Modeling virus transport and dynamics in viscous flow medium

Abstract

In this paper, we developed a mathematical model to simulate virus transport through a viscous background flow driven by the natural pumping mechanism. Two types of respiratory pathogens viruses (SARS-Cov-2 and Influenza-A) are considered in this model. The Eulerian–Lagrangian approach is adopted to examine the virus spread in axial and transverse directions. The Basset–Boussinesq–Oseen equation is considered to study the effects of gravity, virtual mass, Basset force, and drag forces on the viruses transport velocity. The results indicate that forces acting on the spherical and non-spherical particles during the motion play a significant role in the transmission process of the viruses. It is observed that high viscosity is responsible for slowing the virus transport dynamics. Small sizes of viruses are found to be highly dangerous and propagate rapidly through the blood vessels. Furthermore, the present mathematical model can help to better understand the viruses spread dynamics in a blood flow.

Keywords:

• Virus spread dynamics
• blood flow transport
• Stokes flow
• Basset force

Nomenclature

$\mathbf{v}$	=	Velocity vector of fluid($\text{m}/\text{s}$)
${\mathbf{v}}_{\mathbf{p}}$	=	Velocity vector of particle($\text{m}/\text{s}$)
u, v	=	Velocity Component($\text{m}/\text{s}$)
c	=	wave velocity($\text{m}/\text{s}$)
a	=	Width of the wave(L)
x, y	=	Coordinate axes
W	=	Transverse channel length scale($\text{m}$)
p	=	Pressure(${\text{kg.m}}^{-1}.s^{-2}$)
$m_p$	=	Mass of the particle($\text{kg}$)
$m_f$	=	Mass of the fluid($\text{kg}$)
g	=	Gravity($\text{m}.{\text{s}}^{-2}$)
t	=	Time($\text{s}$)
r	=	Radius of the particle($\text{m}$)
Re	=	Reynolds number
$S_N$	=	Stokes number
S	=	Density ratio
$R{e}_p$	=	Particle Reynolds number
$F_D$	=	Drag force
μ	=	Dynamic viscosity($\text{kg}.{\text{m}}^{-1}.{\text{s}}^{-1}$)
ν	=	Kinematic viscosity($\text{m}.{\text{s}}^{-2}$)
ρ	=	Blood density($\text{kg}/{\text{m}}^3$)
λ	=	Axial channel length scale($\text{m}$)
ϵ	=	Wave number
${\tau }_c$	=	Fluid characteristic time ($\text{s}$)
${\tau }_p$	=	Particle relaxation time ($\text{s}$)
ζ	=	Dynamics shape correction factor

1. Introduction

The dynamics of the spread of infectious diseases are controlled by many factors, including but not limited to the virus type, size, shape, and the fluid medium (air or liquid) around these viruses. Many mathematical models for the spread of infectious diseases in populations have been analysed and proposed to specific diseases including the SARS-CoV-2 pandemic [3, 4, 17]. Several studies reported that SARS-CoV-2 and Influenza-A are both infectious respiratory diseases but they are caused by different viruses [16]. However, sometimes people have other symptoms such as abdominal pain, irritable bowel syndrome (IBS), and diarrhoea due to SARS-CoV-2 which could become more serious [28]. After the vaccination, people seem to be falling sick. In these regards, numerous viewpoints and questions are raised related to the infection spread capability including; what is known/unknown about the virus? and what are the next steps to stop the outbreak? how the healthcare systems should react to minimize the infection risk? How virus reacts inside the blood stream?

Researchers have found that the SARS-CoV-2 is similar in its structure to other pathogenic respiratory viruses such as influenza A, B, C and D variants. The size of these respiratory viruses lies between 80 and $200\text{nm}$ and is a spherical-like in shape [33]. Among of them, the shape and structure of Influenza-A and SARS-CoV-2 are similar, with the same density [32]. An interesting numerical simulation on state-of-the-art for evaporation of respiratory droplets during the cough was reported and noted that small droplets travel less than $2.5\text{m}$ versus more than $7.5\text{m}$ [31]. From the literature, it is reported that the survival of coronavirus is more on impermeable surface as compared to porous medium [9]. In general, the surrounding fluid motion plays a central role in the spread dynamics of the viruses within a given population. For example, the coronavirus and Influenza viruses have shown the capability in both air and liquids media which means that it is very difficult to control its spreading mechanism without a clear understanding of how it transports in a fluid domain [29]. Therefore, the dynamic characteristics of puffs generated by sneezing/coughing are indeed responsible for the spread of virus infection from one to other. The SARS-CoV-2 and Influenza viruses' capability of spread or transport is discussed in a full detail [2, 11, 34, 36]. This study has also addressed the virus replication and pathogenicity different paradigms.

In additional, a review study that focused on summarizing the effects of temperature and relative humidity on survival of airborne and practices employed to mitigate respiratory infections is given [35]. Furthermore, the evaporation and movement of droplets of pure and saline water have been investigated and recommended that droplet's size affects the evaporation and movement of droplets [39]. Some study reported that the respiratory pathogens viruses spread mechanisms are rooted in the field of fluid dynamics [20]. In this regard, the data for the particle emission rate when person is talking, sneezing, coughing which is related to the amplitude of vocalization (approximately 1–50 particles/second and 0.06–3 particles/ ${\mathrm{c}\mathrm{m}}^3$) is analysed [5]. A dynamical model is developed to reduce the spread rate of COVID-19 and reported that social distancing could be greater than $1.6\text{m}$ [26]. Another study reported the fluid dynamics simulation techniques to understand the effectiveness of materials used in the mask design [22]. The possibilities of getting infected from a virus carrier at a certain distance have recently simulated [10].

