Vortex Dynamics in Dusty Plasma Flow Past a Dust Void

Abstract

The beauty in the formation of vortices during flow around obstacles in fluid mechanics has fascinated mankind since ages. To beat the curiosity behind such an interesting phenomenon, researchers have been constantly investigating the underlying physics and its application in various areas of science. Examining the behavior of the flow and pattern formations behind an obstacle renders a suitable platform to realize the transition from laminar to turbulence. A dusty plasma system comprising of micron-sized particles acts as a unique and versatile medium to investigate such flow behavior at the most kinetic level. In this perspective, this chapter provides a brief discussion on the fundamentals of dusty plasma and its characteristics. Adding to this, a discussion on the generation of a dusty plasma medium is provided. Then, a unique model of inducing a dusty plasma flow past an obstacle at different velocities, producing counter-rotating symmetric vortices, is discussed. The obstacle in the experiment is a dust void, which is a static structure in a dusty plasma medium. Its generation mechanism is also discussed in the chapter.

1. Introduction

Vortices are common in fluid motion that originates due to the rotation of fluid elements. They occur widely and extensively in a broad range of physical systems from the earth’s surface to interstellar space. A few examples include spiral galaxies in the universe, red spots of Jupiter, tornadoes, hurricanes, airplane trailing vortices, swirling flows in turbines and in different industrial facilities, vortex rings formed by the firing of certain artillery or in the mushroom cloud resulting from a nuclear explosion. The physical quantity that characterizes the rotation of fluid elements is the vorticity ω = ∇ × u where u is the fluid velocity. Qualitatively, it can be said that in the region of vortex formation, the vorticity concentration is high compared with its surrounding fluid elements. Vortices formed behind obstacles to a fluid flow are also an interesting observation in various aspects of daily life. Study on the fluid flow around obstacles dates back to the fifteenth century when Leonardo da Vinci drew some sketches of vortex formation behind obstacles in flowing fluids. It has been an interesting and challenging problem in fluid mechanics and is of basic importance in several areas such as the study of aircraft designing, oceanography, atmospheric dynamics, engineering, human blood circulation [1, 2, 3, 4]. Analyzing the behavior of flow around such obstacles also provides a medium to study the physical mechanism of transition from laminar to turbulent flow. If a stationary solid boundary lies in the path of a fluid flow, the fluid stops moving on that boundary. Thereby, a boundary layer is formed and its separation from the solid boundary generates various free shear layers that curl into concentrated vortices. These vortices then evolve, interact, become unstable and detach to turbulence. The dynamics of fluids is very diverse and the detail characteristics of transition to turbulence are quite complicated, which also differ from flow to flow. Such understandings can only be realized by experiments and computational models. However, there are a few unifying themes in the theory and a few routes to turbulence that are shared by many flows. One such theme is that when the Reynolds number (the parameter measuring the speed of a class of similar flows with steady configuration) increases, the temporal and spatial complexity of the flows increases eventually leading to turbulence. At a low Reynolds number, a pair of counter-rotating vortices forms behind the cylinder. As the Reynolds number increases, the vortices become unstable and gradually evolve into a von Karman vortex street [5, 6, 7, 8, 9]. The topic of flow past an obstacle is of utmost importance from the experimental point of view also. Its understanding is applicable in the stability of submerged structures, vortex-induced vibrations, etc. [10].

