1 Introduction

Corona discharge is a phenomenon that occurs when there exists a high enough potential difference between a sharp conductive electrode (i.e., thin wire, needle, or sharp-tipped objects) and a ground electrode. This is a low-current electrical discharge that creates ions around the sharp (corona-generating) electrode, carrying them toward the ground electrode. The corona discharge has been an important research interest since the late nineteenth century [1, 2] followed by further growth and a more comprehensive understanding of its underlying physics in the twentieth century [3,4,5,6]. The ionization of the gas (i.e., air or other inert gases) medium via the corona discharge enables its application in combustion [7, 8], ozone production [9, 10], reduction of gas pollutants [11, 12], and surface treatment [13] among others. Other applications of the corona discharge arise from its ability to manipulate particles in electrostatic precipitators [14,15,16] or to excite flow in air [17, 18]. This is often referred to as the electrohydrodynamic (EHD) property of the corona discharge. Previous studies have focused on the traveling wave induction pumps [19] and ion drag pumps [20] which are all examples of a single-phase EHD pump.

However, recent studies have explored a two-phase EHD pump, which is essentially a gas–liquid phase pump [21, 22]. In the two-phase EHD pump, placing the ground electrode inside a dielectric medium (i.e., silicone oil) while maintaining the corona-generating (i.e., needle) in air leads to a momentum transfer at the gas–liquid interface [23]. This transfer of momentum results in the dragging and pumping the dielectric liquid [20]. An advantage of the two-phase dielectric fluid pump over a one-phase one is its contact-less nature. It means there is no need for any moving mechanical component which might intrusively perturb the flow during the liquid manipulation. While this is a power-efficient fluid manipulation, it has certain disadvantages as well [23, 24]. For example, working with the two-phase EHD pump adds the complexity of evaluating a Coulombic force [25] as well as forces arising from the interfacial momentum transfer effects [26]. Previous studies that discussed the two-phase EHD pumps have focused on transferring the fluid outside of the pumping container with potential application in a contact-less transfer of hazardous waste [20,21,22]. However, characteristics of the flow and its dependence on the fluid properties and the operating parameters of the corona discharge have not been fully investigated.

Here, a better understanding of fluid characteristics by the EHD pump induced via the corona discharge is investigated using particle image velocimetry (PIV). In this study, a circular transparent container made in-house is used to examine the characteristics of silicone oils with varying viscosities under different corona operating voltages. A power law correlation between fluid characteristics (i.e., Reynolds number) and corona operating voltage is found. In addition, an analytical solution is derived, verifying the acquired PIV data. The results of this study pave the path for controlled fluid manipulation (EHD pumping) via corona discharge for its potential application in emulsion formation and separation.

2 Materials and methods

Silicone oils with kinematic viscosities (\(\nu\)) of 50 cSt (SO50) and 100 cSt (SO100) from \(\mu\)MicroLubrol are used as the dielectric fluids of which their properties are provided in Table 1. The electrical conductivity values were verified in Ref. [27] with a rudimentary setup using standard 4-mm electrophoresis cuvettes as test cells. A pumping container is made of two petri dishes with diameters of 60 mm and 90 mm. The smaller petri dish is inverted and placed concentrically inside the larger one to create a circular channel within which the silicone oil will flow. Figure 1 shows a schematic of the setup.

Fig. 1
figure 1

A schematic of the experimental setup a top view showing all components along with the 2D polar coordinate system (\(r,\theta\)) originating from the center of the circular pumping container, flow profile and direction, and b showing the side view, for clarification. The needle is 7 mm away from the oil surface, and the oil has a thickness of 14 mm (for all experiments). Schematic is not to scale

