Research article

Modified Newton-EHS method for solving nonlinear problems with complex symmetric Jacobian matrices

  • Received: 09 May 2023 Revised: 08 July 2023 Accepted: 27 July 2023 Published: 14 August 2023
  • MSC : 65F10, 65F50, 65H10

  • This manuscript is devoted to the study of numerical methods for a class of nonlinear problems. Instead of the standard Newton method, an efficient nonlinear solver is suggested to be used, and it is referred to as the Newton-EHS method, where "EHS" stands for Euler-extrapolated Hermitian-skew-Hermitian splitting. We construct this modified Newton-EHS method by utilizing a modified Newton method as the outer iteration and the EHS method as the inner iteration. Furthermore, we give the derivations of the local and semilocal convergence properties of the proposed method under the Hölder condition. Finally, in order to show the feasibility and validity of our new method, we compare it with some other iterative methods in two numerical examples.

    Citation: Lv Zhang, Qingbiao Wu. Modified Newton-EHS method for solving nonlinear problems with complex symmetric Jacobian matrices[J]. AIMS Mathematics, 2023, 8(10): 24233-24253. doi: 10.3934/math.20231236

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  • This manuscript is devoted to the study of numerical methods for a class of nonlinear problems. Instead of the standard Newton method, an efficient nonlinear solver is suggested to be used, and it is referred to as the Newton-EHS method, where "EHS" stands for Euler-extrapolated Hermitian-skew-Hermitian splitting. We construct this modified Newton-EHS method by utilizing a modified Newton method as the outer iteration and the EHS method as the inner iteration. Furthermore, we give the derivations of the local and semilocal convergence properties of the proposed method under the Hölder condition. Finally, in order to show the feasibility and validity of our new method, we compare it with some other iterative methods in two numerical examples.



    Nomenclature
    a coefficient of velocity
    cp specific heat at constant pressure
    C nanoparticles concentration
    Cfx skin friction coefficient
    Cw concentration of the fluid at the surface
    C ambient concentration
    D the diffusion coefficient
    De1 the thermal Deborah number
    De2 the Deborah number of concentration field
    Ec the Eckert number
    f dimensionless stream function
    j the mass flux
    k permeability of porous medium
    k1 the rate of chemical reaction
    k mean absorption coefficient
    K chemical reaction parameter
    Nux local Nusselt number
    Pr Prandtl number
    q the heat flux
    qr radiative heat flux
    R radiation parameter
    Re local Reynolds number
    Sc Schmidt number
    Shx the local Sherwood number
    T the temperature of the fluid
    Tw sheet temperature
    T fluid temperature away the sheet
    u the component of the velocity in the x direction
    v the component of the velocity in the y direction
    We the local Weissenberg number
    x,y Cartesian coordinates
    Greek symbols
    μ the viscosity coefficient
    μ the ambient viscosity
    ρ the fluid density
    ρ the ambient fluid density
    α the viscosity parameter
    γ1 the relaxation time for heat flux
    γ2 the relaxation time for mass flux
    Γ Williamson parameter
    κ the fluid thermal conductivity
    ε thermal conductivity parameter
    ϕ dimensionless fluid concentration
    θ dimensionless fluid temperature
    λ1 slip velocity factor
    λ slip velocity parameter
    ν the kinematic viscosity
    ν the ambient kinematic viscosity
    Δ the porous parameter
    η similarity variable
    σ Stefan-Boltzmann constant
    Superscripts
    differentiation with respect to η
    w wall condition
    free stream condition

