1. Gupta, R. and Meheboob Alam, Knudsen paradox in rarefied gases and the roles of thermal and athermal walls,
(2017).
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The type of wall-conditions is shown to have a profound impact on the well-known ``Knudsen paradox'' [which refers to the decrease of the mass flow rate of a gas with increasing Knudsen number $Kn$, reaching a minimum at $Kn\sim O(1)$ and increasing logarithmically with $Kn$ as $Kn\to \infty$] in the acceleration-driven Poiseuille flow of rarefied gases. The non-monotonic-variation of the flow-rate with $Kn$ occurs even in a granular/dissipative gas in contact with both diffuse and specular thermal-walls. This result is in contradiction with a recent work (Alam et al, {\it J. Fluid Mech.}, vol. 782, 2015, pp.~99-126) that revealed that the gas flow-rate decreases at large values of $Kn$. The above conundrum is resolved by distinguishing between ``thermal'' and ``athermal'' walls, and it is shown that, for both molecular and granular gases, the momentum-transfer to athermal-walls is much lower than that to thermal-walls which is directly responsible for the ``anomalous'' flow-rate-variation with $Kn$ in the rarefied regime. In the continuum limit of $Kn\to 0$, the athermal walls are in fact closely related to `no-flux/adiabatic' walls for which the Knudsen-minimum does not exist too. The underlying mechanistic arguments lead to Maxwell's slip boundary condition and a possible characterization of athermal walls in terms of an effective specularity coefficient is discussed.

2. Ansari I H, Rivas N and Meheboob Alam, Phase-coexisting patterns with segregation and controlled convection in vibrated binary granular mixtures,
(2017).
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Understanding segregation-driven patterns and finding ways to control them is a key challenge in granular physics research. Here we report novel patterns displaying the coexistence of sub-harmonic/harmonic and asynchronous states (for example, a granular gas co-existing with (i) bouncing bed, (ii) undulatory waves and (iii) Leidenfrost-like state) along with a {\it partial} convection state, in experiments on vertically vibrated binary mixtures of different density particles. The segregation of heavier and lighter particles along the horizontal direction is shown to be the progenitor of such {\it phase-coexisting} patterns, having different temporal and spatial symmetries. A simple method to control the buoyancy-driven granular convection is demonstrated. The non-equipartition of granular energy between two species seems to be responsible for the observed phenomena as confirmed in particle-dynamics simulations.

3. Saha, S. and Meheboob Alam, Revisiting ignited-quenched transition and the non-Newtonian rheology of a sheared dilute gas-solid suspension,Journal of Fluid Mechanics,
(2017).
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The hydrodynamics and rheology of a sheared dilute gas-solid suspension, consisting of inelastic hard spheres suspended in a gas, are analysed using anisotropic Maxwellian as the single particle distribution function. For the simple shear flow, the second-moment tensor ${\mathsfb M}$ is assumed to be anisotropic, characterized by three parameters: (i) the non-coaxiality angle ($\phi$, the angle between the principal eigen-directions of ${\mathsfb M}$ and the shear tensor ${\mathsfb D}$), (ii) the shear-plane temperature-anisotropy ($\eta$, the difference between the principal eigenvalues of ${\mathsfb M}$ on the shear plane, $\propto T_x-T_y$) and (iii) the excess temperature ($\lambda^2\propto T -T_z$) along the vorticity direction. The closed-form expressions for three invariants ($\phi$, $\eta$, $\lambda^2$) and granular temperature ($T$) are obtained as functions of the Stokes number ($St$), the mean density ($\nu$) and the restitution coefficient ($e$) by solving the second-moment balance equation. Multiple states of high and low temperatures are found when the Stokes number is small, thus recovering the ``ignited'' and ``quenched'' states, respectively, of Tsao \& Koch (J. Fluid Mech.,1995, vol. 296, pp. 211-246). The phase diagram is constructed in the three-dimensional ($\nu, St, e$)-space that delineates the regions of the ignited and quenched states and their coexistence. Analytical expressions for the shear viscosity and the normal stress differences are obtained. At any $e$, the shear-viscosity undergoes a discontinuous jump with increasing shear rate ($St$) at the ``quenched-ignited'' transition. The first (${\mathcal N}_1$) and second (${\mathcal N}_2$) normal-stress differences also undergo similar first-order transitions: (i) ${\mathcal N}_1$ jumps from large to small positive values and (ii) ${\mathcal N}_2$ from positive to negative values with increasing $St$, with the sign-change of ${\mathcal N}_2$ being identified with the system making a transition from the quenched to ignited states. For both granular and gas-solid suspensions, the excess temperature along the vorticity direction is responsible for ${\mathcal N}_2\neq 0$, while the non-coaxiality angle and the shear-plane temperature-anisotropy are responsible for ${\mathcal N}_1\neq 0$. The superior prediction of the present theory over the standard Grad's method and the Chapman-Enskog solution is demonstrated via comparisons of transport coefficients with simulation data for a range of Stokes number and restitution coefficient.

A recently proposed beyond-Navier-Stokes order hydrodynamic theory for dry granular fluids is revisited by focussing on the behaviour of the stress tensor and the scaling of related transport coefficients in the dense limit. For the homogeneous shear flow, it is shown that the eigen-directions of the second-moment tensor and those of the shear tensor become co-axial, thus making the first normal stress difference (${\mathcal N}_1$) to zero in the same limit. In contrast, the origin of the second normal stress difference (${\mathcal N}_2$) is tied to the `excess' temperature along the mean-vorticity direction and the imposed shear field, respectively, in the dilute and dense flows. The scaling relations for transport coefficients are suggested based on the present theory.

Granular flows around an object have been the focus of numerous analytical, experimental and simulation studies. The structure and nature of the oblique shock wave developed when a quasi-two dimensional flow of spherical granular particles streams past a fixed cylindrical obstacle forms the focus of this study. The binary granular mixture, consisting of particles of the same diameter but different material properties, is investigated via soft-particle DEM. Variations of the solid fraction and granular temperature are analysed and the shock-front is identified. The local Mach number is calculated to distinguish between the subsonic and the supersonic regions of the bow shock.

Hydrodynamic fields, macroscopic boundary conditions and non-Newtonian rheology of the acceleration-driven Poiseuille flow of a dilute granular gas are probed using `direct simulation Monte Carlo' (DSMC) method for a range of Knudsen numbers ($Kn$, the ratio between the mean free path and the macroscopic length) spanning the rarefied regime of slip and transitional flows. It is shown that the ``dissipation-induced clustering'' (for $1-e_n>0$, where $e_n$ is the restitution coefficient), leading to inhomogeneous density profiles along the transverse direction, competes with ``rarefaction-induced declustering'' (for $Kn>0$) phenomenon, leaving seemingly ``anomalous'' footprints on several hydrodynamic and rheological quantities; one example is the well-known rarefaction-induced temperature bimodality which could also result from inelastic dissipation that dominates in the continuum limit ($Kn\to 0$) as found recently (M. Alam et al., 2015, J.~Fluid Mech., vol. 782, pp.~99-126). The simulation data on the slip-velocity and the temperature-slip are contrasted with well-established boundary-conditions for molecular gases. A modified Maxwell-Navier-type boundary condition is found to hold in granular Poiseuille flow, with the velocity slip-length following a power-law relation with Knudsen number $Kn^\delta$, with $\delta\approx0.95$, for $Kn\leq 0.1$. Transverse profiles of both first [${\mathcal N}_1(y)$] and second [${\mathcal N}_2(y)$] normal stress differences seem to correlate well with respective density profiles at small $Kn$; their centreline values [${\mathcal N}_1(0)$ and ${\mathcal N}_2(0)$] can be of ``odd'' sign with respect to their counterparts in molecular gases. The phase-diagrams are constructed in the ($Kn, 1-e_n$)-plane that demarcates the regions of influence of inelasticity and rarefaction which compete with each other resulting in the sign-change of both ${\mathcal N}_1(0)$ and ${\mathcal N}_2(0)$. The results on normal stress differences are rationalized via a comparison with a Burnett-order theory (N. Sela and I. Goldhirsch, 1998, J. Fluid Mech., vol. 361, pp.~41-75) which is able to predict their correct behaviour at small values of the Knudsen number. Lastly, the Knudsen paradox and its dependence on inelasticity are analysed and contrasted with related recent works.

7. Gera, B, Singh, R K and Meheboob Alam, Numerical study of the effect of density and aspect-ratio on oscillatory exchange flow through a circular opening in horizontal partition.,Proceedings of Computational Mechanics and Simulation (ICCMS2016),
246 - 249 (2016).
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An interesting transport phenomenon is observed through openings between two compartments separated by a thin, vented, horizontal partition. A heavier fluid located on the top of a lighter fluid and separated by a horizontal vent constitutes a gravitationally unstable system and produces an unstable flow with irregular oscillatory behaviour. In the present work CFD simulations have been performed to simulate such type of flow across a circular opening in a horizontal partition. The effect of density ratio and opening aspect ratio on the oscillation frequency and flow coefficient through the opening has been investigated. An in-house FVM based CFD code was developed to solve unsteady, axisymmetric Navier-Stokes equations along with realizable k-ε turbulence model and species transport for salt-water mass fraction. In terms of temporal differencing a second order accurate Crank-Nicolson scheme was used. Interpolation to cell faces for the convective terms was performed using a third-order QUICK scheme and a second-order central differencing was used for viscous terms. The upper-chamber was filled with salt water and the lower-chamber with fresh water, creating a density differential between the two chambers. The code was validated against reported experiments of this nature. The flow coefficients and pulsation frequency have been determined. Various cases were studied by varying the density ratio from 1.012 to 1.2 while the opening aspect ratio was varied from 0.008 to 0.9. The effect of these parameters was investigated on oscillation frequency and flow coefficient.

