# Thermally-driven microfluidic swirler for flow manipulation

Filippo Azzini, Gian Luca Morini, Beatrice Pulvirenti, Massimiliano Rossi, Marcos Rojas-Cárdenas

► To cite this version:

Filippo Azzini, Gian Luca Morini, Beatrice Pulvirenti, Massimiliano Rossi, Marcos Rojas-Cárdenas. Thermally-driven microfluidic swirler for flow manipulation. International Journal of Thermal Sciences, 2025, 215, pp.109944. 10.1016/j.ijthermalsci.2025.109944. hal-05087864

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Submitted on 28 May 2025

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# Thermally-driven microfluidic swirler for flow manipulation

Filippo Azzini<sup>a, ID</sup>, Gian Luca Morini<sup>a, ID</sup>, Beatrice Pulvirenti<sup>a</sup>, Massimiliano Rossi<sup>a</sup>,
Marcos Rojas-Cárdenas<sup>b, *</sup>

## ARTICLE INFO

**Keywords:**

Microfluidics

Thermally-driven mixer

Mixed convection

Flow manipulation

Swirl

Heat transfer enhancement

## ABSTRACT

In this work, an original thermally-driven micro-fluidic swirler for flow manipulation applications is proposed. The swirling mechanisms are thermally activated only. Since the device does not require any moving parts, it is robust and needs extremely low maintenance. The swirling effect is triggered by activating a temperature gradient transversely to the stream-wise direction of the flow. The swirling mechanism corresponds to a combined effect of advection and natural convection inside a squared cross-section straight micro-channel. We here offer a complete characterization of the micro-fluidic system both from an experimental and numerical point of view. A first numerical design and optimization of the device was realized in respect to the swirling effect for parameters such as the Reynolds number and hydraulic diameter of the channel. Subsequently, the study explored the impact of the flow swirling on the heat transfer mechanisms along the channel. Results are proposed in terms of the Nusselt number for a wide range of channel dimensions. On these basis, a physical microfluidic swirler prototype was developed and was experimentally characterized. The experiments where performed via micro particle image velocimetry (μ – PIV) and velocity fields results were compared to numerical experiments with excellent agreement.

## 1. Introduction

In recent times, the advent of new micro-fabrication techniques has radically widened the spectrum of possibilities for novel and breakthrough micro-fluidic applications [1,2]. Micro-fluidic devices are today used for bioanalysis and diagnostics [3], precision medicine [4], passive particle separation [5], drug nano-particle preparation [6], nano-particle synthesis [7], rheology [8], temperature management and heat dissipation [9], mixing [10], etc. The main advantages of micro-fluidic devices are related to low fabrication costs, low weight, low size, low energy consumption, reduced sample volumes for diagnostics, quick response time, high sensitivity and selectivity, automation of processes, high repeatability, etc. [11].

In the last decades, microfluidic devices have radically improved mixing and heat transfer enhancement for liquid flows. Micro-channels-based systems are characterized by the possibility of generating high heat-fluxes in laminar flow regimes, promoting both a rapid dissipation of heat [12,13] and a controlled mixing processes [14,15], respectively. These characteristics are achieved mainly due to the high surface-to-volume ratio in such devices. However, at the micro-scale rapid mixing can be difficult to achieve since surface forces (viscous forces) are

dominant in respect to volume forces (inertial forces), thus making flow perturbation necessary for fast mass transport. For this reason, several proposals can be found in the literature and their focus relies on how to improve the mixing capabilities inside micro-devices, or, in other words, how to enhance mass and/or heat transport in the fluid region. In this work, we present a mixing micro-device capable of generating a swirl flow inside a straight micro-channel by applying a temperature gradient transversal to the axial component of the flow velocity. By definition, swirl flows are laminar flows composed of a tangential and axial velocity component that produces a helicoidal or spiral flow around an axis [16].

Generally speaking, micro-mixing devices are mainly classified as passive and active systems [10]. Passive systems do not require external energy sources in order to work. However, they are strongly dependent on the geometrical configuration of the devices, which often needs to be of a complex nature in order to achieve significant enhancement. It is also important to underline that these kind of devices cannot be remotely controlled, since it is not possible to change their geometries during operation (single modus operandi). Additionally, the presence of obstacles or variations in shape leads to relatively high increase in

<footnote><sup>a</sup> Department of Industrial Engineering, Alma Mater Studiorum Università di Bologna, Viale Risorgimento 2, Bologna, 40136, Italy</footnote>

<footnote><sup>b</sup> Institut Clément Ader (ICA), Université de Toulouse, CNRS, INSA Toulouse, ISAE-SUPAERO, IMT Mines Albi, UT3, 3 rue Caroline
Aigle, Toulouse, 31400, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: marcos.rojas@insa-tonouse.fr (M. Rojas-Cárdenas).</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 1. Geometry of the micro-channel and section of the channel with temperature boundary conditions.</figcaption></figure>

pressure drops, in respect to straight channel configuration devices. Some interesting experimental work coupled with numerical simulations has been performed for complex geometries, such as string of round cavities along channels [17], gear-shaped [18] or serpentine [19] micro-channels, sharp corners [20], baffles [21], and obstacles or vortex promoters [15]. From a thermal perspective, the objective of this approach is to disrupt the thermal boundary layer close to the solid walls, thereby augmenting heat transfer. Similarly, by promoting vortex formation, it is possible to enhance also mass transport mixing capabilities. Both experimental and numerical literature studies illustrate how these devices can effectively enhance heat and mass transfer. To our knowledge, only two experimental works exist in respect to swirling formation in passive micro-devices and they are based on the positioning of twisted tape within the micro-channel. For these cases, the fluid is forced to follow the helicoidal path created by the tape. Huang et al. [22] used an ad-hoc geometry based on protrusions and indentations of the channel walls to induce the swirl flow, demonstrating an increase in mixing for Reynolds number in the 0.01–10 range. It was demonstrated that the use of a swirl flow can increase the fluid mixing while maintaining a limited system volume and a more compact design compared to other systems adopted to perturb the fluid motion. Later, Lan et al. [23] experimentally demonstrated that swirl flows generated via twisted tape could generate an enhancement of heat transfer by mitigating the rapid growth of bubbles in the case of flow boiling inside micro-devices.

