Journal of the Taiwan Institute of Chemical Engineers 135 (2022) 104369

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Journal of the Taiwan Institute of Chemical Engineers
journal homepage: www.elsevier.com/locate/jtice

Flow transition of magnetohydrodynamic bubbly jet driven by Lorentz
force
Jia-Hong Cheng, Li-Wei Cheng, Ching-Yao Chen*
Department of Mechanical Engineering, National Yang Ming Chiao Tung University, Taiwan R.O.C., 1001 University Road, Hsinchu, Taiwan 300, R.O.C.

A R T I C L E

I N F O

Article History:
Received 2 December 2021
Revised 12 April 2022
Accepted 5 May 2022
Available online xxx
Keywords:
MHD JetLorentz force
Reynolds number
Turbulence breakup

A B S T R A C T

Background: Magnetohydrodynamic (MHD) jet in saltwater possesses potential applications to improve
marine vehicle propulsion and hydrogen collection. In the present paper, pattern and ﬂow transition of MHD
bubbly jets are studied. Bubbles of oxygen and hydrogen, produced by electrolysis in saltwater and displacedby the Lorentz force, form a typical multiphase MHD jet ﬂow.
Methods: Taking advantage of the brightness of gaseous bubbles, emergence of bubbly jet can be directly
observed. Pattern of the bubbly jet is experimentally conﬁrmed to resemble the saltwater jet, and used forfurther analysis.
Findings: A Lorentz-force based Reynolds number ReL is proposed to demonstrate the pattern similarity of
the MHD jet, and thus applied to categorize the transition of ﬂow regimes from laminar to turbulent. For ajet of lower ReL , the pattern of the bubbly jet appears as a continuous, wavy and laminar stream. Turbulent
breakups start to evolve at farther downstream of the bubbly jet once the ReL exceeds a critical value. The
critical value of ReL  1500 is veriﬁed both by visual observation and relevant quantitative measures of the
bubbly jet.
© 2022 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

1. Introduction
Magnetohydrodynamic (MHD) concerns the dynamics of conducting or magnetic ﬂuids driven by magnetic and electric ﬁelds. A branchof MHD is driving conducting ﬂuids, e.g., plasma, liquid metal andsaltwater etc., by the Lorentz force generated by perpendicularly
placed magnetic ﬁeld and electric current. Applications of this particular branch of MHD had been interests of engineers since its initiation in 1940s [1], e.g., propulsion systems [2,3], pumps [4,5],
motors [6] and silicon growth [7]. More recently, MHD is also actively
studied for the enhancement of heat transfer in conductive nanoﬂuid
ﬂows [812]. The research for these practical applications mainly
focused on the internal ﬂow phenomena bounded by electric ﬁeld
where the Lorentz force is generated. In the meantime, the MHD jet
ﬂow driven by the Lorentz force is also an important issue in astrophysics [13,14] and atmospheric pressure plasma jet [15,16], inwhich the conductive medium is plasma.
On the other hand, the conventional jet ﬂows, which neither electro nor magneto effects are involved, had been thoroughly investigated for decades, e.g., reviewed in Refs. [17,18]. The formation of
turbulent jet is generally described as three main regions, i.e., thepotential core, the developing region and the self-preservation* Corresponding author. Tel.: 886-3-5712121ext55125.
E-mail address: chingyao@mail.nctu.edu.tw (C.-Y. Chen).

region [19,20]. When the jet ﬂow discharges immediately from thenozzle exit, the ﬂow is mostly unperturbed to form a potential core.
The region from the exit to the end of the potential core is also called
the zone of ﬂow establishment (ZFE). As interaction with the ambient
ﬂuid starts, the well-known Kelvin-Helmholtz instability emerges on
the mixing layers. This region is referred to as the developing region.
Signiﬁcant turbulence is triggered in the developing region, andeventually results in the region of self-preservation, a state that theﬂow ﬁeld can be solely determined by local behaviors. The developing region and the self-preservation region are commonly referred toas the zone of established ﬂow. In addition, the Reynolds number(Re), given by Re ¼ VL=n, had been well veriﬁed to dominate thedevelopment of ﬂow regimes in two classic works [19,21]. Here, V, L
and n represent the jet exit velocity, the jet diameter and the kinematic viscosity of the discharge ﬂuid, respectively. By visuallyinspecting the laminar length [19], fully laminar jet develops for
Re < 500, while the laminar length completely disappears for
Re > 2500. Determined by the turbulence dilution represented bytemperature, a critical Reynolds number of Re  1500 is quantitatively concluded [21]. Beyond this critical Reynolds number, dependence of the turbulence dilution on the Reynolds number isinsigniﬁcant. Numerous follow-up studies regarding the role of the
Reynolds number to the formation of jet ﬂow had been presented in
the last decades. For instance, the decay rate of the mean ﬂow
appears nearly independent on the Reynolds number for Re104 in a

https://doi.org/10.1016/j.jtice.2022.104369
1876-1070/© 2022 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

