The sandfish lizard's aerodynamic filtering system

Five different grids were created and simulations were performed for steady boundary conditions for the temporal average volumetric flux during inhalation of 92.8 mm³ s⁻¹. Grid data, the results for five different flow quantities as well as the corresponding uncertainty estimates are presented in table 1. Grid convergence is mostly monotonic. Since curvature based grid refinement was used in the grid creation, local grid sizes in areas of complex geometry can differ significantly from the average cell edge length given in table 1, this is especially true for the coarser grids. Since the flow is laminar, the dependence of the results on wall resolution is not particularly pronounced. Uncertainty measures for the medium grid with approximately 400 000 cells are around 5% or lower. This mesh was chosen for all further simulations.

Table 1. Results of the grid resolution study for five quantities: $\Delta p_{tot}$ denotes the total pressure difference between inlet and outlet, $\bar{\tau}_w$ is the average wall shear stress, $\bar{v}_{cf}$ denotes the average cross-flow velocity in the cut-plane, $\max(v_{cf})$ is the maximum cross-flow velocity in the cut-plane, and $\max(v_n)$ denotes the maximum normal velocity in the cut-plane. The cut-plane is located 2.2 mm downstream of inlet; $\bar{h}$ is the average cell edge length, U is the estimate of uncertainty with 95% coverage and has been calculated by the procedure presented in Eça and Hoekstra (2014).

Mesh	Very coarse	Coarse	Medium	Fine	Very fine
Index	1	2	3	4	5
Number of cells	100 640	200 005	401 730	1202 333	8010 356
$\bar{h}_i$ (mm)	0.1251	0.0995	0.0790	0.0550	0.0294
$\frac{\bar{h}_i}{\bar{h}_5}$	4.255	3.384	2.687	1.871	1.000
$\Delta p_{tot,i}$ (Pa)	1.6866	1.7112	1.7664	1.8087	1.8314
$U_{\Delta p_{tot},i}$ (Pa)	0.1981	0.1275	0.0965	0.0449	0.0104
$\frac{U_{\Delta p_{tot},i}}{\Delta p_{tot,5}}$	0.1081	0.0696	0.0527	0.0245	0.0057
$\bar{\tau}_{w,i}$ (Pa)	0.011 33	0.011 48	0.011 80	0.012 03	0.012 15
$U(\bar{\tau}_w,i)$ (Pa)	$1.11\times 10^{-3}$	$6.93\times 10^{-4}$	$5.06\times 10^{-4}$	$2.26\times 10^{-4}$	$3.87\times 10^{-5}$
$\frac{U(\bar{\tau}_w,i)}{\bar{\tau}_{w,5}}$	0.0914	0.0570	0.0416	0.0186	0.0032
$\bar{v}_{cf,i}$ (m s−1)	0.030 63	0.030 95	0.031 35	0.031 64	0.031 82
$U(\bar{v}_{cf},i)$ (m s−1)	$1.60\times 10^{-3}$	$1.01\times 10^{-3}$	$6.93\times 10^{-4}$	$3.24\times 10^{-4}$	$6.96\times 10^{-5}$
$\frac{U(\bar{v}_{cf},i)}{\bar{v}_{cf,5}}$	0.0501	0.0317	0.0218	0.0102	0.0022
${\rm Max}(v_{cf})_i$ (m s−1)	0.059 62	0.060 42	0.061 57	0.062 24	0.062 16
$U(\max(v_{cf})_i)$ (m s−1)	$3.43\times 10^{-3}$	$1.84\times 10^{-3}$	$1.03\times 10^{-3}$	$1.96\times 10^{-4}$	$3.08\times 10^{-4}$
$\frac{U(\max(v_{cf})_i)}{\max(v_{cf})_i}$	0.0551	0.0297	0.0165	0.0032	0.0049
${\rm Max}(v_{n})_i$ (m s−1)	0.2533	0.2550	0.2589	0.2627	0.2655
$U(\max(v_{n})_i)$ (m s−1)	0.0195	0.0147	0.0123	0.0078	0.0038
$\frac{U(\max(v_{n})_i)}{\max(v_{n})_i}$	0.0734	0.0555	0.0465	0.0292	0.0144

