Open access peer-reviewed chapter

Energy Transfer in Mixed Convection MHD Flow of Nanofluid Containing Different Shapes of Nanoparticles in a Channel Filled with Saturated Porous Medium

By Aaiza Gul, Ilyas Khan and Sharidan Shafie

Submitted: May 2nd 2016Reviewed: December 28th 2016Published: March 15th 2017

DOI: 10.5772/67367

Downloaded: 1228

Abstract

Energy transfer in mixed convection unsteady magnetohydrodynamic (MHD) flow of an incompressible nanofluid inside a channel filled with a saturated porous medium is investigated. The walls of the channel are kept at constant temperature, and uniform magnetic field is applied perpendicular to the direction of the flow. Three different flow situations are discussed on the basis of physical boundary conditions. The problem is first written in terms of partial differential equations (PDEs), then reduces to ordinary differential equations (ODEs) by using a perturbation technique and solved for solutions of velocity and temperature. Four different shapes of nanoparticles inside ethylene glycol (C2H6O2) and water (H2O)‐based nanofluids are used in equal volume fraction. The solutions of velocity and temperature are plotted graphically, and the physical behavior of the problem is discussed for different flow parameters. It is evaluated from this problem that viscosity and thermal conductivity are the dominant parameters responsible for different consequences of motion and temperature of nanofluids. Due to greater viscosity and thermal conductivity, C2H6O2‐based nanofluid is regarded as better convectional base fluid assimilated to H2O.

Keywords

  • mixed convection
  • nanofluid
  • heat transfer
  • cylindrical‐shaped nanoparticles
  • MHD flow
  • porous medium
  • analytical solutions

1. Introduction

1.1. Nanofluids

Thermal conductivity is the important thermophysical property that plays an essential role in heat transfer enhancement. The internal property of the material is depended on the nature of the fluids. Changing the nature of the fluids or materials can also be affected. In contrast, by applying external force, thermal convection can also be affected on the fluids. Thermal convections are depended on the nature of the fluids as well as on the fluid flow, fluid regime, geometry, etc. Heat transfer conventional fluids, such as ethylene glycol (EG), water, and oils, which are lubricant and kerosene oil have poor thermal conductivities compared to solids. On the other hand, solid particles exhibited rich thermal conductivities than those of conventional heat transfer fluids. The significant improvements in the thermal conductivity of the fluids, which are conventional base by adding up the nanosized particles, were considered by Choi [1] for the first time. The mixture of nanosized particles containing conventional‐based fluids is usually known as nanofluids. More accurately, nanofluids are suspensions of nanosized particles in base fluids. The most ordinary nanoparticles used in nanofluids are oxides, carbides, and metals. Nanofluids are mainly used in electronic equipment, power generation, energy supply, production, and air conditioning. Vajjha and Das [2] investigated, for the first time, EG (60%) and water (40%) mixture that is a base fluid for the preparation of alumina (Al2O3), copper oxide (CuO), and zinc oxide (ZnO)nanofluids. At the equal temperature and concentration, CuOnanofluids have higher thermal conductivity than those of Al2O3and ZnOnanofluids. Naik and Sundar [3] have prepared (CuO)nanofluids with 70% propylene glycol and 30% water. As estimated, they found that CuOnanofluids have rich thermal conductivity and viscosity as compared to the other fluids that are base. Recently, Mansur et al. [4] have studied magnetohydrodynamic (MHD) stagnation‐point flow of nanofluids over a permeable sheet for stretching and shrinking cases. They used bvp4c program in MATLAB to obtain the numerical solutions and computed results for different parameters.

The quality of nanofluids in terms of heat transfer performance depends on the volume fraction and types of nanoparticles as well as also depends on the shapes of nanoparticles. Many researchers have used nanoparticles in spherical shapes. However, the nanofluids containing nonspherical shapes of nanoparticles have higher thermal conductivities as compared to spherical ones. Hence, nonspherical shapes of nanoparticles are most suitable to be used. Nanoparticles other than spherical shapes have been used in this research due to their above‐mentioned properties. More closely, in this study we discuss four different types of nanoparticles, namely, cylinder, platelet, blade, and brick. Nanoparticles of nonspherical shapes with desirable properties are the main focus of present study. Recently, the development in the field of nanotechnology has shown that cylindrical‐shaped nanoparticles are seven times more harmful than spherical‐shaped nanoparticles in the deliverance of drugs to breast cancer. In the literature survey of nanofluids, we found that analytical studies for the suspension of a variety of shapes of nanoparticles containing EG and water‐based fluids are not described yet. Although Timofeeva et al. [5] study the problem of Al2O3nanofluids which are of different shaped nanoparticles, but they performed experimentally with theoretical modeling together in this study. More closely, they studied various shapes of Al2O3nanoparticles in a base fluid mixture of EG and water of similar volumes. Improvement in the effective thermal conductivities due to particle shapes was mentioned by Timofeeva et al. Loganathan et al. [6] considered nanoparticles that are sphere shaped and analyzed radiation effects on an unsteady natural convection flow of nanofluids over an infinite vertical plate. They have found that the velocity of spherical silver (Ag)nanofluids is less than that of copper (Cu),titanium dioxide (TiO2), and Al2O3spherical nanofluids. Recently, Asma et al. [7] found exact solutions for free‐convection flow of nanofluids at ramped wall temperature by using five various kinds of spherical‐shaped nanoparticles.