Active systems require an external energy source for activation and they can be classified as acoustically, electrically, magnetically and thermally driven. Endaylalu et al. [24] investigated acoustic streaming within a T-junction, noting that the vortex generated increases mixing efficiency, especially at low Reynolds numbers. Zhao et al. used microelectrodes to induce the electric-driven micromixing, this approach offers enhanced efficiency and integration for rapid mixing [25]. Zhou et al. studied the mixing of ferrofluid and distilled water suggesting its potential for biological applications due to its compact magnetic field and minimal heat generation [26]. It is important to underline that none of the previous active systems were designed to generate swirl flows.

To the best of our knowledge, just a few works have investigated the impact of using thermal activation to generate a swirl flow in microchannels. Ishii et al. have investigated the thermal induced spiral flow inside a T-shape micro-junction from an experimental [27] and numerical [28] point of view. The experimental work examined the mechanism of spiral flow for a fixed hydraulic diameter of 1 mm, not covering the micrometric scale. The experimental velocity mapping was performed via micro Particle Image Velocimetry (μ-PIV) for three different inlet velocities, that is 50, 75, and 100 µm/s. The work focused on the flow stratification at the junction where the hot and cold fluids came into contact, resulting in a S-shaped velocity distribution. Numerically, the authors found that the intensity of the spiral movement was strongly dependent to the temperature differences imposed and independent on the inlet velocity. The study also analyzed the effects

of the channel width and depth of the T-junction, finding that the depth is the most relevant parameter. They observed that no mixing enhancement was present for sub-millimetric channel depth, that is 0.25 mm, and only a clear swirling activation was present in millimetric and centimetric channel depths.

Still, an important lack of information exists in the literature on thermally driven swirl flows for microfluidic applications. Up to the present moment, this system was investigated experimentally only until the millimetric scale in a T-junction configuration. In this work, we proved that swirling motion activation is possible also in micro-metric scale devices by conceiving and manufacturing an original thermally driven device for swirl flow generation of simple geometry, that is by using a single straight micro-channel configuration. Flow measurement was performed via μ-PIV to determine the velocity profiles of the swirling motion at different planes of the micro-device. Furthermore, we performed a systematic and in-depth numerical analysis of heat transport enhancement via flow swirling. A thorough parametric study of the Nusselt number and the performance evaluation criteria (PEC) was performed by studying different hydraulic diameters of the device for a wide variety of imposed flow rates.

## 2. State of the problem

The first step of this work was to study numerically the possibility to activate swirl flows in microchannels by using natural convection. The micro-device numerically investigated consisted of a long squared cross-section ($H \times H$) straight microchannel composed of two passive regions at the inlet and outlet and one active region at the center (Fig. 1). At the inlet of the passive region, a stationary Poiseuille flow of water in laminar regime conditions was generated along the x-axis direction. The inlet region of length $L_{in} = 15$ mm was designed to reach fully developed flow conditions before fluid activation, while the outlet region of length $L_{out} = 5$ mm helped to avoid reverse-flow at the outlet. The central active region of length $L = 50$ mm is the part of the device used for flow manipulation. Here the swirling flow mechanism can be activated by imposing a transversal temperature gradient to the flow direction. This region consists of two heat-exchangers placed parallel to the channel axis at opposite sides of the channel. The exchangers provided constant temperatures in time along the full length of the active region. The hot wall ($z = -H/2$) was maintained at $T_h = 60$ °C, while the cold wall ($z = +H/2$) was maintained at $T_c = 20$°C. This temperature difference induced a constant temperature gradient along the z-direction, that is perpendicular to the main direction of the flow (Fig. 1).

It is of great importance to recall that gravity plays a crucial role in the activation of the swirling phenomenon: fluid density variations induced via organized temperature distributions drive the actuation of natural convection inside the channel. The combination of natural convection and advection finally generates the activation of the swirling effect. In the cases studied hereafter, the direction of the gravitational acceleration is opposite to the y-axis direction.
<figure><img src="image_4.png" /><figcaption>Fig. 2. Stream-lines of the Swirling flow mechanisms for different hydraulic diameter dimensions $D_h$ and same Reynolds number ($Re = 2.5$).</figcaption></figure>

# 3. Numerical work

## 3.1. Methodology

Simulations were carried out for the case of water, that is an incompressible Newtonian fluid, in laminar regime, and steady-state flow conditions. The simulations were performed via STARCCM+, which is a commercially available computational fluid dynamics (CFD) finite-volume-based software used for solving the Navier-Stokes-Fourier (NSF) equations. The set of three-dimensional balance equations used in this work are the following:

$$ \nabla \cdot (\rho \mathbf{v}) = 0 \qquad (1) $$

$$ \nabla \cdot (\rho \mathbf{v} \mathbf{v}) = -\nabla p + \mu \nabla^2 \mathbf{v} + \rho g \qquad (2) $$

$$ \nabla \cdot (\rho c_p T \mathbf{v}) = -\nabla \cdot (k \nabla T) \qquad (3) $$

