Velocity Discontinuity Research Articles

Artificial stress method resistance to tensile instability in simulating metal microspalling

<p indent=0mm>Metal microspalling is a phenomenon caused by the tensile failure of near-surface materials in molten metals. To study the dynamics of metal microspalling, the simulation results should be quantitatively compared with the experimental results. However, because of the problem’s complexity, the quantitative simulation is challenging. In this paper, the finite element method is used to simulate the detonation-driven metal tin microspalling phenomenon. Because of some nonphysical phenomena in the simulation, it is challenging to compare the velocity of the free surface and the density distribution of the microspalling region to the displacement pin system (DPS) velocity and X-ray results in the experiment. Nonphysical phenomena, such as density and velocity discontinuity in simulating the microspalling region, can be reduced or eliminated via an artificial stress method that can resist tensile instability in simulation. An artificial stress limiter is used to limit its action range to the metal tensile region. The results show that the nonphysical phenomena in the tensile region of microspalling can be effectively restrained using the artificial stress method. The density and velocity continuity of the microspalling region are considerably improved, and the mesh separation of the free surface is effectively suppressed. The simulation results in the density distribution and free surface velocity correlate well with the tin microspalling experiment driven by detonation. The analysis shows that although the artificial stress method’s resistance to tensile instability is false stress that does not physically exist, it can achieve stable calculation, weaken or eliminate nonphysical phenomena. Moreover, it has the characteristics of strong action range limitation, small amplitude, and does not affect the original conservation. The work in this paper breaks through the bottleneck of quantitative comparison between simulation and experiment of metal microspalling; in addition, it provides an important simulation technique support for dynamic analysis and physical model research of metal microspalling.

Evidence from the Vredefort Granophyre Dikes points to crustal relaxation following basin-size impact cratering

The timescale of the modification stage of basin-sized impact structures is not well understood. Owing to ca. 10 km of erosion since its formation, the Vredefort impact structure, South Africa, is an ideal testing ground for deciphering post-impact modification. Here, we present geophysical and geochemical evidence from the Vredefort Granophyre Dikes, which were derived from the - now eroded - Vredefort impact melt sheet. The dikes have been studied mostly in terms of their composition, while the timing and duration of their emplacement remain controversial. We examined the modern depth extent of five dikes, with three from the inner crystalline core of the central uplift, and two from the boundary between the core and the supracrustal collar of the central uplift, using two-dimensional electrical resistivity tomography. We found that the core dikes terminate near the present erosion surface (i.e., <5 m depth). In contrast, the dikes at the core-collar boundary extend to a depth ≥ 9 m. These observations suggest that the core dikes are exposed near their lowermost terminus. In addition, we obtained bulk geochemical composition of the dikes, finding that the andesitic composition phase is present in the core-collar dikes that is not found in the core dikes. The presence of this phase indicates the episodic emplacement of impact melt into subvertical crater floor fractures.We conclude that the dike formation was protracted and occurred over a time span of at least 104 years. The sequential formation of the Vredefort Granophyre Dikes points to horizontal extension of crust below the impact melt sheet above a kinematic velocity discontinuity, a crustal instability resulting from the dynamic collapse of the transient cavity.

Ultra-Thin Liquid Film with Shear and the Influences on Thermal Energy Transfer at Solid–Liquid Interfaces of Simple Liquid Methane in Contact With (110) Surface Structure of Face-Centred Cubic Lattice (FCC)

Thermal energy transfer (TET) is the main performance of contact interfaces which has been studied at a molecular level. Several investigations on TET were accomplished, however, the influences of liquid film thickness on TET have not been sufficiently examined. Thus, this paper analyses the influences of liquid film thickness on TET across solid–liquid (S-L) interfaces. Two liquid film thicknesses (Lz) of 30 Å and 60 Å have been evaluated, and two shear directions (x- and y-directions) have been tested in the simulation system. It has been found that there is no significant difference in the density distribution of liquid regardless of the shear directions for the same Lz. However, there are differences in the density distribution of liquid between Lz of 30 Å and 60 Å. Based on the results its suggests that, the cut-off of the temperature and velocity at the contact interfaces of solid and liquid is substantially influences by the liquid thickness of the simulation system. It is found that, there are a significant different in the thermal boundary resistance (TBR) for Lz of 30 Å and 60 Å for cases liquid sheared in the x-direction. Whereas TBR for Lz of 30 Å and 60 Å sheared in the y-direction have no significant difference. In conclusion, the TET is affected by the velocity cut-off at the contact interfaces of solid and liquid where larger velocity discontinuity exhibits higher TBR.