Sebdani et al. [8] studied heat transfer of Al2O3water nanofluid in mixed convection flow inside a square cavity. Fan et al. [9] described mixed convection heat transfer in a horizontal channel filled with nanofluids. Tiwari and Das [10] and Sheikhzadeh et al. [11] investigated laminar mixed convection flow of a nanofluid in two‐sided lid‐driven enclosures. Furthermore, a magnetic field in nanofluids has diverse applications such as in the metallurgy and polymer industry where hydromagnetic techniques are used. Nadeem and Saleem [12] presented the unsteady flow of a rotating MHD nanofluid in a rotating cone in the presence of magnetic field. Al‐Salem et al. [13] examined MHD mixed convection flow in a linearly heated cavity. The effects of variable viscosity and variable thermal conductivity on the MHD flow and heat transfer over a nonlinear stretching sheet was studied by Prasad et al. [14]. The problem of Darcy Forchheimer mixed convection heat and mass transfer in fluid‐saturated porous media in the presence of thermophoresis was discussed by Rami et al. [15]. The effect of radiation and magnetic fields on the mixed convection stagnation‐point flow over a vertical stretching sheet in a porous medium bounded by a stretching vertical plate was analyzed by Hayat et al. [16]. Some other studies on mixed convection flow of nanofluids are mentioned in Refs. [1728]. Nanofluids due to the higher conduction or heat transfer rate of together with other several applications in the field of nanoscience have attracted the attention of researchers to perform future research. Altogether, several researchers are working experimentally, some of them are using numerical simulation. However, very limited research studies are obtainable on analytical side. Experimental research mainly focuses on the enhancement of heat transfer rate of nanofluids through thermal conduction. And this study also includes heat transfer through mixed convection. Several other efforts made on nanofluids are presented in Refs. [2939].

In order to encounter the importance of MHD in nanofluids, Mansur et al. [4] explored the MHD stagnation point flow of nanofluids over a stretching/shrinking sheet with suction. Colla et al. [21] conducted water‐based nanofluids characterization, thermal conductivity and viscosity measurements, and correlation. Abareshi et al. [40] investigated fabrication, characterization, and measurement of thermal conductivity of nanofluids. Borglin et al. [41] examined experimentally the flow of magnetic nanofluids in porous media. MHD effect on nanofluid with energy and hydrothermal behavior between two collateral plates was presented by Sheikholeslami et al. [42]. Sheikholeslami et al. [43] studied forced convection heat transfer in a semiannulus under the influence of a variable magnetic field. Some other research studies on electrically conducting nanofluids are mentioned in Refs. [4450].

In view of the above literature, the existing research is concerned with the radiative heat transfer in mixed convection MHD flow of different shapes of Al2O3in EG base nanofluid in a channel filled with saturated porous medium. The foremost focus of this study is the importance of cylindrical‐shaped nanofluids on several flow parameters. The uniform constant magnetic field is applied at 90° to the flow and the nanofluids are supposed electrically conducting. No slip condition is taken at the boundary walls of the channel. Three different flow cases on the basis of appropriate boundary conditions are explored. Both of the boundary walls of the channel are at rest in the first case, while motion in the fluids is induced due to buoyancy force and external pressure gradient. In the second case, the right wall of the channel is oscillating in its own plane whereas in the third case, both of the bounding walls of the channel are set into oscillatory motions. The analytical solutions of velocity and temperature are obtained by using a perturbation technique. Graphs are plotted, and the physical behavior of the problem is discussed for different parameters of interest. Nusselt number and skin friction are also computed.

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2. Derivation and solutions of governing equations

Consider flow in oscillating form of an incompressible nanofluid inside vertical porous channel. Flow of the nanofluids is supposed electrically conducting under the influence of uniform constant magnetic field. The uniform magnetic field is applied perpendicular to the direction of the flow. Reynolds number is supposed small to ignore the effect of evoked magnetic field. The external electric and electric fields due to polarization of charges are taken zero in order to ignore the influence of electric force in Lorentz force. It is assumed that at time t=0, the flow is at constant temperature θ0.Right boundary of the channel (y=d)is maintained at constant temperature θwR, while the left boundary (y=0)has a uniform temperature θwL. Coordinate axis is considered where flow of fluids is moving in the xaxis direction, while the y‐axis is taken perpendicular to the velocity of the flow direction.

After derivation of governing equations for quantity of motion and energy under the assumption of Boussinesq approximation are as follow:

ρnfut=px+μnf2uy2(σnfB02+μnfκ1)u+(ρβ)nfg(θθ0),E1
(ρcp)nfθt=κnf2θy2qy,E2

where u=u(y,t),θ=θ(y,t),ρnf,μnf,σnf,κ1>0,(ρβ)nf,g,(ρcp)nf,κnfqspecify the fluid velocity, temperature, density, dynamic viscosity, electrical conductivity, permeability of the porous medium, thermal expansion coefficient,gravitational acceleration, heat capacitance, thermal conductivity of nanofluids, and radiative heat flux in the x‐direction. p/x=λexp(iωt)represents the pressure gradient of the flow in oscillatory form where ωis the oscillation parameter and λis the amplitude of oscillation.