The sandfish lizard has a characteristic respiratory pattern that was described in our previous work (Stadler et al 2016, 2018, Stadler et al 2019). For simulation, each phase of the ventilation cycle (exhalation, inhalation, relaxation and pause) was integrated and interpolated by quadratic and cubic spline functions in MATLAB (R2016a, MathWorks, USA), and subsequently programmed in C to create a user defined function (UDF), which was compiled and integrated in ANSYS Fluent.

To understand if the respiratory patterns are a necessary requirement for the filtering system to work, we also performed a simulation with constant velocities, being equal to 80% of the maximum inhalation and exhalation velocity ($v = 0.8\;v_{max}$), respectively, which is the temporal average of the inhalation and exhalation velocity profile.

To analyze the trajectories of individual particles, an Euler–Lagrangian modeling approach was used. It allows the observer to follow a single particle as it moves through space and time (Batchelor 2001). For a detailed description of the capabilities of ANSYS Fluent discrete phase model (DPM) to predict the trajectories of discrete Lagrangian particles, see the ANSYS Fluent Theory Guide (ANSYS 2016).

In ANSYS Fluent the particle trajectories are calculated by integrating the following force balance (Newton's second principle):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{d\vec {v_p}}{dt} = \vec {F_D} + \vec {F_G} + \vec {F_P}, \nonumber \end{align} \tag{ 1 }$$

where $\vec {v_p}$ is the particle velocity, $\vec {F_D}$ the particle drag force per unit mass, $\vec {F_G}$ the effective gravitational force and $\vec {F_P}$ represents additional forces like virtual mass and Saffman force. In this study, the additional forces could be neglected, since the density difference between sand particles and air is high and the particle size is relatively large.

The drag force per unit mass is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \vec {F_D} = \frac{\vec {v}-\vec {v_p}}{\tau_r}, \nonumber \end{align} \tag{ 2 }$$

where $\vec {v}$ denotes the flow velocity, and with $\tau_r$ being the particle relaxation time

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \tau_r = \frac{\rho_pd_p^2}{18\mu}\frac{24}{C_D{\rm Re}_p}.\label{eq:tau_r} \nonumber \end{align} \tag{ 3 }$$

Here, $\mu$ denotes the molecular viscosity of the fluid, $\rho_p$ the density of the particle, d_p the particle diameter, and C_D stands for the particle's drag coefficient. Re_p is the particle Reynolds number, which is defined as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm Re}_p = \frac{\rho d_p|\vec {v_p}-\vec {v}|}{\mu}, \nonumber \end{align} \tag{ 4 }$$

with $\rho$ being the fluid density. The highest particle Reynolds numbers were estimated to be approximately 5 during inhalation and 250 during exhalation (for d_p  =  0.3 mm).

For spherical particles, the relationships provided by Morsi and Alexander (1972) can be applied to determine the relationship between drag coefficient and Reynolds number for various flow regimes. Morsi and Alexander Morsi and Alexander (1972) describe that the response of a particle depends on the relative velocity of the particle and the fluid. The relative velocity specifies the drag. If we assume that particles do not interact with each other and have no impact on the basic flow pattern, the drag is the only force determining the trajectory of the particle.

Since the particles are relatively large, we included the particle torque balance in our simulation. The rotational lift force was taken into account; this lift is caused by a pressure differential along a particle's surface. To determine the rotational lift coefficient the Oesterle & Bui Dinh formulation was used, which has been obtained from experiments of $10<{\rm Re}_p<140$ (Oesterlé and Dinh 1998), and is suggested by ANSYS (2016) Theory Guide to be used for Re_p  <  2000.