Estimating the static shear stiffness of joints by wave attenuation

Recent studies on the shear behavior of joints demonstrated that the joint seismic stiffness value measured using seismic wave propagation is different from the static stiffness values determined by static stress-displacement measurements. Several experimental studies on mated fractures quantified this difference and revealed that the seismic stiffness is almost two to eight times larger than the static one. However, the underlying physical mechanisms responsible for this discrepancy are still poorly understood. In this study, the difference between the seismic and static shear stiffnesses was attributed to the frequency dependence of joint stiffness. A velocity discontinuity model composed of Hooke and Newton elements was used to interpret experimentally collected shear waves transmitted through a rock joint to predict the static shear stiffness at different strain levels. Specifically, the rate-independent stiffness of this model was associated with joint deformation responses at low frequencies, providing an estimation of the static stiffness. Existing experimental results support that the excess stress model enables the estimation of the values of static stiffnesses of rough joints, as indicated by a simple shear wave transmission experiment. Moreover, the results indicate that strain rate effects and frequency effects are intrinsically related.

Possibilities and limitations of the sequential kinematic method for simulating evolutionary plasticity problems

Evolutionary plasticity processes, such as ploughing and penetrating, widely exist in many geotechnical engineering applications. The simulation of these processes poses considerable challenges due to the occurrence of large deformation, unsteady nature of the material free surface, and inherent coupling between mechanical response and material geometries. This paper explores the possibility of simulating the first-order response of these processes by using sequential kinematic method (SKM) in combination with simple deformation mechanism. The mechanism consists of rigid elements separated by velocity discontinuities. Computations based on the kinematic approach of limit analysis are sequentially performed to evaluate the most likely deformation mode and update material geometries. An r-adaptive kinematic formulation is used that captures versatile velocity fields by optimizing the geometries of simple kinematic mechanism. The modelling methodology is studied in detail for two archetypal evolutionary plasticity problems: wedge ploughing Tresca material and cylinder penetrating undrained clay. The numerical results obtained by using the SKM are compared against existing analytical and numerical solutions, as well as experimental evidence. The paper demonstrates that evolutionary plasticity problems can be simulated in a conceptually simple way using SKM and highlights the potential pitfalls of this technique.

Experimental Investigation on Mass Loss Characteristics of Broken Rocks with Discontinuous Gradation

This study mainly focuses on the transport process of broken rocks in geological fractures, faults, and karst conduits. Because previous research mainly focused on the influencing factors of adverse geological conditions, there is no clear understanding of the migration behavior of fillings and the formation process of pathways in the process of water inrush. An experimental testing apparatus was designed and developed to perform systematic research on the transport process of granular soils subjected to seepage flow. This system was comprised of a water supply system, acrylic glass holding device, and collection system, which could effectively be used to study the influence of discontinuous gradation and flow velocity on the permeability characteristics of loosening fillings. The experimental results show that the water inrush induced by seepage failure results from the gradual migration and loss of particles by water seepage. According to the fine-grain content and loss characteristics of the samples, the samples are divided into three types: those with a nonskeleton structure, with a dense-suspension structure, and with a void-skeleton structure. With an increase in the content of fine particles, the porosity and permeability decrease whereas the time required for water inrush of the sample gradually increases. With an increase in flow velocity, the time required for water inrush and its rate of change gradually decrease. However, the mass loss of particles does not increase with the increasing flow velocity during the same period. The research results provide a valuable reference with which to further study the mechanism of variable mass seepage in faulted and fractured rock.

