A super-stable granular heap is a pile of grains whose free surface is inclined above the angle of repose, and which forms when particles are poured onto a plane that is confined laterally by frictional sidewalls that are separated by a narrow gap. During continued mass supply, the heap free surface gradually steepens until all the inflowing grains can flow out of the domain. As soon as the supply of grains is stopped, the heap is progressively eroded, and if the base of the domain is inclined above the angle of repose, then all the grains eventually flow out. This phenomenology is modelled using a system of two-dimensional width-averaged mass and momentum balances that incorporate the sidewall friction. The granular material is assumed to be incompressible and satisfy the partially regularized $\mu (I)$-rheology. This is implemented in OpenFOAM$^{\circledR}$ and compared against small-scale experiments that study the formation, steady-state behaviour and drainage of a super-stable heap. The simulations accurately capture the dense liquid-like flows as well as the evolving heap shape. The steady uniform flow that develops along the heap surface has non-trivial inertial number dependence through its depth. Super-stable heaps are therefore a sensitive rheometer that can be used to determine the dependence of the friction $\mu$ on the inertial number $I$. However, these flows are challenging to simulate because the free-surface inertial number is high, and can exceed the threshold for ill-posedness even for the partially regularized theory.

We propose a novel approach for non-Newtonian viscoelastic steady flows based on a decomposition of the rate-of-deformation tensor which here, in a simplified version, leads to an anisotropic generalised Newtonian fluid-like model with separated treatment of kinematics pertaining to shear and extensional flows. Care is taken to assure that the approach is objective and does not introduce spurious effects due to dependence on a specific reference frame. This is done by separating the rate-of-deformation tensor into shear and elongational components, with the local scalar shear and elongation rates being the second invariant of each tensor separately, and by modifying the method used to separate the rate-of-deformation tensor so that it becomes independent of superimposed rigid rotations, thus satisfying the principle of material indifference. We assess the model with two test cases: planar contraction flow, often employed to evaluate numerical methods or constitutive equations and for which experimentally observed corner vortex enhancement and pressure drop increase are seldom found in numerical simulations; and flow past confined or unconfined cylinders, for which experiments indicate a drag increase due to elasticity and most predictions give a drag decrease. With our anisotropic model, incorporating additional elongational-flow-related terms, large vortices and accentuated pressure drop coefficients can be predicted for the contraction problem and enhanced drag coefficient for the cylinder problems. This is the first work where problems of non-Newtonian fluid mechanics are solved numerically with separation of strain rate into shear and elongational components.

The marginal ice zone represents the periphery of the sea ice cover. In this region, the macroscale behaviour of the sea ice results from collisions and enduring contact between ice floes. This configuration closely resembles that of dense granular flows, which have been modelled successfully with the $\mu (I)$ rheology. Here, we present a continuum model based on the $\mu (I)$ rheology that treats sea ice as a compressible fluid, with the local sea ice concentration given by a dilatancy function $\varPhi (I)$. We infer expressions for $\mu (I)$ and $\varPhi (I)$ by nonlinear regression using data produced with a discrete element method (DEM) that considers polygon-shaped ice floes. We do this by driving the sea ice with a one-dimensional shearing ocean current. The resulting continuum model is a nonlinear system of equations with the sea ice velocity, local concentration and pressure as unknowns. The rheology is given by the sum of a plastic term and a viscous term. In the context of a periodic patch of ocean, which is effectively a one-dimensional problem, and under steady conditions, we prove this system to be well-posed, present a numerical algorithm for solving it, and compare its solutions to those of the DEM. These comparisons demonstrate the continuum model's ability to capture most of the DEM results accurately. The continuum model is particularly accurate for ocean currents faster than 0.25 m s$^{-1}$; however, for low concentrations and slow ocean currents, the continuum model is less effective in capturing the DEM results. In the latter case, the lack of accuracy of the continuum model is found to be accompanied by the breakdown of a balance between the average shear stress and the integrated ocean drag extracted from the DEM. Since this balance is expected to hold independently of our choice of rheology, this finding indicates that continuum models might not be able to describe sea ice dynamics for low concentrations and slow ocean currents.

