This paper is devoted to the study of the peristaltic motion of non-Newtonian fluid with heat and mass transfer through a porous medium in the channel under the effect of magnetic field. A modified Casson non-Newtonian constitutive model is employed for the transport fluid. A perturbation series’ method of solution of the stream function is discussed. The effects of various parameters of interest such as the magnetic parameter, Casson parameter, and permeability parameter on the velocity, pressure rise, temperature, and concentration are discussed and illustrated graphically through a set of figures.
1. Introduction
Peristaltic motion is a phenomenon that occurs when expansion and contraction of an extensible tube in a fluid generate progressive waves which propagate along the length of the tube, mixing and transporting the fluid in the direction of wave propagation. In some biomedical instruments, such as heart-lung machines, peristaltic motion is used to pump blood and other biological fluids [1]. Peristaltic pumping is a form of fluid transport generally from a region of lower to higher pressure, by means of a progressive wave of area contraction or expansion, which propagates along the length of a tube like structure. Some electrochemical reactions are held responsible for this phenomenon. This mechanism occurs in swallowing of food through oesophagus, in the ureter, the gastro intestinal tract, the bile duct, and even in small blood vessels. It has now been accepted that most of the physiological fluids behave like a non-Newtonian fluids. The peristaltic flows have attracted a number of researchers because of wide applications in physiology and industry. The theoretical work of peristaltic transport primarily with the inertia free Newtonian flow driven by a sinusoidal transverse wave of small amplitude is investigated by Fung et al. [2]. Burns and Parkes [3] studied the peristaltic motion of a viscous fluid through a pipe and channel by considering sinusoidal variations at the walls. A mathematical study of the peristaltic transport of Casson fluid is given by Mernone and Mazumdar [4, 5]; they used the perturbation method to solve the problem. Mekheimer [6, 7] studied the peristaltic transport of MHD flow. Peristaltic transport of Casson fluid in a channel is discussed by Nagarani and Sarojamma [8, 9]. El Shehawy et al. [10] Studied the peristaltic transport in a symmetric channel through a porous medium. Finite element solutions for non-Newtonian pulsatile flow in a non-Darcian porous medium are given by Bharagava et al. [11]. Mekheirmer and Abd elmaboud [12] discussed the influence of heat transfer and magnetic field on peristaltic transport. Nadeem et al. [13] have discussed the influence of heat and mass transfer on peristaltic flow of third order fluid in a diverging tube. Abdelmaboud and Mekheimer [14] analyzed the transport of second order fluid through a porous medium. Abd Elmaboud [15] studied the heat transfer characteristics of micropolar fluid through an isotropic porous medium in a two-dimensional channel with rhythmically contracting walls. El-dabe et al. [16] studied the effects of radiation on the unsteady flow of an incompressible non-Newtonian (Jeffrey) fluid through porous medium. Mustafa et al. [17] studied the peristaltic transport of nanofluid in a channel with complaint walls. Anwr Beg and Tripathi [18] introduced a theoretical study to examine the peristaltic pumping with double-diffusive convection in nanofluids through a deformable channel. El-dabe et al. [19] have discussed the effects of heat and mass transfer on the MHD flow of an incompressible, electrically conducting couple stress fluid through a porous medium in an asymmetric flexible channel over which a traveling wave of contraction and expansion is produced, resulting in a peristaltic motion. El-dabe et al. [20] studied the peristaltic motion of incompressible micropolar fluid through a porous medium in a two-dimensional channel under the effects of heat absorption and chemical reaction in the presence of magnetic field. Ebaid and Emad Aly [21] showed the mathematical model describing the slip peristaltic flow of nanofluid application to the cancer treatment. Emad and Ebaid [22] applied two different analytical and numerical methods to solve the system describing the mixed convection boundary layer nanofluids flow along an inclined plate embedded in a porous medium. Abd Elmaboud [23] investigated the magneto thermodynamic aspects micropolar fluid (blood model) through an isotropic porous medium in a nonuniform channel with rhythmically contracting walls. Noreen et al. [24] studied the mathematical model to investigate the mixed convective heat and mass transfer effects on peristaltic flow of magnetohydrodynamic pseudoplastic fluid in a symmetric channel. Hayat et al. [25] discussed the effects of heat and mass transfer on the peristaltic flow in the presence of an induced magnetic field. Noreen [26] consider the peristaltic flow of third order nanofluid in an asymmetric channel with an induced magnetic field.