Disentangling of Local and Wide-Field Motion Adaptation.

Motion adaptation has been attributed in flying insects a pivotal functional role in spatial vision based on optic flow. Ongoing motion enhances in the visual pathway the representation of spatial discontinuities, which manifest themselves as velocity discontinuities in the retinal optic flow pattern during translational locomotion. There is evidence for different spatial scales of motion adaptation at the different visual processing stages. Motion adaptation is supposed to take place, on the one hand, on a retinotopic basis at the level of local motion detecting neurons and, on the other hand, at the level of wide-field neurons pooling the output of many of these local motion detectors. So far, local and wide-field adaptation could not be analyzed separately, since conventional motion stimuli jointly affect both adaptive processes. Therefore, we designed a novel stimulus paradigm based on two types of motion stimuli that had the same overall strength but differed in that one led to local motion adaptation while the other did not. We recorded intracellularly the activity of a particular wide-field motion-sensitive neuron, the horizontal system equatorial cell (HSE) in blowflies. The experimental data were interpreted based on a computational model of the visual motion pathway, which included the spatially pooling HSE-cell. By comparing the difference between the recorded and modeled HSE-cell responses induced by the two types of motion adaptation, the major characteristics of local and wide-field adaptation could be pinpointed. Wide-field adaptation could be shown to strongly depend on the activation level of the cell and, thus, on the direction of motion. In contrast, the response gain is reduced by local motion adaptation to a similar extent independent of the direction of motion. This direction-independent adaptation differs fundamentally from the well-known adaptive adjustment of response gain according to the prevailing overall stimulus level that is considered essential for an efficient signal representation by neurons with a limited operating range. Direction-independent adaptation is discussed to result from the joint activity of local motion-sensitive neurons of different preferred directions and to lead to a representation of the local motion direction that is independent of the overall direction of global motion.

Sandbox Modeling of Transrotational Tectonics With Changeable Poles: Implications for the Yinggehai Basin

The main formation of the Yinggehai Basin has been related to the rotation of the Indochina block, resulting in large-scale strike-slip motion along the Red River Fault Zone (RRFZ). Transrotational tectonics played a key role in the evolution of the Yinggehai Basin. In this study, we present analog experiments with a preexisting basal velocity discontinuity boundary, rotation of crustal blocks concerning vertical axes, and syntectonic sedimentations to evaluate how the transrotational tectonics controls the evolutionary process of the Yinggehai Basin. Particle image velocimetry (PIV) was used to monitor the deformation of the model surface. Four successive poles of rotation have been applied to the model. The basin evolution underwent two phases. An early phase of deformation is characterized by the nucleation of the main internal faults above the velocity discontinuity boundary and segmented en echelon border fault systems. In the early phase, the internal and boundary faults mainly accommodated large-scale strike-slip displacement. During progressive extension, the main internal faults deactivated, and tectonic activity is localized along the boundary and secondary internal faults in the late phase. The boundary faults in the rotating block play a dominant role in the widening and deepening of the rift zone at an accelerating rate. The model surface morphology shows similarities to the Yinggehai Basin, which is wide in the middle and converges toward the northwest and southeast. In addition, experimental profiles have been compared with seismic profiles in the Yinggehai Basin. The model results also indicate that the rotation of the Indochina block combines with strong strike-slip motion. The similarities between modeling and nature provide support for ∼250 km sinistral displacement along the RRFZ between ∼32 and ∼21 Ma.