Several mathematical models were derived to study the motion of airborne droplets aiming to propose potential solution on how to prevent the outbreaks [13, 14]. In most of the above mathematical models, Lagrangian and Eulerian hypothesis were used to examine the trajectories of droplets in fluid medium [25]. This technique can be useful to study other types of flu-like viruses. Hence, the virus trajectories in a fluid medium can be studied using particle-based modelling approaches [12, 15, 19, 21].

From the previous studies and computational simulations, it is unidentified that the physiological factors are also responsible for the epidemics of airborne infectious disease. Due to the high risk of the SARS-CoV-2 and Influenza viruses, it is essential to understand its transmission modes within a specific fluid medium. For example, the virus can be transported in the urine, cerebral spinal fluid, blood, washings of the respiratory tract, and stool. This indicates that better understanding the transportation of virus in a fluid medium might help to proposing rational solution to protect the vulnerable and limit the virus spread capability. In point of view, Tripathi et al. [37] addressed how blood vessel response when virus enter in the blood stream. Further, discussed that the temperature effect on virus emission in the blood vessel. Ram et al. [30] studied the H1N1 virus infection in the oesophagus move through saliva. In their study it has been found that the infection rate is higher for men than women.

In this study, we derive a mathematical model to study the virus motion in a viscous fluid medium. The model is developed to simulate the nature of virus transport through the blood flow driven by a natural pumping mechanism. SARS-CoV-2 and Influenza-A are considered to examine the virus spread dynamics in blood flow by adopting the Eulerian–Lagrangian approach. The Basset–Boussinesq–Oseen (BBO) equation is also considered to discuss the effects of gravity, virtual mass, Basset force, and drag forces on flow field. We believe that it is very essential to understand how different class of viruses are transmitted in a viscous fluid medium (internal flow regimes), which in turn might provide possible methods for future virus outbreak prevention and control.

2. Methods

Consider the Influenza-A and SARS-CoV-2 propagation through blood stream (red blood cells, white blood cells, proteins, etc.) in a capillary blood vessel (channel) driven by the natural pumping mechanism as shown schematically by Figure 1. The various cells, protein improve our immune system which fight against the viruses and bacteria to reduce the infection. In the sever condition, these cells are damages so that our immune system become weak. In the current study, the sever stage is considered and questioned on how these virus behave in the blood vessel. The structure, shape, density, diameter of both the viruses are tabulated in Table 1. In the next section, the Navier–Stokes equation is consider to derive the behaviour of viscous fluid medium that describing the virus motions.

Figure 1. Schematic representation of the virus transport in peristaltic flow medium.

Table 1. Specifications of viruses based on literature.

2.1. Fluid flow analysis

Consider the motion of a virus in a two-dimensional incompressible viscous flow at low Reynolds regime. The flow is governed by the peristaltic pumping mechanism in a confined domain where the lubrication theory can be applied. The governing equations can be derived as: (1) $\begin{array}{rl}& \mathrm{\nabla }\cdot \mathbf{v}=0,\end{array}$(1) (2) $\begin{array}{rl}& \rho (\frac{\mathrm{\partial }}{\mathrm{\partial }t}+\mathbf{v}\cdot \mathrm{\nabla })\mathbf{v}=-\mathrm{\nabla }p+\mu {\mathrm{\nabla }}^2\mathbf{v},\end{array}$(2) where, ρ is the fluid density, $\mathbf{v}=(u,v,0)$ is a velocity vector field subject to the no-slip boundary conditions on the walls, p is the pressure, and μ is the fluid viscosity.

