Thermo-electrohydrodynamic convection in a dielectric fluid layer with volumetric heating

Abstract

Thermo-electrohydrodynamic (TEHD) convection is investigated in a dielectric fluid layer with internal volumetric heating and boundary heating. The experiment considered a stable stratified fluid layer with an a.c. voltage applied at the heated boundary to destabilise the flow to investigate the evolving TEHD convection by synthetic Schlieren. The voltage magnitude had a range of 2.5 to 17.5 kV with a frequency of 50Hz. The results suggest that TEHD convec-tion onsets at below 7.5 kV that is in line with literature. When the voltage magnitude is increased , flow patterns emerge that are similar to those of the classical Rayleigh-Bénard con-vection cell. However, at higher voltages the volumetric heating increases and thus the flow become chaotic which could lead to large refraction angles that may not be captured by the particular measurement system.

Full text

Fachtagung “Experimentelle Strömungsmechanik”
6.–8. September 2022, Ilmenau
Thermo-Elektro-Hydrodynamische Konvektion in einer
dielektrischen Fluidschicht mit volumetrischer Erwärmung
Thermo-electrohydrodynamic convection in a dielectric fluid layer with volu-
metric heating
Matthias Strangfeld, Oguzhan Bölük, Peter S. B. Szabo, Martin Meier, Antoine Meyer,
Yaraslau Sliavin, Vasyl Motuz, Christoph Egbers
Department of Aerodynamics and Fluid Mechanics, BTU Cottbus-Senftenberg, Cottbus, Germany
Thermische Konvektion, Elektrohydrodynamik, Dielektrophoretische Kraft, Plattenspalt
Thermal convection, Electrohydrodynamics, Dielectrophoretic force, Rectagular cavity
Abstract
Thermo-electrohydrodynamic (TEHD) convection is investigated in a dielectric fluid layer with
internal volumetric heating and boundary heating. The experiment considered a stable strati-
fied fluid layer with an a.c. voltage applied at the heated boundary to destabilise the flow to
investigate the evolving TEHD convection by synthetic Schlieren. The voltage magnitude had
a range of 2.5 to 17.5 kV with a frequency of 50Hz. The results suggest that TEHD convec-
tion onsets at below 7.5 kV that is in line with literature. When the voltage magnitude is in-
creased, flow patterns emerge that are similar to those of the classical Rayleigh-Bénard con-
vection cell. However, at higher voltages the volumetric heating increases and thus the flow
become chaotic which could lead to large refraction angles that may not be captured by the
particular measurement system.
Introduction
Thermal convection of a dielectric fluid layer in a horizontal plate cavity is investigated under
the influence of an electrical force field. To provide a stable satisfied fluid layer the top and
bottom plates of the cavity are maintained at two different temperatures T1 and T2, respect-
ively. This provides a temperature difference of
ΔT =T1– T 2>0
within the fluid layer of thick-
ness d=10 mm. The effect of TEHD convection is provided by the so called dielectrophoretic
effect that is induced in a dielectric fluid when an inhomogeneous electric field is applied in
the presence of a temperature gradient (Pohl, 1978). Up till now scientises have been invest-
igating this effect for several decades in different geometries such as in fluid layers (Roberts,
1969; Stiles, 1991; Stiles and Kagan, 1993), differentially heated cylinders (Chandra and
Smylie, 1972; Futterer et al., 2016; Meier et al., 2018; Szabo et al., 2021; Yoshikawa et al.,
2013) and spherical shells (Busse et al., 2003; Erdogdu et al., 2021; Futterer et al., 2012;
Travnikov et al., 2003; Zaussinger et al., 2020). While the inhomogeneous electric field trig-
gers the convection, the effect on the electric body force may differ between each geometry.
Thus, the inhomogeneity in the electric field for cylindrical and spherical shells is majorly
caused by the curvature whereas for the plate cavity an intrinsic temperature difference
causes the inhomogeneous electric field (Mutabazi et al., 2016). In any case the force dens-
ity is given by (Landau and Lifshitz, 1984) and written as
fEHD =ρeE − ∇
[
ρ
2
(
∂ ϵ
∂ ρ
)
T
E2
]
−E2
2∇ϵ
(1)
where ρe is the density of free charges, E the electric field, ϵ the electric permittivity and T the
temperature. The first term on the right-hand side is the Columb force which can be neg -