A new DNS formalism dedicated to turbulent two-phase flows with phase change

Phase change in multiphase flows occurs in many natural and industrial applications, e.g., rain formation, internal combustion engines, heat exchangers, multiphase reactors, etc. In dense two-phase flows, quantitative experimental results are scarce due to the complexity of the configuration and experimental techniques limitations. Hence, the interest in the direct numerical simulation of such flows has grown recently to better understand and access quantitative data in this kind of flows.When simulating phase change in multiphase flows, advanced numerical method are needed to consider the jump conditions at the interface, and to ensure mass, energy, momentum and species conservation. In the literature, this problem is mainly investigated with an incompressible formalism. However, this assumption is not longer suitable for simulation of multiphase flows with phase change in enclosed environment or in atomized/aerated flows.The purpose of this work is to present a numerical formalism dedicated to turbulent two phase flows, including acoustics and compressible effects with a proper treatment of the jump conditions at the interface due to phase change. To achieve this task, first, the incompressible level-set method for vaporizing two-phase flows proposed by Tanguy et al. (2007) is revisited and adapted to a mass conservative interface representation: The Coupled Level-set/Volume of Fluid method. In this context, evaporating static cylinder with a constant vaporization rate and a droplet vaporization (D2 law) have been performed as validation cases. Both cases illustrate the method accuracy and robustness in presence of velocity discontinuities at the interface due to the presence of the Stefan flow. The D2 law configuration is used as validation of the implementation of the heat and mass transfer transport equations with their jump conditions and the coupling of the evaporation rate with the flow dynamic.Then, the numerical method is extended to compressible flows using the framework dedicated to the pressure based method proposed by Duret et al. (2018). The main advantage of this framework is the ability to consider acoustics effects, variable density and multiple gas inclusions with its own thermodynamic pressure. A modified Volume of Fluid transport equation is presented, including phase change and compressibility effects. Navier–Stokes and heat and mass transfer transport equation are solved using compressible assumptions. A validation case of a static evaporating cylinder in an enclosed environment is studied to illustrate and quantify the mass conservation properties of the method and the mass transfer between the two phases. Finally, a 3D two-phase Homogeneous Isotropic Turbulence (HIT) configuration is presented to demonstrate the potential of this method in presence of breakup, gas encapsulation, coalescence and evaporation processes.

Seismic Discontinuities Across the North American Caribbean Plate Boundary From S‐to‐P Receiver Functions

AbstractThe evolution of the Caribbean plate has resulted in the formation of volcanic arcs, the Caribbean Large Igneous Province (CLIP) and micro‐plates across the plate boundary zones. The northern plate boundary with the North American plate has been particularly segmented with the transition from oblique subduction to oblique collision moving from east to west. However, there are few constraints on the seismic structure of the upper mantle across the plate boundary. Here we use S‐to‐P receiver functions to map seismic velocity discontinuities across the plate boundary, placing constraints on crustal and lithospheric thicknesses, as well as the structures associated with subduction and collision. We image a velocity increase with depth, consistently seen at 28–34 ± 4 km along the plate boundary, which corresponds to the Moho. A second strong velocity increase with depth is observed at depths of 64–66 ± 5 km, which is related to the presence of subducting slabs and anisotropic effects. We image a velocity decrease with depth at 95–135 ± 7 km, which reflects a lithosphere‐asthenosphere boundary that varies in depth across the plate boundary. The deepest negative discontinuity spatially maps to the CLIP. We suggest that a deep melting depth at 135 km, associated with an elevated potential mantle temperature of 1585 ± 20°C during CLIP formation, caused a depleted and dehydrated root to the base of melting, thus thickening the lithosphere.

Effect of the transition layer on the stability of a fluid-porous configuration: Impact on power-law rheology

The role of the transition layer (existing at the fluid-porous interface) on the hydrodynamic stability of a fluid-porous channel configuration has been investigated, particularly in the context of a non-Newtonian (power-law) fluid. The inherent relationships among the transition layer, the velocity discontinuity at the interface, and the flow criticality are explored in detail. Unlike shear-thickening, the effect of shear-thinning behavior on flow criticality is found to be non-monotonic, owing to an intricate interplay between the transition layer and the fluid rheology. Possible avenues to exploit these flow transition characteristics are also discussed.

