**4.3 Numerical modelling of the virus behaviour in the straight pore**

In addition to the mechanisms of filtration described by the classical filtration theory the filtration caused by laminar flow through pores is described. A pore in the mask inner layer is approximated by a tube or a channel. The inner layer is formed in complicated way - it is composed of 37 sub-layers. Each of these sub-layers is composed of fibres randomly placed in random direction, making a pore meandering through all sub-layers a complex path for air to take, see figure 4. We can suppose that:


Based on the presumptions about shape of the pore the simple numerical model was built in order to show different possibilities of filtration and to asses some postulates of the classical filtration theory. The shape of a virus is idealised with the sphere/circle.

A simple 2D computational fluid dynamics (CFD) model was used for the simulation purposes of the virus behaviour in the laminar fluid flow in straight tube as a first

Fig. 7. 2D model of the virus in the pore. The size of the largest virus is magnified 10 times

Novel Theoretical Approach to the Filtration of Nano Particles Through Non-Woven Fabrics 219

Fig. 8. CFD mesh. The size of the largest virus is magnified 10 times here for the presentation

here for the presentation purpose only.

purpose only.

approximation. A virus, represented by a circle, is placed in a rectangular, which is representing 2D model of a tube (pore), figure 7. The top and bottom walls are defined as no-slip wall where fluid velocity is zero due to the surface roughness. The virus boundary is defined in roughly the same way. The radial velocity of the fluid at the virus boundary is zero (no fluid penetration is allowed) and the circumference fluid velocity at the virus boundary is set to an arbitrary value to simulate possible virus rotation. The initial velocity with the uniform profile is prescribed at the inlet (left tube boundary) and the environment pressure is prescribed at the outlet (right tube boundary), figure 7. The results of the laminar model obtained with our target solver (ANSYS-FLUENT) were checked with the results of the turbulent model of the same solver and with the laminar model results of the 3D model in ANSYS-CFX solver. The results of the different models/solvers were in the excellent agreement and we proceeded with the 2D laminar modelling in ANSYS-FLUENT solver.

The aim the numerical modelling is gaining a qualitative picture of the mechanical system that is governing the particle flow through a medical mask. On this base the mesh of the model was relatively course, figure 8. The boundary layer at the tube walls were not in focus of this analysis and so the free meshing was used at whole region. The virus surroundings are meshed with the structured mesh with the smallest elements at the virus surface.

The tube diameter is 30*μ*m, which correspond to the maximal pore hydraulic diameter. The length of the tube is 120*μ*m, which is enough for the complete laminar velocity profile formation before striking the virus in the middle of the tube, figure 9. The laminar velocity profile, equation (8), was established approximately one tube diameter from the inlet.

$$v(r) = \frac{3}{2} v\_o \left( 1 - \frac{r^2}{R^2} \right) \tag{8}$$

where *vo* stands for the initial velocity at the inlet, *r* for the radius of the streamline and *R* for the radius of the tube Widden (1966). The maximal velocity is *vm* = 1.5*vo*.

#### **4.3.1 Fluid induced motion of the particle in the axial direction**

The fluid induced force was estimated for the stationary virus at the centre of the tube cross-section (*r* = 0*μ*m) for different initial velocities as a function of the maximal flow velocity. The circumferential velocity of the fluid at the virus surface is set to zero value no virus rotation allowed. The fluid induced force is actually the driving force in this case, since the fluid is moving and the virus is stationary. It was found out that the fluid induced force *Fa* scales linearly with the fluid velocity *vm*.

$$F\_a = \mathbb{C} \ v\_m \tag{9}$$

where *<sup>C</sup>* stands for the constant that depends on the virus size. *<sup>C</sup>*<sup>300</sup> <sup>=</sup> 4.8 · <sup>10</sup>−12Ns/m for the maximal virus size of 300nm, figure 10, and *<sup>C</sup>*<sup>10</sup> <sup>=</sup> 32.5 · <sup>10</sup>−15Ns/m for the minimal virus size of 10nm. It is obvious that the forces acting on a virus are extremely small even if the virus is kept stationary.

