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An unsteady MHD Maxwell nanofluid flow with convective boundary conditions using spectral local linearization method

  • Hloniphile M. Sithole , Sabyasachi Mondal EMAIL logo , Precious Sibanda and Sandile S. Motsa
Published/Copyright: November 6, 2017
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Open Physics
From the journal Open Physics Volume 15 Issue 1

Abstract

The main focus of this study is on unsteady Maxwell nanofluid flow over a shrinking surface with convective and slip boundary conditions. The objective is to give an evaluation of the impact and significance of Brownian motion and thermophoresis when the nanofluid particle volume fraction flux at the boundary is zero. The transformed equations are solved numerically using the spectral local linearization method. We present an analysis of the residual errors to show the accuracy and convergence of the spectral local linearization method. We explore the effect of magnetic field and thermophoresis parameters on the heat transfer rate. We show, among other results, that an increase in particle Brownian motion leads to a decrease in the concentration profiles but concentration profiles increase with the increasing value of thermophoresis parameter

Keywords: Unsteady Maxwell nanofluid; Navier slip; Ohmic dissipation; Spectral local linearization method
PACS: 47.11.Kb

1 Introduction

There have been only a few studies on Maxwell nanofluid flow for a shrinking sheet in the recent past. However, in general, research on non-Newtonian fluid flow has gained sizeable attention because of the multiplicity of its applications in the biomedical and chemical industries [1]. A non-Newtonian fluid is a fluid whose viscosity varies with the applied stress. The relation between the strain rate and the shear stress is nonlinear, and can be time-dependent [2]. The constitutive equations tend to be highly nonlinear and intricate in comparison with those of a Newtonian fluid. A Maxwell fluid is a viscoelastic material having the properties of elasticity and viscosity [3]. Unlike the Newtonian model, the upper convected Maxwell (UCM) model incorporates relaxation time.

The UCM model has been studied by many researchers, for example, Choi et al. [4] gave an analysis of incompressible steady two-dimensional UCM fluid flow in a porous channel. Their study included a consideration of inertia and fluid elasticity. Nandy [5] focused on the unsteady boundary layer flow of a Maxwell nanofluid over a permeable shrinking sheet with a Navier slip condition at the surface. The flow equations were solved using the shooting method. The homotopy analysis method was used by Rashidi et al. [6] to find solutions to the conservation equations for heat and mass transfer in a two-dimensional steady magnetohydrodynamic fluid flow in a porous medium [6]. Awais [7] investigated heat absorption and generation in steady flow over a surface stretched linearly in its own plane using the UCM fluid model. A nanofluid is defined as a fluid with suspended solid nanoparticles that are less than 100nm in size, and solid volume fraction less than 4%, [8]. Even at low nanoparticle volume concentration, nanofluids have been shown to have improved conductivity and thermal the performance compared to base fluids such as water and oil, [9].

The unsteady Maxwell fluid flow over a stretching surface subject to constructive/destructive chemical reaction was studied by Mukhopadhyay and Bhattacharyya [10]. They showed that for a constructive chemical reaction the concentration field increased but decreased for a destructive chemical reaction [10]. Also, in line with physical expectations, the concentration boundary-layer decreased for a destructive chemical reaction. Nandy et al. [11] studied forced convection in an unsteady nanofluid flow past a permeable shrinking sheet subject to heat loss due to thermal radiation. They explored the simultaneous impact of a magnetic field, thermal radiation, and unsteadiness on the heat transfer and flow properties of the fluid. Das et al. [12] presented simulated results for heat and mass transfer in an electrically conducting incompressible nanofluid flow near a heated stretching sheet with a convective boundary condition. The impact of an inclined magnetic field in the flow of a fluid with variable thermal conductivity was studied by Hayat et al. [13, 14]. An analysis of the significance of a heat source/ sink and temperature dependent thermal conductivity was given. Qasim and Hayat [15] investigated the impact of heat loss through thermal radiation in unsteady magnetohydrodynamic flow of a micropolar fluid. The influence of Joule heating and thermophoresis in a Maxwell fluid was studied in [16].

Som et al. [17] studied Ohmic dissipation and thermal radiation in flow over a stretching sheet embedded in a porous media. The results indicated that fluid injection causes a reduction in heat transfer whereas fluid suction raises the heat transfer coefficient. Hsiao [18] studied conjugate heat transfer with Ohmic dissipation in an incompressible Maxwell fluid close to a stagnation point. Mahapatra et al. [19] gave an analysis of stagnation point fluid flow over a stretching surface.

