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Counter-rotating suspension Taylor–Couette flow: pattern transition, flow multiplicity and the spectral evolution
- Suryadev Pratap Singh, Manojit Ghosh, Meheboob Alam
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- Journal:
- Journal of Fluid Mechanics / Volume 944 / 10 August 2022
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- 27 June 2022, A18
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Inertial transitions in a suspension Taylor–Couette flow are explored for the first time via experiments in the counter-rotation regime (i.e. the rotation ratio $\varOmega =\omega _o/\omega _i<0$, where $\omega _i$ and $\omega _o$ are the angular speeds of the inner and outer cylinders, respectively) up to a particle volume fraction $\phi \leq 0.2$. The primary bifurcation from the circular Couette flow (CCF) is found to yield patterns similar to those in a particle-free Newtonian fluid, and is supercritical at $|\varOmega |\leq 0.5$ but becomes hysteretic at large $|\varOmega |=1$. It is shown that the states with different numbers of vortices can coexist over a large range of shear Reynolds number $\mbox {Re}_s(\phi )$, confirming multi-stable behaviour of the primary and higher-order states for any particle loading at $|\varOmega |\leq 0.5$. A quasi-periodic state characterized by two incommensurate frequencies, namely, the modulated wavy vortices (MWV), is found at $\varOmega =-0.5$ as a tertiary bifurcation from the wavy Taylor vortices (WTV), with the stationary Taylor vortex flow (TVF) being the primary bifurcating state from CCF at $\phi \geq 0$. A novel sequence of transitions ${\rm TVF}\to {\rm MWV}_1\to {\rm WTV}\to {\rm MWV}$, with another variant of modulated vortices (MWV$_1$) appearing directly from TVF as a secondary bifurcation, may also occur even for the particle-free ($\phi =0$) case; the coexistence/non-uniqueness of WTV and MWV states is demonstrated over a range of $\mbox {Re}_s$ values spanning the secondary and tertiary bifurcation loci. At $\varOmega =-1$, the primary bifurcation yields an oscillatory state (spiral/helical vortex flow) that gives birth to another oscillatory state (interpenetrating spiral vortices) and a non-periodic state (non-propagating interpenetrating spirals, NIS) as secondary and tertiary bifurcations, respectively, with NIS being characterized by the absence of frequency peaks in the power spectrum of the scattered light intensity. For all transitions at $\varOmega < 0$, the critical values of $\mbox {Re}_s^c(\phi )$ decrease with increasing $\phi$, with more destabilizing effects of particles being found at larger $|\varOmega |$. The effect of particle loading is found to (i) decrease the amplitude and (ii) increase the wavelength of wavy vortices, with the latter seeming to be responsible for the decreased propagation frequencies of azimuthal waves with increasing $\phi$. The normalized rotation frequency of spiral vortices also decreases with increasing $\mbox {Re}_s$ and $\phi$, and a scaling relation in terms of the relative viscosity is found to collapse the frequency data for all $\phi$.
Nonlinear axisymmetric Taylor–Couette flow in a dilute gas: multiroll transition and the role of compressibility
- Pratik Aghor, Meheboob Alam
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- Journal:
- Journal of Fluid Mechanics / Volume 908 / 10 February 2021
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- 08 December 2020, A24
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The compressible Navier–Stokes–Fourier equations are numerically solved for axially bounded Taylor–Couette flow (TCF) of an ideal gas, with rotating inner cylinder and stationary outer cylinder, to understand the roles of compressibility and finite aspect ratio on the genesis of nonlinear Taylor vortices and the related bifurcation scenario. Restricting to the axisymmetric case in the wide-gap limit, the aspect ratio $\varGamma =h/\delta$ (where $\delta$ is the annular gap between two cylinders and $h$ is the height of the cylinders) is changed quasistatically by varying the height of the cylinders, following the experimental protocol of Benjamin (Proc. R. Soc. Lond. A, vol. 359, 1978b, pp. 27–43). The symmetric even-numbered ($2,4,6,\ldots$) vortices are found at $\varGamma \geq 2$, whereas the asymmetric 2-roll modes (called the single-roll or ‘anomalous’ modes), that break the midplane $Z_{2}$-symmetry, are uncovered at $\varGamma \leq O(1)$. The phase boundaries of both symmetric and asymmetric rolls and the coexisting regions of different number of rolls are identified in the ($\varGamma , Re$)-plane. These phase diagrams, along with related bifurcation diagrams and various patterns, at finite Mach numbers ($Ma$) are contrasted with their incompressible ($Ma\to 0$) analogues. It is shown that the ‘$1\leftrightarrow 2$’-roll transition in small aspect ratio cylinders is subcritical in compressible TCF in contrast to the supercritical nature of bifurcation in its incompressible counterpart. In general, the gas compressibility has a stabilizing effect on nonlinear Taylor vortices as the underlying phase diagrams in the ($\varGamma , Re$)-plane are shifted to larger values of $Re$ with increasing $Ma$. The stabilizing role of compressibility can be tied to the weakening of the outward jets, which is further aided by the strengthening of the Ekman vortices with increasing Mach number.
