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Numerical investigation of magnetohydrodynamic slip flow of power-law nanofluid with temperature dependent viscosity and thermal conductivity over a permeable surface

  • Sajid Hussain , Asim Aziz EMAIL logo , Chaudhry Masood Khalique and Taha Aziz
Published/Copyright: December 29, 2017
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Open Physics
From the journal Open Physics Volume 15 Issue 1

Abstract

In this paper, a numerical investigation is carried out to study the effect of temperature dependent viscosity and thermal conductivity on heat transfer and slip flow of electrically conducting non-Newtonian nanofluids. The power-law model is considered for water based nanofluids and a magnetic field is applied in the transverse direction to the flow. The governing partial differential equations(PDEs) along with the slip boundary conditions are transformed into ordinary differential equations(ODEs) using a similarity technique. The resulting ODEs are numerically solved by using fourth order Runge-Kutta and shooting methods. Numerical computations for the velocity and temperature profiles, the skin friction coefficient and the Nusselt number are presented in the form of graphs and tables. The velocity gradient at the boundary is highest for pseudoplastic fluids followed by Newtonian and then dilatant fluids. Increasing the viscosity of the nanofluid and the volume of nanoparticles reduces the rate of heat transfer and enhances the thickness of the momentum boundary layer. The increase in strength of the applied transverse magnetic field and suction velocity increases fluid motion and decreases the temperature distribution within the boundary layer. Increase in the slip velocity enhances the rate of heat transfer whereas thermal slip reduces the rate of heat transfer.

Keywords: Non-Newtonian nanofluids; Power-law model; Brickman nanofluid Model; Temperature dependent viscosity; Temperature dependent thermal conductivity; Partial slip; Magnetohydrodynamics
PACS: 44.20.+b; 44.40.+a; 47.10.-g; 47.15.-x

1 Introduction

In recent years, gradual development in fluid dynamics has resulted active studies of nanofluids, due to their applications in different industrial sectors. Choi [1] first introduced the term nanofluids and demonstrated theoretically the validity of the concept by including nano-sized particles in ordinary fluids. Eastman et al. [2] observed an unusual thermal conductivity enhancement in copper-water nanofluids at small nanoparticle volume fraction. Experiments performed by [3, 4, 5] confirm that the thermal conductivity of nanofluids is higher than the thermal conductivity of ordinary fluids. Buongiorno [6] theoretically observed that the properties of nanofluids like wetting, spreading and dispersion on a solid surface are better and more stable when compared to ordinary fluids. A comprehensive literature survey on slip flow of nanofluids under different thermo-physical situations is presented in[7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. In addition to the above studies Vahabzadeh et al. [19] obtained the analytical solutions of nanofluid flow over a horizontal stretching surface with variable magnetic field and viscous dissipation effects. They employed the homotopy perturbation method, Adomian decomposition method, and variational iteration method to determine the solution for the nanofluid flow and heat transfer characteristics within the boundary layer. Comparisons are also drawn with solutions obtained through numerical techniques. Similar to Vahabzadeh, Anwar et al. [20] used the variation iteration method to study the flow of nanofluid over an exponentially stretching surface within a porous medium.

The aforementioned studies and comprehensive literature review on convective transport of nanofluids reveal that non-Newtonian models for nanofluids have not received much attention. Some authors considered non-Newtonian models for the study of nanofluids, for example, Santra et al. [21] numerically studied forced convective flow of Cu-water nanofluid in a horizontal channel. They considered the power-law model for a non-Newtonian nanofluid and concluded that the rate of heat transfer increases due to increase in the volume of nanoparticles. Ellahi et al. [22] presented series solutions for flow of third-grade nanofluids considering both the Reynolds and Vogels model. Flow and heat transfer analysis of Maxwell nanofluids over a stretching surface was considered by Nadeem et al. [23]. Ramzan and Bilal [24] studied unsteady MHD second grade incompressible nanofluid flow towards a stretching sheet. Hayat et al. [25] examined the influence of convective boundary conditions on power-law nanofluid. Khan and Khan [26] presented a numerical investigation on MHD flow of power-law nanofluid induced by nonlinear stretching of a flat surface. Moreover, Aziz et al.[27] found exact solutions for the Stokes’ flow of a non-Newtonian third grade nanofluid model using a Lie symmetry approach. In addition to the above, the models for non-Newtonian nanofluid are well discussed in [28, 29, 30].

