Stress-strain relations for random packings of entangling chains under triaxial compression can exhibit strain stiffening and sustain stresses several orders-of-magnitude beyond typical granular materials. X-ray tomography reveals the transition to this strong strain stiffening occurs when chains are long enough to entangle an average of about one chain each, which results in system-filling clusters of entangled chains, similar to the Erdös-Rényi model for randomly connected graphs. The number of entanglements is nearly proportional to the area surrounded by entangling particles with an excluded volume effect, thus the existence of system-filling clusters of entanglements can be predicted assuming random particle positions and orientations with an excluded volume effect if the particle shapes in the packing are known. A tendency was found for chain links to stretch when the packing was strained. This suggests that the strength of these packings comes from stretching of the links of chains, but only when the system-filling network of entanglements provides constraints that prevents failure by shear banding, so that particles must be deformed to move further under strain. The slope of the stress-strain relation of a packing can be calculated from a mean-field model consisting of the product of the effective extensional modulus of the chain, packing fraction, probability of stretched links, and the ratio of strain of stretched links to packing strain. In this model, the increasing slope of the stress-strain curve is mainly due to the fraction of stretched links increasing with strain, and assuming the fraction of stretched links is proportional to strain results in a quadratic prediction for the stress-strain curve. The stress-strain model requires as input measurements of the ratio between local particle deformation and global average strain, and the probability of stretching for nonrigid particles, resulting in a quadratic curvature that agrees with experiments within the run-to-run variation (30%). This model for the stress-strain relation is shown to be generalizable to different shapes of entangling particles by applying it to staples, where the packing strength comes from the bending of staples instead of stretching links. The permanent plastic deformation of staples allows measuring statistical quantities from inspection of a poured-out sample after a triaxial compression, without the need for in situ imaging. Both the probability of staples bending and the average bend angle of the arms were found to increase with strain, and these inputs into the model result in a quadratic curvature of the stress-strain that agrees with experiments within the model uncertainties (37%).

Rayleigh-B{\'e}nard convection experiments were done with two adjacent cubic cells with a partial wall in between to force the generation of two interacting convection rolls. Observed stable states include both counter-rotating and co-rotating states. The stability of each of these states and their dynamics were modeled by stochastic ordinary differential equations of motion in terms of the orientation, amplitude, and mean temperature of each convection roll. The form of the interaction terms is predicted based on an effective turbulent diffusion of temperature between the adjacent rolls. Predictions are made for stable fixed points of the co- and counter-rotating states. This suggests that the same turbulent thermal diffusivity that describes macroscopically averaged heat transport also controls the interactions between neighboring convection rolls. The surprising stability of co-rotating states is due to the temperature difference between the neighboring rolls becoming large enough that the heat flux between the rolls stabilizes the temperature profile of aligned co-rotating states. This temperature difference can be driven by heating the plates of the two cells to different mean temperatures. This shifts the orientations of the rolls of counter-rotating states in opposite directions, and for large temperature differences only co-rotating states are stable Spontaneous switching between co-rotating and counter-rotating states is also observed. Switching to counter-rotating states occurs mainly due to cessation (a significant weakening of a convection roll), which reduces damping on changes in orientation, allowing the orientation to change rapidly due to diffusive fluctuations. Switching to co-rotating states is mainly driven by smaller diffusive fluctuations, which have a positive feedback that destabilizes the counter-rotating state.

We investigate dilation-induced surface deformations in a discontinuous shear thickening (DST) suspension to determine the relationship between dilation and stresses in DST. Video is taken at two observation points on the surface of the suspension in a rheometer while shear and normal stresses are measured. A roughened surface of the suspension is observed as particles poke through the liquid-air interface, an indication of dilation in a suspension. These surface roughening events are found to be intermittent and localized spatially. Shear and normal stresses also fluctuate between high- and low-stress states, and surface roughening is observed frequently in the high-stress state. On the other hand, a complete lack of surface roughening is observed when the stresses remain at low values for several seconds. Surface roughening is most prominent while the stresses grow from the low-stress state to the high-stress state, and the roughened surface tends to span the entire surface by the end of the stress growth period. Surface roughening is found only at stresses and shear rates in and above the shear thickening range. These observed relations between surface roughening and stresses confirm that dilation and stresses are coupled in the high-stress state of DST.

