# Dielectrophoretic force-enhanced thermal convection within a horizontal cylindrical annulus

M H Hamede, J. Roller, Antoine Meyer, V. Heuveline, C. Egbers

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M H Hamede, J. Roller, Antoine Meyer, V. Heuveline, C. Egbers. Dielectrophoretic force-enhanced thermal convection within a horizontal cylindrical annulus. Physics of Fluids, 2024, 36 (12), 10.1063/5.0234772. hal-05000229

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# Dielectrophoretic force-enhanced thermal convection within a horizontal cylindrical annulus 📦

M. H. Hamede (محمد حسین حماده) ; J. Roller, A. Meyer; V. Heuveline; C. Egbers

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Physics of Fluids 36, 124106 (2024)

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# Dielectrophoretic force-enhanced thermal convection within a horizontal cylindrical annulus EP

Cite as: Phys. Fluids 36, 124106 (2024); doi: 10.1063/5.0234772

Submitted: 23 August 2024 · Accepted: 4 November 2024 ·

Published Online: 3 December 2024

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M. H. Hamede (محمد حسین حماده),¹,ª J. Roller,² A. Meyer,³ V. Heuveline,² and C. Egbers¹

## AFFILIATIONS

¹Department of Aerodynamics and Fluid Mechanics, BTU Cottbus-Senftenberg, Cottbus, Germany

²Engineering Mathematics and Computing Lab, Heidelberg University, Heidelberg, Germany

³Laboratory of Waves and Complex Media, Université Le Havre Normandie, CNRS-UMR6294, Le Havre, France

ª) Author to whom correspondence should be addressed: Hamede@b-tu.de

## ABSTRACT

This study investigates, both experimentally and numerically the flow of a dielectric fluid confined between two concentric, differentially heated, horizontally aligned cylinders subjected to a 200 Hz alternating radial electric field. A wide-gap annular setup with a length 20 times larger than the gap size is utilized in this investigation. The study focuses exclusively on the outward heating configuration, meaning the inner cylinder is hotter than the outer one. The electric field, in conjunction with the temperature gradient, triggers thermal electro-hydrodynamic instability caused by the application of dielectrophoretic force. When the applied electric tension exceeds a critical value for specific temperature gradients between the cylinders, the flow symmetry in the gap is disturbed. The instability manifests as periodically oscillating vortices occurring on top of the gap. A notable increase in heat transfer efficiency accompanies the onset of instability. The experimental and numerical results demonstrate quantitative and qualitative agreement.

© 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0234772

## I. INTRODUCTION

Many technical applications rely on flow management to improve heat transfer efficiency. Examples include using water circulation to improve the effectiveness of solar water heaters,¹ applying an external magnetic field to reduce the natural convection of a conductive fluid,² and inducing thermomagnetic convection in a ferrofluid.³ Another possibility is to apply an electric field to a dielectric fluid.⁴ Consider such a fluid with kinematic viscosity, $ν$, thermal diffusivity $κ$, density $ρ$ and electric permittivity $ε$. When an electric field $E$ is applied to the fluid, it feels an electric body force called the electrohydrodynamic (EHD) force, which reads:⁵,⁶

$$ 
\begin{aligned}
F_{EHD} &= F_C + F_{DEP} + F_{ES} \\
&= \rho_e E - \frac{1}{2} E^2 \nabla \epsilon + \frac{1}{2} \nabla \left[ \rho E^2 \left( \frac{\partial \epsilon}{\partial \rho} \right)_T \right],
\end{aligned} 
\quad (1) 
$$

where $ρ_e$ is the electric charge density and $T$ is the temperature. The first term on the right-hand side (RHS) of Eq. (1) is the electrophoretic

force, arising from the Coulomb forces that the field exerts on free charges. The second term is the dielectrophoretic (DEP) force. It corresponds to the effect of a non-uniform electric field on the differential polarization of the fluid particles. The third term relates to electrostatic interaction. It represents a fluid strain due to its polarization that can be lumped with the pressure,⁴ and will not affect the flow dynamics for incompressible fluids. The coexistence of the electrophoretic force and the DEP force must be nuanced with the fact that an experiment will usually see the prevalence of one or the other term depending on the electric field's frequency. The electric charge density of a dielectric fluid submitted to an external electric field grows with the charge relaxation time $τ_e = σ/ε$, where $σ$ is the electric conductivity of the fluid. In the problem under investigation, an alternating current electric field is applied with a frequency $f \gg \tau_e^{-1}$. The mechanism building up free charges in the dielectric fluid's bulk is interrupted so $ρ_e = 0$. As a result, the DEP force becomes the main contributor of the EHD force and creates momentum when the electric permittivity is inhomogeneous. The temperature gradient of the fluid can result in inhomogeneity of the fluid electric permittivity. For small temperature
differences, it can be considered that both the density and the electric permittivity are linearly decreasing functions of the temperature: $ρ(T) = ρ_{ref}[1 - αθ]$ and $ε(T) = ε_{ref}[1 - eθ]$, with α the thermal expansion coefficient, e the coefficient of the thermal variation of permittivity, and $θ = T - T_{ref}$ the temperature deviation from the reference temperature.⁷ The quantities with the subscript “ref” denote the quantity measured at the reference temperature. Using the above approximation of ε, the DEP force in a differentially heated annular geometry can be formulated as:⁸

$$F_{\text{DEP}} = \nabla \left( \frac{e\theta\varepsilon_{\text{ref}}E^2}{2} \right) - \rho\alpha\theta g_e. \qquad (2)$$

The first term on the RHS contributes to the pressure balance, and the second term is analog to the Archimedean buoyancy due to an effective electric gravity $g_e$, written as

$$g_e = \frac{e\varepsilon_{\text{ref}}}{\alpha\rho_{\text{ref}}} \nabla \left( \frac{E^2}{2} \right). \qquad (3)$$

Researchers have been drawn to the topic of dielectrophoretic force-driven thermal convection in different geometries as a method for enhancing heat transfer through the triggering of thermo-electro hydrodynamic (TEHD) instability. The annular geometry constituted a significant proportion of these studies, which used various numerical, analytical, and experimental techniques to investigate this particular type of instability in earth gravity in vertical configuration⁹,¹⁰ and microgravity conditions.⁸,¹¹,¹² The studies demonstrate that at specific critical values of the temperature difference between the inner and outer cylinders and electric potential, the DEP force results in the formation of stationary counter-rotating pairs of vortices. Under microgravity conditions, the TEHD instability manifests in helicoidal modes.⁸,¹¹,¹³ However, in laboratory conditions with a vertically aligned cylindrical annulus, the instability occurs through the formation of columnar vortices.⁹,¹⁰ In both conditions, the onset of instability was accompanied by an increase in heat transfer.⁹,¹¹ Notably, the onset of instability is influenced by various factors, including the radius ratio of the annular geometry, the fluid properties, and the direction of the temperature gradient.⁸

