Hydrodynamic particle removal from surfaces

tags: research

• Hydrodynamic particle removal from surfaces
  • Background
  • ABAQUS simulations
  • Theoretical model
  • Reomval mechanisms
  • Results
  • Discussion

Background

The coronavirus SARS-CoV-2 is the cause of the COVID-19 pandemic which has emerged as a serious public health threat. The schematic of its structure is shown in the figure below^[1]. Its diameter is of approximately 60–140 nm^[2].

Airborne droplets from an infected person's cough, sneeze, or even talking are a major source of viral spread. These droplets land on the mucous membranes of potential hosts (mouth, noses, eyes) or on hard surfaces. There the virus will be dispersed for next infection.

A (scientific) question we may ask is if the virus could be removed by mucus secreted by mucous membranes under some conditions. This problem is similar to the hydrodynamic particle removal from surfaces by a liquid flow.

ABAQUS simulations

We simulate the case where the fluid flows over an elastic cylinder, which is adhered to a rigid substrate. This simulation is just an demonstration, and the parameters used here need to change in order to simulate the case of mucus flow.

Geometry:
fluid domain: rectangle, 1 m

$\times$ 3 m
solid domain: cylinder, radius = 0.25 m

Material properties:

• Fluid: density
  $\rho =1000$ kg/m
  ${}^3$, dynamic viscosity
  $\mu =0.001$ Pa
  $\cdot$s
• Solid: density
  $\rho =1100$ kg/m
  ${}^3$, elastic modulus
  $E=1$ MPa, Poisson's ratio
  $\nu =0.45$

Boundary conditions:
fluid domain:

• Left, Top: velocity
  $V_1=0.1$ m/s,
  $V_2=V_3=0$
• Right: pressure
  $P=0$
• Bottom: no-slip

solid domain:

• Bottom: fixed displacement
• Sphere surface: fluid-structure interaction

The video below shows the velocity field of the fluid.

The video below shows the deformation of the cylinder when the fluid flows over it.

The figure below shows the pressure distribution along the cylinder surface at different times.

Theoretical model

As a starting point, we model the virus to be an elastic sphere adhered on the mucous membrane. The mucus is modeled as a slow (quasi-static), linear shear flow on the membrane, as shown in the figure below.

The thickness of the mucus typically ranges from tens of

$\mu$m^[3] to several mm at different organs in human body. Close to the mucous membrane, the velocities of the mucus flow is small, a linear velocity profile can be assumed in the boundary layer as a first order approximation. Modeling the mucus as a linear shear flow would be reasonable if the thickness of the boundary layer of the flowing mucus,

$\delta$, is larger than the virus' characteristic length scale, diameter

$2R$.

There are three different kinds of forces exerted on the virus when it is passed by the mucus flow. (We ignore the gravitational force here.)

The first one is the adhesive force,

$F_A$, due to the adhesive interaction between the virus and the mucous membrane. We adopt the Johnson–Kendall–Roberts (JKR) theory^[4] is to describe such adhesive interaction. The contact region formed between the virus and the membrane is circular with radius

$a$. As per the JKR theory, the adhesive force is given by

${\mathit{F}}_A=-F_A{\mathit{e}}_y$, where

$$F_A=\frac{3}{2}\pi wR,$$

where

$w$ is the work of adhesion and

$R$ is the radius of the virus.

The second one is the drag force originating from a pressure difference across the sphere. For a slow, linear shear flow, the drag force has been computed analytically^[5][6] and given as

${\mathit{F}}_D=F_D{\mathit{e}}_x$, where

$$F_D=1.7\times 6\pi \mu u_{0}R\approx 32\mu u_{0}R,$$

where

$\mu$ is the viscocity of the mucus,

$u_0$ is the flow velocity at the center of the sphere. The pre-factor 1.7 corrects for the effect of the sphere surface, which changes the flow pattern around the sphere and thus results in the drag force.

