Sciresol Sciresol https://indjst.org/author-guidelines Indian Journal of Science and Technology 0974-5645 10.17485/IJST/v14i13.1796 A Two-phase mixture model of Atmospheric Aerosols in the channel with an applied electric and magnetic field Meenapriya P 1 Uma Maheswari K umamohansusd@gmail.com 1 Nirmala P Ratchagar 1 Department of Mathematics, Annamalai University Chidambaram, 608 002 India 14 13 2021 Abstract

Objectives: A mathematical model is constructed to investigate the concentration of aerosol mixture in the atmosphere. Aerosol is the significant constituent of the atmosphere, as it is playing a key role in environmental effects including air quality degradation, smoke-fog related accidents, acid rain formation, and visibility issues. The objective of this paper is to formulate the effects of aerosol mixture, and it is monitored through its concentration in the presence and absence of chemical reaction. The effects of concentration of aerosol mixture has to be reduced so as to enhance visibility. Methodology: A two-dimensional schematic geometry is considered which comprises the channel extended to inﬁnity in the x-axis bounded by porous layers, also electric and magnetic field are applied transversely. A general exact solution of the dimensionless governing partial differential equations are obtained using perturbation technique with appropriate boundary conditions. Findings: Detailed results of velocity and concentration of the aerosol mixture were presented graphically. The graphical representation validated the accuracy of attained results with an encouraging level of agreement. As a measure to enhance the effects of air visibility, significant conclusions are specified using the obtained results.

Keywords atmospheric fluid coagulation concentration porous layers mixture theory None
Introduction

Atmospheric aerosols are extremely small, finely distributed, liquid or solid particles suspended in a gaseous medium. Both natural and anthropogenic aerosol particles have been playing a major role in the global climate for several years. Due to coagulation, aerosols bind each other and deforms in the atmosphere resulting in either larger or smaller particles than its original size and shape. The particles that are formed due to coagulation, have a direct impact on fog formation, cloud physics, reduction of visibility, etc. Thus, there is a requirement of evaluating the concentration of the mixture of aerosols and atmospheric fluid using mixture theory.

A handful of research work was attempted in the field of mixture theory. Barry et al1 developed the governing equations for flow in porous layers and obtained solutions using mixture theory. Bowen 2 established the use of thermodynamics of mixture to porous media for an incompressible fluid. Labonokitov et al3 used Cauchy-Born hypothesis to show that there is an interaction force between solid and fluid and provided linear boundary conditions to check the consistency of porosity theory.

Many studies were conducted in the coagulation of aerosols in particular. Anand et al4 bring forth a numerical study of the coagulation of aerosols and its number concentration in air. The concentration and source strength of vapors has been measured using aerosol condensation analytically by Dal Maso et al 5. Gan Fu Zan et al6 elucidated the role of the fluid shear rate on the particle coagulation using LES technique and Taylor– series expansion moment method. The mathematical model for concentration of pollution is developed based on the coagulation and deposition process by Keller and Siegmann 7. The results of both theoretical and experimental analysis of nanoparticle aerosols in the atmosphere were studied by Seipenbusch et al 8.

Different analytical approaches are existing to solve the derived partial differential equations. Dettman9 exhibited a perturbation technique in elliptic and hyperbolic problems and showed how the method tends to unify the behavior of variety of problems in perturbation theory. Rudraiah et al10 solved the problem of dispersion of aerosol mixture using the perturbation technique.

Meenapriya and Ratchagar 11 attempted to study coagulation of aerosol in couple stress fluid using generalised dispersion method. The concentration and dispersion of air pollutants was evaluated in an applied electric and magnetic field by Meenapriya et al 12, 13. Dhange and Sankad14 investigated the dispersion and chemical effects of the sinusoidal stream of an MHD couple stress fluid analytically. Varaksin15 reviewed the current state of aerosol process modeling which pertains to gas-aerosol partitioning, and highlighted the gap between current understanding of gas-aerosol partitioning from measurements, comprehensive box models and parameterizations for large-scale modelling.