To the best of the authors’ knowledge no research has been conducted to study MHD slip flow of power-law nanofluids having variable thermo-physical properties over a permeable flat surface. In the present research, we investigate slip flow of an electrically conducting power-law nanofluid over a permeable flat surface with power-law wall slip condition and temperature dependent viscosity and thermal conductivity. The forth order RK method with shooting technique is employed to obtain solutions numerically. The influence of various physical parameters on the behavior of velocity and temperature of Cu-water nanofluid, with skin friction coefficient and the Nusselt number is discussed in detail.

2 Physical model and formulation

Consider the steady, laminar flow of an incompressible electrically conducting non-Newtonian nanofluid over a permeable flat surface. The power-law non-Newtonian fluid model is assumed for the water based nanofluid and power-law velocity slip condition is employed at the boundary. A uniform magnetic field is applied in the transverse direction to the flow. The viscosity and the thermal conductivity vary with temperature T. The porosity of the surface is considered as uniform and the uniform magnetic field of strength Bo is applied in the direction perpendicular to sheet. The induced magnetic field is considered negligible when compared to the applied magnetic field. In view of the above assumptions, as well as of the usual boundary layer approximations the governing equations are thus:

∂u∂x+∂v∂y=0,(1)
u∂u∂x+v∂u∂y=1ρnf∂∂yμnf(T)|∂u∂y|n−1∂u∂y−σnfρnfB02(u−U∞),(2)

and

u∂T∂x+v∂T∂y=1(ρCp)nf∂∂yκnf(T)∂T∂y.(3)

Here u is a component of velocity along the direction of the flow and v is the velocity perpendicular to it. μnf(T), ρnf, σnf, (Cp)nf, κnf(T) refer to nanofluid dynamic viscosity, density, electrical conductivity, specific heat capacity of nanofluid at fixed pressure and thermal conductivity respectively. Moreover, U∞ is the velocity far away from the surface known as the free stream velocity and n is the power-law index.

The appropriate conditions for the modeled problem are:

u(0)=L1(∂u∂y)n,v(0)=Vw,uy→∞=U∞(4)
T(0)=Tw+D1∂T∂y,Ty→∞=T∞,(5)

where Tw is the surface temperature and T∞ is the temperature outside the boundary layer, L1=L0Rex and D1=D0Rex are the velocity and thermal slip factors with L0 as initial velocity slip, D0 as initial thermal slip and Rex=ρfxnμfU∞(n−2) is the local Reynolds number. Finally, Vw is the constant suction/injection velocity across the surface. Commonly used property relations for nanofluids are presented as (for details see [31, 32, 33]):

μnf(T)=μnf∗a+b(Tw−T),κnf(T)=κnf∗1+ϵT−T∞Tw−T∞,(6)
ρnf=(1−ϕ)ρf+ϕρs,(ρCp)nf=(1−ϕ)(ρCp)f+ϕ(ρCp)s,(7)
μnf∗=μf(1−ϕ)2.5,κnf∗κf=(κs+2κf)−2ϕ(κf−κs)(κs+2κf)+ϕ(κf−κs),(8)

and

σnfσf=1+3(σsσf−1)ϕ(σsσf−2)−(σsσf−1)ϕ.(9)

In Eqs. (6)-(9), μnf∗ is the effective dynamic viscosity of the nanofluid, κnf∗ is a constant thermal conductivity with ϵ as a parameter, a and b are positive constants, ϕ is the volume fraction of solid nanoparticles in the base fluid, ρf, (Cp)f, μf, κf and σf are the density, specific heat capacity, coefficient of viscosity, thermal conductivity and the electrical conductivity of the base fluid. Whereas ρs, (Cp)s, μs, κs and σs are the density, specific heat capacity, coefficient of viscosity, thermal conductivity and electrical conductivity of the nanoparticles respectively.