8. Gera, B, Singh, R K and Meheboob Alam, CFD simulation of combined buoyancy and pressure driven hot-gas flow through square-opening with salt-water analogy.,Proceedings of Computational Mechanics and Simulation (ICCMS2016),
273 - 277 (2016).
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Movement of hot gases/smoke generated from accidental fire through openings in the horizontal partition between two compartments is crucial for proper ventilation design. Current transport calculations often use vent-flow model and consider it as pressure-driven flow and do not account for the effect of buoyancy on the flow. Use of salt-water/fresh water analogy to model such type of flow is an excellent alternative approach as this system requires less cost and the flow visualization is easy. In the present work the CFD simulations of such experiments have been conducted. In the reported experiments a scaled-compartment of size 0.375 m X 0.25 m X 0.25 m was placed in a large compartment connected through a square opening in horizontal partition. Both the compartments were filled with fresh water; a plume of salt water was injected vertically downward in the small compartment to simulate the heated plume. Salt-water layer height, flow-rate through opening, stratification in the compartment was recorded. Various experiments were conducted by varying the size of opening and density of injected salt water. 3D transient CFD simulations have been performed to validate the in-house CFD code against this experiment. Standard k-ε model was used for modelling turbulence. Salt-water layer height, exchange flow coefficients were determined through CFD simulations and compared with available experimental data.

9. Reddy, M H L and Meheboob Alam, Plane shock wave structure in a dilute granular gas,AIP Conference Proceedings,
1786, 120001 (2016).
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We analyse the early time evolution of the Riemann problem of planar shock wave structures for a dilute granular gas by solving Navier-Stokes equations numerically. The one-dimensional reduced Navier-Stokes equations for plane shock wave problem are solved numerically using a relaxation-type numerical scheme. The results on the shock structures in granular gases are presented for different Mach numbers and restitution coefficients. Based on our analysis on early time shock dynamics we conclude that the density and temperature profiles are “asymmetric”; the density maximum and the temperature maximum occur within the shock layer; the absolute magnitudes of longitudinal stress and heat flux which are initially zero at both end states attain maxima in a very short time and thereafter decrease with time.

Experiments are conducted in a two-dimensional monolayer vibrofluidized bed of glass beads, with a goal to understand the transition scenario and the underlying microstructure and dynamics in different patterned states. At small shaking accelerations (\Gamma = Aω2/g < 1, where A and ω = 2πf are the amplitude and angular frequency of shaking and g is the gravitational acceleration), the particles remain attached to the base of the vibrating container; this is known as the solid bed (SB). With increasing (at large enough shaking amplitude A/d) and/or with increasing A/d (at large enough $\Gamma$), the sequence of transitions/bifurcations unfolds as follows: SB (“solid bed”) to BB (“bouncing bed”) to LS (“Leidenfrost state”) to “2-roll convection” to “1-roll convection” and finally to a gas-like state. For a given length of the container, the coarsening of multiple convection rolls leading to the genesis of a “single-roll” structure (dubbed the multiroll transition) and its subsequent transition to a granular gas are two findings of this work. We show that the critical shaking intensity (\GammaLS ) for the BB → LS transition has BB a power-law dependence on the particle loading (F = h0/d, where h0 is the number of particle layers at rest and d is the particle diameter) and the shaking amplitude (A/d). The characteristics of BB and LS states are studied by calculating (i) the coarse-grained density and temperature profiles and (ii) the pair correlation function. It is shown that while the contact network of particles in the BB state represents a hexagonal-packed structure, the contact network within the “floating cluster” of the LS resembles a liquid-like state. An unsteadiness of the Leidenfrost state has been uncovered wherein the interface (between the floating cluster and the dilute collisional layer underneath) and the top of the bed are found to oscillate sinusoidally, with the oscillation frequency closely matching the frequency of external shaking. Therefore, the granular Leidenfrost state is a period-1 wave as is the case for the BB state.

The rheology of the steady uniform shear flow of smooth inelastic spheres is analysed by choosing the anisotropic/triaxial Gaussian as the single-particle distribution function. An exact solution of the balance equation for the second-moment tensor of velocity fluctuations, truncated at the ‘Burnett order’ (second order in the shear rate), is derived, leading to analytical expressions for the first and second (N 1 and N 2 ) normal stress differences and other transport coefficients as functions of density (i.e. the volume fraction of particles), restitution coefficient and other control parameters. Moreover, the perturbation solution at fourth order in the shear rate is obtained which helped to assess the range of validity of Burnett-order constitutive relations. Theoretical expressions for both N1 and N2 and those for pressure and shear viscosity agree well with particle simulation data for the uniform shear flow of inelastic hard spheres for a large range of volume fractions spanning from the dilute regime to close to the freezing-point density (ν ∼ 0.5). While the first normal stress difference N1 is found to be positive in the dilute limit and decreases monotonically to zero in the dense limit, the second normal stress difference N2 is negative and positive in the dilute and dense limits, respectively, and undergoes a sign change at a finite density due to the sign change of its kinetic component. It is shown that the origin of N1 is tied to the non-coaxiality (φ\neq 0) between the eigendirections of the second-moment tensor M and those of the shear tensor D. In contrast, the origin of N2 in the dilute limit is tied to the ‘excess’ temperature (Tex = T − Tz, where Tz and T are the z-component and the average of the granular temperature, respectively) along the mean vorticity (z) direction, whereas its origin in the dense limit is tied to the imposed shear field.

The numerical simulation of gravity-driven flow of smooth inelastic hard-disks through a channel, dubbed `granular' Poiseuille flow, is conducted using event-driven techniques. We find that the variation of the mass-flow rate ($Q$) with Knudsen number ($Kn$) can be non-monotonic in the elastic limit (i.e.~the restitution coefficient $e_n\to 1$) in channels with very smooth walls. The {\it Knudsen-minimum} effect (i.e.~the minimum flow rate occurring at $Kn\sim O(1)$ for the Poiseuille flow of a molecular gas) is found to be absent in a granular gas with $e_n < 0.99$, irrespective of the value of the wall roughness. Another rarefaction phenomenon, the {\it bimodality} of the temperature profile, with a local minimum ($T_{\min}$) at the channel centerline and two symmetric maxima ($T_{\max}$) away from the centerline, is also studied. We show that the inelastic dissipation is responsible for the onset of temperature bimodality [i.e.~the `excess' temperature, $\triangle T= (T_{\max}/T_{\min}-1)\neq 0$] near the continuum limit ($Kn\sim 0$), but the rarefaction being its origin (as in the molecular gas) holds beyond $Kn\sim O(0.1)$. The dependences of the excess temperature $\triangle T$ on the restitution coefficient is compared with the predictions of a kinetic model, with reasonable agreement in the appropriate limit. The competition between dissipation and rarefaction seems to be responsible for the observed dependence of both the mass-flow rate and the temperature bimodality on $Kn$ and $e_n$ in this flow. The validity of the Navier-Stokes-order hydrodynamics for granular Poiseuille flow is discussed with reference to the prediction of bimodal temperature profiles and related surrogates.

The Riemann problem of planar shock waves is analysed for a dilute granular gas by solving Euler- and Navier-Stokes-order equations numerically. The density and temperature profiles are found to be asymmetric, with the maxima of both density and temperature occurring within the shock-layer. The density-peak increases with increasing Mach number and inelasticity, and is found to propagate at a steady speed at late times. The granular temperature at the upstream end of the shock decay according to Haff's law [$\theta(t)\sim t^{-2}$], but the downstream temperature decays faster than its upstream counterpart. The Haff's law seems to hold inside the shock up-to a certain time for weak shocks, but deviations occur for strong shocks. The time at which the maximum temperature deviates from Haff's law follows a power-law scaling with upstream Mach number and the restitution coefficient. The origin of the continual build-up of density with time is discussed, and it is shown that the granular energy equation must be `regularized' to arrest the maximum density.

14. Reddy, M. H. L., Ansumali, S. and Meheboob Alam,
Shock Waves in a Dilute Granular Gas,AIP Conf. Proc.,
1628, 480 - 487 (2014).
, Read abstract.

We study the evolution of shock waves in a dilute granular gas which is modelled using three variants of hydrodynamic equations: Euler, 10-moment and 14-moment models. The one-dimensional shock-wave problem is formulated and the resulting equations are solved numerically using a relaxation-type scheme. Focusing on the specific case of blast waves, the results on the density, the granular temperature, the skew temperature, the heat flux and the fourth moment are compared among three models. We find that the shock profiles are smoother for the 14-moment model compared to those predicted by the standard Euler equations. A shock-splitting phenomenon is observed in the skew granular temperature profiles for a blast wave.

The nonlinear instability of the density-inverted granular Leidenfrost state and the resulting convective motion in strongly shaken granular matter are analysed via a weakly nonlinear analysis of the hydrodynamic equations. The base state is assumed to be quasi-steady and the effect of harmonic shaking is incorporated by specifying a constant granular temperature at the vibrating plate. Under these mean-field assumptions, the base state temperature decreases with increasing height away from from the vibrating plate, but the density profile consists of three distinct regions: (i) a collisional dilute layer at the bottom, (ii) a levitated dense layer at some intermediate height and (iii) a ballistic dilute layer at the top of the granular bed. For the nonlinear stability analysis [Shukla & Alam, J.~Fluid Mech., vol.~672, 2011b, pp.~147--195], the nonlinearities up-to cubic order in the perturbation amplitude are retained, leading to the Landau equation, and the related adjoint stability problem is formulated taking into account appropriate boundary conditions. The first Landau coefficient and the related modal eigenfunctions (the fundamental mode and its adjoint, the second harmonic and the base-flow distortion, and the third harmonic and the cubic-order distortion to the fundamental mode) are calculated using a spectral-based numerical method. The genesis of granular convection is shown to be tied to a supercritical pitchfork bifurcation from the density-inverted Leidenfrost state. Near the bifurcation point the equilibrium amplitude ($A_e$) is found to follow a square-root scaling law, $A_e ~ \sqrt{\triangle}$, with the distance ${\triangle}$ from the bifurcation point. We show that the strength of convection (measured in terms of velocity circulation) is maximal at some intermediate value of the shaking strength, with weaker convection both at weaker and stronger shaking. Our theory predicts that at very strong shaking the convective motion remains concentrated only near the top surface, with the bulk of the expanded granular bed resembling the conduction state of a granular gas, dubbed as floating convection state. The linear and nonlinear patterns of density and velocity fieldsare analysed and compared with experiments qualitatively. The evidence of $2:1$ resonance is shown for certain parameter combinations. The influences of bulk viscosity, effective Prandtl number, shear work and free-surface boundary conditions on nonlinear equilibrium states are critically assessed.