These studies urge to study about aerosol mixture in a channel bounded by porous beds with the externally applied electric and magnetic fields. The concentration of the aerosol mixture is evaluated both in the presence and absence of homogeneous first-order chemical reaction. Numerical calculations are computed using MATHEMATICA and the results are portrayed graphically by varying parameters like Hartmann number, electric number, reaction rate parameter, porous parameter. The potrayed graphical results helps to analyze the effects of air visibility through the concentration of aerosol mixture.

Mathematical Formulation

A two-dimensional turbulent flow of chemically reactive aerosols in a poorly conducting atmospheric fluid bounded by porous layers is modelled as shown in Figure 1 . Assuming at each point in the channel is simultaneously occupied by both aerosols and atmospheric fluid. The component is represented by β, where β=s for solid phase (aerosols) and β=f for fluid phase (atmospheric fluid). Both solid and fluid are assumed to be intrinsically incompressible.

Uniform magnetic field of strength B0 and electric field through electrodes with electric potentials φ=Vhx at y=-h,  φ=(Vhx-x0) at y=h are applied transversely through porous boundaries. The channel is symmetric about x–axis which extends to infinity in both directions with applied pressure gradient pt=G. The porous medium considered here is homogenous and isotropic. The concentration of aerosols with and without chemical reaction both in the solid phase and fluid phase are calculated respectively.

Physical Configuration

Basic governing equations are based on poro-elasticity and are derived using mixture theory and Reynolds averaging procedure.

Equation of continuity

.φβqiβ=0

Equation of Momentum

ρβqiβt+qjβqiβxj=-φβpxi+μβ 2qiβx2+2qiβy2±K(qis-qif)

Where qiβis the velocity of aerosols (if β=s) and qiβis the velocity of the atmospheric fluid (if β=f) with i=1,2 and j=1,2. Also  p is the pressure of the fluid, K is the linear drag coefficient that is, Darcy resistance offered by solid to fluid. The term pxi is the pressure gradient,  φs is the aerosol volume fraction and φf is the fluid volume fraction, ρβ is the density of the mixture and μβ is the viscosity of the mixture.

Equation of Species

Cit+uβCix+vβCiy=De2Cix2+2Ciy2-RCi

For i=1,2 let C1 denotes mean aerosol concentration with chemical reaction, C2 signifies mean aerosol concentration without chemical reaction, De is eddy diffusion coefficient and R represents first-order chemical reaction.

To derive a cartesian form of governing equations let the components of velocity be q1s=us, is the x-component of aerosol, q1f=uf, is the x- component of the atmospheric fluid,  q2s=vs, is the y-component of aerosol, q2f=vf, is y-component of atmospheric fluid. Let the components of xi are x1=x, x2=y.

The Cartesian form of equations (1) to (3) are,

Continuity equation

x(φsvs+φfvf)+yφsvs+φfvf=0

x-momentum equation

ρβuβt+uβuβx+ vβuβy=-φβpx+μβ2uβx2+2uβy2      -Kus-uf+ρeEx-σ0B02uβ

y-momentum equation

ρβvβt+uβvβx+ vβvβy=μβ2vβx2+2vβy2    -Kvs-vf

Species equation

Cit+uβCix+vβCiy=D2Cix2+2Ciy2-RCi

For  De=D+Kxx=D+Kyy, where the eddy diffusivity coefficients Kyy,Kyy are assumed to be small , hence eddy diffusion coefficient (De) is molecular diffusion coefficient (D). To solve the governing equations Beaver Joseph slip boundary conditions16 are used to study the boundary effects of velocity.