In order to solve the governing boundary value problem (1)-(3), the stream function ψ(x, y) is introduced which identically satisfies Eq. (1) with

u=∂ψ∂yandv=−∂ψ∂x.(10)

Equations (2)-(3) after utilizing Eqs.

∂ψ∂y∂2ψ∂x∂y−∂ψ∂x∂2ψ∂y2=μnf∗ρnf−b∂T∂y|∂2ψ∂y2|n−1∂2ψ∂y2−σnfρnfB2(∂ψ∂y−u∞)+μnf∗ρnfa+b(Tw−T){n−1|∂2ψ∂y2|n−1(∂2ψ∂y2)2∂3ψ∂y3+|∂2ψ∂y2|n−1∂3ψ∂y3}(11)

and

∂ψ∂y∂T∂x−∂ψ∂x∂T∂y=κnf∗(ρCp)nfϵTw−T∞∂T∂y2+κnf∗(ρCp)nf1+ϵT−T∞Tw−T∞∂2T∂y2.(12)

Boundary conditions (4) are likewise transformed into

∂ψ∂y=L1∂2ψ∂y2n,∂ψ∂x=−Vw,aty=0;∂ψ∂y→0asy→∞.(13)

The following dimensionless similarity variable and similarity transformations are introduced into Eqs. (11)-(12)

η=Rex/L1n+1yL,ψ(x,y)=LU∞(x/LRe)1n+1f(η),θ(η)=T−T∞Tw−T∞,(14)

where θ is the dimensionless temperature and Re=ρfLnμfU∞(n−2) is the generalized Reynolds number.

The system (11)-(12) is reduced into a self-similar system of ordinary differential equations

n(a+A−Aθ)|f″|n−1f‴+(ϕ2/ϕ1)1n+1ff″−Aθ′|f″|n−(ϕ4/ϕ1)M(f′−1)=0,(15)
(1+ϵθ)θ″+1n+1(ϕ3/ϕ5)Prfθ′+ϵθ′2=0.(16)

Here M=σfB2xρfU∞ is the magnetic parameter, A = b(Tw − T∞) is the viscosity parameter,

Prx=(ρCp)fL2U∞κfx is the local Prandtl number and

ϕ1=(1−ϕ)2.5,ϕ2=(1−ϕ)+ϕρsρf,ϕ3=(1−ϕ)+ϕ(ρCp)s(ρCp)f,ϕ4=1+3(r−1)ϕ(r−2)−(r−1)ϕ,ϕ5=(ks+2kf)−2ϕ(kf−ks)(ks+2kf)+ϕ(kf−ks),r=σsσf.

Similarly, the boundary conditions in equation(13) are transformed to

f(0)=S,f′(0)=1+δ(f″(0))n,θ(0)=1+γθ′(0),(17)
f′(η)→1,θ(η)→0,asη→∞,(18)

where S=−(n+1)xnn+1(U∞)2n−1n+1(ρfμf)1n+1Vw is the suction/injection parameter, δ=Ln+22(U∞)n(n−5)2n+2(ρfμf)3n+12n+2 is the velocity slip parameter and γ=D(Ln(n+2)x)1n+1(ρfμf)n+3n+1 is the thermal slip parameter.

3 Numerical solution

The governing equation (15)-(16) with their associated boundary conditions (17)-(18) are solved numerically using the fourth order Runge-Kutta method with a shooting method. For the present, flow and heat transfer analysis, system (15)-(16) is first reduced to

f′=g,g′=h,h′=1n(a+A−Aθ)|h|n−1−1n+1(ϕ2/ϕ1)fh+Az|h|n+(ϕ4/ϕ1)M(g−1),θ′=p,p′=11+ϵθ−1n+1(ϕ3/ϕ5)Prfp−ϵp2

where n ≠ 1,

f(0)=S,g(0)=δ(h(0))n,θ(0)=1+γp(0),h(0)=G1,p(0)=G2,

where G1 and G2 are missing values which are chosen by a hit and trial method (as guesses) such that the boundary conditions g(∞) and θ(∞) are satisfied.