The majority of the studies mentioned above examined the flow configuration in which the induced electric gravity is mostly aligned with the temperature gradient. However, in the framework of industrial applications, it is of best interest to investigate various orientations of the system. The convection induced by the DEP force has potential applications for heat evacuation devices. Such heat exchangers would have the advantage of being effective in any orientation and without any moving parts. In particular, the DEP torque applied in a planar cavity highlighted an improvement of 8% in heat transfer compared to classical thermal buoyancy.¹⁴ Although the improvement was modest, optimizing the geometry might have led to further heat transfer enhancement. In this study, we focused on the horizontal configuration of the annular setup, which has rarely been examined. In contrast to the vertical orientation, the horizontally aligned gap exhibits a break in the axial symmetry of the problem. The temperature gradient of the base flow is no longer aligned with the electric gravity field, resulting in a finite DEP torque acting in the fluid’s bulk (see Sec. III B). As a result, a flow modification could potentially be observed at any value of the applied electric field. To investigate the stability of the flow and the

impact of varying flow parameters on heat transfer, we conducted both experimental and numerical simulations. The investigation was performed using the thermal Rayleigh number $Ra$ and the electric Rayleigh number $L$ to characterize the Archimedean buoyancy and the thermoelectric buoyancy, respectively:

$$Ra = \frac{\alpha g \Delta T d^3}{\nu \kappa} \quad \text{and} \quad L = \frac{\alpha \bar{g}_e \Delta T d^3}{\nu \kappa}, \qquad (4)$$

where $d$ is the gap size, $ΔT$ is the temperature difference between the two surfaces, $g$ is the gravitational acceleration of the Earth, and $\bar{g}_e$ is the electric gravity at the center of the gap.

The following is a description of the paper’s structure. Section II presents the experimental setup and the measurement techniques that were used. Section III reviews the flow equations that govern the system, along with the equations characterizing the base flow and the numerical scheme used to resolve these equations. The results are presented in Sec. IV. The paper concludes with a short discussion and conclusion in Sec. V.

## II. EXPERIMENTAL SETUP AND MEASUREMENT TECHNIQUES

### A. Experimental cell

The experimental cell used in this project is composed of two concentric cylinders. The inner cylinder has an outer radius of $R_i = 5$ mm and the outer cylinder has an inner radius of $R_o = 10$ mm. This configuration yields an aspect ratio of $\eta = R_i/R_o = 0.5$ and a gap width of $d = R_i - R_o = 5$ mm. The length of both cylinders is $L = 100$ mm, resulting in an aspect ratio of $\Gamma = L/d = 20$. The inner cylinder is made of aluminum with a wall thickness of $d_i = 2$ mm. The outer surface of the cylinder is anodized to reduce any light reflection. The outer cylinder is constructed from borosilicate glass with a wall thickness of $d_o = 2$ mm. The inner surface of the outer glass cylinder is coated with a transparent conductive oxide, which serves to conduct an electric field when applied to the cylinder. Silicone oil Elbesil B5 [$\nu(25^\circ C) = 5 \times 10^{-6} \text{ m}^2\text{s}^{-1}$, $\kappa(25^\circ C) = 7.74 \times 10^{-8} \text{ m}^2\text{s}^{-1}$, $\rho(25^\circ C) = 925 \text{ kg/m}^3$, $\varepsilon(25^\circ C) = 2.39 \times 10^{-11} \text{ F/m}$] is used as a working fluid filling the gap between the inner and outer cylinder. In the current configuration, an AC high-voltage electric potential $V(t) = V_p \sin(2\pi ft)$ with $f = 200$ Hz is applied to the aluminum inner cylinder, with a peak electric potential of up to $V_p = 20$ kV. The outer cylinder is electrically grounded. The chosen value of $f$ is several orders of magnitude larger than the inverse of the charge relaxation time $\tau_e^{-1} \approx 10^{-4}$ Hz. In addition, the frequency must also be larger than the inverse of the characteristic time of viscous dissipation $\tau_\nu^{-1} = \nu/d^2 = 0.2$ Hz, and of the characteristic time of thermal diffusion $\tau_\kappa^{-1} = \kappa/d^2 = 3.3 \times 10^{-3}$ Hz. Under these considerations, the effect of the electric field oscillation can be disregarded, and the problem can be time-averaged over a period of the electric field oscillation. To apply a temperature gradient to the fluid within the gap, the inner and outer cylinders are maintained at different temperatures by a heating and cooling fluid loop, respectively. Therefore, the outer cylinder is encased and centered by an acrylic box, within which a cooling fluid is continuously pumped to form a cooling loop. The temperatures of the fluid loops are monitored at the inlets and outlets of the cells to regulate the temperature differential between the cylinders. The temperature difference between the two cylinders $ΔT$ corresponds to the difference in temperature between the inlets of the heating and cooling loops. The
reference temperature $T_{ref}$ is set as the cooling loop temperature measured at the inlet of the experimental cell. Silicone oil was used as the cooling fluid, exhibiting dielectric characteristics and an optical index comparable to that of glass. This resulted in minimal light distortion when observing the flow within the gap through the outer cylinder. The experimental cell was sealed by acrylic glass plates at the top and bottom. The transparent top and bottom plates permit optical measurements in the gap in the radial azimuthal plane.

## B. Measurement technique

In this study, the shadowgraph method and particle image velocimetry (PIV) are used to quantitatively and qualitatively analyze the flow. The shadowgraph is a straightforward flow visualization technique that captures the fluid refractive index variation.¹⁵ The temperature gradient and density gradient used in the gap serve to influence the variation of the fluid refractive index. As shown in Fig. 1, a telecentric light illuminator is placed at one end of the cell, opposite to one of the end plates. It illuminates monochromatic red telecentric light with a wavelength of $λ = 635$ nm, which travels through the cell height. The transmitted light through the gap is modified to the incoming light due to the variation of the refractive index, as previously described. Subsequently, the reflected light is captured at the second end of the cell using a telecentric lens connected to a GigE camera (ImagingSource DMK 33GX174), which has a CMOS pixel sensor with a resolution of 1920 × 1200 pixels (2.3 MP). The camera is ultimately responsible for capturing the density variation. As the light traverses the entire height of the annulus, its intensity profile undergoes modification, which is then captured by the camera with an acquisition frequency of 10 fps.