lected if the thermal diffusion, charge relaxation, ion migration and viscose dissipation are
much smaller than the oscillation period of the electric field (Turnbull, 1969). The second
term is the electrostrictive force and has no effects on the flow if the fluid is incompressible
and has no mobile boundaries (Zaussinger et al., 2018). The third term is the dielectrophor-
etic effects which is analogues to the Archimedean buoyancy. For small temperature differ-
ences, given by
θ=T – T 0
, the electric permittivity, as well as the density, can be approxim-
ated by linear functions written as:
ϵ=ϵ0ϵr
(
1−e θ
)
and
ρ=ρr
(
1− α θ
)
where ϵ0 is the electric
permittivity in free space, ϵr the relative permittivity at reference temperature T0 and α the
thermal expansion coefficient. The two body forces may than be written for Archimedean
buoyancy as
Fg=ρrα θ g
and it is equivalent for the TEHD convection as
FDEP=− ρrα θ ge
where ge is expressed as an artificial electric gravity as suggested by (Roberts, 1969):
ge=e
ρrα∇
(
ϵ0ϵrE2
2
)
.(2)
Hence, one can define the classical Rayleigh number as
Ra=α ΔT gd3
ν κ
(3)
and its electrical equivalent as
Rae=α ΔT god3
ν κ
.(4)
Where ν is the kinematic viscosity, κ the thermal diffusivity of the fluid and g0 the reference
value of the artificial electric gravity by replacing the terrestrial gravity to formulate an expres-
sion for the TEHD convection, see Mutabazi et al. (2016).
For certain working fluids an a.c. voltage may cause volumetric heating by dielectric loss
which arises e.g. from friction of the orientation and movement of the polarization state of
molecules and can be defined by a temperature increase given by
ΔT IH=ϵ0ϵrω V RMS
2tan δ
8λ
,(5)
with the thermal conductivity λ. Equation (5) represents the equilibrium between the heating
power – contributed by the dielectric loss – and the thermal diffusion (Yoshikawa et al.,
2020). By replacing the temperature difference given by the boundaries of the cavity with the
volumetric heating further Rayleigh numbers may be defined e.g. for terrestrial gravity a
RaIH =αϵ tan δ g d3ω V rms
2
64 νκλ
(6)
and for the electric equivalent as
Rae , IH =ε3e2tan2δ ω2Vrms
6
256 ρ νκ λ2
(7)
That follows the definition of (Yoshikawa et al., 2020). However, Yoshikawa et al. only invest-
igated the internal heating case while here the case with boundary is investigated in addition.
One can define appropriate relations to all Rayleigh numbers to obtain one characteristic
number such as suggested in (Travnikov and Egbers 2021) as Rayleigh number ratios. With
the above theoretical approaches experiments have been conducted by Turnbull (1969) in
rectangular systems which we have further investigated.
The aim and objectives of this study is to investigate TEHD flow driven in the presence of ter-
restrial gravity and volumetric heating to investigate the potential TEHD flow for heat ex-
changers. In this particular study the onset of convection and the pattern formation is invest-
igated which is captured by synthetic Schlieren technique. In the following section the experi-
ment setup is discussed which is followed by an introduction of the synthetic Schlieren tech-
nique and the experiment procedure to record images. The recordings are than post treated
to evaluate the results which are discussed in the following section. A conclusion is given at
the end of the manuscript.

Experiment setup
To establish a horizontal fluid layer a rectangular cavity is designed to meet several require-
ments for investigating TEHD driven fluid flow with non-invasive optical measurement tech-
niques. The housing of the rectangular cavity is built out of Polyoxymethylene (POM) and ac-
rylic glass PMMA to ensure optical transparency for the light spectrum investigated and to
guarantee electrical and thermal insulation. Furthermore, the field of view (FoV) over the
width of the gap is ensured to provide optical access of the measurement system. The fluid
gap is filled with a dielectric fluid. To investigate different gap properties the cavity can be
geometrically modified in depth, length and width by varying the PMMA-inserts with different
thicknesses between 3 and 10 mm. The bottom and top of the cavity are made out of boro-
silicate glass plates coated with ITO (Indium Tin Oxide) as transparent conductive oxide
(TCO) to provide electrical conductive and thermal boundaries. One plate is connected to the
electrical potential while the other is grounded. A high voltage (HV) generator with a peak
voltage of up to 20 kV with a frequency range of up to 1000 Hz creates an electric field
between the top and bottom plates. To maintain a temperature difference between the top
and bottom plates cooling and heating loops are used which are filled with a silicone oil B5.
The temperature stability within the loops are provided by a thermostat water bath and act-
ively controlled by thermocouples which are located at each inlet and outlet of the cooling
and heating channels. These control loops and a sufficient flowrate enable a homogeneous
and an adequate temperature stability to provide a quasi-stable temperature difference
between the two plates. To provide a uniform pressure onto the sealing a frame made out of
aluminium is added to make sure that the whole experiment is tight. For the synthetic
Schlieren technique a foil with a random dot pattern is used that is illuminated and recorded
by a camera. The power of a planar LED-Array of 5.2 W was sufficient to illuminate the foil.
On the opposite of the cavity a camera is mounted to observe an area of 50 by 40 mm. Thus,
the dotted pattern, the dielectric fluid and the cooling and heating loops are therefore in the
FoV of the synthetic Schlieren technique to investigate the effect of TEHD convection in a
laboratory environment.
Figure 1: Design of the plate cavity experimental cell (left) and the setup of the synthetic Schlieren
technique for the horizontal plate cavity (right)