where $v$ is the fluid velocity, with components $v_x$, $v_y$ and $v_z$, $\rho$ is the fluid density, $p$ is pressure, $T$ is temperature, $\mu$ is the dynamic viscosity, $k$ is the thermal conductivity and $c_p$ is the specific thermal capacity at constant pressure. The numerical domain was divided into three different parts, that is the inlet, outlet and central active region. The numerical grid considered in this work was a structured mesh composed of uniform hexahedral elements. From the sensitivity analysis, the grid was imposed to have 26 × 26 elements on the inlet surface. For what concerns the flow boundary conditions, a Poiseuille velocity distribution was imposed at the inlet ($x = -L_{in}$), while atmospheric static pressure conditions were imposed at the outlet ($x = L + L_{out}$). The flow is set to be at ambient temperature at its inlet ($T_{in} = 30^{\circ}C$). For what concerns the thermal boundary conditions of the channel walls, the inlet and outlet passive regions were set to ambient temperature, while the active region walls were set at $T_h$ ($z = -H/2$) and $T_c$ ($z = +H/2$). For all the simulations $T_c$ and $T_h$ are equal to 20 $^{\circ}C$ and 60 $^{\circ}C$,

respectively. Dirichlet boundary conditions were imposed at the wall in order to generate the transverse temperature gradient. The remaining top ($y = +H/2$) and bottom ($y = -H/2$) channel walls were imposed to be adiabatic surfaces. Finally, let us remind that the thermophysical and transport properties of the working fluid (water) are considered as temperature dependent, following the IAPWS-IF97 model. The water density variation with temperature can be thus considered as the main actuator of the swirling flow [28] as the buoyancy gradients in the flow triggers natural convection.

In this work 36 different cases were numerical studied, that is 6 different hydraulic diameter dimensions in the range $D_h = [0.1, 2]$ mm and 6 different values of Reynolds number in the range $Re = [0.1, 25]$. The Reynolds number was calculated as

$$ Re = \frac{\bar{v}_x D_h}{\nu}, \qquad (4) $$

where $\bar{v}_x$ is the mean velocity component along the flow direction, $D_h = H$ is the hydraulic diameter of the channel and $\nu$ is the kinematic viscosity of water at $T_m = (T_h + T_c)/2$. For these set of parameters, the mean flow velocities were in the $\bar{v}_x = [0.04, 223]$ mm/s range. The full set of parameters used for the calculations are listed in Table 1.

## 3.2. Swirling effect: temperature and velocity field results

The numerical simulations performed served to characterize the swirling activation in the microchannel. The swirling effect (ribbon structures) is generated via the combination of natural convection and advection (flow moving in the x-direction). Natural convection is activated by the lifting movement of the lighter fluid in the y-direction (against the direction of gravity) at the hot-wall of the channel, while the heavier fluid sinks (direction of gravity) at the cold-wall of the channel.

The streamlines obtained in the active region were presented in Figure 2 to visualize at best the swirling effect, the streamlines entering
<figure><img src="image_5.png" /><figcaption>Fig. 3. Cross-sectional velocity and temperature distribution for different hydraulic diameter dimensions $D_h$ and same Reynolds number ($Re = 2.5$).</figcaption></figure>

<table><caption>**Table 1**
Numerical results of $\bar{v}_x$ and $|\bar{v}_z|$ in mm/s and $f_R$, the number of revolutions per mm (Eq. (5)), as a function of $Re$ and $D_h$.</caption><thead><tr><th>↓ Re \ D<sub>h</sub> →</th><th>100 µm</th><th>400 µm</th><th>800 µm</th><th>1200 µm</th><th>1600 µm</th><th>2000 µm</th></tr></thead><tbody><tr><td rowspan="3">0.1</td><td>ν̄<sub>x</sub> |ν̄<sub>z</sub>|</td><td>0.89</td><td>0.22</td><td>0.11</td><td>0.07</td><td>0.06</td><td>0.04</td></tr><tr><td>f<sub>R</sub></td><td>0.0024</td><td>0.038</td><td>0.15</td><td>0.29</td><td>0.40</td><td>0.48</td></tr><tr><td></td><td>0.0172</td><td>0.275</td><td>1.085</td><td>2.198</td><td>2.653</td><td>3.820</td></tr><tr><td rowspan="3">0.25</td><td>ν̄<sub>x</sub> |ν̄<sub>z</sub>|</td><td>2.23</td><td>0.56</td><td>0.28</td><td>0.19</td><td>0.14</td><td>0.11</td></tr><tr><td>f<sub>R</sub></td><td>0.0024</td><td>0.038</td><td>0.15</td><td>0.29</td><td>0.40</td><td>0.47</td></tr><tr><td></td><td>0.0069</td><td>0.108</td><td>0.426</td><td>0.810</td><td>1.137</td><td>1.360</td></tr><tr><td rowspan="3">1</td><td>ν̄<sub>x</sub> |ν̄<sub>z</sub>|</td><td>8.94</td><td>2.23</td><td>1.12</td><td>0.74</td><td>0.56</td><td>0.45</td></tr><tr><td>f<sub>R</sub></td><td>0.0024</td><td>0.038</td><td>0.15</td><td>0.29</td><td>0.40</td><td>0.48</td></tr><tr><td></td><td>0.0017</td><td>0.027</td><td>0.106</td><td>0.208</td><td>0.284</td><td>0.340</td></tr><tr><td rowspan="3">2.5</td><td>ν̄<sub>x</sub> |ν̄<sub>z</sub>|</td><td>22.35</td><td>5.59</td><td>2.79</td><td>1.86</td><td>1.40</td><td>1.12</td></tr><tr><td>f<sub>R</sub></td><td>0.0024</td><td>0.038</td><td>0.15</td><td>0.29</td><td>0.40</td><td>0.48</td></tr><tr><td></td><td>0.0007</td><td>0.011</td><td>0.043</td><td>0.083</td><td>0.114</td><td>0.137</td></tr><tr><td rowspan="3">10</td><td>ν̄<sub>x</sub> |ν̄<sub>z</sub>|</td><td>89.38</td><td>22.35</td><td>11.17</td><td>7.45</td><td>5.59</td><td>4.47</td></tr><tr><td>f<sub>R</sub></td><td>0.0024</td><td>0.038</td><td>0.15</td><td>0.29</td><td>0.40</td><td>0.48</td></tr><tr><td></td><td>0.00017</td><td>0.0027</td><td>0.0107</td><td>0.0207</td><td>0.0285</td><td>0.0342</td></tr><tr><td rowspan="3">25</td><td>ν̄<sub>x</sub> |ν̄<sub>z</sub>|</td><td>223.46</td><td>55.87</td><td>27.93</td><td>18.61</td><td>13.95</td><td>11.16</td></tr><tr><td>f<sub>R</sub></td><td>0.0024</td><td>0.038</td><td>0.15</td><td>0.29</td><td>0.40</td><td>0.48</td></tr><tr><td></td><td>0.00007</td><td>0.0011</td><td>0.00428</td><td>0.00827</td><td>0.01141</td><td>0.01369</td></tr></tbody></table>