In the present work, the tangential (swirl) velocity component is superimposed at the intake of a narrow fluidic cylindrical pipe to achieve the desired mixing of inelastic non-Newtonian fluids/solutes at the outlet. We discuss an analytical method for obtaining the swirl velocity profile, considering the nonlinear viscous effects for both shear-thinning and shear-thickening fluids, represented by the power-law model. We numerically solve the species transport equation, coupled with the analytically derived swirl velocity, using our in-house developed code for the concentration distribution in the flow field. The results show that the inlet swirl and an increase in the shear-thinning fluid property improve advection-dominated mixing. Additionally, higher Reynolds numbers significantly enhance advection's dominance, as more rotation leads to the generation of vortices, resulting in an engulfment flow (chaotic convection) based mixing. We demonstrate that considering the increase in the shear-thinning fluid property with swirl intake reduces the amount of mixing time required in the convective regime.

The dynamic behaviour of granular media can be observed widely in nature and in many industrial processes. Yet, the modelling of such media remains challenging as they may act with solid-like and fluid-like properties depending on the rate of the flow and can display a varying apparent friction, cohesion and compressibility. Over the last two decades, the $\mu (I)$-rheology has become well established for modelling granular liquids in a fluid mechanics framework where the apparent friction $\mu$ depends on the inertial number $I$. In the geo-mechanics community, modelling the deformation of granular solids typically relies on concepts from critical state soil mechanics. Along the lines of recent attempts to combine critical state and the $\mu (I)$-rheology, we develop a continuum model based on modified cam-clay in an elastoplastic framework which recovers the $\mu (I)$-rheology under flow. This model permits a treatment of plastic compressibility in systems with or without cohesion, where the cohesion is assumed to be the result of persistent inter-granular attractive forces. Implemented in a two- and three-dimensional material point method, it allows for the trivial treatment of the free surface. The proposed model approximately reproduces analytical solutions of steady-state cohesionless flow and is further compared with previous cohesive and cohesionless experiments. In particular, satisfactory agreements with several experiments of granular collapse are demonstrated, albeit with shear bands which can affect the smoothness of the surface. Finally, the model is able to qualitatively reproduce the multiple steady-state solutions of granular flow recently observed in experiments of flow over obstacles.

The plane Poiseuille flow of a rarefied gas in a finite length channel, driven by an axial pressure gradient, is analysed numerically to probe (i) the role of ‘dilatation’ ($\varDelta ={\boldsymbol \nabla }\boldsymbol {\cdot }{\boldsymbol u}\neq 0$) on its thermohydrodynamics as well as to clarify (ii) the possible equivalence with its well-studied ‘dilatation-free’ or ‘isochoric’ (${\rm D}\rho /{\rm D}t=0$) counterpart driven by a constant acceleration. Focussing on the mass flow rate ${\mathcal {M}}({Kn})$, which is an invariant quantity for both pressure-driven and acceleration-driven Poiseuille flows, it is shown that while ${\mathcal {M}}\sim \log {{Kn}}$ at ${Kn}\gg 1$ in the acceleration-driven case, the mass flow saturates to a constant value ${\mathcal {M}}\sim {{Kn}}^0$ at ${Kn}\gg 1$ in the pressure-driven case due to the finite length ($L_x<\infty$) of the channel. The latter result agrees with prior theory and recent experiments, and holds irrespective of the magnitude of the axial pressure gradient ($G_p$). The pressure-dilatation cooling ($\varPhi _p=-p\varDelta <0$) is shown to be responsible for the absence of the bimodal shape of the temperature profile in the pressure-driven Poiseuille flow. The dilatation-driven reduction of the shear viscosity and the odd signs of two normal stress differences (${\mathcal {N}}_1$ and ${\mathcal {N}}_2$) in the pressure-driven flow in comparison with those in its acceleration-driven counterpart are explained from the Burnett-order constitutive relations for the stress tensor. While both ${\mathcal {N}}_1$ and ${\mathcal {N}}_2$ appear at the Burnett order $O({{Kn}}^2)$ in the acceleration-driven flow, the leading term in ${\mathcal N}_1$ scales as $(\mu/p)\varDelta$ due to the non-zero dilatation in the pressure-driven Poiseuille flow which confirms that the two flows are not equivalent even at the Navier–Stokes–Fourier order $O({{Kn}})$. The heat-flow rate (${{\mathcal {Q}}_q}_x=\int q_x(x,y) \,{{\rm d} y}$) of the tangential heat flux is found to be negative (i.e. directed against the axial pressure gradient), in contrast to its positive asymptotic value (at ${Kn}\gg 1$) in the acceleration-driven flow. Similar to the scale-dependence of the mass flow rate, ${{\mathcal {Q}}_q}_x({{Kn}}, L_x)$ is found to saturate to a constant value at ${Kn}\gg 1$ in finite length channels. The double-well shape of the $q_x(y)$-profile in the near-continuum limit agrees well with predictions from a generalized Fourier law. On the whole, the dilatation-driven signatures (such as the pressure-dilatation work and the ‘normal’ shear-rate differences) are shown to be the progenitor for the observed differences between the two flows with regard to (i) the hydrodynamic fields, (ii) the rheology and (iii) the flow-induced heat transfer.