Additionally, the gravitational force, i.e. the magnitude (g  =  9.81 m s⁻¹) and direction of the gravity vector, was included.

We strived for a coherent approach to mimic the dynamics of the sandfish lizard's filtering system by simulating two different cases.

• Case (A) Reflection: Particles were reflected at the walls of the respiratory tract. ANSYS Fluent provides a reflection model that can account for wall friction and particle rotation. The standard friction coefficient of 0.2 was applied. The reader should refer to the ANSYS Theory Guide (ANSYS 2016) and references therein for further information regarding this model.
• Case (B) Capture: To mimic the mucus-covered surface, a trapping model was used. Particles were captured as soon as they reached the wall. We then injected the particles at the beginning of the exhalation phase exactly at that location, where they got trapped during the inhalation. This model was also useful for studying the minimum capturable particle size.

Sand grains were approximated to be of spherical shape and modeled as single Lagrangian particles. They were injected at a central position at the naris with a starting velocity of $v_s=0.02$ m s⁻¹; the sand grains were injected with a starting velocity, because it can be assumed that they were already accelerated by the air flow caused by the inhalation. Four particles of different diameters were introduced: 0.1, 0.15, 0.2 and 0.3 mm, respectively. For the determination of the minimum particle size, both starting velocity and particle size were varied (see section 2.3.2).

In the natural habitat of the sandfish lizard, sand particle diameters range between 0.1 and 0.5 mm (Hartmann 1989). Theoretically, all these particles can enter the nasal cavity as it is opening diameter is approximately 0.84 mm. Because the nasal opening can be constricted by an erectile tissue (Stadler et al 2016), particles  >0.3 mm were thus not considered for simulation. For each particle the flow rate ($\dot{m}=dm/dt$), time of injection (at the beginning of the inhalation or exhalation) and the material were defined. We approximated the density of the sand particles by applying the density of quartz (SiO₂, $\rho_Q= 2652$ kg m⁻³).

The density of indoor dust particles is approximately 1000 kg m⁻³ (Molhave 2009). We therefore adapted the particle density to observe particle trajectories with lower density in the sandfish lizard's nasal cavity.

In a previous study (Stadler et al 2016), the above described experiment was conducted with two distinct differences: (a) the model was scaled-up for handling purposes and the experimental setup was adapted (particle magnification, helium atmosphere) to guarantee similar Reynolds numbers, (b) the model was not rinsed with mucus prior to the experiment.

While Reynolds number similarity and hence similar geometric flow behavior was achieved for both experiments, the settling behavior of the particles was not characterized. In a work by Dávila and Hunt (2001), the settling of particles near vortices and in turbulence was studied. The authors state that the trajectories of small heavy particles are determined by a balance between the settling process and the centrifugal effects of the particles' inertia, and defined a particle Froude number as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle F r_p=\frac{{\rm particle~inertia}}{{\rm buoyancy~forces}}. \nonumber \end{align} \tag{ 5 }$$

In the case under investigation, we have stationary flow characteristics with symmetrical separation during the inhalation (the maximum Reynolds number results in Re$_{in} \approx 13$), and we stay in the subcritical range during exhalation (where the maximum Reynolds number results in Re$_{ex} \approx 700$) (Schlichting and Gersten 1997). In both calculations the characteristic length is the diameter of the entrance to the chamber (l_char  =  0.79 mm).

According to Zhu and Fan (2016) in a multiphase gas-solid flow, the motion of a single particle may be described by a spherical particle accelerating and rotating in a flow field. If we assume that particles are not interacting, we can define four different forces acting on a particle: the drag force, Basset force, Saffmann force and Magnus force. In this study, the Basset force is negligible because the gas-particle density ratio is small and the acceleration rate is low. The Saffmann force may not be taken into account, when gradient related forces are neglected. The lift force due to particle spin is small compared to the drag force. The Magnus effect can hence be neglected for similarity estimates, nonetheless it has been taken into account in the particle trajectory simulations (section 2.1.3).

