Modeling Microgravity Using Clinorotation in a Microfluidic Environment: A Numerical Approach

Abstract

Microgravity simulation is essential for studying particle dynamics in space-related applications where traditional gravitational effects are absent. This study presents a numerical investigation of particle behavior in a clinostat-driven microfluidic channel, aiming to simulate microgravity conditions. A computational model was developed in COMSOL Multiphysics to analyze the impact of channel size, particle diameter, and rotational speed on particle trajectories and establish sets of parameters for assuring microgravity conditions. The results revealed that stable microgravity-like conditions could be achieved within specific parameter ranges, e.g., larger channel radii requiring lower rotational velocities for particle suspension. However, the tendency for gravitational settling increased with particle size or under suboptimal rotational speeds. These findings provide insights into the effectiveness of clinorotation as a microgravity simulation method and establish a foundation for optimizing experimental designs in space research and biomedical applications.

1. Introduction

The study of particle behavior in microgravity environments has gained significant attention due to its relevance in a wide range of fields, from medical research in a space environment to biophysical studies [1]. On Earth, gravity dominates most physical phenomena, influencing the behavior of fluids, particles, and biological systems, as well as larger-scale phenomena such as the movement of the oceans [2]. However, when this gravitational force is significantly reduced, such as in the microgravity conditions found in space, the behavior of the particles and fluids changes drastically [3]. Understanding these changes is crucial for space-related research, where conventional models based on terrestrial gravity do not always apply. Mechanical forces, including those governed by gravity, operate as regulatory signals that affect the morphology and function of biological tissues. The absence of these forces can cause significant changes, including bone loss [4], muscle atrophy [5], cardiovascular deconditioning [6], and dysregulation of the immune system [7,8]. These effects reinforce the importance of investigating how microgravity affects cellular and physiological functioning. Thus, research into microgravity is relevant for supporting long-duration space missions or even future colonization efforts on celestial bodies with gravity different from that of Earth [9,10].

One of the main challenges for ground-based research is the simulation of these microgravity conditions. For long-duration studies, actual space-based experiments, such as those on the International Space Station (ISS), face significant challenges, such as excessive costs, limited accessibility, extreme conditions, scarce availability of equipment and long waiting times for results [11,12]. Clinostats offer a cost-effective approach for simulating microgravity [13]. These devices continuously rotate samples, averaging out gravitational effects over time [14], allowing for the free-floating cells/particles to remain in a fixed position relatively to the medium. In simple terms, under Earth′s gravity, when a cell in a static tube is rotated 90°, it settles toward the bottom of the tube due to gravity. This behavior continues throughout a full 360° rotation. However, if the rotation frequency is increased, the traveling distance becomes smaller. Also, if the rotation is constant, it generates a circular path, where the cells are rotated around their own axis [15] and do not settle. The clinostats’ simple design, low cost, and ability to efficiently reproduce microgravity conditions have made them valuable tools in various areas, such as biomedical research, especially in studying the growth of cancer cell populations and as a tool for testing cancer therapies [16].

Clinostats can be divided into 2D or 3D systems, implying 1- or 2-axis rotations, respectively. Two-dimensional clinostats achieve around 10⁻² g by rotating around 1-axis, perpendicular to the direction of gravity, at a rotation rate highly dependent on the fluidic domain (density and viscosity) and on the mass and dimension on the cellular sample being studied (including particles, cells, and bacteria, among others [15]). For instance, the literature reports effective microgravity simulation using 2D clinostats, based on specially designed 1-axis rotating bioreactors called high-aspect-ratio rotating wall vessel bioreactors (HARVs). These have been reported, among other applications, for bacterial studies, particularly for Salmonella [17] bacteria (typically 2–5 μm length, considering 50 mL culture volume), Mycobacterium marinum [18] samples (0.2–0.6 μm width and 1–4 μm length, considering 10 mL and 50 mL culture volumes), both under 25 rpm rotation, and yeast cultures (namely S. cerevisiae cells, typically 3–4 μm diameter, 5–10 μm length) under 30 rpm rotation [19]. Another recognized system is the Synthecon Rotary Cell Culture System (RCCS) (Synthecon, Texas, USA), commonly used in conjunction with HARVs, which promotes cell rotation around a horizontal axis and has been used to study microorganisms [18] and different lines of mammalian cells (at different rotation speeds, from 10 up to 24 rpm [20,21,22]). Other examples of 2D clinostats include the work developed by Wang et al., (2016) [23], which focused on Arabidopsis seedlings, where seeds (mostly between 250 and 600 μm) were embedded in the central 1 cm area (relative to the horizontal rotation axis) of square Petri dishes, under 60 rpm rotation, to create a g-force of 0.02 g. Eiermann et al., (2013) [24], who studied the effects of 1-axis rotation, at 60 rpm (below 0.012 g), on the growth of melanoma adherent cells (around 16 μm diameter but ranging 13–21 μm), seeded onto 9 cm² slide flasks. Overall, the works reported here show that, for most 2D clinostat applications, bigger cells require higher rotation velocities to sustain microgravity conditions. Alternatively, 3D clinostats comprise two frames that rotate independently and at slow rotation speeds (slower than 2D clinostats), commonly under 10 rpm [8,11,16,25,26] (even reported below 2 rpm for one of the axes). Especially for larger samples, 3D clinostats can assure microgravity environments similar to the ISS and with better performance than 2D clinostats. Nevertheless, for small dimension systems, 2D clinostats have demonstrated enough performance for an effective microgravity simulation, combined with a simple configuration and easy fabrication and control.

