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G-Jitter Effects on Chaotic Convection in a Rotating Fluid Layer

Written By

Palle Kiran

Submitted: May 3rd, 2019 Reviewed: December 12th, 2019 Published: February 10th, 2020

DOI: 10.5772/intechopen.90846

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Abstract

The effect of gravity modulation and rotation on chaotic convection is investigated. A system of differential equation like Lorenz model has been obtained using the Galerkin-truncated Fourier series approximation. The nonlinear nature of the problem, i.e., chaotic convection, is investigated in a rotating fluid layer in the presence of g-jitter. The NDSolve Mathematica 2017 is employed to obtain the numerical solutions of Lorenz system of equations. It is found that there is a proportional relation between Taylor number and the scaled Rayleigh number R in the presence of modulation. This means that chaotic convection can be delayed (for increasing value of R) or advanced with suitable adjustments of Taylor number and amplitude and frequency of gravity modulation. Further, heat transfer results are obtained in terms of finite amplitude. Finally, we conclude that the transition from steady convection to chaos depends on the values of Taylor number and g-jitter parameter.

Keywords

  • g-jitter effect
  • nonlinear theory
  • rotation
  • chaos
  • truncated Fourier series

1. Introduction

The study of chaotic convection is of great interest due to its applications in thermal and mechanical engineering and in many other industry applications. It was introduced by Lorenz [1] to illustrate the study of atmospheric three-space model arising from Rayleigh-Benard convection. Some of the applications are production of crystals, oil reservoir modeling, and catalytic packed bed filtration. He developed a simplified mathematical model for atmospheric convection given below:

x=Pryx,E1
y=xRzy,E2
z=xyβz.E3

This model is a system of three ordinary differential equations known as the Lorenz equations. These equations are related to the properties of a two-dimensional Rayleigh-Benard convection. In particular, the system describes the rate of change of three quantities convection, temperature variation vertically with respect to time. These equations are related to the properties of two-dimensional flow model warmed uniformly from below and cooled from above. In particular, the system describes the rate of change of three quantities of time, x is proportional to the rate of convection, y is the horizontal temperature variation, and z is the vertical temperature variation. The constants Pr,R and β are the system parameters proportional to the Prandtl number, Rayleigh number, and certain physical dimensions of the media. If R<1 then there is only one equilibrium point at the origin which is represented as no convection point. Further, all orbits converge to the origin, which is a global attractor. When R = 1, then a pitchfork bifurcation occurs, and for R1, two additional critical points arise and are known as convection points, and there the system loses its stability. In addition to this model, I would like to add the concept of modulation either to suppress or to enhance nonlinearity. The literature shows that there are different types available; some of them are temperature modulation (Venezian [2]), gravity (Gresho and Sani [3] and Bhadauria and Kiran [4, 5]), rotation (Donnelly [6], Kiran and Bhadauria [7]), and magnetic field modulation (Bhadauria and Kiran [8, 9]). Their studies are mostly on thermal convection either considering fluid or porous medium. Their ultimate idea behind the research is to find external regulation to the system to control instability and measure the heat mass transfer in the system. But what happens when we consider the external configuration to system Eq. (1). The external configurations are like thermal, gravity, rotation, and magnetic field modulation. In this direction, no data are reported so far. With this, I would like to extend the work of Lorenz along with modulation.

The studies on chaos with respect to the different types of parameters like Rayleigh number and Prandtl number are mostly investigated by the following studies. The transition from steady convection to chaos occurs by a subcritical Hopf bifurcation producing a solitary cycle which may be associated with a homoclinic explosion for low Prandtl number is investigated by Vadasz and Olek [10]. The work of Vadasz [11] suggests an explanation for the appearance of this solitary limit cycle via local analytical results. The effect of magnetic field on chaotic convection in fluid layer is investigated by Mahmud and Hasim [12]. They found that transition from chaotic convection to steady convection occurs by a subcritical Hopf bifurcation producing a homoclinic explosion which may limit the cycle as Hartman number increases. For the moderate values of Prandtl number, the route to chaos occurs by a period of doubling sequence of bifurcations given by Vadasz and Olek [13]. Feki [14] proposed a new simple adaptive controller to control chaotic systems. The constructed linear structure of controller may be used for chaos control as well as for chaotic system synchronization. Yau and Chen [15] found that the Lorenz model could be stabilized, even in the existence of system external distraction. For non-Newtonian fluid case, Sheu et al. [16] have shown that stress relaxation tends to accelerate onset chaos. A weak nonlinear solution to the problem is assumed by Vadasz [17], and it can produce an accurate analytical expression for the transition point as long as the condition of validity and consequent accuracy of the latter solution is fulfilled. Narayana et al. [18] investigated heat mass transfer using truncated Fourier series method. They have also discussed chaotic convection under the effect of binary viscoelastic fluids. The studies related to gravity modulation are given by Kiran et al. [19, 20, 21, 22, 23, 24, 25]. These studies show that the gravity modulation can be used to control heat and mass transfer in the system in terms of frequency and amplitude of modulation.

