Magnetic field effect on Poiseuille flow and heat transfer of carbon nanotubes along a vertical channel filled with Casson fluid

Applications of carbon nanotubes, single walls carbon nanotubes (SWCNTs) and multiple walls carbon nanotubes (MWCNTs) in thermal engineering have recently attracted significant attention. However, most of the studies on CNTs are either experimental or numerical and the lack of analytical studies limits further developments in CNTs research particularly in channel flows. In this work, an analytical investigation is performed on heat transfer analysis of SWCNTs and MWCNTs for mixed convection Poiseuille flow of a Casson fluid along a vertical channel. These CNTs are suspended in three different types of base fluids (Water, Kerosene and engine Oil). Xue [Phys. B Condens. Matter 368, 302–307 (2005)] model has been used for effective thermal conductivity of CNTs. A uniform magnetic field is applied in a transverse direction to the flow as magnetic field induces enhancement in the thermal conductivity of nanofluid. The problem is modelled by using the constitutive equations of Casson fluid in order to characterize the non-Newtonian fluid behavior. Using appropriate non-dimensional variables, the governing equations are transformed into the non-dimensional form, and the perturbation method is utilized to solve the governing equations with some physical conditions. Velocity and temperature solutions are obtained and discussed graphically. Expressions for skin friction and Nusselt number are also evaluated in tabular form. Effects of different parameters such as Casson parameter, radiation parameter and volume fraction are observed on the velocity and temperature profiles. It is found that velocity is reduced under influence of the exterior magnetic field. The temperature of single wall CNTs is found greater than MWCNTs for all the three base fluids. Increase in volume fraction leads to a decrease in velocity of the fluid as the nanofluid become more viscous by adding CNTs.