Vortices have been extensively studied and explored in the liquid state of matter. However, scientists have also extended their research to study the formation and behavior of vortices in the fourth state of matter, the plasma. Measurements done in space have shown that plasma vortices appear in the earth’s magnetosphere as well as along the Venus wake. On both planets, the solar wind encounters different obstacles. For earth, it is the earth’s magnetic field and for Venus, the interaction takes place with the ionized components in the upper layer of the planet’s atmosphere. Plasma vortices in earth-based laboratories have also been studied theoretically and experimentally [11, 12, 13, 14, 15]. Plasma is said to cover more than 99% of the matter found in the universe and dust particles are the unavoidable, omnipresent ingredients in it. Hence, in most cases, plasma and dust particles exist together, and these particles are massive (billion times heavier than the protons). Their size ranges from tens of nanometer to as large as hundreds of microns. Foreign particles in the plasma environment get charged up by the inflow of electrons and ions present in the plasma. The presence of these charged and massive particles increases the complexity of the plasma environment, and hence, this class of plasma has been named as ‘complex plasmas’ or ‘dusty plasmas’. They involve in a rich variety of physical and chemical processes and are thus investigated as a model system for various dynamical processes [16, 17]. Phase transition is an important and characteristic feature in dusty plasmas, due to which it is considered as a versatile medium to study all the three different phases (solid, liquid and gas) in just a single phase. They also behave as many particle interacting systems and provide a unique platform to study various organized collective effects prevalent in fluids, clusters, crystals, etc., in greater spatial and temporal resolution. With the help of laser light scattering, it is possible to visualize the micrometer or nanometer-sized dust particles through proper illumination. This allows to study the various phenomena in dusty plasma in greater spatial and temporal resolution since they appear in a slower time scale owing to their heavier mass [18, 19]. Along with a variety of dynamic phenomena that includes waves, shocks, solitons, etc., dusty plasma medium also supports the formation of vortices. Self-generated vortices have been observed in many dusty plasma experiments, which have been dealt with significant attention. The main cause of such vortex formation is the nonzero curl of the various forces acting on the electrically charged dust particles that are commonly found in radiofrequency (RF) discharges, microgravity conditions and subsonic dusty plasma flow with low Reynolds numbers [20, 21]. The nonzero component of the curl induces a rotational motion to the charged dust particles, which leads to the formation of the vortices. Depending on the different plasma production mechanisms and dust levitation (floating of dust particles in the plasma medium), the causes of the rotation of dust particles vary accordingly. Most importantly, the problem of fluid flow around obstacles can be investigated at the most elementary individual particle level in dusty plasmas. The existence of a liquid phase of dusty plasmas provides us the suitable conditions for the study. However, the obstacles used for such study in dusty plasmas are different from the solid obstacles in the hydrodynamic fluid medium.

In this chapter, we will concentrate mainly on dusty plasmas, their characteristics and a model system to study fluid flow around an obstacle at the particle level. After the introductory portion in Section 1, the fundamentals of dusty plasma are discussed in Section 2. In Section 3, the production of a dusty plasma medium by RF discharge will be discussed. Then in Section 4, we will discuss about the type and behavior of the obstacle which is used in dusty plasma flows. In Section 5, we will discuss the dusty plasma flow and the pattern formation behind the obstacle. The final section then summarizes the chapter as a whole.

2. Fundamentals of dusty plasma

2.1 Dusty plasma

First, let us start with a very brief explanation of plasma! Basically, plasma is an assembly of a nearly equal number of electrons and ions, and the charge neutrality is sustained on a macroscopic scale. In the absence of any external disturbance, that is, under equilibrium conditions, the resulting total electric charge is zero. The microscopic space-charge fields cancel out inside the plasma and the net charge over a macroscopic region vanishes totally. The quasi-neutrality condition at equilibrium is given by,

$${\mathit{n}}_{\mathit{e}}\approx {\mathit{n}}_{\mathit{i}}$$E1

where ${\mathit{n}}_{\mathit{e}}$ and ${\mathit{n}}_{\mathit{i}}$ are the electron and ion densities, respectively.

The ‘plasma’ state of matter differs from ordinary fluids and solids by its natural property of exhibiting collective behavior. These collective effects result in the occurrence of various physical phenomena in the plasma, resulting in the long-range of electromagnetic forces among the charged particles. The very first example of plasma that is obvious to refer is the Sun, the source of existence of life. The protective layer to the earth’s atmosphere, known as the ionosphere, also remains in the form of ionized particles, that is plasma. Moreover, natural plasmas exist in interstellar space, stars, intergalactic space, galaxies, etc. On earth, the common form of natural plasma is lightning, fire and the amazing Aurora Borealis. Artificial plasmas are generated by applying electric or magnetic fields through a gas at low pressures. These are commonly found in street lights, neon lights, etc. Neon light is a gas discharge light, which is actually a sealed glass tube with metal electrodes at both the ends of the tube and filled with one or several gases at low pressure.