Boundary conditions for velocity are

uβ=0 , vβ=h2G0μβϵeiαx+wt, at  y=0 uβy=-αpkuβ-up,vβ=h2G0μβ(1+ϵeiαx+wt) , at  y=h uβy=αpkuβ-up, at y=-h

For Concentration

Ci=C0ϵeiαx+wt, at  y=0 Ci=C01+ϵeiαx+wt, at  y=h Ci=C0, at  y=-h

Where up is the Darcy velocity of the porous layer, αp is the slip parameter, k is the permeability of the porous layer,  α is the streamwise wave number, w is the frequency parameter, ϵ is the perturbation parameter, i is the imaginary number, C0 is the dimensionless concentration and up=-kμpx represents Darcy law. To make (4) to (13) non-dimension the following dimensionless quantities are introduced.

x*=xh;y*=yh;t*=tt0;us*=μsush2G0;uf*=μfufh2G0;G*=GG0;  vs*=μsvsh2G0; vf*=μfvfh2G0;ρe*=ρeϵ0Vh2;p*=pρVh2;up*=upVh; Ex*=ExVh;   φf*=φfV;φs*=φsV;β2=h2RD;Ci*=CiC0;

The reaction rate parameter is β and V is the applied constant electric potential due to embedded electrodes at the boundaries. The porous parameter σ=hk , the electric number We=ϵ0 V2μβ , Hartmann number is M2=σ0 B02h2μ. After dimensionalising the above governing equations and boundary conditions neglecting asterisk we obtain,

φfμfufx+vfy+φsμs(usx+vsy)=0

x- momentum solid phase

2usx2+2usy2-A1ust-A2ususx+ vsusy-φsA3px-A4us-ufr3+A5WeρeEx-M2us=0

x- momentum fluid phase

2ufx2+2ufy2-A6uft-A7ufufx+ vfufy-φfA3px+A4us-r3uf+A5WeρeEx-M2uf=0

y- momentum solid phase

2vsx2+2vsy2-A1vs t-A2usvsx+ vsvsy-A4vs-vfr3=0

y- momentum fluid phase

2vfx2+2vfy2-A6vf t-A7ufvfx+ vfvfy+A4vs-vfr3=0

Concentration

2Cix2+2Ciy2-c1Cit-c2uβCix+vβCiy-βCi=0

The dimensionless boundary conditions for velocity are given by,

uβ=0 , vβ=ϵeiαx+wt, at  y=0 uβy=-ασuβ-up,vβ=(1+ϵeiαx+wt) , at  y=1 uβy=ασuβ-up, at y=-1

For Concentration

Ci=ϵeiαx+wt, at  y=0 Ci=(1+ϵeiαx+wt), at  y=1 Ci=1, at  y=-1

To determine the velocity of the fluid, we need to evaluate the value of ρeEx in x- momentum equations. Using conservation of charges equation and Maxwell’s equation, the value of ρeEx is calculated as given below.

Conservation of charges

ρet+q.ρe+.J=0

Maxwell’s equation

.E=ρe0  (Gauss law) -B0t=×E   (Faradays law) J=σc(E+u×B0)  (Ohms law)

In a poorly conducting fluid, the induced magnetic field is negligibly small compared to an electric field and hence the current density is J=σcE and the electrical conductivity σc1 and hence any perturbation on it is negligible and so it depends upon the conduction temperature Tb.

So (26) reduces to Jy=0 and the equation of electric potential φ reduces to,

2φy2+1σcφyσcy=0 with σc=σ0(1+αhTb-T0)

Where αh is the volumetric expansion coefficient and Tb is the solution of equation d2Tbdy2=0 which is solved using dimensionless quantities, Tb*=TbT; η=yh; and the boundary conditions are Tb=T0 at  η=-1,  and Tb=T1 at  η=1. Hence we obtain the equation Tb-T0=T(y+h)2h, where T = T1-T0.

Using this value in (31), we get σc=σ0eαc(y+h)   and solving (30) using the boundary condition, φ=x Pe at y=-1,  and  φ=x-x0Pe  at y=1, we get the value φ=x-x0 eαch-e-αcy2sinhαcPe

Now consider, ρeEx=.EEx (from (27)

Since Ex=-1,  and E=-φ  then   ρeEx=-2φ(-1)