I. INTRODUCTION
Non-Newtonian fluids have immense applications in industries such as chemicals, cosmetics, pharmaceuticals.They are attracted for scientists due to their wide applications.Navier Stokes theory is inadequate for such fluids and no single equation can exhibit the properties of all fluids.Thus a number of non-Newtonian fluid models are introduced to explain the characteristics of such fluids.Some of them are power law, second grade, third grade, fourth grade, Brinkman type, micropolar,  12 They presented dual solutions to the problem by comparing the results of Casson fluid with Newtonian fluid numerically by using Matlab bvp4c package.Gentile et al. 13 investigated the transport of nanoparticles in blood vessels with blood considered as Casson fluid (the effect of vessel permeability and blood Rheology), hence contributing in biomedical engineering.Rizwan et al. 14 studied convective heat transfer and MHD effects on Casson nanofluid flow over a shrinking sheet by taking suction/injection effect on the wall.Siddiqui et al. 15 developed a mathematical model to study the transport mechanism of a Casson fluid flow due to metachronal beating of cilia in a cylindrical tube.Nadeem et al. 16 investigated MHD threedimensional boundary layer flow of Casson nanofluid past a linearly stretching sheet with convective boundary condition.MHD stagnation-point flow of Casson fluid and heat transfer over a stretching sheet with thermal radiation was investigated by Krishnendu. 17Reynolds number impact on the flow of electro rheological fluid of a Casson type between fixed sheets of revolution was discussed by Walicka and Falicki. 18The effect of magnetic field and heat source on the steady boundary layer flow and heat transfer of a Casson nanofluid past a vertical exponentially stretching cylinder across its radial direction was discussed by Sarojamma and Vendabai. 19Mustafa and Junaid 20  As compared to other nanostructure materials, CNTs have got the highest conductivity enhancement, opening the door to a wide range of nanotube applications, including use in nanofluids.Their thermal conductivity is much higher than the thermal conductivity of metal nanoparticles or metal oxide nanoparticles.Due to cylindrical carbon molecules origin, CNTs have exceptional mechanical, thermal, electrical and optical properties.They have special thermal properties with very high thermal conductivities.The diameter of CNTs ranges from 1 to 100 nm and lengths in micrometer.They have two hundred times strength and five times elasticity of steel, fifteen times thermal conductivity and 1000 times current capacity of copper.Besides that CNTs are declared the non-hazardous particles for the environment by environmental protection agency.Therefore, these days the researchers are concentrating more on CNTs research to enhance the thermal conductivity as compare to other nanoparticles fluids.Liu et al. 21studied the enhancement of thermal conductivity of ethylene glycol and synthetic engine oil in the presence of MWCNTs.They found that CNTs-ethylene glycol suspensions have higher thermal conductivities than ethylene glycol fluid without CNTs.Similar result was obtained in case of synthetic engine oil.For CNTs-ethylene glycol suspension at a volume fraction of 0.01, thermal conductivity is increased by 12.4% while for CNTs-synthetic engine oil by 30% at a volume fraction of 0.02.Marquis and Chibnate 22 investigated that both types of CNTs (SWCNT and MWCNTs) increased the thermal conductivity of nanofluids and nanolubricants.They used three different types of nanofluids and nanolubricants in their study.The effects of concentration of CNTs and temperature on effective thermal conductivity were investigated by Wen and Ding. 23It was found that effective thermal conductivity increased with increasing concentration of CNTs.Xie et al. 24 studied experimentally the nanofluids containing MWCNTsand their enhanced conductivities.As expected, theyfound that the addition of CNTs into a fluid leads to enhancement of thermal conductivity.The thermal conductivity enhancement increases with the increase in nanotubes loading.Hone 25 and Antar et al. 26 reported that for SWCNTs the thermal conductivity is upto 6,600 W/mK and for MWCNTs is upto 3,000 W/mK.Water driven flow of CNTs in a rotating channel has been investigated by Hussain et al. 27 They observed that SWCNTs show less resistance to the fluid flow as compared to MWCNTs.Khan et al. 28 studied fluid flow and heat transfer of CNTs along a flat plate with Navier slip boundary condition.They found that water-based CNTs have higher densities and thermal conductivities than kerosene and engine oil-based CNTs.The reason for this is the higher thermal conductivity and density of water.Haq et al. 29 studied water based flow in the presence of SWCNT and MWCNT, they observed a higher skin friction and Nusselt number for SWCNT compare to MWCNT.In another study, Haq et al. 30 showed that engine oil based CNT fluid has higher skin friction and heat transfer rate as compared to water and kerosene based CNT fluid.
Recently an immense attention has been paid on coating CNTs on textile fabrics in the field of textile.Another application of CNTs is that in acoustics such as loudspeakers and earphone.Xiao et al. 31 found that very thin CNTs films could emit loud sounds, hence they made practical CNT thin film loudspeakers.They have been suggested as promising absorbents for removal of contaminant in water and wastewater treatment systems.Camilli et al. 32 proposed a three-dimensional carbon nanotube network for water treatment.They demonstrated that a 3-dimensional structure uptakes masses of toxic organic solvent from water about 3.5 times higher than the absorbed by individual CNTs.Moreover, they have a high oil-absorption capacity than the CNT sponges.Halelfadel et al. 33 investigated viscosity of carbon nanotubes water based-nanofluids.They presented experimental result on viscosity of carbon nanotubes considering the influence of temperature and volume fraction.In another study, Halelfadel et al. 34 worked on the efficiency of carbon nanotubes water based nanofluids as coolants.They investigated the effect of low nanoparticles volume fraction (ranging from 0.0055% to 0.278%) on density, thermal conductivity and viscosity of nanofluids.
Kamali and Binesh 35 numerically investigated the convective heat transfer of multi wall CNTsbased nanofluids in a straight tube under constant wall heat flux condition.They found that the heat transfer coefficient is dominated by the wall region due to non-Newtonian behavior of CNT nanofluid.Meyer et al. 36 investigated experimentally the convective heat transfer enhancement of aqueous suspensions of multi-walled CNTs flowing through a straight horizontal tube.They determined the heat transfer coefficients and friction factors are Reynolds number dependent.More exactly, they found that heat transfer was enhanced when comparing the data on a Reynolds Nusselt graph and the increase in viscosity was four times the increase in the thermal conductivity.Kandasamy et al. studied the impact of chemical reaction on Cu, Al2O3, and SWCNTs-nanofluid flow under the slip conditions numerically by using Runge-Kutta-Fehlberg Method with shooting technique.Marangoni convection flow and heat transfer characteristics of water-CNT nanofluid droplets was proposed by Al-Sharafi et al. 38 Kandasamy et al. 39 studied single walled carbon nanotubes on MHD unsteady flow over a porous wedge with thermal radiation with variable stream condition.Xue 40 presented a model for the effective thermal conductivity of carbon nanotube composites.Their model predicts that large length of CNTs plays a key role in the enhancement of their thermal conductivity.Jiang et al. 41 in their experimental work, used CNTs of different diameters and lengths and tested their effect on thermal conductivities.They found that larger the aspect ratio, larger will be the thermal conductivity.An interesting result was found that thermal conductivity of CNTs with smaller diameter is higher than those of larger diameter.Mixed convection occurs in many industrial and technological processes such as chemical processing, nuclear reactors, pollution control in buildings and thermal insulations.X-Hong Ren et al. 42 studied combined convective heat and airborne pollutant removals in a slot vented enclosure under different flow schemes.MHD mixed convection flow in a linearly heated cavity was analyzed by Al-Salem et al. 43 For other studies on mixed convection heat transfer the reader is referred to the references 44-50.
Applications of CNTs in thermal engineering have recently attracted significant attention.However, most of the studies on CNTs are either experimental or numerical and the lack of analytical studies limits further developments in CNTs research particularly in channel flows.From the literature review, it is found that no study has been reported on analytical work for heat transfer analysis of CNTs Casson fluid.In this direction, even numerical or experimental studies for Casson fluid are not available.This work will not only provide a pioneering study but will also help the numerical or experimental investigators to compare their studies with the results reported in this work.
This work aims to study heat transfer analysis in Poiseuille flow of carbon nanotubes (SWCNTs and MWCNTs) with Casson fluid in a vertical channel.The flow is affected by the magnetization of the fluid under the influence of an external magnetic field.The equations are formulated and then transformed to non-dimensionless form.Perturbation method is applied to find the analytical solution of velocity and temperature fields.At the end, skin friction and Nusselt number are evaluated.