As already mentioned before, dust particles in space as well as in earth’s atmosphere, are unavoidable. These particles in plasma form a new field, that is dusty plasma or complex plasma. Dusty plasma is defined as a normal electron-ion plasma with charged dust components added to it. Naturally, dust grains are metallic, conducting, or made of ice particulates. Until and unless these are manufactured in laboratories, their shape and size vary. Depending on the surrounding plasma environments (due to the inflow of electrons and ions), dust particulates are either negatively or positively charged. These charged particles as a whole affect the plasma and result in collective and unusual behavior. When observed from afar, dust particles can be considered as point charges. As they are charged by the plasma species (electrons and ions), the charge neutrality condition is now modified, which is given by,

$${\mathit{Q}}_{\mathit{d}}{\mathit{n}}_{\mathit{d}\mathbf{0}}+\mathit{e}{\mathit{n}}_{\mathit{eo}}=\mathit{e}{\mathit{n}}_{\mathit{io}}$$E2

where ${\mathit{n}}_{\mathit{e}\mathbf{0}}$, ${\mathit{n}}_{\mathit{i}\mathbf{0}}$ and ${\mathit{n}}_{\mathit{d}\mathbf{0}}$ are the equilibrium densities of electrons, ions and dust, respectively, ‘e’ is the magnitude of electron charge, ${\mathit{Q}}_{\mathit{d}}\mathit{=}\mathit{e}{\mathit{Z}}_{\mathit{d}}$ is the charge on the dust’s surface and ${\mathit{Z}}_{\mathit{d}}$ is the dust charge number. It is important to highlight that the charge of the particles depends significantly on the plasma parameters. And the basic physics of the dusty plasma medium entirely rests on the ${\mathit{Q}}_{\mathit{d}}{\mathit{n}}_{\mathit{d}\mathbf{0}}$ term of the charge neutrality condition.

Plasma possesses the fundamental property of shielding any external potential by forming a space charge around it. This particular property provides a measure of the distance over which the influence of the electric field of a charged particle (dust particle in our case) is experienced by other particles (electrons and ions) inside the plasma. Typically, this length is known as the dust Debye length ${\mathit{\lambda }}_{\mathit{D}}$, within which the dust particles can rearrange themselves to shield all the existing electrostatic fields. The negatively charged heavier dust particles are assumed to form a uniform background and the electrons and ions, which are assumed to be in thermal equilibrium, simply obey the Boltzmann distribution. The dust Debye length is given by,

$${\mathit{\lambda }}_{\mathit{D}}=\frac{{\mathit{\lambda }}_{\mathit{De}}{\mathit{\lambda }}_{\mathit{Di}}}{\sqrt{\left({\mathit{\lambda }}_{\mathit{De}}^{\mathbf{2}}+{\mathit{\lambda }}_{\mathit{Di}}^{\mathbf{2}}\right)}}$$E3

where ${\mathit{\lambda }}_{\mathit{De}}$ and ${\mathit{\lambda }}_{\mathit{Di}}$ are electron and ion Debye lengths, respectively. These are expressed as,

$${\mathit{\lambda }}_{\mathit{De}}=\sqrt{\frac{{\mathit{k}}_{\mathit{B}}{\mathit{T}}_{\mathit{e}}}{\mathbf{4}\mathit{\pi }{\mathit{n}}_{\mathit{eo}}{\mathit{e}}^{\mathbf{2}}}}$$E4

$${\mathit{\lambda }}_{\mathit{Di}}=\sqrt{\frac{{\mathit{k}}_{\mathit{B}}{\mathit{T}}_{\mathit{i}}}{\mathbf{4}\mathit{\pi }{\mathit{n}}_{\mathit{io}}{\mathit{e}}^{\mathbf{2}}}}$$E5

${\mathit{T}}_{\mathit{e}\mathit{,}\mathit{i}}$ represents the electron and ion temperatures, respectively, ${\mathit{n}}_{\mathit{eo}\mathit{,}\mathit{io}}$ are the electron and ion densities, respectively, and ${\mathit{k}}_{\mathit{B}}$ is the Botlzmann constant.

A pictorial representation of a dusty plasma medium is shown in Figure 1.

Figure 1.

Schematic of a dusty plasma medium. The pink-shaded portion is the plasma medium. The green ball is the dust particle that is negatively charged. ${\mathit{\lambda }}_{\mathit{D}}$ is the dust Debye length.