The Hamilton and Crosser [17] model for thermal conductivity and the Timofeeva et al. [5] model for calculating dynamic viscosity of nanofluids are used in order to encounter spherical and other than spherical shapes of nanoparticles inside nanofluids. From these models:

μnf=μf(1+aφ+φ2b),E3
κnfκf=κs+(m1)κf+(n1)(κsκs)φκs+(m1)κf(κsκf)φ,E4

where κfand κsare the thermal conductivities of the base fluid and solid nanoparticles, respectively. The density ρnf, thermal expansion coefficient (ρβ)nf, heat capacitance (ρcp)nfand thermal conductivity σnf,of nanofluids are derived by using the relations given by [6, 7]

(ρcp)nf=(1φ)(ρcp)f+φ(ρcp)s,ρnf=(1φ)ρf+φρs,
(ρβ)nf=(1φ)(ρβ)f+φ(ρβ)s,κnf=αnf(ρcp)nf,
σnf=σf[1+3(σ1)φ(σ+2)(σ1)φ],σ=σsσf,E5

where φis the volume fraction of the nanoparticles, ρfand ρsare the density of the base fluid and solid nanoparticles, respectively, μfis the dynamic viscosity of the base fluid, (cp)fand (cp)sdenote the specific heat at constant pressure corresponding to the base fluid and solid nanoparticles and σfand σsare electrical conductivities of base fluids and solid nanoparticles. The total term (ρcp)is known as heat capacitance.

Here aand bare shape constants and different for different shape of nanoparticles as presented in Table 1 [5], μf,κf,and κsare the dynamic viscosity, thermal conductivity of the base fluid, and thermal conductivity of solid nanoparticles, respectively. The empirical shape factor ndefined in Eq. (4) is equal to n=3/Ψ,where Ψis the sphericity. Sphericity given in Hamilton and Crosser model is the ratio of surface area of the sphere to the surface area of real particle with equal volumes. The values of sphericity for different shapes of nanoparticles are given in Table 2 [5].

ModelPlateletBladeCylinderBrick
a37.114.613.51.9
b612.6123.3904.4471.4

Table 1.

Constants aand bempirical shape factors.

ModelPlateletBladeCylinderBrick
Ψ0.520.360.620.81

Table 2.

Sphericity Ψ for different shapes nanoparticles.

In Eqs. (1) and (2), the (ρcp)nf, ρnf,(ρβ)nf, κnf, and σnfof nanofluids are used from the relations given by Asma et al. [7], as

The temperature θ0and θwof both walls of the channel is assumed high and both walls are emitting radiations [20]. Therefore, radiative heat flux as a function of temperature is given by

qy=4α02(θθ0),E6

where α0is the radiation absorption coefficient.

Using Eq. (6) into Eq. (2) gives

(ρcp)nfθt=κnf2θy24α02(θθ0),E7

In order to convert dimensional partial differential equations (PDEs) into dimensionless PDEs, the following nondimensional variable are introduced

x=xd,y=yd,u=uU0,t=tU0d,θ=θθ0θwθ0,p=dμU0p,ω=dω1U0,E8

After dimensionalization (dropping for convenience) Eqs. (1) and (7) give

[(1φ)+φρsρf]Reut=λexp(iωt)+(1+aφ+bφ2)2uy2[1+3(σ1)φ(σ+2)(σ1)φ]M2u1K(1+aφ+bφ2)+[(1φ)+φ(ρβ)s(ρβ)f]Grθ,E9
Peλn[(1φ)+φ(ρcp)s(ρcp)f]θt=2θy2+N2λnθ,E10

where

Re=U0dvf,M2=σfB02d2μf,K=κ1d2Gr=gβfd2(θwθ0)vfU0,Pe=(ρcp)fdU0κf,λn=κnfκf=κs+(m1)κf+(n1)(κsκs)φκs+(m1)κf(κsκf)φ,N2=4d2α02κf.

Here Re,M,Gr,Pe, and Ndenote the Reynolds number, magnetic parameter, the thermal Grashof number, the Peclet number, and the radiation parameter, respectively. Three different flow cases are considered in order to solve Eqs. (9) and (10). These are as follow:

2.1. Case I: Stationary walls of the channel

In this case, the gap between the two plates of the channel is denoted by dand both plates are assumed stationary at y=0and y=d. The flow of nanofluids is unidirectional and moving with velocity in the x‐axis. Both plates of the channel are maintained at constant and uniform temperature θwand θ0. Thus, the appropriate boundary conditions are

u(0,t)=0,u(d,t)=0,E11
θ(0,t)=θ0,θ(d,t)=θw.E12

After reducing Eqs. (11) and (12) into dimensionless form, we get

u(0,t)=0;u(1,t)=0,t>0,E13
θ(0,t)=0;θ(1,t)=1;t>0.E14

Further simplification of Eqs. (9) and (10) after conversion of dimensional PDEs, we get

d0ut=λexp(iωt)+φ22uy2h02u+d1θ,E15
e0θt=2θy2+e1θ,E16

where

d0=φ1Re,φ1=(1φ)+φρsρf,φ2=(1+aφ+bφ2),h02=φ5M2+1/K,
φ5=[1+3(σ1)φ(σ+2)(σ1)φ],d1=φ3Gr,φ3=(1φ)+φ(ρβ)s(ρβ)f,e02=Peφ4λn,φ4=[(1φ)+φ(ρcp)s(ρcp)f],e12=N2λn.