Synthetic Schlieren technique
The synthetic Schlieren technique – also known as Background orientated Schlieren (BOS) –
utilises the changes in the refractive index that are caused by local variations the fluid’s
density. The relation of the refractive index to the density is given by the Lorentz-Lorenz
equation. The light beams get refracted which leads to a refraction angle ε along the optical
axis given by
εx=1
n∫∂ n
∂ x d z
,(8)
εy=1
n∫∂ n
∂ y d z
(9)
where n is the refractive index of the fluid (Settles and Hargather, 2017). The refracted light
beams are captured by a camera mounted in the direction of the undisturbed FoV. With the
distance between the camera and the fluid the absolute distance in refraction is given by Δx
and Δy and can be calculated. To visualise the refraction the illuminated random dot pattern
is used as described in the section above. With the light refraction the doted pattern is
altered to the reference picture that is taken in an undisturbed case. The differences between
both pictures can be calculated with a cross correlation algorithm. This is explained below in
the section post processing. The result is a density gradient of the fluid which can be used to
calculate the divergence field of the density to visualise uprising plumes that are indicative for
convective flow. Hence, the synthetic Schlieren technique is able to produce quantitative
measurement data with the support of the cross-correlation algorithm over a large FoV. In
general, the technical requirement are easy to set up however limited the resolution to the
traditional Schlieren technique that is simply related to the post processing algorithm that av-
erages over a certain window size (Settles and Hargather, 2017).
Procedure
The measurement setup requires two main arrangements. The first deals with the convective
flow which is control by the temperature difference and the a.c. voltage. The boundary tem-
perature difference will be set up with the thermostat water bath while the applied difference
is kept at 1 K for all experiment runs. The adjustment time of the system can be assumed to
be in the range of the thermal diffusion given by
τ=d²/κ
. To ensure that there is a fully de-
veloped thermally stable stratified fluid layer the adjustment time in the experiment should be
at least 1.5 ∙ τ. The high voltage system is triggered by a HV power supply unit. The system
will therefore adjust to the changes. The recording of the measurement starts with the activa-
tion of the high voltage to capture the evolving flow fields. The second arrangement deals
with the synthetic Schlieren setup. For this the LED array and the recording camera is activ-
ated and the focus of the camera is adjusted onto the random pattern foil to be able to take
images that are in focus of the gap with an interval of one frame per second (fps). For the
post processing a reference image of the random dot pattern foil is created for each para-
meter set. After stetting the system, the observation area (FoV) is investigated by capturing
the recorded images for about 15 min for each measurement. The post processing will be
done with a MatPIV algorithm by using MatLab that is explained in the following section.
Post processing
The post processing is realised with a MatLab-code which uses the MatPIV toolbox (Dalziel
et al., 2000; Jongmanns, 2019). A cross correlation algorithm is used that finds it origin in
post processing PIV images. To capture the refractive index changes a reference picture,
see Figure 2 (a) is of need to compare the optical displacement based on the fluid’s refract-
ive index, see Figure 2 (b). The quantitative investigation is realised by creating converting
the pixels (px) in the pictures to a real physical distance. In the next step the MatPIV al-
gorithm is used to calculate the shifting between the flow and the reference picture by a
cross correlation algorithm. Therefore, the picture is split into interrogation windows of the