The non-Newtonian stress tensor, the collisional dissipation rate and the heat flux in the plane shear flow of smooth inelastic disks are analysed from the Grad-level moment equations using the anisotropic Gaussian as a reference. For the steady uniform shear flow, the balance equation for the second moment of velocity fluctuations is solved semi-analytically, yielding closed-form expressions for the shear viscosity ($\mu$), the pressure ($p$), the first normal stress difference (${\mathcal N}_1$) and the dissipation rate (${\mathcal D}$) as functions of (i) the density or the area fraction ($\nu$), (ii) the restitution coefficient ($e$), (iii) the dimensionless shear rate ($R$), (iv) the temperature anisotropy [$\eta$, the difference between the principal eigenvalues of the second moment tensor] and (v) the angle ($\phi$) between the principal directions of the shear tensor and the second moment tensor. The last two parameters are zero at the Navier-Stokes order, recovering the known exact transport coefficients from the present analysis in the limit $(\eta,\phi)\to 0$, and are therefore measures of the non-Newtonian rheology of the medium. An exact analytical solution for leading-order moment equations is given which helped to determine the scaling relations of $R$, $\eta$ and $\phi$ with inelasticity. We show that the super-Burnett order terms must be retained for a quantitative prediction of transport coefficients, especially at moderate-to-large densities for small values of the restitution coefficient ($e << 1$). Particle simulation data for a sheared inelastic hard-disk system is compared with theoretical results, with good agreement for $p$, $\mu$ and ${\mathcal N}_1$ over a range of density spanning from the dilute up-to the freezing point. In contrast, the predictions from a Navier-Stokes order constitutive model are found to deviate significantly from both the simulation and the moment theory even at moderate values of the restitution coefficient ($e\sim 0.9$). Lastly, a generalized Fourier law for the granular heat flux, which vanishes identically in the uniform shear state, is derived for a dilute granular gas by analysing the non-uniform shear flow via an expansion around the anisotropic Gaussian state. We show that the gradient of the deviatoric part of the kinetic stress drives a heat current and the thermal conductivity is characterized by an anisotropic second-rank tensor for which explicit analytical expressions are given.

The orientational/angular correlation between the directions of the translational and rotational motions is analyzed theoretically for the homogeneous cooling state of a rough granular gas. The dynamical equations are derived (via the technique of pseudo-Liouville operator) using an approximate form of the single-particle distribution function that incorporates angular correlations. The goal is to assess the effects of higher-order angular corrections for which both quadratic- and quartic-order terms (in translational and rotational velocities of particles) are retained in the perturbation expansion of distribution function. We show that higher-order corrections can markedly affect steady-state orientational correlation when the normal restitution coefficient is moderate or small and this effect is more prominent for nearly smooth particles. The transient evolution of orientational correlation is found to be significantly affected by higher-order terms. In particular the higher-order orientational correlations can dominate over its leading-order contribution during short times even in the quasi-elastic limit although the steady correlation remains unaffected by such corrections in the same limit. The building-up of correlations during transient stage seems to be closely tied to the evolution of the ratio between the rotational and translational temperatures. It is demonstrated that the transient dynamics of the temperature ratio and its steady state remain insensitive to higher-order angular correlation.

A method is introduced to simulate jamming of polyhedral grains under controlled stress that incorporates global degrees of freedom through the metric tensor of a periodic cell containing grains. Jamming under hydrostatic (isotropic) stress and athermal conditions leads to a precise definition of the ideal jamming point at zero shear stress. The structures of tetrahedra jammed hydrostatically exhibit less translational order and lower jamming-point density than previously described maximally random jammed hard tetrahedra. Under the same conditions, cubes jam with negligible nematic order. Grains with octahedral symmetry having s > 0.5 (where s interpolates from octahedra [s = 0] to cubes [s = 1]) jam with an abundance of face-face contacts in the absence of nematic order. For sufficiently large face-face contact number, percolating clusters form that span the entire simulation box. The response of hydrostatically jammed tetrahedra and cubes to shear-stress perturbation is also demonstrated with the variable-cell method.

19. Shukla, P. and Meheboob Alam, Nonlinear Stability and Shearbanding in a Sheared Granular Fluid,ACFM Conf. Proc.,
5 - 8 (2013).

20. Bharadwaj, S., Gera, B., Sharma, P. and Meheboob Alam, Experiments on Round Turbulent Jet with PIV,ACFM Conference Proceedings,
1 - 4 (2013).
, Read abstract.

The mean-flow characteristics and large- scale vortical structures of a turbulent round jet is analyzed using planar and volumetric particle image velocimetry (PIV). The method of snapshots is used to construct the POD (proper orthogonal decomposition) basis that helped to identify energy-containing large- scale structures of the jet. Two different vorticity detection criteria have been used to correlate the underlying vortical structures with POD-modes. Preliminary data on three-dimensional particle tracks are also discussed.

21. Ansari I. and Meheboob Alam, Patterns and Velocity Field in Vertically Vibrated Granular Materials.,AIP Conf. Proc.,
1542, 775 - 778 (2013).
,

(Editors: A. Yu et al.; ISBN 978-0-7354-1166-1)

, Read abstract.

We report experimental results on pattern formation in vertically vibrated granular materials confined in a quasi-two-dimensional container. For a deep bed of mono-disperse particles, we uncovered a new transition from the bouncing bed to an f /4-wave ( f is the frequency of shaking) which eventually gives birth to an f /2-undulation wave, with increasing shaking intensity. Other patterned states for mono-disperse particles and their transition-route are compared with previous experiments. The coarse-grained velocity field for each patterned state has been obtained which helped to characterize convective rolls as well as synchronous and sub-harmonic waves in this system.

22. Meheboob Alam and Shukla, P., Shearbanding and Inhomogeneous States in a Granular Fluid,AIP Conf. Proc.,
1542, 795 - 798 (2013).
,

(Editors: A. Yu et al.; ISBN 978-0-7354-1166-1)

, Read abstract.

In micro-structural fluids, the homogeneous shear flow breaks into alternate regions of low and high shear rates (i.e., shear localization), respectively, when the applied shear rate exceeds a critical value and this is known as gradient banding. On the other hand, if the applied shear stress exceeds a critical value, the homogeneous flow separates into bands of different shear stresses (having the same shear rate) along the vorticity (spanwise) direction, leading to stress localization, and the resulting pattern is dubbed vorticity banding. Here we provide a brief overview of our recent work on nonlinear order-parameter theory to describe various pattern formation scenario in a sheared granular fluid, with a specific focus on the vorticity-banding phenomena. The analysis holds for any general constitutive model, but the results are presented for a kinetic-theory constitutive model that holds for rapid granular flows. Our theory predicts that the vorticity banding can occur via supercritical/subcritical pitchfork and subcritical Hopf bifurcations in dilute and dense flows, respectively, resulting in an inhomogeneous state of shear stress and pressure.

The rapid granular plane Couette flow is known to be unstable to pure spanwise perturbations (i.e.~perturbations having variations only along the mean vorticity direction) below some critical density (volume fraction of particles), resulting in banding of particles along the mean vorticity direction -- this is dubbed {\it vorticity banding} instability. The nonlinear state of this instability is analysed using quintic-order Landau equation that has been derived from the pertinent hydrodynamic equations of rapid granular fluid. We have found analytical solutions for related modal/harmonic equations of finite-size perturbations up-to quintic-order in perturbation amplitude, leading to an exact calculation of both first and second Landau coefficients. This helped to identify the bistable nature of nonlinear vorticity banding instability for a range of densities spanning from moderately dense to dense flows. For perturbations with small spanwise wavenumbers, the bifurcation scenario for vorticity banding unfolds, with increasing density from the dilute limit, as: {supercritical pitchfork} $\to$ {subcritical pitchfork} $\to$ {subcritical Hopf} bifurcations. The transition from supercritical to subcritical pitchfork bifurcations is found to occur via the appearance of a degenerate/bicritical point (at which both the linear growth rate and the first Landau coefficient are simultaneously zero) that divides the critical line into two parts: one representing the first-order and the other the second-order phase transitions. Both subcritical oscillatory and stationary solutions have also been uncovered for dilute and dense flows, respectively, when the spanwise wavenumber is large. In all cases, the nonlinear solutions correspond to inhomogeneous states of shear stress and pressure along the vorticity direction, and hence are analogs of vorticity banding in other complex fluids. The quartic-order mean-flow resonance is evidenced in the parameter space for which the second Landau coefficient undergoes a jump discontinuity of infinite-order. The importance of retaining higher-order terms to calculate the second Landau coefficient and their possible effects on the nature of bifurcations are elucidated.

The effects of three-dimensional (3D) perturbations, having wave-like modulations along both the streamwise and spanwise/vorticity directions, on the nonlinear states of five types of linear instability modes, the nature of their bifurcations and the resulting nonlinear patterns are analysed for granular plane Couette flow using an order-parameter theory which is an extension of our previous work on two-dimensional (2D) perturbations (Shukla and Alam, J. Fluid Mech., vol. 672, 2011b, pp.~147-195). The differential equations for modal amplitudes (the fundamental mode, the mean-flow distortion, the second harmonic and the distortion of the fundamental mode), up to cubic order in perturbation amplitude, are solved using a spectral-based numerical technique, yielding an estimate of the first Landau coefficient that accounts for the leading-order nonlinear effect on finite-amplitude perturbations. In the near-critical regime of flows, we found evidence of mean-flow resonance, characterized by the divergence of the first Landau coefficient, that occurs due to the interaction/resonance between a linear instability mode and a mean-flow mode. The nonlinear solutions are found to appear via both pitchfork and Hopf bifurcations from the underlying linear instability modes, leading to supercritical nonlinear states of stationary and travelling wave solutions. The subcritical travelling wave solutions have also been uncovered in the linearly stable regimes of flow. It is shown that multiple nonlinear states of both stationary and travelling waves can coexist for a given parameter combination of mean density and Couette gap. The three-dimensional nonlinear solutions persist for a range of spanwise wavenumbers up to $k_z=O(1)$ that originate from 2D instabilities which occur beyond a moderate value of the mean density. For purely 3D instabilities in dilute flows (having no analogue in 2D flows), the supercritical finite-amplitude solutions persist for a much larger range of spanwise wavenumber up to $k_z=O(10)$. For all instabilities, the vortical motion on the cross-stream plane has been characterized in terms of the fixed/critical points of the underlying flow-field: saddles, nodes (sources and sinks) and vortices have been identified. While the cross-stream velocity field for supercritical solutions in dilute flows contains nodes and saddles, the subcritical solutions are dominated by large-scale vortices in the background of saddle-node-type motions. The latter type of flow pattern also persists at moderate densities in the form of supercritical nonlinear solutions that originate from the dominant 2D instability modes for which the vortex appears to be driven by two nearby saddles. The location of this vortex is found to be correlated with the local maxima of the streamwise vorticity.