Evolution of parallel, oblique and triple-junction rifts: Insights from analogue modelling

The study of extensional structures such as normal faults, grabens and rift systems using analogue modelling techniques has been widely used to understand the evolution of sedimentary basins. These have been particularly helpful to understand the generation of oil reservoirs and structural traps. The present work was inspired by the discussion on the origins of two sub-parallel grabens located in the intracratonic Parecis Basin in the centre-west of Brazil. Parallel, oblique or intersecting grabens may form by the reactivation of inherited basement weaknesses that can act as velocity discontinuities, with their geometry and proximity controlling the evolution of the resulting grabens. In this work, we use sandbox analogue experiments to investigate the geometric and kinematic evolution of two nearby, sometimes intersecting, grabens. We do this by varying the distance and the geometry of two basal velocity discontinuities (VDs). In all experiments, an extension of 3 cm (scaling to 12 km) was prescribed to affect a 3 cm thick sand pack. The materials were deformed using a plexiglass box with two basal velocity discontinuities attached to two lateral movable walls. In a first set of experiments, we varied systematically the distance between the VDs. In the second set, we varied the angles between the VDs with different configurations, including situations in which the VDs intersect forming triple junctions. The results show that when the VDs are initially close (<2 cm, corresponding to ~8 km in nature) there is the formation of a wide graben composed of two sub-grabens that interfere with each other, while if the spacing between the VDs is >2 cm, the grabens develop independently and do not interfere. Furthermore, it is possible to conclude that whenever (not interfering) VDs are prescribed to the base of a sand pack it always leads to the formation of asymmetric grabens, except when the two VDs are initially close to each other. In the experiments with conjugate triple-junction VDs, the extension was shared between two independent asymmetric grabens where the VDs were at a certain distance from each other, and always led to the formation of a wide graben where the VDs were close or intersect each other.

Nonequilibrium solidification velocity, recalescence degree and grain refinement of highly undercooled Ni-based single-phase alloys

The solidification velocity-undercooling relationship of infinite miscible solid solution Ni-Cu alloys was established. The research shows that the growth velocity follows four stages of “exponential growth—platform period—linear growth—platform period” within the undercooling range. The solute trapping occurs in the linear growth stage with a discontinuity velocity jump which makes the migration interface smooth and solidification uniform. According to the recalescence degree-undercooling function, the relationship among solute composition, non-equilibrium solidification degree and grain refinement was established. It is found that higher recalescence degree enhances the non-equilibrium solidification effect and the primary solid fraction formed is higher, which makes it easier for grain refinement to occur at high undercooling. By using EBSD characterization, the obvious misorientation and high-density twin boundaries were observed in the crystal structure at high undercooling. The microhardness decreases sharply after critical undercooling, and HRTEM analysis shows that the structural orientation of adjoining grains is totally different, which can provide comprehensive and reliable evidence for recrystallization mechanism.

Experimental Study of Steady Flows in Centrifuged Immiscible Liquids in Rotating Horizontal Cylinder

The steady time-averaged flows in a rotating system of immiscible fluids are studied experimentally. The fluids fill a horizontal cylindrical container and are brought into a centrifuged state, when a less dense fluid forms a cylindrical core, and a denser fluid is distributed in the form of a coaxial layer. Under the action of the gravity field, the core is steadily displaced along the radius in such a way that its axis is oriented parallel to the container axis and remains stationary in the laboratory frame of reference. In the frame rotating with the container, the core and the fluid surrounding it oscillate. This leads to the generation of a steady flow in the form of differential rotation of the fluid with respect to the container. To study the structure of the flow, light-scattering tracers of neutral buoyancy are added to the liquid. The flow rate is calculated from their displacement. Measurements show that at the interface, from the side of the external fluid, a Stokes boundary layer is formed, at the boundaries of which a tangential discontinuity of the averaged velocity is formed.