The above paragraph demonstrated the earlier work on chaotic convection with different configurations and models to control chaos. Recently Vadasz et al. [26] and Kiran et al. [27] have investigated the effect of vertical vibrations and temperature modulation on chaos in a porous media. Their results show that periodic solutions and chaotic solutions alternate as the value of the scaled Rayleigh number changes in the presence of forced vibrations. The root to chaos is also affected by three types of thermal modulations.

The effect of rotation on chaos is investigated by Gupta et al. [28] without any modulation. They found that rotation has delay in chaos and controls nonlinearity. It is also concluded that there are suitable ranges over Ta and R to reduce chaos in the system. Based on the above studies in this chapter, I would like to investigate the study of chaotic convection in the presence of rotation and gravity modulation.

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2. Mathematical model

An infinitely extended horizontal rotating fluid layer about its vertical z-axis is considered. The layer is gravity modulated and the lower plate held at temperature T0 while the upper plate at T0+ΔT. Here ΔT is the temperature difference in the medium. The mathematical equation of the flow model is given by

.q=0,E4
q¯t+2Ωq¯=1ρ0p+ρρ0g¯+νΔ2q¯,E5
Tt+q¯.T=kT2T,E6
ρ=ρ01αTTT0.E7

The thermal boundary conditions are given by

T=T0+ΔTatz=0andT=T0atz=d,E8

where q¯> is the velocity of the fluid, Ω> is the vorticity vector, p> is the fluid pressure, ρ> is the density, ν> is the kinematic viscosity, KT> is the thermal diffusivity ratio, and αt> is the thermal expansion coefficient. We consider in our problem the externally imposed gravitational field (given by Gresho and Sani [3]):

g=g01+δgsinωgtk̂,E9

where δg,ωg are the amplitude and frequency of gravity modulation.

2.1 Basic state

The basic state of the fluid is quiescent and is given by

qb=0,0,0,p=pbz,T=Tbz.E10

Using the basic state Eq. (10) in the Eqs. (4)(6), we get the following relations

q¯bt+2Ωq¯b=1ρ0pb+ρbρ0g¯+νΔ2q¯b,E11
o=1ρ0pb+ρbρ0g¯,E12
pb=ρbg¯,E13
pbz=ρbg¯,E14

and from Eq. (6)

Tbt+q¯b.T=kT2Tb,E15
kT2Tb=0,E16
Tb=T0+ΔT1zd.E17

2.2 Perturbed state

On the basic state, we superpose perturbations in the form

q=qb+q,ρ=ρbz+ρ,p=pbz+p,T=Tbz+TE18

where the primes denote perturbed quantities. Now substituting Eq. (18) into Eqs. (4)(7) and using the basic state solutions, we obtain the equations governing the perturbations in the form

.q¯=,0E19
Tb+Tt+qb+q.ΔTb+T=KT2Tb+T,E20
Tt+q.Tb+T=KT2T,E21
Tt+ux+wzTb+T=KT2T,E22

simplifying the above equation, then we get

TtψxTbz+ψTxz=KT2T.E23

Similarly we can derive the same for momentum equation of the following form

q¯t+2Ωq¯=1ρ0p+ρρ0g¯+νΔ2q¯.E24

We consider only two-dimensional disturbances and define the stream functions ψ and q¯ by

uw=ψzψx,g¯=00g,E25

which satisfy the continuity Eq. (19). While introducing the stream function ψ and non-dimensionalizing with the following nondimensional parameters (x′,y′,z′) = dxyz, t′ = d2KTt, T=ΔTT, and p=μKTd2p, then the resulting Eq. (19) becomes