Numerical computations are performed taking a uniform step size Δη = 0.01 so that smoothness in the convergent solutions is obtained with an error of tolerance 10−6. Errors are inevitable due to the unbounded domain, whereas the computational domain must be finite. The step size and the position of the edge of the boundary layer are adjusted for different values of the governing parameters to maintain convergence of the numerical solution. Details of the solution procedure are not presented here for brevity. To confirm the numerical procedure, using the proposed code, the present results are compared with results available in the literature. The test case is natural convection of MHD boundary layer flow of power-law nanofluid over a surface (flat) having power-law slip. Comparison of our results and results of other investigators is shown in Table 1. These results are obtained for a = 1, ϕ = A = ϵ = S = 0. As can be seen from Table 1, the present results and the results presented by Hirschhorn at el. [34] show excellent agreement, for both the Nusselt number and the skin friction coefficient.

Table 1

Values of skin friction = −f″(0) and Nusselt number = −θ′(O) for natural convective MHD flow of power-law nanofluid with slip at the boundary

nMPrδγ−f″(0)[34]−θ′(O)[34]Present −f″(0)Present −θ′(O)
0.40.60.70.30.30.822690.393190.8226900.393190
1.00.680470.334970.6804710.334972
1.40.676380.314840.6763800.314843
1.40.60.70.30.00.507010.347680.507010.347683
1.40.60.70.30.60.676380.287670.6763800.287672
1.41.00.70.30.30.785860.324100.7858600.324101

4 Results and discussion

A numerical investigation has been carried out to study flow of fluid and variations in temperature of a Cu-water power-law nanofluid within a boundary layer. To demonstrate the meaningful relationship between the parameters, i.e, n, A, ε, ϕ, M, S, δ and γ, numerically obtained results are presented in the form of graphs for variations in velocity f′(η) and temperature θ(η) profiles. The behavior of frictional drag and heat transfer rate at the flat surface are also calculated with variation in the governing parameters and tabulated in Table 1. Material properties of Cu-water nanofluid are tabulated in Table 2 (see for example, B. Shankar and Y. Yirga [35]). Graphs of our results are drawn for fixed values of Pr = 6.2 and a = 1.0, while varying parameters A, M, ϕ, ε, S, δ and γ.

Table 2

Properties of Copper-Water Nanofluid

Material PropertiesUnitsWaterCu(300 − Kelvin)
Density ρ(kg/m3)997.18933
Specific heat Cp(J / kgK)4179385
Thermal conductivity κ(W /mK)0.613401
Electrical conductivity σ(Ω.m)−10.055.96×107

The effect of temperature dependent viscosity on velocity of Newtonian (n = 1), pseudoplastic (n < 1) and dilatant (n > 1) fluids is presented in Figure 1. We have chosen an arbitrary value n = 0.4 for pseudoplastic and n = 1.4 for dilatant nanofluids. The curves in Figures 1 follow the general trend, i.e, fluid motion within the boundary layer will decrease due to an increase in resistance in the fluid caused by increasing the viscosity parameter. Moreover, for a fixed value of the viscosity parameter (say, A = 0.6), the effect of the power-law index n on the velocity profiles of non-Newtonian nanofluids can be observed in Figure 1. Initially fluids with pseudoplastic behavior move fastest within the boundary layer, whereas the velocity of dilatant fluids is the slowest. The reason behind this is the lowest effective viscosity of pseudoplastic fluids. This trend is reversed in the range where shear stress becomes more dominant and the viscosity of pseudoplastic fluids becomes highest. The temperature distribution of a power-law nanofluid effected by variation in the fluid viscosity is shown in Figure 2. The effect of increasing the value of A, leads to thickening of the thermal boundary layer, which results in an increase in temperature and a decrease in heat transfer. This behavior is very noticeable in shear thickening fluids when compared to Newtonian and shear thinning fluids.