In this research article, response surface methodology (RSM) based optimization of three production parameters namely temperature, time and amount of starter culture of Vechur cow milk yoghurt (VCMY) on the basis of sensory evaluation responses comparing cross-bred cow milk yoghurt (CCMY) as the control is reported. The optimized values of production parameters were 2.15 per cent rate of inoculation, 42°C incubation temperature and 4 h incubation period. The optimized product exhibited significantly lower syneresis, a*, b* values and higher L* values than CCMY. Physico-chemical, microbiological, textural and sensory properties of both VCMY and CCMY during room temperature and refrigerated storage were assessed daily until the onset of spoilage (room temperature) or at five day intervals over a period of 15 d (refrigerated). Both room temperature stored products were graded undesirable by the sensory panel upon one day of storage. Significant reduction was observed in the fat, SNF, total solids, protein and pH content and all the tested colour parameters of the optimized product during refrigerated storage. Total viable counts as well as yeast and mould counts and lactic acid bacteria counts of both VCMY and CCMY progressively increased over the 15 d of storage. Significant reductions were observed in the flavour (P < 0.01), body and texture, colour and appearance and overall acceptability (P < 0.05) scores of both the samples over a period of 15 d. During storage, hardness and adhesiveness values showed an increasing trend whereas the cohesiveness showed a decreasing trend. Storage studies revealed significant differences in the acidity, pH, syneresis, tyrosine value, colour parameters and sensorial attributes of both the yoghurt samples. During the 15 d refrigerated storage period, the VCMY exhibited superior technological attributes to CCMY in terms of lower syneresis %, acidity, microbial population, firmer and less cohesive texture, better flavour, colour and appearance scores. Being the first comprehensive study on the utilization of Vechur cow milk for the preparation of yoghurt, the data generated in the current study would provide a solid base for the exploration of fermentation as a means of value addition of milk of this very rare indigenous cattle breed.

Dense mixtures of particles of varying size tend to segregate based on size during flow. Granular size segregation impacts many industrial and geophysical processes, but the development of coupled, continuum models capable of predicting the evolution of segregation dynamics and flow fields in dense granular media across different geometries remains a challenge. One reason is because size segregation stems from two driving forces: pressure gradients and shear-strain-rate gradients. Another reason is the challenge of integrating segregation models with rheological constitutive equations for dense granular flow. In this paper we develop a continuum model that accounts for pressure-gradient-driven and shear-strain-rate-gradient-driven segregation, coupled to rheological modelling of a dense granular medium across the quasi-static and dense inertial flow regimes. To calibrate and test the continuum model, we perform discrete element method (DEM) simulations of dense flow of bidisperse granular systems in two flow geometries in which both segregation driving forces are present: inclined plane flow and planar shear flow with gravity. Steady-state DEM data from inclined plane flow is used to determine the dimensionless material parameters in the pressure-gradient-driven segregation model for both spheres and disks. Then, predictions of the continuum model are tested against DEM data across different cases of inclined plane flow and planar shear flow with gravity, while varying parameters such as the size of the flow geometry, the flow speed and the initial conditions. We find that it is crucial to account for both driving forces to capture segregation dynamics across both flow geometries with a single set of parameters.