Perturb‐type solutions are supposed to convert PDEs in Eqs. (15) and (16) under appropriate boundary conditions, and in Eqs. (11) and (12) into ordinary differential equations (ODEs) are

u(y,t)=[u0(y)+εexp(iωt)u1(y)],E17
θ(y,t)=[θ0(y)+εexp(iωt)θ1(y)].E18

Using Eqs. (17) and (18) into Eqs. (15) and (16), we obtain the following system of ordinary differential equations

d2u0(y)dy2h12u0(y)=d2θ0(y),E19
d2u1(y)dy2h22u1(y)=λφ2,E20
d2θ0(y)dy2e12θ0(y)=0,E21
d2θ1(y)dy2+h32θ1(y)=0,E22

where

h1=h20φ2,d2=d1φ2,h2=h02+iωd0φ2,h3=e1iωe0.

The associated boundary conditions (Eqs. (13) and (14)) are reduce to

u0(0)=0;u0(1)=0,E23
u1(0)=0;u1(1)=0,E24
θ0(0)=0;θ0(1)=1,E25
θ1(0)=0;θ1(1)=0.E26

Perturb solution for temperature in Eqs. (21) and (22) under appropriate boundary conditions (Eqs. (25) and (26)) give

θ0(y)=sin(e1y)sin(e1),E27
θ1(y)=0.E28

Temperature of nanofluids, Eq. (18) using Eqs. (27) and (28) gives

θ(y,t)=θ(y)=sin(e1y)sin(e1).E29

Perturb solution for temperature in Eqs. (19) and (20), using Eq. (27) under appropriate boundary conditions, Eqs. (23) and (24) yield

u0(y)=c1sinh(h1y)+c2cosh(h1y)+d2(e12+h12)sin(e1y)sin(e1).E30
u1(y)=c3sinh(h2y)+c4cosh(h2y)+λh22φ2.E31

Here c1,c2,c3and c4are arbitrary constants given by

c1=d2sinh(h1)(e12+h12),c2=0,c3=λh22φ21sinh(h2)(cosh(h2)1),c4=λh22φ2.E32

Final velocity for nanofluids, substituting Eqs. (30)(32) into Eq. (17), we obtain

u(y,t)=d2sinh(h1y)(e12+h12)sinh(h1)+d2sin(e1y)(e12+h12)sin(e1)+εexp(iωt)[λ(cosh(h2)1)sinh(h2y)h22φ2sinh(h2)+λh22φ2(1cosh(h2y))].E33

2.2. Case 2: right plate of the channel is oscillating in its own plane

In this case, the right wall of the channel (y=d) is locate into oscillatory motion, while on other hand, the left wall (y=0)is taken as stationary. The first boundary condition is the same, while the second boundary condition in dimensionless form modifies to

u(1,t)=H(t)εexp(iωt);t>0,E34

where H(t)is the Heaviside step function.

By using the same procedure as in Case‐1, and the velocity solution is obtained as

u(y,t)=d2sinh(h1y)sinh(h1)(e12+h12)+d2(e12+h12)sin(e1y)sin(e1)+εexp(iωt)[sinh(h2y)sinh(h2)(H(t)+λ(h22φ2)(cosh(h2)1))λ(h22φ2)cosh(h2y)+λ(h22φ2)].E35

2.3. Case 3: both plates of the channel are oscillating in its own plane

In this case, both plates of the channel are chosen into oscillatory motions. The dimensionless form of the boundary conditions is

u(0,t)=u(1,t)=H(t)εexp(iωt);t>0.E36

The resulting expression for velocity is obtained as

u(y,t)=d2sinh(h1y)sinh(h1)(e12+h12)+d2(e12+h12)sin(e1y)sin(e1)+εexp(iωt)[sinh(h2y)sinh(h2)(H(t)(1cosh(h2y))+λ(h22φ2)(cosh(h2)1))+{H(t)λ(h22φ2)}cosh(h2y)+λ(h22φ2)].E37

2.4. Evaluation of Nusselt number and skin‐friction

The dimensionless derivations for Nusselt number and skin‐frictions are evaluated from Eqs. (29), (33), (35) and (37) as follows:

Nu=e1sin(e1),E38
τ1=τ1(t)=d2h1(e12+h12)sinh(h1)+d2b1(e12+h12)sin(e1)+εexp(iωt)[λ(cosh(h2)1)h2φ2sinh(h2)].E39
τ2=τ2(t)=d2h1(e12+h12)sinh(h1)+d2e1(e12+h12)sin(e1)+εexp(iωt)[h2sinh(h2){H(t)+λ(h2φ2)(cosh(h2)1)}],E40
τ3=τ3(t)=d2h1(e12+h12)sinh(h1)+d2b1(e12+h12)sin(e1)+εexp(iωt)[h2sinh(h2){H(t)(1cosh(h2))+λ(h22φ2)(cosh(h2)1)}].E41

3. Graphical consequences and depiction

In this segment, graphical consequences are figured and debated. Similarly, influence of the radiation effect on heat transfer in mixed convection MHD flow of nanofluids inside a channel filled with a saturated porous medium is explored. Three different flow cases on the basis of appropriate boundary conditions are examined. Four dissimilar shapes of Al2O3as solid nanoparticles, which are cylinder, platelet, brick, and blade, are dangling into conventional base fluids, ethylene glycol and water. The physical performance of the exhibited graphs is conferred for various embedded parameters. The numerical values of constants aand b(called shape factors) are chosen from Table 1, and sphericity Ψis given in Table 2. It should be acclaimed that aand bcoefficients vary highly with particle shape. The numerical values for the various shapes of nanoparticles (platelet, blade, cylinder and brick) at equal volumes are presented in Tables 13.