In a dusty plasma medium, the charged particles interact with each other via the electrostatic Coulomb force. However, due to the inherent shielding property of the plasma electrons and ions, the charged particles are shielded and hence, the interaction energy among them is known as Screened Coulomb or Yukawa potential energy. Consider two dust particles having the same charge $Q_d$ and separated by a distance ‘$a$’. The screened Coulomb potential energy is given by,

$$P.E=\frac{Q_d^2}{4\pi {ϵ}_{0}a}{e}^{-\kappa }$$E6

where $\kappa =\frac{a}{{\lambda }_D}$ is the screening strength. The dust thermal energy is given by,

$$K.E=k_B{T}_D$$E7

where $T_D$ is the dust temperature. The ratio of the P.E to the K.E is termed as the Coulomb Coupling parameter, given by

$$\mathrm{\Gamma }=\frac{Q_d^2}{4\pi {ϵ}_{0}a{k}_B{T}_D}{e}^{-\kappa }$$E8

Depending on the coupling parameter, a dusty plasma system remains in a weakly coupled state or a strongly coupled one. When $\mathrm{\Gamma }<1$, the thermal energy of the dust particles is greater than the potential energy of the system and the system is said to be weakly coupled. On the other hand, when the potential energy exceeds the thermal energy, that is $\mathrm{\Gamma }>1$ the system becomes strongly coupled. So, from Eq. (8), we can see that dust charge, screening parameter and the dust temperature play an important role in determining the system’s coupling state. As $\mathrm{\Gamma }$ exceeds a critical value ${\mathrm{\Gamma }}_c$, called the critical coupling parameter, a dusty plasma system attains a crystalline state. However, this critical value for crystallization is dependent on the screening parameter [22]. For $1<\mathrm{\Gamma }<{\mathrm{\Gamma }}_c$, the system remains in a fluid (liquid or gas) state.

Thus, we see that by adjusting the dusty plasma parameters, we can obtain a fluid state of the dusty plasma medium experimentally. This provides us a unique model to study vortex formation behind an obstacle in the particle most level.

3. Production of a dusty plasma medium

Laboratory dusty plasmas differ from space and astrophysical dusty plasmas in a significant manner. The discharges done in the laboratory have geometrical boundaries. The composition, structure, conductivity, temperature, etc., of these geometries affect the formation and transport of the dust grains. Also, the external circuit, which produces and sustains the dusty plasma, imposes boundary conditions on the dusty discharge, which vary spatially as well as temporally. Dusty plasmas in the laboratory are generally produced by two main discharge techniques—direct current (DC) discharge and RF discharge. In this chapter, we will mainly focus on the production technique by RF discharge method in a DUPLEX device.

As the name suggests, DUPLEX is an abbreviation for Dusty Plasma Experimental Chamber. It comprises of a cylindrical glass chamber, 100 cm in length and 15 cm in diameter. The glass chamber configuration of the DUPLEX device provides a suitable and great access for optical diagnostics. One end of the cylindrical chamber is connected to the vacuum pump systems and the other end is closed by a stainless steel (SS) flange with Teflon O-ring between the glass chamber and the SS flange. On this closed end, there are ports for pressure gauge fitting, probe insertion and electrical connections. A radio frequency power generator (frequency: 13.56 MHz, power: 0–300 W) and an RF matching network are used for the plasma discharge. The RF antennas used in this setup are aluminum strips of 2.5 cm width and 20 cm length typically placed on the outer surface of the glass chamber. A schematic of the setup is shown in Figure 2. This strip acts as the live electrodes.

Figure 2.

Schematic of a DUPLEX setup. The pink-shaded portion is the argon plasma.

Initially, the chamber pressure is reduced to a value of about ∼10⁻³ mbar with the help of a rotary pump. Argon is used as the discharge gas, by injecting which the desired chamber pressure can be maintained. A grounded base plate is also inserted into the chamber (about ∼30 cm length, 14.5 cm width and 0.2 cm thickness), which acts as the grounded electrode and the region above it is selected as the experimental region. Applying a radiofrequency power (13.56 MHz and 5 W) between the aluminum strips (working as live electrodes) and the grounded base plate, a capacitively coupled RF discharge plasma is produced. Due to the application of the RF field, initially, the stray electrons inside the chamber get energized and in turn ionize the gas molecules present in the chamber. The aluminum strips used as live electrode outside give the flexibility to change the electrode position whenever required. Also, it facilitates in forming a uniform plasma over an extensive area of the grounded plate, that is the experimental region. The plasma parameters can be varied manually by tuning the discharge conditions, viz. RF power and neutral pressure.