Adjoint Tomography of Ambient Noise Data and Teleseismic P Waves: Methodology and Applications to Central California

AbstractAdjoint tomography has been recently applied to ambient seismic noise and teleseismic P waves separately to unveil fine‐scale lithospheric structures beyond the resolving ability of traditional ray‐based traveltime tomography. In this study, we propose an inversion scheme that alternates between frequency‐dependent traveltime inversions of ambient noise surface waves and waveform inversions of teleseismic P waves to take advantage of their complementary sensitivities to the Earth's structure. We apply our method to ambient noise empirical Green's functions from 60 virtual sources, direct P and scattered waves from 11 teleseismic events recorded by a dense linear array (∼7 km station spacing) and other regional stations (∼40 km average station spacing) in central California. To evaluate the performance of the method, we compare tomographic results from ambient noise adjoint tomography, full‐waveform inversion of teleseismic P waves, and the alternating inversion of the two data sets. Both applications to practical field data sets and synthetic checkerboard tests demonstrate the advantage of the alternating inversion over individual inversions as it combines the complementary sensitivities of the two independent data sets toward a more unified model. The three dimensional model from our alternating inversion not only shows major features of velocity anomalies and discontinuities in agreement with previous studies, but also reveals small‐scale heterogeneities which provide new constraints on the geometry of the Isabella Anomaly and mantle dynamic processes in central California. The proposed alternating inversion scheme can be applied to other regions with similar array deployments for high‐resolution lithospheric imaging.

Seismic crustal structure beneath Jeju Volcanic Island, South Korea from teleseismicP-receiver functions

SUMMARYJeju Island is an intraplate volcanic island with enigmatic origins, located on the continental shelf south of the Korean Peninsula. A dense temporary seismic array, operated on Jeju Island from 2013 to 2015, revealed several important constraints on the magma plumbing system of Jeju Island. In this study, we determined the deep crustal seismic structure beneath Jeju Island from the teleseismic P-to-S converted phases (receiver functions) recorded from 20 temporary and three permanent stations. We removed the contribution of near-surface reverberations in the resulting receiver functions by applying a resonance removal filter. We estimated crustal P-to-S velocity ratio (VP/VS) and discontinuity depth to provide teleseismic constraints on the composition and structure. We observed two major seismic discontinuities, which are the upper boundaries of a mid-to-lower crustal low-velocity zone (LVZ) and the Moho transition zone. The depth to the upper boundary of the LVZ is deep in the west and southeast (24–30 km) and shallow in the northeast (8–11 km). The LVZ can be interpreted as an extensively distributed residual magma plumbing system, with magma batches stalled at various levels and at various degrees of crystallization, consistent with the chemical diversity of Jeju magmas. The seismic characteristics of the Moho transition zone vary greatly among regions. The top interface of the Moho transition zone is at a wide range of depth (26–40 km), and is shallow at 26–29 km depths beneath central Jeju, suggesting a complex Moho topography. The presence of mafic cumulates and partially molten mushes may contribute to the observed shallow seismic discontinuity at a depth of 26–29 km. The lack of obvious crustal thickening below the shield volcano, Mt Halla, may be associated with mantle upwelling or presence of mafic underplating and cumulates below Jeju. Spatial variations of crustal VP/VS represent highly heterogeneous crustal composition, resulting from magma differentiation during the evolution of the island.

Joint Seismic and Gravity Data Inversion to Image Intra-Crustal Structures: The Ivrea Geophysical Body Along the Val Sesia Profile (Piedmont, Italy)

We present results from a joint inversion of new seismic and recently compiled gravity data to constrain the structure of a prominent geophysical anomaly in the European Alps: the Ivrea Geophysical Body (IGB). We investigate the IGB structure along the West-East oriented Val Sesia profile at higher resolution than previous studies. We deployed 10 broadband seismic stations at 5 km spacing for 27 months, producing a new database of ∼1000 high-quality seismic receiver functions (RFs). The compiled gravity data yields 1 gravity point every 1–2 km along the profile. We set up an inversion scheme, in which RFs and gravity anomalies jointly constrain the shape and the physical properties of the IGB. We model the IGB’s top surface as a single density and shear-wave velocity discontinuity, whose geometry is defined by four, spatially variable nodes between far-field constraints. An iterative algorithm was implemented to efficiently explore the model space, directing the search toward better fitting areas. For each new candidate model, we use the velocity-model structures for both ray-tracing and observed-RFs migration, and for computation and migration of synthetic RFs: the two migrated images are then compared via cross-correlation. Similarly, forward gravity modeling for a 2D density distribution is implemented. The joint inversion performance is the product of the seismic and gravity misfits. The inversion results show the IGB protruding at shallow depths with a horizontal width of ∼30 km in the western part of the profile. Its shallowest segment reaches either 3–7 or 1–3 km depth below sea-level. The latter location fits better the outcropping lower crustal rocks at the western edge of the Ivrea-Verbano Zone. A prominent, steep eastward-deepening feature near the middle of the profile, coincident with the Pogallo Fault Zone, is interpreted as inherited crustal thickness variation. The found density and velocity contrasts of the IGB agree with physical properties of the main rock units observed in the field. Finally, by frequency-dependent analysis of RFs, we constrain the sharpness of the shallowest portion of the IGB velocity discontinuity as a vertical gradient of thickness between 0.8 km and 0.4 km.