Symmetry-breaking bifurcations and hysteresis in compressible Taylor–Couette flow of a dense gas: a molecular dynamics study
- Nandu Gopan, Meheboob Alam
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- Journal of Fluid Mechanics / Volume 902 / 10 November 2020
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- 08 September 2020, A18
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Molecular dynamics simulations with a repulsive Lennard-Jones potential are employed to understand the bifurcation scenario and the resulting patterns in compressible Taylor–Couette flow of a dense gas, with the inner cylinder rotating ($\omega _i>0$) and the outer one at rest ($\omega _o=0$). The steady-state flow patterns are presented in terms of a phase diagram in the ($\omega _i,\varGamma$) plane, where $\varGamma =h/\delta$ is the aspect ratio, $h$ is the height of the cylinders and $\delta =R_o-R_i$ is the gap between the outer and inner cylinders, and the underlying bifurcation scenario is analysed as a function of $\omega _i$ for different $\varGamma$. Considerable density stratification is found along both radial and axial directions in the Taylor-vortex regime of a dense gas, which makes the present system fundamentally different from its incompressible analogue. In the circular Couette flow regime, the stratifications remain small and the predicted critical Reynolds number for the onset of Taylor vortices matches well with that of its incompressible counterpart. The emergence of asymmetric Taylor vortices at $\varGamma >1$ is found to occur via saddle-node bifurcations, resulting in hysteresis loops in the bifurcation diagrams that are characterized in terms of the net circulation or the maximum radial velocity or the axial density contrast as order parameters. For $\varGamma \leq 1$ with reflecting axial boundary conditions, the primary bifurcation yields a single-vortex state which is connected to a two-roll branch via saddle-node bifurcations; however, changing to stationary (no-slip) endwalls yields a new state, which consists of two large symmetric vortices near the inner cylinder coexisting with an irregular pattern near the stationary outer cylinder. It is shown that the endwall conditions and the fluid compressibility play crucial roles on the genesis of asymmetric and stratified vortices and the related multiplicity of states in the Taylor-vortex regime of a dense gas.