We report on rheometry measurements to characterize critical behavior in two model shear thickening suspensions: cornstarch in water and glass spheres in oil. The slope of the shear thickening part of the viscosity curve is found to increase dramatically with packing fraction and diverge at a critical packing fraction φc. The magnitude of the viscosity and the yield stress are also found to have scalings that diverge at φc. We observe shear thickening as long as the yield stress is less than the stress at the viscosity maximum. Above this point the suspensions transition to purely shear thinning. Based on these data we present a dynamic jamming phase diagram for suspensions and show that a limiting case of shear thickening corresponds to a jammed state.

We experimentally test the effects of tilting a turbulent Rayleigh-Benard convection cell on the dynamics of the large-scale circulation (LSC) orientation theta(0). The probability distribution of theta(0)is measured and used to obtain a tilt-induced potential acting on theta(0), which is used in a low-dimensional model of diffusion of theta(0)in a potential. The form of the potential is sinusoidal in theta(0)and linear in tilt angle for small tilt angles, which is explained by a simple geometric model of the vector direction of the mean buoyancy force acting on the LSC. However, the magnitude of the tilt-induced forcing is found to be two orders of magnitude larger than previously predicted. When this parameter is adjusted to match the values obtained from the probability distribution of theta(0), the diffusive model can quantitatively predict the effects of tilt on theta(0). In particular, tilt causes a change in potential barrier height between neighboring corners of a cubic cell, and changes in the barrier-crossing rate for theta(0)to escape a corner are predicted with an accuracy of +/- 30%. As a cylindrical cell is tilted, the tilt-induced potential provides a restoring force that induces oscillations when it exceeds the strength of damping; this critical tilt angle is predicted within 20%, and the prediction is consistent with the measured oscillation frequencies. These observations show that a self-consistent low-dimensional model can be extended to include the dynamics of theta(0)due to tilt. However, the underprediction of the effect of tilt on theta(0)warrants revisiting the predicted magnitude.

We investigate dilation-induced surface deformations in a Discontinuous Shear Thickening (DST) suspension to determine the relationship between dilation and stresses in DST. Video is taken at two observation points on the surface of the suspension in a rheometer while shear and normal stresses are measured. A roughened surface of the suspension is observed as particles poke through the liquid-air interface, corresponding to dilation. Dilation events are found to be intermittent and localized spatially. Shear and normal stresses also fluctuate between high- and low-stress states, and dilation is observed frequently in the high stress state. On the other hand, a complete lack of dilation is observed when the stresses remain at low values for a several seconds. Dilation is most prominent while the stresses grow from the low-stress state to the high-stress state, and the dilated region tends to span the entire surface by the end of the stress growth period. Dilation is found only at stresses and shear rates in and above the shear thickening range. These observed relations between surface dilation and stresses confirm that dilation and stresses are coupled in the high-stress state of DST.

We test the ability of a low-dimensional turbulence model to predict how dynamics of large-scale coherent structures such as convection rolls change in different cell geometries. The model consists of stochastic ordinary differential equations, which were derived from approximate solutions of the Navier-Stokes equations. We test the model using Rayleigh-Benard convection experiments in a cubic container, in which there is a single convection roll known as the large-scale circulation (LSC). The model describes the motion of the orientation theta(0) of the LSC as diffusion in a potential determined by the shape of the cell. The model predicts advected oscillation modes, driven by a restoring force created by the noncircular shape of the cell cross section. We observe the corresponding lowest-wave-number predicted advected oscillation mode in a cubic cell, in which the LSC orientation theta(0) oscillates around a corner, and a slosh angle alpha rocks back and forth, which is distinct from the higher-wave-number advected twisting and sloshing oscillations found in circular cylindrical cells. Using the Fokker-Planck equation to relate probability distributions of theta(0) to the potential, we find that the potential has quadratic minima near each corner with the same curvature in both the LSC orientation theta(0) and slosh angle alpha, as predicted. To quantitatively test the model, we report values of diffusivities and damping timescales for both the LSC orientation theta(0) and temperature amplitude for the Rayleigh number range 8 x 10(7) <= Ra <= 3 x 10(9). The new oscillation mode around corners is found above a critical Ra = 4 x 10(8). This critical Ra appears in the model as a crossing of an underdamped-overdamped transition. The natural frequency of the potential, oscillation period, power spectrum, and critical Ra for oscillations are consistent with the model if we adjust the model parameters by up to a factor of 2.9, and values are all within a factor of 3 of model predictions. However, these uncertainties in model parameters are too large to correctly predict whether the system is in the underdamped or overdamped state at a given Ra. Since the model was developed for circular cross sections, the success of the model at predicting the potential and its relation to other flow properties for a square cross section-which has different flow modes than the circular cross section-suggests that such a modeling approach could be applied more generally to different cell geometries that support a single convection roll.

