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It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

The amplitude equations governing the temporal evolution of internal waves forming a resonant triad in a two-dimensional inviscid stably stratified uniform shear flow, bounded between two infinite horizontal parallel plates, are derived in the absence of diffusive effects. The density is considered to be a linear function of the vertical coordinate. The interaction of two vertically confined and horizontally propagating primary internal waves having the same frequency is considered. Specifically, for different local Richardson numbers, we show the existence and stability of the resonant triad formed by three different internal waves having the wave vector and frequency pairs as (k - m,ω), (k - n,ω), and (k - m+k - n,2ω). For each resonant triad, we solve the amplitude equations numerically as an initial value problem. The equilibrium solutions of the amplitude equations are obtained analytically. Furthermore, the linear stability analysis of the resonant triads, around the equilibrium solutions, is carried out for various interaction cases. The triads containing a wave with the lowest mode number are found to be linearly unstable. The exact solutions of the amplitude equations are presented under the pump-wave approximation, in which the amplitude of one of the waves in the triad is larger than the amplitudes of the other two waves initially. From the exact solutions, the amplitudes of the waves forming a resonant triad are found to be unbounded when the wave with the mode number 1 acts as the pump wave. The present study helps one to have a better understanding of the stability of resonant triads of internal waves formed in a stratified shear flow. © 2021 American Physical Society.

World University root node

Department of Mathematics

Physical Review Fluids

Stability analysis of a resonant triad in a stratified uniform shear flow

The hydrodynamic viscous fingering instability can be influenced by a simple viscosity-changing chemical reaction of type A+B→C, when a solution of reactant A is injected into a solution of B and a product C of different viscosity is formed by reaction. We investigate here numerically such reactive viscous fingering in the case of a reaction decreasing the viscosity to define the optimal conditions on the chemical and hydrodynamic parameters for controlling fingering. In particular, we analyze the influence of the Péclet number, Pe, on the efficiency of the chemical control of fingering. We show that the viscosity-decreasing reaction has an increased stabilizing effect when Pe is decreased. On the contrary, fingering is more intense and the system more unstable when Pe is increased. The related reactive fingering patterns cover then respectively a smaller (larger) area than in the nonreactive equivalent. Depending on the value of the Péclet number, a given chemical reaction may thus either enhance or suppress a fingering instability. This stabilization and destabilization at low and high Pe are shown to be related to the Pe-dependent characteristics of a minimum in the viscosity profile that develops around the miscible interface thanks to the effect of the chemical reaction. © 2020 American Physical Society.

Fulltext

Influence of the Péclet number on reactive viscous fingering

We revisit the problem of the weakly nonlinear stability analysis of an immiscible two-fluid viscosity-stratified, density-matched, plane Poiseuille flow (PPF) in a rigid channel. A formal amplitude expansion method, in which the flow variables are expanded in terms of a small amplitude function, is employed to examine the nonlinear development of the uniform wave trains. By employing the Chebyshev spectral collocation method, the linear growth rate and the first Landau coefficient, which determine the weakly nonlinear temporal evolution of a finite amplitude disturbance in the vicinity of linear instability, are computed. The focus is on the parameter regime where the long-waves are stable. The present analysis reveals the existence of both subcritical unstable and supercritical stable bifurcations. It is found that similar to the single fluid PPF, the two-fluid flow remains subcritically unstable at the onset of linear instability. There is a transition from subcritical bifurcation at higher wave numbers to supercritical bifurcation at lower wave numbers. The feedback of the mean flow correction onto the wave is responsible for the subcritical bifurcation. The equilibrium amplitude increases (decreases) as a function of the Reynolds number at a fixed wave number, where the bifurcation is supercritical (subcritical). Similar to the single fluid PPF, there is a reduction in the critical Reynolds number due to even extremely weak but finite amplitude disturbances. Moreover, as the disturbance intensity increases, the percentage reduction in the critical Reynolds number increases. The study helps one to have a better understanding and perspective of the bifurcations that occur close to criticality in the two-fluid interface dominated PPF system. In addition, it calls for devoted experiments supplemented with numerical and theoretical predictions on Poiseuille flow of viscosity and/or density stratified systems that would shed light on the nature of transition. © 2019 Elsevier Ltd

GENERAL

International Journal of Multiphase Flow

Finite amplitude instability in a two-fluid plane Poiseuille flow

One prototypical instability in granular flows is the shear-banding instability, in which a uniform granular shear flow breaks into alternating bands of dense and dilute clusters of particles having low and high shear (shear stress or shear rate), respectively. In this work, the shear-banding instability in an arbitrarily inelastic granular shear flow is analyzed through the linear stability analysis of granular hydrodynamic equations closed with Navier-Stokes-level constitutive relations. It is shown that the choice of appropriate constitutive relations plays an important role in predicting the shear-banding instability. A parametric study is carried out to study the effect of the restitution coefficient, channel width, and mean density. Two global criteria relating the control parameters are found for the onset of the shear-banding instability. © 2019 American Physical Society. UK.