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circular jet from a smoothly contracting nozzle [22]. The length of ZFE
was experimentally conﬁrmed to decrease with increasing Reynolds
number for Re5412 [23]. Besides, the Reynolds number also plays
an important role to the oscillatory behavior of mutually impinging
jets in a mixing chamber. The frequency of the impinging jet oscillations is dictated by the Reynolds number, in which a transition
regime occurs at 1500 < Re < 2000 [24].
Another important issue studied intensively is the two phase jet,e.g., liquid jet associated with atomizing gas stream [2527]. The coexistence of two phases, such as faster gas in a slower liquid jet,results in a shear instability on the interface, and forms a wavy
stream. The wavy stream might further evolve into ﬁlaments, andeventually breaks into numerous droplets. When the atomization isincomplete, the breakup of liquid jet is non-axisymmetric with awavelength larger than the jet diameter [25]. A ﬂapping instability is
recently observed in an incompletely atomized liquid jet, in which
regime of the ﬂapping instability is determined by the wavelengthand liquid jet diameter [26]. Similar instability also occurs in an oscillating jet [28,29], such that the amplitude of oscillation is enhancedby the formation of a low pressure region near the leading edge ofthe deﬂected jet [29]. The control of self-sustained jet oscillations,which is highly relevant to the above literature, is important for
many industrial applications, e.g., steel casting.
The MHD saltwater jet differs from the conventional liquid-gas jetby two features, i.e., driven by the Lorentz force and chemical reactions associated with electrolysis. It possesses potential applications
to improve marine vehicle propulsion and hydrogen collection. In thepresent study, the products of reactions include gaseous bubble(hydrogen and oxygen) by water electrolysis and solid precipitates(aluminum hydroxide) from the aluminum electrodes. It is interesting to understand if the two-phase jet driven by the distinct Lorentz
force evolves similarly with the conventional pressure-driven jet.
Besides, unequal amount of gaseous bubbles are produced at electrodes, which provide perturbations. The perturbation might possibly
lead to the jet oscillation. To our best knowledge, the inﬂuences ofthese factors to the MHD saltwater jet is rarely studied. In this paper,
we focus on the pattern formation of the MHD jet. Parametric analysis is carried out, mainly by experiments, to identify transition of
ﬂow regime.

Z
I¼

Ac

JdA:

Magnetohydrodynamic (MHD) jet ﬂows driven by the Lorentz
force are studied. It is well known that the so-called Lorentz force isgenerated by perpendicularly oriented magnetic ﬁeld and electric
ﬁeld, such that the force density, denoted as F, can be expressed asð1ÞHere, J and B are current density and magnetic ﬁeld, respectively. To
experimentally study the MHD jet, the experimental apparatus isshown in Fig. 1. A pair of aluminum electrodes (drawn in blue), heldby two acrylic plates (yellow) on their sides forming a 55mm-longopen channel with width w in between, are placed inside a rectangular acrylic tank partially ﬁlled with saltwater of depth h ¼ 5mm. The
saltwater used in the experiments is the solution of distilled waterand chemical salt, such that 35g salt is fully mixed in 1000ml water.
Density and viscosity of this saltwater solution is r ¼ 1023kg/m3 andm ¼ 0:00097N ¢ s/m2 , respectively. The electric conductivity of thesaltwater is s ¼ 5 S/m measured by conductivity meter, where S
stands for Siemens. A 40mm20mm10mm permanent NdFeB magnet (red) is placed right beneath the tank to provide an upward magnetic ﬁeld B. Power suppliers are connected to the electrodes togenerate an electric current I through the conductive saltwater andinduce the Lorentz force F. The relation of current I and current density J is expressed asð2Þ