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    Fluids can be categorized into Newtonian and non-Newtonian categories based on shear stress, according to the rheological behavior presented earlier [1]. The linear relationship between stress and strain identifies the Newtonian type. Physical problems involving Newtonian fluids have piqued the interest of scientists and engineers in recent years [2,3,4,5,6]. Unlike Newtonian fluids, the non-Newtonian fluid's viscosity does not remain constant with the shear rate and can be affected by a variety of factors. Slurries, foams, polymer melts, emulsions, and solutions are examples of non-Newtonian fluids. Non-Newtonian fluids have been extensively researched due of their numerous industrial applications. Furthermore, given the wide range of fluids of interest, heat transfer with non-Newtonian fluids is a vast topic that cannot be treated in its entirety in this study. Paint and adhesives industries, nuclear reactors, cooling systems, and drilling rigs are all examples of non-Newtonian fluid applications. To discover applications in new sectors, non-Newtonian fluid flows need a full investigation in terms of analytical, experimental, and numerical features. A variety of models have been used to explain non-Newtonian fluids in a number of works [7,8,9,10,11,12,13,14,15,16,17,18,19,20]. Non-Newtonian fluids were examined further by Megahed [7] and Ahmed and Iqba [8] for the power-law model; Cortell [9], Midya [10] and Megahed [11] for the viscoelastic model; Ibrahim and Hindebu [12], Bilal and Ashbar [13] and Abbas and Megahed [14] for the Powell-Eyring model; Pramanik [15], Rana et al. [16] and Alali and Megahed [17] for the Casson model; and Hayat et al. [18], Prasad et al. [19] and Megahed [20] for the Maxwell model. It is important to note that this study considers the viscous dissipation phenomenon. Only if the friction increases to the point where the fluid temperature field is noticeably warmed should this phenomenon be taken into account. There is a vast amount of information accessible on the viscous dissipation phenomenon and many fluid flow models [21,22,23].

    The Cattaneo-Christov heat transfer model is a modified form of the Fourier law that is used to calculate the characteristics of a heat flux model while considering the relaxation time for heat flux dispersion across the physical model [24]. As a result, several researchers [25,26,27,28,29,30] have conducted more research on this model's most essential characteristics. In many industrial and technical sectors, including natural processes, biomedical engineering, chemical engineering and petroleum engineering, the Cattaneo-Christov heat transfer model makes the heat transfer mechanism and fluid flow in porous media crucially important [31]. So, the cooling process, which is one of the benefits, is one of the most important real life applications of our research, especially in light of the influence of the double diffusive Cattaneo-Christov model on the non-Newtonian Williamson model. Also, this benefit can lower the cost of the finished product and serve as the primary component in preventing product fault. In addition, the novelty of the current work is to examine how Cattaneo-Christov heat mass fluxes affect non-Newtonian dissipative Williamson fluids due to a slippery stretching sheet under the presence of thermal radiation, chemical reactions, and changing thermal characteristics.

    Consider the flow of a non-Newtonian Williamson fluid with the Cattaneo-Christov phenomenon in two dimensions. It is assumed that the stretched sheet, which is embedded in a porous medium with permeability k, is what causes the fluid to flow. Likewise, it is considered that the fluid concentration along the sheet is Cw and that the sheet at y=0 is heated with temperature Tw. The fluid is flowing in a streamlined pattern, and its viscosity is low but not negligible. The modifying impact appears to be restricted within a narrow layer adjacent to the sheet surface; this is called the boundary layer region. Within such a layer, the fluid velocity rapidly changes from its starting value to its mainstream value. We chose this model because it can accurately describe a large number of non-Newtonian fluids, possibly the majority, over a vast scope of shear rates. The x-axis is chosen parallel to the surface of the sheet in the flow direction, with the origin at the sheet's leading edge, and the y-axis is perpendicular to it (Figure 1). Let T be the Williamson fluid temperature and u and v be the x-axial and perpendicular velocity components, respectively.

    Figure 1.  A physical diagram of a boundary layer slip flow system.

    The sheet on which the slip velocity phenomenon occurs is supposed to be rough and exposed to thermal radiation with a radiative heat flux of qr. The concept of the slip phenomenon was put forth by Mahmoud [32], who proved that it linearly depends on the shear stress τw at the surface, i.e.,

    uUw=λτw, (2.1)

    where τw=[μuy+μΓ2(uy)2]y=0 and Uw is the fluid velocity at the sheet. Also, chemical reactions and the viscous dissipation phenomenon are taken into consideration in this study. Similarly, the temperature Tw and concentration Cw are constant on the sheet. We continue to assume that the Williamson fluid thermal conductivity varies linearly with temperature κ=κ(1+εθ) [33], although its viscosity varies nonlinearly with temperature as μ=μeαθ [33], where μ is the Williamson fluid's viscosity at ambient temperature, α is the viscosity parameter, κ is the liquid thermal conductivity at ambient temperature, and ε is the thermal conductivity parameter. According to the aforementioned hypotheses, the fundamental equations governing the flow are

