Low Viscosity Regime Research Articles

Ultrafast measurement of laser-induced shock waves

We present measurements of laser-induced shockwave pressure rise time in liquids on a sub-nanosecond scale, using custom-designed single-mode fiber optic hydrophone. The measurements are aimed at the study of the shockwave generation process, helping to improve the effectiveness of various applications and decrease possible accidental damage from shockwaves. The developed method allows measurement of the fast shockwave rise time as close as 10 µm from an 8 µm sized laser-induced plasma shockwave source, significantly improving the spatial and temporal resolution of the pressure measurement over other types of hydrophones. The spatial and temporal limitations of the presented hydrophone measurements are investigated theoretically, with actual experimental results agreeing well with the predictions. To demonstrate the capabilities of the fast sensor, we were able to show that the shockwave rise time is linked to liquid viscosity exhibiting logarithmic dependency in the low viscosity regime (from 0.4 cSt to 50 cSt). Additionally, the shockwave rise time dependency on propagation distance close to the source in water was investigated, with shock wave rise times measured down to only 150 ps. It was found that at short propagation distances in water halving the shock wave peak pressure results in the rise time increase by approximately factor of 1.6. These results extend the understanding of shockwave behavior in low viscosity liquids.

Precession-driven flows in stress-free ellipsoids

Motivated by modelling rotating turbulence in planetary fluid layers, we investigate precession-driven flows in ellipsoids subject to stress-free boundary conditions (SF-BC). The SF-BC could indeed unlock numerical constraints associated with the no-slip boundary conditions (NS-BC), but are also relevant for some astrophysical applications. Although SF-BC have been employed in the pioneering work of Lorenzani & Tilgner (J. Fluid Mech., vol. 492, 2003, pp. 363–379), they have scarcely been used due to the discovery of some specific mathematical issues associated with angular momentum conservation. We revisit the problem using asymptotic analysis in the low-viscosity regime, which is validated with numerical simulations. First, we extend the reduced model of uniform-vorticity flows in ellipsoids to account for SF-BC. We show that the long-term evolution of angular momentum is affected by viscosity in triaxial geometries, but also in axisymmetric ellipsoids when the mean rotation axis of the fluid is not the symmetry axis. In a regime relevant to planets, we analytically obtain the primary forced flow in triaxial geometries, which exhibits a second inviscid resonance. Then, we investigate the bulk instabilities existing in precessing ellipsoids. We show that using SF-BC would be useful to explore the non-viscous instabilities (e.g. Kerswell, Geophys. Astrophys. Fluid Dyn., vol. 72, 1993, pp. 107–144), which are presumably relevant for planetary applications but are often hampered in experiments or simulations with NS-BC.

Reflections and focusing of inertial waves in a tilted librating cube

A fluid-filled cube rotating about an axis passing through the midpoints of opposite edges is subjected to small librations (i.e. modulation of the mean rotation). Low viscosity regimes, with Ekman number as small as $10^{-8}$ and equally small relative forcing amplitude, are explored numerically. The full inertial range of forcing frequencies, from 0 to twice the mean rotation rate are considered. The response flows are dominated by inertial wavebeams emitted from edges and/or vertices, depending on the forcing frequency. How these reflect on the cube's walls and focus onto edges and vertices lead to intricate patterns. Most of the results can be reconciled using linear inviscid ray-tracing theory with careful attention to wavebeam emissions and reflections. However, even at the low Ekman number and relative forcing amplitude considered, other effects are discernible which are not captured by ray tracing. These include a symmetry breaking due to viscous effects and a progressive wave, retrograde to the mean rotation and localized in the boundary layers of the cube, due to nonlinear effects and the librational forcing.

Ciliary beating patterns map onto a low-dimensional behavioural space

Biological systems are robust to perturbations at both the genetic and environmental levels, although these same perturbations can elicit variation in behaviour. The interplay between functional robustness and behavioural variability is exemplified at the organellar level by the beating of cilia and flagella. Cilia are motile despite wide genetic diversity between and within species, differences in intracellular concentrations of ATP and calcium, and considerable environment fluctuations in temperature and viscosity. At the same time, these perturbations result in a variety of spatio-temporal patterns that span a rich behavioural space. To investigate this behavioural space we analysed the dynamics of isolated cilia from the unicellular algae Chlamydomonas reinhardtii under many different environmental and genetic conditions. We found that, despite large changes in beat frequency and amplitude, the space of waveform shapes is low-dimensional in the sense that two features account for 80% of the observed variation. The geometry of this behavioural space accords with the predictions of a simple mechanochemical model in the low-viscosity regime. This allowed us to associate waveform shape variability with changes in only the curvature response coefficients of the dynein motors.