Here, Ac is the area of cross-section perpendicular to current I. By thissetup, the Lorentz force aligns with the axial direction of channel,
denoted as x-direction as shown in Fig. 2, and drives a jet ﬂow when
the current presents. In the meantime, saltwater undergoes electrolysis, so that oxygen and hydrogen is generated at the anode and thecathode, respectively. It is noticed that, additional chemical reaction
occurs on the aluminum electrodes, in which solid precipitates of aluminum hydroxide particles, i.e., AlðOHÞ3 , are produced. The overall
reaction equation is
8H2 O þ 2Al ! O2 þ þ5H2 þ 2AlðOHÞ3ðsÞ :2. Physical problems and experimental apparatus

F ¼ J  B:

Fig. 1. Principle sketch of experimental apparatus: top and side view are shown on the
top and bottom ﬁgures, respectively.

ð3Þ

The subscript (s) denotes solid precipitates. As a result, the volume ofhydrogen produced at the cathode is 5 times higher than oxygen at
the anode, which is experimentally conﬁrmed by collecting the gases
separately in a pure electrolysis without magnetic ﬁeld. Aluminumhydroxide precipitates, which are turbid particles, are also visibly
observed. It is noteworthy that, the much higher volume of hydrogenproduced at the cathode signiﬁcantly affects the jet formation for relatively lower current strength, which will be presented in the latter
sections.
The magnetic ﬁeld B generated by the permanent magnet can bepredicted by the Maxwell equation, i.e., r ¢ B ¼ 0. It is numericallysolved by the commercially available software ANASYS Maxwell, andshown in Fig. 2 is the simulated result on the bottom plane of thetank. To better illustrate the position, an x  y coordinate, whose origin is located at the center of the channel exit, is deﬁned as marked
in Fig. 2. By this coordinate, the channel is located at 55mm
x0mm (length) and w=2yw=2 (width), while the position ofmagnet is at 40mm x0mm (length) and 10mm y10mm(width). Within the central region above the magnet, i.e., 30mm
x  10mm, the ﬁeld strength varies insigniﬁcantly with a maximum of B  0:19T. Beyond this region, the strength decays rapidly.
Also shown in Fig. 2 is the ﬁeld distribution along the centerline of
the channel, i.e., y ¼ 0 and 55mm x0mm. The measured ﬁeld
strength by Gauss meter along the channel is also shown for

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3

Fig. 2. Left: Numerical simulation of magnetic ﬁeld strength. The positions of magnet, electrodes and acrylic plates referred in Fig. 1 are also drawn. Right: Comparison of measured
magnetic strength with numerical result along the center line of the channel,.i.e., y ¼ 0 and 0:055x0m. Fairly Good agreement is obtained.