Modelρ(Kgm3)cP(kg1Κ1)k(Wm1Κ1)β×105(Κ1)σ(s/m)
H2O997.141790.613215.5 × 10‐6
C2H6O21.1150.580.14906.51.07 × 10‐6
Cu89333854011.6759.6 × 106
TiO24250686.28.95280.92.6 × 106
Ag10,5002354291.896.30 × 107
Al2O33970765400.851.07 × 10‐6
Fe3O451806709.70.512.7

Table 3.

Thermophysical properties of water and nanoparticles.

The geometry of the problem is shown in Figure 1. The impact of dissimilar shapes of alumina Al2O3nanoparticles on the motion of ethylene glycol‐based nanofluids is represented in Figure 2. It is viewed that the blade shape of alumina Al2O3nanoparticles inside ethylene glycol‐based nanofluids has the uppermost velocity chased by brick, platelet, and cylindrical‐shaped nanoparticles in ethylene glycol‐based nanofluids. The impact of the shapes on the motion of nanofluids is because of the strong subjection of viscosity on particle shapes for φ<0.1. It is exonerated from the present consequences that the elongated shaped nanoparticles like cylinder and platelet have larger viscosities as assimilated to nanofluids comprising square‐shaped nanoparticles like brick and blade. The acquired consequences agree well with the experimental consequences predicted by Timofeeva et al.. A very small divergence is perceived in the present research, where the cylindrical‐shaped nanoparticles have the larger viscosity, whereas from the experimental findings perceived by Timofeeva et al., the platelet has the larger viscosity. Timofeeva et al. had assimilated their consequences with the Hamilton and Crosser model and found that their consequences were equivalent with the Hamilton and Crosser model. In the current work, the model of Hamilton and Crosser is applied and found that analytical consequences in this research also identical to the experimental consequences of Timofeeva et al.

Figure 1.

Physical model and coordinates system.

Figure 2.

Velocity profiles for different shapes ofAl2O3nanoparticles in EG‐based nanofluids whenGr=0.1,N=0.1,M=1,λ=1,K=1,t=5,φ=0.04,ω=0.2.

Figure 3 shows the result of dissimilar shapes of alumina Al2O3nanoparticles on the motion of water H2O‐based nanofluids. It is surely noticed that the cylindrical‐shaped alumina Al2O3nanoparticles in ethylene glycol‐based nanofluids have the uppermost velocity chased by platelet, brick, and blade. Thus, in accordance with Hamilton and Crosser model, solution of elongated and thin shaped particles (high shape factor m) should have larger thermal conductivities, if the ratio of knf/kfis higher than 100. It is also predicted by Colla et al. that the thermal conductivity and viscosity increase with the increase of particle absorption due to which motion of nanofluids decreases. For that reason, the cylindrical‐shaped alumina Al2O3nanoparticles have the higher thermal conductivity chased by platelet, brick, and blade. Timofeeva et al. depicted the conclusion that, when the sphericity of nanoparticles is less than 0.6, the negative concession of heat flow resistance at the solid‐liquid interface increases much faster than the particle shape concession. Thus, the inclusive thermal conductivity of solution starts decreasing less than sphericity of 0.6. However, it is increasing in the case of Hamilton and Crosser model because of the only concession of particle shape parameter m.Nevertheless, flow in this work is one‐directional and one‐dimensional; therefore, the negative concession of heat flow resistance is ignored. Timofeeva et al. reported the model knf/kf=1+(ckshape+cksurface)φ,for calculating the thermal conductivity of nanoparticles. In accordance with this model, ckshapeand cksurfacecoefficients reflecting concessions to the effective thermal conductivity because of particle shape (positive influence) and because of surface resistance (negative effect), respectively. Particle shape coefficient ckshapewas also derived from the Hamilton and Crosser's equation (1962).

Figure 3.

Velocity profiles for different shapes ofAl2O3nanoparticles in water‐based nanofluids whenGr=0.1,N=0.1,M=1,λ=1,K=1,t=5,φ=0.04,ω=0.2.

A comparability of alumina Al2O3in C2H6O2‐based nanofluids with alumina Al2O3in H2Obased nanofluids is shown in Figure 4. It is viewed that the motion of H2O‐based nanofluids is larger than the motion of C2H6O2‐based nanofluids. The thermal conductivity and viscosity of C2H6O2‐ and H2O‐based nanofluids are also perceived by the Hamilton and Crosser model for equal value of φ. This consequence shows that C2H6O2‐based nanofluids have larger thermal conductivity and viscosity than H2O‐based nanofluids.

Figure 4.

Comparison of velocity profiles ofAl2O3in EG and water‐based nanofluids whenGr=0.1,N=0.1,M=1,λ=1,K=1,t=5,φ=0.04,ω=0.2.

The influence of different solid nanoparticles on the motion of different nanofluids is shown in Figure 5. From this figure, it is notable that cylindrical‐shaped Al2O3in C2H6O2‐based nanofluids has the uppermost motion chased by Fe3O4, TiO2, Cu, and Agin C2H6O2‐based nanofluids. This indicates that cylindrical‐shaped silver Ag in C2H6O2‐based nanofluids has the higher viscosity and thermal conductivity assimilated to Cu, TiO2, Fe3O4, and Al2O3in C2H6O2‐based nanofluids. One can view from this consequence that cylindrical‐shaped Agin C2H6O2‐based nanofluids has better quality fluids assimilated to Fe3O4cylindrical‐shaped in C2H6O2‐based nanofluids. This consequence is supported by the Hamilton and Crosser model that the viscosity and thermal conductivity of nanofluids are also influenced by nanoparticles φ, i.e., the viscosity and thermal conductivity increase with the increase in φ. Therefore, motion decreases with the increase in φ. This figure further shows that the viscosity of Al2O3in C2H6O2‐based nanofluids at φis below 0.1, which increases nonlinearly with nanoparticles suspension. This consequence is found similar to the experimental consequence predicted by Colla et al.