We present observations of oscillations in the shape of the temperature profile of the large-scale circulation (LSC) of turbulent Rayleigh-B{\'e}nard convection. Temperature measurements are broken down into Fourier moments as a function of $\theta-\theta_0$, where $\theta$ is the azimuthal angle in a horizontal plane at mid-height, and $\theta_0$ is the LSC orientation. The oscillation structure is dominated by a 3rd order sine moment and 3rd order cosine moment in a cubic cell. In contrast, these moments are not found to oscillate in a cylindrical cell. This geometry-dependent behavior can be explained by a model that assumes that the heat transported by the LSC is conducted from the thermal boundary layers, and is proportional to pathlength of the LSC along boundary layers at the top and bottom plates. In a non-circular cross-section cell, oscillations of the LSC orientation $\theta_0$ result in an oscillation in the container shape in the reference frame of the LSC, resulting in an oscillation in the pathlength of the LSC at a given $\theta-\theta_0$. In a square-cross-section cell, this model predicts the dominant 3rd order sine moment and 3rd order cosine moment with magnitudes within 50\% of measured values, when using the amplitude of the oscillation of $\theta_0$ as input. A cylindrical cell is special in that the pathlength is independent of $\theta_0$, and so these oscillating moments are not induced. In a cylindrical cell, the model reproduces the sinusoidal mean temperature profile with a sloshing oscillation dominated by the 2nd order sine moment, consistent with previous observations in that geometry.

While significant research has been dedicated to the simulation of fluids, not much attention has been given to exploring new interesting behavior that can be generated with the different types of non-Newtonian fluids with non-constant viscosity. Going in this direction, this paper introduces a computational model for simulating the interesting phenomena observed in non-Newtonian shear thickening fluids, which are fluids where the viscosity increases with increased stress. These fluids have unique and unconventional behavior, and they often appear in real world scenarios such as when sinking in quicksand or when experimenting with popular cornstarch and water mixtures. While interesting behavior of shear thickening fluids can be easily observed in the real world, the most interesting phenomena of these fluids have not been simulated before in computer graphics. The fluid exhibits unique phase changes between solid and liquid states, great impact resistance in its solid state and strong hysteresis effects. Our proposed approach builds on existing non-Newtonian fluid models in computer graphics and introduces an efficient history-based stiffness term that is essential to produce the most interesting shear thickening phenomena. The history-based stiffness is formulated through the use of fractional derivatives, leveraging the fractional calculus ability to depict both the viscoelastic behavior and the history effects of history-dependent systems. Simulations produced by our method are compared against real experiments and the results demonstrate that the proposed model successfully captures key phenomena observed in shear thickening fluids.

We present a technique for obtaining an effective packing fraction for discontinuous shear thickening suspensions near a critical point. It uses a measurable quantity that diverges at the critical point-in this case the inverse of the shear rate γ[over ̇]_{c}^{-1} at the onset of discontinuous shear thickening-as a proxy for packing fraction ϕ. We obtain an effective packing fraction for cornstarch and water by fitting γ[over ̇]_{c}^{-1}(ϕ) and then invert the function to obtain ϕ_{eff}(γ[over ̇]_{c}). We further include the dependence of γ[over ̇]_{c}^{-1} on the rheometer gap d to obtain the function ϕ_{eff}(γ[over ̇]_{c},d). This effective packing fraction ϕ_{eff} has better resolution near the critical point than the raw measured packing fraction ϕ by as much as an order of magnitude. Furthermore, ϕ_{eff} normalized by the critical packing fraction ϕ_{c} can be used to compare rheology data for cornstarch and water suspensions from different laboratory environments with different temperature and humidity. This technique can be straightforwardly generalized to improve resolution in any system with a diverging quantity near a critical point.