The main aim of this work is to study the peristaltic motion of non-Newtonian fluid with heat and mass transfer through a porous medium in the channel under the effect of magnetic field. A modified Casson non-Newtonian constitutive model is employed for the transport fluid. A perturbation series’ method of solution of the stream function is discussed. The effects of various parameters of interest such as the magnetic parameter, Casson parameter, and permeability parameter on the velocity, pressure rise, and temperature are discussed and illustrated graphically through a set of figures.
2. Mathematical Analysis
Consider the peristaltic motion of non-Newtonian fluid through a porous medium in two-dimensional channel, having width d. A rectangular coordinate system x,y is chosen such that x-axis lies along the direction of wave progression and y-axis normal to it. The fluid is subjected to a constant magnetic field B_=(0,B0,0). Let u and v be the velocity components. The vertical displacements for the upper and lower walls are ζ and -ζ, see Figure 1, where ζ is defined by
(1)ζ(x,t)=acos2πλ(x-ct).λ is the wavelength, t is the time and a is the amplitude of the sinusoidal waves travelling along the channel at velocity c. The constitutive equation for the non-Newtonian Casson fluid can be written as in [27].
Sketch of the problem.
Consider
(2)τij={2(μB+py2π)eij,π>πc,2(μB+Py2πc)eij,π<πc,
where τij is the components of the stress tensor, π=eijeji and eij are the (i,j)th components of the deformation rate, π is the product of the component of deformation rate by itself, πc is a critical value of this product based on the Nakamura-Sawada model [27], μB is the plastic dynamic viscosity of the non-Newtonian fluid, and py is yield stress of slurry fluid.
The equations governing the fluid motion can be written as follows.
The continuity equation is
(3)∂u∂x+∂v∂y=0.
The momentum equations are
(4)ρ(∂u∂t+u∂u∂x+v∂u∂y)=-∂P∂x+(μ+μB)(∂2u∂x2+∂2u∂y2)-(σB02+μk)u,ρ(∂v∂t+u∂v∂x+v∂v∂y)=-∂P∂y+(μ+μB)(∂2v∂x2+∂2v∂y2)-(σB02+μk)v.
The energy equation is
(5)(∂T∂t+u∂T∂x+v∂T∂y)=k1ρcp(∂2T∂x2+∂2T∂y2).
The concentration equation is
(6)(∂C∂t+u∂C∂x+v∂C∂y)=Dm(∂2C∂x2+∂2C∂y2).
Lorentz force:
(7)F_=J_∧B_,J_=σV_∧B_,
where k is the permeability of the medium, J is the current density, cp is the specific heat of the fluid, k1 is the coefficient of heat conduction, T is the temperature of the fluid, Dm is the coefficient of mass diffusivity, C is the concentration of the fluid, μ is the coefficient of viscosity, σ is the electrical conductivity, P is pressure, and Bo is the strength of the applied magnetic field.
The appropriate boundary conditions are
(8)u=0,v=∂ζ∂t,T=T1,C=C1cccccccccccccccccccccciccccciccaty=d+ζ,u=0,v=-∂ζ∂t,T=T0,C=C0cccccccccccccccccccccccicicccccaty=-d-ζ.
Using the following nondimensional variables:
(9)x′=xd,y′=yd,u′=uc,v′=vc,t′=ctd,p′=pρc2,ζ′=ζd,ε=ad,T=(T1-T0)θ+T0,C=(C1-C0)ϕ+C0,
equations (4)–(8) after dropping the stars mark reduce to
(10)(∂u∂t+u∂u∂x+v∂u∂y)=-∂P∂x+(τ0+1Re)(∂2u∂x2+∂2u∂y2)-(M+1K)u,(∂v∂t+u∂v∂x+v∂v∂y)=-∂P∂y+(τ0+1Re)(∂2v∂x2+∂2v∂y2)-(M+1K)v,(∂θ∂t+u∂θ∂x+v∂θ∂y)=k1ρcp(∂2θ∂x2+∂2θ∂y2),(∂ϕ∂t+u∂ϕ∂x+v∂ϕ∂y)=k1ρcp(∂2ϕ∂x2+∂2ϕ∂y2),
with the boundary conditions
(11)u=0,v=αεsinα(x-t),θ=1,ϕ=1ccccccccccccccccccccccccccccccccccicccaty=1+ζ,u=0,v=-αεsinα(x-t),θ=0,ϕ=0ccccccccccccccccccccccccccccccccciccccaty=-1-ζ,
where M=σB02d/ρc is the magnetic parameter, K=ck/ϑd is the permeability parameter, α=2dπ/μ is the wave number, τ0=μB/ρcd is the Casson parameter, and Re=ρcd/μ is the Reynolds number.