Herein, we assume that the fluid flow is governed by the rhythmic propagation of the wall contraction (i.e. peristaltic pumping) and wall motion profile is modelled as: (3) $h(x,t)=W-a{\cos }^2\left[\frac{\pi }{\lambda }\right(x-ct\left)\right].$(3) The above equations can be rewritten in a non-dimensional form by introducing the following non-dimensional scaling parameters: $x^{\ast }=\frac{\pi x}{\lambda }$, $y^{\ast }=\frac{y}{W}$, $t^{\ast }=\frac{c\pi t}{\lambda }$, $u^{\ast }=\frac{u}{c}$, $v^{\ast }=\frac{v}{cϵ}$, $p^{\ast }=\frac{pϵW}{\mu c}$, $h^{\ast }=\frac{h}{a_0}$, $L^{\ast }=\frac{L}{\lambda }$, $a^{\ast }=\frac{a}{a_0}.$ Where $W=a_0+a$ represents the channel width, $Re=\frac{\rho cW}{\mu }$ is the flow Reynolds number, $ϵ=\frac{\pi W}{\lambda }$ is the wave number and ${\mathrm{\nabla }}^2={ϵ}^2\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^2}+\frac{\mathrm{\partial }}{\mathrm{\partial }{y}^2}$ is considered as the Laplacian operator. Substitute back into Equations (1(1) $\begin{array}{rl}& \mathrm{\nabla }\cdot \mathbf{v}=0,\end{array}$(1) )–(2(2) $\begin{array}{rl}& \rho (\frac{\mathrm{\partial }}{\mathrm{\partial }t}+\mathbf{v}\cdot \mathrm{\nabla })\mathbf{v}=-\mathrm{\nabla }p+\mu {\mathrm{\nabla }}^2\mathbf{v},\end{array}$(2) ) and drop$(\ast )$ for convenience (4) $\begin{array}{rl}& \frac{\mathrm{\partial }u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }v}{\mathrm{\partial }y}=0,\end{array}$(4) (5) $\begin{array}{rl}& ϵRe(\frac{\mathrm{\partial }u}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }u}{\mathrm{\partial }y})=-\frac{\mathrm{\partial }p}{\mathrm{\partial }x}+{ϵ}^2\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2},\end{array}$(5) (6) $\begin{array}{rl}& {ϵ}^{2}Re(\frac{\mathrm{\partial }v}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }v}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }v}{\mathrm{\partial }y})=-\frac{\mathrm{\partial }p}{\mathrm{\partial }y}+{ϵ}^3\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{x}^2}+ϵ\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{y}^2}.\end{array}$(6) The flow transport under the lubrication theory (low Reynolds number (Re $\in [0,ϵ]$) and small wavelength number $ϵ\ll 1$) where the viscous effect is dominant. In particular, neglecting all terms multiplied by Re and ϵ or higher, we obtained (7) $\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }v}{\mathrm{\partial }y}=0,\frac{\mathrm{\partial }p}{\mathrm{\partial }x}=\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2},\frac{\mathrm{\partial }p}{\mathrm{\partial }y}=0.$(7) Subject to the boundary conditions: (i) at the wall edge, that is, y = h we have u = 0, $v=\frac{\mathrm{\partial }h}{\mathrm{\partial }t}$, (ii) at the channel centre line, that is, y = 0, we have $\frac{\mathrm{\partial }u}{\mathrm{\partial }y}=0$ and $v=0$. Hence, expressions for the axial velocity, transverse velocity, and pressure gradient yield (8) $\begin{array}{rl}u& =\frac{1}{2}\frac{\mathrm{\partial }p}{\mathrm{\partial }x}(y^2-h^2),\end{array}$(8) (9) $\begin{array}{rl}v& =\frac{y}{6}[6h\frac{\mathrm{\partial }h}{\mathrm{\partial }x}\frac{\mathrm{\partial }p}{\mathrm{\partial }x}-\frac{{\mathrm{\partial }}^{2}p}{\mathrm{\partial }{x}^2}(y^2-3h^2\left)\right],\end{array}$(9) (10) $\begin{array}{rl}\frac{\mathrm{\partial }p}{\mathrm{\partial }x}& =\frac{1}{h^3}\left[G\right(t)+3{\int }_0^x\frac{\mathrm{\partial }h}{\mathrm{\partial }t}\mathrm{d}s],\end{array}$(10) where $G\left(t\right)$ is the arbitrary function of t and can be computed as: (11) $G\left(t\right)=\frac{\mathrm{\Delta }{p}_L-3{\int }_0^L{h}^{-3}\left({\int }_0^x\frac{\mathrm{\partial }h}{\mathrm{\partial }t}\mathrm{d}s\right)\mathrm{d}x}{{\int }_0^L{h}^{-3}\mathrm{d}x},$(11) where $\mathrm{\Delta }{p}_L$ represents the pressure difference along the channel length and it can be computed as: $\mathrm{\Delta }{p}_L=p(L,t)-p(0,t)={\int }_0^L\frac{\mathrm{\partial }p}{\mathrm{\partial }x}\mathrm{d}x$. The stream function is given as (12) $\psi =\frac{\mathrm{\partial }p}{\mathrm{\partial }x}(\frac{y^3}{6}-\frac{h^{2}y}{2}).$(12) Having the background flow field is resolved, we can then model the virus transport within this flow field as follows.

2.2. Virus transport

The virus transport motion is governed by the Basset–Boussinesq–Oseen (BBO) equation at low Reynolds number flow regime [19]. Moreover, additional terms in relation to the BBO equation to account for the motion of virus transport in a non-uniform flow are also considered. The equation of motion of particles (viruses) in two-dimensional form is given as (See Ref [19]): (13) $\begin{array}{rl}{m}_p\frac{\mathrm{d}{v}_p}{\mathrm{d}t}& =(m_p-m_f)g+m_f\frac{Dv}{Dt}-\frac{m_f}{2}\frac{\mathrm{d}}{\mathrm{d}t}(v_p-v-\frac{r^2}{10}{\mathrm{\nabla }}^{2}v)\\ & -6\pi r\mu (v_p-v-\frac{r^2}{6}{\mathrm{\nabla }}^{2}v)-6\pi r^2\mu \left({\int }_{-\mathrm{\infty }}^t\frac{\frac{\mathrm{d}}{\mathrm{d}\tau }(v_p-v)}{\sqrt{\pi \nu (t-\tau )}}\mathrm{d}\tau \right),\end{array}$(13) where $v_p=(u_p,v_p)$ is the particle velocity vector, $g$ is the gravity vector, $m_f$ is the mass of the fluid with the same volume as the virus, $m_p$ is the particle mass, $r=d_p/2$ is the particle radius and $d_p$ is the particle diameter, and ν is the fluid kinematic viscosity. The suffix p and f designate particle and fluid, respectively. The BBO equation represents the sum of steady state drag force, the pressure, buoyancy force, the virtual mass force, and the body force equated to the mass times the acceleration of an isolated particle.