Hydrodynamic limits and numerical errors of isothermal lattice Boltzmann schemes

With the aim of better understanding the numerical properties of the lattice Boltzmann method (LBM), a general methodology is proposed to derive its hydrodynamic limits in the discrete setting. It relies on a Taylor expansion in the limit of low Knudsen numbers. With a single asymptotic analysis, two kinds of deviations with the Navier-Stokes (NS) equations are explicitly evidenced: consistency errors, inherited from the kinetic description of the LBM, and numerical errors attributed to its space and time discretization. The methodology is applied to the Bhatnagar-Gross-Krook (BGK), the regularized and the multiple relaxation time (MRT) collision models in the isothermal framework. Deviation terms are systematically confronted to linear analyses in order to validate their expressions, interpret them and provide explanations for their numerical properties. The low dissipation of the BGK model is then related to a particular pattern of its error terms in the Taylor expansion. Similarly, dissipation properties of the regularized and MRT models are explained by a phenomenon referred to as hyperviscous degeneracy. The latter consists in an unexpected resurgence of high-order Knudsen effects induced by a large numerical pre-factor. It is at the origin of over-dissipation and severe instabilities in the low-viscosity regime.

Viscous effects on plasmoid formation from nonlinear resistive tearing growth in a Harris sheet

In this work, the evolution of a highly unstable m = 1 resistive tearing mode, leading to plasmoid formation in a Harris sheet, is studied in the framework of full MHD model using the Non-Ideal Magnetohydrodynamics with Rotation, Open Discussion simulation. Following the initial nonlinear growth of the primary m = 1 island, the X-point develops into a secondary elongated current sheet that eventually breaks into plasmoids. Two distinctive viscous regimes are found for the plasmoid formation and saturation. In the low viscosity regime (i.e. P r ≲ 1), the plasmoid width increases sharply with viscosity, whereas in the viscosity dominant regime (i.e. P r ≳ 1), the plasmoid size gradually decreases with viscosity. Such a finding quantifies the role of viscosity in modulating the plasmoid formation process through its effects on the plasma flow and the reconnection itself.

Research on the contact time of a bouncing microdroplet with lattice Boltzmann method

The bouncing dynamics of microdroplets with various viscosities on a superhydrophobic surface is numerically investigated. An axisymmetric lattice Boltzmann method is developed on the basis of Zheng et al. capable of handling multiphase flows with a large density ratio, which is implemented to simulate the impact. It is shown that in the low-viscosity regime, the contact time tc remains constant over a wide Weber number range (10 < We < 120), which is consistent with macro-scale bouncing. Nevertheless, in the high-viscosity regime, tc increases with impact velocity. A contact number T≡WeRe−1/2=ρD0 ηU03/σ21/2 is proposed to describe the viscosity effect; meanwhile, a new scaling τ ∼ D0/U0T=ρηD03U0/σ21/2 is deduced to characterize the contact time for this regime, and the simulated results for such droplets agree well with the new scaling. To find out the internal physical mechanism, the evolution of kinetic energy, dissipated energy, and velocity vector fields is studied, which quantifies the impact dynamics. Also, simulation data demonstrate that viscous dissipation is not negligible even for relatively low-viscosity fluids. These findings are highly useful for fundamental understanding of microdroplet dynamics with various viscosities, and it can be used to precisely control the contact time.

Compressible lattice Boltzmann methods with adaptive velocity stencils: An interpolation-free formulation

Adaptive lattice Boltzmann methods (LBMs) are based on velocity discretizations that self-adjust to local macroscopic conditions such as velocity and temperature. While this feature improves the accuracy and the stability of LBMs for large velocity and temperature variations, it also strongly impacts the efficiency of the algorithm due to space interpolations that are required to get populations at grid nodes. To avoid this defect, the present work proposes new formulations of adaptive LBMs that do not rely anymore on space interpolations, hence drastically improving their parallel efficiency for the simulation of high-speed compressible flows. To reach this goal, the adaptive phase discretization is restricted to particular states that are compliant with the efficient “collide-and-stream” algorithm, and as a consequence, it does not require additional interpolation steps. The development of proper state-adaptive solvers with on-grid propagation imposes new restrictions and challenges on the discrete stencils, namely, the need for an extended operability range allowing for the transition between two phase discretizations. Achieving the minimum operability range for discrete polynomial equilibria requires rather large stencils (e.g., D2Q81, D2Q121) and is therefore not competitive for compressible flow simulations. However, as shown in this article, the use of numerical equilibria can provide for overlaps in the operability ranges of neighboring discrete shifts at acceptable cost using the D2Q21 lattice. Through several numerical validations, the present approach is shown to allow for an efficient realization of discrete state-adaptive LBMs for high Mach number flows even in the low-viscosity regime.