Burnett-order constitutive relations, second moment anisotropy and co-existing states in sheared dense gas–solid suspensions
- Saikat Saha, Meheboob Alam
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- Journal of Fluid Mechanics / Volume 887 / 25 March 2020
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- 21 January 2020, A9
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The Burnett- and super-Burnett-order constitutive relations are derived for homogeneously sheared gas–solid suspensions by considering the co-existence of ignited and quenched states and the anisotropy of the second moment of velocity fluctuations ($\unicode[STIX]{x1D648}=\langle \boldsymbol{C}\boldsymbol{C}\rangle ,C$ is the fluctuation or peculiar velocity) – this analytical work extends our previous works on dilute (Saha & Alam, J. Fluid Mech., vol. 833, 2017, pp. 206–246) and dense (Alam et al., J. Fluid Mech., vol. 870, 2019, pp. 1175–1193) gas–solid suspensions. For the combined ignited–quenched theory at finite densities, the second-moment balance equation, truncated at the Burnett order, is solved analytically, yielding expressions for four invariants of $\unicode[STIX]{x1D648}$ as functions of the particle volume fraction ($\unicode[STIX]{x1D708}$), the restitution coefficient ($e$) and the Stokes number ($St$). The phase boundaries, demarcating the regions of (i) ignited, (ii) quenched and (iii) co-existing ignited–quenched states, are identified via an ordering analysis, and it is shown that the incorporation of excluded-volume effects significantly improves the predictions of critical parameters for the ‘quenched-to-ignited’ transition. The Burnett-order expressions for the particle-phase shear viscosity, pressure and two normal-stress differences are provided, with their Stokes-number dependence being implicit via the anisotropy parameters. The roles of ($St,\unicode[STIX]{x1D708},e$) on the granular temperature, the second-moment anisotropy and the nonlinear transport coefficients are analysed using the present theory, yielding quantitative agreements with particle-level simulations over a wide range of ($St,\unicode[STIX]{x1D708}$) including the bistable regime that occurs at $St\sim O(5)$. For highly dissipative particles ($e\ll 1$) that become increasingly important at large Stokes numbers, it is shown that the Burnett-order solution is not adequate and further higher-order solutions are required for a quantitative agreement of transport coefficients over the whole range of control parameters. The latter is accomplished by developing an approximate super-super-Burnett-order theory for the ignited state ($St\gg 1$) of sheared dense gas–solid suspensions in the second part of this paper. An extremum principle based on viscous dissipation and dynamic friction is discussed to identify ignited–quenched transition.
Suspension Taylor–Couette flow: co-existence of stationary and travelling waves, and the characteristics of Taylor vortices and spirals
- Prashanth Ramesh, S. Bharadwaj, Meheboob Alam
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- Journal of Fluid Mechanics / Volume 870 / 10 July 2019
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- 15 May 2019, pp. 901-940
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Flow visualization and particle image velocimetry (PIV) measurements are used to unravel the pattern transition and velocity field in the Taylor–Couette flow (TCF) of neutrally buoyant non-Brownian spheres immersed in a Newtonian fluid. With increasing Reynolds number ($Re$) or the rotation rate of the inner cylinder, the bifurcation sequence in suspension TCF remains same as in its Newtonian counterpart (i.e. from the circular Couette flow (CCF) to stationary Taylor vortex flow (TVF) and then to travelling wavy Taylor vortices (WTV) with increasing $Re$) for small particle volume fractions ($\unicode[STIX]{x1D719}<0.05$). However, at $\unicode[STIX]{x1D719}\geqslant 0.05$, non-axisymmetric patterns such as (i) the spiral vortex flow (SVF) and (ii) two mixed or co-existing states of stationary (TVF, axisymmetric) and travelling (WTV or SVF, non-axisymmetric) waves, namely (iia) the ‘TVF$+$WTV’ and (iib) the ‘TVF$+$SVF’ states, are found, with the former as a primary bifurcation from CCF. While the SVF state appears both in the ramp-up and ramp-down experiments as in the work of Majji et al. (J. Fluid Mech., vol. 835, 2018, pp. 936–969), new co-existing patterns are found only during the ramp-up protocol. The secondary bifurcation TVF $\leftrightarrow$ WTV is found to be hysteretic or sub-critical for $\unicode[STIX]{x1D719}\geqslant 0.1$. In general, there is a reduction in the value of the critical Reynolds number, i.e. $Re_{c}(\unicode[STIX]{x1D719}\neq 0)<Re_{c}(\unicode[STIX]{x1D719}=0)$, for both primary and secondary transitions. The wave speeds of both travelling waves (WTV and SVF) are approximately half of the rotational velocity of the inner cylinder, with negligible dependence on $\unicode[STIX]{x1D719}$. The analysis of the radial–axial velocity field reveals that the Taylor vortices in a suspension are asymmetric and become increasingly anharmonic, with enhanced radial transport, with increasing particle loading. Instantaneous streamline patterns on the axial–radial plane confirm that the stationary Taylor vortices can indeed co-exist either with axially propagating spiral vortices or azimuthally propagating wavy Taylor vortices – their long-time stability is demonstrated. It is shown that the azimuthal velocity is considerably altered for $\unicode[STIX]{x1D719}\geqslant 0.05$, resembling shear-band type profiles, even in the CCF regime (i.e. at sub-critical Reynolds numbers) of suspension TCF; its possible role on the genesis of observed patterns as well as on the torque scaling is discussed.