Hence ρeEx=αc2x0e-αcy2sinhαc=a2e-αcyPe, where  a2=αc2x02sinhαc

After substituting ρeEx, equations (15) and (16) becomes

2usx2+2usy2-A1ust-A2ususx+ vsusy-φsA3px-A4us-ufr3+A5Wea2e-αcy-M2us=0 2ufx2+2ufy2-A6uft-A7ufufx+ vfufy-φfA3px+A4us-r3uf+A5Wea2e-αcy-M2uf=0 2.1 Method of Solution

To evaluate the values of velocity and concentration of the aerosol mixture, Perturbation technique is used. Disintegrating the flow variables into steady base state quantities denoted by upper case and two-dimensional linear perturbed quantities denoted by tilde ( ) symbol as follows,

ux,y=UBy+u~yϵeiαx+ωt+O(ϵ2) vx,y=v~yϵeiαx+ωt+O(ϵ2) px,y=pBy+p~yϵeiαx+ωt+O(ϵ2) Cix,y=CBiy+C~iyϵeiαx+ωt+O(ϵ2)

After decomposing the above equation into base and perturbed parts, the solution of the base part are obtained analytically and that of the perturbed part are obtained numerically. Let the perturbation parameter ϵ=A4r3=μsμfkh2μs is assumed to be so small, where A4r3 . Assuming the flow to be steady, and splitting (14), (17) to (25), (32), (33) and neglecting the higher orders of perturbation parameter ϵ, we obtain the following set of partial differential equations.

Base part equations

2UBsy2-φsA3pBx+A5Wea2e-αcy-M2UBs=0 2UBfy2-φfA3pBx+A5Wea2e-αcy-M2UBf=0 2CBiy2-βCBi=0

Perturbed part equations

vf~y-a3uf~r1+vs~y-a3 us~r2=0 2us~y2+us~A1a4+A2UBs a3-M2-α2-A2  UBsyv~s+A3 φs a3 p~ =0 2uf~y2+uf~A6a4+A7UBf a3-M2-α2-A7 UBfyv~f+A3 φf a3 p~ =0 2vs~y2+vs~A1a4+A2UBs a3-α2=0 2vf~y2+vf~A6a4+A7UBf a3-α2=0 2C~iy2+Ci~c2a3 UBβ+c1a4-α2-β -c2CBiyv~β=0

where  a3=αtan(αx+ωt), and  a4=wtan(αx+ωt),

The base part boundary conditions are

UBβ=0,  CBi=0,  at y=0 CBi=1,  at  y=1 UBβy=-ασUBβ-UpBβ,  at  y=1 UBβy=ασUBβ-UpBβ, CBi=1, at  y=-1

and perturbed part boundary conditions are,

uβ~=0,vβ~=1, C~i=1 ,  at  y=0 vβ~=1,  C~i=1 ,  at  y=1 uβ~y=-ασuβ~-upβ~,  at  y=1 uβ~y=ασuβ~-upβ~,  C~i=0,at  y=-1

Base part solutions

For equations (34) and (35) the solution is

UBs=A cosh[M y]+Bsinh[M y]-A9M2+A8αc2-M2e-αcy UBf=C cosh[M y]+Dsinh[M y]-A9M2+A8αc2-M2e-αcy

Also the derivatives of UBs,UBf are

UBsy=AMsinh[M y]+BMcosh[M y]-αcA8αc2-M2e-αcy UBfy=CMsinh[M y]+DMcosh[M y]-αcA8αc2-M2e-αcy

The values of integration constants A,B,C,D are calculated using (43) to (46). The perturbed part equations (38) to (41) are simplified using (51) to (54) as given below,

2us~y2+us~s1-M2+s2cosh[M y]+s3sinh[M y]-s4e-αcy-v~s(s5sinhM y+s6 coshM y-s4e-αcy)+A11=0 2uf~y2+uf~s8-M2+s9cosh[M y]+s10sinh[M y]-s11e-αcy-vf~(s12sinhM y+s13 coshM y-s14e-αcy)+A13=0 2vs~y2+vs~s15+s2cosh[M y]+s3sinh[M y]+s4e-αcy=0 2vf~y2+vf~s16+s9cosh[M y]+s10sinh[M y]-s11e-αcy=0

Where s1,s2,s3, s14,s15 are constants obtained when simplification.