If the virus is not stationary, the velocity *vm* in equation (9) represents the difference of the velocity of the fluid flow and the virus. The second Newton law is used to compute virus velocity in the axial direction.

$$\mathbf{C}(v\_m - v) = m \frac{\mathbf{d}v}{\mathbf{d}t} \tag{10}$$

14 Will-be-set-by-IN-TECH

approximation. A virus, represented by a circle, is placed in a rectangular, which is representing 2D model of a tube (pore), figure 7. The top and bottom walls are defined as no-slip wall where fluid velocity is zero due to the surface roughness. The virus boundary is defined in roughly the same way. The radial velocity of the fluid at the virus boundary is zero (no fluid penetration is allowed) and the circumference fluid velocity at the virus boundary is set to an arbitrary value to simulate possible virus rotation. The initial velocity with the uniform profile is prescribed at the inlet (left tube boundary) and the environment pressure is prescribed at the outlet (right tube boundary), figure 7. The results of the laminar model obtained with our target solver (ANSYS-FLUENT) were checked with the results of the turbulent model of the same solver and with the laminar model results of the 3D model in ANSYS-CFX solver. The results of the different models/solvers were in the excellent agreement and we proceeded with the 2D laminar modelling in ANSYS-FLUENT solver.

The aim the numerical modelling is gaining a qualitative picture of the mechanical system that is governing the particle flow through a medical mask. On this base the mesh of the model was relatively course, figure 8. The boundary layer at the tube walls were not in focus of this analysis and so the free meshing was used at whole region. The virus surroundings are

The tube diameter is 30*μ*m, which correspond to the maximal pore hydraulic diameter. The length of the tube is 120*μ*m, which is enough for the complete laminar velocity profile formation before striking the virus in the middle of the tube, figure 9. The laminar velocity

where *vo* stands for the initial velocity at the inlet, *r* for the radius of the streamline and *R* for

The fluid induced force was estimated for the stationary virus at the centre of the tube cross-section (*r* = 0*μ*m) for different initial velocities as a function of the maximal flow velocity. The circumferential velocity of the fluid at the virus surface is set to zero value no virus rotation allowed. The fluid induced force is actually the driving force in this case, since the fluid is moving and the virus is stationary. It was found out that the fluid induced

where *<sup>C</sup>* stands for the constant that depends on the virus size. *<sup>C</sup>*<sup>300</sup> <sup>=</sup> 4.8 · <sup>10</sup>−12Ns/m for the maximal virus size of 300nm, figure 10, and *<sup>C</sup>*<sup>10</sup> <sup>=</sup> 32.5 · <sup>10</sup>−15Ns/m for the minimal virus size of 10nm. It is obvious that the forces acting on a virus are extremely small even if the

If the virus is not stationary, the velocity *vm* in equation (9) represents the difference of the velocity of the fluid flow and the virus. The second Newton law is used to compute virus

d*v*

*C*(*vm* − *v*) = *m*

*Fa* = *C vm* (9)

<sup>d</sup>*<sup>t</sup>* (10)

(8)

meshed with the structured mesh with the smallest elements at the virus surface.

profile, equation (8), was established approximately one tube diameter from the inlet.

*<sup>v</sup>*(*r*) = <sup>3</sup> 2 *vo* <sup>1</sup> <sup>−</sup> *<sup>r</sup>*<sup>2</sup> *R*2 

the radius of the tube Widden (1966). The maximal velocity is *vm* = 1.5*vo*.

**4.3.1 Fluid induced motion of the particle in the axial direction**

force *Fa* scales linearly with the fluid velocity *vm*.

virus is kept stationary.

velocity in the axial direction.

Fig. 7. 2D model of the virus in the pore. The size of the largest virus is magnified 10 times here for the presentation purpose only.

Fig. 8. CFD mesh. The size of the largest virus is magnified 10 times here for the presentation purpose only.

Fig. 11. The virus velocity as a function of time.

presented in figure 11.