The focus of this study is to understand the behavior, trajectories, and rotation patterns of particles within a microfluidic channel under 2D clinorotation (1-axis rotation), specifically using a clinostat to simulate microgravity conditions, aiming for future integration with organ-on-a-chip devices under simulated microgravity [27,28]. Microfluidic channels provide a controlled environment for studying microscale interactions, and when combined with a clinostat, they serve as a powerful tool for understanding the physics of particle dynamics in simulated space-like conditions [29]. Figure 1 illustrates the principles behind clinorotation: While on Earth (Figure 1a), particles settle due to gravity, whereas in real microgravity (Figure 1b), particles are uniformly distributed [30,31]. Under 2D clinorotation (Figure 1c), the sample is rotated rapidly around an axis perpendicular to the gravity vector, which effectively cancels out the sedimentation effect over time. In this condition, no consistent directional movement of the particles is observed, simulating a microgravity environment [30,31].

This study uses COMSOL Multiphysics (Burlington, MA, USA), version 5.3, to numerically model particle dynamics within a confined microfluidic channel under simulated microgravity through 2D clinorotation, for channel radii between 1 and 10 mm, and microparticles between 10 and 50 μm, representing the conditions for the development of organ-on-a-chip devices for microgravity simulation. Particularly, the authors’ team has been developing organ-on-a-chip devices in the 8 mm diameter range, for cell culture, growth, and analysis [27,28], considering different types of cell lines with variable diameter (from a few to dozens of micrometers); hence the need to understand, in this work, the microgravity conditions as a function of the particles’ size). This novel approach, which has not been reported in the literature, provides insights into microgravity’s effects on particle motion and interactions, advancing our understanding for space science, fluid mechanics, and the development of space-compatible technologies.

2. Theoretical Background

Gravity is an attractive force that causes objects with mass to accelerate towards each other, according to Newton’s universal law of gravitation, where $F_g$ represents the gravitational force, $m$ and $M$ are the masses of two interacting bodies, $d$ is the distance between them, and $G$ is the gravitational constant, which represents the intensity of this force in the universe (Equation (1)) [32,33].

$$F_g={\displaystyle \frac{GmM}{d^2}}.$$

(1)

The gravitational force attracts all bodies to Earth with an intensity proportional to their mass, causing an acceleration ($a$) described by Newton’s second law, represented by $F=ma$. The magnitude of the gravitational acceleration ($g$) corresponds to the change rate in an object’s velocity due to the gravitational force [34], which, on Earth’s surface, has an average value of 9.8 m/s².

Microgravity occurs in environments where the gravitational force and acceleration are much lower than on the Earth’s surface [2]. In this environment, there is no sedimentation of particles, and the distribution is homogeneous, as they do not feel a preferential direction. It can be achieved in real life, reaching values between approximately 10⁻³ and 10⁻⁵ m/s² or in a simulated form in which the gravitational forces are reduced to an order of magnitude of around 10⁻² to 10⁻⁵ m/s² [8]. Microgravity can be achieved through three different methods: deep space, free fall, or simulation on Earth. In deep space, a body is so far away from any other body with mass that the gravitational force becomes negligible, as it decreases with the square of the distance (Equation (1)). In free fall, the only force acting is gravity. In this condition, the sensation of microgravity is achieved by the absence of normal or contact forces, creating the sensation of weightlessness. On Earth, there are various microgravity simulation devices, such as clinostats, rotating wall vessels, and random positioning machines, which are based on the principle of rotation to alter the perception of the gravitational vector [30].