The pH of Na-saturated, carbonate-containing and carbonate-free Leda clay, at salinities of 2 and 10 g/liter, was decreased from pH 8 to 4 by the addition of HCl. The Bingham yield stress, as determined with a coaxial viscometer, increased in all materials as the pH decreased. Above about pH 7 the 2-g/liter materials had a lower yield stress at any water content than the 10-g/liter materials, whereas, below about pH 6.8 the yield stress of the carbonate-containing soil at a salinity of 10 g/liter was lower. For the carbonate-free material, the change occurred at about pH 6.2. The influence of salinity on the remolded shear strength of these materials was pH-dependent. A yield stress increase with decreasing pH was likely due to a change in ion saturation. The carbonate-free material exhibited a maximum yield stress at about pH 5.5–6.2, depending on salinity. The isoelectric points for oxides and clay mineral edges most probably account for the existence of the maximum.

The prevailing concept that positive-edge to negative-face attraction accounts for the rheological behavior of montmorillonite suspensions at low electrolyte concentration was investigated. In one experiment, Mg2+ released from Na-montmorillonite was measured at several NaCl concentrations; in a second experiment, the viscosity, η, and the extrapolated shear stress, θ, were measured at several clay concentrations, pHs, and NaCl concentrations; and in a third experiment, the absorbance, A, was measured at two wavelengths (450 and 760 nm) at different clay and electrolyte concentrations. The released Mg2+ decreased with increasing NaCl concentration until it became zero at a NaCl concentration between 0.01 and 0.02 M, depending on pH. Thereafter, it increased with increasing NaCl concentration. Both θ and η were highly correlated with the amount of released Mg2+. Also, A remained constant until the NaCl concentration corresponded to that at the minimum of θ. Thereafter, it increased and became linearly related to θ. These results suggest: (1) positive-edge to negative-face interaction cannot solely account for the rheological properties of montmorillonite at low electrolyte concentration, and (2) the release of octahedral Mg2+ from montmorillonite affects θ, because it reduces the negative charge on the particles and, thereby, the repulsive force between them.

To determine the dependency of rheological properties on mineralogical characteristics, eleven organophilic hectorite and nine organophilic saponite samples were prepared from hydrothermally synthesized smectites and a quaternary ammonium salt containing mainly octadecyl groups as alkyl chains and were examined by X-ray powder diffraction (XRD), particle size, chemical, and rheological procedures. The rheological properties of the organophilic products dispersed in toluene were found to depend chiefly on (1) particle size, (2) the expansion behavior of the stacked clay platelets, in which the amount and location of negative charge of the silicate layer affected the expandability, and (3) the amount of methanol added as a polar activator. By the addition of 2–4% methanol, apparent viscosities of the hectorite clays were drastically increased at low shear rate, although even a 10% addition did not significantly affect the saponite clays. Under the most suitable gellation conditions, the saponite clays showed lower apparent viscosity than the hectorite clays. Inasmuch as the former had its negative charge near the surface of the silicate layer and formed strong bonds to the organic ion, the expansion of the platelets was apparently difficult. An optimum layer charge for increasing the viscosity was found; specifically 0.45–0.50 and 0.50–0.55 equivalents per half unit cell for saponite and hectorite clays, respectively. Because the saponite clays contained a denser population of organic ions than the hectorite clays, the ion density in the interlayer of both clays was considered to be the same, i.e., about 0.4 organic ions per unit ab area.

Direct water-activity measurements were made for calcium-smectite-CaCl2 mixtures. The samples used were prepared by high-pressure filtration and had water contents <0.60% by weight of clay. For deionized water, the water activity (aw) ranged between 1 and 0.4 and was independent of the pressure history of the clay cake. Good agreement was found with the results of an adsorption isotherm. Samples were also prepared in 1,3, and 5 M CaCl2 solutions. The final concentration of chloride present in the cakes was measured: substantial anion exclusion was observed. The water activity of the clay-salt mixtures depended on both the water content and the concentration of CaCl2. The data can be well represented by the expression aw = min(aw0, aw*c × aw*s), where aw0 is the water activity of the brine used to prepare the cake, and aw*c and aw*s are water activities calculated for clay (in absence of salt) and salt (in absence of clay), respectively. The compaction and swelling behavior of a core of the same Ca-smectite was also investigated for pressures (P) between 0.5 and 800 bar. In two compaction-swelling cycles total recovery of the void ratio (e) was observed with, however, large hysteresis in the relationship between e and P. The osmotic pressure developed in the core at equilibrium, evaluated using the water-activity data, was intermediate between the compaction and swelling pressures. Apparently, another force, Pm, linked to irreversible changes in the structure of the matrix was contributing to the resistance to compaction. Pm values agreed with literature results for Ca-smectite suspensions.