The effective susceptibility χeff of suspensions of ferromagnetic particles in a liquid was measured using inductance measurements. These measurements were used to test a model that predicts how χeff varies due to demagnetization, as a function of sample aspect ratio, particle packing fraction, and particle aspect ratio [R. Skomski, G. C. Hadjipanayis, and D. J. Sellmyer, IEEE Trans. Magn. 43, 2956–2958 (2007)]. For spherical particles or cylindrical particles forcibly aligned with an external magnetic field, the model can be fitted to the measured data with agreement within 13%. This model predicts suspensions of aligned, large-aspect-ratio particles should have the largest χeff, approaching the particle material susceptibility in the limit of large particle aspect ratio. However, χeff was found to be no larger than about 4 for cylindrical iron particles of various aspect ratios, close to the value obtained for spheres. This results from the random alignment of non-spherical particles relative to the magnetic field naturally found in suspensions, which increases the demagnetization effect and limits χeff. The contribution of random particle alignments to the demagnetization effect and χeff remains to be accounted for in models.

We experimentally characterize the impact response of concentrated suspensions consisting of cornstarch and water. We observe that the suspensions support a large normal stress-on the order of MPa-with a delay after the impactor hits the suspension surface. We show that neither the delay nor the magnitude of the stress can yet be explained by either standard rheological models of shear thickening in terms of steady-state viscosities, or impact models based on added mass or other inertial effects. The stress increase occurs when a dynamically jammed region of the suspension in front of the impactor propagates to the opposite boundary of the container, which can support large stresses when it spans between solid boundaries. We present a constitutive relation for impact rheology to relate the force on the impactor to its displacement. This can be described in terms of an effective modulus but only after the delay required for the dynamically jammed region to span between solid boundaries. Both the modulus and the delay are reported as a function of impact velocity, fluid height, and weight fraction. We report in a companion paper the structure of the dynamically jammed region when it spans between the impactor and the opposite boundary [Allen et al., Phys. Rev. E 97, 052603 (2018)10.1103/PhysRevE.97.052603]. In a direct follow-up paper, we show that this constitutive model can be used to quantitatively predict, for example, the trajectory and penetration depth of the foot of a person walking or running on cornstarch and water [Mukhopadhyay et al., Phys. Rev. E 97, 052604 (2018)10.1103/PhysRevE.97.052604].

The ability of a person to run on the surface of a suspension of cornstarch and water has fascinated scientists and the public alike. However, the constitutive relation obtained from traditional steady-state rheology of cornstarch and water suspensions has failed to explain this behavior. In another paper we presented an averaged constitutive relation for impact rheology consisting of an effective compressive modulus of a system-spanning dynamically jammed structure [R. Maharjan et al., this issue, Phys. Rev. E 97, 052602 (2018)10.1103/PhysRevE.97.052602]. Here we show that this constitutive model can be used to quantitatively predict, for example, the trajectory and penetration depth of the foot of a person walking or running on cornstarch and water. The ability of the constitutive relation to predict the material behavior in a case with different forcing conditions and flow geometry than it was obtained from suggests that the constitutive relation could be applied more generally. We also present a detailed calculation of the added mass effect to show that while it may be able to explain some cases of people running or walking on the surface of cornstarch and water for pool depths H>1.2 m and foot impact velocities V_{I}>1.7 m/s, it cannot explain observations of people walking or running on the surface of cornstarch and water for smaller H or V_{I}.

We characterize how suspensions of magnetic particles in a liquid respond to a magnetic field in terms of the effective magnetic susceptibility χeff using inductance measurements. We test a model that predicts how χeff varies due to demagnetization, as a function of sample aspect ratio, particle packing fraction, and particle aspect ratio [1]. For spherical particles or cylindrical particles aligned with external magnetic field, the model can be fitted to the measured data with agreement within 17%. However, we find that the random alignment of particles relative to the magnetic field plays a role, reducing χeff by a factor of 3 in some cases, which is not accounted for in models yet. While suspensions are predicted to have χeff that approach the particle material susceptibility in the limit of large particle aspect ratio, instead we find a much smaller particle aspect ratio where χeff is maximized. A prediction that χeff approaches the bulk material susceptibility in the limit of the packing fraction of the liquid-solid transition also fails. We find χeff no larger than about 4 for suspensions of iron particles.

A new class of materials is developed that is a liquid with both high conductivity and magnetic susceptibility for magnetohydrodynamic (MHD) applications. We develop a general method for making such suspensions and demonstrate that various magnetic and non-magnetic metal particles, from 40 nm - 500 microns in diameter, can be suspended in liquid gallium and its alloys. The method uses an acid solution to prevent oxidation of the liquid metal and metallic particles, which allows wetting and thus suspending. We can increase the magnetic permeability by a factor of 5.0 by controlling the packing fraction of magnetic particles, which gives these materials the potential to exhibit strong MHD effects on the laboratory scale that are usually only observable in the cores of planets and stars. We can independently tune the viscosity by a factor of 230 by adding non-magnetic particles, which would allow independent control of MHD effects from turbulence.