The purpose of this study is to investigate unsteady two-dimensional boundary-layer flow, heat and mass transfer in a Maxwell nanofluid flow over shrinking sheet with both slip and convective boundary conditions. The objective is to extend the study by Nandy [5] to include an evaluation of the impact and significance of thermophoresis and Brownian motion when the nanofluid particle volume fraction at the boundary is not actively controlled. The conservation equations are solved numerically using the spectral local linearization method, see Motsa [20, 21]. To show the accuracy of the numerical scheme, analysis of the residual errors is given for different physical parameter values. The effect of several physical parameters on the fluid properties are shown in tabular and graphical form. Comparison with published work for special cases shows an excellent agreement.

2 Problem formulations

An unsteady laminar boundary layer flow of an incompressible viscous Maxwell nanofluid over a shrinking surface in two dimensions is considered here. The shrinking sheet velocity is uw(x,t) while the mass transfer velocity is vw(x,t), t denotes time and x is measured along the sheet. Using the nanofluid model proposed by Buongiorno [22], the conservation equations for mass, momentum, thermal energy and nanoparticles for a Maxwell fluid are:

∂u∂x+∂v∂y=0,(1)
∂u∂t+u∂u∂x+v∂u∂y=ν∂2u∂y2−k0u2∂2u∂x2+2uv∂2u∂x∂y+v2∂2u∂y2−σB2ρfu−kv∂u∂y,(2)
∂T∂t+v∂T∂y+u∂T∂x=α∂2T∂y2+1ρfcp∂∂yκ(T)∂T∂y+μρfcp∂u∂y2+σρfcpuB0−E02+Q0ρfcp(T−T∞)+τDB∂C∂y∂T∂y+τDTT∞∂T∂y2+Dmk0cscp∂2C∂y2−1ρfcp∂qr∂y,(3)
∂C∂t+v∂C∂y+u∂C∂x=DB∂2C∂y2+DTT∞∂2T∂y2+Dmk0Tm∂2T∂y2−k1(C−C∞),(4)

where u is the velocity component in the x- direction and v is the velocity component in the y-direction. The kinematic viscosity is ν, the relaxation time of the UCM fluid is k0, the thermal diffusivity is α, the variable thermal conductivity is κ(T) and the chemical reaction parameter is k1. The Brownian diffusion coefficient is DB, while the thermophoresis diffusion coefficients is DT. Here, τ is the effective heat capacity of the nanoparticle material divided by the heat capacity of the ordinary fluid, T is the fluid temperature and C is the nanoparticle volume fraction. The temperature and the nanoparticle concentration at the wall are Tw and Cw, respectively, and T∞ and C∞ denote the ambient temperature and concentration respectively. The velocity slip is proportional to the local shear stress. These equations are subject to the boundary conditions:

u=uw(x,t)+uslip(x,t),v=vw(x,t),−k01−λt∂T∂y=hf(Tw−T),DB∂C∂y+DBT∞∂T∂y=0aty=0,u→0,T→T∞,∂u∂y→0,C→C∞asy→∞,(5)

where k0 = k(1−λ t), uw(x,t) = −ax(1−λ t)−1, uslip(x,t) = νN1∂ u/∂ y, N1=N1−λt is the slip velocity, λ is the unsteadiness parameter and k0(> 0), a(> 0) are positive constants. This assumption is to allow for the possibility of a similarity solution. We introduce the similarity variables:

η=yaν(1−λt),ψ=aν1−λtxf(η),T(x,t)=T∞+θ(η)(Tw−T∞),C(x,t)=C∞+ϕ(η)(Cw−C∞),(6)

where ψ is the stream function, defined by v = −∂ψ/∂x and u = ∂ψ/∂y. Equations (1) - (4) are transformed to

f‴−f′2+ff″−Aη2f″+f′−βf2f‴−2ff′f″−M2f′−M2βff″=0,(7)
1Preffθ″+1Prθ″1+εθ+εθ′2+Ecf″2+EcM2f′−E12+f−Aη2θ′+Dfϕ″+Heθ+Ntθ′2+Nbθ′ϕ′=0,(8)
ϕ″+Scf−Aη2ϕ′+NtNbθ″−γScϕ+NtNbθ″+ScSrθ″=0,(9)