1.6 · <sup>10</sup>−5s and 99.9% in 2.4 · <sup>10</sup>−5s.

from the top or bottom boundary.

(11) once again.

where *v* stands for the virus velocity and *m* for its mass. The mass of the virus of the maximal size can be estimated to *<sup>m</sup>*<sup>300</sup> <sup>=</sup> 17.0 · <sup>10</sup>−<sup>18</sup> kg and the mass for the smallest can be estimated to *<sup>m</sup>*<sup>10</sup> <sup>=</sup> 0.63 · <sup>10</sup>−<sup>21</sup> kg. Equation (10) can be resolved analytically and the results are

Novel Theoretical Approach to the Filtration of Nano Particles Through Non-Woven Fabrics 221

 1 − *e* − *C m t* 

Further on, the axial displacement of the virus can be computed by integrating the equation

The smaller virus reaches 99% of the fluid velocity in 8.9 · <sup>10</sup>−8s and 99.9% in 1.3 · <sup>10</sup>−7s. The larger virus's inertia is larger and so does the time. The 99% of the fluid flow is reached in

It might appear that the predominant direction of motion is in the axial direction since the forces acting on a absolutely stationary virus in the perpendicular direction to the fluid flow are much smaller. The analysis providing us with the axial force on a stationary virus provides us also with the radial one. There is no radial force if the virus is placed in the tube centreline, due to the symmetry. If the stationary virus is moved towards the tube top boundary, the radial force is pointing towards the tube centreline. The same is true if the stationary virus is moved towards bottom tube boundary. The radial forces are at least one order of the magnitude smaller than the axial ones depending on the virus position. It appears that the

(11)

(12)

<sup>4</sup> of the tube diameter

*v*(*t*) = *vm*

 *t* + *m C e* − *C <sup>m</sup> <sup>t</sup>* <sup>−</sup> <sup>1</sup>

*x*(*t*) = *vm*

**4.3.2 Fluid induced motion of the particle in the radial direction**

largest radial force relative to the axial one can be found at roughly <sup>1</sup>

Fig. 9. Velocity contours. The size of the largest virus is magnified 10 times here for the presentation purpose only.

Fig. 10. The fluid induced force on a larger virus.

Fig. 11. The virus velocity as a function of time.

16 Will-be-set-by-IN-TECH

Fig. 9. Velocity contours. The size of the largest virus is magnified 10 times here for the

presentation purpose only.

Fig. 10. The fluid induced force on a larger virus.

where *v* stands for the virus velocity and *m* for its mass. The mass of the virus of the maximal size can be estimated to *<sup>m</sup>*<sup>300</sup> <sup>=</sup> 17.0 · <sup>10</sup>−<sup>18</sup> kg and the mass for the smallest can be estimated to *<sup>m</sup>*<sup>10</sup> <sup>=</sup> 0.63 · <sup>10</sup>−<sup>21</sup> kg. Equation (10) can be resolved analytically and the results are presented in figure 11.

$$v(t) = v\_m \left(1 - e^{-\frac{C}{m}t}\right) \tag{11}$$

Further on, the axial displacement of the virus can be computed by integrating the equation (11) once again.

$$\mathbf{x}(t) = \upsilon\_m \left( t + \frac{m}{\mathbb{C}} \left( e^{-\frac{\mathbb{C}}{m}t} - 1 \right) \right) \tag{12}$$

The smaller virus reaches 99% of the fluid velocity in 8.9 · <sup>10</sup>−8s and 99.9% in 1.3 · <sup>10</sup>−7s. The larger virus's inertia is larger and so does the time. The 99% of the fluid flow is reached in 1.6 · <sup>10</sup>−5s and 99.9% in 2.4 · <sup>10</sup>−5s.