The non-dimensional quantities defined for the ${u_p}^{\ast }=u_p/c$, & ${v_p}^{\ast }=v_p/c$, and the gravity $g^{\ast }=g/g_0$. The fluid characteristic time is defined as ${\tau }_c=\lambda /\pi c$, the particle relaxation time as ${\tau }_p=\left({\rho }_p{d_p}^2\right)/18\mu$. For the case where there is a steady flow prior to t = 0 with an impulse at t = 0, the Basset equation is replaced by (See Ref. [21]) (14) ${\int }_{-\mathrm{\infty }}^t\frac{\frac{\mathrm{d}}{\mathrm{d}\tau }(v_p-v)}{\sqrt{\pi \nu (t-\tau )}}\mathrm{d}\tau ={\int }_0^t\frac{\frac{\mathrm{d}}{\mathrm{d}\tau }(v_p-v)}{\sqrt{\pi \nu (t-\tau )}}\mathrm{d}\tau +\frac{(v_o-v_{po})}{\sqrt{\pi \nu t}},$(14) where $v_0$ & $v_{p0}$ are the initial velocity of the fluid and virus particle. Dropping the (*) symbol, the non-dimensional BBO equation with its force terms per particle mass is given by (15) $\begin{array}{rl}\frac{\mathrm{d}{v}_p}{\mathrm{d}t}& =\frac{1}{S_N}\frac{2S}{(2S+1)}(v-v_p)+\frac{3}{2S+1}\frac{\mathrm{d}v}{\mathrm{d}t}+\frac{{\alpha }^2}{40(2S+1)}\frac{\mathrm{d}}{\mathrm{d}t}\left({\mathrm{\nabla }}^{2}v\right)\\ & +\frac{{\alpha }^2}{24S_N}\left(\frac{2S}{2S+1}\right)\left({\mathrm{\nabla }}^{2}v\right)+\sqrt{\frac{9}{2\pi S{S}_N}}\left(\frac{2S}{2S+1}\right)\\ & \times [{\int }_0^t\frac{\frac{\mathrm{d}}{\mathrm{d}\tau }(v-v_p)}{\sqrt{(t-\tau )}}d\tau +\frac{(v_o-v_{po})}{\sqrt{t}}]+\frac{2(S-1)}{(2S+1)}\frac{{\tau }_c}{c}{g}_{0}g,\end{array}$(15) where $S={\rho }_p/\rho$ is the density ratio, $S_N={\tau }_p/{\tau }_c$ is the Stokes number, $\alpha =d_p/W$ is a size ratio. The terms in right hand side of Equation (13(13) $\begin{array}{rl}{m}_p\frac{\mathrm{d}{v}_p}{\mathrm{d}t}& =(m_p-m_f)g+m_f\frac{Dv}{Dt}-\frac{m_f}{2}\frac{\mathrm{d}}{\mathrm{d}t}(v_p-v-\frac{r^2}{10}{\mathrm{\nabla }}^{2}v)\\ & -6\pi r\mu (v_p-v-\frac{r^2}{6}{\mathrm{\nabla }}^{2}v)-6\pi r^2\mu \left({\int }_{-\mathrm{\infty }}^t\frac{\frac{\mathrm{d}}{\mathrm{d}\tau }(v_p-v)}{\sqrt{\pi \nu (t-\tau )}}\mathrm{d}\tau \right),\end{array}$(13) ) designate the Stokes drag, virtual masses, Faxén, Basset and gravity respectively. For the spherical shape, the drag force and Stokes drag are same however drag force is defined as: $F_d=3\pi d_p\mu \zeta (v-v_p)$, where ζ is dynamic shape correction factor, which is unity for sphere, 1.12 and 1.27 for two chained and three chained cluster of the spheres [19]. For non-spherical particle, Stokes drag can be modified as: $\frac{\zeta }{S_N}\left(\frac{2S}{(2S+1)}\right)(v-v_p)$.