where the prime denotes derivatives with respect to η. The boundary conditions are

f′(0)=−1+δf″(0),f(0)=s,f″(∞)=0,f′(∞)=0,θ′(0)=−Bi(−θ(0)+1),θ(∞)=0,Nbϕ′(0)+Ntθ′(0)=0,ϕ(∞)=0,(10)

where the parameter δ=Naν is the non-dimensional velocity slip and vw(0,t) is the wall mass transfer velocity given by

vw(0,y)=v01−λt,

where v0 is the constant mass flux velocity. Thus

s=−v0aν=f(0),

where wall mass suction occurs when s > 0 while wall mass injection occurs when s < 0. The non-dimensional parameters in the above equations are the Maxwell parameter β( = k0a), the unsteadiness parameter A ( = λ/a), the magnetic field parameter M=σB02/ρfa,He( = Q0/ρf cp) the heat generation parameter and Q( = Q0(1−λ t)), Nb ( = τ DB(Cw−C∞)/ν) is the Brownian motion parameter and Nt( = τ DT (Tw − T∞)/ν T∞) is the thermophoresis parameter. The Prandtl number is Pr( = ν/α), the Schmidt number is Sc=ν/DB,Preff=Pr/(1+43R) is the effective Prandtl number, Bi=hf/k0νa is the Biot number, γ = k1(Cw−C∞) is the reaction parameter where γ < 0 denotes a destructive reaction, γ = 0 indicates that there is no reaction and γ > 0 denotes a generative reaction.

Other important physical parameters are the variable thermal conductivity κ(T), the Eckert number Ec, the local electromagnetic parameter E1, the Soret number Sr, the Dufour number Df and the radiation parameter R. These are defined as

κ(T)=K∞1+εT−T∞ΔT,Ec=x2a2cpΔT(1−λt),(11)
E1=E0(1−λt)aBox,Sr=Dmk0ΔTTmνΔC,Df=Dmk0ΔCcscpΔT,R=4σ1T∞3K1ρfcpαm,(12)

where ε is a small parameter, Δ T = Tw − T∞ and the thermal conductivity parameter is K∞. The important flow attributes, the Nusselt number Nux and skin friction coefficient Cf are described by

Cf=τwρuw2(x),Nux=xqwk(Tf−T∞).(13)

Here qw and τw are the plate heat flux and the skin friction respectively, defined as

qw=−k∂T∂yy=0,τw=μ∂u∂yy=0,(14)

where μ is the coefficient of viscosity. Equations (13) may be written as

Rex12Cf=f″(0),(15)
Rex−12Nux=−θ′(0),(16)

where Rex=uw(x)xν is the local Reynolds number. Here, the Sherwood number is zero, due to the assumption of zero mass flux at the surface.

3 Method of solution

The transport equations have been solved using the iterative spectral local linearization method (SLLM), see Motsa [20]. In principle of the SLLM algorithm is to linearize and decouple the system of equations. Motsa et al. [21] used the spectral local linearization method to solve the equations that model natural convection in glass-fibre production processes. Shateyi and Marewo [23] used the SLLM to solve equations that models an unsteady MHD flow and heat transfer. Nonetheless, this method has only been used in a limited number of studies, hence its general validation in complex systems remains to be made. For the interested reader, the SLLM algorithm is described in [20].

The differential equations arising from the linearization procedure are solved using a Chebyshev pseudo spectral method. The domain of the problem is transformed to the interval [−1, 1] using the transformation (b−a)(τ+1)/2. The differentiation matrix D is used to approximate the derivatives Zi(η) of the unknown variables to the matrix vector product,

dZidη=∑k=0N¯DlkZi(τk)=DZi,l=0,1,…,N¯,(17)

where the vector function at the collocation points is given by Z = [z(τ0), z(τ1), …, z(τN)]T, D = 2D/(b−a) with N+1 collocation points, [20] and

Zj(p)=DpZj.(18)

where the superscript in D denotes higher order derivatives.