As microgravity simulation methods are based on rotation, it is necessary to understand the physical effects that underpin the operation of these systems. A non-inertial reference frame, which includes rotational movement, is accelerated in relation to a fixed reference frame. Under these conditions, bodies do not directly follow Newton’s first law. During rotation, an object has a tangential velocity that is constant in modulus but with a direction that is constantly changing. To describe this behavior, centripetal and centrifugal forces are introduced [35]. The centripetal force, a radial force pointing towards the center of the trajectory, is responsible for keeping the body in circular motion, continuously changing direction instead of following a rectilinear trajectory. This force generates a centripetal acceleration $a_c$ perpendicular to the body’s tangential velocity. It is given by [36,37]

$$a_c={\displaystyle \frac{v^2}{r}}or{a}_c={\omega }^{2}r,$$

(2)

where $\omega$ is the angular velocity, $v$ is the linear velocity, and $r$ is the radius of the trajectory.

Centrifugal force is an apparent force that ‘pulls’ the body away from its center of rotation. This effect results from inertia, and it is proportional to the centripetal force but acts in the opposite direction (Figure 2). Together with this force, centrifugal acceleration arises. Figure 2 illustrates the interaction between centripetal and centrifugal forces in a non-inertial frame of reference associated with the circular movement of an object of mass m. The tangential velocity, v, indicates the instantaneous direction of the body’s movement at any point on the circular path, while R represents the radius of the path.

3. Numerical Simulation

COMSOL Multiphysics software, version 5.3, was used to simulate the particles’ trajectory inside a rotating microfluidic channel with a simple geometry. It was considered a circular channel with different radii, varying from 1 mm to 10 mm, as represented in Figure 3. To apply a clinostat-based single axis rotation (representing a 2D clinostat), and since it is expected the same particles’ trajectory along the devices’ length, the problem can be simplified to a 2D geometry and domain, with the significant advantage, over 3D simulation, of computational efficiency. Particularly, as the rotation occurs over a single horizontal axis, a 2D domain (x, y) accurately represents the acting forces (gravity and rotation) that act in x and y directions, and no significant forces are identified in z direction (the fluid is not under any flow through the channel, so the velocity in z direction is null). In the model, after implementing 1-axis rotation, a 2D velocity field is created. When it reaches stability, sets of particles are released along the diameter of the channel (see the horizontal line in Figure 3 as reference for the particles’ release), making it possible to analyze their movement as well as the forces being applied on them (drag force and gravity force) to confirm if simulated microgravity was properly achieved.

The numerical model for solving the particles’ and fluids’ behavior includes the laminar flow, described by the Navier–Stokes governing equations for the conservation of momentum and mass (through the continuity Equation (4) [38]):

$$\rho {\displaystyle \frac{\partial u}{\partial t}}+\rho (u\cdot \nabla )u=\nabla \cdot [-pI+K]+F+\rho g$$

(3)

$$\rho \nabla \cdot u=0$$

(4)

$$K=\mu (\nabla u+{\left(\nabla u\right)}^T),$$

(5)

where $u$ is the velocity vector of the fluid (m/s), $\rho$ is the fluid density (kg/m³), $p$ is the pressure field (Pa), $I$ is the identity matrix (related to isotropic pressure), $K$ is the stress tensor, representing internal forces within the fluid due to viscosity (Pa), $F$ is an external force applied to the fluid (N/m³), $g$ is the acceleration due to gravity (m/s²), and $\mu$ is the dynamic viscosity of the fluid (Pa·s). The model also includes a set of initial and boundary conditions, including the following points:

(1)

The channel wall is a nonslip boundary (u = 0);

(2)

The acceleration of gravity is applied on the y-axis;

(3)

The fluid pressure and temperature are set to atmospheric values.

The particle tracking module was included to predict the trajectory vectors of the particles in the channel over time. The gravitational force and Stokes drag force applied to the particles are obtained by solving the gravitational field and the flow field, respectively. The governing equations for these forces are given as follows [39]:

$$F_g=m_{P}g{\displaystyle \frac{{\rho }_P-\rho }{{\rho }_P}},$$

(6)

$$F_D={\displaystyle \frac{1}{{\tau }_P}}{m}_{P}M\left(u-v\right),$$

(7)

$$\begin{gathered} {\tau }_P={\displaystyle \frac{{\rho }_P{d}_P^2}{18\mu }}; \\ M={\displaystyle \frac{1}{1-\frac{9}{16}\alpha +\frac{1}{8}{\alpha }^3-\frac{45}{256}{\alpha }^4-\frac{1}{16}{\alpha }^5}}\left(I-P\left(n\right)\right)+{\displaystyle \frac{1}{1-\frac{9}{8}\alpha +\frac{1}{2}{\alpha }^3}}P\left(n\right);\alpha ={\displaystyle \frac{r_P}{L}}, \end{gathered}$$