the active region were colored in blue in the $z > 0$ domain (cold-side), while they are colored in red in the $z < 0$ domain (hot-side). From these results it is possible to readily see that the fluid swirls clockwise with respect to the flow direction inside the channel. The results presented were obtained for constant Reynolds number ($Re = 2.5$) and different channel dimensions.

These results clearly evidence that the dimensions of the channel significantly influence the swirling mechanism. This is due to the fact that the swirling effect is directly related to the buoyancy of the fluid,

which can be considered as proportional to the fluid volume and scales with the third power of the hydraulic diameter. Consequently, in scenarios involving very small channels, the temperature gradient does not induce natural convection within the fluid. More specifically, for the smaller dimension considered ($D_h = 100 \mu m$) the fluid flows through the channel without activation of the swirling mechanism. For the bigger cross-sections, that is $D_h \ge 400 \mu m$, the swirling effect is clearly activated. The swirling frequency, that is the number of swirl revolutions per millimeter ($f_R$), increases as the channel height increases due to the larger swirling magnitude, and decreases as the Re number increases, due to the fastest streamwise transport (see Table 1). $f_R$ was here calculated as

$$f_R = \frac{2|\bar{v}_z|}{\pi D_h \bar{v}_x} \qquad (5)$$

assuming that the average length of one swirl revolution is a circumference with radius $D_h/4$, and the swirl tangential velocity corresponds to the mean value of $|\bar{v}_z|$.

Let us notice that the swirling mechanism is activated almost instantaneously at the entrance of the active region ($x \ge 0$). That means that the swirling phenomenon induced by temperature gradients can be rapidly activated, which makes this mixing technique of a very useful nature also in the case of fast dynamic processes.

The temperature and velocity fields in the cross-section of the channel for the complete set of dimensions investigated at $Re = 2.5$ were presented in Fig. 3. The maximum temperature corresponds to the hot-wall temperature ($T_h = 60^\circ$) while the minimum temperature corresponds to the cold-wall temperature ($T_c = 20^\circ$). The velocity map corresponds to the transversal component of velocity $v_z$. The same color scale is maintained throughout the 6 cross-sections investigated in order to give a representation of the magnitude of natural convection
<figure><img src="image_6.png" /><figcaption>Fig. 4. (a) Geometry of the micro-chip and layout of the experimental apparatus used to impose the temperature gradient and temperature map obtained from infrared camera, (b) optical system used for the measurement.</figcaption></figure>

in the channel. Velocity vectors represent the sum of components $v_z$ and $v_y$ of the flow.

These results evidence once again that the swirling effect is barely activated for $D_h$ = 100 µm and that the temperature distribution corresponds to a pure diffusion of heat case from heated to cooled walls. For increasing diameters, that is $D_h$ ≥ 400 µm, natural convection vorticity is activated in the flow. Moreover, the magnitude of the velocity component in the z-direction $v_z$ increases with the hydraulic diameter, meaning that the swirling effect is more intense by increasing $D_h$. As it can be readily seen, the temperature distributions in the cross-section is also greatly altered by the swirling structures. From a heat transfer perspective, the generated velocity gradients modify the temperature distribution in the flow. As a consequence, the thermal gradient increase near the wall can be directly correlated to a heat transfer enhancement, meaning an increase of the Nusselt number. This point will be discussed extensively in Section 5, where a parametric study on the quantities of interest is presented.

Table 1 summarizes, for the full set of simulations, the values of the mean velocity component in the stream-wise direction $\bar{v}_x$ and the absolute value of the mean velocity in the transversal direction to the flow $|v_z|$. It is of great interest to notice that the transversal velocity component of the flow $|v_z|$ is independent of the stream-wise velocity $v_x$ at constant channel cross-section, meaning that the only driver of the swirling movement is the temperature gradient imposed to the flow as well as the dimension of the channel. The strongest mean transversal velocity was obtained for $D_h$ = 2000 µm and corresponds to $|v_z|$ = 0.48 mm/s, while the weakest value was obtained for $D_h$ = 100 µm, that is $|v_z|$ = 0.0024 mm/s. Let us point out, that for several investigated cases, the transversal velocity is of the same order of magnitude than the stream-wise mean velocity. The ratio between the mean transversal and stream-wise velocity affects the swirling frequency of the flow, or the number of swirl revolutions per millimeter ($f_R$), in the channel.