Now, we shall define a stream function ψ as u=ψy and v=-ψx then (10) can be written as
(12)ψyt+ψyψyx-ψxψyy=-∂p∂x+(τ0+1Re)(ψyyy+ψyxx)-(M+1K)ψy,-ψxt-ψyψxx+ψxψxy=-∂p∂y-(τ0+1Re)(ψxxx+ψyyx)+(M+1K)ψx,(∂θ∂t+ψY∂θ∂x-ψX∂θ∂y)=k1ρcp(∂2θ∂x2+∂2θ∂y2),(∂ϕ∂t+ψY∂ϕ∂x-ψX∂ϕ∂y)=Dm(∂2ϕ∂x2+∂2ϕ∂y2),
with conditions
(13)ψy=0,ψx=-αεsinα(x-t),θ=1,ϕ=1ccccccccccccccccccccccccccccccccicccccccccaty=1+ζ,ψy=0,ψx=αεsinα(x-t),θ=0,ϕ=0cccccccccccccccccccccccccccccccccccccciaty=-1-ζ.
Express a stream function ψ, p, θ, and ϕ as a series in terms of small amplitude ratio ε, we have
(14)ψ(x,y,t)=ψ0+εψ1+⋯,p(x,y)=p0+εp1+⋯,θ(x,y,t)=θ0+εθ1+⋯,ϕ(x,y,t)=ϕ0+εϕ1+⋯,
where ψ0 is a function of y only. Substituting (14) in (12) and collecting the terms in ε, we get the following system of equations.
Coefficient of ε0:
(15)∂p0∂y=0,thisleadstoPoisafunctionofx,p0=c1(x),(16)ψoyyy-λ12ψ0y+L=0,(17)∂2θ0∂y2=0,(18)∂2ϕ0∂y2=0,
where
(19)λ1=(M+(1/K))(τ0+(1/Re)),L=(∂p0/∂x(τ0+(1/Re))).
and coefficient of ε:
(20)ψ1yt+ψ0yψ1yx-ψ1xψ0yy=-∂p1∂x+(τ0+1Re)(ψ1yyy+ψ1yxx)-(M+1K)ψ1y,(21)-ψ1xt-ψ0yψ1xx=-∂p1∂y-(τ0+1Re)(ψ1xxx+ψ1yyx)+(M+1K)ψ1x,(22)(∂θ1∂t+ψ0y∂θ1∂x-ψ1x∂θ1∂y)=k1ρcp(∂2θ1∂x2+∂2θ1∂y2),(23)(∂ϕ1∂t+ψ0y∂ϕ1∂x-ψ1x∂ϕ1∂y)=Dm(∂2ϕ1∂x2+∂2ϕ1∂y2).
Also, boundary conditions (13) can be written after using the Taylor series expansions about y=±1±ζ as follows:
(24)ψ0y(1)=0,θ0(1)=1,ϕ0(1)=1,ψ1y(1)=-ψ0yy(1)cosα(x-t),θ1(1)=-θ0y(1)cosα(x-t),ψ0(-1)=0,θ0(-1)=0,ϕ0(-1)=0,ψ1x(1)=-αsinα(x-t),ψ0y(-1)=0,ϕ1(1)=-ϕ0y(1)cosα(x-t).
Using conditions (24) with (15), the solution of (16) can be written as
(25)ψ0(y)=-Lλ1(y-sinhλ1yλ1coshλ1).
The flow rate, q, is given by
(26)q=∫01udy=∫01∂ψ∂ydy=ψ(1)-ψ(0)=-1λ1(τ0+(1/Re))(1-sinhλ1λ1coshλ1)∂p0∂x.
The pressure rise is given by
(27)Δp0=∫01∂p0∂xdx=-(τ0+(1/Re))λ1q(1-(sinhλ1/λ1coshλ1)).