(8)

where $F_g$ is the gravitational force acting on the particle (N), $m_P$ is the mass of the particle (kg) and ${\rho }_P$ represents the density of the particle (kg/m³). $F_D$ is the drag force acting on the particle (N), $M$ is the wall correction factor, $v$ is the particle velocity vector (m/s), ${\tau }_P$ is the particle response time (s), and $d_P$ is the diameter of the particle ($m$). $L$ is the distance to the nearest wall, $r_P$ is the radius of the particle $m$, $P\left(n\right)$ is the projection onto the vector normal to the wall, and the ratio $\alpha$ is the distance from the wall (it determines how the proximity of the particle to the wall affects the drag force). The corresponding initial and boundary conditions are as follows:

(1)

The channel wall condition is defined as bounce;

(2)

The gravity force is applied on the y-axis;

(3)

The drag force follows Stokes’ law and includes wall corrections.

Using the same module, a procedure was defined for releasing the particles, while keeping consistency for all future studies. To achieve this, a line segment (Figure 3) was drawn along the horizontal diameter of the channel and used as an inlet where all the particles were released at the same time and with a uniform distribution along the length of the line, with an initial velocity equal to the velocity field calculated through the laminar flow.

Additionally, to simulate the effects of microgravity induced by a 2D clinostat (1-axis rotation), a rotating reference frame was implemented using the Arbitrary Lagrangian-Eulerian Formulation (ALE) Moving Mesh method. Specifically, a Rotating Domain feature was applied to account for the continuous rotation of the system. The domain was set to rotate at a constant angular velocity, which will vary depending on the size of the channel and particles (see Section 4—Results). The axis of rotation was aligned with the central axis of the channel, simulating the uniform rotational environment that mimics a microgravity condition by averaging out gravitational forces over time.

Water was selected as the fluid medium in the microfluidic channel, with a 998 kg/m³ density and a 1.009 mPa·s dynamic viscosity, both at room temperature. The particles in the numerical model were assumed as polystyrene spheres, with a density of 1050 kg/m³ and varying sizes from 5 μm to 50 μm diameter.

The 2D domain was refined with a small mesh, formed by triangular and quadrilateral elements, as shown in Figure 4a. From this figure, it is possible to see greater refinement near the borders of the microfluidic channel (where the maximum velocities will be observed, Figure 4b). A mesh study in Figure 4c shows no differences in the maximum rotation velocity in the microchannel for different mesh sizes.

Table 1. The number of mesh elements and quality statistics of the mesh for different microchannel sizes.

Radius of the Channel	Number of Elements	Average Element Quality
1 mm	4902	0.7761
2 mm	4902	0.7761
5 mm	4890	0.7785
10 mm	4890	0.7785

presents the number of elements and the quality of the selected mesh for each microchannel radius.

A time-dependent study with a 10 ms time step and a varying total simulation time (dependent on the time required to achieve velocity stability), was performed to compute and evaluate the velocity magnitude in the channel, using the PARDISO (Parallel Direct Sparse Solver Interface) direct solver due to its low computational cost. The same time-dependent study with the same time step was applied to track the particle trajectories over time, but a GMRES (Generalized Minimal Residuals) iterative solver was used, as the larger-scale problem demanded higher computational resources.

4. Results

This section presents the analysis of the trajectory of the microparticles subjected to the varying conditions within the microchannel, including particle diameter, channel radius, and the corresponding velocity magnitude, needed to achieve the desired trajectories mimicking simulated microgravity. The methodology for structuring the results is organized as follows: first, the geometric parameter of the microchannel, i.e., the channel radius, is defined. Following this, different particle diameters were introduced, which were chosen arbitrarily to explore a wider range of particle behaviors. Several rotation speeds were explored until finding the minimum velocity necessary for the particles to follow a predictable trajectory under simulated microgravity. Finally, in these conditions, the particle trajectories after a five-second period (t = 5 s, for the majority of the microgravity conditions) were presented.

Some considerations were made for studying different velocities and defining their minimum value needed for achieving microgravity conditions. Firstly, even though the analysis of the particles’ trajectory was evaluated, for most cases, for a 5 s period, the total study time was set depending on the rotational velocity in a way that the particles would do an arbitrary number of full rotations plus one-half rotation (explaining why 10 mm radius channels, for low rotation speeds, required longer analysis times—see

Table 2. The required time for the rotating flow to achieve stability and constant velocity for different microchannel sizes and rotation speeds.