Buoyancy driven granular convection is studied for a shallow, vertically shaken granular bed in a quasi 2D container. Starting from the granular Leidenfrost state, in which a dense particle cluster floats on top of a dilute gaseous layer of fast particles [Eshuis et al., Phys. Rev. Lett. 95, 258001 (2005)], we witness the emergence of counter-rotating convection rolls when the shaking strength is increased above a critical level. This resembles the classical onset of convecion - at a critical value of the Rayleigh number - in a fluid heated from below. The same transition, even quantitatively, is seen in molecular dynamics (MD) simulations, and explained by a hydrodynamic-like model in which the granular material is treated as a continuum. The critical shaking strength for the onset of granular convection is accurately reproduced by a linear stability analysis of the model. The results from experiment, simulation, and theory are in excellent agreement. The present paper extends and completes our earlier analysis [Eshuis et al., Phys. Rev. Lett. 104, 038001 (2010)]

26. Meheboob Alam and Shukla, P., Vorticity Banding and Stress Localization in a Granular Fluid.,ICTAM Conference Proceedings,
1 - 2 (2012).
,

The effects of three-dimensional perturbations and particle rotations are analyzed on the non-modal stability characteristics of an unbounded granular shear flow for which the stability problem is solved as an initial value problem. A kinetic-theory constitutive model is used that incorporates the spin degrees of freedom along with certain micro-polar effects of granular materials. The singular values of the underlying {\it non-normal} linear operator play a central role in the non-modal analysis in contrast to the standard modal analysis where the eigenvalues determine the asymptotic (in)stability of the flow. For linearly stable flows, it is shown that the perturbation energy can be amplified by a few orders of magnitude at short times before decaying in the asymptotic time limit. Optimal perturbations, that correspond to maximum energy growth over all possible initial conditions, are found to be two-dimensional for a smooth granular fluid. The effect of particle rotation has been assessed by varying the tangential restitution coefficient ($\beta$) for smooth particles ($\beta=-1$) to perfectly rough particles ($\beta=1$), with significant enhancement of maximum energy for rough particles. Since the non-modal mechanism can significantly amplify perturbation energy, this provides a viable alternate route for pattern formation in a sheared granular fluid.

The granular plane Couette flow is known to be linearly unstable to shear-banding instability beyond a critical density, and our nonlinear analysis suggests that the nature of bifurcation (supercritical/subcritical) depends crucially on the choice of the constitutive model. While the standard Enskog model for nearly elastic particles predicts supercritical bifurcations for moderately to dense systems, a more realistic model with global equation of states (that are likely to hold for the whole range of densities) predicts a subcritical bifurcation in the dense limit. The latter prediction agrees well with recent particle simulations of a sheared inelastic hard-disk system.

The average number of constraints per particle (C_{total}) in mechanically stable amorphous systems of Platonic solids approaches the isostatic value at the jamming point (C_ttotal} -> 12) though average number of contacts are hypostatic. By introducing angular alignment metrics to classify the degree of constraint imposed by each contact, constraints are shown to arise as a direct result of local orientational order reflected in edge-face and face-face alignment angle distributions. With approximately one face-face contact per particle at jamming, chain-like face-face clusters form with finite extent-- a signature of amorphous jammed systems.

From particle simulations of a sheared frictional granular gas, we show that the Coulomb friction can have dramatic effects on orientational correlation as well as on both the translational and angular velocity distribution functions even in the Boltzmann (dilute) limit. The dependence of orientational correlation on friction coefficient ($\mu$) is found to be {\it non-monotonic}, and the Coulomb friction plays a dual role of enhancing or diminishing the orientational correlation, depending on the value of the tangential restitution coefficient (which characterizes the roughness of particles). From the sticking limit (i.e.~with no sliding contact) of rough particles, decreasing the Coulomb friction is found to reduce the density and spatial velocity correlations which, together with diminished orientational correlation for small enough $\mu$, are responsible for the transition from non-Gaussian to Gaussian distribution functions in the double limit of small friction ($\mu \rightarrow 0$) and nearly elastic particles ($e \rightarrow 1$) at any roughness. This double limit in fact corresponds to perfectly smooth particles, and hence the Maxwellian/Gaussian is indeed a solution of the Boltzmann equation for a frictional granular gas in the limit of elastic collisions and zero Coulomb friction at any roughness. The high velocity tails of both distribution functions seem to follow stretched exponentials even in the presence of Coulomb friction, and the related velocity exponents deviate strongly from a Gaussian with increasing friction.

The first evidence of a variety of non-linear equilibrium states of travelling and stationary waves is provided in a two-dimensional granular plane Couette flow via nonlinear stability analysis. The relevant order parameter equation, the Landau equation, has been derived for the most unstable two-dimensional perturbation of finite size. Along with the linear eigenvalue problem, the mean-flow distortion, the second harmonic, the distortion to the fundamental mode and the first Landau coefficient are calculated using a spectral-based numerical method. Two types of bifurcations, Hopf and pitchfork, that result from travelling and stationary instabilities, respectively, are analysed using the first Landau coefficient. The present bifurcation theory shows that the flow is subcritically unstable to stationary finite-amplitude perturbations of long wave-lengths ($k_x\sim 0$, where $k_x$ is the streamwise wavenumber) in the dilute limit that evolve from subcritical shearbanding modes ($k_x=0$), but at large enough Couette gaps there are stationary instabilities with $k_x=O(1)$ that lead to supercritical pitchfork bifurcations. At moderate-to-large densities, in addition to supercritical shearbanding modes, there are long-wave travelling instabilities that lead to Hopf bifurcations. It is shown that both supercritical and subcritical nonlinear states exist at moderate-to-large densities that originate from the dominant stationary and travelling instabilities for which $k_x=O(1)$. Nonlinear patterns of density, velocity and granular temperature for all types of instabilities are contrasted with their linear eigenfunctions. While the supercritical solutions appear to be modulated forms of the fundamental mode, the structural features of unstable subcritical solutions are found to be significantly different from their linear counterpart. It is shown that the granular plane Couette flow is prone to nonlinear resonances in both stable and unstable regimes, the signature of which is implicated as a discontinuity in the first Landau coefficient. Our analysis identified two types of modal resonances that appear at the quadratic order in perturbation amplitude: (i) a `mean-flow resonance' which occurs due to the interaction between a streamwise-independent shear-banding mode ($k_x=0$) and a linear/fundamental mode $k_x\neq 0$, and (ii) an exact `$1:2$ resonance' that results from the interaction between two waves with their wave-number ratio being $1:2$.

A weakly nonlinear theory, in terms of the well-known Landau equation, has been developed to describe the nonlinear saturation of shear-banding instability in rapid granular plane Couette flow using the amplitude expansion method. The nonlinear modes are found to follow certain symmetries of the base flow and the fundamental mode which helped to identify {\it analytical} solutions for the base-flow distortion and the second harmonic, leading to an exact calculation of the first Landau coefficient. The present analytical solutions are further used to validate an spectral-based numerical method for nonlinear stability calculation. The regimes of {\it supercritical} and {\it subcritical} bifurcations for the shear-banding instability are identified, leading to the prediction that the lower branch of the neutral stability contour in the ($H, \phi^0$)-plane, where $H$ is the scaled Couette gap (the ratio between the Couette gap and the particle diameter) and $\phi^0$ is the mean density or the volume fraction of particles, is sub-critically unstable. The predicted finite-amplitude solutions represent shear-localization and density segregation along the gradient direction. Our analysis suggests that there is a sequence of transitions among three types of pitchfork bifurcations with increasing mean density: from (i) the {\it bifurcation from infinity} in the Boltzmann limit to (ii) {\it subcritical} bifurcation at moderate densities to (iii) {\it supercritical} bifurcation at a larger density to (iv) {\it subcritical} bifurcation in the dense limit and finally again to (v) {\it supercritical} bifurcation near the close packing density. It is shown that the appearance of subcritical bifurcation in the dense limit depends on the choice of the contact radial distribution function and the constitutive relations. The scalings of the first Landau coefficient, the equilibrium amplitude and the phase diagram, in terms of mode number and inelasticity, are demonstrated. The granular plane Couette flow serves as a paradigm that supports all three possible types of pitchfork bifurcations, with the mean density ($\phi^0$) being the single control parameter that dictates the nature of bifurcation. The predicted bifurcation scenario for the shearband formation is in qualitative agreement with particle dynamics simulations and experiment in the rapid shear regime of granular plane Couette flow.