SLLM Algorithm

The differential equations (7) - (9) may collectively be stated as,

ωk[F,T,H]=0,fork=1,2,3(19)

where ω1, ω2 and ω3 are non-linear operators and F,H, T are given by

F=f,∂f∂η,∂2f∂η2,∂3f∂η3,(20)
T=θ,∂θ∂η,∂2θ∂η2,(21)
H=ϕ,∂ϕ∂η,∂2ϕ∂η2.(22)

The system of equations (19) can be simplified and decoupled by linearizing the nonlinear terms. The Chebychev pseudo-spectral collocation method is utilized to integrate the decoupled system. The simple algorithm is as follows:

  1. From the first equation, find F while treating H and T as known functions from initial guesses. This gives Fr+1

  2. Solve for T in the second equation while treating F and H as known functions, T is known from 1 above and H is known from the initial guess. This gives Tr+1

  3. Finally, solve for H in the last equation while treating F and T as known functions, T and H are known from 1 and 2 above. We obtain Tr+1.

  4. Repeat steps 1-3 to find the next iterative solutions.

Using these ideas the nonlinear system of equations (7) - (9) are written as:

a1,rfr+1‴+a2,rfr+1″+a3,rfr+1′+a4,rfr+1=a5,r,b1,rθr+1″+b2,rθr+1′+b3,rθr+1=b4,r,c1,rϕr+1″+c2,rϕr+1′+c3,rϕr+1=c4,r,(23)

subject to boundary conditions:

fr+1(0)=s,fr+1′(0)=−1+δfr+1″(0),fr+1″(∞)=0,fr+1′(∞)=0,θr+1′(0)=−Bi(1−θr+1(0)),θr+1(∞)=0,Nbϕr+1′(0)+Ntθr+1′(0)=0,ϕr+1(∞)=0.(24)

where

a1,r=1−βfr2,a2,r=fr+2βfrfr′−Aη2−M2βfr,a3,r=2βfrfr″−2fr′−A−M2,a4,r=fr″−2βfrfr‴+2βfr′fr″−M2βfr″,a5,r=frfr″−fr′2−2βfr2fr‴+4βfrfr′fr″−M2βfrfr″,(25)
b1,r=ϵPrθr+1Preff+1Pr,b2,r=2ϵPrθr′+fr+Nbϕr′+2Ntθr′−Aη2,b3,r=ϵPrθr″,b4,r=He,b5,r=ϵPrθr″θr+ϵPrθr′2−Ecfr″−EcM2fr′−E12−Dfϕr′−Ntθr′2,c1,r=Scfr−ScAη2,c2,r=−γSc,c3,r=−ScSrθr″−NtNbθr″.(26)

The following decoupled matrix system of equations is obtained:

A1Fr+1=B1A2Tr+1=B2A3Hr+1=B3(27)

with corresponding boundary conditions

fr+1(τN)=s,fr+1′(τN)=−1+δfr+1″(τN),fr+1″(τ0)=0,fr+1′(τ0)=0,θr+1′(τN)=−Bi(1−θr+1(τN)),θr+1(τ0)=0,Nbϕr+1′(τN)+Ntθr+1′(τN)=0,ϕr+1(τ0)=0.(28)

where

A1=diaga1,rD3+diaga2,rD2+diaga3,rD+diaga4,r,A1=a5,rA2=diagb1,rD2+diagb2,rD+diagb3,r+b4,rI,A2=c5,rA3=D2+diagc1,rD+c2,rI,A3=c3,r.(29)

I is an identity matrix with (N + 1) rows and columns, F, H and T are approximate values of f, ϕ and θ calculated at the collocation points. Initial approximations are required to start the iteration process, and these can be selected so as to satisfy the boundary conditions and known flow configuration. For our system, the following guesses were used as suitable initial approximations,

f0(η)=α1+δ(−e−η+1)+s,g0(η)=Bi1+Bie−η,θ0(η)=NtBiNb(1+Bi)e−η.(30)
Figure 1 Flow geometry of the problem
Figure 1

Flow geometry of the problem

The boundary conditions are inserted in the matrices in (27) and the approximate solutions at each iteration level are obtained by solving (27).