The last one is the lift force due to the velocity gradient in the shear flow. The analytical expression of the lift force is

${\mathit{F}}_L=F_L{\mathit{e}}_y$, where

$$F_L=6.4\mu u_0{R}^2{\left(k\frac{\rho }{\mu }\right)}^{1/2},$$

in which

$k$ is the magnitude of velocity gradient of the undisturbed flow and

$\rho$ is the density of the mucus^[7][8][9].

Reomval mechanisms

Based on the force diagram of the virus, there exist three possible mechanisms of removing the virus from the mucous membrane: lifting, sliding, and rolling.

Lifting:
The virus can be lifted up and de-adhered from the membrane if the lifting force is larger than the adhesive force, i.e.,

$$F_L\ge F_A\text{or}w\le 1.37\mu u_{0}R{\left(k\frac{\rho }{\mu }\right)}^{1/2}.$$

Sliding:
The virus will be sliding on the surface of the mucous membrane if the lifting condition is not satisfied and

$$F_D\ge {\mu }_f(F_A-F_L),$$

where

${\mu }_f$ is the static friction coefficient between the virus and the membrane.

Rolling:
The virus will start to roll around the contact edge on the membrane if the following criterion is satisfied:

$$F_{D}d\ge (F_A-F_L)a,$$

where

$d$ is the distance from the center of deformed sphere to the top surface of the membrane. As an estimation, the value of

$d$ can be approximated as the virus' radius

$R$.

Results

In order to determine the removal mechanism of the virus by the mucus flow, we need the values for the following physical quantities:

• virus' radius
  $R$,
• mucus' viscosity
  $\mu$,
• mucus' density
  $\rho$,
• mucus' velocity profile
  $u$,
• work of adhesion
  $w$,
• static friction coefficient
  ${\mu }_f$.

Discussion

We can use a more sophisticated theory of cell adhesion^[10][11] than the JKR theory to model the adhesion between the virus and the membrane.

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1. https://www.britannica.com/science/coronavirus-virus-group ↩︎

2. Marco Cascella, et al. Features, evaluation and treatment coronavirus (COVID-19). Statpearls [internet]. StatPearls Publishing, 2020. ↩︎

3. Marlies Elderman, et al. The effect of age on the intestinal mucus thickness, microbiota composition and immunity in relation to sex in mice. PLoS One 12(9), 2017. ↩︎

4. K.L. Johnson, K. Kendall, and A.D. Roberts. Surface energy and the contact of elastic solids. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 324(1558): 301-313, 1971. ↩︎

5. M. E. O'neill. A sphere in contact with a plane wall in a slow linear shear flow. Chemical Engineering Science 23(11): 1293-1298, 1968. ↩︎

6. A.J. Goldman, G. Cox Raymond, and Howard Brenner. Slow viscous motion of a sphere parallel to a plane wall—I Motion through a quiescent fluid. Chemical Engineering Science 22(4): 637-651, 1967. ↩︎

7. P.G.T. Saffman. The lift on a small sphere in a slow shear flow. Journal of Fluid Mechanics 22(2): 385-400, 1965. ↩︎

8. Ahmed Busnaina, Jack Taylor, and Ismail Kashkoush. Measurement of the adhesion and removal forces of submicrometer particles on silicon substrates. Journal of Adhesion Science and Technology 7(5): 441-455, 1993. ↩︎

9. G.M. Burdick, N.S. Berman, and S.P. Beaudoin. Describing hydrodynamic particle removal from surfaces using the particle Reynolds number. Journal of Nanoparticle Research 3.(5-6): 453-465, 2001. ↩︎

10. David Bhella. The role of cellular adhesion molecules in virus attachment and entry. Philosophical Transactions of the Royal Society B: Biological Sciences 370(1661): 20140035, 2015. ↩︎

11. Matthew A. Dragovich, et al. Biomechanical characterization of TIM protein–mediated Ebola virus–host cell adhesion. Scientific Reports 9(1): 1-13, 2019. ↩︎