The axial velocity component of the particle velocity is expressed as: (16) $\begin{array}{rl}\frac{\mathrm{d}{u}_p}{\mathrm{d}t}& ={\text{β}}_1(u-u_p)+{\text{β}}_2\frac{\mathrm{d}u}{\mathrm{d}t}+{\text{β}}_3\frac{\mathrm{d}}{\mathrm{d}t}\left({\mathrm{\nabla }}^{2}u\right)+{\text{β}}_4{\mathrm{\nabla }}^{2}u\\ & +{\text{β}}_5[{\int }_0^t\frac{\frac{\mathrm{d}}{\mathrm{d}\tau }(u-v_p)}{\sqrt{(t-\tau )}}\mathrm{d}\tau +\frac{(u_o-u_{po})}{\sqrt{t}}]\end{array}$(16) The transverse velocity component of the particle velocity is expressed as: (17) $\begin{array}{rl}\frac{\mathrm{d}{v}_p}{\mathrm{d}t}& ={\text{β}}_1(v-v_p)+{\text{β}}_2\frac{\mathrm{d}v}{\mathrm{d}t}+{\text{β}}_3\frac{\mathrm{d}}{\mathrm{d}t}\left({\mathrm{\nabla }}^{2}v\right)+{\text{β}}_4{\mathrm{\nabla }}^{2}v\\ & +{\text{β}}_5[{\int }_0^t\frac{\frac{\mathrm{d}}{\mathrm{d}\tau }(v-v_p)}{\sqrt{(t-\tau )}}\mathrm{d}\tau +\frac{(v_o-v_{po})}{\sqrt{t}}]-{\text{β}}_6\end{array}$(17) where ${\text{β}}_1=\frac{\zeta }{S_N}\left(\frac{2S}{2S+1}\right)$, ${\text{β}}_2=\frac{3}{2S+1}$, ${\text{β}}_3=\frac{{\alpha }^2}{40(2S+1)}$, ${\text{β}}_4=\frac{{\alpha }^2}{24S_N}\left(\frac{2S}{2S+1}\right)$, ${\text{β}}_5=\sqrt{\frac{9}{2\pi S{S}_N}}\left(\frac{2S}{2S+1}\right)$ and ${\text{β}}_6=\frac{2(S-1)}{(2S+1)}\frac{{\tau }_c}{c}$ $g_0$ g.

In this model, the analytical solution of the particle momentum equation for uniform background flows is calculated to comprehend the nature of small size particle flow. The small particle Reynolds number $({Re}_p=|v-v_p|(\rho W)/\mu <<1)$ and small flow Reynolds number $Re=\left(Wc\rho \right)/\mu$ is validate the particle momentum Equation (12(12) $\psi =\frac{\mathrm{\partial }p}{\mathrm{\partial }x}(\frac{y^3}{6}-\frac{h^{2}y}{2}).$(12) ). Here, the convective acceleration (Stokes flow) approximating the substantial derivative (D/Dt) as (d/dt) is exact to the order of approximation due to uniform background flows. The generic term of virtual mass-2 and Faxen forces applied on the non-uniformity of the flow field. These terms are only relevant for background flow fields that present strong velocity gradients. For small size particle (so that $\alpha =d_p/W<<1$) and uniform flows, the virtual mass-2 and Faxen corrections are zero [12]. Therefore, Equations (14(14) ${\int }_{-\mathrm{\infty }}^t\frac{\frac{\mathrm{d}}{\mathrm{d}\tau }(v_p-v)}{\sqrt{\pi \nu (t-\tau )}}\mathrm{d}\tau ={\int }_0^t\frac{\frac{\mathrm{d}}{\mathrm{d}\tau }(v_p-v)}{\sqrt{\pi \nu (t-\tau )}}\mathrm{d}\tau +\frac{(v_o-v_{po})}{\sqrt{\pi \nu t}},$(14) ) and (15(15) $\begin{array}{rl}\frac{\mathrm{d}{v}_p}{\mathrm{d}t}& =\frac{1}{S_N}\frac{2S}{(2S+1)}(v-v_p)+\frac{3}{2S+1}\frac{\mathrm{d}v}{\mathrm{d}t}+\frac{{\alpha }^2}{40(2S+1)}\frac{\mathrm{d}}{\mathrm{d}t}\left({\mathrm{\nabla }}^{2}v\right)\\ & +\frac{{\alpha }^2}{24S_N}\left(\frac{2S}{2S+1}\right)\left({\mathrm{\nabla }}^{2}v\right)+\sqrt{\frac{9}{2\pi S{S}_N}}\left(\frac{2S}{2S+1}\right)\\ & \times [{\int }_0^t\frac{\frac{\mathrm{d}}{\mathrm{d}\tau }(v-v_p)}{\sqrt{(t-\tau )}}d\tau +\frac{(v_o-v_{po})}{\sqrt{t}}]+\frac{2(S-1)}{(2S+1)}\frac{{\tau }_c}{c}{g}_{0}g,\end{array}$(15) ) may further be expressed as: (18) $\begin{array}{rl}\frac{\mathrm{d}{u}_p}{\mathrm{d}t}& ={\text{β}}_1(u-u_p)+{\text{β}}_2\frac{\mathrm{d}u}{\mathrm{d}t}+\frac{{\text{β}}_5}{\sqrt{0.5t}}(u-u_p)\end{array}$(18) (19) $\begin{array}{rl}\frac{\mathrm{d}{v}_p}{\mathrm{d}t}& ={\text{β}}_1(v-v_p)+{\text{β}}_2\frac{\mathrm{d}v}{\mathrm{d}t}+\frac{{\text{β}}_5}{\sqrt{0.5t}}(v-v_p)-{\text{β}}_6\end{array}$(19) Thus, the solution of Equations (16(16) $\begin{array}{rl}\frac{\mathrm{d}{u}_p}{\mathrm{d}t}& ={\text{β}}_1(u-u_p)+{\text{β}}_2\frac{\mathrm{d}u}{\mathrm{d}t}+{\text{β}}_3\frac{\mathrm{d}}{\mathrm{d}t}\left({\mathrm{\nabla }}^{2}u\right)+{\text{β}}_4{\mathrm{\nabla }}^{2}u\\ & +{\text{β}}_5[{\int }_0^t\frac{\frac{\mathrm{d}}{\mathrm{d}\tau }(u-v_p)}{\sqrt{(t-\tau )}}\mathrm{d}\tau +\frac{(u_o-u_{po})}{\sqrt{t}}]\end{array}$(16) ) and (17(17) $\begin{array}{rl}\frac{\mathrm{d}{v}_p}{\mathrm{d}t}& ={\text{β}}_1(v-v_p)+{\text{β}}_2\frac{\mathrm{d}v}{\mathrm{d}t}+{\text{β}}_3\frac{\mathrm{d}}{\mathrm{d}t}\left({\mathrm{\nabla }}^{2}v\right)+{\text{β}}_4{\mathrm{\nabla }}^{2}v\\ & +{\text{β}}_5[{\int }_0^t\frac{\frac{\mathrm{d}}{\mathrm{d}\tau }(v-v_p)}{\sqrt{(t-\tau )}}\mathrm{d}\tau +\frac{(v_o-v_{po})}{\sqrt{t}}]-{\text{β}}_6\end{array}$(17) ) are obtained subjected to zero initial condition i.e. $v_p(x,t=0)=0$ as: (20) $\begin{array}{rl}{u}_p& ={\mathrm{e}}^{-M}{\int }_0^t{\mathrm{e}}^M({\text{β}}_{1}u+{\text{β}}_2\frac{\mathrm{d}u}{\mathrm{d}t}+0.5k{\text{β}}_5\frac{u}{\sqrt{t}})\mathrm{d}t\end{array}$(20) (21) $\begin{array}{rl}{v}_p& ={\mathrm{e}}^{-M}{\int }_0^t{\mathrm{e}}^M({\text{β}}_{1}v+{\text{β}}_2\frac{\mathrm{d}v}{\mathrm{d}t}+0.5k{\text{β}}_5\frac{v}{\sqrt{t}}-{\text{β}}_6)\mathrm{d}t\end{array}$(21) where $M={\text{β}}_{1}t+k{\text{β}}_5\sqrt{(}t)$, k = 2.82.