size of 64 px x 64 px with an overlapping of 50 % afterwards the windows for the correlation
gets iteratively smaller by half the size up to a size of 16 px by 16 px. A field of the refraction
vectors can be calculated for each second of the measurements. To obtain the divergence
field several filters are used to count for the errors. This is in particular the signal-to-noise ra -
tio (snrfilt) which calculates the mean power of the used signal and divides it by the power of
the background noise. Vectors smaller than a factor multiplied by the noise power will be
changed to „NaN“ and thus no longer considered. A common factor is 1.3 which is used for
our calculations (Keane and Adrian, 1992). The global histogram operator (globfilt) removes
vectors with values significantly higher or smaller than the mean size of vectors. For realisa-
tion the global limits are determined by calculation of the mean value of all vectors plus or
minus the standard deviation multiplied by a factor of 3.5. Similar to the global filter is the
local filter that compares the vector size with those in adjacent direction. Therefore, a Kernel
size – standard is 3x3 vectors – around the vector under investigation is created. If the value
overrides the meridian plus or minus the threshold times the standard deviation of the Kernel
the vector will be filtered out. The errors and out-filtered vectors are replaced with interpola-
tion techniques due to the completeness of the vector field. Therefore, the “naninterp” filter
function is used. The linear interpolation in the experiments sorts all outliers based on the
number of spurious adjacent vectors and interpolates them from a low amount to the highest
number of adjacent errors. So the places of NaN‘s can be filled. These filters are used to op-
timise the quality of the density gradient field which forms the base for the subsequent calcu-
lation of the divergence field.
Figure 2: Recorded raw pictures shown in (a) as a reference image with the undisturbed case and in
(b) with displacement at Vpeak =10 kV.
Results
The results are obtained by the working fluid with 1-Nonanol with fluids properties shown in
Table 1. Figure 3 shows the post processed divergence fields for different voltages applied.
Due to the constant temperature difference the Rayleigh number for natural convection is
kept constant at Ra = 7254 for all measurements. However, when applying the voltage a
density change maybe visualised. For small voltages, see Figure 3 (a) no significant diver-
gences are observed which is indicative that TEHD convection is unable to destabilise the
stratified fluid layering. However, small disturbances are recorded in the density which may
refer to the fluid flowing in the cooling and heating loops. When increasing the voltage the di-
vergence is visible seen in Figure 3 (b) as not only the dielectrophoretic force but as well the
volumetric heating increases and perturbates the fluid causing convective flow within the cav-
ity that increase with voltage seen in Figure 3 (c) and (d). The effect is qualitatively observ-
able by certain cell formation which are reminiscent of Marangoni-like patterns that can be
described by arising Rayleigh-Bénard convections cells. Due to the transition in the voltages
in the range of 2.5 kV to 7.5 kV a critical point can be assumed at which the flow becomes
unstable and a dielectric driven convective flow under terrestrial gravity becomes visible. For
voltages in the range of 7.5 kV < Vpeak < 15 kV a quadrangular and pentagonal structures are
visible in the divergence field, which is indicative of rising plumes of warmer fluid. Therefore,
a wavelength λp can be introduced which describes the distance of the central points of two
neighboured structures. For 7.5 kV to 15 kV the wavelength is about
λp≈ d /2
.
a) b)

Table 1: Fluid properties of 1-Nonanol for T=293K at a frequency of 50 Hz (Zaussinger et al., 2018)
Property Value Dimension
Density ρ829.1 kg ∙ m-³
Kinematic viscosity ν1.4 ∙ 10-5 m² ∙ s-1
Thermal diffusivity κ7.94 ∙ 10-8 m² ∙ s-1
Thermal conductivity λ0.16 W ∙ K-1 ∙ m-1
Heat capacity cp2.470 ∙ 103J ∙ kg-1 ∙ K-1
Coefficient of thermal expansion α8.2344 ∙ 10-4 K-1
Prandtl number Pr 176 -
Thermal permittivity e1.021 ∙ 10-2 K-1
Energy dissipation factor tan δ 14.63 -
Refractive Index n1.43 -
Relative electric permittivity ϵr9.056 -
For larger voltages in the range of 15 kV to 17.5 kV the polygonal pattern appears to de-
crease and chaotic structures appear that are not clearly distinguishable as quadrangular
and pentagonal structures. The reason could be related to a higher refraction angle that is
present when the flow increases. However, the cross-correlation algorithm may run into diffi-
culties that can be caused by the refraction of the dot pattern which is critical when individual
dots are not any more clearly visible and thus are impossible to be tracked correctly. The in-
fluence of the internal heating is important to considered for this particular working fluid and
of course cannot be neglected for higher voltages. This can be shown with the dimensionless
number ϒ for the comparison volumetric heat generation see (eq. 5) with the diffusion of
thermal energy, given by (Yoshikawa et al., 2020). For ϒ < 1 the heat generation is smaller
than the diffusion of thermal energy and therefore the influence of the volumetric heating is
negligible. For ϒ ≥ 1 the internal heating may influence the flow significantly for this particular
working fluid. To provide an overview the particular number is shown in Table 2 for each
Figure 3: Divergence field of the refractive index gradients for the range of applied voltages in
(a) Vpeak =2.5 kV, (b) Vpeak=5 kV, (c) Vpeak=10 k and V; (d) Vpeak=17.5 kV.