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The setup housing the corona-generating tungsten needle/electrode and the copper ground electrode is placed inside a transparent acrylic box to provide optical access while enhancing the safety of the experiments. The corona-generating needle is positioned at a vertical distance of 7 mm from the silicone oil and at a horizontal distance of 15 mm from the copper ground electrode, to avoid any electric arc formation. Such a placement prevents the formation of cone deformation [27], and instead results in an almost unnoticeable dip. Furthermore, the needle tip was polished using a 2000 grit sandpaper every six months. Figure S1 shows the needle tip images taken 6 months apart, using an optical microscope. The tip diameter decreased from 50 to 44 \(\upmu\)m, which can be attributed to the polishing with sandpaper. The power source encompasses a ± 10 kV TREK Model 10/10B high-voltage amplifier, which is coupled to a Keithley 2100 digital multimeter for recording voltages and currents (which were actively monitored and recorded at the needle electrode). The amplifier mentioned above is connected to a benchtop DC power supply (BK 1698).

For the seeding purpose, the silicone oil is mixed with 10 \(\upmu\)m diameter hollow glass spheres from Potters Industries. Such seeding material has no/minimal effect on the motion of the dielectric fluid. Gouriou et al studied the effect of three different seeding materials (glass, PMMA, PTFE) in silicone oil [28]. Although glass seeding was shown to hold the highest maximum charge (based on Pauthenier theory [29]), the mean velocity measured using all three particle types was consistent, revealing that the velocity is not sensitive to the particle type. Prior to any experiment, the mixture of oil and seeding particles is sonicated in a water bath for 10 min followed by manual shaking for 5 min. This ensures homogeneous seeding distribution in the oil. For illuminating the seeded oil, a light sheet is formed using a 4-W CW laser at 532 nm. A set of spherical and cylindrical lenses from Edmund Optics are used to ensure a collimated laser sheet of approximately 1 mm in thickness. For imaging, a high-speed camera (Olympus TR i-Speed) with a sensor size of 1280 \(\times\) 1024 pixels is used at a frame rate of 30 Hz. Due to the size of the high-speed camera, it was not possible to mount it vertically (looking down) on the flow. Instead, a circular mirror (Plymor Round 3mm Beveled Glass Mirror, 3-inch \(\times\) 3-inch) at 45\(^{\circ }\) position is used, while keeping the high-speed camera fixed on a tripod. The SO50 and the SO100 are poured into the circular pumping container at a thickness of 14 mm and their pumping characteristics are examined at four different DC corona operating voltages of 7, 8, 9, and 10 kV. Here, a positive corona is chosen to induce streamers in front of the needle electrode, which does not occur when using a negative discharge [30]. As a result, more inertia is added to the fluid, allowing for a better pumping performance/higher induced velocities. In addition, the current that is measured at different applied voltages follows a polynomial fit of second order (Fig. S2), confirming a Townsend regime [31]. The consumed power is also obtained by multiplying the current and voltage as shown in Fig. S3. The imaging location is chosen 180\(^\circ\) away from the corona-generating needle/electrode to avoid interference of the oil deformation (right underneath the needle) on the fluid characterization [27]. Due to such deformation and sloshing motion of the oil surface at higher voltages, the laser sheet is adjusted to be 2 mm below the top surface of the oil. Prior to imaging, the oil is left to circulate for 3 min in order to avoid any transitional effects. For each experimental run, a sequence of 2048 frames is imaged at a frequency of 30 Hz, and then the velocity statistics are calculated. The PIV analysis is performed using PIVlab (an opensource MATLAB package) [32]. All data is processed using Python 3.9.5 in a Jupyter–Notebook environment [33] and plotted using Python’s Matplotlib library [34].

Table 1 Properties of the SO50 and SO100 silicone oils
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3 Results and discussion

3.1 Experimental

Figure 2 shows azimuthal velocity contour plots and their corresponding vector fields at corona voltages of 7, 8, 9, and 10 kV for 50 cSt silicone oil (SO50).