#### **4.3.2 Fluid induced motion of the particle in the radial direction**

It might appear that the predominant direction of motion is in the axial direction since the forces acting on a absolutely stationary virus in the perpendicular direction to the fluid flow are much smaller. The analysis providing us with the axial force on a stationary virus provides us also with the radial one. There is no radial force if the virus is placed in the tube centreline, due to the symmetry. If the stationary virus is moved towards the tube top boundary, the radial force is pointing towards the tube centreline. The same is true if the stationary virus is moved towards bottom tube boundary. The radial forces are at least one order of the magnitude smaller than the axial ones depending on the virus position. It appears that the largest radial force relative to the axial one can be found at roughly <sup>1</sup> <sup>4</sup> of the tube diameter from the top or bottom boundary.

where *M* stands for the fluid induced moment acting on virus, *ω* for the angular velocity of the virus and *a*, *b*, *c* for the approximation parameters. The second Newton's low for rotation

Novel Theoretical Approach to the Filtration of Nano Particles Through Non-Woven Fabrics 223

where *J* stands for the virus moment of inertia and *A*, *B*, *C* for the approximation parameters divided by *<sup>J</sup>*. The moment of inertia *<sup>J</sup>* of the largest virus is 1.53 · <sup>10</sup>−31kgm2 and for the

arctanh *<sup>B</sup>*

where *t* stands for time, *A*, *B*, *C* for the approximation parameters divided by *J* and *D* for the discriminate, *<sup>D</sup>* <sup>=</sup> <sup>√</sup>*B*<sup>2</sup> <sup>−</sup> <sup>4</sup>*AC*. The virus terminal angular velocity is reached in the limit case

*<sup>ω</sup>max* <sup>=</sup> <sup>−</sup>*<sup>D</sup>* <sup>−</sup> *<sup>B</sup>*

The terminal circumferential velocity is computed as *vcmax* = *ωmaxR*, where *R* stands for the virus radius. The values of the terminal (maximal) circumferential velocity for the larger virus is *vcmax* = −0.0386m/s and for the smaller one *vcmax* = −0.0146m/s which is around 4% and 1.5% respectively of the air velocity at the inlet. What is really important is that viruses begins to rotate with the terminal angular velocity in a very short time, almost instantly, see figure

The radial force was computed and approximated in the same way as the moment, see figure 12. Hence, by combining it with the angular velocity, the radial force can be plotted as function of time, figure 14. The force becomes constant with the virus reaching the terminal angular

Although figure 14 presents the resultant force, we can say that there are two competing mechanisms of delivering the radial driving force to the virus. It has been stated that the virus is driven towards the tube centre if it does not rotate due to the fluid flow velocity profile. The same velocity profile also drives the rotation of the virus which in turn adds radial force (Magnus effect, where the rotation is self induced - we named it as J-effect) in the direction of the tube/channel boundary. If the rotation is large enough, which is the case here, the radial force due to the rotation is larger than one due to the fluid profile. The force partition described here is a simplification of a complex fluid velocity and pressure distribution in the vicinity of a virus and serves only as an intuitive description of the physical behaviour due to the fact that an object in the tube/channel distorts the fluid flow profile and the way that fluid

terminal radial forces are presented with dots for both viruses in figures 15 and 16 together

The analysis were repeated for other virus initial positions - position at <sup>1</sup>

*D* − *t D* 2 <sup>−</sup> *<sup>B</sup>*

*<sup>J</sup>* <sup>=</sup> *<sup>A</sup>ω*<sup>2</sup> <sup>+</sup> *<sup>B</sup><sup>ω</sup>* <sup>+</sup> *<sup>C</sup>* (14)

<sup>2</sup>*<sup>A</sup>* (16)

<sup>2</sup>*<sup>A</sup>* (15)

<sup>8</sup> of the tube diameter

<sup>8</sup> of the tube diameter from the top boundary. The

*Fr*(*y*) = *ay*<sup>2</sup> + *by* + *c* (17)

of the rigid body is used to generate the equation of motion for the virus rotation.

<sup>d</sup>*<sup>t</sup>* <sup>=</sup> *<sup>M</sup>*(*ω*)

d*ω*

smallest is 6.3 · <sup>10</sup>−39kgm2. Equation (14) can be resolved analytically.