On peculiar behaviours at critical volumes of a three-dimensional bubble rising in viscoelastic fluids

In this paper, we carried out an investigation of the dynamics of an air bubble rising in viscoelastic liquids with increasing volumes via volume-of-fluid based direct numerical simulations (DNS) with local adaptive mesh refinement techniques. The exponential Phan-Thien Tanner (EPTT) model is adopted for describing the rheological behaviours of the shear-thinning viscoelastic fluid. The well-known jump discontinuity in the terminal rise velocity at a critical bubble volume is captured, which agrees qualitatively with the experimental observation in the literature. On this basis, the numerical simulations have been performed for different rheological properties in wider flow regimes, providing new insights into the local flow field and the stress distribution around the deformable rising bubbles. A characteristic dimensionless regime map has also been proposed to distinguish the peculiar behaviours of a bubble rising in viscoelastic fluids at different Galilei (Ga) and Eötvös (Eo) numbers. We found that at low Ga and low Eo numbers, the large elastic stresses concentrate in the small region near the bubble’s trailing end, leading to the formation of bubble cusp. The appearance of the negative wake is the main reason for the velocity jump to occur. However, for higher Ga of 10, the viscous forces become less dominating. The fluctuations of the polymeric stresses are substantially intensified with increasing bubble volume, and large bubbles experience a pulsating rising velocity with oscillation shapes. On the other hand, at intermediate Ga and high Eo of 10, the modest surface tension induces a cap with a thin skirt trailing the bubble main body at early times. Due to the polymer stretching, the strong extensional flow behind the small bubble eventually forces the bubble tail to break up. The above analyses on the bubble dynamics at constant Morton (Mo) numbers facilitate the extrapolation of the volume effects at different flow regimes and, therefore, will guide the applications in the future.

A positive and stable L2-minimization based moment method for the Boltzmann equation of gas dynamics

We consider the method-of-moments approach to solve the Boltzmann equation of rarefied gas dynamics, which results in the following moment-closure problem. Given a set of moments, find the underlying probability density function. The moment-closure problem has infinitely many solutions and requires an additional optimality criterion to single-out a unique solution. Motivated from a discontinuous Galerkin velocity discretization, we consider an optimality criterion based upon L2-minimization. To ensure a positive solution to the moment-closure problem, we enforce positivity constraints on L2-minimization. This results in a quadratic optimization problem with moments and positivity constraints. We show that a (Courant-Friedrichs-Lewy) CFL-type condition ensures both the feasibility of the optimization problem and the L2-stability of the space-time discrete moment approximation. We provide an extension of our method to multi-dimensional space-velocity domains and perform several numerical experiments to showcase its accuracy.

Lid-driven Stokes slip flow in a rectangular cavity

Slip flow in a rectangular cavity is studied. Using slip eigenfunctions, an exact series solution is found for a lid sliding longitudinally. For a transversely sliding lid, the stream function for Stokes flow is determined by a new set of slip eigenfunctions and supplemented by point match on the boundary. It is found that slip has considerable effect on the flow properties. Analyses of the region very close to the corner of the velocity discontinuity show the stress singularity of no-slip flow can be eliminated by slip for both longitudinal flow and transverse flow, except when the latter corner is a right angle.