Fig. 2
figure 2

Azimuthal velocity contours and vector field at different corona voltages of 7, 8, 9, and 10 kV for silicone oil with 50 cSt kinematic viscosity (SO50). The increase in the operating voltage leads to higher velocities in the oil with enhanced fluctuations at the center of the channel. For a better visibility, only every third vector is in shown in both directions

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Video S1 shows azimuthal movement of SO50 under corona voltage 10 kV. It can be seen from the four subplots that SO50 experiences highest overall velocity under 10 kV voltage, while its velocity descends monotonically when the voltage drops from 9 to 7 kV. This is due to the speed at which the ionized particles exchange charges with neighboring particles, which are later injected into the dielectric fluid (silicone oil) [35]. With a higher voltage, the EHD pump can pump and circulate the fluid inside the pumping container much more efficiently. In addition, the injection of such charged particles into the dielectric fluid at high voltages allows for disturbing the interface in the outwards plane. The contour subplot at 10 kV shown in Fig. 2 indicates concentric circles of high magnitude in the middle of the circular channel. The same behavior with less intensity is observed when the oil is pumped with 9 kV corona voltage. Despite constant input voltage, flow accelerates towards the needle, and such concentric regions are consistent in location at the highest voltage. Flow is not fully developed as the channel does not have a large enough length. These concentric regions are sources of high fluid fluctuations. This can be seen in Fig. S7. Similar to Fig. 2, the contour plots of the azimuthal velocity are shown, but instead of overlaying the vector fields on top, the standard deviation of the fluctuations, also referred to as rms (root mean square), is normalized by the bulk velocity \(V_b\) and plotted as iso-lines. It can be seen for high voltages (i.e., 9 and 10 kV) that the high rms values exist around the regions of high-velocity magnitudes (concentric regions mentioned earlier), whereas the rms for lower voltages (i.e., 7 and 8 kV) despite it being high, is mostly contaminated by the near-wall noise.

Additionally, a Fast Fourier Transform (FFT) is conducted on the collected data. Fig. 3 shows the power spectral density (PSD) of the azimuthal velocity component for SO50 and SO100 under various applied voltages of 7, 8, 9 and 10 kV.

Fig. 3
figure 3

The normalized power spectral density (PSD) plots of azimuthal velocity at four voltages for SO50 and SO100. Both subplots show high-magnitude frequencies below 2 Hz which contribute to the sloshing motion. The 11.3 Hz peak associated with the noise floor becomes more prominent for the lower viscosity oil (SO50)

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It is to be noted that the maximum retrievable frequency (Nyquist frequency) is 15 Hz since the flow is imaged at 30 Hz. Both normalized subplots exhibit high magnitudes at frequencies less than 2 Hz. This is attributed to the sloshing motion of the oil surface as it circulates under high applied voltages (i.e., 9 kV or higher). These low-frequency peaks start to diminish by a decrease in the operating voltage, which agrees with the previously observed contour figures, where there is a decrease in the concentric fluid motion at lower voltages. It can be seen that the overall magnitudes of normalized PSD of SO50 are higher than those of SO100, which is expected as SO100 is twice as viscous as SO50. At higher frequencies, a discernible peak is observed at \(\sim 11.3\) Hz in all cases; however, such a peak is much more prominent for SO50 than SO100. The consistent appearance of this peak suggests that its origin might be due to some component in the setup that is vibrating at \(\sim 11.3\) Hz, despite ensuring its firm fixture. To verify this, the silicone oil is imaged in its natural state, without applying any voltage/corona discharge. Figure S4 shows the normalized PSD of the data collected from a stationary SO50 without applying any corona discharge using the velocity field as shown in Fig. S5. U and V are representative of the velocities in the X and Y directions, respectively. Similarly, a strong peak at \(\sim \, 11.3\) Hz is observed only for the velocity component in the Y direction (V). This PSD plot agrees with the initially proposed hypothesis that the source of this peak is of a certain natural frequency. The normalized PSD was plotted for azimuthal and radial directions, but both velocity components showed a peak at \(\sim \, 11.3\) Hz, and unlike Fig. S4, there was no observable difference between the two PSD plots of azimuthal and radial velocity components. Thus, it was necessary to plot the stationary SO50 PSD plot in Cartesian coordinates as that proved to be more insightful. Additionally, because of the dominating viscous forces, there is no sloshing motion, which is evident in the absence of any noise below 2 Hz.