A mechanically-based structural optimization method is utilized to explore the phenomena of jamming for assemblies of frictionless Platonic solids. Systems of these regular convex polyhedra exhibit mechanically stable disordered phases with density substantially less than optimal for a given shape, revealing that thermal motion is necessary to access high density phases. We confirm that the large system jamming threshold of 0.623 ± 0.003 for tetrahedra is consistent with experiments on tetrahedral dice. Also, the extremely short-ranged translational correlations of packed tetrahedra observed in experiments are confirmed here, in contrast with those of thermally simulated glasses. Though highly ordered phases are observed to form for small numbers of cubes and dodecahedra, the short correlation length scale suppresses ordering in large systems, resulting in packings that are mechanically consistent with ‘orientationally-disordered’ contacts (point-face and edge-edge contacts). Mild nematic ordering is observed for large systems of cubes, whereas angular correlations for the remaining shapes are ultra short-ranged. In particular the angular correlation function of tetrahedra agrees with that recently observed experimentally for tetrahedral dice. Power-law scaling exponents for energy with respect to distance from the jamming threshold exhibit a clear dependence on the ‘highest order’ percolating contact topology. These nominal exponents are 6, 4, and 2 for configurations having percolating point-face (or edge-edge), edge-face, and face-face contacts, respectively. Jamming contact number is approximated for small systems of tetrahedra, icosahedra, dodecahedra, and octahedra with order and packing representative of larger systems. These Platonic solids exhibit hypostatic behavior, with average jamming contact number between the isostatic value for spheres and that of asymmetric particles. These shapes violate the isostatic conjecture, displaying contact number that decreases monotonically with sphericity. The common symmetry of dual polyhedra results in local translational structural similarity. Systems of highly spherical particles possesing icosahedral symmetry, such as icosahedra or dodecahedra, exhibit structural behaviour similar to spheres, including jamming contact number and radial distribution function. These results suggest that though continuous rotational symmetry is broken by icosahedra and dodecahedra, the structural features of these particles are well replicated by spheres. Octahedra and cubes, which possess octagonal symmetry, exhibit similar local translational ordering, despite exhibiting differences in nematic ordering. In general the structural features of systems with tetrahedra, octahedra and cubes differ significantly from those of sphere packings.

Strongly vertically shaken granular matter can display a density inversion: A high-density cluster of beads is elevated by a dilute gas-like layer of fast beads underneath ("granular Leidenfrost effect"). For even stronger shaking the granular Leidenfrost state becomes unstable and granular convection rolls emerge. This transition resembles the classical onset of convection in fluid heated from below at some critical Rayleigh-number. The same transition is seen in molecular dynamics (MD) simulations of the shaken granular material. The critical shaking strength for the onset of granular convection can be calculated from a linear stability analysis of a hydrodynamic-like model of the granular flow. Experiment, MD simulations, and theory quantitatively agree.

Probability distribution functions of fluctuation velocities ($P(u_x)$ and $P(u_y)$, where $u_x$ and $u_y$ are the fluctuation velocities in $x$ and $y$-directions, respectively; the gravity is acting along the periodic $x$-direction and the flow is bounded by two walls parallel to $y$-direction) and the density and the spatial velocity correlations are studied using event-driven simulations for an inelastic smooth hard-disk system undergoing gravity-driven Poiseuille flow (GPF). It is shown that for GPF with smooth and/or perfectly-rough walls the Maxwellian/Gaussian is the leading order distribution over a wide range of densities in the quasi-elastic limit which is a surprising result, especially for a dilute granular gas, for which the Knudsen number belongs to the transitional-flow regime. The signature of wall-roughness-induced dissipation mainly shows up in the $P(u_x)$ distribution in the form of a sharp peak for negative velocities in the near-wall region. Both $P(u_x)$ and $P(u_y)$ distributions become asymmetric with increasing dissipation at any density, and the emergence of density-waves, that appear in the form of sinuous-wave/slug at low-to-moderate values of mean density, makes these asymmetries stronger, especially in the presence of a slug. At high densities, the flow degenerates into a dense plug (where the density approaches its maximum limit and the shear-rate is negligibly small) around the channel centerline and two shear-layers (where the shear rate is high and the density is low) near the walls. The distribution functions within the shear-layer follow the characteristics of those at moderate mean densities. Within the dense plug, the high-velocity tails of both $P(u_x)$ and $P(u_y)$ appear to undergo a transition from Gaussian in the quasi-elastic limit to power-law distributions. For dense flows, it is shown that although the density correlations play a significant role in enhancing the velocity correlations when the collisions are sufficiently inelastic, they do not induce velocity correlations when the collisions are quasi-elastic for which the distribution functions are close to Gaussian. The combined effect of enhanced density and velocity correlations around the channel-centerline with increasing inelastic dissipation seems to be responsible for the emergence of non-Gaussian high-velocity tails of distribution functions.

36. Smith KC, TS Fisher and Meheboob Alam, Quasi-static Compaction of Polyhedra by the Discrete Element Method,AIP Conference Proceedings,
1145, 90 - 93 (2009).
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(Editors: M. Nakagawa and S. Luding; ISBN 978-0-7354-0682-7)

The formation of density waves and the effect of wall roughness on them are studied using molecular dynamics simulations of gravity-driven granular Poiseuille flow. Three basic types of structures are found in moderately dense flows: a plug, a sinuous wave and a slug; a new varicose wave mode has been identified in dense flows with channels of large widths at moderate dissipations; only clump-like structures appear in dilute flows. The simulation results are contrasted with the predictions of a linear stability analysis of the kinetic-theory continuum equations for granular Poiseuille flow. The theoretical predictions on the form of density waves are in qualitative agreement with simulations in denser flows, however, there are discrepancies between simulation and theory in dilute flows.

We show that a Landau-type `order-parameter' equation describes the onset of shear-band formation in granular plane Couette flow wherein the flow undergoes an ordering transition into alternate layers of dense and dilute regions of low and high shear rates, respectively, parallel to the flow-direction. Even though the linear theory predicts the stability of the homogeneous shear solution in {\it dilute} flows, our analytical bifurcation theory suggests that there is a sub-critical {\it finite-amplitude} instability that is likely to lead to shearband formation in dilute flows which is in agreement with previous numerical simulations.

Event-driven simulations of inelastic smooth hard-disks are used to probe the slip velocity and rheology in gravity-driven granular Poiseuille flow. It is shown that both the slip velocity ($U_w$) and its gradient ($dU_w/dy$) depend crucially on the mean density, wall-roughness and inelastic dissipation. While the gradient of slip velocity follows a single power law relation with Knudsen number, the variation of $U_w$ with $Kn$ shows three distinct regimes in terms of Knudsen number. An interesting possibility of Knudsen-number-dependent specularity coefficient emerges from a comparison of our results with a first-order transport theory for the slip velocity. Simulation results on stresses are compared with kinetic-theory predictions, with reasonable agreement of our data in the quasi-elastic limit. The deviation of simulations from theory increases with increasing dissipation which is tied to the increasing magnitude of the first normal stress difference (${mathcal N}_1$) that shows interesting non-monotonic behavior with density. As in simple shear flow, there is a sign-change of ${mathcal N}_1$ at some critical density, and its collisional component and the related collisional anisotropy are responsible for this sign-reversal.

40. Malik M., J Dey and Meheboob Alam, Spatial modal stability of compressible plane Couette flow,Proc. of 12th Asian Congress of Fluid Mechanics,
1 - 4 (2008).
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(Editor: H. J. Sung)

41. Shukla P. and Meheboob Alam, Nonlinear Stability of Granular Shear Flow: Landau Equation and Shearbanding,Proc. of 12th Asian Congress of Fluid Mechanics,
1 - 4 (2008).
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(Editor: H. J. Sung)

42. Malik A. and Meheboob Alam, Large-scale Structures and Fluctuations in 3D granular Poiseuille Flow,Proc. of 12th Asian Congress of Fluid Mechanics,
1 - 4 (2008).
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(Editor: H. J. Sung)

43. Meheboob Alam, Dynamics of Sheared Granular Fluid,Proc. of 10th Asian Congress of Fluid Mechanics,
1 - 6 (2008).
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(Editor: H. J. Sung; Inaugural ``Zhao-Sato-Narasimha' Lecture)

44. Meheboob Alam and P. Shukla, Nonlinear Stability of Granular Shear Flow: Landau Equation, Shearbanding and Universality,ICTAM Conference Proceedings,
1 - 2 (2008).
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The mean flow and the linear stability characteristics of a two-dimensional particulate suspension, driven horizontally via harmonic oscillation, are analyzed. A constitutive model based on the kinetic theory of granular materials, that takes into account the dissipative collisional interactions among particles as well as their interactions with the interstitial fluid, is used; the effects of the interstitial fluid are incorporated in the balance equations for the particle phase. Assuming that the suspension is thin along the vertical direction, the effects of driving are incorporated into the governing equations in a mean-field manner. Using Floquet theory, a linear stability analysis of the time-periodic mean flow indicates that the oscillatory suspension supports stationary- and traveling-wave instabilities that corresponds to particle-banding patterns that are aligned parallel or orthogonal or at an oblique angle to the driving direction. The phase diagram of instabilities is studied as a function of external driving parameters and various system parameters. It is shown that the fluid-particle interaction is responsible for the onset of travelling instabilities in this flow.

Linear stability and the non-modal transient energy growth in compressible plane Couette flow are investigated for two prototype mean flows: (a) the {\it uniform shear} flow with constant viscosity, and (b) the {\it non-uniform shear} flow with {\it stratified} viscosity. Both mean flows are linearly unstable for a range of supersonic Mach numbers ($M$). For a given $M$, the critical Reynolds number ($Re$) is significantly smaller for the uniform shear flow than its non-uniform shear counterpart; for a given $Re$, the {\it dominant} instability (over all stream-wise wavenumbers, $\alpha$) of each mean flow belongs different modes for a range of supersonic $M$. An analysis of perturbation energy reveals that the instability is primarily caused by an excess transfer of energy from mean-flow to perturbations. It is shown that the energy-transfer from mean-flow occurs close to the moving top-wall for ``mode I'' instability, whereas it occurs in the bulk of the flow domain for ``mode II''. For the non-modal transient growth analysis, it is shown that the maximum temporal amplification of perturbation energy, $G_{\max}$, and the corresponding time-scale are significantly larger for the uniform shear case compared to those for its non-uniform counterpart. For $\alpha=0$, the linear stability operator can be partitioned into ${\cal L}\sim \bar{\cal L} + Re^2{\cal L}_p$ and the $Re$-dependent operator ${\cal L}_p$ is shown to have a negligibly small contribution to perturbation energy which is responsible for the validity of the well-known quadratic-scaling law in uniform shear flow: $G(t/{\it Re}) \sim {\it Re}^2$. In contrast, the dominance of ${\cal L}_p$ is responsible for the invalidity of this scaling-law in non-uniform shear flow. An inviscid reduced model, based on Ellingsen-Palm-type solution, has been shown to capture all salient features of transient energy growth of full viscous problem. For both modal and non-modal instability, it is shown that the {\it viscosity-stratification} of the underlying mean flow would lead to a delayed transition in compressible Couette flow.