voltage. The occurrence of volumetric heating increases the convective flow and related to
an internal temperature increase above the boundary temperature that can in addition cause
convection within the system.
Table 2: Characteristic numbers for the range of high voltages applied to the experimental system at a
temperature difference of 1 K.
Vpeak 2500 V 5000 V 7500 V 10000 V 12500 V 15000 V 17500 V
Rae28 113 255 453 708 1019 1387
TIH 0.9 3.5 7.9 14.15 22.1 31.9 43.35
RaIH 802 3207 7217 12831 20048 28870 39295
Rae,IH 6 355 4040 22702 86601 2.59 ∙ 1056.52 ∙ 106
ϒ7 29 65 115 180 259 353
Discussion
The experiment shows a clear development of convection within the thermally stable strati-
fied fluid layering. This is due to the conducting TCO coating of the glass boundaries and the
applied electric field inducing the dielectrophoretic force field together with volumetric dielec -
tric heating of the working fluid 1-Nonanol fluid. This provides a destabilising effect and a de-
velopment of convective flow within a stratified fluid by buoyancy. This effect leads to a tem-
perature gradient and increase the heat transfers between the boundaries. The results show
that the a.c. voltage can induce a force field that is strong enough to overcome the thermal
stratification and is therefore enable to induce a convective flow. Thus, light refraction ap-
pears that is caused by the inhomogeneous density gradient cause by a temperature distri-
bution. The recorded images provide a qualitative understanding of the flow and the resulting
pattern that are comparable to Rayleigh-Bénard convection. For a quantitative statement the
correlation between the refraction and the density gradient should be used in future experi-
ments to calculate the temperatures out of the divergence field (Hayasaka et al., 2016). The
qualitative results can be improved by increasing the frame rate of the camera and thus cap-
ture the movement of the polygonal structures in a better shape to make them traceable.
Conclusion
A high voltage is applied between the boundaries of an externally heated fluid layer to estab-
lish a dielectrophoretic volumetric force with volumetric internal heating. This force was able
to destabilise the thermally stratified dielectric fluid layer and observed by synthetic Schlieren
technique to visualise the evolving temperature fields by light refection. To investigate differ-
ent cases an a.c. voltage of 50Hz was applied with its magnitude ranging form 1.5 kV to 17.5
kV with increment of 2.5 kV. Within the voltage range of 2.5 kV to 7.5 kV a critical point is
found where the fluid layer destabilises due to the increasing dielectrophoretic force field.
The recorded and post treated images show above the critical threshold polygonal structured
pattern which are reminiscent to Rayleigh-Bénard convection cells. An interpretation of the
pattern movement is not possible due to the low acquisition frequency. However, it is ob-
served for voltage higher than 15 kV that polygonal structures decreases and the refraction
becomes more chaotic. In addition, the introduced heating number ϒ indicated that volumet-
ric heating is not negligible for high voltage and shows some significant influences on the
flow as was suggested by (Yoshikawa et al., 2020). The observed chaotic flow may occur
due to the post processing progress and the large refractions of light that are not easy to be
traced by the cross correlation algorithm. To determine the cause of the deceasing structures
by higher voltages as well as the quantitative investigation of the temperature field by the di -
vergence of the density gradient field needs to be further investigated.
Acknowledgements
The project “Thermoelektrische Konvektion unter Schwerelosigkeit (TEKUS)” is supported by
the BMWi via the space administration of the Deutsches Zentrum für Luft und Raumfahrt
(DLR) under Grant No. 50WM1944.

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