# 4. Experimental work

## 4.1. Proof-of-concept device

The thermally-driven microfluidic swirler was designed to have the simplest configuration possible (Fig. 4). The simplicity of the design underlines the robustness of the proposed thermally driven swirling micro-fluidic device for mixing applications. The device was composed of a commercially available squared cross-section long micro-channel made of glass. The capillary dimensions were height $H$ = 800 µm, length $L$ = 50 mm and wall thickness $w$ = 160 µm.

A specifically designed 3D-printed aluminum casing of dimensions 20 mm × 50 mm × 3 mm hosted the device. The main role of the casing was to provide the correct transversal temperature gradient to the micro-capillary. This was realized via two macro-channels, that served as heat-exchangers, positioned parallel to the direction of the flow. The dimensions of the macro-channels of rectangular cross-section were 3.5 mm width and 1.5 mm height and they were positioned at 1.5 mm from the hot and cold walls of the micro-capillary. The cold and hot water flowing in the macro-channels was provided by means of two thermostatic baths connected to the chip. The temperature of the hot and cold fluids was controlled using T-type thermocouples (with an uncertainty of ±0.5 K) at the outlet of the channels.

The measured stationary temperatures at the outlet of the heat exchangers were fed in the numerical simulations, that is $T_h$ = 60° for the hot-wall condition and $T_c$ = 20° for the cold-wall condition. The temperature was uniformly distributed across the active region of the microchannel due to the high conductivity of the aluminum casing (237.0 W/mK), as verified by temperature field measurements obtained with infrared imagery (NEC Handy Thermo TVS-2000EX). Fig. 4.a shows the isothermal temperature conditions of two distinguishable hot and cold regions separated neatly by the micro-capillary. The actual temperatures of the inner walls of the micro-capillary, which are directly in contact with the fluid, might be slightly different from the values measured by the thermocouple. However, the excellent
<figure><img src="image_7.png" /><figcaption>Fig. 5. Velocity profiles evaluated in three different planes (y = 130 µm, y = 390 µm and y = 650 µm) and for two different mass flow rates (Q = 0.5 ml/h and Q = 0.1 ml/h) the red and blue bar are referred to the hot and cold side respectively. The light-black rectangle indicates the portion of the domain used to calculate the average velocities. A central section was used to avoid bias caused by the misalignment of the channel.</figcaption></figure>

agreement between the numerical and experimental results presented in Section 4.3, indicate that this difference is rather small.

Preliminary CFD simulations were performed to design the thermal architecture of the device. Via this modeling, it was possible to define the shape and the dimensions of the heating and cooling macrochannels in order to obtain isothermal temperatures at both the cold and hot sides of the channel. Let us remind here that the proposed heating and cooling design is just one of the many solutions that could be used in order to impose a stable transversal temperature gradient along the micro-capillary

## 4.2. Experimental setup and methodology

Two flow cases were studied, that is $\dot{Q} = 0.5$ ml/h, which corresponds to $Re = 1$ and $\bar{v}_x = 0.3$ mm/s, and $\dot{Q} = 0.1$ ml/h, which corresponds to $Re = 0.2$ and $\bar{v}_x = 0.06$ mm/s, respectively. A syringe pump (Harvard PHD 400) was used to impose the pressure-driven flow inside the capillary, using distilled water as the working fluid.

Micro particle image velocimetry (μ-PIV) [29] was used to characterize the swirling flow inside the device. μ-PIV provides a two-dimensional two-component (2D2C) measurement of the flow field at the focal plane of the microscope objective. A scanning procedure was used to characterize the flow field across the entire channel depth. Polystyrene microspheres with a diameter of 1.19 µm (Microparticles GmbH) were used as tracer particles in the flow with a concentration of 20% in volume. The μ-PIV system consisted of a Nikon Eclipse TE2000-U inverted microscope coupled to a high-speed camera (Olympus I-Speed). The illumination was provided by a high-power LED. The microfluidic device was housed in a 3D-printed PLA (Polylactic Acid) support and observed through a 10 × objective lens with a numerical aperture NA = 0.25. The images were recorded at 200 frames per second with a resolution of 800 × 600 pixels, with a corresponding scaling

factor of 1.42 µm/pixel. With these settings, the expected thickness of the measurement plane (depth of correlation) was 34 µm. The main components of the experimental setup are presented in Fig. 4.b.

The μ-PIV analysis was performed using the open-source software DefocusTracker [30]. The velocity fields were calculated using interrogation windows of 16 × 32 pixels in the z and x direction, respectively, with a 50% overlap, corresponding to a spatial resolution of 8 in the x direction, and 16 in the z direction. The stationary velocity profiles were obtained from time averaging the measurements over 200 consecutive frames (1 s). For each configuration, eight measurement planes were taken across the channel depth (y-direction), at a distance of 88 µm between each other, to characterize the entire flow field. The estimated uncertainty in the velocity measurement was 0.14 µm for the instantaneous values, and 9.9 · 10⁻³ µm for the averaged value.

## 4.3. Velocity field results

In Fig. 5, the velocity vectors obtained from the μ-PIV analysis were presented. The images showcase the x-z plane of the channel at different channel depths, i.e. y = 130 µm (bottom), y = 390 µm (middle), y = 650 µm (top). The results are presented for the two analyzed cases, that is $\dot{Q} = 0.5$ ml/h ($Re = 1$) and $\dot{Q} = 0.1$ ml/h ($Re = 0.2$). The scanning μ-PIV approach allows to analyze the flow pattern in a quasi-three-dimensional fashion. From these results, one can readily see the swirling pattern inside the channel generated via an imposed temperature gradient. At the bottom (y = 130 µm), the flow velocity field deviates towards the hot side (left), while at the top (y = 650 µm), the flow deviates towards the cold side (right). At approximately the center of the channel (y = 390 µm), the flow pattern does not deviate and follows the standard Poiseuille flow profile. The flow here rotates in the clockwise direction in the y-z plane. Let us point out for the lower
<figure><img src="image_8.png" /><figcaption>Fig. 6. Comparison between numerical and experimental results for one flow rate $\dot{Q} = 0.5$ ml/h. Left columns: the axial direction component ($v_x$). Right columns: the normal velocity component ($v_z$). The numerical results with swirl are represented by the red solid line, while without swirl are represented by the dashed blue lines. The experimental results are represented by the black dots.</figcaption></figure>

flow rate case ($\dot{Q} = 0.1$ ml/h), that the measurements clearly reveal a transversal velocity component $v_z$ much stronger than the stream-wise direction component $v_x$.