Efficient supersonic flow simulations using lattice Boltzmann methods based on numerical equilibria.

A double-distribution-function based lattice Boltzmann method (DDF-LBM) is proposed for the simulation of polyatomic gases in the supersonic regime. The model relies on a numerical equilibrium that has been extensively used by discrete velocity methods since the late 1990s. Here, it is extended to reproduce an arbitrary number of moments of the Maxwell–Boltzmann distribution. These extensions to the standard 5-constraint (mass, momentum and energy) approach lead to the correct simulation of thermal, compressible flows with only 39 discrete velocities in 3D. The stability of this BGK-LBM is reinforced by relying on Knudsen-number-dependent relaxation times that are computed analytically. Hence, high Reynolds-number, supersonic flows can be simulated in an efficient and elegant manner. While the 1D Riemann problem shows the ability of the proposed approach to handle discontinuities in the zero-viscosity limit, the simulation of the supersonic flow past a NACA0012 aerofoil confirms the excellent behaviour of this model in a low-viscosity and supersonic regime. The flow past a sphere is further simulated to investigate the 3D behaviour of our model in the low-viscosity supersonic regime. The proposed model is shown to be substantially more efficient than the previous 5-moment D3Q343 DDF-LBM for both CPU and GPU architectures. It then opens up a whole new world of compressible flow applications that can be realistically tackled with a purely LB approach.This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.

Predicting the Observational Signature of Migrating Neptune-sized Planets in Low-viscosity Disks

Abstract The migration of planetary cores embedded in a protoplanetary disk is an important mechanism within planet-formation theory, relevant for the architecture of planetary systems. Consequently, planet migration is actively discussed, yet often results of independent theoretical or numerical studies are unconstrained due to the lack of observational diagnostics designed in light of planet migration. In this work we follow the idea of inferring the migration behavior of embedded planets by means of the characteristic radial structures that they imprint in the disk’s dust density distribution. We run hydrodynamical multifluid simulations of gas and several dust species in a locally isothermal α-disk in the low-viscosity regime (α = 10−5) and investigate the obtained dust structures. In this framework, a planet of roughly Neptune mass can create three (or more) rings in which dust accumulates. We find that the relative spacing of these rings depends on the planet’s migration speed and direction. By performing subsequent radiative transfer calculations and image synthesis we show that—always under the condition of a near-inviscid disk—different migration scenarios are, in principle, distinguishable by long-baseline, state-of-the-art Atacama Large Millimeter/submillimeter Array observations.

Effects of diamagnetic drift on magnetohydrodynamic explosive bursts in reversed magnetic shear tokamak plasmas

The explosive bursts, induced by self-organized double tearing modes (DTMs) in reversed magnetic shear tokamak plasmas, are numerically investigated based on a set of reduced two-fluid equations. It is found that the diamagnetic drift has a destabilizing effect on the mode in the Rutherford phase but can effectively stabilize the explosive burst of the mode in the following nonlinear phase. The plasma viscosity µ is found to have an important effect on the critical ion skin depth for completely suppressing the explosive bursts. In the low viscosity regime (i.e. , where is the critical viscosity), increases with increasing µ, because the increase of µ can effectively reduce the velocity difference of islands between the two rational surfaces, which mainly result from zonal flow and electron diamagnetic flow. In the high viscosity regime (i.e. ), however, decreases with increasing µ, because the increase of µ can greatly stabilize the DTM instability. In addition, the dependences of critical viscosity on viscosity and resistivity are obtained and the relevant mechanisms are analyzed.

Tides Between the TRAPPIST-1 Planets

Abstract The TRAPPIST-1 system is sufficiently closely packed that tides raised by one planet on another are significant. We investigate whether this source of tidal heating is comparable to eccentricity tides raised by the star. Assuming a homogeneous body with a Maxwell rheology, we find that energy dissipation from stellar tides always dominates over that from planet–planet tides across a range of viscosities. TRAPPIST-1 g may experience the greatest proportion of planet–planet tidal heating, where it can account for between 2% and 20% of the total amount of tidal heating, for high-viscosity (1021 Pa s) and low-viscosity (1014 Pa s) regimes, respectively. If planet–planet tidal heating is to exceed that from stellar eccentricity tides, orbital eccentricities must be no more than e = 10−3–10−4 for most of the TRAPPIST-1 planets.