Figure 1 Influence of A on velocity of a power-law nanofluid when a = 1,M = 0.2,ϕ = 0.2,ε = 0.3,Pr = 6.2,S = 0.4,δ = 0.1,γ = 0.6
Figure 1

Influence of A on velocity of a power-law nanofluid when a = 1,M = 0.2,ϕ = 0.2,ε = 0.3,Pr = 6.2,S = 0.4,δ = 0.1,γ = 0.6

Figure 2 Influence of A on temperature of a power-law nanofluid when a = 1,M = 0.2,ϕ = 0.2,ε = 0.3,Pr = 6.2,S = 0.4,δ = 0.1,γ = 0.6
Figure 2

Influence of A on temperature of a power-law nanofluid when a = 1,M = 0.2,ϕ = 0.2,ε = 0.3,Pr = 6.2,S = 0.4,δ = 0.1,γ = 0.6

Figures 3 and 4 depict the velocity and temperature profiles of power-law non-Newtonian nanofluid for variation in ε when M = ϕ = A = 0.2, S = 0.4,δ = 0.1,γ = 0.6. It is clear from Figure 3, that the thermal conductivity parameter has no effect on the velocity profiles of nanofluid. However, increasing values of the index n, show thickening of the momentum boundary layer in some initial range of η and subsequently thinning of the momentum boundary layer thickness is observed. The reason for this behavior has already been explained in the preceding discussion of Figure 1. Analysis of curves in Figure 4 illustrates that nanofluid temperature rises within the boundary layer with a rise in thermal conductivity and tends asymptotically to zero as the distance from the boundary increases. This fact is clear from our definition, κnf>κnf∗ for ε > 0, i.e., the thermal conductivity of a nanofluid increases with increasing values of ε.

Figure 3 Influence of ε on velocity of a power-law nanofluid
Figure 3

Influence of ε on velocity of a power-law nanofluid

Figure 4 Influence of ε on the temperature of a power-law nanofluid
Figure 4

Influence of ε on the temperature of a power-law nanofluid

The variation in M (strength of applied transverse magnetic field) and its effect on nanofluid velocity is presented in Figure 5 for parameter values A = ϕ = 0.2,ε = 0.3,S = 0.4,δ = 0.1,γ = 0.6. The curves with index n = 1.4, indicate that thickness of the momentum boundary layer decreases with increasing value of M. In other words, the increasing strength of the magnetic parameter M boosts flow of the nanofluid within the boundary layer. In this case, the Lorentz force (generated due to application of a transverse magnetic field) counteracts the viscous forces and enhances nanofluids motion. Figure 6 presents variation in temperature profiles under slip conditions for different values of M. As the parameter M increases the temperature of the power-law nanofluid decreases within the boundary layer. This is because increasing the value of the strength of the transverse magnetic field boosts the fluid velocity within the boundary layer.

Figure 5 Influence of M on velocity of a power-law nanofluid
Figure 5

Influence of M on velocity of a power-law nanofluid

Figure 6 Influence of M on temperature of a power-law nanofluid
Figure 6

Influence of M on temperature of a power-law nanofluid

Figures 7 and 8 display that both the nanofluid velocity and temperature within the boundary layer increase with increasing values of ϕ when A = M = 0.2, ε = 0.3, S = 0.4, δ = 0.1, γ = 0.6. These figures are in good agreement with the physical behavior of nanofluids namely that the volume fraction of denser nanoparticles causes thinning of the momentum boundary layer. The rate of heat transfer is also reduced within the momentum boundary layer. The reason for this trend is that an increase in the volume of nanoparticles increases the overall thermal conductivity of nanofluids since the solid particles have higher thermal conductivity when compared to the base fluid.

Figure 7 Influence of ϕ on velocity of a power-law nanofluid
Figure 7

Influence of ϕ on velocity of a power-law nanofluid

Figure 8 Influence of ϕ on temperature of a power-law nanofluid
Figure 8

Influence of ϕ on temperature of a power-law nanofluid

The variation in δ and its effect on nanofluid velocity is shown in Figure 9 for A = M = ϕ = 0.2,ε = 0.3, S = 0.4,γ = 0.6. For the parametric value δ =0.2, 0.6 it can be seen that the velocity for shear thinning, Newtonian and shear thickening fluids increases with increasing values of the velocity slip at the boundary. This expected behavior is due to positive values of nanofluid velocity at the boundary. As a result, thickness of the boundary layer will decrease when parameter δ increases. On the other hand, it is evident from Figure 10 that the surface slipperiness affects the temperature of the fluid inversely at the boundary, i.e., an increase in slip parameter tends to decrease the temperature of the power-law nanofluid and enhance the rate of heat transfer.