Dust particles used in the experiment are gold-coated silica dust particles (∼ 5 micron diameter). These are initially put inside a buzzer that is fitted to the grounded base plate. After the production of the plasma, a direct current (DC) voltage of ∼ (6–12)V is applied to the buzzer, which ejects the dust particles from it through a hole. When these dust particles enter into the plasma environment, electrons and ions flow towards it and charge up the particles. In the laboratory, the dust particles are usually negatively charged as the electrons are lighter and highly mobile than the ions. These negatively charged dust particles are acted upon by two forces mainly, the upward electric field force (Q_dE) due to the sheath electric field (E) of the grounded base plate and the downward gravitational force (m_dg). Dust particles levitate at the position where these two forces exactly balance. The dust particles are illuminated by laser light scattering, and the dust dynamics are recorded in high-speed cameras. Figure 3 shows the levitation of dust particles in a plasma medium. Above the dust layer, the purple color signifies argon discharge plasma. The dark region below the dust layer and above the plate is the sheath (where ionization does not take place) where a strong electric field (E) is present. The charged dust particles levitate at the interface region (∼ 0.8 cm above the plate) between plasma and the sheath where the force balance occurs. This is shown by a dashed line.

Figure 3.

Photograph of a dust layer levitation in plasma.

4. Obstacle in dusty plasma flows

The obstacle used in dusty plasma flow experiments is actually a dust void. A void is a dust-free region, which is encountered spontaneously in certain experimental conditions or can be produced externally also [23, 24, 25, 26, 27, 28]. In the past couple of decades, there have been a few studies on the interaction of a dusty plasma medium with dust voids. In 2004, Morfill et al. studied a laminar flow of liquid dusty plasma with a velocity ∼ 0.8 cms⁻¹ around a spontaneously generated lentil-shaped void [29]. They observed the formation of a wake behind the void that is separated from the laminar flow region by a mixing layer. The flow also exhibited stable vortex flows adjacent to the boundary of the mixing layer. Another study was made in 2012 by Saitou et al. where they externally placed a thin conducting wire of 0.2 mm diameter and 2.5 cm length. They made the dust particles flow with velocity in the range ∼ (5–15) cms⁻¹ but did not observe any vortex formation behind the obstacle. What they observed was a bow shock in front of it [30]. In the very next year itself (2013), Meyer et al. also did a similar experiment with a different configuration and dust flow mechanism (velocity ∼ 10–25 cms⁻¹) than Saitou’s [21]. They produced a dust void by placing a 0.5-mm-diameter cylindrical wire transverse to the flow. They too observed a bow shock and a tear-shaped wake in front and behind the obstacle, respectively. Moreover, Charan et al. in 2016 did a molecular dynamics simulation study where they used a square obstacle and observed von Karman vortex street at low Reynolds number (i.e. low velocity) compared with normal hydrodynamic fluids [31]. Then in 2018, Jaiswal et al. investigated dust flow towards a spherical obstacle over a range of flow velocities ∼ (4–15)cms⁻¹ and different obstacle biases [32]. The spherical obstacle also generated a dust-free area in its vicinity. They too observed bow shock formation in front of the obstacle but no vortex formation behind it. In 2020, Bailung et al. also investigated the study of dust flow around a dust void with a unique flow mechanism (dust flow velocity ∼ 3–10 cms⁻¹) in a DUPLEX setup [33]. Dust particles are allowed to flow towards an already existing stationary dust layer. They could observe the formation of a counter-rotating pair of vortices behind the obstacle in a particularly narrow range of velocity ∼ (4–7) cms⁻¹. Above and below this range, their vortices are not observed. Due to the interplay between these two forces, a circular void is generated around the pin. At the void boundary, these two forces equate with each other.

In the next section, we will study the results of Bailung et al. in detail, but before that let us understand the mechanism of the formation of dust void due to the insertion of an external cylindrical wire. A cylindrical pin inside the plasma attains a negative potential for the plasma and a sheath is formed in its vicinity. Due to the negative potential of the pin, ions try to drift towards it giving rise to a force on the dust particles named as ion drag force. This force is directed radially inward with the pin as the centre. Also, the negatively charged dust particles experience a repulsive electrostatic force from the pin which is directed radially outward. The interplay between these two forces generates a circular void around the pin. At the void boundary, these two forces equate with each other. A typical configuration of pin insertion through the grounded plate of a DUPLEX chamber is shown in Figure 4. The pin is externally connected to a DC bias voltage. By varying the bias voltage, the size of the void can be altered according to experimental requirements. Typically, at a RF power of 5 W and chamber pressure ∼ 0.02 mbar, the diameter of the dust void in floating condition (i.e. no external bias) is ∼1.7 cm. A typical example of a dust void is shown in Figure 5. However, unlike the solid obstacles in the case of hydrodynamic fluids, the dust void is not a rigid kind of obstacle. As already seen, the boundary of the void is maintained by a delicate force balance between the outward electrostatic force and inward ion drag force. An incoming dust flow, depending on the velocity of the flow, would cross the void boundary and penetrate into the void.

