Nonlinear instability and convection in a vertically vibrated granular bed

Shukla, Priyanka; Ansari, Istafaul H.; van der Meer, Devaraj; Lohse, Detlef; Alam, Meheboob

dc.contributor.author	Shukla, Priyanka
dc.contributor.author	Ansari, Istafaul H.
dc.contributor.author	van der Meer, Devaraj
dc.contributor.author	Lohse, Detlef
dc.contributor.author	Alam, Meheboob
dc.date.accessioned	2017-02-21T07:03:03Z
dc.date.available	2017-02-21T07:03:03Z
dc.date.issued	2014
dc.identifier.citation	Shukla, P; Ansari, IH; van der Meer, D; Lohse, D; Alam, M, Nonlinear instability and convection in a vertically vibrated granular bed. Journal of Fluid Mechanics 2014, 761, 123-167, http://dx.doi.org/10.1017/jfm.2014.624	en_US
dc.identifier.citation	Journal of Fluid Mechanics	en_US
dc.identifier.citation	761	en_US
dc.identifier.issn	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) similar to root Delta, with the distance 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.
dc.identifier.uri	https://libjncir.jncasr.ac.in/xmlui/10572/2440
dc.description	http://dx.doi.org/10.1017/jfm.2014.624	en_US
dc.description.abstract	@Cambridge Univ Press, 2014	en_US
dc.description.uri	0022-1120	en_US
dc.description.uri	1469-7645	en_US
dc.publisher	English	en_US
dc.rights	Cambridge Univ Press	en_US
dc.subject	Mechanics	en_US
dc.subject	Fluids & Plasmas Physics	en_US
dc.subject	Complex Fluids	en_US
dc.subject	Convection	en_US
dc.subject	Nonlinear Instability	en_US
dc.subject	Plane Couette-Flow	en_US
dc.subject	Shear-Banding Instability	en_US
dc.subject	Unstable Parallel Flows	en_US
dc.subject	Non-Linear Mechanics	en_US
dc.subject	Poiseuille Flow	en_US
dc.subject	Dynamics Simulations	en_US
dc.subject	Boundary-Conditions	en_US
dc.subject	Stability Analysis	en_US
dc.subject	Wave Disturbances	en_US
dc.subject	Inelastic Spheres	en_US
dc.title	Nonlinear instability and convection in a vertically vibrated granular bed	en_US
dc.type	Article	en_US