Figure S5 shows the velocity contour plot of SO50 without applying any corona discharge. There is minimal circulation, as the fluid is traveling in both azimuthal and radial directions, with the counterclockwise direction (towards the needle) being slightly more dominant. It is, however, traveling exclusively in the radial direction, from the inner edge of the channel to the outer side (Video S2). The remaining charged ions in the fluid are speculated to cause this natural convection motion. At a high enough corona voltage applied, the oil acquires enough inertia to overcome this radial convection. This \(\sim \, 11.3\) Hz peak is speculated to arise from conditions of the laboratory, upon which these experiments are conducted. Figure S6 shows the azimuthal velocity contour plots of a 100 cSt silicone oil (SO100) under the same corona voltages of 7, 8, 9, and 10 kV. Similar to the SO50 silicone oil, the SO100 experiences a circular motion that is intensified by an increase in the corona voltage. However, the maximum velocities experienced by SO100 are lower than those of the SO50 under the same operating voltage. As mentioned earlier, this is due to SO100 being twice as viscous as SO50, while having comparable dielectric constants of 2.73 and 2.71, respectively. The previously discussed fluctuations of the silicone oil velocity at higher corona voltages are observed for SO100 as well, however with less intensity. Similarly, the normalized rms by the bulk velocity \(V_b\) is plotted as contour lines over the filled contour plots of azimuthal velocity in Fig. S8. Like Fig. S7, high rms values exist around the regions of high-velocity magnitudes (concentric circles), whereas the rms for the lower voltages (i.e., 7 and 8 kV) despite it being high, is mostly contaminated by the near-wall noise.

The mass conservation at two different locations of the circular channel is used to examine the accuracy of the PIV data. Two distinct lines (line 1 and line 2) are arbitrarily chosen across the channel as shown in Fig. 4a. Figure 4b and c show the plots of flow rates at line 2 (FR2) versus line 1 (FR1) for SO100 and SO50 under various corona voltages, respectively.

Fig. 4
figure 4

a Top view of the circular pumping channel with two lines (line 1 and line 2) arbitrarily chosen at different downstream locations. b, c Plots of flow rates at line 2 (FR2) vs. line 1 (FR1) for SO100 and SO50 silicone oils under various corona voltages, respectively. The FR2 = FR1 line is indicative of the mass conservation, and the accuracy of the PIV measurements in the absence of any sloshing (out-of-plane) motions

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Each point in the scatter plots corresponds to the flow rate passing through a cross section plotted against that of the other cross section and calculated from one PIV realization for SO50 and SO100 under corona voltages 7, 8, 9 and 10 kV. Appearance of the measured (scattered) data along the FR2 = FR1 line is indicative of the mass conservation and the accuracy of the PIV measurements under no sloshing (out-of-plane) motions.

To examine the fluid characteristics further, Reynolds number, \(Re=\frac{V_b h}{\nu }\), is calculated for each of the experiments, taking the oil depth (h) which remained constant for every experimental run (14 mm) as the characteristic length scale, and the velocity as the bulk velocity (\(V_b\)). Table S1 lists the bulk velocity as well as the Reynolds number in each experiment. Plotting Reynolds number as a function of corona voltage exhibits a power law fit, as plotting both axes in a logarithmic scale showed a linear trend with \(R^2=0.967\) and \(R^2=0.992\) for SO50 and SO100, respectively. This is suggestive of \(Re \propto mV^n\) (as shown in Fig. 5), where Re is the Reynolds number, V is the corona operating voltage, and m and n are constants of approximate values \(1.31\times 10^{-6}\) and \(2.99\times 10^{-6}\) for m, and 5.84 and 5.07 for n. Such values correspond to SO50 and SO100, respectively. Of note, it is evident from Fig. 5 that a threshold voltage is required to induce an azimuthal motion in the silicone oils depending on the oil viscosity. In addition, by extrapolation, this power law correlation may be cautiously used to obtain the required operating voltages to induce desired velocities in given silicone oils/dielectric fluids.