Revisiting ignited–quenched transition and the non-Newtonian rheology of a sheared dilute gas–solid suspension
- Saikat Saha, Meheboob Alam
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- Journal:
- Journal of Fluid Mechanics / Volume 833 / 25 December 2017
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- 03 November 2017, pp. 206-246
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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 an anisotropic Maxwellian as the single particle distribution function. For the simple shear flow, the closed-form solutions for granular temperature and three invariants of the second-moment tensor are obtained as functions of the Stokes number ($St$), the mean density ($\unicode[STIX]{x1D708}$) and the restitution coefficient ($e$). 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., vol. 296, 1995, pp. 211–246). The phase diagram is constructed in the three-dimensional ($\unicode[STIX]{x1D708},St,e$)-space that delineates the regions of ignited and quenched states and their coexistence. The particle-phase shear viscosity and the normal-stress differences are analysed, along with related scaling relations on the quenched and ignited states. At any $e$, the shear viscosity undergoes a discontinuous jump with increasing shear rate 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 ${\mathcal{N}}_{2}$ identified with the system making a transition from the quenched to ignited states. The superior prediction of the present theory over the standard Grad’s method and the Burnett-order Chapman–Enskog solution is demonstrated via comparisons of transport coefficients with simulation data for a range of Stokes number and restitution coefficient.
Normal stress differences, their origin and constitutive relations for a sheared granular fluid
- Saikat Saha, Meheboob Alam
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- Journal of Fluid Mechanics / Volume 795 / 25 May 2016
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- 19 April 2016, pp. 549-580
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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 ($\unicode[STIX]{x1D615}_{1}$ and $\unicode[STIX]{x1D615}_{2}$) 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 $\unicode[STIX]{x1D615}_{1}$ and $\unicode[STIX]{x1D615}_{2}$ 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 (${\it\nu}\sim 0.5$). While the first normal stress difference $\unicode[STIX]{x1D615}_{1}$ is found to be positive in the dilute limit and decreases monotonically to zero in the dense limit, the second normal stress difference $\unicode[STIX]{x1D615}_{2}$ 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 $\unicode[STIX]{x1D615}_{1}$ is tied to the non-coaxiality (${\it\phi}\neq 0$) between the eigendirections of the second-moment tensor $\unicode[STIX]{x1D648}$ and those of the shear tensor $\unicode[STIX]{x1D63F}$. In contrast, the origin of $\unicode[STIX]{x1D615}_{2}$ in the dilute limit is tied to the ‘excess’ temperature ($T_{z}^{ex}=T-T_{z}$, where $T_{z}$ 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.
On Knudsen-minimum effect and temperature bimodality in a dilute granular Poiseuille flow
- Meheboob Alam, Achal Mahajan, Deepthi Shivanna
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- Journal of Fluid Mechanics / Volume 782 / 10 November 2015
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- 06 October 2015, pp. 99-126
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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 $e_{n}\rightarrow 1$) in channels with very smooth walls. The 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 bimodality of the temperature profile, with a local minimum ($T_{\mathit{min}}$) at the channel centerline and two symmetric maxima ($T_{\mathit{max}}$) 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, ${\rm\Delta}T=(T_{\mathit{max}}/T_{\mathit{min}}-1)\neq 0$) 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 dependence of the excess temperature ${\rm\Delta}T$ 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.
Plane shock waves and Haff’s law in a granular gas
- M. H. Lakshminarayana Reddy, Meheboob Alam
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- Journal of Fluid Mechanics / Volume 779 / 25 September 2015
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- 18 August 2015, R2
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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 decays according to Haff’s law (${\it\theta}(t)\sim t^{-2}$), but the downstream temperature decays faster than its upstream counterpart. 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 the 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.
