Buoyancy driven convection in vertically shaken granular matter: experiment, numerics, and theory

Abstract

Buoyancy driven granular convection is studied for a shallow, vertically shaken granular bed in a quasi 2D container. Starting from the granular Leidenfrost state, in which a dense particle cluster floats on top of a dilute gaseous layer of fast particles (Meerson et al. in Phys Rev Lett 91:024301, 2003; Eshuis et al. in Phys Rev Lett 95:258001, 2005), we witness the emergence of counter-rotating convection rolls when the shaking strength is increased above a critical level. This resembles the classical onset of convection—at a critical value of the Rayleigh number—in a fluid heated from below. The same transition, even quantitatively, is seen in molecular dynamics simulations, and explained by a hydrodynamic-like model in which the granular material is treated as a continuum. The critical shaking strength for the onset of granular convection is accurately reproduced by a linear stability analysis of the model. The results from experiment, simulation, and theory are in good agreement. The present paper extends and completes our earlier analysis (Eshuis et al. in Phys Rev Lett 104:038001, 2010).

Appendices

Appendix A: Alternative models for the shear viscosity $\mu$

There is quite some discussion on the shear viscosity $\mu$ in granular systems and consequently various expressions have been proposed in the literature. Brey et al. [72] give the following relation for two dimensions and for a dilute granular gas:

$$\begin{aligned} \mu (T)=\frac{1}{2d}\sqrt{\frac{mT}{\pi }}\mu ^*(e), \end{aligned}$$

(61)

where $\mu ^{*}(e)$ is a function of the restitution coefficient $e$.

Ohtsuki and Ohsawa [59] deduce an expression for $\mu$ including a dependence on the density $n$ to account for excluded volume effects:

$$\begin{aligned} \mu (n,T)=\left\{ \frac{1}{4}n^2d^3 + \frac{1}{2\pi d}\left( 1+\frac{\pi }{4}nd^2 \right) ^2 \right\} \sqrt{\pi mT}. \end{aligned}$$

(62)

He et al. [55] propose that the shear viscosity should be equal to the thermal conductivity $\kappa$:

$$\begin{aligned} \mu (n,T)=\kappa (n,T), \end{aligned}$$

(63)

In the present paper we have found good correspondence between experiment and theory using a more general form based on dimension analysis:

$$\begin{aligned} \mu (n,T)=m\text {Pr}\,\kappa (n,T), \end{aligned}$$

(64)

where $\text {Pr}$ is the Prandtl number. We used it as a fit parameter for the phase diagram of Fig. 9 and found that $\text {Pr}=1.7$ gave good agreement.

Figure 13 shows the influence of $\mu$ on the resulting growth rate $\gamma (k_x)$, comparing the results obtained if one uses the expression by Brey et al. (61) with those obtained for expression (64). It is seen that the viscosity definition of (64) has a stabilizing effect on the Leidenfrost state with increasing number of particle layers $F$, in agreement with the experimental observations, whereas (61) has a destabilizing effect. We show in the $(S,F)$-phase diagram of Fig. 9 that (64) yields qualitative and quantitative agreement with the experimental results.

Fig. 13

Theory: Influence of the choice for the shear viscosity $\mu$ on the growth rate $\gamma (k_x)$ for two Leidenfrost states at the same shaking strength $S=200$: a For $F=6$ layers the region of instability of the Leidenfrost state is significantly reduced by going from the expression for $\mu (T)$ by Brey et al. [(61), black dots] to $\mu (n,T)$ as defined by (64) with $\text {Pr}=1.7$ (grey crosses. b For $F=11$ layers the stabilizing effect is even stronger. Note that the range of unstable $k_x$-values for the black dots has increased compared to the $F=6$ Leidenfrost state, whereas the opposite is true for grey crosses

Appendix B: Relations for the pressure, dissipation, and transport coefficients

For the matrix problem (51) we need to specify the elements of the matrices $\mathbf A ,\,\mathbf B$, and $\mathbf C$ of (52)–(54), which contain $p,\,I$, and the transport coefficients and their derivatives. These are given below:

First of all, we have the equation of state for the pressure $\widetilde{p}$ and its derivatives:

$$\begin{aligned}&\widetilde{p}_L = \widetilde{n}_L\widetilde{T}_L \frac{1+\widetilde{n}_L}{1-\widetilde{n}_L},\end{aligned}$$

(65)

$$\begin{aligned}&\frac{\partial \widetilde{p}}{\partial \widetilde{n}}\Big |_L = \widetilde{T}_L \frac{1+2\widetilde{n}_L-\widetilde{n}_L^2}{(1-\widetilde{n}_L)^2}, \end{aligned}$$