Figure 4.

A typical configuration for insertion of a pin through a grounded base plate in DUPLEX chamber.

Figure 5.

Snapshot of a dust void formed in DUPLEX chamber. The bright spot in the Centre is the reflection of laser light from the cylindrical pin. The photograph is taken from the top of the chamber.

5. Vortices in the wake of a dust void

Due to the non-rigidity of the dust void boundary, the behavior of the flow near the obstacle is somewhat different than conditions of hydrodynamic fluid with a rigid obstacle. Despite this difference, the transition from laminar to turbulence is observed in the wake of the obstacle in the case of dusty plasma flow also. As the flow approaches the void boundary, the middle section of the flow slightly penetrates into the void region and slips through the void boundary layer on both sides. The trajectory of the flow (in the mid-section) is deflected in front of the void due to the repulsive force exerted by the sheath electric field of the void and then flows downstream surrounding the void. The curved dust flow again meets behind the void and continues with the flow. As observed by Bailung et al. at a very narrow range of velocity ∼ (4–7) cms⁻¹, a counter-rotating vortex pair is seen to appear. Below and above this range, the dust particles do not form any vortices. A typical example of three different conditions is shown in Figure 6.

Figure 6.

Typical snapshots showing structures formed behind the void at (a) 3.5cms⁻¹, (b) 4.5cms⁻¹, (c) 8 cm⁻¹.

In each of the images, dust particles flow from right to left shown by dashed arrows. The top image (a) depicts a flow with dust flow velocity ∼ 3.5 cms⁻¹ and the snapshot is at time t = 1370 ms from a reference time (t = 0, when dust flow reaches the right edge of the images). The middle image (b) shows dust fluid flow velocity ∼ 4.5 cms⁻¹ at t = 1033 ms showing a vortex pair formation behind the void. The vortices are shown by the two arrow marks. It is observed that vortices are not formed for larger flow velocity ∼ 8 cms⁻¹ (image (c)). For such high velocities, flow trajectories behind the void are elongated and dynamics in the wake is rather complex due to cross-flow at high speed. The bright illuminated point at the centre of each image is the reflection of laser light from the pin. Two horizontal lines that appear in all the images are due to laser reflection from the wall of the glass chamber. It is noted that dust flow with unsteady laminar velocity, which is (4–7)cms⁻¹, and optimum dust density in the experimental region above the grounded plate is required to generate the vortex behind the void.

For a better understanding, a pictorial representation showing the dusty plasma streamlines around dust void at three different velocities are shown in Figure 7(a)–(c) of Figure 7 corresponds to the observation shown in (a), (b) and (c) of Figure 6.

Figure 7.

A pictorial illustration of the dusty plasma flow interaction with the dust void at different flow velocities. (a) Laminar flow, (b) unsteady laminar flow with filamentary vortex-type structure in the upstream and vortex pair in the downstream and (c) turbulent flow.

At a lower dust flow velocity, the void in the upstream is slightly compressed and trajectories of the streamlines flowing close to the void (boundary layer) curl behind the void. However, no structure formation in the wake appears here. Dust particles, after meeting behind the void, just continue with the flow smoothly. For critical flow speed (b), flow dynamics in the upstream void boundary is quite different. Streamlines that hit perpendicularly at the void flow some distance into the void region. They reconstruct the boundary during the flow and get ejected backward making the streamline bifurcation to occur much ahead of the void boundary. The curved streamlines, which are ejected backward, again flow along with the incoming dust flow close to the boundary layer. This critical reorientation in the front of the void generates a suitable condition for the formation of the vortex pair behind the void. Particles get slowed down in this region and these slower particles flow close to the boundary layer around the void and contribute in the formation of the vortex pair. At higher velocities (c), that is above the critical range for vortex formation, all the particles that hit the upstream void boundary are flushed away by the flow along with it. The streamlines intersect and crossover at a distance far behind the void and there is no formation of any stable structure. It is well known that in hydrodynamic fluids, at much higher velocities, vortex streets are observed. However, here such streets are not observed to form. This may be due to the restriction of the experimental geometry. The transition from laminar to turbulence is well known in fluid dynamics. But studying it in dusty plasma provides the chance to observe the individual particle-level trajectory. In turn, the dynamics can be studied in greater detail.