Using particle simulations of the uniform shear flow of a rough dilute granular gas, we show that the translational and rotational velocities are strongly correlated in direction, but there is no orientational correlation-induced singularity at perfectly smooth ($\beta=-1$) and rough ($\beta=1$) limits for elastic collisions ($e=1$); both the translational and rotational velocity distribution functions remain close to a Gaussian for these two limiting cases. Away from these two limits, the orientational as well as spatial velocity correlations are responsible for the emergence of non-Gaussian high velocity tails. The tails of both distribution functions follow stretched exponentials, with the exponents depending on normal ($e$) and tangential ($\beta$) restitution coefficients.

The linear stability analysis of an uniform shear flow of granular materials is revisited using several cases of a Navier-Stokes'-level constitutive model in which we incorporate the global equation of states for pressure and thermal conductivity (which are accurate up-to the maximum packing density $\nu_{m}$) and the shear viscosity is allowed to diverge at a density $\nu_\mu$ ($< \nu_{m}$), with all other transport coefficients diverging at $\nu_{m}$. It is shown that the emergence of shear-banding instabilities (for perturbations having no variation along the streamwise direction), that lead to shear-band formation along the gradient direction, depends crucially on the choice of the constitutive model. In the framework of a dense constitutive model that incorporates only collisional transport mechanism, it is shown that an accurate global equation of state for pressure or a viscosity divergence at a lower density or a stronger viscosity divergence (with other transport coefficients being given by respective Enskog values that diverge at $\nu_m$) can induce shear-banding instabilities, even though the original dense Enskog model is stable to such shear-banding instabilities. For any constitutive model, the onset of this shear-banding instability is tied to a {\it universal} criterion in terms of constitutive relations for viscosity and pressure, and the sheared granular flow evolves toward a state of lower ``dynamic'' friction, leading to the shear-induced band formation, as it cannot sustain increasing dynamic friction with increasing density to stay in the homogeneous state. A similar criterion of a lower viscosity or a lower viscous-dissipation is responsible for the shear-banding state in many complex fluids.

49. Meheboob Alam and A. Khalili, Mean flow and linear stability of an oscillatory particulate suspension.,Proc. of 2nd Intl Conf. on Porous Media and its Applications,
1 - 6 (2007).
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(Editors: K. Vafai et al.)

, Read abstract.

The mean flow and the linear stability characteristics of a particle-fluid mixture, driven uniformly via harmonic oscillation, are analyzed, using a constitutive model based on the kinetic theory of granular materials. This constitutive model incorporates both the dissipative collisional interactions among particles and their interactions with the interstitial fluid. A linear stability analysis of the underlying mean flow indicates that the oscillatory particulate mixture supports stationary- and travelling-wave instabilities that lead to band-like patterns that are aligned parallel or orthogonal or at an oblique angle to the driving direction. The phase diagram of instabilities is studied as a function of external driving parameters and various system parameters. It is shown that fluid-particle interaction plays an important role for the onset of instabilities in this flow.

The effect of Prandtl number on the linear stability of a plane thermal plume is analyzed under quasi-parallel approximation. At large Prandtl numbers ($Pr>100$), we found that there is an additional unstable loop whose size increases with increasing $Pr$. The origin of this new instability mode is shown to be tied to the coupling of the momentum and thermal perturbation equations. Analyses of the perturbation kinetic energy and thermal energy suggest that the buoyancy force is the main source of perturbation energy at high Prandtl numbers that drives this instability.

From event-driven simulations of a gravity-driven channel flow of inelastic hard-disks, we show that the velocity distribution function remains close to a Gaussian for a wide range densities (even when the Knudsen number is of order one) if the walls are smooth and the particle collisions are nearly elastic. For dense flows, a transition from a Gaussian to a power-law distribution for the high velocity tails occurs with increasing dissipation in the center of the channel, irrespective of wall-roughness. For a rough wall, the near-wall distribution functions are distinctly different from those in the bulk even in the quasielastic limit.

52. Malik M., J Dey and Meheboob Alam, Transient growth, optimal perturbation and energy budget in compressible plane Couette flow.,Proc. of Eleventh Asian Congress of Fluid Mechanics,
1 - 4 (2006).
, Read abstract.

Transient energy growth and optimal perturbation studies have been done for the compressible plane Couette flow configuration. The streaks are found to be the optimal patterns of the perturbation velocity. The maxi- mum of the energy amplification factor, , during the transient growth, is found to decrease monotonically with increasing Mach number for all values of streamwise and spanwise wavenumbers. This is shown to be a fact linked to the Mach number dependence of viscosity. In the constant viscosity case ’s trend with Mach number is sensitive to the energy norm chosen. An energy budget study also has been made for the time interval during the transient to- tal energy amplification which reveals the respective roles of energy transfer the from mean flow, viscous dissipation, thermal diffusion and shear work.

Based on a micropolar continuum of rough granular particles that takes into account the balance equations for the spin (/rotational) velocity and the spin granular temperature, the linear stability characteristics of an unbounded shear flow (${\bf u}\equiv (u_x, u_y, u_z) = (\dot\gamma y, 0, 0)$, where $x$, $y$ and $z$ are the streamwise, transverse and spanwise directions, respectively, and $\dot\gamma$ is the shear rate) are analysed. For pure spanwise perturbations ($k_z\neq 0$, with $k_x=0=k_y$, where $k_i$ is the wavenumber in the $i$-th direction), we show that the streamwise translational velocity and the transverse spin velocity modes are subject to linear growths, due to an inviscid `algebraic' instability (that grows linearly with time). This algebraic instability is shown to be tied to a hidden mechanism of momentum transfer from the translational to the rotational modes, via pure spanwise perturbations to the transverse velocity-- in short, we have uncovered an `instability-induced rotational-driving' mechanism. Pure spanwise ($k_z\neq 0$, with $k_x=0=k_y$) and pure transverse ($k_y\neq 0$, with $k_x=0=k_z$) perturbations give rise to `exponential' instabilities (that grow exponentially with time) which are related to similar stationary instabilities in the shear flow of smooth, inelastic particles. Interestingly, both these instabilities also survive in the limiting case of perfectly elastic but rough particles. The scalings of hydrodynamic modes with wavenumbers have been obtained via the respective longwave expansion. Perturbations with modulations in all three directions are shown to be stable in the asymptotic time limit, but there could be short-time `exponential' growth of these general perturbations in the longwave limit for both travelling and stationary waves. The growth rate of all instabilities is maximum at intermediate values of the tangential restitution coefficient ($\beta$), and decreases in both the perfectly smooth ($\beta\to -1$) and rough ($\beta\to 1$) limits; the associated instability length scale is minimum at intermediate $\beta$, and increases in both the perfectly smooth and rough limits. In the perfectly smooth limit, there is a window of particle volume fraction ($\phi$), $\phi_c^s <\phi < \phi_c^t$, over which the flow remains stable to all perturbations. With the inclusion of spin fields, the size of this window decreases and at moderate dissipations with $\beta>0.5$ the flow becomes unstable at all $\phi$.

Based on the Boltzmann-Enskog kinetic theory, we develop a hydrodynamic theory for the well known (reverse) Brazil nut segregation in a vibro-fluidized granular mixture. Under strong shaking conditions, the granular mixture behaves in some ways like a fluid and the kinetic theory constitutive models are appropriate to close the continuum balance equations for mass, momentum and granular energy. Using this analogy with standard fluid mechanics, we have recently suggested a novel mechanism of segregation in granular mixtures based on a {\it competition between buoyancy and geometric forces}: the Archimedean buoyancy force, a pseudo-thermal buoyancy force due to the difference between the energies of two granular species, and two geometric forces, one compressive and the other-one tensile in nature, due to the size-difference. For a mixture of perfectly hard-particles with elastic collisions, the pseudo-thermal buoyancy force is zero but the intruder has to overcome the net compressive geometric force to rise. For this case, the geometric force competes with the standard Archimedean buoyancy force to yield a threshold density-ratio, $R_{\rho 1}=\rho_l/\rho_s < 1$, above which the {\it lighter intruder sinks}, thereby signalling the {\it onset} of the {\it reverse buoyancy} effect. For a mixture of dissipative particles, on the other hand, the non-zero pseudo-thermal buoyancy force gives rise to another threshold density-ratio, $R_{\rho 2}$ ( $> R_{\rho 1}$), above which the intruder rises again. Focussing on the {\it tracer} limit of intruders in a dense binary mixture, we study the dynamics of an intruder in a vibrofluidized system, with the effect of the base-plate excitation being taken into account through a `mean-field' assumption. We find that the rise-time of the intruder could vary {\it non-monotonically} with the density-ratio. For a given size-ratio, there is a threshold density-ratiofor the intruder at which it takes the maximum time to rise, and above(/below) which it rises faster, implying that {\it the heavier (and larger) the intruder, the faster it ascends}. The peak on the rise-time curve decreases in height and shifts to a lower density-ratio as we increase the pseudo-thermal buoyancy force. The rise (/sink) time {\it diverges} near the threshold density-ratio for reverse-segregation. Our theory offers a {\it unified} description for the (reverse) Brazil-nut segregation and the non-monotonic ascension dynamics of Brazil-nuts.

The rheological properties for dilute and moderately dense granular binary mixtures of smooth, inelastic hard disks/spheres under uniform shear flow in steady state conditions are reported. The results are based on the Enskog kinetic theory, numerically solved by a dense gas extension of the Direct Simulation Monte Carlo method for dilute gases. These results are confronted to the ones also obtained by performing molecular dynamics (MD) simulations with good agreement for the lower densities and higher coefficients of restitution. For increasing density and dissipation, the Enskog equation applies qualitatively, but the quantitative differences increase. Possible reasons for deviations of Enskog from MD results are discussed, indicating non-Newtonian flow behavior and anisotropy as the most likely direction in which the theory has to be extended.