Eliminating the pressure terms in (20) and (21), we have
(28)∇2ψ1t+ψ0y∇2ψ1x-ψ1x∇2ψ0y=(τ0+1Re)∇4ψ1-(M+1K)∇2ψ1.
From conditions (24) and (25) we can write ψ1, θ1, and ϕ1 in the form
(29)ψ1(x,y,t)=f(y)cosα(x-t)+g(y)sinα(x-t).θ1(x,y,t)=h1(y)cosα(x-t),ϕ1(x,y,t)=h2(y)cosα(x-t).
Equation (28) can be simplified by using (29) and assuming that the wave number α=2πd/λ is small, so the terms of α2and higher can be neglected.
We get
(30)α[((1-Lλ1)+coshλ1ycoshλ1)f′′+Lcoshλ1ycoshλ1f((1-Lλ1)+coshλ1ycoshλ1)]sinα(x-t)-α[((1-Lλ1)+coshλ1ycoshλ1)g′′ccccccccc-Lcoshλ1ycoshλ1g((1-Lλ1)+coshλ1ycoshλ1)]cosα(x-t)=[(τ0+1Re)f′′′′-(M+1Re)f′′]cosα(x-t)ccccccc+[(τ0+1Re)g′′′′-(M+1K)g′′]sinα(x-t).
Collecting coefficients of cosα(x-t) and sinα(x-t) on either side of (30), two differential equations for f(y) and g(y) are obtained as follows:
(31)-α[((1-Lλ1)+coshλ1ycoshλ1)g′′+Lcoshλ1ycoshλ1g]=[(τ0+1Re)f′′′′-(M+1K)f′′],α[((1-Lλ1)+coshλ1ycoshλ1)f′′+Lcoshλ1ycoshλ1f]=[(τ0+1Re)g′′′′-(M+1K)g′′].
Equation (31) can be simplified by assuming that
(32)f(y)=f0+αf1+⋯,g(y)=g0+αg1+⋯.
Substituting (32) into (31), and equating terms in α, the following ordinary differential equations are obtained for fo, f1, g0, and g1, respectively:
(33)f0′′′′-λ12f0′′=0,g0′′′′-λ12g0′′=0,-[((1-Lλ1)+coshλ1ycoshλ1)g0′′-Lcoshλ1ycoshλ1g0]=[(τ0+1Re)f1′′′′-(M+1K)f1′],[((1-Lλ1)+coshλ1ycoshλ1)f0′′-Lcoshλ1ycoshλ1f0]=[(τ0+1Re)g1′′′′-(M+1K)g1′′]
with boundary conditions:
(34)f0(1)=1,f0′(1)=-A,f0(-1)=-1,f0′(-1)=-A,g0(1)=0,g0′(1)=0,g0(-1)=0,g0′(-1)=0,f1(1)=0,f1′(1)=0,f1(-1)=0,f1′(-1)=0,g1(1)=0,g1′(1)=0,g1′(-1)=0,g1(-1)=0.
Solving (33) by using (34), we get
(35)f0(y)=λ2y-λ3sinhλ1y,(36)f1(y)=0,g0(y)=0,(37)g1(y)=λ7ycoshλ1y+(λ8+λ9y2)sinhλ1y+λ10sinh2λ1y+λ11y.
Substituting (35) in (29), we get
(38)ψ1(x,y,t)=(λ2y-λ3sinhλ1y)cosα(x-t)+αsinα(x-t)×(λ7ycoshλ1y+(λ8+λ9y2)sinhλ1yccc+λ10sinh2λ1y+λ11y(λ8+λ9y2)).
Substituting (25) and (38) in (14), we get
(39)ψ(x,y,t)=-Lλ1(y-sinhλ1yλ1coshλ1)+ε((((λ8+λ9y2)))(λ2y-λ3sinhλ1y)cosα(x-t)cccc+αsinα(x-t)cccc×(λ7ycoshλ1y+(λ8+λ9y2)sinhλ1yccccccccc+λ10sinh2λ1y+λ11y(λ8+λ9y2))).