To see the dust dynamics in greater detail, let us look at the vortex formation behind the dust void step by step. At the outset of the formation, the slower particles moving along the curved boundary layer interact with the stationary particles behind the void and start to swirl on each side. The flow front then meets in the wake region behind the void (Figure 8(a)) and gradually traverses a swirling circular path. This is evident in the dotted arrow marks in (Figure b). After duration of 966 ms from the start of the flow, two counter-rotating vortices complete their formation (Figure c). Only the slower particles flowing close to the boundary layer participate in this swirling motion due to the nonzero curl of the forces. Those particles away from the boundary layer move faster and do not contribute to the swirling. With increasing time and inflow of more particles, the swirling finally grows into a distinct pair of the vortex with an eye in the middle (Figure (d)). As the flow progresses by maintaining a constant inflow of particle flux, the vortex pair sustains till 1167 ms. The one shot of dust flow in the experiment done by Bailung et al. lasted for about 2 sec.

Figure 8.

The parallel arrows depict the direction of the dust flow. (a) when both the oppositely curling flow front meet behind the void. Dotted curve traces in (b) indicate flow trajectories. The arrows in (c) - (g) show the vortex pair. the vortices vanish with time when flow is nearly over (h).

Hence, gradually when the particle flux started decreasing, the vortex pair starts to die out. It is faintly visible till 1233 ms (Figure g). The time for the growth of the vortex pair is ∼200 ms (from the time the particles meet behind the void) and survives for duration of 200 ms (depending on the duration of accelerated dusty plasma fluid flow). Finally, they disappear after 1300 ms. The rotational frequency measured for the vortices is about ∼3 Hz.

It is already mentioned that the advantage of studying vortex dynamics in dusty plasma lies in the fact that particles can be individually tracked. Different particle tracking software and computational models are available, which can generate the velocity vectors of the trajectory of the particles and hence can give a quantitative interpretation of the experimentally observed results. One such particle tracking platform is OpenPIV (Open Particle Image Velocimetry) in MATLAB [34]. This helps to study the evolution of the vortex pair along with its vorticity. But to perform successful PIV from images, the recorded videos of the dust flow dynamics should have a high-quality resolution and must be in high speed. A PIV analysis performed on a video recorded at 100 frames per second is shown in Figure 9.

Figure 9.

PIV analysis showing the time evolution of the vortices for a duration of 1 sec. The velocity vectors and vorticity profile drawn from (a) (0.53-0.62) sec (b) (0.63-0.72) sec (c) (0.73-0.82) sec (d) (0.83-0.92) sec (e) (0.93-1.02) sec (f) (1.03-1.12) sec (g) (1.13-1.22) sec (h) (1.23-1.32) sec (i) (1.33-1.42) sec (j) (1.43-1.52) sec are shown. The color bar shows the value of vorticity in 1/s. The dotted circle in (a) shows the original position of the void boundary before the flow and the dot at the center of the circle depicts the pin position.

Each image in the figure is an average PIV result of 10 consecutive image frames. The position of the void and the pin position are drawn by a red-dashed circle and a red dot, respectively. The velocity vectors show the trajectory of the dust particles and the color code gives the value of the vorticity at different times in units of s⁻¹. The slowing down of the particles in front of the void is clearly seen by comparing the velocity vectors’ lengths in Figure 9(a) and (b). The backflow of the incoming dust particles mentioned earlier (due to repulsive sheath electric field force of the pin) is also observed in (b). The curling of dust particles leading to vortex formation is evident from (c) and (d). The vorticity of the fully formed vortex pair is about ∼ (20–25)s⁻¹, which is shown by the color bar in (e) and (f)). This is nearly equal to twice the measured angular frequency. With the decrease of the dust flow influx, the vortex structure deforms (vorticity ∼15 s⁻¹) and breaks away into smaller vortices (vorticity ∼10s⁻¹) as seen from (g) to (i). Vortices finally disappear in (j), evident from the vorticity value which almost tends to 0.