Fig. 5
figure 5

Reynolds number versus the imposed voltage for both oils (SO50 and SO100), indicating a power law correlation

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3.2 Analytical

To scale the proposed setup, it is imperative to derive an analytical solution of the corona-induced EHD flow inside the channel. The highest Re number measured is 0.99, which is for SO50 under 10 kV corona voltage (Table S1). This gives a maximum Dean number of \(De=Re \sqrt{\frac{r_2-r_1}{r_2}} = 0.52\), where Re is the Reynolds number, and \(r_1\) and \(r_2\) are the inner and outer radii of the curved channel, respectively. This value of Dean number indicates almost no secondary flow (i.e., Dean vortices) [36]. A polar coordinate system (\(r,\theta ,z\)) is chosen to derive the analytical solution as shown in Fig. 1. The radial direction (r) originates from the center of the circular channel, and the azimuthal direction (\(\theta\)) is around the pumping container, and the normal direction (z) points out of plane.

Starting with the continuity equation with three velocity components \(\overrightarrow{V}=v_r,v_{\theta },v_z\)

$$\begin{aligned} \frac{1}{r}\frac{\partial (rv_r)}{\partial r} + \frac{1}{r}\frac{\partial v_{\theta }}{\partial \theta } + \frac{\partial v_z}{\partial z} = 0, \end{aligned}$$
(1)

and simplified momentum equation

$$\begin{aligned} \rho \frac{D\overrightarrow{V}}{Dt} = -\nabla P + \rho \overrightarrow{g} + \mu \nabla ^2 \overrightarrow{V}, \end{aligned}$$
(2)

where \(\rho\) is the density, P is the pressure, g is the gravitational acceleration, D is the total derivative operator, and \(\mu\) is the dynamic viscosity. To simplify the equations, the flow is assumed to be incompressible and to have reached steady state. The centripetal velocity component (i.e., the radial velocity) \(v_r\) is set to be zero as it has negligible effect on the flow which is predominantly azimuthal. The out-of-plane velocity component can be also neglected. However, it is to be noted that this assumption is not valid on the oil surface due to the sloshing motion observed in the oil under corona voltages of 9 and 10 kV. Lastly, the change of pressure (P) in the azimuthal direction (\(\theta\)) is assumed to be constant due to the constant DC nature of the corona voltage. Given very small Reynolds numbers, the pressure gradient along the radial direction induced by the azimuthal velocity is neglected. Below is the summary of the assumptions mentioned earlier:

$$\begin{aligned}{} & {} \frac{\partial v_{\theta }}{\partial \theta } = \frac{\partial v_{\theta }}{\partial z} = \frac{\partial v_{\theta }}{\partial t} = 0, \end{aligned}$$
(3)
$$v_z = v_r = 0,$$
(4)
$$\begin{aligned}{} & {} \frac{\partial P}{\partial r} = \frac{\partial P}{\partial z} = 0, \end{aligned}$$
(5)
$$\begin{aligned}{} & {} v_{\theta },\frac{\partial v_{\theta }}{\partial r},\frac{\partial P}{\partial \theta } \ne 0. \end{aligned}$$
(6)

After applying the above assumptions, the momentum equations in the r and z directions are eliminated with the following simplified momentum equation in the \(\theta\) direction:

$$\begin{aligned} \frac{d^2 v_{\theta }}{dr^2} + \frac{1}{r}\frac{dv_{\theta }}{dr}-\frac{v_{\theta }}{r^2} = \frac{1}{r\mu }\frac{dP}{d\theta }. \end{aligned}$$
(7)

This is an Euler–Cauchy differential equation that can be resolved analytically, with the following solution for the velocity:

$$\begin{aligned} v_{\theta }(r) = c_1r+\frac{c_2}{r} + \frac{r}{2\mu }\frac{dP}{d \theta } \Big ( \ln {r} - \frac{1}{2} \Big ). \end{aligned}$$
(8)

Assuming no-slip condition at the channel walls, the following boundary conditions can be imposed to solve equation 8 and to obtain the two constants \(c_1\) and \(c_2\):

$$\begin{aligned} v_{\theta }(r_1) = v_{\theta }(r_2) = 0. \end{aligned}$$
(9)