<sup>2</sup>*<sup>A</sup>* tanh

*<sup>ω</sup>*(*t*) = *<sup>D</sup>*

*t* → ∞ as

13.

velocity.

flows around a virus.

from the top boundary and for position at <sup>3</sup>

with their quadratic approximations, equation (17).

So far the numerical results showed that the filtering in the straight tube is not possible because the stationary viruses are forced toward the fluid flow centreline. A look at moments acting on the virus reveals that the moment (torque) is present due to fluid viscosity. The torque is acting in the negative (CW - clock wise) direction if the virus is placed between the centreline and top boundary and in the positive (CCW - counter clock wise) direction if the virus is placed between the centreline and bottom tube boundary, figure 18. The virus is actually free to rotate and the redial force changes its direction towards tube boundaries, figure 14, due to the Magnus effect. The model shows this phenomenon, figure 14, and hence the filtering is possible even if the tube is perfectly straight.

The natural way of causes and consequences tells us that the moment drives the virus's rotation. The model is set in a way to address this phenomenon in the inverse approach. The circumferential velocity of the virus is prescribed as boundary condition for the fluid and the moment acting on the virus is estimated in the analysis. The results for the large virus placed at <sup>1</sup> <sup>4</sup> of the tube diameter from the top boundary, figure 12.

Fig. 12. The fluid induced moment acting on large virus as a function of the virus rotation.

The moment is at first negative and quite constant. It increases with the increasing circumferential velocity of the virus. It is obvious that it cannot reach the positive value at high circumferential velocities, due to the simple fact that the moment is driving the virus rotation and the change in the moment sign indicates, that the fluid flow, that generates the moment and was previously acceleration the rotation, is slowing down the virus rotation in such a case. The expectation is that the virus will rotate with the constant angular velocity, which value is defined by the zero moment. The moment is approximated with the polynomial of the second order, equation (13), as presented in figure 12.

$$M(\omega) = a\omega^2 + b\omega + c \tag{13}$$

18 Will-be-set-by-IN-TECH

So far the numerical results showed that the filtering in the straight tube is not possible because the stationary viruses are forced toward the fluid flow centreline. A look at moments acting on the virus reveals that the moment (torque) is present due to fluid viscosity. The torque is acting in the negative (CW - clock wise) direction if the virus is placed between the centreline and top boundary and in the positive (CCW - counter clock wise) direction if the virus is placed between the centreline and bottom tube boundary, figure 18. The virus is actually free to rotate and the redial force changes its direction towards tube boundaries, figure 14, due to the Magnus effect. The model shows this phenomenon, figure 14, and hence

The natural way of causes and consequences tells us that the moment drives the virus's rotation. The model is set in a way to address this phenomenon in the inverse approach. The circumferential velocity of the virus is prescribed as boundary condition for the fluid and the moment acting on the virus is estimated in the analysis. The results for the large virus

Fig. 12. The fluid induced moment acting on large virus as a function of the virus rotation. The moment is at first negative and quite constant. It increases with the increasing circumferential velocity of the virus. It is obvious that it cannot reach the positive value at high circumferential velocities, due to the simple fact that the moment is driving the virus rotation and the change in the moment sign indicates, that the fluid flow, that generates the moment and was previously acceleration the rotation, is slowing down the virus rotation in such a case. The expectation is that the virus will rotate with the constant angular velocity, which value is defined by the zero moment. The moment is approximated with the polynomial of

*M*(*ω*) = *aω*<sup>2</sup> + *bω* + *c* (13)

<sup>4</sup> of the tube diameter from the top boundary, figure 12.

the filtering is possible even if the tube is perfectly straight.

the second order, equation (13), as presented in figure 12.

placed at <sup>1</sup>

where *M* stands for the fluid induced moment acting on virus, *ω* for the angular velocity of the virus and *a*, *b*, *c* for the approximation parameters. The second Newton's low for rotation of the rigid body is used to generate the equation of motion for the virus rotation.