    .U_=0, (2.2)
    ρ(U_.)U_=p+.τ+r_, (2.3)
    ρcp(U_.)T=.q.qr+Φ, (2.4)
    (U_.)C=.jk1(CC), (2.5)

    where U_=(u,v,0), p is the pressure, r_=μkU_ is the Darcy impedance for a Williamson fluid, Φ is the function of dissipation, q is the heat flux, and j is the mass flux. Also, the heat flux q and the mass flux j satisfy the following relations, respectively [34]:

    q+γ1(U_.qq.U_)=κT, (2.6)
    j+γ2(U_.jj.U_)=DC. (2.7)

    Then, using the conventional boundary layer approximation, we can show that the equations below control the flow and heat mass transfer in our physical description [35].

    ux+vy=0, (2.8)
    uux+vuy=1ρy{μuy+μΓ2(uy)2}μρku, (2.9)
    uTx+vTy+γ1ΩE=1ρcpy(κTy)1ρcpqry+μρcp{(uy)2+Γ2(uy)3} (2.10)
    uCx+vCy+γ2ΩC=D2Cy2k1(CC). (2.11)

    In terms of boundary conditions, the fluid flow model is subjected to the following [36]:

    u=ax+λ1μ(μuy+μΓ2(uy)2),v=0,T=Tw,C=Cw at y=0, (2.12)
    u0,TT,CC, at y, (2.13)

    where

    ΩE=u22Tx2+v22Ty2+uuxTx+vvyTy+uvxTy+vuyTx+2uv2Txy, (2.14)
    ΩC=u22Cx2+v22Cy2+uuxCx+vvyCy+uvxCy+vuyCx+2uv2Cxy. (2.15)

    In addition, ρ represents the ambient density, and μ represents the Williamson viscosity. Γ stands for a Williamson time constant, k is the permeability of the porous medium, cp signifies the specific heat at constant pressure, k1 is the reaction rate, D is the diffusion coefficient, γ1 is the time it takes for the heat flux to relax, γ2 is the time it takes for the mass flux to relax, and C is the ambient fluid concentration. Now, we utilize suitable similarity transforms, such as [35]:

    u=axf(η),v=aνf(η),η=yaν,θ(η)=TTTwT,ϕ(η)=CCCwC. (2.16)

    When the last postulate is taken into account, Equation (2.8) is consequently confirmed, and the other equations are reduced to the following form:

    ((1+Wef)fαθf(1+We2f))eαθf2+ffΔeαθf=0, (2.17)
    1Pr((1+R+εθ)θ+εθ2)+fθDe1(ffθ+f2θ)+Ec(f2+We2f3)eαθ=0, (2.18)
    1Scϕ+fϕDe2(ffϕ+f2ϕ)Kϕ=0. (2.19)

    The relevant physical boundary conditions are also modified as a result of the invocation of the prior dimensionless transformations as

    f=0,f=1+λ(f+We2f2)eαθ,θ=1,ϕ=1 at η=0, (2.20)
    f0,θ0,ϕ0 at η. (2.21)

    The dimensionless controlling factors that have emerged are the local Weissenberg number We=Γx2a3ν, the porous parameter Δ=νak, the radiation parameter R=16σT33kκ, the slip velocity parameter λ=λ1aν, the thermal Deborah number De1=γ1a, the Eckert number Ec=u2wcp(TwT), the Deborah number which related to the mass transfer field De2=γ2a, the chemical reaction parameter K=k1a and the Prandtl number Pr=μcpκ. It is interesting to see that when We=0, the nature of the non-Newtonian Williamson fluid model transforms into a Newtonian fluid model.

    Furthermore, the local Nusselt number Nux, the local Sherwood number Shx, and the drag force coefficient in terms of Cfx are determined by [37]:

    CfxRe12=(f(0)+We2f2(0))eαθ(0),NuxRe121+R=θ(0),ShxRe12=ϕ(0), (2.22)

    where Re=uwxν is the local Reynolds number.

    Table 1 compares the values of the local skin-friction coefficient (CfxRe12) with those from Andersson's earlier work [38] to validate the current findings acquired by the shooting technique. This comparison is carried out for various slip velocity parameter values. We can confidently state that our results are in good accord with those references based on this comparison. Furthermore, the collected results show that the proposed strategy is both reliable and efficient.