The final solution for \(v_{\theta }(r)\) becomes:

$$\nu _{\theta } \left( r \right) = \left( {\alpha\, r^{2} \ln r - r_{1}^{2} \left( {r^{2} - r_{2}^{2} } \right)\ln\, r_{1} + r_{2}^{2} \left( {r^{2} - r_{1}^{2} } \right)\ln\, r_{2} } \right)\frac{G}{\alpha\, r},$$
(10)

where \(\alpha =r_1^2-r_2^2\) and \(G=\frac{1}{2\mu }\frac{dP}{d\theta }\). The constant G depends on the fluid’s viscosity and the pressure gradient (\(\frac{dP}{d\theta }\)) that is induced by the corona discharge. To find G for a given corona voltage, \(v_{\theta }(r)\) from Eq. 10 is integrated from \(r_1\) to \(r_2\) to calculate the flow rate across the channel width per unit centerline length and equated with the experimental value obtained from the PIV measurements. The azimuthal velocity \(v_{\theta }(r)\) from Eq. 10 is then normalized by the bulk velocity \(V_b\) and is plotted as a function of (\(r-r_1\)), as shown in Fig. 6 (black dotted line). The same is applied for the velocities obtained from the PIV measurements for SO50 and SO100 at corona voltages 7, 8, 9, and 10 kV. The normalized velocity profiles for SO50 and SO100 all have a peak of approximately 1.5 at the core of the channel. This is in close agreement with the analytical solution. The collapse of the parabolic velocity profiles for all cases (SO50 and SO100 under various corona voltages) is indicative of a scaling effect of the corona voltage on the oil medium. The slight difference between the analytical solution and the PIV data can be attributed to ignoring the centripetal velocity component \(v_r\) and accordingly, \(\frac{dP}{dr}\). The centripetal component of the flow might increase the model’s accuracy and skew it slightly to the left, as shown in Fig. 6. Another potential factor for this slight difference is that the flow is not fully steady-state as it slightly accelerates away from the ground electrode and moves towards the corona-generating needle. It is to be noted that this parabolic shape might not be fully preserved when the corona voltage is high due to the possibilities of inducing secondary flows and Dean vortices.

Fig. 6
figure 6

Azimuthal velocity profiles normalized by the bulk velocity for SO50 (solid lines) and SO100 (dashed lines), at 4 different corona voltages, compared with the normalized ODE solution. A clear collapse is seen with a peak-to-bulk ratio of approximately 1.5 at the center of the channel

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4 Conclusion

A corona discharge induced EHD flow inside a circular channel was studied using PIV. A voltage parametric study was performed for silicone oils with two different kinematic viscosities of 50 cSt (SO50) and 100 cSt (SO100). As hypothesized, a higher voltage led to higher velocity flows in the circular channel, with SO100 exhibiting lower bulk velocity measurements for all voltages, as compared to SO50. A clear scaling effect was noticed when velocity profiles of both SO50 and SO100 were plotted across the channel width with a consistent profile (parabola) for all cases. At high corona voltages, the azimuthal velocity contour plots showed a pattern of concentric circles of high-velocity values, which was also contributing to the fluctuations in the flow. This was shown by the high rms values around the concentric circles of high velocity-values.

Moreover, a Fourier analysis of the measured velocities was performed. Normalized PSD plots showed prominent frequencies below 2 Hz, for both SO50 and SO100. Such frequencies were attributed to the sloshing motion of the fluid, and hence, the concentric circles of high-velocity values. Additionally, a peak at \(\sim \, 11.3\) Hz was also present for SO50, including in the stationary fluid. Although its source was not exactly pinpointed, it is conjectured to originate from natural convection or vibrations in the setup. An analytical model is also derived and plotted against the PIV data, with a very minute difference. The Re versus voltage plot showed a power law correlation. The results of this study may be used as a guide for the corona operating parameters to induce desired pumping behavior of dielectric liquids in a contact-less manner.