Radius of the Channel (mm)	Rotational Speed (rpm)	Duration of the Rotation (s) Until Flow Stabilization
1	30	0.27
	100	0.31
	150	0.34
2	20	1.32
	60	1.4
	200	1.48
5	15	11.65
	30	9.97
	75	8.52
10	5	70.3
	10	50.53
	50	46.23

and Section 4.2.4.). Lastly, the minimum velocity required to achieve a stable microgravity environment was iteratively determined by evaluating the particles’ trajectories, ensuring, as a decision criterium, that the maximum downward displacement per particle rotation did not exceed a 1% tolerance criterium. There is no known reference guideline on the tolerance/threshold to consider, so the tolerance value was selected empirically, aiming at the most accurate prediction of the particle trajectories, only assuming that microgravity would be kept under minimal variations in the particles’ path.

4.1. Analysis of the Velocity Field over Time

As previously mentioned, the release of the particles into the channel occurred only after achieving a stable fluid velocity, which varied according to the channel dimension and rotation speed. To assess this velocity variation over time, a time-dependent study was performed to analyze the velocity magnitude as a function of the X position over the diameter of the microchannel, considering, as an example, the microchannel with 1 mm radius, rotated at 30 rpm, with the velocity profiles exemplified at different time instants (0, 0.05, 0.1, 0.2, 0.3 and 5 s), as represented in Figure 5.

The velocity at the six different time stamps of operation is presented to highlight the various stages of the velocity field distribution, revealing that, for the presented channel radius, the velocity starts around zero at 0 s (blue line) and rapidly increases over time, reaching stability before 0.3 s (specifically at 0.27 s). In particular, the plot shows that within 0.2 s, the system has almost reached its final state, achieving a stable fluid velocity that enables the release of the particles. This velocity increase is due to the influence of fluid dynamics, which introduce complexities in the flow behavior as the system evolves. Additionally, this increase is more pronounced at the barriers, which can be explained by the simple $v=\omega r$ equation, where $\omega$ represents the angular velocity and $r$ is the radial distance. Since the barriers are located farther from the center of rotation, their radial distance is higher, leading to a greater velocity at these points. Furthermore, based on this expression, a linear behavior for achieving a stable velocity was expected, as the angular velocity tends toward the imposed rotational speed and the radial distance increases linearly.

As previously referred, this data was obtained by rotating a microchannel with 1 mm radius filled with water, at 30 rpm, starting from a resting state. An analysis similar to this was made for each microchannel dimension (up to 10 mm) at different rotation speeds to establish the optimal time instant to release the particles and to grant the best stability to assure a microgravity environment. The stabilization time required for each rotating flow condition, and consequently the optimal particle release time, is presented in Table 2. The Reynolds number (Re) was also calculated for each condition (

Table 3. Reynolds number (Re) calculated for each simulated condition, considering different microchannel sizes and rotation speeds.

Radius of the Channel (mm)	Rotational Speed (rpm)	Re
1	30	6.21
	100	20.71
	150	31.08
2	20	16.58
	60	49.73
	200	165.73
5	15	77.64
	30	155.39
	75	388.42
10	5	103.66
	10	201.12
	50	1035.78

), considering the maximum flow velocity (located next to the channel periphery). It was observed that, even for higher rotation speeds, all flows occur under a laminar flow regime (no turbulence is predicted based on the Re).

4.2. Effect of the Channel Dimensions and Particle Size

This section presents the evaluation of the effect of different channel diameters and particles’ sizes on the rotating speed required to guarantee a simulated microgravity environment within the fluidic domain.

4.2.1. Microchannel Radius of 1 mm

Starting with a microchannel radius of 1 mm and a particle diameter of 10 μm, it was found, by successive iterations with increasing rotating speeds, that the minimum velocity needed to sustain the trajectory of the particles and fitting the selection criterium was 30 rpm. Figure 6a shows the distribution of the velocity magnitude profile after five seconds of rotation. Figure 6b shows the trajectory of the particles after the same period. The color scale in the y-axis represents the total forces (N) applied to the particles, combining the forces resultant from the gravity and rotation. As the total forces approach 0 N (represented by a green color), the more stable is the simulated microgravity, as the total forces applied to the particles in x and y directions mostly annul.

As shown in Figure 6b, the particles were able to sustain the gravitational force and keep a consistent trajectory. Overall, it is observed that the forces applied to the particles are virtually zero (given by the bright green color), validating the simulated microgravity environment.

For the same channel radius but with a particle of double the size (diameter 20 μm), there is the need to increase the velocity of the fluid to keep the desirable trajectory of the particles, since the intensity of the gravity force will increase. For a 5 s simulation period, the trajectory does not change much for fluid velocities of 100 rpm and 150 rpm. In Figure 7, images (a) and (c) depict the fluid’s velocity magnitude required to maintain rotational motion at 100 rpm and 150 rpm, respectively. Images (b) and (d) show the particles’ trajectory for those same rotating speeds; correspondingly, the bar in the y-axis indicates the total force acting on the particles, subjected to rotating speeds of 100 rpm and 150 rpm, respectively.