4 Results and discussion

The solutions of the differential equations are given Tables 1 - 2 and Figures 2 - 16. In Table 1 we establish the reliability of the numerical scheme by a comparative analysis of the skin friction coefficient Cf and Nusselt number Nux with results reported by Hayat et al. [24] when β = 0.2, α = 0.3, Pr = 1 and s = 0.5. The other parameters have been set to zero. A good agreement is observed with the previously published work. To gain further insights as to the accuracy and convergence of the method used in this study, we have calculated residual errors as shown in Figures 2 to 4. These are calculated for different values of A, s and β. In general, the solutions have converged with an absolute residual error ∥Res∥ ≈ 10−11 after five iterations. These results sufficiently demonstrate to the accuracy and convergence of the SLLM.

Figure 2 Residual errors for θ and ϕ when A = −2.5, −2.0, −1.5
Figure 2

Residual errors for θ and ϕ when A = −2.5, −2.0, −1.5

Figure 3 Residual errors for θ and ϕ when s = 2, 2.2, 2.3
Figure 3

Residual errors for θ and ϕ when s = 2, 2.2, 2.3

Figure 4 Residual errors for θ and ϕ when β = 0.1, 0.15, 0.2
Figure 4

Residual errors for θ and ϕ when β = 0.1, 0.15, 0.2

Figure 5 Effect of magnetic field parameter M on the velocity profile f′(η)
Figure 5

Effect of magnetic field parameter M on the velocity profile f′(η)

Figure 6 Effect of A on the velocity profile f′(η)
Figure 6

Effect of A on the velocity profile f′(η)

Figure 7 Effect of β on the velocity profile f′(η)
Figure 7

Effect of β on the velocity profile f′(η)

Figure 8 Effect of s on the velocity profile f′(η)
Figure 8

Effect of s on the velocity profile f′(η)

Figure 9 Effect of A on the temperature profile θ(η)
Figure 9

Effect of A on the temperature profile θ(η)

Figure 10 Effect of β on the temperature profile θ(η)
Figure 10

Effect of β on the temperature profile θ(η)

Figure 11 Effect of Nt on the temperature profile θ(η)
Figure 11

Effect of Nt on the temperature profile θ(η)

Figure 12 Effect of M on the temperature profile θ(η)
Figure 12

Effect of M on the temperature profile θ(η)

Figure 13 Effect of unsteadiness parameter A on the concentration profile ϕ(η)
Figure 13

Effect of unsteadiness parameter A on the concentration profile ϕ(η)

Figure 14 Effect of β on the concentration profile ϕ(η)
Figure 14

Effect of β on the concentration profile ϕ(η)

Figure 15 Effect of Nb on the concentration profile ϕ(η)
Figure 15

Effect of Nb on the concentration profile ϕ(η)

Figure 16 Effect of Nt on the concentration profile ϕ(η)
Figure 16

Effect of Nt on the concentration profile ϕ(η)

Table 1

Comparison of the skin friction coefficient and Nusselt number between the results of present study and reported by Hayat et al. [24] when β = 0.2, α = 0.3, Pr = 1, s = 0.5 and the other parameters are set to zero

Iterations[24] −f″(0)Current study −f″(0)[24] −θ′(0)Current study −θ′(0)
10.28299000.281390170.43000000.40401296
2-0.28149501-0.40465707
3-0.28149504-0.40465726
4-0.28149505-0.40465726
50.28149820.281495050.40648110.40465726
6-0.28149505-0.40465726
7-0.28149505-0.40465726
8-0.28149505-0.40465726
9-0.28149505-0.40465726
100.28149500.281495050.40479230.40465726
200.28149500.281495050.40465870.40465726
350.28149500.281495050.40465720.40465726
400.28149500.281495050.40465720.40465726

The numerical computations have been done when A = −1.0, s = 2, δ = 0.25, Nb = Nt = 0.1, Sc = 0.8, Pr = 1.0, M = 0.3, ε = 0.1, Ec = 0.1, E1 = 1.0, Df = 0.1, He = 0.5, Sr = 0.1, Preff = 1.0, γ = 0.1, and Bi = 0.1. For numerical simulations, the parameter values are chosen from the previous literature on nanofluid flow such as [5, 10, 12, 16, 24] etc. Table 2 shows the Nusselt number for different Nb, M and Nt, the other parameters as stated above. The results show that Nux decreases with increasing Nb and M whereas the opposite is observed for increasing values of Nt.

