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Register a new userThe Knudsen temperature jump and the Navier-Stokes hydrodynamics of granular gases driven by thermal walls
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Thermal wall is a convenient idealization of a rapidly vibrating plate used for vibrofluidization of granular materials. The objective of this work is to incorporate the Knudsen temperature jump at thermal wall in the Navier-Stokes hydrodynamic modeling of dilute granular gases of monodisperse particles that collide nearly elastically. The Knudsen temperature jump manifests itself as an additional term, proportional to the temperature gradient, in the boundary condition for the temperature. Up to a numerical pre-factor of order unity, this term is known from kinetic theory of elastic gases. We determine the previously unknown numerical pre-factor by measuring, in a series of molecular dynamics (MD) simulations, steady-state temperature profiles of a gas of elastically colliding hard disks, confined between two thermal walls kept at different temperatures, and comparing the results with the predictions of a hydrodynamic calculation employing the modified boundary condition. The modified boundary condition is then applied, without any adjustable parameters, to a hydrodynamic calculation of the temperature profile of a gas of inelastic hard disks driven by a thermal wall. We find the hydrodynamic prediction to be in very good agreement with MD simulations of the same system. The results of this work pave the way to a more accurate hydrodynamic modeling of driven granular gases.
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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 buoyancy-driven convection (BD-TC). In an experiment where BD-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.
On the basis of the intimate relation between Nambu dynamics and the hydrodynamics, the hydrodynamics on a non-commutative space (obtained by the quantization of space), proposed by Nambu in his last work, is formulated as ``hydrodynamics of granular material. In Part 1, the quantization of space is done by Moyal product, and the hydrodynamic simulation is performed for the so obtained two dimensional fluid, which flows inside a canal with an obstacle. The obtained results differ between two cases in which the size of a fluid particle is zero and finite. The difference seems to come from the behavior of vortices generated by an obstacle. In Part 2 of quantization, considering vortex as a string, two models are examined; one is the ``hybrid model in which vortices interact with each other by exchanging Kalb-Ramond fields (a generalization of stream functions), and the other is the more general ``string field theory in which Kalb-Ramond field is one of the excitation mode of string oscillations. In the string field theory, Altarelli-Parisi type evolution equation is introduced. It is expected to describe the response of distribution function of vortex inside a turbulence, when the energy scale is changed. The behaviour of viscosity differs in the string theory, being compared with the particle theory, so that Landau theory of fluid to introduce viscosity may be modified. In conclusion, the hydrodynamics and the string theory are almost identical theories. It should be noted, however, that the string theory to reproduce a given hydrodynamics is not a usual string theory.
A numerical study is presented to analyze the thermal mechanisms of unsteady, supersonic granular flow, by means of hydrodynamic simulations of the Navier-Stokes granular equations. For this purpose a paradigmatic problem in granular dynamics such as the Faraday instability is selected. Two different approaches for the Navier-Stokes transport coefficients for granular materials are considered, namely the traditional Jenkins-Richman theory for moderately dense quasi-elastic grains, and the improved Garzo-Dufty-Lutsko theory for arbitrary inelasticity, which we also present here. Both solutions are compared with event-driven simulations of the same system under the same conditions, by analyzing the density, the temperature and the velocity field. Important differences are found between the two approaches leading to interesting implications. In particular, the heat transfer mechanism coupled to the density gradient which is a distinctive feature of inelastic granular gases, is responsible for a major discrepancy in the temperature field and hence in the diffusion mechanisms.
We consider solutions to the 2d Navier-Stokes equations on $mathbb{T}timesmathbb{R}$ close to the Poiseuille flow, with small viscosity $ u>0$. Our first result concerns a semigroup estimate for the linearized problem. Here we show that the $x$-dependent modes of linear solutions decay on a time-scale proportional to $ u^{-1/2}|log u|$. This effect is often referred to as emph{enhanced dissipation} or emph{metastability} since it gives a much faster decay than the regular dissipative time-scale $ u^{-1}$ (this is also the time-scale on which the $x$-independent mode naturally decays). We achieve this using an adaptation of the method of hypocoercivity. Our second result concerns the full nonlinear equations. We show that when the perturbation from the Poiseuille flow is initially of size at most $ u^{3/4+}$, then it remains so for all time. Moreover, the enhanced dissipation also persists in this scenario, so that the $x$-dependent modes of the solution are dissipated on a time scale of order $ u^{-1/2}|log u|$. This transition threshold is established by a bootstrap argument using the semigroup estimate and a careful analysis of the nonlinear term in order to deal with the unboundedness of the domain and the Poiseuille flow itself.
As a natural and functional behavior, various microorganisms exhibit gravitaxis by orienting and swimming upwards against gravity. Swimming autophoretic nanomotors described herein, comprising bimetallic nanorods, preferentially orient upwards and swim up along a wall, when tail-heavy (i.e. when the density of one of the metals is larger than the other). Through experiment and theory, two mechanisms were identified that contribute to this gravitactic behavior. First, a buoyancy or gravitational torque acts on these rods to align them upwards. Second, hydrodynamic interactions of the rod with the inclined wall induce a fore-aft drag asymmetry on the rods that reinforces their orientation bias and promotes their upward motion.
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