Figure 5.

Velocity profiles of different nanoparticles in EG‐based nanofluids whenGr=0.1,N=0.1,M=1,λ=1,K=1,t=5,φ=0.04,ω=0.2.

Different φof nonspherical cylindrical‐shaped alumina Al2O3nanoparticles on the motion of alumina Al2O3in C2H6O2‐based nanofluids is exhibited in Figure 6. It is view from this figure that with the increase of φthe motion of nanofluids is decreased. Due to this reason, the nanofluid becomes more viscous with the increase of φ, which governs to decrease the motion of nanofluids. The thermal conductivity of nanofluids also increases with the increase of φ. The experimental research by Colla et al. also supports these results.

Figure 6.

Velocity profiles for different values ofφofAl2O3in EG‐based nanofluids whenGr=0.1,N=0.1,M=1,λ=1,K=1,t=5,ω=0.2.

Consequences for various values of radiation parameter Nof Al2O3in H2O‐based nanofluids are displayed in Figure 7. It is observed that motion increases with the increase of N.This consequence agrees well with the consequence reported by Makinde and Mhone. Physically, this means that with the increase of N,the amount of heat energy transfers to the fluids also increases.

Figure 7.

Velocity profiles for different values ofNofAl2O3in EG‐based nanofluids whenGr=0.1,M=1,λ=1,K=1,t=5,φ=0.04,ω=0.2.

The graphical outcomes of the motion of nanofluids for several values of magnetic parameter Mof alumina Al2O3in H2O‐based nanofluids are exhibited in Figure 8. Increasing magnetic parameter, M,results in the decrease of the motion of the alumina nanofluids. Increasing perpendicular magnetic field on the electrically conducting fluid imparts to a resistive force called Lorentz force, which is identical to drag force, and upon increasing the value of magnetic parameter, M,the drag force rises which has the tendency to reduce the motion of the nanofluid. The resistive force is maximum near the plates of the channel and minimum in the middle of the plates. Therefore, motion on the alumina nanofluids is maximum in the middle of the plates and minimum at the plates. The motion of alumina nanofluids for various numerical values of Grashof number, Grof Al2O3in H2O‐based nanofluids is shown in Figure 9. It is concluded that an increase in Grashof number, Gr,governs to an increase in the motion of alumina nanofluids. An increase in Grashof number, Gr,rises temperature of alumina in ethylene glycol nanofluids, which govern to an increase in the upward buoyancy force. Therefore, motion in alumina nanofluids increases with Gr,because of the increment of buoyancy force. Figure 10 shows permeability parameter K.It is view that motion of Al2O3in C2H6O2‐based nanofluids increases with increasing values of permeability parameter Kof Al2O3in C2H6O2‐based nanofluids because of the small friction force. Similarly, increasing Kreduces the nanofluids friction within the plate walls and motion of alumina nanofluids enhances. In the second case, Figures 1118 show the flow condition when the left wall is oscillating in its own plane and the right wall is stationary. Under the last conditions, when both plates of the channel are oscillating in their planes (Figures 1926). From all these graphs, we found that they are qualitatively identical but different quantitatively to Figures 210. It can also viewed from Figures 11, 13, and 14 that the motion of Al2O3in C2H6O2‐based nanofluids at the right wall (y= 1) is not equal to zero. It is because of the reason of oscillating right plate of the channel. It should also be investigated from Figures 20, 24, and 26 that only the magnitude of motion of nanofluids is considered; therefore, the negative sign in motion is ignored in these figures and only shows that the motion of Al2O3in C2H6O2‐based nanofluids is reduced.

Figure 8.

Velocity profiles for different values ofMofAl2O3in EG‐based nanofluids whenGr=0.1,N=0.1,λ=1,K=1,t=5,φ=0.04,ω=0.2.

Figure 9.

Velocity profiles for different values ofGrofAl2O3in EG‐based nanofluids whenN=0.1,M=1,λ=1,K=1,t=10,φ=0.04,ω=0.2.

Figure 10.

Velocity profiles of different values ofKofAl2O3in EG‐based nanofluids whenGr=0.1,N=0.1,M=1,λ=1,t=5,φ=0.04,ω=0.2.

Figure 11.

Velocity profiles for different shapes ofAl2O3nanoparticles in EG‐based nanofluids whenGr=0.1,N=0.1,M=1,λ=1,K=1,t=5,φ=0.04,ω=0.2.

Figure 12.

Velocity profiles for different shapes ofAl2O3nanoparticles in water‐based nanofluids whenGr=0.1,N=0.1,M=1,λ=1,K=1,t=5,φ=0.04,ω=0.2.

Figure 13.

Comparison of velocity profiles ofAl2O3in EG‐ and water‐based nanofluids whenGr=0.1,N=0.1,M=1,λ=1,K=1,t=5,φ=0.04,ω=0.2.

Figure 14.

