Thermal Convection in Granular Gases with Dissipative Lateral Walls

Abstract

We consider a granular gas under the action of gravity, fluidized by a vibrating base. We show that a horizontal temperature gradient, here induced by limiting dissipative lateral walls (DLW), leads always to a granular thermal convection (DLW-TC) that is essentially different from ordinary bulk-buoyancy-driven convection (BBD-TC). In an experiment where BBD-TC is inhibited, by reducing gravity with an inclined plane, we always observe a DLW-TC cell next to each lateral wall. Such a cell squeezes towards the nearest wall as the gravity and/or the number of grains increase. Molecular dynamics simulations reproduce the experimental results and indicate that at large gravity or number of grains the DLW-TC is barely detectable.Shaken granular media escape most of the laws of equilibrium thermodynamics and statistical mechanics [1], ranging from slow plastic flows [2] to fast gas-like dynamics [3][4][5]. In the wide granular phenomenology [6], an ubiquitous pattern is the convective cell. Notwithstanding its widespread occurrence, many different mechanisms lead to granular convection, and their relevance depends on the granular state under scrutiny.At high packing fraction and low fluidization, "dense convection" is observed [7][8][9][10]. A convincing explanation of dense convection comes from the asymmetric tangential friction at the lateral walls that results in a net downward shear force [11][12][13]. Dense granular convection can also involve more complex mechanisms, including the formation of unstable heaps at the free surface [14].In highly fluidized states (granular gases), the only known theoretical mechanism is bulk buoyancy-driven thermal convection (BBD-TC), first observed in simulations [15,16]. In analogy to molecular liquids [17,18], it originates from the buoyancy force associated to temperature/density gradients that, because of the intrinsic bulk inelasticity, emerge spontaneously [19], even with an open or reflecting top boundary [20]. BBD-TC is an instability of the hydrostatic state which requires a combination of parameters (including inelasticity, gravity and dimensions) to overcome a certain threshold [21,22]. A further confirmation that BBD-TC is essentially a "bulk" effect comes from simulations [15] and theory [21,22] where lateral walls are not required to observe it. Convective circulation in granular gases has been seen also in experiments, where lateral walls are always inelastic [23][24][25][26], and successive simulations with elastic [27] and also inelastic walls [28].The role of lateral walls in dilute granular convection has not been fully understood yet. Some of the mentioned studies recognize that lateral walls influence the observations. For instance, a downward flow velocity is always observed near lateral walls, perhaps because of a reduced buoyancy originated from enhanced dissipation [15,23,28]. However, quite contrasting results emerge in simulations and experiments concerning the necessity of dissipation at the lateral walls. In some simulations convection is strong even w...

Dissipative lateral walls are sufficient to trigger convection in vibrated granular gases

“…For all values of g e f f the width of the convective cell increase with v 2 0 , even if such a growth slows down at high v 2 0 . Consistently with previously reported data [28], the width of the cell decreases monotonically with g e f f for low values of v 0 : on the contrary, it presents non-monotonicy with an initial growth at the first studied values of g e f f followed by a decrease in g e f f , for larger values of v 0 .…”

“…This is in perfect agreement with the idea that DLW-TC is a "boundary effect", but nev- ertheless it spans a macroscopic portion of the system, i.e. a large number of mean free path (see [28] and its Supplemental Materials for a discussion of Knudsen numbers in our system). In Fig.…”

“…DLW-TC is replaced by usual thermal convection when the parameters are chosen in the region of instability [21,22], e.g. for high enough bulk inelasticity, as shown in the Supplemental Materials of [28]. We remark that while many studies in dense vibrated granular materials have shown boundary-induced convection, and while other more recent studies in vibrated granular gases have shown the existence of an important role of the lateral wall, the study in [28] is the first to demonstrate the existence of a wall-induced convection in gases which is well distinct from the bulk-inelasticity-induced effect.…”

“…When a disk collides with the lateral walls, an inelastic collision with an infinite mass and with restitution coefficient α w ∈ [0, 1] is performed. Some simulations with the presence of friction along the ballistic flight (introduced by time-discretized dynamics) -which would be analogous to the rolling/sliding friction between spheres and the setup's floor -have also been performed and discussed in [28], giving very similar results. The simulations clearly show that with the parameters of the experiment and the artificial choice of elastic lateral walls the convection is totally suppressed, as expected for experiments at low gravity [21,22].…”

“…In [28] we have discussed an experiment (inspired on previous works [30,31]) under low-gravity conditions (see Fig. 1 and [28], including Supplemental Materials, for details) with N spherical steel beads (diameter d = 1 mm) moving inside a cuboid of dimensions L x = 175 mm, L y = 600 mm and L z = 1.5 mm = 1.5d, thus assuring for a quasi-2D dynamics restricted to the xy plane. A small tilt angle θ with respect to the horizontal guarantees an effective low value for gravity g e f f ≈ (5/7)g sin(θ), where g is Earth's gravity acceleration [32].…”