    Table 1.  Comparison of CfxRe12 with the results of Andersson [38] when We=α=Δ=0.
    λ Andersson [38] Present work
    0.0 1.0000 1.0000000000
    0.1 0.8721 0.8720029514
    0.2 0.7764 0.7763995210
    0.5 0.5912 0.5911972051
    1.0 0.4302 0.4301859007
    2.0 0.2840 0.2839991098
    5.0 0.1448 0.1447985799

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    The simulation studies of a non-Newtonian Williamson fluid over a stretching surface are presented in this section. The momentum field considers the slip velocity phenomenon, the energy equation includes the viscous dissipation and thermal radiation effects, and the mass transport equation includes the chemical process. Via the shooting approach, the governing equations are numerically solved using supported dimensionless transformation. All the governing physical parameters are utilized in the following ranges: 0.0Δ0.5,0.0α0.5, 0.0λ0.4,0.0We0.5,0.0ε0.5,0.0De10.4,0.0De21.0,0.0K0.5 and 0.0Ec0.5. Thus, the values of the parameters with fixed values which are used for graphical display can be chosen as Δ=0.2,We=0.4,Sc=4.0, R=0.5,α=0.2,K=0.2,ε=0.2,λ=0.2,De1=0.2,Pr=7.0,De2=0.2 and Ec=0.2. This section discusses the graphical effects of physical dimensionless amounts on complicated profiles. The effect of porous parameter Δ on the Williamson fluid velocity is tested and introduced by means of Figure 2(a). When we assume Δ=0.0,0.2 and 0.5, both the velocity of the Williamson fluid and the thickness of the boundary layer are dramatically reduced. The porous parameter exhibits this behavior due to its direct presence in the velocity field. Figure 2(b) shows the impact of the porous parameter on both the fluid temperature and the fluid concentration. We discovered that the Williamson fluid temperature θ(η) rises as the porous parameter rises. Likewise, the Williamson fluid concentration ϕ(η) is subject to the same phenomenon. Physically, the porous parameter causes a restriction in the flow of the fluid, which reduces the fluid's velocity and raises the temperature and concentration distributions.

    Figure 2.  (a) f(η) for chosen Δ. (b) θ(η) and ϕ(η) for chosen Δ.

    The velocity, temperature and concentration profiles for the viscosity parameter α are shown in Figure 3. It's worth noting that α has a greater impact on velocity profiles than on temperature or concentration profiles. Increases in the viscosity parameter α result in decreases in the velocity profile f(η) and associated boundary layer thickness, whereas the temperature θ(η) and concentration ϕ(η) fields show the opposite tendency. Physically, a barrier type of force will be produced in the Williamson flow due to the fluid viscosity's reliance on temperature. The fluid velocity can be slowed down by this force. The fluid layers consequently acquire little thermal energy via the same force.

    Figure 3.  (a) f(η) for chosen α. (b) θ(η) and ϕ(η) for chosen α.

    Figure 4 shows the velocity, temperature, and concentration for the slip velocity parameter λ. When the slip velocity parameter is increased, both the temperature θ(η) and concentration ϕ(η) profiles increase, and as a result, the thermal boundary layer thickness increases. With the same parameter, however, the reverse tendency is observed for both the velocity distribution f(η) and the sheet velocity f(0). In terms of physics, the existence of the slip phenomenon produces a resistance force between the fluid layers, which in turn reduces the fluid's velocity and enhances the fluid temperature.

    Figure 4.  (a) f(η) for chosen λ. (b) θ(η) and ϕ(η) for chosen λ.

    The influence of the local Weissenberg number We on Williamson fluid velocity, Williamson fluid temperature, and Williamson fluid concentration is depicted in Figure 5. The graph illustrates that raising the local Weissenberg number We causes temperature and concentration distributions to rise, whereas the same value of We causes fluid velocity to decrease. Physically, a high local Weissenberg number increases the viscous forces that hold the Williamson fluid layers together, which lowers the fluid's velocity and improves the fluid's heat distribution through the boundary layer.

    Figure 5.  (a) f(η) for chosen We. (b) θ(η) and ϕ(η) for chosen We.