The experimental results were compared to CFD numerical simulations for the full range of measurements performed at different x-z planes (Fig. 6). Two velocity components of the flow were analyzed, that is the axial $v_x$ and the normal $v_z$ velocity components. Let us notice that the temperature gradient activates thermophoretic effects on the tracer particles. However, for the experimental case at hand, the estimated thermophoretic velocity of the tracer particles is about 25 times lower than the $\bar{v}_z$ velocity induced by the buoyancy forces [31]. Therefore, this effect was neglected in the µ-PIV measurement and numerical simulations. Since the flow is stationary, the experimental measurements shown in this figure have been averaged in the streamwise direction. To calculate the average velocity profiles, only the central region of the channel, highlighted in Fig. 5 with a light-black rectangle, was considered. This approach avoids potential bias in the velocity calculation that could arise from channel misalignment. Specifically, if the channel is not perfectly parallel to the focal plane, the velocity profiles in the peripheral regions along the x-axis may be altered. Let us remind here that in these plots the hot-wall is positioned at $z = -400$ µm while the cold wall is at $z = +400$ µm. As it can be readily seen, a very satisfactory agreement was obtained between simulations and experiments for all the investigated planes. From these results it is once again possible to clearly distinguish a strong transversal velocity component ($v_z$) in the flow, which is negative for the lower planes ($y < 400$ µm) and positive for the upper planes ($y > 400$

µm). Around the center of the channel the normal velocity component tends to zero, while on the same plane the stream-wise velocity is at its maximum intensity. The stream-wise and transversal velocity components tend to zero close to the walls.

It is interesting to notice that for all planes the axial component of velocity ($v_x$) corresponds to a typical Poiseuille flow distribution, such as the one to be expected for flows without swirl. This numerical comparison between “swirl on” (simulations in gravity) and “swirl off” (simulations in micro-gravity) allows to deduce that the natural convection created by the temperature gradient is completely decoupled from the advection in the channel. These results give a more precise information of the clock-wise swirling velocity field in the cross section of the channel with normal velocity component magnitudes as high as $v_z = 0.5$ mm/s which for this case scenario are actually slightly larger than the $v_x$ maximum values.

In this work we have also experimentally characterized the stability of the swirling effect in time. The results presented in Fig. 7 correspond to the maximum values of velocity $v_x$ max and $v_z$ max and are not averaged in time, which was the case for the results presented in Fig. 6. The results are here shown again for two different flow rates $\dot{Q} = 0.1$ ml/h ($Re = 0.2$) and $\dot{Q} = 0.5$ ml/h ($Re = 1$). As it can be seen, the normal velocity component $v_z$ is very stable in time, giving a constant value at a fixed plane within the experimental measurement uncertainty. It is once again important to notice that for two different experimental flow rates the transversal component of velocity is constant. As indicated previously by the numerical results, one can state that the swirling effect depends only on the intensity of the temperature gradient applied, the working fluid and the dimension of the channel.
<figure><img src="image_9.png" /><figcaption>Fig. 7. Experimental velocity components in time for two different flow rates, i.e. $\dot{Q} = 0.1$ ml/h (Re = 0.2) and $\dot{Q} = 0.5$ ml/h (Re = 1). Top: the maximum axial direction velocity component ($v_{x,max}$). Bottom: the maximum transversal flow direction velocity component ($v_{z,max}$).</figcaption></figure>

For these experiments, the thermally-driven swirling effect is even more stable than the axial velocity component of the flow, which is affected at this very low flow rates by the oscillations induced by the working mechanisms of the syringe pump. As it can be seen from the right column of Fig. 7, if the syringe pump oscillations are high in respect to the absolute velocity mean value of the stream-wise velocity component $v_x$, the experimental measurements can be largely affected, thus diverging from the theoretical ones. This is especially true for low flow rates. However, it is possible to appreciate that the experimental transversal velocity component $v_z$ still has a very satisfactory trend with respect to numerical results. This is due to the fact that the natural convection-related velocity component $v_z$ is decoupled from the axial component $v_x$.

## 5. Influence of swirl flow on pressure drop and heat transfer

To characterize the heat transfer enhancement created by the swirling effect, parameters of interest, such as the pressure drop ($\Delta p$), and the Nusselt number (Nu), were analyzed for different geometries ($D_h$) and Reynolds number (Re). In this work, the “swirl-on” pressure drop and Nusselt number results were compared with the respective “swirl-off” reference case obtained by blocking the swirl formation inside the channel, which corresponds to deactivating the buoyancy or gravitational terms in the CFD simulations (micro-gravity conditions). The “swirl-off” parameters were indicated as $\Delta p_0$ and $Nu_0$ in the text.

For example, the pressure drop ratio $\Delta p/\Delta p_0$ versus the hydraulic diameter $D_h$ highlights the role of the channel geometry on the friction losses (Fig. 8). One can readily observe from these results that the activation of swirl greatly affects the pressure drop along the channel. Thus, when the natural convection inside the capillary increases, higher pumping power is needed to obtain the same flow rate in the device, especially at low Reynolds numbers.