Figure 9 Influence of δ on velocity of a power-law nanofluid
Figure 9

Influence of δ on velocity of a power-law nanofluid

Figure 10 Influence of δ on temperature of a power-law nanofluid
Figure 10

Influence of δ on temperature of a power-law nanofluid

The variation in S and its influence on motion and temperature of power-law nanofluids are shown in Figures 11-14, respectively. The momentum boundary layer thickness decreases with increasing values of S > 0 (suction parameter). This is because more fluid is transferred into the boundary layer through the porous surface. From Figure 13, it may be noted that the boundary layer thickness increases with increase in injection S < 0. The imposition of suction on the surface causes reduction in the thermal boundary layer thickness and injection causes an increase in the thermal boundary layer thickness.

Figure 11 Influence of S > 0 on velocity of a power-law nanofluid
Figure 11

Influence of S > 0 on velocity of a power-law nanofluid

Figure 12 Influence of S > 0 on temperature of a power-law nanofluid
Figure 12

Influence of S > 0 on temperature of a power-law nanofluid

Figure 13 Effect of S < 0 on velocity of a power-law nanofluid
Figure 13

Effect of S < 0 on velocity of a power-law nanofluid

Figure 14 Effect of S < 0 on temperature of a power-law nanofluid
Figure 14

Effect of S < 0 on temperature of a power-law nanofluid

Table 3 presents the nature of the skin friction coefficient of the wall (velocity gradient) and Nusselt number (temperature gradient) at the boundary for the MHD slip flow of a power-law nanofluid. Similarly the nature of the skin friction coefficient and Nusselt number is observed for pseudoplastic, newtonian and dilatant nanofluids. Similar nature is observed for pseudoplastic, newtonian and dilatant nanofluids. Therefore, numerical computations are only presented for n = 1.4 in Table 3. It can be seen that pseudoplastic fluids have the highest values and the dilatant fluids have the lowest values of the skin friction coefficient and Nusselt number. The lowest rate of effective viscosity for pseudoplastic fluids is the reason behind this. Moreover, it is evident that the velocity gradient at the boundary increases with increasing values of parameters A, ϕ, M, and S; whereas a reduction in the skin friction coefficient is observed for increasing values of δ and γ. Increase in the velocity gradient at the boundary corresponds to thinning of the momentum boundary layer thickness and decrease in the velocity of the nanofluids. The decreasing trend shows that the fluid velocity is approaching the free stream velocity. It is observed from Table 3, that the Nusselt number increases with enhancement in the strength of applied transverse magnetic field M, δ the velocity slip coefficient of the surface and the S the suction/injection parameter. Increasing values of parameters A, ϕ, ε and γ decrease the Nusselt number.

Table 3

Nature of the skin friction coeflcient and the Nusselt number with a = 1.0, Pr = 6.2

nAϕεMSδγ−(f″(0))n−θ′(0)
0.40.10.200.30.60.20.10.11.40060.9700
1.01.05200.7444
1.40.89730.6679
1.40.10.89730.6679
0.60.91500.6636
1.60.94210.6568
1.40.050.77090.7966
0.100.81210.7483
0.150.85440.7059
0.200.89750.6680
1.40.10.89750.7395
0.30.89750.6680
0.60.89750.5894
1.40.20.69170.6446
0.60.83730.6540
1.20.95910.6611
1.40.00.65110.5169
0.20.89750.6680
0.41.15020.8215
1.40.00.94840.6389
0.20.84370.6913
0.40.65420.7510
1.40.00.90220.7088
0.20.89350.6311
0.30.88990.5976

5 Concluding remarks and future work

In this work the principal effects of variable viscosity, variable thermal conductivity and applied transverse magnetic field on slip flow and heat transfer characteristics of a power-law nanofluid over a porous flat surface have been analyzed numerically. To the best of the authors’ knowledge no such study has previously been presented in the literature. The governing system of PDEs was transformed into a system of ODEs and then solved numerically. Numerical computations performed for temperature and velocity variation of Cu-water nanofluid within the boundary layer were presented through graphs and tables. Parametric effects of different governing parameters on velocity and temperature profiles were discussed. It is concluded that:

  1. Pseudoplastic nanofluids have the highest skin friction coefficient and dilatant fluids have the lowest skin friction coefficient. The highest rate of heat transfer is observed for n < 1 and the lowest for the case n > 1

  2. Increase in nanofluid viscosity and volume fraction of nanoparticles decreases the velocity and increases the temperature of nanofluids within the boundary layer. This will reduce the heat transfer rate and enhance thickness of the momentum boundary layer

  3. Increase in strength of the applied transverse magnetic field and suction velocity increases the velocity and decreases the temperature distribution in the boundary layer

  4. Increase in both velocity slip and thermal slip reduces thickness of the momentum boundary layer, whereas increase in slip velocity enhances and thermal slip reduces the rate of heat transfer.

The simplified model for flow and heat transfer analysis of a power-law nanofluid presents a qualitative analysis that can be quantified to calculate the thermal efficiency of the system. Results can be generalized to include the effects of variable viscosity, variable porosity and heat transfer of non-Newtonian nanofluids (see for example [36, 37]). Comparisons can be drawn between the results of the presented model and nano scale flow. Alternatively, the present model can be solved by employing semi-analytical methods [19, 36, 38].

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Received: 2017-8-3
Accepted: 2017-10-16
Published Online: 2017-12-29

© 2017 S. Hussain 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-2017-0104
Keywords for this article
Non-Newtonian nanofluids; Power-law model; Brickman nanofluid Model; Temperature dependent viscosity; Temperature dependent thermal conductivity; Partial slip; Magnetohydrodynamics
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  48. Lie algebraic discussion for affinity based information diffusion in social networks
  49. Regular Articles
  50. Radiation dose and cancer risk estimates in helical CT for pulmonary tuberculosis infections
  51. Regular Articles
  52. A comparison study of steady-state vibrations with single fractional-order and distributed-order derivatives
  53. Regular Articles
  54. Some new remarks on MHD Jeffery-Hamel fluid flow problem
  55. Regular Articles
  56. Numerical investigation of magnetohydrodynamic slip flow of power-law nanofluid with temperature dependent viscosity and thermal conductivity over a permeable surface
  57. Regular Articles
  58. Charge conservation in a gravitational field in the scalar ether theory
  59. Regular Articles
  60. Measurement problem and local hidden variables with entangled photons
  61. Regular Articles
  62. Compression of hyper-spectral images using an accelerated nonnegative tensor decomposition
  63. Regular Articles
  64. Fabrication and application of coaxial polyvinyl alcohol/chitosan nanofiber membranes
  65. Regular Articles
  66. Calculating degree-based topological indices of dominating David derived networks
  67. Regular Articles
  68. The structure and conductivity of polyelectrolyte based on MEH-PPV and potassium iodide (KI) for dye-sensitized solar cells
  69. Regular Articles
  70. Chiral symmetry restoration and the critical end point in QCD
  71. Regular Articles
  72. Numerical solution for fractional Bratu’s initial value problem
  73. Regular Articles
  74. Structure and optical properties of TiO2 thin films deposited by ALD method
  75. Regular Articles
  76. Quadruple multi-wavelength conversion for access network scalability based on cross-phase modulation in an SOA-MZI
  77. Regular Articles
  78. Application of ANNs approach for wave-like and heat-like equations
  79. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  80. Study on node importance evaluation of the high-speed passenger traffic complex network based on the Structural Hole Theory
  81. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  82. A mathematical/physics model to measure the role of information and communication technology in some economies: the Chinese case
  83. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  84. Numerical modeling of the thermoelectric cooler with a complementary equation for heat circulation in air gaps
  85. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  86. On the libration collinear points in the restricted three – body problem
  87. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  88. Research on Critical Nodes Algorithm in Social Complex Networks
  89. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  90. A simulation based research on chance constrained programming in robust facility location problem
  91. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  92. A mathematical/physics carbon emission reduction strategy for building supply chain network based on carbon tax policy
  93. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  94. Mathematical analysis of the impact mechanism of information platform on agro-product supply chain and agro-product competitiveness
  95. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  96. A real negative selection algorithm with evolutionary preference for anomaly detection
  97. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  98. A privacy-preserving parallel and homomorphic encryption scheme
  99. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  100. Random walk-based similarity measure method for patterns in complex object
  101. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  102. A Mathematical Study of Accessibility and Cohesion Degree in a High-Speed Rail Station Connected to an Urban Bus Transport Network
  103. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  104. Design and Simulation of the Integrated Navigation System based on Extended Kalman Filter
  105. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  106. Oil exploration oriented multi-sensor image fusion algorithm
  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
  109. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  110. Expanded Study on the accumulation effect of tourism under the constraint of structure