The velocity components can be written as
(40)u(x,y,t)=-Lλ1(1-coshλ1ycoshλ1)+ε(((λ8+λ9y2)(λ2-λ3λ1coshλ1y)cosα(x-t)ccc+αsinα(x-t)ccc×(λ7coshλ1y+λ1λ7ysinhλ1ycccccc+λ1(λ8+λ9y2)coshλ1ycccccc+2λ1λ10cosh2λ1ycccccc+λ11+2λ9ysinhλ1y)((λ8+λ9y2)).v(x,y,t)=αε[(λ2y-λ3sinhλ1y)sinα(x-t)]
Substituting (39) in (17), (18), (22), and (23) and using (24) we get
(41)θo(y)=y+12,φo(y)=y+12,θ1(x,y,t)=((6λ12)-1-6λ13sinhλ1yλ12λ12y3)(-λ12λ12y+λ12λ12y3+6λ13ysinhλ1cc-6λ13sinhλ1yλ12λ12y3)(6λ12)-1)×cosα(x-t),φ1(x,y,t)=((6λ12)-1(λ12λ12y3-λ12λ14y+λ12λ12y3+6λ15ysinhλ1ccc-6λ15sinhλ1yλ12λ12y3)(6λ12)-1)cosα(x-t).
Substituting (41) in (14), we get
(42)θ(x,y,t)=(y+12)+εα((6λ12)-1(-λ12λ12y+λ12λ12y3cccccc+6λ13ysinhλ1-6λ13sinhλ1yλ12λ12y3)(6λ12)-1)×cosα(x-t),(43)ϕ(x,y,t)=y+12+εα((6λ12)-1(-λ12λ14y+λ12λ14y3+6λ15ysinhλ1-6λ15sinhλ1yλ12)×(6λ12)-1)cosα(x-t),
where
(44)A=ψ0yy(1)=Lsinhλ1λ1coshλ1,λ2=Asinhλ1+λ1coshλ1-sinhλ1+λ1coshλ1,λ3=1+A-sinhλ1+λ1coshλ1,λ4=λ12λ3(1-(L/λ1))(τ0+(1/Re)),λ5=(L+λ12)λ2(τ0+(1/Re))coshλ1,λ6=Lλ2coshλ1,λ7=((24λ12λ4+60λ1λ6)(24λ1λ4+60λ6)sinhλ1λ7=-(24λ12λ4+60λ1λ6)coshλ1)λ7=×(48λ14(λ1coshλ1-sinhλ1))-1,λ8=((24λ12λ4+48λ1λ6)sinhλ1λ8=-12λ12λ6coshλ1-2λ5sinh2λ1λ8=+4λ1λ5cosh2λ1(24λ12λ4+48λ1λ6))λ8=×(48λ14(λ1coshλ1-sinhλ1))-1,λ9=(12λ12λ6)coshλ1-12λ1λ6sinhλ148λ14(λ1coshλ1-sinhλ1),λ10=(2λ5)sinhλ1-2λ1λ5coshλ148λ14(λ1coshλ1-sinhλ1),λ11=((24λ12λ4)-12λ1λ5sinh2λ1+3λ1λ5sinhλ1λ11=-λ1λ5sinh3λ1+48λ1λ6λ11=+(12λ1λ6-30λ6)cosh2λ1(24λ12λ4))λ11=×(48λ14(λ1coshλ1-sinhλ1))-1,λ12=λ2tanα(x-t)Cp,λ13=λ3tanα(x-t)Cp,λ14=λ2sinα(x-t)Dmcosα(x-t),λ15=λ3sinα(x-t)Dmcosα(x-t).