The second measurement technique used in this study to quantify flow velocities is PIV. The fundamental principle of PIV is seeding the flow with tiny tracing particles designed to follow the flow precisely.

Subsequently, the fluid volume is illuminated and the movement of the particle is captured in defined time intervals, resulting in the calculation of its velocity, which describes the flow velocity. In this study, the field velocity was measured at the annulus midheight in the radial-azimuthal plane. The PIV setup comprises a continuous monochromatic green (532 nm) diode laser used to generate a laser sheet in the radial-azimuthal plane, and a GigE camera (ImagingSource DMK 33GX174). It is capable of acquiring images at a rate of up to 50 frames per second. The camera is positioned in a manner that permits the capture of light reflected from the illuminated particle and transmitted through the end wall, as illustrated in Fig. 1.

The annulus is illuminated by two distinct light sources, each with a distinct wavelength, for specific purposes. To eliminate any uncertainty that may be caused by the presence of multiple light sources, each of which is used to offset the reflected light, the light exiting the annulus is split using a beam splitter into two beams, and then each of these beams is filtered using a bandpass filter. The light directed toward the telecentric lens was filtered using a red light bandpass filter with a bandpass range of 625–640 nm, while the light directed toward the PIV camera passed through a green light bandpass filter with a bandpass range of 500–555 nm. This was done to ensure that the light captured was solely reflected from the laser-illuminated sheet for the PIV.

The utilization of the PIV in this study is associated with the complications that arise in the selection of suitable tracing particles. Additionally, sedimentation caused by density variations between particles and the fluid must be taken into account. Another specific issue in this study is the sedimentation velocity of the particles due to polarization under the application of the electric potential. The ability of the tracer particles to adhere to the flow field in a configuration similar to the one under investigation was studied in detail by Seelig et al.⁹ In this study, we selected ORGASOL 2001 EXD NAT 1 spherical particle from the ARKEMA company. The P12 polyamide particles are

<figure><img src="image_10.png" /><figcaption>**FIG. 1.** The schematic diagram of the experimental apparatus, including the PIV and shadowgraph setup, is presented above. An inner heat fluid loop was used to heat the inner cylinder's walls, while the outer cylinder was cooled by a surrounding acrylic-glass box filled with cool fluid. The gap between the inner and outer cylinder was filled with a working fluid mixed with flow tracer particles.</figcaption></figure>
primarily used for coating and manufacturing parts through laser sintering. However, due to their electric and mechanical characteristics, they were selected for use as tracer particles in this study. The ORGAZOL particles utilized in this study have an average radius of $R_p = 5$ µm, a density of $\rho_p = 1020$ kg/m³, and a permittivity of $\epsilon_p = 2.67 \times 10^{-11}$ F/m. The sedimentation velocity of a spherical particle is expressed as:⁹

$$v_{\text{sed}} = \frac{\varepsilon R_p^2 V_p^2}{6\rho\nu(\ln \eta)^2} \frac{(\varepsilon_p - \varepsilon)}{(\varepsilon_p + 2\varepsilon)} e_r + \frac{2gR_p^2}{9\nu} \left(\frac{\rho_p}{\rho} - 1\right) e_z. \quad (5)$$

When no electric field is applied, the maximum axial sedimentation velocity of the particles is $v_{\text{sed}} = 1.04 \times 10^{-6}$ m/sec, which can be considered negligible. Considering microgravity conditions, i.e., in the absence of axial sedimentation, the application of a temperature difference of $\Delta T = 9$ K and a maximum electric potential of 18 kV leads to a particle sedimentation velocity in the radial direction of $5.99 \times 10^{-7}$ mm/s at the inner cylinder where the electric gravity field is the largest. Consequently, the particle sedimentation velocity in both cases is considered to be negligible when the particles are used in PIV as flow-tracing particles. The single frame mode is used with a sampling frequency of 10 Hz, resulting in 10 frames/second. The MATLAB MatPIV toolbox¹⁶ was used to post-process the PIV images with an initial interrogation window size of 64 × 64 and a final size of 16 × 16 with 50% of overlap. The measurements are performed with a spatial resolution of 0.033 mm/pixel. Subsequently, the data sets were calibrated using a method based on detecting the cylinders' walls from the PIV images. For a detailed explanation of the calibration method, the reader is referred to.¹⁷,¹⁸

### III. MODELING AND NUMERICAL SIMULATION

#### A. Fundamental equations

As a mathematical description of the experiment, we consider the equations of the conservation of mass, momentum, energy, and electric flux adopting the Boussinesq approximation:⁹,¹⁰

$$\begin{align} \frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla)\mathbf{v} - \nu \Delta \mathbf{v} + \nabla \hat{p} = f_{\text{BUO}}, \nonumber \\ \nabla \cdot \mathbf{v} = 0, \nonumber \\ \frac{\partial \theta}{\partial t} + (\mathbf{v} \cdot \nabla)\theta - \kappa \Delta \theta = 0, \nonumber \\ -\nabla \cdot (\varepsilon_{\text{ref}}(1 - e\theta)\nabla\Phi) = 0, \tag{6} \end{align}$$

where $\mathbf{v}$ is the velocity field, $\Phi$ is the electric tension, and $\hat{p}$ is the reduced pressure, which includes the static pressure, the electrostrictive term of Eq. (1) and the conservative part of the DEP force from Eq. (2). The external forces acting on the dielectric fluid are lumped on the buoyancy term in the right-hand side of Eq. (6):

$$f_{\text{BUO}} = -\alpha(\mathbf{g} + \mathbf{g}_e)\theta. \quad (7)$$

The heat source due to the viscous dissipation and due to the oscillation of the electric field¹⁹ are neglected in this study.