$$\frac{d\omega}{dt} = \frac{M(\omega)}{J} = A\omega^2 + B\omega + \mathbb{C} \tag{14}$$

where *J* stands for the virus moment of inertia and *A*, *B*, *C* for the approximation parameters divided by *<sup>J</sup>*. The moment of inertia *<sup>J</sup>* of the largest virus is 1.53 · <sup>10</sup>−31kgm2 and for the smallest is 6.3 · <sup>10</sup>−39kgm2. Equation (14) can be resolved analytically.

$$
\omega(t) = \frac{D}{2A} \tanh\left(\operatorname{arctanh}\left(\frac{B}{D}\right) - t\frac{D}{2}\right) - \frac{B}{2A} \tag{15}
$$

where *t* stands for time, *A*, *B*, *C* for the approximation parameters divided by *J* and *D* for the discriminate, *<sup>D</sup>* <sup>=</sup> <sup>√</sup>*B*<sup>2</sup> <sup>−</sup> <sup>4</sup>*AC*. The virus terminal angular velocity is reached in the limit case *t* → ∞ as

$$
\omega\_{\text{max}} = \frac{-D - B}{2A} \tag{16}
$$

The terminal circumferential velocity is computed as *vcmax* = *ωmaxR*, where *R* stands for the virus radius. The values of the terminal (maximal) circumferential velocity for the larger virus is *vcmax* = −0.0386m/s and for the smaller one *vcmax* = −0.0146m/s which is around 4% and 1.5% respectively of the air velocity at the inlet. What is really important is that viruses begins to rotate with the terminal angular velocity in a very short time, almost instantly, see figure 13.

The radial force was computed and approximated in the same way as the moment, see figure 12. Hence, by combining it with the angular velocity, the radial force can be plotted as function of time, figure 14. The force becomes constant with the virus reaching the terminal angular velocity.

Although figure 14 presents the resultant force, we can say that there are two competing mechanisms of delivering the radial driving force to the virus. It has been stated that the virus is driven towards the tube centre if it does not rotate due to the fluid flow velocity profile. The same velocity profile also drives the rotation of the virus which in turn adds radial force (Magnus effect, where the rotation is self induced - we named it as J-effect) in the direction of the tube/channel boundary. If the rotation is large enough, which is the case here, the radial force due to the rotation is larger than one due to the fluid profile. The force partition described here is a simplification of a complex fluid velocity and pressure distribution in the vicinity of a virus and serves only as an intuitive description of the physical behaviour due to the fact that an object in the tube/channel distorts the fluid flow profile and the way that fluid flows around a virus.

The analysis were repeated for other virus initial positions - position at <sup>1</sup> <sup>8</sup> of the tube diameter from the top boundary and for position at <sup>3</sup> <sup>8</sup> of the tube diameter from the top boundary. The terminal radial forces are presented with dots for both viruses in figures 15 and 16 together with their quadratic approximations, equation (17).

$$F\_r(y) = ay^2 + by + c \tag{17}$$

Fig. 15. The radial force at virus positions form the centreline up for the small virus.

Novel Theoretical Approach to the Filtration of Nano Particles Through Non-Woven Fabrics 225

Fig. 16. The radial force at virus positions form the centreline up for the large virus.

Fig. 13. The virus circumferential velocity as a function of time.

Fig. 14. The radial force as a function of time.

20 Will-be-set-by-IN-TECH

Fig. 13. The virus circumferential velocity as a function of time.

Fig. 14. The radial force as a function of time.

Fig. 15. The radial force at virus positions form the centreline up for the small virus.

Fig. 16. The radial force at virus positions form the centreline up for the large virus.

**4.4 Criticism of the classical theory of filtration**

the flow in a tube.

summarized in following points.

J-effect.

**4.5 Mechanisms of filtration of the medical masks**

curved around the fibre, get together again behind the fibre.

reason that the porosity measurement takes into account split pores.

The shortcomings of the classical filtration theory can be summarized as follows:

is never a significant air flow, are not explained by the classical filtration theory.

important role in the caverns, where there is predominantly still air.