comparison. Fairly good agreement is obtained between the numerical simulation and the actual measurement. We take the average ofmeasured ﬁeld strength of B0 ¼ 0:147T within the length covered bymagnet (40mm x0mm) as a representative strength for the latter analysis.
Productions of gases and solid precipitates result in the present
problem a typical multiphase ﬂow. To visualize the jet pattern, we
take advantage of the motion of gaseous bubbles, i.e., the bubbly jet,to represent the formation of MHD jet ﬂow. When the Lorentz force
is generated by electric current and magnetic ﬁeld, top view (x  yplane) of the bubbly jet ﬂow is recorded by a CCD camera (GoPro
Action with a fame speed of 60 fps) placed at the exit of the open
channel. The pictures taken are imported into the Matlab software tofurther enhance the contrast, and presented in the latter sections.
New saltwater solution is used in every experiment to ensure thesame conductivity of the electrolyte.
3. results and discussion
In this study, the pattern formation of MHD bubbly jet is presented and discussed. Important parameters to the jet ﬂow, including
current strength I and channel width w, are varied to evaluate theireffects.
3.1. Pattern formation
The MHD bubbly jets, represented by motion of oxygen andhydrogen produced respectively at the anode (top electrode) and thecathode (bottom electrode), are ﬁrstly presented in a series of experiments with increasing current strength I for w ¼ 4mm. Shown in
Fig. 3 is the experimental snapshots of I ¼ 1A. Under the present condition of relatively weak Lorentz force, i.e., low electric current, mixing of the produced gases at early time is insigniﬁcant, so that two
apparent layers, i.e., top oxygen layer and bottom hydrogen layer, aredistinguishable inside the main structure of jet. As a result, the top
and bottom interfaces of bubbly jet are mainly oxygen-saltwater andhydrogen-saltwater interface, respectively, and referred to as theoxygen interface and the hydrogen interface hereafter. At t ¼ 1:73s
shown in Fig. 3(a), a pair of counter-rotating vortices are formed at
the front of jet by these two interfaces. Because of the unequally produced amount and different molecular weights of oxygen and hydrogen, the vortex pair results in an asymmetric bow-shaped front asshown at t ¼ 3:47s shown in Fig. 3(b). In the meantime, the gaseous
bubbles ﬂoat upward to saltwater/air surface because of gravity.
These bubbles are no longer visible once they escape from the saltwater. This explains the fewer amount of bubbles (represented byweaker brightness of image) at the downstream, especially for thehydrogen interface, where hydrogen is much lighter than oxygen.
The jet appears as a slim, wavy, laminar and continuous bubbly
stream at t5:20s shown in Fig. 3(c). It is noticed that, the pattern ofwavy jet shown in Fig. 3 resembles the results reported in the literatures, e.g., experiments in incompletely atomized liquid jet [25] andsimulations of spatially oscillating planar two-phase liquid jet [29].
The pattern resemblance suggests that production of unequalamount of oxygen and hydrogen results in an effect similar withincompletely atomization or spatial perturbation. For the case ofstronger Lorentz force of I ¼ 4A shown in Fig. 4, the main body of jet
is much thicker due to larger amount of gases produced. The jet isstrongly dispersive forming a bubbly stream with signiﬁcant oscillation, i.e., shorter wave length and larger amplitude, at its upstream
near the exit. On the other hand, typical turbulent feature of jet
breakup evolves at farther downstream. As a result, similar with theconventional jet, the formation of the present MHD bubbly jet can be
categorized into two regions by visual observation, i.e., (I) laminar
bubbly stream near the exit, and (II) turbulent breakup at farther
downstream. The length of nearly continuous laminar bubbly stream,
denoted as Lc , is cutoff because of the downstream breakup.
The ﬂows presented above indicate the evolution of jet from fullylaminar bubbly stream to partially turbulent breakup by increasing
the Lorentz force. A Reynolds number, the dimensionless parameter
determining the tendency toward turbulence, based on the control
input parameters is proposed to determine the transition. If length,
qﬃﬃﬃﬃﬃﬃﬃﬃ3ﬃ qﬃﬃﬃﬃﬃﬃﬃ
time, velocity and pressure are scaled by DH , rBD0HI , rBD0 HI , and BD0HI,

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J.-H. Cheng et al. / Journal of the Taiwan Institute of Chemical Engineers 135 (2022) 104369

Fig. 3. Temporal evolution of MHD bubble jet for I ¼ 1A, w ¼ 4mm and h ¼ 5mm, whose correspondent Reynolds number deﬁned by (3.4) is ReL ¼ 842. The top-viewed images
shown are mainly the gaseous bubbles, including oxygen and hydrogen respectively generated at the anode (top electrode) and the cathode (bottom electrode), by electrolysis ofsaltwater. The overall formation of bubble jet remains laminar.

Fig. 4. Temporal evolution of MHD bubble jet for I ¼ 4A, w ¼ 4mm and h ¼ 5mm (ReL ¼ 1685). Turbulent breakups are triggered at farther downstream.

respectively, the Reynolds number based on the Lorentz force ReL isdeﬁned aspﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
rB0 DH I
ReL ¼
:
ð4Þ

m

Here, DH is the hydraulic diameter of the channel. The dimensionlessqﬃﬃﬃﬃﬃﬃﬃﬃ3ﬃ
time t  and length x are obtained accordingly as t  ¼ t= rBD0HI andx ¼ x=DH , respectively. By the scaling, the jets presented in Figs. 3

and 4 correspond to ReL ¼842 and 1685, respectively. It is noticed
that, the images shown in Fig. 3(a-e) and Fig. 4(a-e) are at about thesame dimensionless times t  for direct comparison. The longest visible positions of jet front at the downstream, denoted as xF , are very
close at near dimensionless times for all the cases. As expected, thejet ﬂow evolves from fully laminar bubbly stream at low ReL ¼ 842 to
turbulent breakup for a sufﬁciently high ReL ¼ 1685. For an evenhigher ReL ¼ 2064 (I ¼ 6A) as shown in Fig. 5, in which the image istaken when xF  165mm, Lc is greatly shortened because of strong

J.-H. Cheng et al. / Journal of the Taiwan Institute of Chemical Engineers 135 (2022) 104369

5

Fig. 5. (Left) Highly turbulent MHD bubble jet of I ¼ 6A, w ¼ 4mm and h ¼ 5mm (ReL ¼ 2064). (Middle) Correspondent binary image indicating the length of continuous gaseous
stream (Lc ) and spread angles (u1 and u2 ). (Right) Image of red-dyed experiment with the same experimental parameters.