TtψxTbz+ψTxz=KT2T,

after simplifying the above equation, we get

t2T=ψxψTxz.E26

Similarly while eliminating the pressure term and using the dimensionless quantities, from the momentum equation (24), we get the following:

1Prt222+Ta2z2ψx=Ra1+δgsinωgt2x21Prt2T,E27

where Pr = νKT is the Prandtl number, Ta=4d4Ω2ν2 is the Taylor number, and Ra=αΔTd3g0νKT is the Rayleigh number. The assumed boundaries are stress free and isothermal; therefore, the boundary conditions are given by

w=2wz2=T=0atz=0andz=1.E28

The set of partial differential Eqs. (26) and (27) forms a nonlinear coupled system of equations involving stream function and temperature as a function of two variables in x and z. We solve these equations by using the Galerkin method and using Fourier series representation.

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3. Truncated Galerkin expansion

To obtain the solution of nonlinear coupled system of partial differential equations (26) and (27), we represent the stream function and temperature in the form

ψ=A1sinaxsinπz,E29
T=B1cosaxsinπz+B2sin2πzE30

The above are the Galerkin expansion of stream function and temperature. Now substituting these equations in Eqs. (26) and (27) and applying the orthogonal conditions to Eqs. (30) and (31) and finally integrating over the domain [0,1] × [0,1] yield a set of equations:

B1tcosaxsinπz+B2tsin2πz+k2B1cosaxsinπz+4B2π2sin2πzE31
=A1acosaxsinπzA1B1cosπzsinπzE32
2A1B2cos2πzcosaxsinπz.E33

Now multiply with cosaxsinπz on both sides, and apply integration from 0 to 1 with respect to x and 0 to 2πa:

B1t0102πacos2axsin2πzdxdz+B2t0102πacosaxsinπzsin2πzdxdzE34
+k2B10102πacos2axsin2πzdxdzE35
+4B2π20102πasin2πzcosaxsinπzdxdzE36
=A1a0102πacos2axsin2πzdxdzE37
A1B10102πacosaxcosπzsin2πzdxdzE38
2A1B20102πacos2πzcos2axsin2πzdxdz.E39
B1tπ2a+k2B1π2a=A1aπ2a2A1B2π2a,E40
B1t=A1a+A1B2k2B1.E41

Now we consider τ=k2tt=τk2.

B1τ=A1ak2+k2A1B2B1.E42

Now let us consider Eq. (30) and multiply with sin2πz on both sides of the equation and apply integration from 0 to 1 with respect to x and 0 to 2πa:

B1t0102πacosaxsinπzsin2πzdxdz+B2t0102πasin22πzdxdzE43
+k2B10102πacosaxsinπzsin2πzdxdzE44
+4B2π20102πasin22πzdxdzE45
=A1a0102πacosaxsinπzsin2πzdxdzE46
0102πa0102πaA1B1cosπzsinπzsin2πzdxdzE47
0102πa2A1B2aπcos2πzcosaxsinπzsin 2πzdxdz,E48

then by simplifying the above equation, we get

B2τ=4π2k2B22k2A1B1.E49

Similarly from Eq. (50)

2A1τ2=2PrA1τ+aK6a2Ra1+δgsinωgtπ2TaPrk6PrA1+πa2PrRak6A1B2+aRaPrPr1k4B1,E50

where k2=π2+a2 is the total wavenumber and τ=k2t is the rescaled time.

