Analytic structure of a drag-driven confined dust vortex flow in plasma

Flow structure of a dust medium electrostatically suspended and confined in a plasma presents a unique setup where the spatial scale of a volumetric drive by the plasma flow might exceed that of the boundaries confining the dust. By means of a formal implementation of a two-dimensional hydrodynamic model to a confined dust flow and its analytic curvilinear solutions, it is shown that the eigenmode spectrum of the dust vortex flow can lose correlations with the driving field even at the low dust Reynolds numbers as a result of strong shear and finer scales introduced in the equilibrium dust vorticity spectrum by the boundaries. While the boundary effects can replace the desired turbulent processes unavailable in this regime, the shear observable in most of the dust vortex flows is identified to have a definite exponent of dependence on the dust viscosity over a substantially large range of the latter. These results and scalings allow quantification of the notion of dusty plasma medium as a paradigm for a wide range of natural flow processes having scales inaccessible to ordinary laboratory experiments.

Figure 1

Schematic of the setup in the cylindrical geometry. Surface plot of an example effective confining potential $V(r,z)$ for the dust fluid (surface) is presented along with the flux conserving velocity field vector $\mathbf{v}$ of the unconfined driving fluid in the $r\text{−}z$ plane (arrows) and radial profile of its $z$ component $v_z\left(r\right)$ (2D plot).

Figure 2

Radial functions for the flow profiles with $I=1$ at $z=z_0=0$. (a) Driver vorticity ${\omega }_s$, (b) driver velocity $v_z$, (c) dust stream function $\psi$, and (d) dust flow velocity $u_z$ for $\mu =0.1U_0{L}_r$, $\xi ={10}^{-5}{U}_0/L_r$, and $\nu =0.1U_0/L_r$.

Figure 3

2D dust stream function $\psi (r,z)$ for single-eigenmode source vorticity with (a) $I=1$, (c) $I=3$, and (e) $I=5$. Dust flow stream lines, or the contours of the product $r\psi$, for single-eigenmode source vorticity with (b) $I=1$, (d) $I=3$, and (f) $I=5$.

Figure 4

Source flow velocity profile at $z=0$ for (a) $I=1$, (d) $I=3$, and (g) $I=5$. Dust flow velocity profile at $z=0$ for (b) $I=1$, (e) $I=3$, and (h) $I=5$. Intensity spectrum for the range of mode number $m$ at $z=0$ for (c) $I=1$, (f) $I=3$, and (i) $I=5$.

Figure 5

Radial profiles of dust velocity $u$ normalized to the corresponding maximum values $u_{\max }$ appearing at the the edge of the boundary layer whose width shrinks with decreasing $\mu$ values, ranging from ${10}^{-1}$ to ${10}^{-7}{U}_0{L}_r$ arranged on a logarithmic scale.

Figure 6

Intensity spectrum for the solutions with $I=1$, normalized to the intensity $a_1^2$ of the dust vorticity mode number $m=1$ for the cases with the value of viscosity coefficient $\mu ={10}^{-1},{10}^{-3},{10}^{-5}$, and ${10}^{-7}$.

Figure 7

(a) Width $\mathrm{\Delta }{r}_b$ of the boundary layer and (b) values of the Reynolds number $\mathrm{Re}$ for $I=1$ and the cases with various values of $\xi$.

Figure 8

(a) Dependence on $\mu$ of width $\mathrm{\Delta }{r}_b$ of the boundary layer with various values of $\xi$ for the cases with the driver mode number (a) $I=1$, (b) $I=3$, and (c) $I=5$. The broken line represents the dependence $\mathrm{\Delta }{r}_b\propto {\mu }^{1/3}$.

Figure 9

(a) Dependence on $\mu$ of the Reynolds number $\mathrm{Re}$ with various values of $\xi$ for the cases with the driver mode number (a) $I=1$, (b) $I=3$, and (c) $I=5$. The broken line represents the dependence $\mathrm{Re}\propto {\mu }^{-2/3}$.