Fig. 6. Binary images of I ¼ 1A (ReL ¼ 842), 2A (ReL ¼ 1191), 3A (ReL ¼ 1459) and 4A (ReL ¼ 1685). Shorter continuous gaseous stream (Lc ) associated with larger total spread angle(u ¼ u1 þ u2 ) for higher current strength (or ReL ) is observed.

Fig. 7. Jet formation of wider channel w ¼ 6mm and h ¼ 5mm for various I ¼ 3A (ReL ¼ 1615), 4A (ReL ¼ 1865) and 5A (ReL ¼ 2086).

turbulent breakup. Besides, the spreading area of gaseous bubbles isdramatically increased.
Another important issue is the appropriateness of the above
visualization, based on bubbly stream, to represent the jet formation. The bubbles ﬂoat vertically toward the saltwater surface
because of buoyance, so that numerous bubbles will eventually disperse into the air once they reach beyond the water surface. The
prominence, or brightness, of jet formation is expected to be
weaker at farther downstream. In addition, bubbles might burst or
coalesce. These phenomena result in much fewer bubbles remainedin the downstream region. The jet ﬂow by the conventional reddyed saltwater for the highly turbulent case of ReL ¼ 2064 is also
shown in Fig. 5. Overall pattern of the red-dyed jet shows strong
similarity with the image represented by bubbly stream. It isnoticed that, the red-dyed jet ﬂows of various current strengths,
e.g., I ¼ 1 » 6A, have all experimented, and similar patterns withbubbly jets are conﬁrmed.

To more easily identify some interesting phenomena of MHD jets,
the original bubble image of ReL ¼ 2064 is converted to binary format
by Matlab as also shown in Fig. 5. More binary images of lower current strengths taken when xF  165mm are demonstrated in Fig. 6,
i.e., ReL ¼ 842, 1191, 1459 and 1685. It can be seen that the conversion to binary format accurately preserves the main pattern of bubbly
jet. Consequently, useful quantitative measures, such as the length ofcontinuous gaseous stream Lc and the spreading angles, top and bottom angle respectively denoted as u1 and u2 and depicted in Fig. 5,
can be obtained. It is noticed that, no exact value of Lc can be determined considering the dispersive nature of bubbles. We ﬁrst calculate
the bubble density, represented by number of dark pixel in the binary
image, along the jet. The length of Lc is then approximated when the
bubble density drops dramatically. By visually inspecting thesebinary images, ﬂow transition to trigger turbulent breakup by
increasing ReL is qualitatively categorized. For cases of ReL 1191, thejet formation appears a nearly continuous bubbly stream. Even

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J.-H. Cheng et al. / Journal of the Taiwan Institute of Chemical Engineers 135 (2022) 104369

Fig. 8. (a) Areas occupied by gaseous bubbles BA measured at t ¼ 1s for various current strength I, and (b) dimensionless bubble area BA =D2H obtained at t   70 versus ReL . The
growth of dimensionless bubble area is much higher for ReL > 1500.

Fig. 9. (a): Spread angles u of MHD bubble jet at xF  165mm for various current strength I, and (b) u versus ReL at dimensionless position of x  30. The growth of u shows rapid
increase for ReL > 1500.