Magnetic quenching of the inverse cascade in rapidly rotating convective turbulence

We present results from an asymptotic magnetohydrodynamic model that is suited for studying the rapidly rotating, low viscosity regime typical of the electrically conducting fluid interiors of planets and stars. We show that the presence of sufficiently strong magnetic fields prevents the formation of large-scale vortices and saturates the inverse cascade at a finite length-scale. This saturation corresponds to an equilibrated state in which the energetics of the depth-averaged flows are characterized by a balance of convective power input and ohmic dissipation. A quantitative criteria delineating the transition between finite-size flows and domain-filling (large-scale) vortices in electrically conducting fluids is found. By making use of the inferred and observed properties of planetary interiors, our results suggest that convection-driven large-scale vortices do not form in the electrically conducting regions of many bodies.

Generalized Warped Disk Equations

Abstract The manner in which warps in accretion disks evolve depends on the magnitude of the viscosity. For small viscosity (α < H/R), the warp evolves in a wave-like manner; for large viscosity, H/R < α ≪ 1, it evolves diffusively. Here, α is the viscosity parameter and H/R is the disk aspect ratio. Currently there is no simple set of equations that describes the evolution in both regimes. In this paper, we describe a possible solution to this problem and introduce a set of one-dimensional equations that describe the evolution of a warped disk that are applicable in both high- and low-viscosity regimes for arbitrary tilts, but small warps.

Precessing spherical shells: flows, dissipation, dynamo and the lunar core

SUMMARYPrecession of planets or moons affects internal liquid layers by driving flows, instabilities and possibly dynamos. The energy dissipated by these phenomena can influence orbital parameters such as the planet’s spin rate. However, there is no systematic study of these flows in the spherical shell geometry relevant for planets, and the lack of scaling law prevents convincing extrapolation to celestial bodies. We have run more than 900 simulations of fluid spherical shells affected by precession, to systematically study basic flows, instabilities, turbulence and magnetic field generation. We observe no significant effects of the inner core on the onset of the instabilities. We obtain an analytical estimate of the viscous dissipation, mostly due to boundary layer friction in our simulations. We propose theoretical onsets for hydrodynamic instabilities, and document the intensity of turbulent fluctuations. We extend previous precession dynamo studies towards lower viscosities, at the limits of today’s computers. In the low viscosity regime, precession dynamos rely on the presence of large-scale vortices, and the surface magnetic fields are dominated by small scales. Interestingly, intermittent and self-killing dynamos are observed. Our results suggest that large-scale planetary magnetic fields are unlikely to be produced by a precession-driven dynamo in a spherical core. But this question remains open as planetary cores are not exactly spherical, and thus the coupling between the fluid and the boundary does not vanish in the relevant limit of small viscosity. Moreover, the fully turbulent dissipation regime has not yet been reached in simulations. Our results suggest that the melted lunar core has been in a turbulent state throughout its history. Furthermore, in the view of recent experimental results, we propose updated formulas predicting the fluid mean rotation vector and the associated dissipation in both the laminar and the turbulent regimes.

Deep and wide gaps by super Earths in low-viscosity discs

Planets can open cavities (gaps) in the protoplanetary gaseous discs in which they are born by exerting gravitational torques. Viscosity counters these torques and limits the depletion of the gaps. We present a simple one-dimensional scheme to calculate the gas density profile inside gaps by balancing the gravitational and viscous torques. By generalizing the results of Goodman & Rafikov (2001), our scheme properly accounts for the propagation of angular momentum by density waves. This method allows us to easily study low-viscosity discs, which are challenging for full hydrodynamical simulations. We complement our numerical integration by analytical equations for the gap's steady-state depth and width as a function of the planet's to star's mass ratio $\mu$, the gas disc's aspect ratio $h$, and its Shakura & Sunyaev viscosity parameter $\alpha$. Specifically, we focus on low-mass planets ($\mu<\mu_{\rm th}\equiv h^3$) and identify a new low-viscosity regime, $\alpha<h(\mu/\mu_{\rm th})^5$, in which the classical analytical scaling relations are invalid. Equivalently, this low-viscosity regime applies to every gap that is depleted by more than a factor of $(\mu_{\rm th}/\mu)^3$ relative to the unperturbed density. We show that such gaps are significantly deeper and wider than previously thought, and consequently take a longer time to reach equilibrium.