For particles with a diameter of 20 μm, flowing in a 1 mm radius microchannel, it was evident that the particles were struggling to keep the same trajectory without falling due to the gravity force at speeds already extremely high, as demonstrated in Figure 7b,d, where the presence of forces applied to the particles (the trajectories are not plotted in plain green) is shown. But even in these conditions, the maximum total forces applied to the particles, as seen in the color bars, are in the ≈2 × 10⁻¹² N range, showing that the resultant forces, even in these cases, are residual.

Due to the extremely high rotation speeds already required to achieve microgravity conditions, the particles’ diameters above 20 μm were not considered for this 1 mm microchannel radius. Notice that the line segment where the particles are released always starts horizontally, but after the 5 s interval, they may appear with different orientations, depending on the rotational velocity, without affecting the overall trajectory observed.

4.2.2. Microchannel Radius of 2 mm

When simulating a 2 mm radius microchannel, a rotational speed of 20 rpm was enough to achieve microgravity trajectories for particles with 10 μm diameter. Figure 8a,b shows the results of both velocity magnitude and particle trajectories, respectively, under these conditions.

Compared to the trajectory previously observed for the 1 mm radius microchannels (Section 4.2.1), for the same particle size, it is observed that a better and more stable particle trajectory can be achieved at lower rotational velocities of the fluid, revealing a simpler way to achieve the representation of the expected microgravity environment.

By increasing the particles’ diameter to double (20 μm), and in opposition to what was observed in the 1 mm radius (Section 4.2.1.), a minimum velocity of 60 rpm was enough to achieve microgravity at this channel size. Figure 9 shows the obtained results, where (a) represents the velocity magnitude and (b) shows the particles’ trajectory and forces applied to the particles (N).

Again, compared to the particles’ trajectory reported for a microchannel with 1 mm of radius and particles with a diameter of 20 μm, it is observed that a wider channel leads to a minor required rotational velocity, which is responsible for a shorter number of total rotations performed by the particles and for assuring a stable trajectory of the particles. By increasing the particles’ diameter up to 50 μm, the system was not able to sustain the desired trajectory (not complying with the 1% tolerance), even with a rotational velocity of 200 rpm, which is already substantially high for a standard clinostat. Figure 10 shows the particles’ behavior observed, presenting the velocity magnitude and particles’ trajectory, respectively. As can be observed in Figure 10b, the total forces applied to the particles are in the 10 × 10⁻¹¹ N range, which is significantly higher than in the previous cases where microgravity conditions were achieved (here, the sum of the total forces cannot be neglected).

4.2.3. Microchannel Radius of 5 mm

Once again, by increasing the radius of the microchannel up to 5 mm, the analysis for the same particles’ diameter (10 μm and 20 μm) was repeated. Figure 11 shows both the fluid’s velocity magnitude and the particles’ trajectory for 10 μm diameter particles at 15 rpm rotational speed and for 20 μm diameter particles at 30 rpm, respectively.

For this channel radius, the velocity magnitude takes a longer time to stabilize (slightly above 8 s for these rotational speeds). This stabilization time was assessed previously to evaluate the particles’ trajectories. Nevertheless, even if the full stabilization was not achieved at 5 s, the velocity field was almost established, and by maintaining the analysis of the particles’ trajectory on a duration of 5 s, it helped ensure consistency throughout the studies. As seen in Figure 10, for the 5 mm radius channel size, the particles could sustain the same microgravity trajectory with a lower rotational velocity than the two previous microchannels (1 mm and 2 mm radius). After successful evaluation, the achievement of simulated microgravity trajectories in the 5 mm radius microchannel for 10 and 20 μm particles at velocities lower than 30 rpm, the particles’ diameter was increased to 50 μm to see if the system was still capable of sustaining the trajectory for this particle size. In Figure 12, the velocity magnitude and particles’ trajectory are presented for a rotational velocity of 75 rpm, where it is evident that the microchannel is still not capable of sustaining the expected trajectory, revealing a different behavior, depending on the initial position of the particles. Figure 12b shows that the leftmost particles, which start their motion upwards (opposite of the gravitational vector) can maintain the ideal trajectory, averaging out the gravity force inherently applied to it. On the other hand, the rightmost particles reveal a clear descent in every rotation (above the defined 1% criterium for maintaining a microgravity trajectory), confirming that, for bigger particles (with higher mass), the downward force, due to gravity, is dominating the net force applied to the particles. A similar behavior was observed for other rotational speeds above 75 rpm.