(66)

$$\begin{aligned}&\frac{\partial }{\partial \widetilde{y}}\left( \frac{\partial \widetilde{p}}{\partial \widetilde{n}}\Big |_L \right) = \frac{(1-\widetilde{n}_L)(1+2\widetilde{n}_L-\widetilde{n}_L^2) \frac{\partial \widetilde{T}_L}{\partial \widetilde{y}} + 4 \widetilde{T}_L\frac{\partial \widetilde{n}_L}{\partial \widetilde{y}}}{(1-\widetilde{n}_L)^3},\nonumber \\ \end{aligned}$$

(67)

$$\begin{aligned}&\frac{\partial \widetilde{p}}{\partial \widetilde{T}}\Big |_L = \widetilde{n}_L \frac{1+\widetilde{n}_L}{1-\widetilde{n}_L}, \end{aligned}$$

(68)

$$\begin{aligned}&\frac{\partial }{\partial \widetilde{y}} \left( \frac{\partial \widetilde{p}}{\partial \widetilde{T}}\Big |_L \right) = \left[ \frac{1+2\widetilde{n}_L - \widetilde{n}_L^2}{(1 - \widetilde{n}_L)^2} \right] \frac{\partial \widetilde{n}_L}{\partial \widetilde{y}}. \end{aligned}$$

(69)

The expressions for the energy dissipation rate $\widetilde{I}$ read as follows:

$$\begin{aligned}&\widetilde{I} = \frac{\varepsilon }{\gamma } \frac{\widetilde{n}\widetilde{T}^{3/2}}{\widetilde{\ell }},\end{aligned}$$

(70)

$$\begin{aligned}&\frac{\partial \widetilde{I}}{\partial \widetilde{n}}\Big |_L = \frac{\varepsilon }{\gamma }\widetilde{T}^{3/2} \left( \frac{\widetilde{\ell } - \widetilde{n} \frac{\partial \widetilde{\ell }}{\partial \widetilde{n}}}{\widetilde{\ell }^2} \right) , \end{aligned}$$

(71)

$$\begin{aligned}&\frac{\partial \widetilde{I}}{\partial \widetilde{T}}\Big |_L = \frac{3\varepsilon }{2\gamma } \frac{\widetilde{n} \sqrt{\widetilde{T}}}{\widetilde{\ell }}. \end{aligned}$$

(72)

The mean free path $\widetilde{\ell }$ and its derivatives are given by:

$$\begin{aligned}&\widetilde{\ell } = \sqrt{\frac{3}{32}}\left[ \frac{1}{\widetilde{n}} \left( \frac{1-\widetilde{n}}{1-a\widetilde{n}}\right) \right] ,\end{aligned}$$

(73)

$$\begin{aligned}&\frac{\partial \widetilde{\ell }}{\partial \widetilde{n}} = \sqrt{\frac{3}{32}} \left( \frac{-a\widetilde{n}^2 + 2a\widetilde{n}-1}{\widetilde{n}^2\left( 1-a\widetilde{n} \right) ^2} \right) ,\end{aligned}$$

(74)

$$\begin{aligned}&\frac{\partial ^2\widetilde{\ell }}{\partial \widetilde{n}^2} = 2\sqrt{\frac{3}{32}} \left( \frac{-a^2\widetilde{n}^3 + 3a^2\widetilde{n}^2 - 3a\widetilde{n} + 1}{\widetilde{n}^3 \left( 1-a\widetilde{n} \right) ^3}\right) . \end{aligned}$$

(75)

We continue with the transport coefficient for the thermal conductivity $\widetilde{\kappa }$ and its derivatives:

$$\begin{aligned}&\widetilde{\kappa }_L = \frac{\left( \alpha \widetilde{\ell } +1 \right) ^2}{\widetilde{\ell }} \widetilde{n} \sqrt{\widetilde{T}},\end{aligned}$$

(76)

$$\begin{aligned}&\frac{\partial \widetilde{\kappa }_L}{\partial \widetilde{y}} = \frac{\partial \widetilde{\kappa }_L}{\partial \widetilde{n}} \frac{\partial \widetilde{n}}{\partial \widetilde{y}} + \frac{\partial \widetilde{\kappa }_L}{\partial \widetilde{T}} \frac{\partial \widetilde{T}}{\partial \widetilde{y}},\end{aligned}$$

(77)

$$\begin{aligned}&\frac{\partial \widetilde{\kappa }}{\partial \widetilde{n}}\Big |_L = \sqrt{T} \left[ \frac{(\alpha \widetilde{\ell } + 1)^2}{\widetilde{\ell }} + \widetilde{n} \frac{\alpha ^2 \widetilde{\ell }^2 - 1}{\widetilde{\ell }^2} \frac{\partial \widetilde{\ell }}{\partial \widetilde{n}} \right] , \end{aligned}$$