We investigated the transient relaxation of a Discontinuous Shear Thickening suspension of cornstarch in water. We performed 2 types of relaxation experiments starting from a steady shear in a parallel plate rheometer, followed by either stopping the top plate rotation and measuring the transient torque relaxation, or removing the torque on the plate and measuring the transient tool rotation. We found that at low weight fraction $\phi_{eff}<58.8\pm0.4\%$, the suspensions exhibited a relaxation behavior consistent with a generalized Newtonian fluid. However, for larger weight fraction $58.8\% < \phi_{eff} < 61.0\%$, near the liquid-solid transition $\phi_c=61.0\pm0.7\%$, we found relaxation behaviors different from the generalized Newtonian model. The relaxation time in this range scales with the inverse of the critical shear rate at the onset of shear thickening. In this range the relaxation time was the same in both stress and rate controlled experiments, rather than the viscosity calculated from the relaxation time which is expected to be intrinsic material parameter in the generalized Newtonian model. The discrepancy between the measured relaxation times and the generalized Newtonian prediction was found to be up to $10^4$, and extrapolations diverge in the limit of $\phi_c$ as the generalized Newtonian prediction approaches 0. At the highest weight fractions, the relaxation time scales were measured to be on the order of $\sim 1$ s. The fact that this timescale is resolvable by the naked eye may be important to understanding some of the dynamic phenomenon commonly observed in these systems. We also showed that using the critical shear rate $\dot\gamma_c$ at the onset of shear thickening to characterize the effective weight fraction can more precisely characterize material properties near the critical point $\phi_c$, allowing us to resolve this transition so close to $\phi_c$.

We experimentally study the impact response of concentrated suspensions consisting of cornstarch and water to identify the structure of the dynamically jammed region that appears and propagates ahead of the impactor once it spans from the impactor to a solid boundary. Using particle tracking at the boundary opposite the impactor, we observed a dead zone about the same size as the impactor cross-section with no particle flow in the central part of the dynamically jammed region. In the outer part of the dynamically jammed region at the bottom boundary, we observed particle flow with a net transverse displacement of up to about 5% of the impactor displacement, which indicates shear. Direct imaging to the surface of the outer part of the dynamically jammed region reveals dilation as in a dense granular flow, and cracks like a solid. This shear and dilation are in contrast to the dynamically jammed region as it propagates through the bulk, where it is argued to exhibit no shear or decrease in packing fraction. The particle flow at the boundary occurs with a delay after impact, at about the same time as the the strong stress response, confirming that the strong stress response is a consequence of this dynamically jammed structure spanning between the impactor and a solid boundary. These observations suggest the dynamically jammed structure can temporarily support stress like a soil or dense granular material along a network of frictional contacts between the impactor and solid boundary.

We investigated the transient relaxation of a Discontinuous Shear Thickening (DST) suspension of cornstarch in water. Starting from a steady shear in a parallel plate rheometer, we stopped the top plate rotation and measured the transient stress relaxation. We found that at low effective packing fraction $\phi_{eff}$, the suspensions exhibited a relaxation behavior consistent with a rheometric fluid in which the relaxation is determined by the steady-state viscosity. However, for larger $\phi_{eff}$, we find up to two exponential relaxation regimes, which both become distinct from the rheometric model. The discrepancy between the measured relaxation times and the rheometric prediction was found to be as large as 4 orders of magnitude and diverges in the limit as $\phi_{eff} \rightarrow \phi_c$, corresponding to the liquid solid transition, as the measured relaxation times diverge to infinity while the rheometric prediction approaches 0. In this limit, the measured relaxation time scales are on the order of $\sim 1$ s, which may be important to understanding the dynamic phenomenon exhibited by DST suspensions. We also showed that using the shear rate $\dot\gamma_c$ at the onset of shear thickening to characterize the effective packing fraction $\phi_{eff}$ can more precisely characterize material properties near $\phi_c$. This conversion to $\phi_{eff}$ can also be used to compare experiments done in other laboratories or under different temperature and humidity conditions on a consistent $\phi_{eff}$ scale at our reference temperature and humidity environment.