Taking into account the above mentioned considerations, we defined a modified particle Froude number $F r_p^*$ as a measure of the ratio of the horizontal to the vertical acceleration forces experienced by a particle. The modified particle Froude number is hence defined as the ratio between drag force and effective gravitational force. In fluid dynamics the drag force can be expressed as a function of the particle drag coefficient C_D in the following way:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle F_D = \frac{1}{2}v_{rel}^2\rho C_D A_p, \nonumber \end{align} \tag{ 6 }$$

where $v_{rel}$ is the flow velocity relative to the particle, $\rho$ the density of the fluid, and A_p denotes the projected area of the particle. If we assume that the particle is of spherical shape, $A_p = \frac{d_p^2}{4}\pi$.

The effective gravitational force (Archimedes' principle) denotes the difference between gravitational and buoyancy force and is defined as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle F_G = V_p\Delta \rho g = d_p^3\frac{\pi}{6}\Delta \rho g. \nonumber \end{align} \tag{ 7 }$$

$V_p$ denotes the particle volume. The modified particle Froude number results therefore in

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle F r_p^* = \frac{3}{\pi} \frac{\rho}{\Delta \rho} \frac{v_{rel}^2 C_D A_p}{d_p^3 g}. \nonumber \end{align} \tag{ 8 }$$

The sandfish filters all sand grains (0.1–0.5 mm) of its natural habitat. We estimated the minimal particle size that could still be filtered by the geometry of the sandfish lizard's nasal cavity. A previous study Stadler et al (2018) shows that both gravitational and inertial forces are relevant in order to trap sand particles on a sticky, mucus-covered surface. Both mechanisms depend on particle size and density; smaller and lighter particles would closely follow the flow streamlines into the trachea. A rough estimate of the critical size of captured particles can be performed using the relationships presented in Kaufmann et al (1993).

Gravity-dominated separation

For gravity- and buoyancy-dominated separation a particle drift velocity can be calculated by equating the buoyancy and the particle friction forces: if the particle Reynolds number is sufficiently small (Re$_p \leqslant 1.4$), Stokes' law can be applied, where the dependency between particle Reynolds number and the drag coefficient is inversely proportional ($C_D = 24/{\rm Re}_p$). The critical particle diameter that is still separated can be determined by the following equation (Kaufmann et al 1993):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle d_{pg} = \sqrt{\frac{18v_d\rho\nu}{g\Delta\rho}},\label{eq:dpg_Stokes} \nonumber \end{align} \tag{ 9 }$$

where d_pg denotes the minimum particle size for gravity-driven separation, $v_d$ the critical drift velocity, $\nu$ the kinematic viscosity of the fluid, g the magnitude of gravity, and $\Delta\rho$ is the difference between particle and fluid density. For a general drag law $C_D({\rm Re}_p)$ equation (9) reads

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle d_{pb} = \frac{3}{4}\frac{v_d^2\rho}{g\,\Delta\rho}C_D({\rm Re}_p),\label{eq:dpb_CD} \nonumber \end{align} \tag{ 10 }$$

which is an implicit relationship due to the diameter-dependency of the particle Reynolds number.

The critical drift velocity is $v_d=\dot{V}/A$, where $\dot{V}$ denotes the volume flux. A is the lower surface area of the characteristic section, where mucus and cilia are present, subsequently referred to as the ' chamber'. We can assume here that particle deposition occurs only in the lower surface area of the chamber due to both gravitational forces and the strong cross-flow velocities acting on the particles. This chamber is about 1.50 mm long and has a cross-section area of $A_{cross}\approx 0.70$ mm² (figure 2). The lower part of the surface area is hence estimated as $A\approx 2.22$ mm² (Stadler et al 2016). For steady flow situations, $\dot{V}$ could be approximated as tidal volume divided by total inhalation time. Since the inhalation velocity is however time dependent, we estimated $\dot{V}$ with 80% of the maximum outflow velocity $v_{max}$.