(78)

$$\begin{aligned}&\frac{\partial }{\partial \widetilde{y}} \left( \frac{\partial \widetilde{\kappa }}{\partial \widetilde{n}}\Big |_L \right) = \frac{\partial }{\partial \widetilde{n}} \left( \frac{\partial \widetilde{\kappa }}{\partial \widetilde{n}}\Big |_L \right) \frac{\partial \widetilde{n}}{\partial \widetilde{y}} + \frac{\partial }{\partial \widetilde{T}} \left( \frac{\partial \widetilde{\kappa }}{\partial \widetilde{n}}\Big |_L \right) \frac{\partial \widetilde{T}}{\partial \widetilde{y}},\nonumber \\ \end{aligned}$$

(79)

$$\begin{aligned}&\frac{\partial }{\partial \widetilde{n}} \left( \frac{\partial \widetilde{\kappa }}{\partial \widetilde{n}}\Big |_L \right) \nonumber \\&\quad \!=\! \sqrt{\widetilde{T}} \left[ \frac{ 2( \alpha ^2\widetilde{\ell }^2\!-\!1) \frac{\partial \widetilde{\ell }}{\partial \widetilde{n}} + \frac{2\widetilde{n}}{\widetilde{\ell }}\left( \frac{\partial \widetilde{\ell }}{\partial \widetilde{n}} \right) ^2 \!+\! \widetilde{n}( \alpha ^2\widetilde{\ell }^2\!-\!1)\frac{\partial ^2 \widetilde{\ell }}{\partial \widetilde{n}^2} }{\widetilde{\ell }^2} \right] .\nonumber \\ \end{aligned}$$

(80)

$$\begin{aligned}&\frac{\partial }{\partial \widetilde{T}} \left( \frac{\partial \widetilde{\kappa }}{\partial \widetilde{n}}\Big |_L \right) \!=\! \frac{1}{2\sqrt{\widetilde{T}}} \left[ \frac{(\alpha \widetilde{\ell } \!+\! 1)^2}{\widetilde{\ell }} \!+\! \widetilde{n} \frac{\alpha ^2 \widetilde{\ell }^2 \!-\! 1}{\widetilde{\ell }^2} \frac{\partial \widetilde{\ell }}{\partial \widetilde{n}} \right] ,\end{aligned}$$

(81)

$$\begin{aligned}&\frac{\partial \widetilde{\kappa }}{\partial \widetilde{T}}\Big |_L \!=\! \frac{1}{2\sqrt{\widetilde{T}_L}} \widetilde{n} \frac{(\alpha \widetilde{\ell } \!+\! 1)^2}{\widetilde{\ell }}, \qquad \end{aligned}$$

(82)

$$\begin{aligned}&\frac{\partial }{\partial \widetilde{y}} \left( \frac{\partial \widetilde{\kappa }}{\partial \widetilde{T}}\Big |_L \right) \!=\! \frac{\partial }{\partial \widetilde{n}} \left( \frac{\partial \widetilde{\kappa }}{\partial \widetilde{T}}\Big |_L \right) \frac{\partial \widetilde{n}}{\partial \widetilde{y}} \!+\! \frac{\partial }{\partial \widetilde{T}} \left( \frac{\partial \widetilde{\kappa }}{\partial \widetilde{T}}\Big |_L \right) \frac{\partial \widetilde{T}}{\partial \widetilde{y}},\qquad \end{aligned}$$

(83)

$$\begin{aligned}&\frac{\partial }{\partial \widetilde{n}} \left( \frac{\partial \widetilde{\kappa }}{\partial \widetilde{T}}\Big |_L \right) \!=\! \frac{1}{2\sqrt{\widetilde{T}}} \left[ \frac{(\alpha \widetilde{\ell } \!+\! 1)^2}{\widetilde{\ell }} \!+\! \widetilde{n} \frac{\alpha ^2\widetilde{\ell }^2 - 1}{\widetilde{\ell }^2} \frac{\partial \widetilde{\ell }}{\partial \widetilde{n}} \right] ,\end{aligned}$$

(84)

$$\begin{aligned}&\frac{\partial }{\partial \widetilde{T}} \left( \frac{\partial \widetilde{\kappa }}{\partial \widetilde{T}}\Big |_L \right) = - \frac{1}{4\widetilde{T}\sqrt{\widetilde{T}}} \widetilde{n} \frac{(\alpha \widetilde{\ell } + 1)^2}{\widetilde{\ell }}. \end{aligned}$$

(85)