The Nusselt number, $Nu$, which serves to quantify in a non-dimensional manner the heat transfer between the working fluid and the solid walls of the microchannel, was here defined as the average $Nu$ across the perimeter of the microchannel

$$Nu = - \frac{D_h}{P(T_w - T_b)} \int_P \nabla T \cdot \hat{n} dl, \qquad (6)$$

where $P = 4D_h$ is the perimeter of the square section, $\int_P \nabla T \cdot \hat{n} dl$ is the thermal gradient normal to the wall, integrated along the curvilinear

<figure><img src="image_10.png" /><figcaption>Fig. 8. Pressure drop ratio $\Delta p/\Delta p_0$ versus the hydraulic diameter $D_h$ for different Re. $\Delta p$ refers to the swirl-on condition case, while $\Delta p_0$ refers to the swirl-off condition case.</figcaption></figure>

coordinate $l$ relative to the channel edge for a fixed $x$-position, $T_w$ is the wall temperature and $T_b$ is the bulk temperature which can be calculated as

$$T_b = \frac{1}{A\bar{\varepsilon}_x} \int_A v_x T dA \qquad (7)$$

where $A$ is the surface of the cross-section of the capillary. In order to validate our numerical runs, we have run two specific heat transfer numerical benchmark problems of a thermally developing and hydrodynamically developed laminar flow inside a square section microchannel. We have compared the local Nusselt number along the length of the channel as a function of the reciprocal of the Graetz number ($Gz = D_h Re Pr x^{-1}$). The first case of study (case 1) corresponded to the work of Ozisik et al. [32] which consisted of a channel with walls at constant temperature. The second case of study (case 2) consisted in a channel with upper and lower horizontal adiabatic walls and two isothermal vertical walls at same temperature. For this configuration only the asymptotic value of $Nu$ is present in the literature and corresponds to 3.703 [33]. For both cases we have set the temperature of the channel walls at 40°C, the temperature of the fluid at the inlet of the channel at 30°C and we used $1.13 \cdot 10^6$ elements for the simulations. The good agreement between literature and the current results validates the numerical procedure of this work (Fig. 9).
<figure><img src="image_11.png" /><figcaption>Fig. 9. Nusselt number trend along the stream-wise direction of the channel for thermally developing and hydrodynamically developed laminar flow inside a square section channels as a function of $Gz^{-1}$. Case 1: walls at same temperature (◦) this work, (−) Ozisik et al. [32]. Case 2: adiabatic upper and lower walls and vertical walls at same temperature (Δ) this work, (−) Shah et al. [33].</figcaption></figure>

For the specific case of thermally-driven swirl formation analyzed via experimental and numerical runs, the upper and lower horizontal walls were adiabatic, while the two vertical walls had different temperature. In Fig. 10 we presented the Nusselt number as a function of the normalized stream-wise direction along the capillary ($x^* = x/L$). The position $x^* = 0$ corresponds to the starting point of the active region (where the temperature gradient was applied), while $x^* = 1$ corresponds to the end of the active region. The comparison was realized for the “swirl-on” and “swirl-off” cases for the totality of geometries studied and three different $Re$ number.

From these results it is possible to analyze the evolution of the Nusselt number in the thermal entrance region of the active region of the channel. As expected, for higher Reynolds number, or higher stream-wise velocities, longer is the length of the thermal development in a specific channel of constant hydraulic diameter. For the smaller hydraulic diameter cases ($D_h \le 400 \mu m$), that is when the thermally-driven natural convection is weak, the $Nu$ evolution coincides for the “swirl-on” and “swirl-off” cases. For cases where natural convection is stronger ($D_h \ge 800 \mu m$), the $Nu$ evolution changes in the stream-wise direction of the channel being the condition $Nu(x^*) > Nu_0(x^*)$ always respected.

Towards the exit of the active region of the channel, it is possible to notice that for the “swirl-off” cases, the Nusselt number tends to an asymptotic value ($Nu = 2$). However, when the swirling motion (“swirl-on”) is activated, the asymptotic Nusselt number increases considerably. This is the case for all the geometries considered, expect for the smaller one, where natural convection cannot be triggered. This result directly implies that the heat transfer was enhanced due to swirling motion along the channel. The limit value of $Nu = 2$ can be easily explained by analyzing Eq. (6): without the transversal component of velocity ($v_z$), which is activated by the swirling motion of the flow, the Nusselt number solution corresponds asymptotically to the one to be found for a pure diffusion of heat problem, where a linear temperature gradient is present transversal to the flow, that is in the hot to cold wall direction. Furthermore, for a given geometry, the $Nu$ number at the exit of the channel is always independent of the $Re$ number imposed, since what determines the heat transfer mechanism of the flow is the natural convection induced in the channel, for which the tangential velocity $v_z$ is always constant for a given channel dimension (Table 1).

To give a clearer picture of the swirling potential in terms of heat transfer enhancement, in Fig. 11.a the ratio $Nu/Nu_0$ is presented as a function of the channel hydraulic diameter, where $Nu_0$ relates to the swirl-off cases. The $Nu$ values studied here correspond to the asymptotic values of thermal fully developed flow. It is thus possible to verify that there is a net enhancement of the heat transfer in the flow for all cases where swirl is triggered, the value of the ratio can go

as far as 2 folds in respect to the case without swirl activation for the bigger channel dimensions.

Finally, it is of interest to compare the heat transfer enhancement in respect to the mechanical energy needed in order to engender the flow (Fig. 11.b). As it was previously presented, even if the heat transfer enhancement due to the swirling motion of the flow is evident, higher pressure drops are to engendered in the flow. Here we analyzed the performance efficiency parameter (PEC)

$$ PEC = \frac{Nu}{Nu_0} \left( \frac{\Delta p}{\Delta p_0} \right)^{1/3}, \qquad (8) $$

as given by [34,35].