#### B. Base flow

We define the base state of the flow as the stationary 2D flow that consists of a hot outward vertical jet on the top of the gap and a cold inward vertical jet on the bottom of the gap. This crescent-shaped flow, which occurs when a temperature difference is applied, has been

demonstrated to bifurcate into secondary flows when the temperature difference reaches a critical value. The secondary flow and its corresponding critical temperature difference depend on the gap size, the curvature of the system, and the Prandtl number.²⁰⁻²² For the studied configuration, the central electric gravity field superimposes the Earth's gravity, which may result in modifications to the flow of the base state and alterations to the stability conditions of the fluid. For small temperature differences, the temperature profile does not involve large heat fluxes. In that case, the electric gravity, which varies as the gradient of the square of the electric field, is mainly driven by the curvature of the cylindrical annulus, rather than by thermal fluctuations. Under this approximation, the base convective temperature profile, which has no analytical solution, can be reduced to its purely conductive regime case:

$$\theta_b = \Delta T \frac{\ln r/R_o}{\ln \eta}, \quad (8)$$

where the temperature of the cylindrical surfaces is assumed to be fixed, which hypothesis will not be adopted in the next sections (see Sec. III C). In this case, the electric gravity is always centripetal and reads:⁸,¹²

$$g_e = F(\gamma_e, \eta, r) \frac{e\epsilon_f}{\alpha\rho r^3} \left(\frac{V_0}{\ln(\eta)}\right)^2, \quad (9)$$

with

$$F(\gamma_e, \eta, r) = \frac{[\gamma_e / \ln(1 - \gamma_e)]^2 [1 - e\theta_b + \gamma_e / \ln \eta]}{[1 - e\theta_b]^3} \quad (10)$$

in the polar coordinate system $(r, \phi)$. We introduced the thermoelectric parameter $\gamma_e = e\Delta T$, which characterizes the coupling between the electric potential and the temperature. For small temperature differences between the two cylinders, we have for most dielectric fluids $\gamma_e \ll 1$ so that the coefficient $F$ defined in (10) approaches $F \sim 1$. The governing equations for the base state consist of the continuity equation, the momentum equation, and the energy equation, written in the Cartesian coordinates:

$$\begin{align} \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0, \nonumber \\ u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = -\frac{\partial \pi}{\partial x} + \nu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) + \alpha \theta G_x, \nonumber \\ u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} = -\frac{\partial \pi}{\partial y} + \nu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} \right) + \alpha \theta G_y, \tag{11} \\ u \frac{\partial \theta}{\partial x} + v \frac{\partial \theta}{\partial y} = \kappa \left( \frac{\partial^2 \theta}{\partial x^2} + \frac{\partial^2 \theta}{\partial y^2} \right), \nonumber \end{align}$$

where $G_x$ and $G_y$ are the two components of the total gravity field $\mathbf{G} = G_x \mathbf{e}_x + G_y \mathbf{e}_y$ given by;

$$G_x = g_e(r) \cos \phi \quad \text{and} \quad G_y = g + g_e(r) \sin \phi. \quad (12)$$

As the gravity field (12) is an even function to the x-axis $[g_e(-x) = g_e(x)]$, we can conclude that the x-component of the velocity is an odd function $[u(-x) = -u(x)]$, while the temperature and
the y-component of the velocity are even functions of the x-axis. The axial component of the base state vorticity $\omega_z$ satisfies the following equation:

$$u \frac{\partial \omega_z}{\partial x} + v \frac{\partial \omega_z}{\partial y} = \nu \left( \frac{\partial^2 \omega_z}{\partial x^2} + \frac{\partial^2 \omega_z}{\partial y^2} \right) + \tau_F, \quad (13)$$

where the torque $\tau_F$ produced by the external forces is given by:

$$\tau_F = \alpha \left( G_x \frac{\partial \theta}{\partial y} - G_y \frac{\partial \theta}{\partial x} \right). \quad (14)$$

One can find that $\omega_z$ and $\tau_F$ are odd functions along the x-axis at the base state.

To identify the flow characteristics of the system, a 2D stationary numerical simulation is done using the software COMSOL Multiphysics based on the finite element method (FEM). The base state of the flow is then solved for a small temperature difference ($\Delta T = 2$ K), assuming the absence of coupling between the temperature and the electric potential [see Eq. (11)]. The torque produced by the misalignment between the heat flux and the Earth's gravity is shown in Fig. 2(a) when no electric potential is applied. The base state torque changes its sign at the ax $x = 0$ and highlight the production of vorticity of the dielectric fluid. When an electric tension is applied to the fluid, the electric gravity contributes to the torque by increasing it. The torque average along the radial direction is defined as:

$$\langle \tau_F \rangle = \int_{R_i}^{R_o} \tau_F dr / \int_{R_i}^{R_o} r dr, \quad (15)$$

and is plotted along the azimuthal direction in Fig. 2(b) for various applied electric potentials. The torque is enhanced by about 19% when going from no applied potential to an applied electric potential of 12 kV. Although the approximation made for that analysis is strong, it is expected that the application of an external electric potential will produce an intensification of the flow inside the gap.

### C. Boundary conditions

To adopt a numerical model as close as possible to the experimental configuration, the boundary conditions are set to fit the thermal property of the cylindrical walls. No-slip boundary conditions

<figure><img src="image_11.png" /><figcaption>FIG. 2. (a) surface of constant torque, and (b) torque averaged over the radial position for different electric potential.</figcaption></figure>

$v = 0$ are assumed on all surfaces. The lateral lids of the cylindrical annulus are considered electrically and thermally insulated:

$$\frac{\partial \theta}{\partial z} = \frac{\partial \Phi}{\partial z} = 0 \quad \text{on } \{z=0\} \text{ and } \{z=h\}. \quad (16)$$

The electric potential is fixed at the cylindrical surfaces:

$$\begin{align}
\Phi &= V_0 \quad \text{at} \quad \{r = R_i\}, \nonumber \\
\Phi &= 0 \quad \text{at} \quad \{r = R_o\}. \tag{17}
\end{align}$$

Because the inner cylinder is made of aluminum, the heat transfer from the fluid in the heating loop to the outer surface of the cylinder in contact with the working fluid is driven by conduction, so it can be assumed that the temperature of the inner cylinder is fixed:

$$\theta = \Delta T \quad \text{at} \quad \{r = R_i\}. \quad (18)$$

However, the outer cylinder made of borosilicate glass does not transport heat as efficiently as aluminum. Therefore, we work with an altered thermal boundary condition at the inner wall of the outer cylinder, known as Newton's law of cooling:²³

$$-\kappa \frac{\partial \theta}{\partial r} = \tilde{h} \theta \quad \text{at} \quad \{r = R_o\}, \quad (19)$$

with the heat transfer coefficient $h = \tilde{h} \rho_{ref} c_p$, where $c_p$ is the specific heat capacity of the fluid. Therefore, the effective temperature difference between the outer wall of the inner cylinder and the inner wall of the outer cylinder is always smaller than the target $\Delta T$, except when $\Delta T = 0$ K.

Figure 3 schematically represents the model used to define the thermal boundary conditions at the cylindrical surfaces.