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Inertial impaction

When particles are transported in a moving fluid, they do not perfectly follow the flow streamline when the fluid is accelerated. Due to inertia, particle trajectories do not coincide with streamlines in areas of high streamline curvature. Inertial impaction is described by the Stokes number

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle St = \frac{\tau_r\,v}{L},\label{eq:St} \nonumber \end{align} \tag{ 11 }$$

where $\tau_r$ is the particle relaxation time, $v$ is the undisturbed fluid velocity and L a characteristic length scale. For impactors with circular nozzles, L is equal to the nozzle radius and the Stokes number for 50% impaction efficiency is 0.24 (Zhang 2005). Adopting this criterion, in the Stokes regime, where

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \tau_r = \frac{\rho_p\,d_p^2}{18\rho\nu},\label{eq:taur} \nonumber \end{align} \tag{ 12 }$$

the critical particle size can be estimated from

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle d_{pi} = \sqrt{\frac{4.32 \rho \nu r}{\rho_p v_{ref}}}, \nonumber \end{align} \tag{ 13 }$$

where d_pi denotes the minimum particle size for inertia-driven separation, r the radius of the chamber, $\rho_p$ the particle density and $v_{ref}$ is the reference velocity where $v_{ref}=0.8v_{max}$ ($v_{ref} = 0.24$ m s⁻¹). For higher particle Reynolds numbers, equation (3) must be applied using a Reynolds number dependent drag law $C_D({\rm Re})$.

Based on the results of the theoretical approach, we injected particles of various sizes ranging from 20 $\mu$m–90 $\mu$m (step size 10 $\mu$m) with different particle densities (for dust and sand) and a particle starting velocity of $v_s = 0.02$ m s⁻¹. The starting velocity considers not only the dynamics of the particles, but also the dynamics of the sandfish lizard, that moves through sand with velocities of up to 0.5 m s⁻¹, while breathing and therefore possibly simultaneously inhaling particles.

During the experiments, we did not only observe strong adhesion forces between the sand particles and the mucus, but also between the sand particles and the artificial resin (the material used for the 3D model). The reason for this lies in the properties of the artificial resin and the sand grains. We noticed that the adhesion and friction forces are high enough to let particles settle in the chamber during inhalation. During the strong exhalation process, particles are blown out of the cavity, and the bond between the resin and the sand particles is overcome.

The experiments showed that all particles, if present, were found in the front section of the nasal cavity (vestibulum and chamber, see figure 2(a)). No particles were observed in the posterior section. The detailed results are listed in table 2.

The mucus enhances the adhesion forces between the sand particles and the wall, and is therefore indispensable for the sandfish lizard's filtering mechanism.

For the design of a biomimetic filter device the results are promising, because a liquid-covered surface such as mucus would not be feasible for a particle filtering unit. A surface with specific properties to enhance adhesion and friction forces between particulate matter and the wall however is achievable.

The experiment with the scaled-up model (Stadler et al 2016) had the advantage that adhesion forces such as electrostatic forces were irrelevant. The disadvantage though was that the modified particle Froude number $F_p^*$ was not similar; gravitational forces were hence more dominant and particles tended to settle with higher probability. The modified particle Froude number was approximately 400 times higher in the experiment than in reality. It is therefore not surprising that the results of the experiments with the original-sized model (see table 2) are in agreement with the results of the experiment with the scaled-up model (Stadler et al 2016).

The nasal cavity of the sandfish lizard has the ability to filter sand particles without a specific physiological filtering system that hinders particles from entering the lungs. Instead, the animal filters particles with an aerodynamic filtering system. The morphology of the upper nasal cavity in combination with characteristic respiration patterns and a mucus-covered wall is designed to aerodynamically filter particles of a certain size.