Kaolin floc structures of commercial utility are produced by adsorption of hexamethylenediamine. These flocs are largely face-to-face structures with small diameter platelets of colloidal kaolinite aggregated on larger platelets of kaolinite and muscovite. Because of its higher charge density, muscovite is more effective than kaolinite in scavenging colloidal particles.

Although earlier workers have been at odds as to where amines adsorb on kaolinite, the present work indicates that both edge and basal surfaces are involved. Evidence for this contention is two-fold: (1) Face-to-face and edge-to-face flocculation of hexamethylenediamine kaolins has been observed. (2) Polyesters have been observed adsorbed on basal and edge surfaces of ethylenediamine-treated kaolins.

The mechanism by which kaolinite particles are bonded together into floc structures is unclear. Bridging by hexamethylenediammonium ions is a probable mechanism up to satisfaction of the cation-exchange capacity; further amine addition continues to promote flocculation. From this observation it is inferred that, in addition to bridging, hydrogen bonding and, perhaps, van der Waals forces are instrumental in floc formation. Although rheologically inferior to the untreated equivalent, these chemically induced flocs of kaolinite provide loosely packed, randomly oriented structures in paper coatings. Improved brightness, opacity, and printability of paper coatings arise from this structure.

The rheological behavior of concentrated lateritic suspensions from Cuba is affected by mineral composition and particle size. Electrophoretic mobility and yield stress were considered. The lateritic samples were found to be mostly composed of mixtures of serepentine and goethite in varying proportions. The flow properties of the lateritic suspensions are strongly affected by the mineral composition and particle size. This result was determined by comparison of flow properties of the bulk sample and the colloidal fraction. The electrokinetic curves suggest that heterocoagulation is present in all samples, with a zeta potential minimum at the isoelectric point (IEP), which varies with the serpentine to goethite ratio. A relationship between yield stress (τ0) and the sample volume fraction (ϕ) and particle size (d) was obtained at the IEP from the expression τ0 = kϕ3/d0.5, with the constant k dependent on the sample serpentine to goethite ratio.

The colloidal state (stable, coagulated, or gel-like) and the rheological properties of Na-rich montmorillonite (Wyoming) dispersions are strongly influenced by organic cations. This effect is shown for homologous organic cations: alkyl trimethylammonium ions, paraquat, diquat, alkyl bispyridinium ions, and the triphenylmethane dyes crystal violet, methyl green, and tris (tri-methylammonium phenyl) methane chloride. The critical coagulation concentrations, cK, are small (often < 1 mmol/L) because the cations are enriched in the Stern layer and influence the solvent structure near the surface. The strong adsorption of the counterions at the clay-mineral surface causes cK values to increase with the solid content. Charge reversal (recharging) of the particles was observed with the longer chain alkyl trimethyl-ammonium ions, dodecyl bispyridinium ions, and crystal violet. Other cations reduced the electrophoretic mobility to zero but positive particle charges were not observed.

The plastic viscosity increased sharply at the critical coagulation concentration and showed a minimum slightly below cK, which was caused by the electroviscous effect. Yield values were developed at concentrations above cK. In most cases, yield values reached a plateau where the amount of organic cations was ∼0.5 mmol/g, i.e., about half of the cation-exchange capacity. The cK values decreased with increasing hydrophobicity of homologous compounds, but the yield value showed maxima at intermediate chain lengths. The yield value of several 0.5% dispersions was high, e.g., dodecyl trimethylammonium ions, 71 Pa; paraquat, 100 Pa; diquat, 42 Pa; hexyl bispyridinium ions, 53 Pa (vs. Ca2+, 0.2 Pa; Al3+, 0.7 Pa). The storage modulus as a function of the number of organic cations changed in a similar way as the yield value, and high values were observed (e.g., dodecyl trimethylammonium ions, hexyl bispyridinium ions: 1000 Pa, paraquat: >4000 Pa). Thus, dispersions with high viscosity, yield value, and pronounced viscoelasticity are obtained by coagulating Na-rich montmorillonite dispersions with organic cations.