### D. Numerical method

We solve the initial-boundary value problem (6, 16, 17, 19) using the finite element method (FEM). After semi-discretization in time by a linearized BDF2 scheme, we approximate $(v, p)$ in space with the Taylor-Hood finite element $Q_2 \times Q_1$. For $T$ and $\Phi$, we use the $Q_2$ element. The resulting linear systems of equations are solved using a preconditioned GMRES method. A detailed description of the

<figure><img src="image_12.png" /><figcaption>FIG. 3. Schematic representation of the model used for the thermal boundary conditions. The upper scheme corresponds to the experimental configuration while the lower one is a simplified case for the numerical simulations.</figcaption></figure>
numerical scheme can be found in Ref. 24 The circular annular mesh has 66 360 spatial degrees of freedom in 2D. In 3D, we work on a cylindrical mesh with $4 \cdot 10^6$ degrees of freedom. Both meshes are anisotropic, with smaller cell diameters close to the boundaries. The temporal discretization uses a constant time step length of $\tau = 0.01$ s. Larger values of $\tau$ lead to physically infeasible oscillations in the numerical solution for $L \ge L_c$. As initial conditions, we use the stationary conduction solution of (11) when $G_x = G_y = 0$. The simulation is stopped at $t = 200$ s. While the solutions for the base state already converge to a steady state by $t = 120$ s, the longer time interval allows us to acquire more data on the unsteady solution in the case of unstable flows. All of the following numerical results have been generated with the open-source finite element software HiFlow$^{2,5}$. The computationally expensive 3D simulations were accomplished with up to 768 parallel CPU threads, while only four threads were used for the 2D geometry.

## E. Model calibration

Since $\tilde{h}$ cannot be measured experimentally, we use the stochastic collocation method described in Ref. 27 to estimate $\tilde{h}$ from experimental data. Let $\tilde{\xi}$ denote the experimental measurements of $v_r$ and $v_\phi$ from the experiments with $V_p = 0, \dots, 16$ kV. We use a surrogate model, enabling us to interpolate between model evaluations of $(v_r, v_\phi)$ along a wide interval of candidate values for $\tilde{h}$. Assuming a uniform prior distribution $p(\tilde{h})$, we use the surrogate model to approximate the likelihood $p(\tilde{\xi}|\tilde{h})$. Following Bayes' rule

$$p(\tilde{h}|\tilde{\xi}) \propto p(\tilde{\xi}|\tilde{h})p(\tilde{h}),$$

we can then generate samples from the posterior distribution $p(\tilde{h}|\tilde{\xi})$. We consequently take the posterior mean as an estimate for $\tilde{h}$. As shown in Fig. 4, this corresponds to an estimate of $\tilde{h} \approx 3 \times 10^{-4}$ Wm/J or $\tilde{h} \approx 451$ W/(m$^2$ K). This value of $\tilde{h}$ is used in all of our

<figure><img src="image_13.png" /><figcaption>FIG. 4. Approximated posterior density $p(\tilde{h}|\tilde{d})$. The high-density interval (HDI) is narrowly concentrated around the mean, indicating only minor uncertainty. PyMC (Ref. 26) was used to sample from this distribution.</figcaption></figure>

simulations. To further motivate the boundary conditions, consider the heat transfer coefficient $h$ to be given by

$$\frac{1}{hR_o} = \frac{1}{\lambda_o} \ln \left( \frac{R_o + d_o}{R_o} \right) + \frac{1}{h_{cl}(R_o + d_o)} \quad (20)$$

The first term on the right-hand side of (20) accounts for heat conduction through the cylindrical glass wall$^{2,3}$ while the parameter $h_{cl}$ models the convective heat transfer inside the cooling loop. If one were to model the overall heat transfer through the outer cylinder by $h$, the right-hand side of (20) would contain an additional term corresponding to the convective heat transfer at $r = R_o$ due to the thermal convection of silicone oil in the gap. However, since our high-resolution simulations resolve the thermal and viscous boundary layers, there is no convection at $r = R_o$: the no-slip velocity is applied and conduction is the heat transfer mechanism at $r = R_o$. Solving for $h_{cl}$ using the numerically determined value of $h$ results in $h_{cl} \approx 1194$ W/(m$^2$ K). As the heating loop is a flow in a tube, i.e., the inner cylinder, its heat transfer coefficient $h_{hl}$ can be computed by Colburn's semi-empirical formula$^{28}$

$$h_{hl} = \frac{0.023 \mathrm{Re}^{0.8} \lambda_i}{2(R_i - d_i)} \left(\frac{\nu}{\kappa}\right)^{0.33} \left(1 + \left(\frac{2(R_i - d_i)}{L}\right)^{0.7}\right) \quad (21)$$

Using the velocity of the heating loop fluid $v_{hl} \approx 0.354$ m/s we calculate a corresponding Reynolds number $Re = v_{hl}2(R_i - d_i)/\nu \approx 383$, so Eq. (21) yields $h_{hl} \approx 278$ W/(m$^2$ K). Similarly, we define a heat-transfer coefficient $h_i$ for the inner cylinder by

$$\frac{1}{h_i R_i} = \frac{1}{\lambda_i} \ln \left( \frac{R_i}{R_i - d_i} \right) + \frac{1}{h_{hl}(R_i - d_i)} \quad (22)$$

which can be solved for $h_i \approx 166$ W/(m$^2$ K). If one defines the Biot number $Bi$ for the inner cylinder as

<figure><img src="image_14.png" /><figcaption>FIG. 5. Representative comparisons between the experimental and numerically obtained profiles of: (a) the spatiotemporal averaged radial profile $U_\phi$, (b) the temporally averaged radial profiles of the radial velocity at the top of the cell ($\pi/2$), for $\Delta T = 4$ k and $V_p = 2$ kV. The error bars represent the standard deviations of the experimental data.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>FIG. 6. Representative comparisons between the experimental and numerically obtained profiles of: (a) the spatiotemporal averaged radial profile $U_φ$, (b) the temporally averaged radial profiles of the radial velocity at the top of the cell (π/2), for ΔT = 7k and $V_p = 12$ kV. The error bars represent the standard deviations of the experimental data.</figcaption></figure>

$$Bi_i = \frac{h_i d_i}{\lambda_i}, \qquad (23)$$

we arrive at $Bi_1 \approx 2 \times 10^{-3}$. At a Biot number lower than 0.1, a body is commonly considered isothermal. This justifies our use of fixed-temperature boundary conditions at $r = R_i$. A similar calculation for the outer cylinder gives

$$Bi_o = \frac{hd_o}{\lambda_o} \approx 0.75, \qquad (24)$$

indicating that a non-isothermal condition on the outer cylinder is more accurate. As presented in Figs. 5 and 6, there is a strong qualitative and quantitative agreement between the numerical velocity profiles with the experimental data for a wide range of values in both ΔT and $V_p$. In particular, we compare the mean radial jet velocities at the top of the annulus and the mean azimuthal velocities at the mid-axial position, i.e., 50 mm. Due to the symmetry of the flow (see, e.g., Fig. 8) in the steady case, we only compute the mean of the azimuthal velocity on the interval (−π/2, π/2). We will limit our comparison to steady flow, as the radial jet velocity cannot be clearly defined in the unsteady regime. We acknowledge that the quantitative fit with the mean measurements is not perfect. This can be explained first and foremost by the fact that the experimental measurements did not cover the entire radial extent of the domain. The calibration could only take incomplete data into account.