The filtering system of the sandfish lizard not only prevents sand particles from reaching the trachea but also from reaching the olfactory tract. The geometric properties of the respiratory tract produce a flow-field which hinders particles from entering the olfactory tract. This filtering mechanism comprises a combination of various factors: size and shape of the nasal cavity and the corresponding flow-field, the properties of thin layers of mucus and the respiration pattern. This enables sand grains to be trapped when flow velocities are low during inhalation, and to be released during a fast and intense exhalation.

Dust particles have a significantly lower density of 1000 kg m⁻³ (Molhave 2009) than sand particles (2652 kg m⁻³). Therefore, we also estimated the critical particle diameter for particulate matter. The theoretical approach (Kaufmann et al 1993) lead to a critical particle diameter of 63 $\mu$m and 9 $\mu$m for gravity-driven and inertia-driven separation, respectively. The simulation results show that particles up to 70 $\mu$m are filtered.

The Stokes number defined equation (11) has been evaluated for the resulting minimum diameter estimates based on the velocity and length scales of the chamber. The results are also provided in table 3. A comparison of results of particle deposition in the nasopharyngeal region of mice and humans has been conducted by Kolanjiyil et al (2019). Despite the size differences of the respective organs they detected a good agreement of deposition fractions of particles of different sizes based on the Stokes number, especially during inhalation. Kolanjiyil et al (2019) also provide a correlation formula for the deposition rate

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle DF = 1- 0.88\,e^{-1049\,St^{2.5}}. \nonumber \end{align} \tag{ 14 }$$

Evaluating this relation for the Stokes number estimates for inertial impaction of the sandfish lizard leads to deposition rates of 0.98 and 0.94, respectively, indicating that these would be reasonable estimates of critical particle diameters, if inertial impaction was the primary separation mechanism. This further demonstrates that the sandfish lizard mainly relies on gravitational separation to prevent sand grains from entering its lungs and cannot effectively filter smaller aerosols in its upper respiratory tract.

A common definition for the sphericity of a particle, sometimes called true sphericity (e.g. Rorato et al (2019)) is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \phi = \frac{A_{sphere,eqV}}{A_{particle}}, \nonumber \end{align} \tag{ 15 }$$

with A_particle as the particle's surface area and $A_{sphere,eqV}$ as the surface area of a sphere of equal volume. The sphericity of sand grains can deviate significantly from the ideal value of one. The minimum sphericity out of samples of two different sand specimens measured by Rorato et al (2019) is 0.5075. Particle drag rises with decreasing sphericity. A comprehensive analytical equation for $C_D({\rm Re}_p,\phi)$ has been developed by Haider and Levenspiel (1989) based on measurements for particles of non-spherical isometric shapes and on free-fall experiments of disk-shaped particles:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle C_D=\frac{24}{{\rm Re}_p}\left(1+A(\phi)\,{\rm Re}_p^B(\phi)\right)+\frac{C(\phi)}{1+D(\phi)/{\rm Re}_p}. \nonumber \end{align} \tag{ 16 }$$

The particle Reynolds number is calculated using the equivalent spherical diameter (diameter of a sphere of equal volume), and the coefficients A, B, C and D are empirical functions of sphericity. The result of Haider and Levenspiel (1989) correlation shows that for low particle Reynolds numbers, relevant during the sandfish lizard's inhalation, the drag coefficient increase of compact particles $\phi>0.6$ is relatively low. Hence, non-spherical sand grain shapes have little impact on particle separation in the chamber of the sandfish's respiratory tract. Estimates of the critical equivalent spherical diameter for buoyancy driven separation according to equation (10) are shown in figure 6. The critical diameter increases with decreasing sphericity. In the sphericity range relevant for sand grains ($0.5<\phi\leqslant 1$) the critical diameter estimates change by less than 18%.

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