Table 2

Computed values of the Nusselt number for different values of Nb, M and Nt

NbMNt|−θ′(0)|
0.20.06758610
0.40.30.10.06758600
0.80.06758579
0.10.09046540
0.20.30.10.08959087
0.50.08728149
0.050.08031901
0.20.30.070.08434208
0.100.09125982

Figure 5 shows that the velocity reduces with increasing magnetic field strength. This is an indication of an increase in the Lorentz force that creates a resistance to the fluid flow near the boundary slowing down the fluid motion. Figure 6 shows that the boundary layer thickness increases as the unsteadiness parameter increases. The velocity profiles decrease with increasing value of A.

Figure 7 shows that as β increases, the boundary layer thickness increases with a cross-over of profiles near the surface. The physical interpretation of this behavior is that an increase in β reduces the fluid in the flow leading to the boundary layer thickness increasing near the surface but a contrary trend is recognized away from the surface.

Figure 8 shows a reduction in the velocity with an increase in the mass suction parameter s. Figure 9 demonstrates the effect of A on the temperature profile. An increase in unsteadiness causes an increase in the solute concentration. Figure 10 shows how the temperature profiles change with respect to variations in the Maxwell parameter β. The temperature profiles increase with increasing β values. This is to be expected since higher Maxwell parameters generally suggest a more solid material able to conduct and retain heat better.

Figure 11 shows the change in temperature profiles with respect to variations in the thermophoresis parameter values. As thermophoresis increases, the temperature profiles decrease near the surface. Figure 12 shows the effect of the magnetic field parameter on θ(η) the temperature profiles. We observe that the temperature profiles increase with increasing values of the magnetic field parameter. The existence of a magnetic field in an electrically conducting fluid produces a body force that decelerates the fluid flow which in turn has the effect of retaining more heat within the boundary layer.

The impact of the unsteadiness parameter, Maxwell parameter, Brownian motion parameter, thermophoresis parameter and the Schmidt number on the concentration profiles is depicted in Figures 13 - 16. In Figures 13 and 14 an increase in the unsteadiness parameter and Maxwell parameter causes an increase in the concentration profiles near the surface but the opposite movement is observed far from the surface. Figure 15 shows the impact that the Brownian motion parameter has on the concentration profiles. An increase in the Brownian motion parameter causes a decrease in the concentration profiles. The Brownian motion tends to intensify particle displacement away from the fluid flow regime onto the surface; this phenomenon accounts for a decrease in the concentration of the nanoparticles far from the surface, thus resulting in a decrease in the nanoparticle concentration boundary layer thickness.

Figure 16 depicts the effect of the thermophoresis parameter on the concentration profiles. Thermophoresis is associated with the movement of nanoparticles from a hot to a cold wall, and since it is generated by temperature gradients, this creates a fast flow away from the moving surface. Consequently more fluid is heated away from the surface leading to an increase in the temperature within the thermal boundary layer.

5 Conclusion

In this study, we have investigated unsteady Maxwell nanofluid flow over a shrinking sheet with convective and slip boundary conditions. The conservation equations were solved using an iterative spectral local linearization method. We have given an error analysis to establish the accuracy and convergence of the method. The impact and significance of various physical parameters on the fluid properties has been demonstrated both qualitatively and quantitatively. The findings can be briefly outlined as follows;

  1. The concentration and velocity profiles decrease whereas the temperature profile increases with increasing unsteadiness parameter.

  2. Increasing particle Brownian motion leads to a reduction in the concentration profiles but concentration profiles increase with increasing thermophoresis.

  3. In terms of heat transfer coefficients, increasing value of particle Brownian motion and magnetic field strength reduces the heat transfer coefficient but the opposite is observed in the case of increased thermophoresis parameter.

Acknowledgement

The authors are grateful to the University of KwaZulu-Natal and the Claude Leon Foundation, South Africa for financial support.

  1. Conflict of interest

    Conflict of interests: The authors declare that there is no conflict of interests regarding the publication of this article.