From an energy balance point of view, when $PEC = 1$ (dashed line) one can establish that the heat transfer enhancement balances the mechanical loss given by higher pressure losses in the flow. A good example can be given by the $Re = 1$ case, where the results are stabilized around the $PEC = 1$ value. Nevertheless, for all dimensions at this given $Re$, it is possible to appreciate that the heat transfer is greatly enhanced, especially for higher values of $D_h$ (Fig. 11.a). For the $Re > 1$ configurations the net efficiency increases always ($PEC > 1$), thus a clear net energy balance benefit can be obtained for heat transfer enhancement in respect to mechanical energy needed to move the swirling flow in the stream-wise direction. Instead, for the $Re < 1$ configurations the net efficiency decreases always ($PEC < 1$), which means that even if the heat transfer is enhanced, the mechanical power needed to move the flow is greater than the heat transfer enhancement benefit.

## 6. Conclusions

In this work, an original thermally-driven micro-fluidic swirler for flow manipulation applications was proposed. The flow swirl was triggered by activating a temperature gradient transversely to the stream-wise direction of the flow. The swirling mechanism is a combination of advection and thermally-driven natural convection inside a squared cross-section straight micro-channel. We offered a complete characterization of the micro-fluidic system both from an experimental and numerical point of view.

The experimental and numerical results validated that the swirling motion could be easily activated inside a straight micro-channel by imposing temperature differences of only 40° C between hot and cold walls of the channel. However, the swirling effect was triggered only for relatively large channel hydraulic diameters ($D_h \ge 100 \mu m$). It was thus verified that the channel geometry had a very important impact on the thermally-driven swirl formation.

In this respect, a numerical parametric study on the device was realized via CFD in order to optimize the heat transfer enhancement
<figure><img src="image_12.png" /><figcaption>**Fig. 10.** Local Nusselt number (Nu) evolution inside the channel as a function of $x^* = x/L$ for 6 different $D_h$ and 3 different $Re$. Comparison between the “swirl-on” and “swirl-off” cases.</figcaption></figure>

behavior. A large spectrum of numerical runs were realized, that is a study on 6 different Reynolds number and 6 different hydraulic diameters, for a total of 36 numerical experiments. The numerical study explored kinematic and thermodynamic parameters, such as velocity and temperature fields, but also studied the impact of the swirling mechanism in terms of heat transfer along the channel. The results were proposed in terms of the Nusselt number and the methodology was validated against benchmark cases described in the literature. From this study we found that the thermally-driven swirling motion always enhances the heat transfer in the fluid flow.

In addition, an analysis of the performance efficiency coefficient (PEC) of net energy balance gain for heat transfer enhancement in respect to mechanical power needed to move the flow in the stream-wise direction was proposed. Results suggest that under several Re and hydraulic diameter configurations there is an overall net energy gain by using the thermally-driven swirling motion for heat transfer enhancement. Despite the fact that Nu always increases with swirling motion

intensity, the pressure drop along the channel increases drastically for low stream-wise velocities (low Re). As a result, a clear energy gain can be achieved only for the specific range $D_h > 400$ µm and $Re > 1.0$.

Furthermore, a physical microfluidic swirler prototype was developed and was experimentally characterized. Via experiments we could validate the numerical velocity fields results. The experiments were performed via micro-particle-image-velocimetry (μ-PIV) to obtain velocity field measurements in different planes of the channel. The scanning μ-PIV approach allowed to analyze the flow pattern in a quasi-three-dimensional fashion. inside the channel. The velocity field results were compared to numerical simulations with excellent agreement. Via this experimental work, it was also possible to prove the stability of the engendered swirling motion which was perfectly constant and reproducible in time. The vorticity generated in the flow was even more stable than the stream-wise component of the flow, which was subjected to oscillations induced by the syringe pump. Furthermore, it was proven that the convective motion and the advection motion
<figure><img src="image_13.png" /><figcaption>**Fig. 11.** Left: Nusselt number ratio $Nu/Nu_0$ versus the hydraulic diameter $D_h$ for different Re. Nu refers to the “swirl-on” condition case, while $Nu_0$ refers to the “swirl-off” condition case. Right: performance efficiency coefficient (PEC, Eq. (8)) as a function of the hydraulic diameter for different Reynolds number.</figcaption></figure>

of the flow are completely decoupled in the thermally-driven swirling mechanism.

To conclude, through this prototype we demonstrated that the original idea was of quite simple implementation. In the future, we will explore the feasibility of manufacturing, via novel micro-fabrication technologies, a functioning compact microfluidic chip, which can result in a very small, robust and low power consumption device. In order to achieve this goal, specific thermal architectures of the system will be investigated and designed to simplify the temperature gradient generation and regulation in the system. Also, more complex geometries will be explored for particle manipulation in the flow.

### CRediT authorship contribution statement

**Filippo Azzini:** Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.
**Gian Luca Morini:** Writing – review & editing, Validation, Supervision, Resources, Project administration, Funding acquisition.
**Beatrice Pulvirenti:** Supervision, Resources.
**Massimiliano Rossi:** Writing – review & editing, Validation, Supervision, Software, Resources, Project administration, Methodology, Conceptualization.
**Marcos Rojas-Cárdenas:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Project administration, Methodology, Funding acquisition, Formal analysis, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgment

Marcos Rojas-Cárdenas gratefully acknowledges the University of Bologna, Italy for the “2022 DIN Young Visiting Professor Fellowship” mobility award, which facilitated this fruitful collaboration.

### Data availability

Data will be made available on request.

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