Velocity profiles for different values ofφofAl2O3in EG‐based nanofluids whenGr=0.1,N=0.1,M=1,λ=0.01,K=0.3,t=10,ω=0.2.

Figure 15.

Velocity profiles for different values ofNofAl2O3in EG‐based nanofluids whenGr=1,M=1,λ=0.01,K=0.2,t=10,φ=0.04,ω=0.2.

Figure 16.

Velocity profiles for different values ofMofAl2O3in EG‐based nanofluids whenGr=1,N=0.1,λ=0.001,K=1,t=10,φ=0.04,ω=0.2.

Figure 17.

Velocity profiles for different values ofGrofAl2O3in EG‐based nanofluids whenN=0.1,M=1,λ=0.01,K=0.2,t=10,φ=0.04,ω=0.2.

Figure 18.

Velocity profiles for different values ofKofAl2O3in EG‐based nanofluids whenGr=0.1,N=0.1,M=1,λ=0.01,t=10,φ=0.04,ω=0.2.

Figure 19.

Velocity profiles for different shapes ofAl2O3nanoparticles in EG‐based nanofluids whenGr=1,N=0.1,M=1,λ=0.01,K=1,t=10,φ=0.04,ω=0.2.

Figure 20.

Velocity profiles for different shapes ofAl2O3nanoparticles in water‐based nanofluids whenGr=1,N=0.1,M=1,λ=0.01,K=1,t=10,φ=0.04,ω=0.2.

Figure 21.

Comparison of velocity profiles ofAl2O3in EG‐ and water‐based nanofluids whenGr=1,N=0.1,M=1,λ=0.01,K=1,t=10,φ=0.04,ω=0.2.

Figure 22.

Velocity profiles for different values ofφofAl2O3in EG‐based nanofluids whenGr=1,N=0.1,M=1,λ=0.01,K=1,t=10,ω=0.2.

Figure 23.

Velocity profiles for different values ofNofAl2O3in EG‐based nanofluids whenGr=1,M=1,λ=0.01,K=1,t=10,φ=0.04,ω=0.2.

Figure 24.

Velocity profiles for different values ofMofAl2O3in EG‐based nanofluids whenGr=1,M=1,λ=0.01,K=1,t=10,φ=0.04,λ=0.01,ω=0.2.

Figure 25.

Velocity profiles for different values ofGrofAl2O3in EG‐based nanofluids whenGr=1,M=1,λ=0.01,K=1,t=10,φ=0.04,ω=0.2.

Figure 26.

Velocity profiles for different values ofKofAl2O3in EG‐based nanofluids whenGr=1,M=1,λ=0.01,K=1,t=10,φ=0.04,ω=0.2.

The influence of dissimilar particle shapes on the temperature of Al2O3in H2Oand C2H6O2‐based nanofluids is presented in Figures 27 and 28. The temperature of both types of nanofluids is different for dissimilar shapes because of the various viscosity and thermal conductivity of these nanoparticles. It should be concluded that the influence of thermal conductivity increases with the increase of temperature. However, the viscosity decreases with the increase of temperature. It is view that an elongated shape of nanoparticles inside H2O‐based nanofluids like cylinder and platelet has small temperature due to the larger viscosity and thermal conductivity while the blade shape of nanoparticles has the uppermost temperature because of least viscosity and thermal conductivity. The brick shape nanoparticles suspended fluids are lowest in the temperature range; however, it has low viscosity. This is because of the shear thinning behavior with temperature of nanofluids. Moreover, cylindrical‐shaped nanofluids also show shear thinning behavior. However, the effect is less dominant. All the other dissimilar shapes like platelet and blade show Newtonian behavior and independence of viscosity on shear rate. This shear thinning behavior is also analyzed experimentally by Timofeeva et al.

Figure 27.

Temperature profiles for different shapes ofAl2O3nanoparticles in EG‐based nanofluids whenN=1.5,t=1.

Figure 28.

Temperature profiles for different shapes ofAl2O3nanoparticles in water‐based nanofluids whenN=1.5,t=1.

A comparatively study of Al2O3in H2O‐ and C2H6O2‐based nanofluids is displayed in Figure 29. It is analyzed that both nanofluids are temperature dependent and the variation is found at the same rate for both types of nanofluids. This means that the influence of temperature of nanofluids on the thermal conductivity and viscosity of two types of base nanofluids may cause at the same rate. Figure 30 exhibits the influence of φon the temperature of Al2O3in C2H6O2‐based nanofluids. It is evaluated that with the increase of φtemperature of the nanofluid increases because of the shear thinning nature. The viscosity of cylindrical‐shaped nanoparticles inside water and C2H6O2‐based nanofluids shows shear thinning nature at the highest suspension. This was also experimentally investigated by Timofeeva et al.

Figure 29.

Comparison of temperature profiles ofAl2O3in EG‐ and water‐based nanofluids whenN=1.5,t=1.

Figure 30.

Temperature profiles for different values ofφofAl2O3in EG‐based nanofluids whenN=1.5,t=1.

The graphical consequences of temperature of nanofluids for several values of Nare displayed in Figure 31. It is indicated that the temperature of the cylindrical‐shaped alumina Al2O3nanoparticles in C2H6O2‐based nanofluids shows larger oscillation with the increase of N.It is evaluated in the solution of the problem that temperature of the alumina nanofluids is oscillating and the influence of oscillation is increase with the increase of N.The increasing Nmeans cooler or dense nanofluids or reduce the influence of energy transport to the nanofluids. The cylindrical‐shape nanofluids have temperature dependent viscosity because of the shear thinning nature.