3. Results and discussion

This section presents the computational results on axial and transverse particle velocities along with their contour plots for Influenza-A and SARS-CoV-2. The aim of the present model to understand how rapidly the virus particles are transmit in the blood vessel and harm the tissues. This includes the coronavirus and Influenza-A floating in a blood vessel (channel) governed by the natural pumping. Kim et al. [21] reported that there are several fluid forces involving buoyancy force, Basset force, Stokes drag force that can affect the movement of the particle. In this regard, a comparative discussion for both the viruses' transmission under the effects of various forces like blood viscosity, Stokes drag, and Basset are presented. The parametric values for the medium wavelength, wave velocity, viscosity, length of the channel, particle characteristic time, gravity considered for the simulation of the graphical results are listed in Table 2. The standard formula for computing the Stokes number, densities ratio, and gravity parameter are used to find their values for all viruses is listed in Table 3.

Table 2. Values of basic parameters used in the simulation [18, 19, 23].

References	Parameters	Unit
Value
Wavelength (λ)	$\text{mm}$	120
Wave velocity (c)	$\text{mm}/\text{s}$	300−400
Blood density (ρ)	$\text{kg}/{\text{m}}^3$	1050
Fluid characteristic time (${\tau }_c$)	$\text{s}$	0.127
Gravity ($g_0$)	$\text{m}/{\text{s}}^2$	1

Table 3. Computed values of Stokes number, density ratio, and gravity of the viruses as well as the estimated values of coefficients of Stokes drag and Virtual mass.

				${\text{β}}_1\times {10}^{-8}$	${\text{β}}_2$
Viruses	$S_N=\frac{{\tau }_p}{{\tau }_c}$	$S=\frac{{\rho }_p}{\rho }$	${\text{β}}_1\times {10}^{-8}$	$(\zeta =1)$	$(\zeta =1.12)$	${\text{β}}_5$	${\text{β}}_6$
Coronavirus	$8.44\times {10}^{-10}$	1.100	8.145	9.122	0.937	2700	25.92
Influenza-A	$7.30\times {10}^{-10}$	1.104	9.428	10.559	0.935	2902	26.85