  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
  118. Physical-chemical properties studying of molecular structures via topological index calculating
  119. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  120. Local kernel nonparametric discriminant analysis for adaptive extraction of complex structures
  121. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  122. City traffic flow breakdown prediction based on fuzzy rough set
  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
  132. Study on maintaining formations during satellite formation flying based on SDRE and LQR
  133. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  134. Relationship between solitary pulmonary nodule lung cancer and CT image features based on gradual clustering
  135. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  136. A novel fast target tracking method for UAV aerial image
  137. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  138. Fuzzy comprehensive evaluation model of interuniversity collaborative learning based on network
  139. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  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
  158. Analysis on trust influencing factors and trust model from multiple perspectives of online Auction
  159. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  160. Coupling of two-phase flow in fractured-vuggy reservoir with filling medium
  161. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  162. Production decline type curves analysis of a finite conductivity fractured well in coalbed methane reservoirs
  163. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  164. Flow Characteristic and Heat Transfer for Non-Newtonian Nanofluid in Rectangular Microchannels with Teardrop Dimples/Protrusions
  165. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  166. The size prediction of potential inclusions embedded in the sub-surface of fused silica by damage morphology
  167. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  168. Research on carbonate reservoir interwell connectivity based on a modified diffusivity filter model
  169. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  170. The method of the spatial locating of macroscopic throats based-on the inversion of dynamic interwell connectivity
  171. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  172. Unsteady mixed convection flow through a permeable stretching flat surface with partial slip effects through MHD nanofluid using spectral relaxation method
  173. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  174. A volumetric ablation model of EPDM considering complex physicochemical process in porous structure of char layer
  175. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  176. Numerical simulation on ferrofluid flow in fractured porous media based on discrete-fracture model
  177. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  178. Macroscopic lattice Boltzmann model for heat and moisture transfer process with phase transformation in unsaturated porous media during freezing process
  179. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  180. Modelling of intermittent microwave convective drying: parameter sensitivity
  181. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  182. Simulating gas-water relative permeabilities for nanoscale porous media with interfacial effects
  183. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  184. Simulation of counter-current imbibition in water-wet fractured reservoirs based on discrete-fracture model
  185. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  186. Investigation effect of wettability and heterogeneity in water flooding and on microscopic residual oil distribution in tight sandstone cores with NMR technique
  187. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  188. Analytical modeling of coupled flow and geomechanics for vertical fractured well in tight gas reservoirs
  189. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  190. Special Issue: Ever New "Loopholes" in Bell’s Argument and Experimental Tests
  191. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  192. The ultimate loophole in Bell’s theorem: The inequality is identically satisfied by data sets composed of ±1′s assuming merely that they exist
  193. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  194. Erratum to: The ultimate loophole in Bell’s theorem: The inequality is identically satisfied by data sets composed of ±1′s assuming merely that they exist
  195. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  196. Rhetoric, logic, and experiment in the quantum nonlocality debate
  197. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  198. What If Quantum Theory Violates All Mathematics?
  199. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  200. Relativity, anomalies and objectivity loophole in recent tests of local realism
  201. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  202. The photon identification loophole in EPRB experiments: computer models with single-wing selection
  203. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  204. Bohr against Bell: complementarity versus nonlocality
  205. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  206. Is Einsteinian no-signalling violated in Bell tests?
  207. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  208. Bell’s “Theorem”: loopholes vs. conceptual flaws
  209. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  210. Nonrecurrence and Bell-like inequalities
  211. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  212. Three-dimensional computer models of electrospinning systems
  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
  226. Detection of inter-turn short-circuit at start-up of induction machine based on torque analysis
  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
  230. Modelling axial vibration in windings of power transformers
  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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