A three dimensional linear stability analysis has been carried out to understand the origin of vortices and related density patterns in bounded uniform shear flow of granular materials, using a kinetic-theory constitutive model. This flow is found to be unstable to pure spanwise stationary perturbations ($k_z\neq 0$, $k_x=0$ and $\pl/\pl y(.)=0$, where $k_i$ is the wavenumber for the $i$-th direction) if the solid fraction is below some critical value $\nu < \nu_{3D}$. The growth rates of these spanwise instabilities are an order of magnitude larger than those of the two-dimensional (2D, $k_z=0$) streamwise-independent ($k_x=0$) instabilities that occur if the solid fraction is above some critical value $\nu>\nu_{2D}$ ($>\nu_{3D}$). The spanwise instabilities give birth to new 3D travelling wave instabilities at non-zero values of the streamwise wavenumber ($k_x\neq 0$) in dilute flows ($\nu < \nu_{3D}$). For moderate-to-large densities with $k_x\neq 0$, there are additional 3D instability modes in the form of both stationary and travelling waves, whose origin is tied to the corresponding 2D instabilities. While the 2D streamwise-independent modes lead to the formation of stationary streamwise vortices for moderately dense flows ($\nu>\nu_{2D}$), the pure spanwise modes are responsible for the origin of such vortices in the dilute limit ($\nu<\nu_{3D}$). For more general kind of perturbations ($k_x\neq 0$ and $k_z\neq 0$), the `modulated' streamwise vortices are born which could be either stationary or travelling depending on control parameters.The rolling motion of vortices will lead to a major redistribution of the streamwise velocity and hence such vortices can act as potential progenitors for the mixing of particles. The effect of non-zero wall-slip has been investigated, and it is shown that some dilute-flow instabilities can disappear with the inclusion of the wall-slip. Even though the streamwise granular vortices have similarities to the well-known stationary Taylor-Couette vortices (which are `hydrodynamic' in origin), their origin is, however, tied to `constitutive' instabilities, and hence they belong to a different class.

Nonmodal transient growth studies and estimation of optimal perturbations have been made for the compressible plane Couette flow with three-dimensional disturbances. The steady mean flow is characterized by a non-uniform shear-rate and a varying temperature across the wall-normal direction for an appropriate perfect gas model. The maximum amplification of perturbation energy over time, $G_{\max}$, is found to increase with increasing Reynolds number ${\it Re}$, but decreases with increasing Mach number $M$. More specifically, the optimal energy amplification $G_{\rm opt}$ (the supremum of $G_{\max}$ over both the streamwise and spanwise wavenumbers) is maximum in the incompressible limit and decreases monotonically as $M$ increases. The corresponding optimal streamwise wavenumber, $\alpha_{\rm opt}$, is non-zero at $M=0$, increases with increasing $M$, reaching a maximum for some value of $M$ and then decreases, eventually becoming zero at high Mach numbers. While the pure streamwise vortices are the optimal patterns at high Mach numbers (in contrast to incompressible Couette flow), the modulated streamwise vortices are the optimal patterns for low-to-moderate values of the Mach number. Unlike in incompressible shear flows, the streamwise-independent modes in the present flow do not follow the scaling law $G(t/{\it Re}) \sim {\it Re}^2$, the reasons for which are shown to be tied to the dominance of some terms (related to density and temperature fluctuations) in the linear stability operator. Based on a detailed nonmodal energy analysis, we show that the transient energy growth occurs due to the transfer of energy from the mean flow to perturbations via an inviscid {\it algebraic} instability. The decrease of transient growth with increasing Mach number is also shown to be tied to the decrease in the energy transferred from the mean flow ($\dot{\mathcal E}_1$) in the same limit.

58. Meheboob Alam, Universal unfolding of pitchfork bifurcations and shearband formation in rapid granular Couette flow.,Trends in Applications of Mathematics to Mechanics,
11 - 20 (2005).
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(Editors: Y. Wang and K. Hutter; ISBN 3-8322-3600-7)

, Read abstract.

A numerical bifurcation analysis is carried out to understand the role of gravity on the shear-band formation in rapid granular Couette flow. At {\it low} shear rates, there is a unique solution with a {\it plug} near the bottom wall and a {\it shear-layer} near the top-wall; this solution mirrors typical shearbanding-type profiles in earth-bound shear-cell experiments. Interestingly, a {\it stable} plug near the top-wall is also a solution of these equations at {\it high} shear rates; there is a multitude of other plugged states, with the plugs being located in an ordered fashion within the Couette gap. The origin of such shearbanding solutions is tied to the spontaneous symmetry-breaking {\it shearbanding} instabilities of the gravity-free uniform shear flow, leading to both subcritical and supercritical pitchfork bifurcations. In the language of singularity theory, we have established that this bifurcation problem admits {\it universal unfolding} of pitchfork bifurcations.

59. Meheboob Alam and S. Luding, Non-Newtonian granular fluid: Simulation and theory.,Powders and Grains 2005,
1141 - 1145 (2005).
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(Editors: R. Garcia-Rojo, S. McNamara and HJ Herrmann; ISBN 041538348X)

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Rheological properties of granular fluids are probed via event-driven simulations of the inelastic hard-sphere model. We find that granular fluids support large normal stress differences for the whole range of densities, clearly indicating their non-Newtonian rheology. Interestingly, both first (N_1) and second (N_2) normal stress differences undergo sign-reversals with density. While {\mathcal N}_1 changes its sign in the dense limit, {\mathcal N}_2 changes its sign in the dilute limit. The origin of such sign-reversals has been tied to the microstructural reorganization of particles. We briefly outline a viscoelastic constitutive model for granular fluids which allows the sign-reversals of both first and second normal stress differences.

Event driven simulations of smooth inelastic hard-disks are used to probe the transport properties and the microstructure of bidisperse granular mixtures. A generic feature of such mixtures is that the two species have different levels of fluctuation kinetic energy ($T_l\neq T_s$) in contrast to their elastic counterpart. The microscopic mechanism for this energy non-equipartition is shown to be directly tied to the {\it asymmetric} nature of collisional probabilities between the heavier and lighter species, compared to their purely elastic counterpart. The degree of collisional asymmetry increases with both increasing inelasticity and mass-disparity, thereby increasing the energy ratio $T_l/T_s$ in the same limit. A phenomenological constitutive model, allowing energy {\it non-equipartition}, is proposed which captures the {\it non-monotonic} behaviour of the transport coefficients, in agreement with simulation results, whereas the standard constitutive model with equipartition assumption predicts {\it monotonic} variations. The sheared granular mixture readily forms clusters, having striped-patterns along the extensional-axis of the flow. The microstructural flow-features are extracted by measuring the cluster-size distributions, the pair correlation function and the collision-angle distribution. While the inelastic dissipation is responsible for the onset of clustering, we have found that the mass-disparity between the two species enhances the degree of clustering significantly in the sense that the size of the largest cluster increases with increasing mass-disparity. At the microscopic-level, the particle motion becomes more and more {\it streamlined} (i.e. {\it ordered} along the streamwise direction) with increasing dissipation and mass-disparity, which is responsible for the enhanced first normal stress difference in the same limit.

The linear stability theory and bifurcation analysis are used to investigate the role of gravity on shear-band formation in granular Couette flow considering a kinetic-theory rheological model. We show that the only possible state, at low shear rates, corresponds to a `plug' near the bottom wall, in which the particles are densely packed and the shear rate is close to zero, and an uniformly sheared dilute region above it. The origin of such plugged-states is shown to be tied to the spontaneous symmetry-breaking instabilities of the gravity-free uniform shear flow, leading to the formation of ordered bands of alternating dilute and dense regions in the transverse direction, via an infinite hierarchy of pitchfork bifurcations. Gravity plays the role of an `imperfection', thus destroying the `perfect' bifurcation-structure of uniform shear. The present bifurcation problem admits {\it universal unfolding} of pitchfork bifurcations which subsequently leads to the formation of a sequence of a countably infinite number of `isolas', with the solution structures being a modulated version of their gravity-free counterpart. While the solution with a plug near the bottom wall looks remarkably similar to the shear-banding phenomenon in dense slow granular Couette flows, a `floating' plug near the top-wall is also a solution of these equations at high shear rates. A two-dimensional linear stability analysis suggests that these floating plugged-states are unstable to longwave travelling disturbances. The unique solution having a bottom-plug can also be unstable to longwaves, but remains stable at sufficiently low shear rates. The implications and realizability of the present results are discussed in the light of shear-cell experiments under `microgravity' conditions.

62. Arakeri V.H., A. Pal, SN Goswami and Meheboob Alam, Observations on some bubble driven plane laminar flows,ACFM Conference Proceedings,
1 - 4 (2004).
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We report observations on some bubble driven plane laminar flows. These flows are not only of interest due to their unique characteristics but can also serve as benchmarks for developing computational methods to handle broader class of two-phase flows. Two classes of bubble driven flows are considered here in some detail; one is inter- action of bubble plumes with different boundary conditions and the second is analysis of bubble plume driven flow in a closed environment. From bubble plume interaction observations, we have been able to estimate the value for the entrainment coefficient.

Starting from hydrodynamic equations of binary granular mixtures, we derive an evolution equation for the relative velocity of the intruders, which is shown to be coupled to the inertia of the smaller particles. The onset of Brazil-nut segregation is explained as a competition between the buoyancy and geometric forces: the Archimedean buoyancy force, a buoyancy force due to the difference between the energies of two granular species, and two {\it geometric} forces, one {\it compressive} and the other-one {\it tensile} in nature, due to the {\it size-difference}. We show that inelastic dissipation strongly affects the phase diagram of the Brazil nut phenomenon and our model is able to explain the experimental results of Breu et al. (2003, Phys Rev Lett).