Visco-resistive MHD study of internal kink (m = 1) modes

We have investigated the effect of sheared equilibrium flows on the m = 1, n = 1 resistive internal kink mode in the framework of a reduced magnetohydrodynamic model in a periodic cylindrical geometry. Our numerical studies show that there is a significant change of the scaling dependence of the mode growth rate in the Lundquist number in the presence of axial flows compared to the no flow case. Poloidal flows do not influence the scaling. We have further found that viscosity strongly modifies the effect of flows on the (1,1) mode both in the linear and nonlinear regimes. Axial flows increase the linear growth rate for low viscosity values, but they decrease the linear growth rate for higher viscosity values. In the case of poloidal flows, the linear growth rate decreases in all cases. Additionally at higher viscosity, we have found strong symmetry breaking in the behaviour of linear growth rates and in the nonlinear saturation levels of the modes as a function of the helicities of the flows. For axial, poloidal, and most helical flow cases, there is flow induced stabilisation of the nonlinear saturation level in the high viscosity regime and destabilisation in the low viscosity regime.

Electromagnetic-radiation absorption by water.

Why does a microwave oven work? How does biological tissue absorb electromagnetic radiation? Astonishingly, we do not have a definite answer to these simple questions because the microscopic processes governing the absorption of electromagnetic waves by water are largely unclarified. This absorption can be quantified by dielectric loss spectra, which reveal a huge peak at a frequency of the exciting electric field of about 20 GHz and a gradual tailing off toward higher frequencies. The microscopic interpretation of such spectra is highly controversial and various superpositions of relaxation and resonance processes ascribed to single-molecule or molecule-cluster motions have been proposed for their analysis. By combining dielectric, microwave, THz, and far-infrared spectroscopy, here we provide nearly continuous temperature-dependent broadband spectra of water. Moreover, we find that corresponding spectra for aqueous solutions reveal the same features as pure water. However, in contrast to the latter, crystallization in these solutions can be avoided by supercooling. As different spectral contributions tend to disentangle at low temperatures, this enables us to deconvolute them when approaching the glass transition under cooling. We find that the overall spectral development, including the 20GHz feature (employed for microwave heating), closely resembles the behavior known for common supercooled liquids. Thus water's absorption of electromagnetic waves at room temperature is not unusual but very similar to that of glass-forming liquids at elevated temperatures, deep in the low-viscosity liquid regime, and should be interpreted along similar lines.

Fluorescent Polystyrene Films for the Detection of Volatile Organic Compounds Using the Twisted Intramolecular Charge Transfer Mechanism.

Thin films of styrene copolymers containing fluorescent molecular rotors were demonstrated to be strongly sensitive to volatile organic compounds (VOCs). Styrene copolymers of 2-[4-vinyl(1,1′-biphenyl)-4′-yl]-cyanovinyljulolidine (JCBF) were prepared with different P(STY-co-JCBF)(m) compositions (m% = 0.10–1.00) and molecular weights of about 12,000 g/mol. Methanol solutions of JCBF were not emissive due to the formation of the typical twisted intramolecular charge transfer (TICT) state at low viscosity regime, which formation was effectively hampered by adding progressive amounts of glycerol. The sensing performances of the spin-coated copolymer films (thickness of about 4 µm) demonstrated significant vapochromism when exposed to VOCs characterized by high vapour pressure and favourable interaction with the polymer matrix such as tetrahydrofurane (THF), CHCl3 and CH2Cl2. The vapochromic response was also reversible and reproducible after successive exposure cycles, whereas the fluorescence variation scaled linearly with VOC concentration, thus suggesting future applications as VOC optical sensors.

High aspect ratio metal microcasting by hot embossing for X-ray optics fabrication

Metal microstructured optical elements for grating-based X-ray phase-contrast interferometry were fabricated by using an innovative approach of microcasting: hot embossing technology with low melting temperature (280°C) metal alloy foils and silicon etched templates. A gold-tin alloy (80wt% Au/20wt% Sn) was used to cast micro-gratings with pitch sizes in the range of 2 to20μm and depth of the structures up to 80μm. The metal filling of the silicon template strongly depends on the wetting properties of the liquid metal on the groove surface. A thin metal wetting layer (20nm of Ir or Au) was deposited before the casting in order to turn the template surface into hydrophilic with respect of the melted metal alloy. Temperature and pressure of the hot embossing process were optimized for a complete filling of the cavities in a low viscosity regime of the liquid metal, and for minimizing the shear force that might damage the silicon structures for small pitch grating. The new method has relevant advantages, such as being a low cost technique, fast and easily scalable to large area fabrication.