Nonlinear instability and convection in a vertically vibrated granular bed
- Priyanka Shukla, Istafaul H. Ansari, Devaraj van der Meer, Detlef Lohse, Meheboob Alam
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- Journal of Fluid Mechanics / Volume 761 / 25 December 2014
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- 17 November 2014, pp. 123-167
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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 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, $A_{e}\sim \sqrt{{\it\Delta}}$, with the distance ${\it\Delta}$ 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 at both 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 a floating-convection state. The linear and nonlinear patterns of the density and velocity fields are analysed and compared with experiments qualitatively. 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.
Non-Newtonian stress, collisional dissipation and heat flux in the shear flow of inelastic disks: a reduction via Grad’s moment method
- Saikat Saha, Meheboob Alam
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- Journal of Fluid Mechanics / Volume 757 / 25 October 2014
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- 19 September 2014, pp. 251-296
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The non-Newtonian stress tensor, collisional dissipation rate and 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 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 $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mu $, pressure $p$, first normal stress difference ${\mathcal{N}}_1$ and dissipation rate ${\mathcal{D}}$ as functions of (i) density or area fraction $\nu $, (ii) restitution coefficient $e$, (iii) dimensionless shear rate $R$, (iv) temperature anisotropy $\eta $ (the difference between the principal eigenvalues of the second-moment tensor) and (v) 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 terms at super-Burnett order must be retained for a quantitative prediction of transport coefficients, especially at moderate to large densities for small values of the restitution coefficient ($e \ll 1$). Particle simulation data for a sheared inelastic hard-disk system are compared with theoretical results, with good agreement for $p$, $\mu $ and ${\mathcal{N}}_1$ over a range of densities spanning from the dilute to close to the freezing point. In contrast, the predictions from a constitutive model at Navier–Stokes order 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.
Nonlinear vorticity-banding instability in granular plane Couette flow: higher-order Landau coefficients, bistability and the bifurcation scenario
- Priyanka Shukla, Meheboob Alam
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- Journal of Fluid Mechanics / Volume 718 / 10 March 2013
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- 08 February 2013, pp. 131-180
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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 the banding of particles along the mean vorticity direction: this is dubbed ‘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 $\rightarrow $ subcritical pitchfork $\rightarrow $ 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 analogues 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.
Nonlinear stability, bifurcation and vortical patterns in three-dimensional granular plane Couette flow
- Meheboob Alam, Priyanka Shukla
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- Journal of Fluid Mechanics / Volume 716 / 10 February 2013
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- 25 January 2013, pp. 349-413
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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 & 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 3D 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.
Nonlinear stability and patterns in granular plane Couette flow: Hopf and pitchfork bifurcations, and evidence for resonance
- PRIYANKA SHUKLA, MEHEBOOB ALAM
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- Journal of Fluid Mechanics / Volume 672 / 10 April 2011
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- 18 February 2011, pp. 147-195
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The first evidence of a variety of nonlinear 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 wavelengths (kx ~ 0, where kx is the streamwise wavenumber) in the dilute limit that evolve from subcritical shear-banding modes (kx = 0), but at large enough Couette gaps there are stationary instabilities with kx = O(1) that lead to supercritical pitchfork bifurcations. At moderate-to-large densities, in addition to supercritical shear-banding 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 kx = 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 counterparts. 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 (kx = 0) and a linear/fundamental mode kx ≠ 0, and (ii) an exact ‘1 : 2 resonance’ that results from the interaction between two waves with their wavenumber ratio being 1 : 2.