The above four perturbed equations (55) to (58) are solved for us~,uf~,vs~,vf ~ numerically using MATHEMATICA, and hence velocities for both solid and fluid phases are plotted as graphs respectively.

2.1.1 Case 1: Concentration of smog (C<sub id="s-7885a6f1e43e">1</sub>) (when <inline-formula id="if-0268f39c1b6d"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mi mathvariant="bold-italic">β</mml:mi><mml:mo>≠</mml:mo><mml:mn mathvariant="bold">0</mml:mn></mml:math></inline-formula> )

The base part of  C1 is CB1 and its solution is obtained from (36) is given by,

CB1=Ecoshβ y+F sinh[β y]

Using boundary conditions (43) to (46) the values E=0,  F=1sinh[β] are obtained. so,

CB1=sinhβysinhβ CB1y=βcoshβysinhβ

For solid phase:

Equation (42) becomes,

2C~1y2+C1~c2a3 UBs+c1a4-α2-β -c2CB1yv~s=0

Using (51) and (60), the above equation becomes,

2C~1y2+C1~A15+s17 coshM y+s18 sinhM y-s19+ s20 e-αcy-A16 coshβ yv~s=0

For fluid phase:

Equation (42) becomes,

2C~1y2+C1~c2a3 UBf+c1a4-α2-β -c2CB1yv~f=0

Using (52) and (60), the above equation becomes,

2C~1y2+C1~A15+s17 coshM y+s18 sinhM y-s19+ s20 e-αcy-A17 coshβ yv~f=0

These perturbed equations (63), (65) are solved numerically subject to the boundary conditions prescribed in (47) to (50). Graphs are plotted for concentration with chemical reaction ( C1) for both solid and fluid phase using MATHEMATICA.

2.1.2 Case 2: Concentration of Haze C<sub id="s-0d2878f02c2a">2</sub> (when <inline-formula id="if-32593c3cfa92"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="bold-italic">β</mml:mi><mml:mo mathvariant="bold">=</mml:mo><mml:mn mathvariant="bold">0</mml:mn></mml:math></inline-formula>)

The base part of  C2 is CB2 whose equation from (36) is

2CB2y2=0

and its solution is given by,

CB2=Gy+H

Using boundary conditions (43) to (46) we get, G=1,  H=0. so,

CB2=y CB2y=1

For solid phase:

Equation (42) becomes,

2C~2y2+C2~c3a3 UBs+c1a4-α2 -c3CB2yv~s=0

Using (51) and (69), the above equation becomes,

2C~2y2+C2~A18+s17 coshM y+s18 sinhM y+s19+ s20 e-αcy-c2 v~s=0

For fluid phase:

Equation (42) becomes,

2C~2y2+C2~c3a3 UBf+c1a4-α2 -c3CB2yv~f=0

Using (52) and (69), the above equation becomes,

2C~2y2+C2~A18+s21 coshM y+s22 sinhM y-s23+ s24 e-αcy-c3 v~f=0

These perturbed equations (71) and (73) are solved numerically subject to the boundary conditions prescribed in (47) to (50). Graphs are plotted for the concentration without the effects of chemical reaction ( C2), for both solid and fluid phase using MATHEMATICA.

Results and Discussion

The velocity directions are exposed to get a better perception of the flow. The velocity profile for aerosols and atmospheric fluid for parameters including Hartmann number ( M), electric number ( We), and porous parameter (σ) are displayed in Figure 2, Figure 3. Figure 2 depicts that increasing electric number and porous parameter, the velocity of aerosol increases respectively. But results vary when increasing the values of Hartmann number, where aerosol velocity decreases. Figure 3illustrates the same result as in Figure 2. Figure 3shows that increasing electric number and porous parameter, velocity of atmospheric fluid increases, but by increasing Hartmann number fluid velocity decreases.

<bold id="s-16d118116395"/>Velocity profile of aerosols

<bold id="s-fa3981603935"/>Plots of atmospheric fluid velocity

Concentration profiles for different values of parameters are discussed as two cases in the proceeding sub-sections.