Physical Review E

Shear-banding instability in arbitrarily inelastic granular shear flows

Grad’s method of moments is employed to develop higher-order Grad moment equations – up to the first 26 moments – for dilute granular gases within the framework of the (inelastic) Boltzmann equation. The homogeneous cooling state of a freely cooling granular gas is investigated with the Grad 26-moment equations in a semi-linearized setting and it is shown that the granular temperature in the homogeneous cooling state still decays according to Haff’s law while the other higher-order moments decay on a faster time scale. The nonlinear terms of the fully contracted fourth moment are also considered and, by exploiting the stability analysis of fixed points, it is shown that these nonlinear terms have a negligible effect on Haff’s law. Furthermore, an even larger Grad moment system, which includes the fully contracted sixth moment, is also scrutinized and the stability analysis of fixed points is again exploited to conclude that even the inclusion of the scalar sixth-order moment into the Grad moment system has a negligible effect on Haff’s law. The constitutive relations for the stress and heat flux (i.e. the Navier–Stokes and Fourier relations) are derived through the Grad 26-moment equations and compared with those obtained via the Chapman–Enskog expansion and via computer simulations. The linear stability of the homogeneous cooling state is analysed through the Grad 26-moment system and various subsystems by decomposing them into longitudinal and transverse systems. It is found that one eigenmode in both longitudinal and transverse systems in the case of inelastic gases is unstable. By comparing the eigenmodes from various theories, it is established that the 13-moment eigenmode theory predicts that the unstable heat mode of the longitudinal system remains unstable for all wavenumbers below a certain coefficient of restitution, while any other higher-order moment theory shows that this mode becomes stable above some critical wavenumber for all values of the coefficient of restitution. In particular, the Grad 26-moment theory leads to a smooth profile for the critical wavenumber, in contrast to the other considered theories. Furthermore, the critical system size obtained through the Grad 26-moment theory is in excellent agreement with that obtained through existing theories. © 2017 Cambridge University Press

Journal of Fluid Mechanics

Higher-order moment theories for dilute granular gases of smooth hard spheres

A fingering instability can develop at the interface between two fluids when the more mobile fluid is injected into the less-mobile one. For example, viscous fingering appears when a less viscous (i.e., more mobile) fluid displaces a more viscous (and hence less mobile) one in a porous medium. Fingering can also be due to a local change in mobility arising when a precipitation reaction locally decreases the permeability. We numerically analyze the properties of the related precipitation fingering patterns occurring when an A+B→C chemical reaction takes place, where A and B are reactants in solution and C is a solid product. We show that, similarly to reactive viscous fingering patterns, the precipitation fingering structures differ depending on whether A invades B or vice versa. This asymmetry can be related to underlying asymmetric concentration profiles developing when diffusion coefficients or initial concentrations of the reactants differ. In contrast to reactive viscous fingering, however, precipitation fingering patterns appear at shorter time scales than viscous fingers because the solid product C has a diffusivity tending to zero which destabilizes the displacement. Moreover, contrary to reactive viscous fingering, the system is more unstable with regard to precipitation fingering when the high-concentrated solution is injected into the low-concentrated one or when the faster diffusing reactant displaces the slower diffusing one. © 2016 American Physical Society.

Fingering dynamics driven by a precipitation reaction: Nonlinear simulations

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 (Ae) is found to follow a square-root scaling law, Ae ∼ Δ, with the distance Δ 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. © 2014 Cambridge University Press.

Nonlinear instability and convection in a vertically vibrated granular bed

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 &amp; Alam, J.&nbsp;Fluid Mech., vol.&nbsp;672, 2011b, pp.&nbsp;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 𝑘𝑧=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 𝑘𝑧=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, bifurcation and vortical patterns in three-dimensional granular plane Couette flow

The rapid granular plane Couette flow is known to be unstable to pure spanwise perturbations (i.e.Â 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 ˚ightarrow subcritical pitchfork ˚ightarrow 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. © 2013 Cambridge University Press.

Nonlinear vorticity-banding instability in granular plane Couette flow: Higher-order Landau coefficients, bistability and the bifurcation scenario

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.

AIP Conference Proceedings

Shearbanding and inhomogeneous states in 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) in dense flows depends strongly on the choice of the constitutive model. While the standard Enskog model for nearly elastic hard-disks predicts supercritical bifurcations formoderate to dense systems, amore realistic model with global equation of states for hard-disks (that are likely to hold for the whole range of densities) predicts a subcritical bifurcation in the dense limit. The latter prediction agrees with recent particle simulations of a sheared inelastic hard-disk system. © Springer-Verlag 2012.

Granular Matter

Origin of subcritical shear-banding instability in a dense two-dimensional sheared granular fluid

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. Copyright © Cambridge University Press 2011.

Nonlinear stability and patterns in granular plane Couette flow: Hopf and pitchfork bifurcations, and evidence for resonance

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,)-plane, where H is the scaled Couette gap (the ratio between the Couette gap and the particle diameter) and 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 () 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. © 2010 Cambridge University Press.

Weakly nonlinear theory of shear-banding instability in a granular plane Couette flow: Analytical solution, comparison with numerics and bifurcation

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 dilute flows, our analytical bifurcation theory suggests that there is a subcritical finite-amplitude instability that is likely to lead to shear-band formation in dilute flows, which is in agreement with previous numerical simulations. © 2009 The American Physical Society.

Physical Review Letters

Landau-type order parameter equation for shear banding in granular couette flow

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 νm (&lt;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. © 2008 Cambridge University Press.

Universality of shear-banding instability and crystallization in sheared granular fluid

This investigation presents a fully spectral method for solving coupled hyperbolic partial differential equations. The spectral method is based on the Galerkin-collocation technique. Two different preconditioners the Preissmann and upwind schemes are evaluated for their performance in solving the discretized equations. It has been found, for the cases considered, that the upwind scheme is a viable preconditioner for the fully spectral discretization of hyperbolic PDEs. Its performance as a preconditioner is in every way superior to that of the Preissmann scheme. It is established that the relative accuracy of different numerical solutions is reliably indicated by the root-mean-square average of their residuals obtained by the discretization. It ts also established that the scheme gives much better accuracy than the finite-difference Preissmann scheme, for the same amount of computational effort, for both linear and non-linear problems.