Theoretical Analysis of the Induction of Forced Resonance Mechanical Oscillations to Virus Particles by Microwave Irradiation: Prospects as an Anti-virus Modality

The induction of acoustic-mechanical oscillations to virus particles by illuminating them with microwave signals is analyzed theoretically. Assuming the virus particle being of spherical shape, its capsid consisting primarily of glycoproteins, a viscous fluid model is adopted while the outside medium of the sphere is taken to be ideal fluid. The electrical charge distribution of virus particle is assumed to be spherically symmetric with a variation along the radius. The generated acousticmechanical oscillations are computed by solving a boundary value problem analytically, making use of the Green’s function approach. Resonance conditions to achieve maximum energy transfer from microwave radiation to acoustic oscillation to the particle is investigated. Estimation of the feasibility of the technique to compete virus epidemics either for sterilization of spaces and/or use for future therapeutic applications is examined briefly.


Introduction
The study of physical properties of various type viruses has attracted significant interest by several interdisciplinary research groups during the last 10 years 1  Also, Elasticity Theory methods applied to draw conclusions on the mechanical properties of virus particles 4 . Electric charge distributions of virus have been also studied by several researchers 5,6,7,8,9 . It is observed that under physiological conditions of salinity and acidity, virus capsid assembly requires the presence of genomic material that is oppositely charged to the core proteins 10 . Furthermore, few researchers have focused their research on the possibility of inducing photon-phonon interactions 11 in virions, which are the virus causing infections 12,13,14,15,16,17 . minutes. This principle allows, to propose in present case short duration pulsed -periodic high intensity microwave signals 28,29 . This is expected to alleviate to some degree the penetration problem of electromagnetic energy to human body.
In all the mentioned publications of this resonance phenomenon, the virus particle is assumed of being an elastic particle, as was modelled by H. Lamb in 1887 30 for the oscillations of an ideal spherical isolated in space. In present article, the mathematical analysis is carried out also considering the surrounding medium of the virus particle and taking into account the interaction of external microwave radiation with the electric charge distribution of the virus particle. Based on the recently published data on the structure 31 of the Covid-19 virion of being of 100-150nm in diameter, because of its reach liposome capsid with few proteins on it, in the present work lead us to adopt the model the spherical virus particle as a viscous fluid while the outer space is taken to be an ideal fluid, with different acoustic characteristics of the spherical particle. Furthermore, vortex phenomena in modelling the viscous virus structure are neglected, since it assumed these are very weak and they have no effect on the resonance phenomenon to be studied.

Mathematical Formulation of the Phenomenon.
A spherical particle of radius α, shown in Fig.1, is assumed to poses a continuous electric charge distribution with spherical symmetry defined with the equation Where Q is the total positive electric charge in the center of the sphere, above eq.(1) was selected to have the total charge of the particle to be zero, that is to have a balance between the positive (inner r<α(3/5) 1/2 region) and negative (towards the external surface) charge distributions.
It is evident that the particle could have the opposite charge distribution and the same analysis is valid. The proposed method is extendable to the case of non-symmetric charge distribution, then higher order modes will be excited.
The spherical model of the virus is assumed to be a compressible fluid, characterized with its homogenous mass density ρ1 acoustic wave propagation speed c1 and total viscosity constant (dynamic and bulk) χ.
Then, assuming a a time dependence, the propagation of acoustic wave phenomena are described by the following field equations 32 : where is the velocity, 1 the pressure field and is the force density (N/m 3 ) because of the electric charge distribution inside the sphere.

Mass continuity equation
The force density term in eq. (2), taken into account the charge distribution given in eq.(1) and the incident electric field ( ) = ̂ of the microwave radiation propagating along the x axis and polarized parallel to z axis (see Fig.1), considering the size of the particle to be extremely small compared to microwave radiation wavelength, is obtained to be: Operating on the eq.(2) the ∇. operator from the left hand side, substituting eq.(3) , eq.(4) and rearranging the terms the following wave equation is obtained: ∇ .
The pressure field outside of the particle assuming an ideal fluid, is described by two respective equations of eqs. (2) and (3): where ( ) is the pressure, ( ) the velocity, the mass density and the acoustic speed. Also combining eqs. (7) and (8): where = / is the wave constant of the infinite space outside of the sphere.

Solution of the boundary value problem.
The field expression inside the sphere r<α The acoustic pressure inside spherical particle being excited by the interaction of microwave electric field component acting to electric charges, being inside the spherical volume could be described in terms of the Primary ( 10 ( )) and secondary ( 11 ( )) pressure fields 33 .
In the analysis to follow spherical coordinates are used r, θ and φ being radial distance from the origin, θ being the angle measured from z axis and φ the azimuth angle.
Noticing that the primary field depending only to 1 ( ) = cos( ) angular function (n=1and m=0 terms), the secondary pressure field is written easily: The field expression outside the sphere ( r>α) Considering the excitation only the wave with cos(θ) dependence and the necessity of radiation condition to be valid for → +∞ we can write easily: In eqs. (14), (15)

Numerical Calculations
After computing the pressure field as given in eq. (16) it is shown that the "form factor" W is a function of the dimensionless which corresponds to "dipole mode" of the spherical particle.

Conclusions
A rather simple model of a virus particle allowed to analyze the coupling phenomena between microwave (electromagnetic) radiation and acoustic waves generated inside the particle. Based