The fluid flow that flows around a single fibre is the base upon the classical filtration theory is built, as presented in figure 6. The flow is divided by the fibre and the steam lines that are

Novel Theoretical Approach to the Filtration of Nano Particles Through Non-Woven Fabrics 227

The situation, described by the classical filtration theory, is actually occurring during the air flow through the mask. However, the amount of air experiencing it is much smaller than the amount of air that is flowing through pores; see figures 2, 3 and 4, particularly due to the

1. The fibres have a role of the pore boundary at non-woven fabrics. There are caverns in between the fibres where the air does not flow, see figure 5. The virtual surface is formed between the fluid flow and the void when fluid is in motion. The pore caverns, where there

2. The classical filtration theory foresees the Brownian motion with one predominant direction, see figure 6, for particles smaller than 0.15*μ*m. Such particles supposedly hit the fibre by being thrown out of the flow by the diffusion mechanism. They may be forced by a collision with a larger particle or supposedly by air molecules. The mechanism of the collision with the larger particle is feasible. On the contrary, the model of the mechanics of continuum does not support a mechanism of filtration based on the Brownian motion of the small particle in the fluid flow. The Brownian motion of the small particles plays an

3. The classical filtration theory considers the diameter of a fibre as the only characteristic linear measure when computing the Reynolds number. It has been shown that, if the maximal pore diameter is taken into account, the Reynolds number is the order of the magnitude larger. The Reynolds number for the flow around an object in the flow is much smaller, but the limit where turbulent flow starts is also much smaller than in the case of

The main difference between the classical filtration theory and the theory presented here is that the air flowing through the medical mask is supposed to flow through channels. This claim is supported by the theoretical and experimental findings of the J-method for the flat textile porosity assessment. The laminar flow through a channel (or a tube) establishes parabolic fluid flow velocity profile. The velocity profile enables the self-induced rotation of the spherical or near-spherical particles. The angular velocity reaches its limit value almost instantly causing the radial force to point towards the channel boundary due to the Magnus effect. Thus, the effective filtering is enabled and the whole phenomenon is named as the

Based on the developed theory, numerical simulations and observations of the filtration of the industry waste gases, where needled flat textile is used, the mechanisms of filtration can be

1. The particle, which is positioned at the centerline of the channel (tube), stays there until it

collides with another particle or until the channel changes the direction.

where *Fr* stands for the radial force on virus, *y* for the virus displacement in the radial direction and *a*, *b*, *c* for the approximation parameters. This data is used to compute the virus kinematics in the radial direction. The second Newton's law was used again.

$$\frac{\mathrm{d}^2 y}{\mathrm{d}t^2} = \frac{F\_r(y)}{m} = Ay^2 + By + \mathbb{C} \tag{18}$$

where *m* stands for the virus mass and *A*, *B*, *C* for the approximation parameters divided by *m*. The equation can be solved analytically. Nevertheless, the solution is awkward to use and the numerical integration of the differential equation (18) presented in figure 17 was used.

Fig. 17. The radial virus path.

It is interesting to compare figures 11 and 17. We can conclude for the larger virus that it is captured by the tube wall in 5.1 · <sup>10</sup>−7s and for the smaller virus that its capturing time is approximately 2.7 · <sup>10</sup>−8s. The axial velocity of the larger virus at the capture time is only 0.15m/s, which is about 10% of its maximal axial velocity and its axial displacement is approximately 4.0 · <sup>10</sup>−8m, which is three orders of the magnitude less than its displacement in the radial direction. The picture is somewhat different with the smaller virus. It reaches 0.85m/s, which is already about 57% of its maximal axial velocity. It travels approximately 1.4 · <sup>10</sup>−8m in the axial direction, which is again three orders of the magnitude less than its displacement in the radial direction.

The mechanism of filtration presented here, which is based on fluid induced rotation of a round object, is closely tied to the Magnus effect. It is valid also in the case of the curved tube, due to the laminar nature of the fluid flow. Hence it is an efficient way of filtration of viruses and microbes.