4.2.4. Microchannel Radius of 10 mm

Lastly, by increasing the radius of the microchannel to 10 mm, the analysis for the same particles’ diameter was repeated. Starting with a particle diameter of 10 μm, the trajectories of the particles revealed distinct asymmetries. In Figure 13, the velocity magnitude and particles’ trajectory are shown for two different rotational velocities (5 rpm and 10 rpm). Unexpectedly, the particles released on the left side of the channel exhibit trajectories that progressively flatten over time, while, in contrast, the trajectories of the particles released on the right side expand over time.

For this study, due to the low rotational velocity required for this channel size, the condition of analyzing the trajectory after a period of 5 s was no longer kept, since there was not enough time to complete a single rotation. Instead, a longer period of time was used (54 s) for a better visualization of the trajectory.

The same behavior was evident for bigger particles. For 50 μm diameter particles, it a trajectory similar to the one reported for the 5 mm radius channel was observed (Figure 12). As presented in Figure 14, the velocity magnitude and particle trajectories for this scenario (rotational velocity of 50 rpm) showed, again, the ideal trajectory on the particles released on the left side of the channel, while the right side was clearly dominated by the gravity force, descending considerably after every rotation.

5. Discussion

From the obtained results, several key takeaways can be drawn regarding the dynamic of particle behavior under clinorotation. The simulations demonstrate that it is possible to achieve a microgravity-like environment through clinorotation by finding the ideal rotational speed and balancing it with both the fluidic channel diameter and the particles’ size. At low rotational velocities, particles succumb to sedimentation under gravitational forces, revealing a constant downward motion in every rotation. As the rotational speed increases to an optimal range, the gravitational vector is effectively neutralized, allowing the particles to remain suspended and with a nearly fixed trajectory. However, at excessive speeds, centrifugal forces, due to the channel rotation, dominate, leading to a rotational motion of particles around the channel’s perimeter. The critical role of rotational speed in governing particle behavior is presented in Figure 15, exemplified for a 5 mm microchannel radius with 20 μm diameter particles.

The displacement was measured on the leftmost particle by calculating the difference between the top of the second and first rotations, indicating that, if the gravitational forces dominated, the particles would shift downward and the displacement would be negative. In case of tending to microgravity, the displacement would be close to zero. Lastly, if centrifugal forces dominated, the displacement would be positive. This way, Figure 15 illustrates the expected trend: As the rotational speed increases, the particles transition from moving toward the bottom of the channel to mimicking a microgravity environment and finally circulating around the channel’s perimeter.

Furthermore, the relationship between particle size, channel radius, and rotational velocity plays a pivotal role in stabilizing trajectories. Larger channel radii allow for a stable trajectory at lower speeds, as described by the following equation:

$$F_c=mr{\omega }^2,$$

(9)

where $F_c$ is the centrifugal force, $m$ is the particle’s mass, $r$ is the distance between the center of rotation and the particle (radius), and $\omega$ is the angular velocity. This means that, for the same particle mass (same gravitational force), increasing the radius reduces the required rotational speed to balance gravitational effects.

In addition to channel radius, particle size has a significant impact on the rotational speed required to sustain stable trajectories. Larger particles have greater mass, resulting in a stronger gravitational force acting on them. To counteract this, higher rotational velocities are required, making stabilization more challenging as particle size increases. Figure 16 illustrates the relationship between particle size and microchannel rotational velocity for three different channel radii. The results, obtained for particles with diameters of 5 μm, 10 μm, 15 μm, and 20 μm, confirm the expected behavior, demonstrating how larger particles and smaller channel radii demand higher rotational speeds for stabilization.

As shown in Figure 16, for smaller particles (5 μm), the required rotational velocity to stabilize trajectories is low across all channel radii. At this size, the centrifugal force needed to counteract gravity is minimal and the differences between the channel radii are less pronounced.

As the particle size increases, the rotational velocity required to maintain stability rises significantly, particularly for smaller-channel radii. For instance, for a 1 mm channel radius, the velocity increases steeply to accommodate particles of 15 μm and 20 μm, exceeding 80 rpm for the largest size. This sharp increase highlights the greater influence of gravitational force on larger particles, necessitating higher centrifugal force for stabilization. In contrast, larger channel radii, such as 5 mm, shows a more gradual increase in rotational velocity as particle size grows. This demonstrates the mitigating effect of a larger radius on the rotational velocity requirements, where even the largest particles (20 μm) can achieve stable trajectories at around 30 rpm. The difference between radii becomes more pronounced for larger particles, emphasizing the importance of selecting an optimal radius based on the particle size and desired operating velocity. The balance between channel radius, particle size, and rotation velocity is also critical regarding possible secondary flow effects. Particularly, Coriolis effects may become significant for higher rotation speeds where, because of the fluid’s inertia, Coriolis forces may slightly affect the particles’ trajectories within the fluid [40]. Nevertheless, for the specific conditions studied here (laminar flows based on the calculated Re), these effects were never observed, as bigger-channel radii implied lower rotation velocities, significantly decreasing the risk of Coriolis secondary effects.