apparent oscillation occurs at the downstream region, the ﬂow
remains laminar. If the Reynolds number is increased to ReL 1459,
chaotic turbulent breakup evolves at downstream, while oscillationof the remaining laminar bubbly stream at upstream is more signiﬁcant. A general trend of shorter Lc associated with larger total spreadangle (u1 þ u2 ) for larger ReL is conﬁrmed. It is noteworthy the transition of turbulent breakup occurs at ReL 1459 is closed to the criticalvalue of Re ¼ 1500 reported in Ref. [19]. More discussion regarding
the quantitative analysis of the ﬂow transition will be presented inlatter sections.
Change of channel width w results in different hydraulic diameter
DH , so that the correspondent ReL is altered. It is expected that the jet
would behave more unstable for a wider w if the rest of parameters
remain unchanged. The jet formation of I ¼ 3A (ReL ¼ 1615), 4A(ReL ¼ 1865) and 5A (ReL ¼ 2086) with w ¼ 6mm are shown in Fig. 7.
Turbulent breakup starts to evolve for ReL ¼ 1615 at downstream .
The turbulent breakup is more prominent for the cases of ReL ¼ 1865
and 2086, which is consistent with the observation described in theabove paragraphs. It is interesting to compare the cases of close Reynolds number but different combination of control parameters, e.g., R
eL ¼ 1685 (Fig. 4d) and 1615 (Fig. 7), ReL ¼ 2064 (Fig. 5) and 2086
(Fig. 7). The jet formation for these two pairs of close ReL appears
very similarly. The pattern similarity supports the appropriatenessfor applying the proposed Reynolds number based on Lorentz force
to consider the pattern formation of MHD jet. Flow transition, i.e.,
downstream turbulent breakup, occurs at a critical Reynolds number
of ReL  1500.

3.2. Quantitative analysis
In this section, we turn to more quantitative analysis to the MHD
bubbly jet. The ﬁrst measure presented in Fig. 8(a) is the observed bubble area BA at a representative time of t ¼ 1s for various I. To obtain thearea, number of dark pixel of the binary image is ﬁrst counted, andconverted to area. On the other hand, the bubbles, including oxygenand hydrogen, are produced by the water electrolysis, so that theamount can be calculated by the total electricity applied. If the gas is
assumed to fully occupy the channel, the theoretical area is also shownin Fig. 8(a) as a reference. Fairly good agreement is obtained for sufﬁciently weak current strength of I ¼ 1A. Because of the buoyancy anddispersion, bubbles spread and ﬂow upward, so that the observed BA ismuch larger than the theoretical prediction for I2A. The BA grows
much more rapidly for increasing I, despite of a liner proportionality
by theoretical prediction. BA for wider w ¼ 6mm is generally larger
than the cases of w ¼ 4mm, which reﬂects stronger ﬂow dispersion toenhance the spread of bubbles. To obtain a more general behavior ofobserved bubble area, dimensionless results are shown in Fig. 8(b).
The dimensionless area, measured at an identical dimensionless timet   70 and scaled by D2H , is plotted versus ReL . The dimensionless area
appears a nearly monotonic growth with ReL . In addition, the growthincreases rapidly for ReL > 1500, which indicates ﬂow transition to turbulence of strong dispersion.
Another measure of interest is the spreading angle u, i.e., the top
spreading angle u1 and the bottom spreading angle u2 as indicated inFig. 5. To make direct comparison, all the measures are taken at

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7

Fig. 10. (a) Applied electric potential c for various current strength I, and (b) effective electric resistance R versus ReL . Theoretical values predicted by the Ohm’s Law are also shownby dash lines.