## IV. RESULTS

This study investigated the flow for four different temperature difference values between the concentric cylinders, designated as ΔT = 2, 4, 7, and 9 K. where the reference temperature is fixed at the value $T_{ref} = 20°$ C. After establishing the thermal boundary conditions, each case was investigated for different electric potential values,

taking values up to $V_p = 18$ kV. The investigation was conducted both experimentally and numerically. Below, the results of the study are shown in different sections.

### A. Flow visualization

To obtain qualitative information about the flow field and its adaptation to different applied parameters, the shadowgraph technique is used. For the various experiments, the temperature difference is set in the gap first, and each set of measurements is terminated after 15 minutes of switching on the high voltage. The procedure was repeated, with the voltage increasing in a stepwise manner from one experiment to the next.

Figure 7 depicts shadowgraphs of the flow with ΔT = 7 K ($Ra = 2.16 \times 10^4$) and different applied electric voltage. In the natural convective state, where there is no applied electric field, Fig. 7(a) shows a shadow of a symmetrical pair of convective cells with a crescent shape, which corresponds to the base flow of the system. As illustrated in the figure, the two pairs of convective cells are detached from the top and bottom jets. Upon the application of an electric potential of up to $V_p = 15$ kV to the dielectric fluid in the gap, no discernible qualitative difference can be observed in the flow field for the current ΔT = 7 K case as shown in Figs. 7(b)–7(d). The symmetrical pairs of the convective cell remain in existence, although it can be observed that for higher electric potential values, the light intensity is increasing, which is indicative of the reinforcement of the convective cells. For $V_p \ge 16$ kV, a notable alteration in the flow is discernible. The flow is disturbed in the upper half-domain (0–π), and the upper flow jet has disappeared, rendering it no longer feasible to distinguish or capture the existence of the convective cell pair. This behavior is exclusively observed on the upper side, while on the lower half-domain (π–2π), no notable deviation from the $V_p \le 15$ kV cases is evident unless the cells are reinforced further. The shadows of the convective cells persist, and the bottom jet remains intact, as illustrated in Fig. 7(f). It is noteworthy that the observed behavior for ΔT = 7 K is consistent with that observed for ΔT = 2, 4, and 9 K. However, a notable distinction is that the flow tends to become unstable at lower electric potential for lower temperature gradients. The investigation revealed that the flow was disrupted at $V_p = 15$ kV for ΔT = 2 K and 16 kV for ΔT = 4 and 9 K. A further quantitative study of the flow velocity field will be essential to describe more fully the flow behavior and the existence of any instability.

### B. Flow velocity field

To gain further insight into the flow field, PIV measurements are conducted to capture the radial and azimuthal flow velocity components at the apparatus midheight. Figure 8 depicts the temporally averaged resultant velocity of the radial and azimuthal components for ΔT = 2 K and varying electric potential values. To facilitate the identification of any potential vortical patterns, the velocity magnitude is multiplied by the sign of the temporal-averaged azimuthal velocity component. It is important to note that due to the constraints of the experimental setup, it was not possible to measure the flow at the entire gap as depicted in the figure. When no electric potential is applied, Fig. 8(a) illustrates the existence of a symmetrical pair of convective cells, which correspond to the base flow state. The fluid flows downward adjacent to the outer cylinder, which has cooled walls.
Conversely, in the region adjacent to the heated inner cylinder, the
fluid is rising.

These convective cell pairs are distinguished at the top and bot-
tom of the gap, where the radial velocity region appears. At the top, an
outward jet forms where the two cells separate, while at the bottom, an
inward jet is observed. In agreement with the results observed in Fig. 7,
applying an electric potential to the flow and increasing it to a specific
value resulted in no quantitative difference in the flow behavior. Up to
Vp = 14 kV, Fig. 8(d) shows that the symmetrical pair cells are main-
tained, although the mean-driven velocity of the cells increases with
increasing electric potential. For Vp ≥ 15 kV, a disparity between the
pair of cells emerges, accompanied by the emergence of distinct out-
flow regions from the inner cylinder wall within the gap. As Vp
increases, the disparity intensifies, giving rise to the development of
inflow and outflow regions within the gap, thereby exacerbating
flow instability. The disturbance in the flow is confined to the upper
half-domain of the cell, while the lower half remains in its base flow
state, as illustrated in Figs. 8(e) and 8(f). Another noteworthy observa-
tion when comparing the flow behavior in Figs. 7 and 8 is that the flow
asymmetry, or instability, appears for lower Vp for the lower ΔT cases.
Thus, the instability is observed first at Vp = 15 kV for ΔT = 2 K,
while for ΔT = 7 K, it is observed to start at Vp = 16 kV.
Furthermore, for the results not shown here for ΔT = 4 and 9 K, the
instability was observed to begin at Vp = 16 kV. For ΔT = 7 K and
the same parameters illustrated in Fig. 7, the resulting flow velocity
field is presented via Fig. 13 in the Appendix.

To evaluate the temporal behavior of the flow, space-time dia-
grams are built showing the variation of the radial velocity measured
at the midgap in the (t, φ) plane. Such diagrams were obtained numer-
ically and experimentally for ΔT = 9k and Vp = 17 kV is shown in

Fig. 9, considering the investigated case with the highest Ra and L val-
ues. The two diagrams indicate a quantitative and qualitative correla-
tion, providing further substantiation for the consistency between our
experimental and numerical findings. The plotted velocity is generated
at the time when the flow reaches saturation, indicating that growth in
the perturbations has ceased. From the data presented in the figure, it
can be observed that the instability is of a temporal non-stationary
nature, manifesting in oscillating modes. As can be observed in Figs.
7–9, the secondary flow only emerges in the upper half-domain of the
gap, although it penetrates briefly to the lower half-domain. The
observed instability modes cannot be described as any coherent flow
pattern; however, they can be interpreted as intermittent bursts and
plumes with a constant frequency, as illustrated in the figure. Similarly,
the dependence of the number of modes on the flow forcing condition,
observed in the vertical configuration,⁹ was also evident in this
instance. By applying the same electric potential (e.g., Vp = 17 kV), it
was found that the frequency of the oscillating modes decreased as the
temperature difference increased.