Introducing the following dimensionless quantities

R=a2RaK6,T=π2Tak6 and γ=4π2k2,σ=Pr,E51

and rescale the amplitudes in the form of

X=πak22A1,Y=πR2B1andZ=πRB2.E52

To provide the following set of equations, we consider the following equations γ=4π2k2,1k2=γ4π2

B1τ==γa4π2A1γaπ4π2A1B2B1,E53
τY2πR=γaR4π2Xk22πaRγa4πXk22πazπRY2πR,E54

and then simplifying the above equation, we get

Y=RXXZY,E55

now from the Eq. (50)

B2τ=γB212γ4π2πaA1B1,E56
τZπR=γzπR12γ4π2πaXk22πRY2πR,E57
Z=γZ+XY.E58

Similarly from Eq. (28),

X=W,E59
W=2σw+σR1+δgsinωgtσT+1XσXZ+σσ1Y,E60

where the symbol (/) denotes the time derivative d. Eqs. (56), (59), and (61) are like the Lorenz equations (Lorenz (13), sparrow (14)), although with different coefficients. The final nonlinear differential equations are given by

X=W,E61
Y=RXXZY,E62
Z=γZ+XY,E63
W=2σW+σR1+δgsinωgτσT+1XσXZ+σσ1Y.E64
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4. Stability analyses

To understand the stability of the system, we determine the fixed points of the system and will try to find the nature of these fixed points through eigen equation. The nonlinear dynamics of Lorenz-like system (62)(65) has been analyzed and solved for σ = 10, γ=83 corresponding to convection. The basic properties of the system to obtain the eigen function are described next.

4.1 Dissipation

The system of Eqs. (62)(65) is dissipative since

V=XX+YY+ZZ+WW=2σ+1γ<0.E65

If the set of initial solutions is the region of V(0), then after some time t, the endpoints of the trajectories will decrease to a volume:

Vt=V0exp2σ+1γt.E66

The above expression shows that the volume decreases exponentially with time.

4.2 Equilibrium points

System (62)(65) has the general form, and the equilibrium (fixed or stationary) points are given by:

X=W,E67
W=0.E68

From Eq. (83) we got

X=YRZ,E69

and similarly we also got the following from Eq. (64):

Z=Y2γRZ,E70

and similarly we also got the following from Eq. (65) for the momentum case:

R=T+1,E71

then we get a relation

X2,3=±T+1RγT+1E72

the remaining Y2,3,Z2,3 will be accessed. The fixed points of rescaled system for modulated case are X1Y1Z1=0,0,0 corresponding to the motionless solution and X2,3Y2,3Z2,3=±Zc±cZcRI1cR12 corresponding to the convection solution. The critical value of R, where the motionless solution loses their stability and the convection solution takes over, is obtained as Rcr=cI1, which corresponds to Ra=4π2cI1 where c=1+Cπ2γ and I1=01sin2πzf2dz. This pair of equilibrium points is stable only if R<Zc; beyond this condition the other periodic, quasi-periodic, or chaotic solutions take over at R>Zc. The corresponding stability of the fixed points associated with the motionless solution X1Y1Z1=0,0,0 is controlled by the zeros of the following characteristic polynomial:

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5. Stability of equilibrium points

The Jacobian matrix of Eqs. (62)(65) is as follows:

J=DFXYZW=0000RZ1X0YXγ0σRσT+1Zσσ1σX2σ.

The characteristic values of the above Jacobian matrix, obtained by solving the zeros of the characteristic polynomial, provide the stability conditions. If all the eigenvalues are negative, then the fixed point is stable (or in the case of complex eigenvalues, they have negative real parts) and unstable, when at least one eigenvalue is positive (or in the case of complex eigenvalues, it has positive real part):

DF0,0,0,0=0000R10000γ0σRσT+1σσ102σ.

The characteristic equation for the above system at origin is given by AλI=0 which implies the following

γ=λ,λ3+2σ+1λ2+2Rσ+σ2T+1λ+σ2TR+1=0.

The first eigenvalue γ is always negative as γ=83, but the other three eigenvalues are given by equation

λ3+2σ+1λ2+2Rσ+σ2T+1λ+σ2TR+1=0.