xF  165mm as the sample images shown in Figs. 5 and 6. As shownin Fig. 9(a), the spreading angles are widened as the current strengthis raised. The top and bottom spreading angle appears nearly symmetric for higher current strengths, which indicates the pattern ismainly dominated by ﬂow rather than the local production of gases.
The spreading angle for w ¼ 6mm is generally wider than the case of
w ¼ 4mm under the same I. If u is plotted versus the ReL at an identical dimensionless position at x  32 as shown in Fig. 9(b), monotonic increase with ReL is nearly maintained. The u also starts to show
a rapid growth for ReL  1500. Overall, the dependence of u on ReL issimilar with dimensionless BA shown in Fig. 8(b), which suggests thetransition of ﬂow regime is dominated by ReL and strong turbulent
breakup is triggered at ReL  1500.
Before concluding this section, we discuss an interesting effect of
the solid precipitates of AlðOHÞ3 . When considering the MHD applications, the input power is an important factor often discussed. In thepresent study, the direct control parameter is current strength I (orcurrent density). Since gases and precipitates are produced, the conductivity of the solution would not be the same as the pure saltwater.
As a result, the applied electric ﬁeld, represented by the electric
potential c, can be roughly used to assess the effect of individualcomponent. Production of gaseous bubbles increases the total electric
resistance of solution, so that the applied electric potential to maintain the same current strength is expected higher than the case ofpure saltwater. The actual electrical potential c applied to maintain
constant current strength I is plotted in Fig. 10(a). The potential
needed for the case of pure saltwater, whose linear dependence canbe obtained theoretically by the Ohm’s Law, such as c ¼ Iw=s Ac , isalso shown for comparison. As expected, higher potential is needed
for a wider channel of w ¼ 6mm because of more amount of saltwater between the electrodes. It is interesting to notice that, the applied
potential c appears slightly lower than the theoretical value forintermediate current strength, e.g., 2A I5A and 1A I2A for
w ¼ 4mm and 6mm, respectively. Only for sufﬁciently weak andstrong current strength, higher potential than theoretical value isneeded as one might expect. The reason for this inconsistent behavior
is due to the presence of solid precipitate AlðOHÞ3 . Since AlðOHÞ3 ismore conductive than saltwater, so that production of AlðOHÞ3 could
reduce the overall electric resistance. To conﬁrm the effect ofAlðOHÞ3 , a reference experiment of I ¼ 1A and w ¼ 4mm is performedby using saltwater solution originally containing AlðOHÞ3 . As also
shown in Fig. 10, the potential needed and the correspondent resistance, obtained by the Ohm’s Law, is lower than the experiment ofpure saltwater, which conﬁrms the enhancement of conductivity byAlðOHÞ3 .
According to the above results, two opposite effects compete todetermine the needed electrical potential, e.g., conductive solid precipitates and non-conductive gaseous bubbles. It can be seen in thereaction Eq. (3), the amount of conductive solid precipitate is lessthan the non-conductive bubbles. For conditions of weak current, theenhancement of resistance by bubbles in the channel prevails, so thatthe needed potential is higher. As the current is raised, more amountof bubbles are displaced out of the channel, either by the ﬂux orbuoyancy, than solid precipitates. As a result, the effect of conductive
solid precipitates becomes more signiﬁcant to reduce the needed
potential. However, for an even stronger current, turbulent ﬂow
evolves. The chaotic turbulent ﬂow may trap the bubbles inside thenarrow channel more easily, so that the effective resistance starts torise again. The resistance R in different ReL is plotted in Fig. 10(b) torealize the inconsistent behavior affected by ﬂow regime. R is initiallyhigher than the pure saltwater for ReL < 1000, and decreases forhigher ReL . Nevertheless, R reverses to rise for ReL ¼ 1500 » 1600,
which is in line with the change of ﬂow pattern presented above.
4. Conclusion
Pattern formation and ﬂow transition of magnetohydrodynamic(MHD) bubbly jet driven by the Lorentz force in conductive saltwater
is studied. The Lorentz force generated by perpendicularly placed
magnetic ﬁeld and electric ﬁeld displaces the saltwater in an open
channel bounded by the two plates of aluminum electrodes. In themeantime, chemical reactions take place to produce gaseous bubbles(oxygen and hydrogen) and solid precipitates (aluminum hydroxide).
The displaced ﬂux of the saltwater/bubbles/precipitates solution into
the ambient saltwater forms a typical three-phase MHD jet ﬂow. Taking advantage of the brightness of gaseous bubble, the pattern of jet
ﬂow is mainly represented by the bubbly ﬂow recorded directly.
Flow transition from laminar to turbulent is categorized.

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The Lorentz force is directly proportional to the current density, sothat a representative series of experiments, by varying the input electric current strength, is ﬁrstly presented to illustrate the jet ﬂow
affected by the Lorentz force. By raising the current strength, patternof MHD bubbly jet evolves from a slightly wavy, nearly continuous,
laminar gaseous stream toward chaotic turbulent ﬂow, which meets
common expectation. A new Reynolds number ReL , based on the Lorentz-force and deﬁned by the strength of magnetic ﬁeld and electric
current, is proposed to categorize the ﬂow regime. Turbulent breakup
starts to evolve once the ReL exceeds a critical value at ReL  1500.
Dependence of relevant quantitative measures, e.g., spreading area,
angles of bubbly jet and needed electric potential, on the control
parameters are also studied. These dimensionless measures generally
depend monotonically on the ReL . Nevertheless, rapid growth ofthese measures occurs for the cases of ReL > 1500, which indirectly
conﬁrms the transition of ﬂow regime at ReL  1500.
Declaration of Competing Interest
Study the MHD bubbly jet experimentally.
Propose a new Lorentz-force based Reynolads number ReL capable
to demonstrate the pattern similarity, and determine the ﬂow transition.
Verify the critical value of ReL  1500 to trigger turbulence
breakup.
Acknowledgements
Supports by the Ministry of Science and Technology, Taiwan(MOST 110-2221-E-A49 -045 -MY3) is acknowledged.
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