3. Results and Discussion
In this work, we have studied the effect of different parameters of the considered problem on the solutions of the momentum, heat, and mass equations. This discussion is illustrated graphically through a set of Figures 2–28. Since Figures 2 and 3 illustrated the influence of the Casson parameter τ0 on the velocity component v, hence we noticed that the velocity component v increases with the increase of the Casson parameter τ0 for 0≤y≤1 and decreases for -1≤y≤0. The effect of the magnetic parameter M on the velocity component v is shown in Figures 4 and 5. These figures reveal that the velocity component v decreases with the increase of M at 0≤y≤1 and increases at -1≤y≤0. Figures 6 and 7 depicted the behavior of permeability parameter K on the velocity component v. It is noticed that the velocity component v decreases with K in the region -1≤y≤0, and it increases in the region 0≤y≤1. Figures 8 and 9 showed the effect of τ0 on the velocity component u; it is clear that the velocity component u increases with increasing of τ0. Also, the velocity component u decreases when the magnetic parameter M increases, then shown through Figures 10 and 11. From Figures 12 and 13, since the motion is sinusoidal, we have seen that the longitudinal velocity u increases or decreases as the permeability parameter K increases. Figures 14 and 16 illustrated the influence of τ0 and K on the pressure rise Δp0. These figures show that the pressure riseΔp0 decreases with the increase of both τ0 and K. Figure 15 displayed the effected of the magnetic parameter M on the pressure rise Δp0; it is noticed that the magnitude of Δp0 increases with M. We can see from Figures 17 and 18 that the temperature θ decreases when M increases in the interval -1≤y≤0, while it increases in the interval 0≤y≤1. We observed from Figures 19, 20, 21, and 22 that the temperature θ increases when K and τ0 increase in the interval -1≤y≤0, and it decreases in the interval 0≤y≤1. In Figures 23 and 24, it is seen that the concentration distribution ϕ decreases with M in the region -1≤y≤0, but it increases in the region 0≤y≤1. The concentration distribution ϕ increases when K and τ0 increase in the interval -1≤y≤0, and it decreases in the interval 0≤y≤1, this is shown in Figures 25, 26, 27, and 28.
The velocity v is plotted against the distance for different values of τo at α=0.1,ε=0.1,M=5,k=0.9.
The velocity is plotted against the time for different values of τo at α=0.1,ε=0.1,M=5,k=0.9.
The velocity v is plotted against the distance for different values of M at α=0.1,ε=0.1,τo=2,k=0.9.
The velocity v is plotted against the time t for different values of M at α=0.1,ε=0.1,τo=2,k=0.9.
The velocity v is plotted against the distance for different values of k at α=0.1,ε=0.1,M=5,τo=2.
The velocity v is plotted against the time t for different values of k at α=0.1,ε=0.1,M=5,τo=2.
The velocity u is plotted against the distance for different values of τo at α=0.1,ε=0.1,M=5,k=0.9.
The velocity u is plotted against the time t for different values of τo at α=0.1,ε=0.1,M=5,k=0.9.
The velocity u is plotted against the distance for different values of M at α=0.1,ε=0.1,τo=2,k=0.9.
The velocity u is plotted against the time t for different values of M at α=0.1,ε=0.1,τo=2,k=0.9.
The velocity u is plotted against the distance for different values of k at α=0.1,ε=0.1,M=5,τo=2.
The velocity u is plotted against the time t for different values of k at α=0.1,ε=0.1,M=5,τo=2.
Δpo is plotted against q for different values of τo at α=0.1,ε=0.1,M=5,k=0.2.
Δpo is plotted against q for different values of τo at α=0.1, ε=0.1, M=5, k=0.2.
Δpo is plotted against q for different values of k at α=0.1, ε=0.1, M=5, τo=2.
The temperature θ is plotted against the distance for different values of M at α=0.1, ε=0.1, τo=2, k=0.9.
The temperature θ is plotted against the time t for different values of M at α=0.1, ε=0.1, τo=2, k=0.9.
The temperature θ is plotted against the distance for different values of k at α=0.1, ε=0.1, τo=2, M=5.
The temperature θ is plotted against the time t for different values of k at α=0.1, ε=0.1, τo=2, M=5.
The temperature θ is plotted against the distance for different values of τo at α=0.1, ε=0.1, k=0.9, M=5.
The temperature θ is plotted against the time t for different values of τo at α=0.1, ε=0.1, k=0.9, M=5.
The concentration distribution ϕ is plotted against the distance for different values of M at α=0.1, ε=0.1, τo=2, k=0.9.
The concentration distribution ϕ is plotted against the time t for different values of M at α=0.1, ε=0.1, τo=2, K=0.9.
The concentration distribution ϕ is plotted against the distance for different values of k at α=0.1, ε=0.1, τo=2, M=5.
The concentration distribution ϕ is plotted against the time t for different values of k at α=0.1, ε=0.1, τo=2, M=5.
The concentration distribution ϕ is plotted against the distance for different values of τo at α=0.1, ε=0.1, M=5, k=0.9.
The concentration distribution ϕ is plotted against the time t for different values of τ0 at α=0.1, ε=0.1, K=0.9, M=5.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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