The stability of the fixed points corresponding to the convection solution X2,3Y2,3Z2,3 is controlled by the following equation for the eigenvalues λi,=1,2,3,4:

λ4+λ32σ+1γ+λ22σγ2γσγσT+σ2Tσ+σ2+X2+λ(X2σT+1E73
σγ+Tσγσ2γTσ2γ)+2X2σ2T+1=0,E74
λ4+λ32σ+1γ+λ2+γRT+1+2σ1γ+σσ1T+1λ2E75
+2σγRT+1+σγ2σT+1Rλ+2σ2YT+1R=0,E76
σγ2T+31γσσTT+12R2σγ[2σ+1γ{γ2σ+2σ1γT+3T+1E77
+σT+3σ12σ2σ+1γ}2σγT+32σ]R,E78
+σ2γT+12σ2σ+1γ21γ+1σT+1γT+12σ=0.E79

The loss of stability of the convection fixed points for σ=10,γ=83 using Eq. (80) is evaluated to be Rc2=25.75590 for system parameters T = 0, Rc2 for T = 0.1, Rc2=25.75590 for T = 0.2, Rc2=29.344020 for T = 0.45, and Rc2=32.775550 for T = 0.6.

5.1 Nusselt number

According to our problem, the horizontally averaged Nusselt number for an oscillatory mode of convection is given by

Nuτ=conduction+convectionconduction.E80
=ac2π02πacTbz+T2zdxz=0ac2π02πacTbzdxz=0.E81
=1+ac2π02πacT2zdxz=0ac2π02πacTbzdxz=0.E82

In the absence of the fluid motions, the Nusselt number is equal to 1. And simplifying the above equation, we will get the expressions for heat transfer coefficient:

Nu=12πB2τ.E83
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6. Result and discussion

In this section we present some numerical simulation of the system of Eqs. (62)(65) for the time domain 0τ40. The computational calculations are obtained by using Mathematica 17, fixing the values σ=10,γ=8/3, and taking in the initial conditions X(0) = Y(0) = 0.8, Z(0) = 0.9. In the case of T = 0, it is found that at Rc1=1, obtained from Eq. (80), the motionless solution loses stability, and the convection solution occurs. Also the eigenvalues from Eq. (80) become equal and complex conjugate when R varies from 24.73684209 to 34.90344691 given by Gupta et al. [28]. The evolution of trajectories over a time domain in the state space for increasing the values of scaled Rayleigh number and modulation terms is given in the figures. The projections of trajectories onto Y-X, Z-Y, Z-Y, and W-Z planes are also drawn (Figure 1). In Figure 2, we observe that the trajectory moves to the steady convection points on a straight line for a Rayleigh number (R = 1:1) just above motionless solutions. It is clear from Figure 3a that the trajectories of the solutions approach the fixed points at R = 12, which means the motionless solution is moving around the fixed points. As the value of R changes around R = 25.75590, there is a sudden change and transition to chaotic solution (in Figure 3b).

Figure 1.

Physical configuration of the problem.

Figure 2.

Phase portraits for the evolution of trajectories over time in the state space for increasing the value of rescaled Rayleigh number (R). The graphs represent the projection of the solution data points onto Y-X, Z-X, Z-Y, and W-Z planes for γ = −8/3; σ = 10, T = 0.1, R = 1.1 ωg = 0, δg = 0.0.

Figure 3.

(a) Phase portraits for the evolution of trajectories over time in the state space Y-X, Z-X, Z-Y, and W-Z planes for γ = −8/3, σ = 10, T = 0.1, R = 12, ωg = 2, δg = 0.0. (b) Phase portraits for the evolution of trajectories over time in the state space Y-X, Z-X, Z-Y, and W-Z planes for γ = −8/3, σ = 10, T = 0.1, R = 25.75590, ωg = 2, δg = 0.0.

In the case of gravity modulation in Figure 4, just keeping the values δg=0.05,ωg=10 in connection with Figure 3, the motionless solution loses stability, and convection solution takes over. Even at the subcritical value of R = 25.75590, transition to chaotic behavior solution occurs, but one can develop fully chaotic nature with suitably adjusting the modulation parameter values δg=0.05,ωg=10.

Figure 4.

Phase portraits for the evolution of trajectories over time in the state space modulation. The graphs represent the projection of the solution data points onto Y-X, Z-X, Z-Y, and W-Z planes for γ = −8/3; σ = 10, T = 0.1, R = 1.1, ωg = 10, δg = 0.05.