    The impact of the thermal conductivity parameter ε has been depicted in Figure 6. The larger values of ε correspond to a broader temperature distribution θ(η) and a marginal increase in the concentration field ϕ(η), but an increase in the same parameter results in a slight decrease in the velocity graphs f(η). Physically, higher values of the thermal conductivity characteristic indicate a fluid's ability to gain higher temperatures, which may cause the fluid's temperature to rise through the thermal layer.

    Figure 6.  (a) f(η) and ϕ(η) for chosen ε. (b) θ(η) for chosen ε.

    The effect of the thermal Deborah number De1 on velocity distribution, temperature distribution and concentration distribution is depicted in Figure 7. The Deborah number is a rheological term that describes the fluidity of materials under specified flow circumstances. Low-relaxation-time materials flow freely and exhibit quick stress decay as a result. The fluid motion f(η) scarcely improves with the bigger De1 parameter, but the concentration profiles ϕ(η) are slightly reduced. Furthermore, extended values of the thermal Deborah number De1 reduce both the thickness of the thermal region and the profiles of temperature θ(η).

    Figure 7.  (a) f(η) and ϕ(η) for chosen De1. (b) θ(η) for chosen De1.

    Figure 8 depicts the dimensionless velocity, dimensionless concentration, and thermal profiles for different Eckert number Ec estimates. It is evident that as the Eckert number is increased, the dimensionless velocities f(η) diminish modestly, while the dimensionless concentration ϕ(η) increases slightly. Furthermore, the increase in Eckert number Ec obviously increases both the temperature profile and the thickness of the thermal region. Physically, the Williamson fluid moves quickly, causing some kinetic energy to be transformed into thermal energy as a result of the viscous dissipation phenomenon. This promotes the fluid distributing heat more through the thermal layer.

    Figure 8.  (a) f(η) and ϕ(η) for chosen Ec. (b) θ(η) for chosen Ec.

    Figure 9 shows the effects of the Deborah number De2, which is related to mass transfer, and the chemical reaction parameter K on Williamson fluid concentration. Because both the concentration characteristics ϕ(η) and the concentration boundary thickness of the Williamson fluid decrease when both the chemical reaction parameter and the Deborah number increase, the mass transfer rate increases as well.

    Figure 9.  (a) ϕ(η) for chosen De2. (b) ϕ(η) for chosen K.

    From an engineering standpoint, we now concentrate on the fluctuations of physical quantities of interest. For all regulating factors of our model, the local skin-friction Cfx(Rex)12, the local Nusselt number NuxRe12x1+R, and the local Sherwood number ShxRe12x are introduced in Table 2. Skin friction coefficient values decrease when the viscosity parameter, slip velocity parameter, and local Weissenberg number increase, lowering both the local Nusselt number and the local Sherwood number. The skin friction coefficient and the local Nusselt number, respectively, grow and shrink uniformly with the thermal Deborah number and the Eckert number. Furthermore, increasing the chemical reaction parameter or the Deborah number increases the rate of mass transfer, while increasing the thermal conductivity parameter reduces it significantly.