Furthermore, a general overview of the flow field can be obtained
by studying the velocities' mean profiles. Figure 10(a) shows the tem-
poral averaged radial profile of the radial velocity at the exact topside
of the gap, which we point out as the jet (U<sub>r</sub><sup>jet</sup>). The profiles were plot-
ted for different V<sub>p</sub> values up to 15 kV. The reason for not plotting the
profiles for higher values is that for V<sub>p</sub> ≥ 16 kV, the asymmetry of the
flow is triggered, and it is no longer possible to recognize the jet posi-
tion, as there will be different inflow and outflow regions in the gap.
Figure 10(a) which corresponds to ΔT = 4 K shows that as the applied
electric potential is increased, the value of (U<sub>r</sub><sup>jet</sup>) also increases, in
accordance with the observations presented in Fig. 8. The same behav-
ior is observed for the other studied ΔT cases. Figure 10(b) presents

<figure><img src="image_16.png" /><figcaption>FIG. 7. Shadowgraphs of the flow with ΔT = 7 K (Ra = 2.16 × 10⁴) and different values of the applied voltage:(a) Vp = 0 kV (L = 0), (b) Vp = 6 kV (L = 5 × 10³), (c) Vp = 14 kV (L = 2.7 × 10⁴), (d) Vp = 15 kV (L = 3.1 × 10⁴), (e) Vp = 16 kV (L = 3.56 × 10⁴), and (f) Vp = 18 kV (L = 4.5 × 10⁴).</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>**FIG. 8.** Flow velocity field evaluated by PIV measurements at the radial-azimuthal plane in the apparatus mid-height for $\Delta T = 2$ K ($Ra = 6.17 \times 10^3$). The velocity field was evaluated for different values of the applied voltage: (a) $V_p = 0$ kV ($L = 0$), (b) $V_p = 6$ kV ($L = 5 \times 10^3$), (c) $V_p = 14$ kV ($L = 2.7 \times 10^4$), (d) $V_p = 15$ kV ($L = 3.1 \times 10^4$), (e) $V_p = 16$ kV ($L = 3.56 \times 10^4$), and (f) $V_p = 18$ kV ($L = 4.5 \times 10^4$). The plots present the temporal averaged velocity magnitude multiplied by the sign of the temporal averaged azimuthal velocity component. The superimposed arrows indicate the direction of the flow and its magnitude.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>FIG. 9. Space time plots of the radial velocity $U_r$ at the midgap ($r=7.5$ mm), for different azimuthal positions [$-\pi/2, \pi/2$]. The plots are related to flows with $\Delta T = 9$ k (Ra = 2.78 × 10⁴), and $V_p = 17$ kV ($L = 5.18 × 10^4$). (a) The plotted velocities are calculated from numerical simulations, (b) the velocities are obtained from experiments.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>FIG. 10. (a) The temporal averaged radial profiles of the radial velocity at the top of the cell ($\pi/2$) referred to as jet velocity ($U_r^{jet}$), for $\Delta T = 4$ K (Ra = 1.23 × 10⁴), (b) the peak value of the $<U_r^{jet}>_t$ radial profile for different $\Delta T$ and $V_p$ values.</figcaption></figure>

the peak value of the $U_r^{jet}$ radial profile for all the studied flow parameters. It clearly demonstrates that $U_r^{jet}$ increases first by increasing $\Delta T$, and for the fixed $\Delta T$ cases how it increases by increasing $V_p$. Moreover, the temporal-azimuthal averaged radial profile of the azimuthal velocity of one pair of cells is defined as:

$$ \langle U_{\phi} \rangle_{t, \phi} = \int_{-\pi/2}^{\pi/2} \int_{t_0}^{t_f} U_{\phi}(r, \phi, z) d(t, \phi), \quad (25) $$

is presented in Figs. 11(a) for $\Delta T = 4$ K, and Fig. 11(b) for $\Delta T = 9$ K, for different $V_p$ values. The same behavior was observed for $U_r^{jet}$ is observed with the $\langle U_{\phi} \rangle_{t, \phi}$. The $\langle U_{\phi} \rangle_{t, \phi}$ profiles show increasing values with increasing $V_p$ until reaching a maximum velocity for $V_p = 17$ kV, after which they decrease for higher $V_p$ values. This phenomenon can be attributed to the fact that high values of $V_p$, where the instability in the gap is more pronounced, the appearance of the radial velocity is not limited to the jets but rather manifests in a variety of in- and out-flow regions within the gap, particularly in the upper half-domain. As the radial velocity contribution increases gradually, the azimuthal velocity values decrease. From Fig. 11, it can be observed that the profiles associated with the base states for the various $V_p$ cases converge to a point approximately in the midgap. This behavior is markedly altered once the instability buffer, where the generated disturbance in the flow is reflected in the $\langle U_{\phi} \rangle_{t, \phi}$ profiles.

### C. Heat transfer

The enhancement of heat transfer by convective forces is commonly described by the Nusselt number $Nu$, which is defined as the ratio between the heat transfer $ht(T(t))$ at a time $t$ when the flow is fully developed and the heat transfer of the conductive state $ht(T_{cond})$, both evaluated at the inner cylinder. $Nu$ is defined as:
<figure><img src="image_20.png" /><figcaption>FIG. 11. The spatial-temporal averaged radial profiles of the azimuthal velocity along one pair of the convective cell (3π/2 ≤ φ ≤ π/2), for (a) ΔT = 4 K (Ra = 1.23 × 10⁴), and (b) ΔT = 9 K (Ra = 2.78 × 10⁴).</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>FIG. 12. Nusselt numbers computed from numerical simulations by formulas (27) and (28) for the range Vp = 0, ..., 17 kV. The dashed lines represent data obtained from 2D simulations, while the dotted lines represent data obtained from 3D simulations.</figcaption></figure>

$$Nu = \frac{ht(T(t))}{ht(T_{cond})}, \qquad (26)$$

with

$$ht(T) = \frac{1}{0.05d} \int_{R_i}^{R_i+0.05d} \int_{0}^{2\pi} \int_{20\text{mm}}^{80\text{mm}} \lambda \frac{\partial T}{\partial r} dr^3. \qquad (27)$$