The first normal stress difference (${\mathcal N}_1$) and the microstructure in a dense sheared granular fluid of smooth inelastic hard-disks are probed using event-driven simulations. While the anisotropy in the second moment of fluctuation velocity, which is a Burnett-order effect, is known to be the progenitor of normal stress differences in {\it dilute} granular fluids, we show here that the collisional anisotropies are responsible for the normal stress behaviour in the {\it dense} limit. As in the elastic hard-sphere fluids, ${\mathcal N}_1$ remains {\it positive} (if the stress is defined in the {\it compressive} sense) for dilute and moderately dense flows, but becomes {\it negative} above a critical density, depending on the restitution coefficient. This sign-reversal of ${\mathcal N}_1$ occurs due to the {\it microstructural} reorganization of the particles, which can be correlated with a preferred value of the {\it average} collision angle $\theta_{av}=\pi/4 \pm \pi/2$ in the direction opposing the shear. We also report on the shear-induced {\it crystal}-formation, signalling the onset of fluid-solid coexistence in dense granular fluids. Different approaches to take into account the normal stress differences are discussed in the framework of the relaxation-type rheological models.

The bulk rheology of bidisperse mixtures of granular materials is examined under homogeneous shear flow conditions using the event-driven simulation method. The granular material is modelled as a system of smooth inelastic disks, interacting via the hard-core potential. In order to understand the effect of size- and mass-disparities, two cases were examined separately, namely, a mixture of different sized particles with particles having the same mass and the other with particles having the same material density. The relevant macroscopic quantities are the pressure, the shear viscosity, the granular energy (fluctuating kinetic energy) and the first normal stress difference.

Numerical results on pressure, viscosity and granular energy are compared with a kinetic-theory constitutive model with excellent agreement in the low dissipation limit even at large size-disparities. Systematic quantitative deviations occur for stronger dissipations. Mixtures with equal-mass particles show a stronger shear resistance as compared to an equivalent monodisperse system; in contrast, however, mixtures with equal-density particles show a reduced shear resistance. The granular energies of the two species are unequal, implying that the equipartition principle assumed in most of the constitutive models does not hold. Inelasticity is responsible for the onset of energy non-equipartition, but mass-disparity significantly enhances its magnitude. This lack of energy equipartition can lead to interesting non-monotonic variation for the pressure, viscosity and granular energy with the mass-ratio if the size-ratio is held fixed, while the model predictions (with equipartition assumption) suggest a monotonic behaviour in the same limit. In general, the granular fluid is non-Newtonian with a measurable first normal stress difference (which is {\it positive} if the stress is defined in the {\it compressive} sense), and the effect of bidispersity is to increase the normal stress difference, thus enhancing the non-Newtonian character of the fluid.

Kinetic-theory, with the assumption of equipartition of granular energy, suggests that the pressure and viscosity of a granular mixture vary monotonically with the mass-ratio. Our simulation results show a non-monotonic behaviour that can be explained qualitatively by a simple model allowing for non-equipartition of granular energy between the species with different mass.

Corrections are provided for two transport coefficients derived by Willits and Arnarson [Phys. Fluids 11, 3116 (1999)] for a binary mixture of nearly elastic, circular disks using the revised Enskog theory. The corrected viscosity coefficient is compared to the shear viscosity obtained from simulation results of a bidisperse mixture of inelastic, hard disks, undergoing uniform shear flow. The agreement is good for a wide range of sizes and masses.

Previous investigations have shown that inelastic collapse is a common feature of inelastic, hard-sphere simulations of nondriven or unforced flows, provided that the coefficient of restitution is small enough. The focus of the current effort is on a driven system, namely, simple shear flow. Two-dimensional, hard-sphere simulations have been carried out over a considerable range of restitution coefficients r, solids fractions , and numbers of particles N. The results indicate that inelastic collapse is an integral feature of the sheared system. Similar to nondriven systems, this phenomenon is characterized by a string of particles engaging in numerous, repeated collisions just prior to collapse. The collapsed string is typically oriented along a 135° angle from the streamwise direction. Inelastic collapse is also found to be more likely in systems with lower r, higher , and higher N, as is true for unforced systems. Nonetheless, an examination of the boundary between the collapsed and noncollapsed states reveals that the sheared system is generally more ‘‘resistant’’ to inelastic collapse than its nondriven counterpart. Furthermore, a dimensionless number V* is identified that represents the magnitude of the initial fluctuating velocities relative to that of a characteristic steady-state velocity i.e., the product of shear rate and particle diameter. For values of V*O(1), the transient portion of the simu- lation is found to be more reminiscent of a nondriven system i.e., isotropic particle bunching is observed instead of diagonal particle bands.

This article is a review of recent theoretical work on shear flow instabilities of particulate suspensions and dry granular medium. Attention is devoted largely to steady homogeneous unbounded simple shearing flows, as a generalization of the classical Kelvin problem for Newtonian fluids, with a view towards identifying material or constitutive instabilities arising from the coupling of stress to particulate concentration and temperature fields. After reviewing the most common constitutive models, a unified linear-stability treatment is given for suspensions and granular media, based on an assumed ‘short-memory’ response of stress and various fluxes to perturbations on materially steady and uniform base states. A two-dimensional stability analysis of inertialess suspension flow indicates the possibility of particle-depleted shear bands. A comprehensive three-dimensional analysis of rapid granular flow reveals transverse ‘layering and spanwise ’corrugations’ as possible modes of instability. The latter appear to result from a kind of material instability, although not the simple short-wavelength instability found for suspensions. Based on the current theoretical treatments, several new studies are recommended, including the effects of granular dilatancy and yield stress, three-dimensional disturbances in suspensions and the effects of gravity in granular now.

The tendency of granular materials in rapid shear flow to form non-uniform structures is well documented in the literature. Through a linear stability analysis of the solution of continuum equations for rapid shear flow of a uniform granular material, performed by Savage (1992) and others subsequently, it has been shown that an infinite plane shearing motion may be unstable in the Lyapunov sense, provided the mean volume fraction of particles is above a critical value. This instability leads to the formation of alternating layers of high and low particle concentrations oriented parallel to the plane of shear. Computer simulations, on the other hand, reveal that non-uniform structures are possible even when the mean volume fraction of particles is small. In the present study, we have examined the structure of fully developed layered solutions, by making use of numerical continuation techniques and bifurcation theory. It is shown that the continuum equations do predict the existence of layered solutions of high amplitude even when the uniform state is linearly stable. An analysis of the effect of bounding walls on the bifurcation structure reveals that the nature of the wall boundary conditions plays a pivotal role in selecting that branch of non-uniform solutions which emerges as the primary branch. This demonstrates unequivocally that the results on the stability of bounded shear flow of granular materials presented previously by Wang et al. (1996) are, in general, based on erroneous base states.

This paper presents a linear stability analysis of plane Couette flow of a granular material using a kinetic-theory-based model for the rheology of the medium. The stability analysis, restricted to two-dimensional disturbances, is carried out for three illustrative sets of grain and wall properties which correspond to the walls being perfectly adiabatic, and sources and sinks of fluctuational energy. When the walls are not adiabatic and the Couette gap H is sufficiently large, the base state of steady fully developed flow consists of a slowly deforming ‘plug’ layer where the bulk density is close to that of maximum packing and a rapidly shearing layer where the bulk density is considerably lower. The plug is adjacent to the wall when the latter acts as a sink of energy and is centred at the symmetry axis when it acts as a source of energy. For each set of properties, stability is determined for a range of H and the mean solids fraction ν. For a given value of ν, the flow is stable if H is sufficiently small; as H increases it is susceptible to instabilities in the form of cross-stream layering waves with no variation in the flow direction, and stationary and travelling waves with variation in the flow and gradient directions. The layering instability prevails over a substantial range of H and ν for all sets of wall properties. However, it grows far slower than the strong stationary and travelling wave instabilities which become active at larger H. When the walls act as energy sinks, the strong travelling wave instability is absent altogether, and instead there are relatively slow growing long-wave instabilities. For the case of adiabatic walls there is another stationary instability for dilute flows when the grain collisions are quasi-elastic; these modes become stable when grain collisions are perfectly elastic or very inelastic. Instability of all modes is driven by the inelasticity of grain collisions.

Some recent studies have considered the stability of unbounded rapid granular shear flow, with the sole mechanism for stress generation being instantaneous inelastic collisions between grains. This paper extends these studies by presenting a linear stability analysis in which stress generation due to grain friction is also accounted for. This is accomplished by using the ‘frictional–kinetic’ model, which integrates in a simple manner the stress arising from the two mechanisms. Solution of the linearized equations of motion is obtained by allowing the wavenumber vector of the disturbances to rotate as a function of time. As in the case of a purely kinetic stress, it is found that the flow is stable to non-layering disturbances. Disturbances in the form of layering modes may lead to instability, depending on the solids fraction and material parameters. Instability is absent altogether if the balance of fluctuational energy is not considered or if the material is assumed to be incompressible. Friction may stabilize or destabilize the flow, depending on the inelasticity of grain collisions and the effective roughness of the medium. When a purely frictional stress is considered, it is found that the system is always neutrally stable. Even if the flow is asymptotically stable, there may be significant transient growth of disturbances due to the non-normality of the associated linear operator. The initial transient growth rate, as well as the temporal maximum of transient growth is enhanced by friction.

73. Meheboob Alam and Arakeri V. H., Observations on interaction of laminar bubble plumes,Proc. of ASME Conference (Fluid Engineering Division),
153, 63 - 65 (1993).

Flow visualization studies of plane laminar bubble plumes have been conducted to yield quantitative data on transition height, wavelength and wave velocity of the most unstable disturbance leading to transition. These are believed to be the first results of this kind. Most earlier studies are restricted to turbulent bubble plumes. In the present study, the bubble plumes were generated by electrolysis of water and hence very fine control over bubble size distribution and gas flow rate was possible to enable studies with laminar bubble plumes. Present observations show that (a) the dominant mode of instability in plane bubble plumes is the sinuous mode, (b) transition height and wavelength are related linearly with the proportionality constant being about 4, (c) wave velocity is about 40% of the mean plume velocity, and (d) normalized transition height data correlate very well with a source Grashof number. Some agreement and some differences in transition characteristics of bubble plumes have been observed compared to those for similar single-phase flows.