To analyse the behaviour of the virus spread in the blood stream (red blood cells, white blood cells, proteins, etc.) it is important to examine the blood viscosity effect in the blood flow analysis. Typically, the blood is composition of the red blood cells, white blood cells, and platelets in blood plasma. So, the blood viscosity is around $(3.3-5.5)\times {10}^{-3}\text{Ns}/{\text{m}}^2$ which treat as Newtonian fluid when the shear stress rate is above $100{\text{s}}^{-1}$, while the non-Newtonian behaviour is attained when shear rate is less $100{\text{s}}^{-1}$ [38]. The cardiovascular system maintains an adequate blood flow to all tissues/cells in the body which is depends upon the peristaltic pumping mechanism of the heart. In case of infection or diseases, the cardiovascular system has been associated with blood stream in the blood vessel [24]. By using the Stokes equation (i.e. $u=1/2\mu \frac{\mathrm{\partial }p}{\mathrm{\partial }x}(y^2-h^2)$), the viscosity effect on the axial blood velocity is depicted in Figure 2. This figure illustrates the streamlines of axial velocity induced by the pressure gradient which is generated by the peristaltic motion of the blood vessel at fixed value of amplitude (a = 0.3) and time (t = 0.1). The stream lines of the axial velocity are being circulated and trapped the fluids in the form of bolus for control movement of the fluids in circulatory systems. It is further noted that axial velocity range varies from 0 to 53.25 at blood viscosity $\mu =3.3\times {10}^{-3}$, while axial velocity range varies from 0 to 31.9 at blood viscosity $\mu =5.5\times {10}^{-3}$. It means that, the high blood viscosity is responsible for the decrement of the axial velocity , that is, viscosity resist to fluid flow in the blood vessel. To see the virus transmission through blood flow driven by natural pumping, the following results are presented.

In Figure 3(a,b), we draw the axial and transverse particle velocity profiles at a fixed time t = 0.5 for an averaged diameter of SARS-CoV-2 and Influenza-A during the fluid flow in the medium. The results show that axial particle velocity for both viruses performs sinusoidal waves with achieved peaks near the centre due to the contraction region of the bloodstream. It is clear that the spread of viruses will be at a higher rate if the blood flow is increased. The results also show that the influenza-A virus is having higher particle velocity as compared to SARS-CoV-2 in the bloodstream as seen in Figure 3(a,b). It means that the infection rate of Influenza-A is higher than SARS-CoV-2. It is important to mention that the Stokes number relies on the assumption of a Stokes drag. In fact, as inertia is increased, the drag on settling particles deviates from the Stokes law. The particle velocities distribution magnitude is attained in the range $(-10-5)\times {10}^6$ due to the Stokes number [12, 19]. These results suggested that the largest difference in the particle velocity occurs when the Stokes number is considered in force calculation involving Stokes drag. The streamlines of the SARS-CoV-2 and Influenza-A are shown in Figure 4 at fixed time point t = 0.1, respectively. From Figure 4, it is concluded that the flow field is reducing in the relaxation region, while more flow is developed in the contraction region along the bloodstream. In other words, the movement of the virus particle in the bloodstream is increased cause of the natural pumping mechanism. However, the movement of the flow field is almost similar i.e. the propagation of respiratory pathogens virion is almost similar in the circulatory system. As SARS-CoV-2 and Influenza-A are tiny viruses with a range from 20 to $140\text{nm}$ in diameter. To estimate the probability that a droplet would contain at least one virus particle, Tripathi et al. [37] illustrated how much virus can exist in the microdroplet of mucus of an infected person. It was reported that about 1950–3000 virus particles can be infected a person [6]. Once it enters the human body it produces many viruses over time. The spread of the SARS-CoV-2 and Influenza-A viruses along the axial length of the blood vessel is depicted through Figures 5–6.

Figure 2. Contour lines that represent the axial velocity field for the background flow with different viscosity (a) $\mu =3\times {10}^{-3}$, (b) $\mu =5.5\times {10}^{-3}$ at fixed value of $a=0.3$.

Figure 3. Combined effects of Stokes drag, virtual mass-1, Basset and gravity on (a) SARS-CoV-2 and (b) Influenza-A virus velocities.

Figure 4. The streamlines of the (a) SARS-CoV-2 and (b) Influenza virus in the bloodstream with the Combined effects of Stokes drag, virtual mass-1, Basset, and gravity.

Figure 5. Contour plots of axial velocity components for (a) SARS-CoV-2 (b) Influenza-A in the presence of Stokes drag force.

Figure 6. Contour plots of axial velocity components for (a) SARS-CoV-2 (b) Influenza-A in the presence of Basset force.

The blood vessel trapped the fluids in the form of bolus due to two different phases (contraction and relaxation phase) which control the movement of the fluid particles in the circulatory system. The motion of virus particle may be similar to the fluid particle motion however, the movement rate may differ due to various forces acting on the virus as compared to the fluid particles. The magnitude of axial velocity for SARS-CoV-2 varies from $(-13.67-7.03)\times {10}^6$ under Stokes drag, while the axial velocity for Influenza-A varies from $(-2.66-1.70)\times {10}^7$ as shown in Figure 5(a,b). This particular result revealed that the Stokes drag is highly responsible for the transmission of SARS-CoV-2. Since the Stokes drag on the particle is dominating as compared to the other forces. Another hand, the magnitude of axial velocity for SARS-CoV-2 varies from $(-2.89-1.60)\times {10}^2$ under Basset force, while the axial velocity for Influenza-A varies from $(-4.14-2.21)\times {10}^2$ as presented in Figure 6(a,b). The velocity is significantly reduced from ${10}^7$ to ${10}^2$ due to the absence of Stokes number in the Basset force. A large number of virus particles responsible for damaging the cell lead to the signs and symptoms of an illness appearing [6]. The body reacts against the high level of infection, which can cause organ damage and even death.