Thus, we compute the heat transfer as a volume integral of the radial heat flux density over an annulus with a near-unity radius ratio. Approximation of the heat transfer by volume integrals was found to yield more accurate results compared with surface integrals in Ref. 9. We also restrict the axial integration to a smaller subinterval of the full extents (0, 100 mm) to discard any potential boundary layer effects. In the absence of applied electric tension, the computed Nusselt number is found to increase with the temperature difference. The dependence of Nu with the thermal Rayleigh number agrees with the numerical and experimental measurements of Kraussold²⁹ and Tsui and Tremblay.³⁰ The significant impact of the electric field on the temperature sensors, which increases the uncertainty of the measured values, made it challenging to calculate Nu experimentally. However, the satisfactory agreement between the experimental and numerically obtained velocities indicates that the numerically calculated Nu can be applied for analyzing the experimental results. Figure 12 shows that there is a monotonic increase in Nu with both Vp and ΔT. In addition, it can be

seen that for all values of ΔT there is a certain threshold around Vp = 15 kV where there is no longer just an increase in Nu, but also its derivative. The latter increase correlates with a stronger tendency of the flow toward destabilization at very high voltages. We therefore conclude that destabilization offers a substantial benefit in terms of heat transfer in scenarios such as ours. Figure 12 illustrates that for Vp < 15 kV, heat transfer calculations for 2D (circular) and 3D annular geometries with a temperature difference of 2 and 9 K align on a single line. This provides further evidence that the steady flow is 2D and that, for Vp ≤ 14 kV, the calculated heat transfer for the circular geometry is sufficient to describe that in the cylindrical annulus. For Vp ≥ 15 kV, the full 3D model displays a more pronounced rise in Nu compared to the 2D model. This observation suggests the presence of a notable axial velocity contribution, which is absent in the steady regime. To compute the heat transfer in the 2D annulus, we adapt (27) accordingly, i.e.:

$$ht_{2D}(T) = \frac{1}{0.05d} \int_{R_i}^{R_i+0.05d} \int_{0}^{2\pi} \lambda \frac{\partial T}{\partial r} r dr d\phi. \qquad (28)$$

## V. DISCUSSION AND CONCLUSION

The results above show that the destabilization of the flow occurs at relatively high values of the applied electric field, i.e., at Vp = 15 kV at the earliest. However, this does not imply that the electric field for values below the threshold has no influence on the flow. As shown in Figs. 7 and 8, the mean velocity driving the convective cell exhibited an increase as the value of Vp was elevated. This behavior was translated into Fig. 12, which illustrate how the Nusselt number increases as we increase Vp for values before the threshold. While the increase rate is relatively modest compared to the cases after the instability occurs, it is nevertheless discernible. This evident correlation between the enforcement of the convective cells and the
enhancement of the heat transfer is a notable observation. Another noteworthy flow behavior was observed in this study: in the unstable cases, the flow disturbance was predominantly confined to the upper half-domain of the gap, while in the lower half-domain, it was largely stable. A comparable behavior was observed when studying Taylor-Couette flow with differentially heated cylinders in horizontal alignment.³¹ This behavior can be attributed to the fact that as the inner cylinder is heated and the outer one cooled, a stable temperature stratification is achieved at certain points in the lower half-domain, while an unstable stratification is maintained in the upper half-domain. This configuration can suppress any initial asymmetry while supporting subsequent ones. This is consistent with the observations made in our results.

This study uses a combination of numerical and experimental techniques to examine the impact of electric field application on flow dynamics within a confined system comprising two horizontally aligned, differentially heated cylinders. In the absence of an electric potential and a fixed temperature difference, two symmetrical pairs of crescent-shaped convective cells emerge in the gap. As the value of $V_p$ increases, the convective cells become increasingly robust. For the specified temperature difference cases, up to $V_p = 14$ kV, no asymmetry or stagnation in the flow is observed. The strength of the convective cell is responsible for an increase in the rate of heat transfer. For $V_p \ge 15$ kV, the destabilizing effect of the DEP force on the flow became evident for temperature differences of 2 K, while for temperature differences of 4, 7, and 9 K, this effect emerged at a later stage at

<figure><img src="image_22.png" /><figcaption>FIG. 13. Flow velocity field evaluated by PIV measurements at the radial-azimuthal plane in the apparatus mid-height for $\Delta T = 7$K ($Ra = 2.16 \times 10^4$). The velocity field was evaluated for different values of the applied voltage:(a) $V_p = 0$ kV ($L = 0$), (b) $V_p = 6$ kV ($L = 1.4 \times 10^5$), (c) $V_p = 12$ kV ($L = 5.7 \times 10^3$), (d) $V_p = 14$ kV ($L = 7.75 \times 10^3$), (e) $V_p = 15$ kV ($L = 8.9 \times 10^3$), and (f) $V_p = 17$ kV ($L = 1.14 \times 10^4$). The plots present the temporal averaged velocity magnitude multiplied by the sign of the temporal averaged azimuthal velocity component. The superimposed arrows indicate the direction of the flow and its magnitude.</figcaption></figure>
$V_p$ = 16 kV. The disturbed flow was observed to be confined to the upper half of the gap, manifesting as intermittent bursts and plumes. The flow instability was found to exert a pronounced influence on the enhancement of heat transfer within the gap.

## ACKNOWLEDGMENTS

This work was supported by the German Research Foundation (DFG) grant 'Numerische und experimentelle Untersuchungen zur thermoelektrohydrodynamischen Konvektion in einem Zylinderspalt mit unterschiedlicher Orientierung und mit rotierendem Innenzylinder (Nos. EG 100/20-3 and HE 4760/11-3). The authors acknowledge support by the state of Baden-Württemberg through bwHPC and the German Research Foundation (DFG) through grant INST No. 35/1597-1 FUGG. We also want to sincerely acknowledge Dr. Olivier Crumeyrolle for the fruitful discussions.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**M. H. Hamede:** Conceptualization (lead); Data curation (equal); Formal analysis (equal); Investigation (equal); Project administration (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). **J. Roller:** Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (lead); Writing – original draft (equal); Writing – review & editing (equal). **A. Meyer:** Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). **V. Heuveline:** Funding acquisition (equal); Project administration (equal); Writing – original draft (supporting); Writing – review & editing (supporting). **C. Egbers:** Funding acquisition (equal); Project administration (supporting); Writing – original draft (supporting); Writing – review & editing (supporting).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## APPENDIX: ADDITIONAL FLOW VELOCITY FIELDS

Flow velocity field evaluated by PIV measurements at the radial-azimuthal plane in the apparatus mid-height for $\Delta T = 7K$ ($Ra = 2.16 \times 10^4$).

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