-significant trend for rate of torque development during -1800.s-1 to be lower 100ms after onset of contraction (p=0.064). Biceps femoris long head muscle activation was lower at 100ms at both contraction speeds (-600.s-1, p=0.009; -1800.s-1, p=0.009). Medial hamstring activation did not differ between limbs in the injured group. Comparisons in the uninjured group showed no significant between limbs difference for any variables. Conclusion: Previously injured hamstrings displayed lower rate of torque development and impulse during eccentric contraction. Lower muscle activation was confined to the biceps femoris long head. Regardless of whether these deficits are the cause of or the result of injury, these findings have important implications for hamstring strain injury and re-injury and suggest greater attention be given to neural function of the knee flexors.

Bubble extinction in Hele-Shaw flow with surface tension and kinetic undercooling regularisation

We perform an analytic and numerical study of an inviscid contracting bubble in a two-dimensional Hele-Shaw cell, where the effects of both surface tension and kinetic undercooling on the moving bubble boundary are not neglected. In contrast to expanding bubbles, in which both boundary effects regularise the ill-posedness arising from the viscous (Saffman-Taylor) instability, we show that in contracting bubbles the two boundary effects are in competition, with surface tension stabilising the boundary, and kinetic undercooling destabilising it. This competition leads to interesting bifurcation behaviour in the asymptotic shape of the bubble in the limit it approaches extinction. In this limit, the boundary may tend to become either circular, or approach a line or "slit" of zero thickness, depending on the initial condition and the value of a nondimensional surface tension parameter. We show that over a critical range of surface tension values, both these asymptotic shapes are stable. In this regime there exists a third, unstable branch of limiting self-similar bubble shapes, with an asymptotic aspect ratio (dependent on the surface tension) between zero and one. We support our asymptotic analysis with a numerical scheme that utilises the applicability of complex variable theory to Hele-Shaw flow.

Mathematical models of bubble evolution in a Hele-Shaw Cell

This thesis concerns the mathematical model of moving fluid interfaces in a Hele-Shaw cell: an experimental device in which fluid flow is studied by sandwiching the fluid between two closely separated plates. Analytic and numerical methods are developed to gain new insights into interfacial stability and bubble evolution, and the influence of different boundary effects is examined. In particular, the properties of the velocity-dependent kinetic undercooling boundary condition are analysed, with regard to the selection of only discrete possible shapes of travelling fingers of fluid, the formation of corners on the interface, and the interaction of kinetic undercooling with the better known effect of surface tension. Explicit solutions to the problem of an expanding or contracting ring of fluid are also developed.

Contraction-induced changes in TNFα and Akt-mediated signalling are associated with increased myofibrillar protein in rat skeletal muscle

Resistance training results in skeletal muscle hypertrophy, but the molecular signalling mechanisms responsible for this altered phenotype are incompletely understood. We used a resistance training (RT) protocol consisting of three sessions [day 1 (d1), day 3 (d3), day 5 (d5)] separated by 48 h recovery (squat exercise, 4 sets × 10 repetitions, 3 min recovery) to determine early signalling responses to RT in rodent skeletal muscle. Six animals per group were killed 3 h after each resistance training session and 24 and 48 h after the last training session (d5). There was a robust increase in TNF? protein expression, and IKKSer180/181 and p38MAPK Thr180/Tyr182 phosphorylation on d1 (P < 0.05), which abated with subsequent RT, returning to control levels by d5 for TNF? and IKK Ser180/181. There was a trend for a decrease in MuRF-1 protein expression, 48 h following d5 of training (P = 0.08). Notably, muscle myofibrillar protein concentration was elevated compared to control 24 and 48 h following RT (P < 0.05). AktSer473 and mTORSer2448 phosphorylation were unchanged throughout RT. Phosphorylation of p70S6k Thr389 increased 3 h post-exercise on d1, d3 and d5 (P < 0.05), whilst phosphorylation of S6Ser235/236 increased on d1 and d3 (P < 0.05). Our results show a rapid attenuation of inflammatory signalling with repeated bouts of resistance exercise, concomitant with summation in translation initiation signalling in skeletal muscle. Indeed, the cumulative effect of these signalling events was associated with myofibrillar protein accretion, which likely contributes to the early adaptations in response to resistance training overload in the skeletal muscle.