The normalized axial velocity with time for the SARS-CoV-2 and Influenza-A at fixed position x = 0.3 are computed in Figure 7. From this result it is found that the axial velocity is does not vary at particular position. The velocity of Influenza-A is maximum than SARS-CoV-2 and rapidly decreasing at the initial time. This means that the movement of virion at particular position does not oscillate through the time (i.e. contraction and relaxation of the blood vessel). From the previous studies, it is found that the human physiological system such as living organisms (cells, viruses) have the ions which are affected by the magnetic field [1, 8]. Therefore, we can control the propagation of the virus particle by magnetic therapy as the Lorentz force resist the movement of the fluid flow due to the magnetic field [7]. Further, it is mentioned in various literature that where the effect of magnetic field is more, the possibility of the reduction of the virus infection in the body [27, 30].

It is well known that the Stokes number characterizes the behaviour of virus particles suspended in a fluid flow. When the particles are spherical or non-spherical in shape, their drag differs from the Stokes drag reported in the literature [19]. This can be accounted for with the help of shape correction factor. The effects of the shape of SARS CoV-2 on axial and transverse particle velocity can be observed in Figure 8(a b). Along the axial direction, SARS-CoV-2 exhibits a considerable difference in the particle paths of spherical and non-spherical virus particles. Non-spherical particle exhibits a slightly higher particle velocity along the axial and transverse direction due to the shape correction factor. It is inferred that the effect of the shape of the SARS-CoV-2 may play a significant role in the transmission during the viscous fluid flow. The results suggest that the non-spherical particles may move faster as compared to spherical particles in the fluid flow driven by peristaltic pumping.

Human blood has a range of viscosity ($\mu =(3.5-5.5)\times {10}^{-3}$). Due to the viscosity variation, there is much chance of the spread of the virus rapidly/slowly in the blood vessel, which may be caused of a severe case of the patient or damage to the vessel part. Figure 9 (a,b) illustrates the behaviour of axial velocity and transverse velocity of the coronavirus particle in the blood vessel. This figure illustrates the influence of the blood viscosity on the axial and transverse flow of coronavirus particle. Here, high viscosity is responsible for decrementing the axial and transverse flow in the blood vessel. In another word, we can say that the patient of high blood viscosity has low transmission of SARS Cov-2 virus as compared to the less blood viscosity patient [37]. Although, the range of the axial flow is higher than the transverse flow in the blood vessel.

The influence of the diameter of the coronavirus particle on the axial flow at a fixed value of blood viscosity $=3.3\times {10}^{-3}$ is demonstrated in Figure 10 (a,b). This result concluded that the small diameter of the viruses increases the axial flow. Thus, the small particle rapidly moves rather than the large particle in the blood flow through blood vessel. The range of axial velocity field of coronavirus for $d_p=80\text{nm}$ is vary from $-2.3\times {10}^7$ to $1.04\times {10}^7$. Another hand, the range varies from $-7.51\times {10}^6$ to $3.39\times {10}^6$ for $d_p=140\text{nm}$. From these results, it is concluded that the flow of small size of virus particle is high and it is dangerous for the human circulatory system that can damage the blood vessel, capillaries rapidly.

Figure 7. The normalized axial velocity with time for Influenza-A and SARS-CoV-2 with the combined effect of all forces.

Figure 8. Effect of Stokes drag on spherical and non-spherical particle of coronavirus virus in axial and transverse particle velocity.

Figure 9. Effect of blood viscosity of coronavirus in (a) axial and (b) transverse particle velocity.

Figure 10. Contour plots of the axial particle velocity of Coronavirus for (a) $d_p=80\text{nm}$ (b) $140\text{nm}$.

4. Conclusions

A mathematical model based on the Eulerian–Lagrangian formulation is considered to study virus transport in a viscous medium flow (blood flow) driven by natural pumping. We examined the transmission of two types of respiratory pathogens viruses such as SARS-CoV-2 and Influenza-A in blood flow. These viruses are treated as particles with spherical and non-spherical shapes. The axial and transverse particles motion under the effects of various applicable forces such as the gravity, virtual mass 1, Basset force, and drag force are investigated. The keyout comes of the current paper as follows:

• The movement of both viruses in the blood vessel is very similar, however the infection rate of Influenza-A is higher than SARS-CoV-2.

• High viscosity is reduces the velocity of the virus transmission in the viscous fluid medium.

• The nano-size virions are propagated rapidly in the blood vessel, capillaries which lead to high rate of infection.

• A large number of virus particles (i.e. 1950 to 3000 virus particles) responsible for damaging the cell lead to the signs and symptoms of an illness appearing.

It is further concluded that the virus may be circulated in the blood vessel where the flow is governed by the natural pumping with more circulations of the viruses are reported in contracted region as compared to the relaxed region. Although the present analysis ignored the biophysical details of the real viruses, it may define the importance of the fluid dynamics in the prevention of the COVID-19 spread and other similar viruses. Further research is indeed required to focus on (i) including the biological aspects of viruses, (ii) study the rheological effects on transmission of viruses within a non-Newtonian background flow, and (iii) thermal effects on the transmission of viruses.

Author's Contributions

All authors contributed equally to this work.

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The authors report no conflict of interest.

Data Availability

The data that supports the findings of this study are available within the article.

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

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