    Table 2.  Values for Cfx(Rex)12, NuxRe12x1+R and ShxRe12x for various values of Δ, α, λ,ε,We,De1,Ec,De2 and K with Pr=7.0, Sc=4.0 and R=0.5.
    Δ α λ We ε De1 Ec De2 K Cfx(Rex)12 NuxRe12x1+R ShxRe12x
    0.0 0.2 0.2 0.4 0.2 0.2 0.2 0.2 0.2 0.693231 0.728441 1.549131
    0.2 0.2 0.2 0.4 0.2 0.2 0.2 0.2 0.2 0.742281 0.691424 1.521591
    0.5 0.2 0.2 0.4 0.2 0.2 0.2 0.2 0.2 0.804956 0.642281 1.485710
    0.2 0.0 0.2 0.4 0.2 0.2 0.2 0.2 0.2 0.797036 0.703410 1.535110
    0.2 0.2 0.2 0.4 0.2 0.2 0.2 0.2 0.2 0.742281 0.691424 1.521591
    0.2 0.5 0.2 0.4 0.2 0.2 0.2 0.2 0.2 0.658089 0.670908 1.496560
    0.2 0.2 0.0 0.4 0.2 0.2 0.2 0.2 0.2 0.902041 0.698173 1.604453
    0.2 0.2 0.2 0.4 0.2 0.2 0.2 0.2 0.2 0.742281 0.691424 1.521591
    0.2 0.2 0.4 0.4 0.2 0.2 0.2 0.2 0.2 0.629907 0.673791 1.458850
    0.2 0.2 0.2 0.0 0.2 0.2 0.2 0.2 0.2 0.796081 0.705117 1.537281
    0.2 0.2 0.2 0.3 0.2 0.2 0.2 0.2 0.2 0.758219 0.695903 1.526750
    0.2 0.2 0.2 0.5 0.2 0.2 0.2 0.2 0.2 0.723105 0.685343 1.514782
    0.2 0.2 0.2 0.4 0.0 0.2 0.2 0.2 0.2 0.743328 0.753634 1.521621
    0.2 0.2 0.2 0.4 0.2 0.2 0.2 0.2 0.2 0.742281 0.691424 1.521591
    0.2 0.2 0.2 0.4 0.5 0.2 0.2 0.2 0.2 0.740897 0.619058 1.521451
    0.2 0.2 0.2 0.4 0.2 0.0 0.2 0.2 0.2 0.741341 0.667262 1.521580
    0.2 0.2 0.2 0.4 0.2 0.2 0.2 0.2 0.2 0.742281 0.691424 1.521591
    0.2 0.2 0.2 0.4 0.2 0.4 0.2 0.2 0.2 0.743587 0.716488 1.521622
    0.2 0.2 0.2 0.4 0.2 0.2 0.0 0.2 0.2 0.744102 0.827635 1.521640
    0.2 0.2 0.2 0.4 0.2 0.2 0.2 0.2 0.2 0.742281 0.691424 1.521591
    0.2 0.2 0.2 0.4 0.2 0.2 0.5 0.2 0.2 0.739576 0.487913 1.521520
    0.2 0.2 0.2 0.4 0.2 0.2 0.2 0.0 0.2 0.742281 0.691424 1.497912
    0.2 0.2 0.2 0.4 0.2 0.2 0.2 0.5 0.2 0.742281 0.691424 1.559190
    0.2 0.2 0.2 0.4 0.2 0.2 0.2 1.0 0.2 0.742281 0.691424 1.619098
    0.2 0.2 0.2 0.4 0.2 0.2 0.2 0.2 0.0 0.742281 0.691426 1.238491
    0.2 0.2 0.2 0.4 0.2 0.2 0.2 0.2 0.2 0.742281 0.691426 1.521591
    0.2 0.2 0.2 0.4 0.2 0.2 0.2 0.2 0.5 0.742281 0.691426 1.872482

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    A description of the heat and mass transport features of a viscous non-Newtonian Williamson fluid across a stretching sheet that is embedded in a porous medium was attempted here. Heat and mass transfer in the Williamson fluid in the presence of thermal radiation, slip velocity, variable thermal characteristics, Cattaneo-Christov heat mass fluxes, chemical reaction and viscous dissipation are all investigated in this study. The governing equations have been converted via dimensionless transformations into a set of coupled nonlinear ordinary differential equations, which are then numerically solved using the shooting technique. The following are the major characteristics of the current work.

    1) A concentration field grows as the viscosity and slip parameters increase, but as the chemical reactions and Deborah number increase, the concentration field decreases.

    2) The Sherwood number decreases as the Eckert number rises, whereas the Deborah number and chemical parameter increase it.

    3) The velocity profile is reduced by the viscosity and slip velocity parameters, whereas a minor increase in velocity profile occurs due to the Deborah number being increased.

    4) The Deborah number has a lower temperature profile than the thermal conductivity parameter and the Eckert number.

    5) The heat transfer rate is reduced by both the thermal conductivity parameter and the viscous dissipation phenomenon.

    6) Skin-friction coefficient values increase with the porous parameter, but they decrease with an increase in either the slip velocity or the viscosity parameter.

    7) When the local Weissenberg number is higher, both the skin-friction coefficient and the local Nusselt number's magnitude fall.

    The authors are thankful to the honorable editor and anonymous reviewers for their useful suggestions and comments which improved the quality of this paper.

    The authors declare that they have no competing interests.



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