Figure 31.

Temperature profiles for different values ofNofAl2O3in EG‐based nanofluids whent=1.

4. Conclusions

In this chapter, the influence of radiative heat transfer on mixed convection MHD flow of different shapes of Al2O3in C2H6O2, and H2Obase nanofluids in a channel filled with a saturated porous medium is analyzed. The two plates of the channel at finite distance with nonuniform wall temperature are chosen in a vertical direction under the influence of a perpendicular magnetic field. The governing PDEs are solved by the perturbation method for three different flow cases, and analytic solutions are evaluated. The influence of the dissimilar shapes of nanoparticles, namely, platelet, blade, cylinder, and brick of the same volume, on the motion of nanofluids and temperature of nanofluids is examined with different consequences. An elongated shape of nanoparticles inside base fluids like cylinder and platelet results in greater viscosity at the equal volume fraction due to structural limitation of rotational and transitional Brownian motion. The shear thinning nature of cylinder and blade shape of nanoparticles inside H2Oand C2H6O2is also investigated in this research. Viscosities and thermal conductivities of nanofluids are viewed depending on nanoparticle shapes, suspension of volume fraction, and base fluid of solid nanoparticles. The concluded results are as follow:

  1. The motion of nanofluid decreases with the increase of volume fraction of nanoparticles because of the increment in viscosity and thermal conductivity.

  2. The motion of C2H6O2base nanofluid is found smaller than H2Obase nanofluid because the viscosity of base fluid influences the Brownian movement of the nanoparticles.

  3. Elongated nanoparticles inside water and C2H6O2like cylinder and platelet shapes have smaller motion as assimilated to blade and brick shapes of nanoparticles because of greater viscosity.

  4. The motion of the nanofluid decreases with the enhancement of magnetic parameter because of the increase of the resistive type drag force that has the tendency to reduce the motion of the nanofluids.

  5. The motion of the nanofluid also decreases with the increases of thermal Grashof number. The increase of Grincreases the temperature of nanofluids, which governs to an increase in the upward buoyancy force. Therefore, the motion in nanofluids increases with increasing Grbecause of the increment of buoyancy force.

Roman letters
a,bConstants depend on shape of nanoparticles
B0Applied magnetic field
B0Magnitude of applied magnetic field
BTotal magnetic field
b_Induced magnetic field
(cp)sHeat capacity of solid nanoparticles
(cp)fHeat capacity of base fluids
(cp)nfHeat capacity of nanofluids
DRate of strain tensor
dpDiameter of solid nanoparticles
ETotal electric field
eInternal energy per unit volume
expExponential function
FForce
fFunction of temperature and volume fraction, etc.
GrThermal Grashof number
gGravitational acceleration
H(.)Heaviside function
HTotal momentum of the system
IIdentity tensor
iCartesian unit vector in the x‐direction
JCurrent density
J×BLorentz force
jCartesian unit vector in the y‐direction
KDimensionless permeability parameter
ksThermal conductivity of solid nanoparticles
kfThermal conductivity of base fluids
knfThermal conductivity of nanofluids
kbBoltzmann constant
kCartesian unit vector in the z‐direction
MMagnetic parameter
mMass of the flow of fluids
NRadiation parameter
NuNusselt number
nEmpirical shape factors
pPressure
phHydrostatic pressure
pdDynamic pressure
PePeclet number
QHeat generation parameter
Radiant flux vector
qrMagnitude of radiant heat flux
q"Heat conduction per unit area
q"Magnitude of heat conduction per unit area
ReReynolds number
rcRadius of gyration
SSurface of the control volume
TCauchy stress tensor
TTemperature
tTime
uVelocity in x‐direction
U0Reference velocity
VDyadic tensor
VVelocity vector field
VControl volume
VMagnitude of velocity
v0Constant velocity in y‐direction
WWork done
Greek letters
ρsDensity of solid nanoparticles
ρfDensity of base fluids
ρnfDensity of nanofluids
βsVolumetric coefficient of thermal expansion of solid nanoparticles
βfVolumetric coefficient of thermal expansion of base fluids.
βnfVolumetric coefficient of thermal expansion of nanofluids
μsDynamic viscosity of solid nanoparticles
μfDynamic viscosity of base fluids
μnfDynamic viscosity of solid nanofluids
β_Modeling function
ϕVolume fraction of solid nanoparticles
ΨSphericity
ωOscillating parameter
εPerturbed parameter
λWilliamson parameter
α0Mean absorption coefficient
σnfElectrical conductivity of nanofluids
μmMagnetic permeability
Delta function
τ1Skin friction
τViscous stress tensor

Acknowledgments

The authors are grateful to the reviewers for their excellent comments to improve the quality of the present article. The authors would also like to acknowledge the Research Management Center‐UTM for the financial support through vote numbers 4F109 and 03J62 for this research.

Conflicts of interest

The authors declare that they have no conflicts of interest.

Nomenclature

© 2017 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Aaiza Gul, Ilyas Khan and Sharidan Shafie (March 15th 2017). Energy Transfer in Mixed Convection MHD Flow of Nanofluid Containing Different Shapes of Nanoparticles in a Channel Filled with Saturated Porous Medium, Nanofluid Heat and Mass Transfer in Engineering Problems, Mohsen Sheikholeslami Kandelousi, IntechOpen, DOI: 10.5772/67367. Available from:

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