3.1 Concentration of aerosol mixture in the presence of chemical reaction (when <inline-formula id="if-78639bc1c643"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mi mathvariant="bold-italic">β</mml:mi><mml:mo>≠</mml:mo><mml:mn mathvariant="bold">0</mml:mn></mml:math></inline-formula>):

The concentration of aerosols and atmospheric fluid in the presence of chemical reaction by varying some parameters are illustrated in Figure 4 to Figure 7.

Figure 4 depicts the results of aerosol concentration for some values of reaction rate parameter and Hartmann number. It reveals that increasing the rate of the reaction, aerosol concentration increases but when maximizing Hartmann number the concentration of aerosols decreases. Figure 5 represents the plots of aerosol concentration for some values of an electric and porous parameter. It reveals that increasing both electric number and porous parameter, the concentration of aerosol decreases.

Concentration of aerosols for different values of reaction rate and parameter andHartmann number

Concentration of aerosols for various values of electric number and porous parameter

Figure 6 illustrates the plots of fluid concentration for different values of an electric number and porous parameter. It gives a clear picture that increasing both electric number and porous parameter, fluid concentration decreases. The results of atmospheric fluid concentration for various values of reaction rate parameter and Hartmann number are presented in Figure 7. It reveals that increasing both reaction rate parameter and Hartmann number, the concentration of fluid decreases.

Concentration of atmospheric fluid for some values of electric number and porous parameter

Concentration of atmospheric fluid for various values of reaction rate parameter and Hartmann number 3.2 Case(2) : Concentration of aerosol mixture in the absence of chemical reaction (when <inline-formula id="if-da6f814e6840"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="bold-italic">β</mml:mi><mml:mo mathvariant="bold">=</mml:mo><mml:mn mathvariant="bold">0</mml:mn></mml:math></inline-formula> ):

The concentration of aerosols and atmospheric fluid without the effect of a chemical reaction by varying some parameters are depicted in Figure 8, Figure 9.

Figure 8 shows the plots of aerosol concentration for different values of electric and Hartmann number. It is noted from the figure that while increasing electric number aerosol concentration maximizes. Also, increasing Hartmann number, the concentration of aerosol declines. The results of fluid concentration for various values of electric and Hartmann number are presented in Figure 9. It is observed that increasing both Hartmann number and electric number, concentration of atmospheric fluid without chemical reaction reduces.

Concentration of aerosol for different values of Hartmann number (when <inline-formula id="inline-formula-830dc68e59f74337a8349475935c9618"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>)

Concentration of atmospheric fluid for some values of Hartmann number and electric number (when <inline-formula id="inline-formula-eae9cf865fe747b595af4308d00c1de1"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>) 3.3 Discussion

In the presence of chemical reaction,

The concentration of aerosols in the aerosol mixture is reduced by enhancing electric, magnetic and porous effects.

The concentration of atmospheric fluid in the aerosol mixture is reduced by increasing the rate of the reaction, and by improving electric, magnetic and porous effects.

In the absence of chemical reaction,

The concentration of aerosols in the aerosol mixture is reduced by enhancing magnetic field effects.

The concentration of atmospheric fluid in the aerosol mixture is reduced by increasing electric, magnetic field effects.

Conclusion

The concentration of the aerosol mixture is evaluated both in the presence and absence of homogeneous first-order chemical reaction under the presence of electric and magnetic field. The concentration of aerosols in the mixture when chemically reactive, reduces when enhancing the electric, magnetic, and porous effects. The same results hold for the concentration of atmospheric fluid in the presence of chemical reaction. Atmospheric fluid concentration in the mixture reduces while increasing the rate of the reaction, electric, magnetic and porous effects. In the absence of chemical reaction, aerosol concentration reduces when improving the magnetic field and fluid concentration reduces when maximizing the effects of both electric and magnetic fields. Hence it is concluded that both in the presence and absence of chemical reaction concentration of the mixture is minimized when electric, magnetic and porous effects are enhanced.

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