In this paper a fully coupled particle-size segregation model for granular flows (Barker et al., J. Fluid Mech., vol. 909, 2021, p. A22) is used to simulate the development of the patterns in a triangular rotating drum. The results are compared with the experimental patterns formed with bidisperse and tridisperse granular mixtures, and with varying compositions and fill heights. In all cases the agreement between the simulations and experiments is remarkably good. The experimental patterns are generated in a narrow gap between transparent front and back sidewalls. These prevent three-dimensional motion, but also impose friction on the flow, making it thinner and faster than it would otherwise be. This promotes segregation, as it simultaneously increases the shear rate and reduces the local pressure. To obtain the correct flow dynamics and segregation, width-averaged sidewall friction is incorporated into the two-dimensional simulations, which are performed in OpenFOAM$^{\circledR}$. The free-surface avalanche forms a boundary layer within which all the segregation occurs. Material in the lower reach of the avalanche is continuously deposited into an underlying solid body of grains, which rotates with the drum, and is eventually re-entrained into the avalanche along its upper reach. The changing geometry of the granular region (as the drum rotates) implies that the avalanche is constantly adjusting its length, position and depth. This generates a complex quasi-periodic flow, which when combined with particle-size segregation generates amazing patterns in the solid rotating granular body after only two drum rotations.

Gravure printing is a type of printing method that uses metal cylinders with engraved cells that hold ink. The ink is transferred directly to the paper or other material by pressing it against the cylinder. The flow associated with gravure printing includes a flow in a liquid bridge formed in the contact region of the cylinders and a thin-film coating flow of the ink. The flow is governed by viscous and capillary forces. In many cases, the flow is unstable, which leads to the formation of instability patterns on the printed surfaces. The analysis of these instabilities is a very challenging problem, especially since industrial inks are usually rheologically complex. In this experimental and theoretical study, the flow of inks on a rotating cylinder is analysed, accounting for the shear-rate-dependent liquid viscosity. A theoretical solution for the film flow allows us to predict the width of the liquid bridge between two cylinders. Moreover, it is shown that the measured characteristic size of the printed pattern is of the same order as the predicted liquid bridge width. We observe a nearly linear dependence of pattern size and liquid bridge width.

The finitely extensible nonlinear elastic with Peterlin closure (FENE-P) and simplified Phan-Thien–Tanner (sPTT) viscoelastic models are used widely for modelling of complex fluids. Although they are derived from distinct micro-structural theories, these models can become mathematically identical in steady and homogeneous flows with a particular choice of the values of the model parameters. However, even with this choice of parameter values, the model responses are known to differ from each other in transient flows. In this work, we investigate the responses of the FENE-P and sPTT constitutive models in large-amplitude oscillatory shear (LAOS). In steady shear, the shear stress scales with the non-dimensional group $Wi/(aL)$ ($Wi\,\sqrt {\epsilon }$) for the FENE-P (sPTT) model, where $Wi$ is the Weissenberg number, $L^2$ is the limit of extensibility in the FENE-P model ($a$ being $L^2/(L^2-3)$), and $\epsilon$ is the extensibility parameter in the sPTT model. Our numerical and analytical results show that in LAOS, the FENE-P model shows this universality only for large values of $L^2$, whereas the sPTT model shows it for all values of $\epsilon$. In the strongly nonlinear region, there is a drastic difference between the responses of the two models, with the FENE-P model exhibiting strong shear stress overshoots that manifest as self-intersecting secondary loops in the viscous Lissajous curves. We quantify the nonlinearity exhibited by each constitutive model using the sequence of physical processes framework. Despite the high degree of nonlinearity exhibited by the FENE-P model, we also show using fully nonlinear one-dimensional simulations that it does not shear band in LAOS within the range of conditions studied.