Weakly nonlinear theory of shear-banding instability in a granular plane Couette flow: analytical solution, comparison with numerics and bifurcation
- PRIYANKA SHUKLA, MEHEBOOB ALAM
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- Journal:
- Journal of Fluid Mechanics / Volume 666 / 10 January 2011
- Published online by Cambridge University Press:
- 16 November 2010, pp. 204-253
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A weakly nonlinear theory, in terms of the well-known Landau equation, has been developed to describe the nonlinear saturation of the shear-banding instability in a 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 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 used to validate a spectral-based numerical method for the nonlinear stability calculation. The regimes of supercritical and subcritical bifurcations for the shear-banding instability have been identified, leading to the prediction that the lower branch of the neutral stability contour in the (H, φ0)-plane, where H is the scaled Couette gap (the ratio between the Couette gap and the particle diameter) and φ0 is the mean density or the volume fraction of particles, is subcritically 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 bifurcation from infinity in the Boltzmann limit to (ii) subcritical bifurcation at moderate densities to (iii) supercritical bifurcation at larger densities to (iv) subcritical bifurcation in the dense limit and finally again to (v) supercritical bifurcation near the close packing density. It has been 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, have been demonstrated. The granular plane Couette flow serves as a paradigm that supports all three possible types of pitchfork bifurcations, with the mean density (φ0) being the single control parameter that dictates the nature of the bifurcation. The predicted bifurcation scenario for the shear-band formation is in qualitative agreement with particle dynamics simulations and the experiment in the rapid shear regime of the granular plane Couette flow.
Velocity distribution function and correlations in a granular Poiseuille flow
- MEHEBOOB ALAM, V. K. CHIKKADI
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- Journal of Fluid Mechanics / Volume 653 / 25 June 2010
- Published online by Cambridge University Press:
- 06 May 2010, pp. 175-219
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Probability distribution functions of fluctuation velocities (P(ux) and P(uy), where ux and uy are the fluctuation velocities in the x- and y-directions, respectively; the gravity is acting along the periodic x-direction and the flow is bounded by two walls parallel to the 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 granular 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(ux) distribution in the form of a sharp peak for negative velocities in the near-wall region. Both P(ux) and P(uy) distributions become asymmetric with increasing dissipation at any density, and the emergence of density waves, which 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 centreline 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(ux) and P(uy) appear to undergo a transition from Gaussian in the quasi-elastic limit to power-law distributions at large inelasticity of particle collisions. 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 centreline with increasing inelastic dissipation seems to be responsible for the emergence of non-Gaussian high-velocity tails of distribution functions.
Universality of shear-banding instability and crystallization in sheared granular fluid
- MEHEBOOB ALAM, PRIYANKA SHUKLA, STEFAN LUDING
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- Journal of Fluid Mechanics / Volume 615 / 25 November 2008
- Published online by Cambridge University Press:
- 25 November 2008, pp. 293-321
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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 νm) and the shear viscosity is allowed to diverge at a density νμ (<νm), with all other transport coefficients diverging at νm. 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 νm) 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 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.
Effects of Prandtl number and a new instability mode in a plane thermal plume
- R. LAKKARAJU, MEHEBOOB ALAM
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- Journal of Fluid Mechanics / Volume 592 / 10 December 2007
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- 14 November 2007, pp. 221-231
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The effect of Prandtl number on the linear stability of a plane thermal plume is analysed 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.
Streamwise structures and density patterns in rapid granular Couette flow: a linear stability analysis
- MEHEBOOB ALAM
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- Journal:
- Journal of Fluid Mechanics / Volume 553 / 25 April 2006
- Published online by Cambridge University Press:
- 06 April 2006, pp. 1-32
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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 $\partial/\partial y(.)\,{=}\,0$, 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 ($k_z\,{=}\,0$) 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 three-dimensional 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 three-dimensional instability modes in the form of both stationary and travelling waves, whose origin is tied to the corresponding two-dimensional instabilities.
While the two-dimensional 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 kinds of perturbations ($k_x\,{\neq}\, 0$ and $k_z\,{\neq}\, 0$), ‘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.
Instability-induced ordering, universal unfolding and the role of gravity in granular Couette flow
- MEHEBOOB ALAM, V. H. ARAKERI, P. R. NOTT, J. D. GODDARD, H. J. HERRMANN
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- Journal of Fluid Mechanics / Volume 523 / 25 January 2005
- Published online by Cambridge University Press:
- 21 January 2005, pp. 277-306
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Linear stability theory and bifurcation analysis are used to investigate the role of gravity in 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 a 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 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 long-wave travelling disturbances.The unique solution having a bottom plug can also be unstable to long waves, 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.