Reynolds number is the characteristic parameter that helps to predict flow patterns. It is the ratio of the inertial forces to the viscous forces and is given by,

$$\mathit{Re}=\frac{\rho v_{d}L}{\eta }$$E9

where $\rho$ is mass density, $v_d$ is dust velocity, $L$ is the obstacle dimension, that is the void diameter and $\eta$ is the viscosity of the dust fluid.

In case of dusty plasma fluids, the viscosity is estimated from the formula,

$$\eta =\sqrt{3}\widehat{\eta }{m}_d{n}_d{\omega }_E{a}^2$$E10

where $\widehat{\eta }$ is the normalized shear viscosity, $m_d$ is the dust mass, $n_d$ is the dust density, ${\omega }_E=\left({\omega }_{\mathit{pd}}/\sqrt{3}\right)$ is the Einstein frequency, ${\omega }_{\mathit{pd}}$ is the dust plasma frequency and $a$ is the interparticle distance. The normalized shear viscosity in dusty plasma fluid is a function of the Coulomb coupling parameter $\mathrm{\Gamma }$, which has been estimated for a range of coupling parameters in different conditions via simulation [35, 36]. For typical plasma parameters of DUPLEX chamber, that is,

$$m_d=1.7\times {10}^{-13}\mathit{kg}$$

$$n_d=9\times {10}^9{m}^{-3}$$

$${\omega }_{\mathit{pd}}=247.5s^{-1}$$

$$a=3\times {10}^{-4}m$$

The viscosity is calculated to be $9\times {10}^{-9}$ Pas.

Thus, the Reynolds number for dust flow velocity ∼ (3–10) cms⁻¹ is estimated to be lying in the range 50–190. The vortex pair formation appears in a critical range of 60–90.

In the case of hydrodynamic fluid, the range of Reynolds number for vortex formation is 5–40, which is much lower compared with that in dusty plasma fluid. This is because the ratio $\rho /\eta$ (which is the kinematic viscosity) is one order larger in the case of dusty plasma fluids than that in hydrodynamic fluids. The estimated kinematic viscosity for dusty plasma fluids is ∼0.088 cm²s⁻¹, whereas the kinematic viscosity for water is ∼0.008 cm²s⁻¹_.

6. Conclusion

The study of vortices in the problem of flow past an obstacle is significant as it provides a platform to investigate the transition from laminar to turbulence. Formation of vortices in the wake region behind an obstacle appears in the unsteady laminar regime of flows and has been widely studied in hydrodynamic fluids. However, dusty plasma medium, which is a component of the fourth state of matter, provides a unique stage to study such phenomena at the particle level. A special property of this medium is that it can remain in both fluids (liquid- or gas-like) as well as the crystalline state. By mere adjustment of plasma conditions, the desired state can be obtained. The individual tracking of micron-sized dust particles by methods such as PIV (Particle Image Velocimetry) yields the particle trajectory in form of velocity vector fields. This gives a very clear picture of the behavior of flow near obstacle boundaries. However, the obstacles used in dusty plasma flow experiments differ from those in hydrodynamic fluid experiments in the sense that unlike those in hydrodynamics, the obstacle boundaries in dusty plasma are non-rigid. Any foreign pin or wire inserted into the plasma would possess a negative potential with respect to the plasma. Dust particles in its vicinity are repelled due to electrostatic force and form a dust-free region around it, called the dust void. This dust void, whose boundary is delicately maintained by dusty plasma forces, acts as a non-rigid type of obstacle. Dusty plasma flows also generate counter-rotating vortices in the wake region behind a dust void at a particular range of velocities. Below and above this range, no structure formation is seen to appear. The particle behavior causing the formation of the vortices is better understood by tracking particles in consecutive frames. The estimated Reynolds number value for vortices to appear in the wake of a void in a dusty plasma medium is estimated to lie in the range 60–90. This is quite larger than the Reynolds number range for hydrodynamic fluids which is roughly about 5–40. This higher range in dusty plasma medium is attributed to the higher kinematic viscosity of dusty plasma fluids. However, in dusty plasma experiments, Von Karman vortex streets (observed in the turbulent regime of hydrodynamic fluids) are not yet explored. If such experiments could be successfully performed, then there will be immense scope of understanding turbulence at the particle-most level and with a better perspective. Although to study turbulent dynamics, high-speed cameras with high-quality resolution would be necessary.