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Received: 2017-1-30
Accepted: 2017-7-3
Published Online: 2017-11-6

© 2017 H.M. Sithole et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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https://doi.org/10.1515/phys-2017-0074
Keywords for this article
Unsteady Maxwell nanofluid; Navier slip; Ohmic dissipation; Spectral local linearization method
Creative Commons
BY-NC-ND 4.0

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  107. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  108. Analysis of Product Distribution Strategy in Digital Publishing Industry Based on Game-Theory
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  111. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  112. Unstructured P2P Network Load Balance Strategy Based on Multilevel Partitioning of Hypergraph
  113. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  114. Research on the method of information system risk state estimation based on clustering particle filter
  115. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  116. Demand forecasting and information platform in tourism
  117. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
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  119. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
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  121. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
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  123. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  124. Conservation laws for a strongly damped wave equation
  125. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  126. Blending type approximation by Stancu-Kantorovich operators based on Pólya-Eggenberger distribution
  127. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  128. Computing the Ediz eccentric connectivity index of discrete dynamic structures
  129. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  130. A discrete epidemic model for bovine Babesiosis disease and tick populations
  131. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
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  133. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
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  135. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
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  137. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
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  140. Conservation laws, classical symmetries and exact solutions of the generalized KdV-Burgers-Kuramoto equation
  141. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  142. After notes on self-similarity exponent for fractal structures
  143. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  144. Excitation probability and effective temperature in the stationary regime of conductivity for Coulomb Glasses
  145. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  146. Comparisons of feature extraction algorithm based on unmanned aerial vehicle image
  147. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  148. Research on identification method of heavy vehicle rollover based on hidden Markov model
  149. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  150. Classifying BCI signals from novice users with extreme learning machine
  151. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  152. Topics on data transmission problem in software definition network
  153. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  154. Statistical inferences with jointly type-II censored samples from two Pareto distributions
  155. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  156. Estimation for coefficient of variation of an extension of the exponential distribution under type-II censoring scheme
  157. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
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  159. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
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  161. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
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  169. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
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  189. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
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  193. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
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  195. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
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  199. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
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  205. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
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  207. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
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  209. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
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  211. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
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  213. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  214. Electric field computation and measurements in the electroporation of inhomogeneous samples
  215. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  216. Modelling of magnetostriction of transformer magnetic core for vibration analysis
  217. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  218. Comparison of the fractional power motor with cores made of various magnetic materials
  219. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  220. Dynamics of the line-start reluctance motor with rotor made of SMC material
  221. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  222. Inhomogeneous dielectrics: conformal mapping and finite-element models
  223. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  224. Topology optimization of induction heating model using sequential linear programming based on move limit with adaptive relaxation
  225. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
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  227. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  228. Current superimposition variable flux reluctance motor with 8 salient poles
  229. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
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  231. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  232. Field analysis & eddy current losses calculation in five-phase tubular actuator
  233. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  234. Hybrid excited claw pole generator with skewed and non-skewed permanent magnets
  235. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  236. Electromagnetic phenomena analysis in brushless DC motor with speed control using PWM method
  237. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  238. Field-circuit analysis and measurements of a single-phase self-excited induction generator
  239. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  240. A comparative analysis between classical and modified approach of description of the electrical machine windings by means of T0 method
  241. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  242. Field-based optimal-design of an electric motor: a new sensitivity formulation
  243. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  244. Application of the parametric proper generalized decomposition to the frequency-dependent calculation of the impedance of an AC line with rectangular conductors
  245. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  246. Virtual reality as a new trend in mechanical and electrical engineering education
  247. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  248. Holonomicity analysis of electromechanical systems
  249. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  250. An accurate reactive power control study in virtual flux droop control
  251. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  252. Localized probability of improvement for kriging based multi-objective optimization
  253. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  254. Research of influence of open-winding faults on properties of brushless permanent magnets motor
  255. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  256. Optimal design of the rotor geometry of line-start permanent magnet synchronous motor using the bat algorithm
  257. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  258. Model of depositing layer on cylindrical surface produced by induction-assisted laser cladding process
  259. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  260. Detection of inter-turn faults in transformer winding using the capacitor discharge method
  261. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  262. A novel hybrid genetic algorithm for optimal design of IPM machines for electric vehicle
  263. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  264. Lamination effects on a 3D model of the magnetic core of power transformers
  265. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  266. Detection of vertical disparity in three-dimensional visualizations
  267. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  268. Calculations of magnetic field in dynamo sheets taking into account their texture
  269. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  270. 3-dimensional computer model of electrospinning multicapillary unit used for electrostatic field analysis
  271. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  272. Optimization of wearable microwave antenna with simplified electromagnetic model of the human body
  273. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  274. Induction heating process of ferromagnetic filled carbon nanotubes based on 3-D model
  275. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  276. Speed control of an induction motor by 6-switched 3-level inverter
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