To see the effect of rotation on chaotic convection for the value of T = 0.45, we get Rc1=1.45 from Eq. (80), which concludes that the motionless solution loses stability at this stage and the convection solution takes over. The other second and third eigenvalues become equal and complex conjugate at R = 31.44507647. In this state the convection points lose their stability and move onto the chaotic solution. The corresponding projections of trajectories and evolution of trajectories are presented in Figure 5a and b, planes Y-X, Z-X, Z-Y, and W-Z. At the subcritical value of R = 31.44507647, transition to chaotic behavior solution occurs. Observing Figure 5b it is clearly evident that in the presence of modulation σ=20,δg=0.2,ωg=20, the trajectories are manifolds around the fixed points. Which are the interesting results to see that the system is unstable mode with rotation and buoyancy. But with gravity modulation, the system becomes stable mode.

Figure 5.

(a) Phase portraits for the evolution of trajectories over time in the state space Y-X, Z-X, Z-Y, and W-Z planes for γ = −8/3, σ = 10, T = 0.2, R = 31.44507647, ωg = 2, δg = 0.0. (b) Phase portraits for the evolution of trajectories over time in the state space Y-X and Z-Y planes for γ = −8/3, σ = 20, T = 0.2, R = 31.44507647, ωg = 25, δg = 0.2.

For the value of T = 0.6, we obtain the motionless solution (where the system loss stability) given in Figure 5b. The values of the second and third eigenvalues become equal and complex conjugate when the value of R = 24.73684209; at this point the convection points lose their stability, and chaotic solution must occur. But due to the presence of modulation, the trend is reversed given in Figure 6. Observing that in the presence of modulation δg=0.1,ωg=2, the system will come to stable mode for large values of R. The effect of frequency of modulation for the values ωg=2 and ωg=20 on chaos is presented in Figure 7a and b. It is clear that low-frequency-modulated fluid layer is in stable mode and high-frequency-modulated fluid layer in unstable mode. The reader may have look on the studies of [29, 30, 31, 32, 33] for the results corresponding to the modulation effect on chaotic convection.

Figure 6.

Phase portraits for the evolution of trajectories over time in the state space Y-X and Z-Y planes for γ = −8/3, σ = 20, T = 0.6, R = 24.73684209, ωg = 10, δg = 0.1.

Figure 7.

(a) For γ = −8/3, σ = 20, T = 0.2, R = 34.90344691, ωg = 2, δg = 0.1. (b) Phase portraits for the evolution of trajectories over time in the state space Y-X and Z-Y planes for γ = −8/3, σ = 20, T = 0.2, R = 34.90344691, ωg = 20, δg = 0.1.

Finally we also derived the heat transfer coefficient (Nuτ) given by Eq. (83) and verified the rate of transfer of heat under the effect of gravity modulation. It is clear from Figure 8 that heat transfer in the system is high for low-frequency modulation and for δg values varies from 0.1 to 0.5. The results corresponding to the gravity modulation may be observed with the studies of [19, 20, 21, 22, 23, 24, 25, 26].

Figure 8.

Effect of ωg and δg on Nu.

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7. Conclusions

In this chapter, we have studied chaotic convection in the presence of rotation and gravity modulation in a rotating fluid layer. It is found that chaotic behavior can be controlled not only by Rayleigh or Taylor numbers but by gravity modulation. The following conclusions are made from the previous analysis:

  1. The gravity modulation is to delay the chaotic convection.

  2. Taking the suitable ranges of ωg, δg, and R, the nonlinearity is controlled.

  3. The chaos in the system are controlled by gravity modulation either from stable to unstable or unstable to stable depending on the suitable adjustment of the parameter values.

  4. The results corresponding to g-jitter may be compared with Vadasz et al. [27], Kiran [31] and Bhadauria and Kiran [33].

  5. It is found that heat transfer is enhanced by amplitude of modulation and reduced by frequency of modulation.

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Acknowledgments

The author Palle Kiran is grateful to the college of CBIT for providing research specialties in the department. He also would like to thank Smt. D. Sandhya Shree (Board member of CBIT) for her encouragement towards the research. He also would like to thank the HOD Prof. Raja Reddy, Dept. of Mathematics, CBIT, for his support and encouragement. Finally the author PK is grateful to the referees for their most valuable comments that improved the chapter considerably.

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Conflict of interest

The authors declare no conflict of interest.

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Written By

Palle Kiran

Submitted: May 3rd, 2019 Reviewed: December 12th, 2019 Published: February 10th, 2020