When comparing the main numerical findings against the available literature, based only on experimental data, we observe a good qualitative agreement in the achieved results, as the reported clinostats’ studies (see Section 1—Introduction) show that, overall, cells with larger sizes (and higher masses) require higher rotational velocities to sustain microgravity conditions. Some of the works analyzed, for instance, Mycobacterium marinum [18] samples, which are normally 0.2–0.6 μm in width and 1–4 μm in length, and Salmonella [17] bacteria, which are typically 2–5 μm in length; both rotated under 25 rpm. With S. cerevisiae cells, by increasing the cell size to 3–4 μm in diameter and 5–10 μm in length, the rotation speed increased to 30 rpm [19] rotation speed. Finally, for the bigger melanoma adherent cells (around 16 μm in diameter but ranging from 13 to 21 μm), the rotation speed was significantly increased up to 60 rpm [24]. The results on the effect of the channel radius could not be compared against the literature due to the lack of information regarding the dimensions of the channels and tubes in the reported studies above.

The results of this study highlight several limitations in achieving a consistent microgravity-like environment using 1-axis clinorotation. The effectiveness of microgravity simulation was highly dependent on the combination of particle size, channel radius, and rotational velocity, with clear constraints observed for specific configurations. For a 1 mm channel radius, only particles up to 15 μm in diameter maintained a stable trajectory. Particles larger than this exhibited significant gravitational settling, even at high rotational speeds. In a 2 mm channel, 20 μm particles could remain stable, but only at speeds of at least 60 rpm. When using a 5 mm channel, particles up to 50 μm could be assessed; however, asymmetries in trajectories became more pronounced. The 10 mm channel allowed for lower rotational velocities, but inconsistencies in particle motion emerged, particularly for larger particles, where left-positioned particles maintained stability while right-positioned ones displayed a downward drift. This implies, for each target application, a careful balance between the variables under analysis, including the diameter of the cells to be cultured and the minimum channel diameter to keep those cells under rotation. Another point that needs to be addressed as a limitation of the current study is the need for a stable velocity field prior to the particles’ release, which may be significant as the channel radius increases. This could be impactful as, in an experimental scenario, the particles will not stay in the release central line during the time required for the velocity to stabilize. Instead, they will start moving immediately, which could affect their trajectory during this intermediate phase by increasing the probability of the particles to reach the microchannel wall and stay adhered instead of staying in the small rotation loops. Additionally, if the flow velocity does not stabilize, it will be not possible to achieve a stable particle trajectory path and, thus, no microgravity conditions can be reached. Nevertheless, in this numerical study, we aimed to present a wide range of parameter combinations (diameters, velocities, and particles dimensions) to show how these variables depend on each other when the goal is to achieve a simulated microgravity environment.

Finally, it is important to note that this model is valid only for strictly diluted suspensions, as particle–particle interactions are not considered in the numerical model and they can significantly affect particle dynamics and trajectories. This model can currently predict the effect of the particle size and mass as well as the effect of the number of particles flowing in the channel (even combining particles with different diameters). Nevertheless, future work should address particle interactions and how they can affect the resultant microgravity trajectories.

6. Conclusions

This study explored the behavior of particles in a rotating microfluidic channel under clinorotation, aiming to simulate microgravity conditions. By varying the microchannel radius, particle size, and rotational velocity, it revealed the critical conditions required to achieve stable particle trajectories that effectively counteract gravitational forces. The findings demonstrate that, while clinorotation can effectively simulate microgravity, its success is highly dependent on the interplay between these parameters.

Larger-channel radii facilitated stable trajectories at lower rotational velocities, while increasing particle size required higher rotation speeds to compensate for gravitational effects. However, as particle size increased beyond a critical threshold, achieving consistent microgravity-like conditions became increasingly challenging due to gravitational settling and trajectory asymmetries.

The results emphasize the importance of optimizing system parameters to achieve reliable microgravity simulation in ground-based experiments. Future work could further refine these conditions, incorporating more complex fluid interactions and experimental validation to enhance the accuracy of clinorotation-based microgravity models. These insights contribute to the ongoing development of space research methodologies and the design of more effective simulation systems for biological and physical experiments conducted in reduced gravity environments. Additionally, this research provides valuable advancements in the field of microfluidics, offering a deeper understanding of particle behavior in confined microenvironments under dynamic conditions.