II. THEORETICAL MODELS FOR EFFECTIVE THERMAL CONDUCTIVITIES
Many theoretical models are available in literature, which can be used to predict the effective thermal conductivity enhancement of CNTs suspensions.Maxwell 51 proposed an explicit relation for the effective thermal conductivity in terms of the thermal conductivity ratio σ = k CNT k f , and the volume fraction φ : Jeffery 52 and Davis 53 proposed the following two theoretical models for higher order of volume fractions: respectively, where λ = (σ − 1)/(σ + 2).The higher order terms represent pair interactions of randomly dispersed spheres.The above three models give identical predictions of the effective thermal conductivity for low-volume fractions.However those models do not account for the shape factor of the particles.Hamilton and Crosser 54 developed a theoretical model which accounts for the shapes factor of the particles: i) Spherical particles are considered.ii) Accurate to φ 1 , applicable to φ < 1 Jeffery iii) Accurate to order φ 2 ; higher order terms represent pair interactions of randomly dispersed spheres. Devis Accurate to order φ 2 ; higher order terms represent pair interactions of randomly dispersed spheres.ii) f (σ) = 2.5 for σ = 10 ; f (σ) = 0.5 for σ = ∞

Hamilton and Crosser
i) Spherical and non-spherical particles are considered: n=3 for spheres, n=6 for cylinders.ii) Accurate to φ 1 ,applicable to φ < 1 where n is the shape factor of the particle given by n = 3 ϕ ω , where ϕ = 1 for spheres and 0.5 for cylinders and ω ranges from 1 to 2. Hamilton and Crosser (HC) model reduces to Maxwell model when ϕ = 1.Choi 55 shows that for cylindrical shape nanoparticles HC underestimate the experimental results by comparing measurements to the existing theoretical predictions.The enhancement of thermal conductivity is 160 % for 1% vol nanotubes in oil, while that is by theoretical model are not more than 10% as listed in Table I.
Xue 40 observed that the existing models are only valid for spherical or rotational elliptical particles with small axial ratio.Another limitation of those models is that they do not account for the effect of space distribution of CNTs on thermal conductivity.He proposed a theoretical model based on Maxwell theory considering rotational elliptical nanotubes with very large axial ratio and compensating the effects of the space distribution on CNTs We used Xue 40 model to predict thermal conductivity and dimensionless heat transfer rate of nanofluids in our study.We will solve the proposed model analytically by using perturbation technique.In the end, we will present the physical interpretation of our results.

III. FORMULATION AND SOLUTION OF THE PROBLEM
Consider MHD flow in a vertical channel of width a, with static walls at y = 0 and y = a, with heat transfer in a Casson fluid (three different fluids i.e.Water, Engine oil, Kerosene oil-based nanofluids) containing SWCNTs and MWCNTs.The flow is assumed to be incompressible and unsteady.A uniform magnetic field of strength B 0 is applied in a perpendicular direction to the flow (along xaxis).Mixed convection is generated by external pressure gradient together with buoyancy force.The viscous dissipation effect is neglected in the energy equation.In vertical channel, T 0 and T a show lower and upper plate temperatures.The Reynolds number is assumed small.The governing equations of momentum and energy are as under: where u = u(y, t), T = T (y, t) , ρ nf , µ nf , σ nf , β T g , ρc p nf , k nf , q r , are respectively fluid velocity in the x− direction, temperature, density, the dynamic viscosity, electrical conductivity of the base fluid, volumetric thermal expansion coefficient, gravitational acceleration, heat capacitance of nanofluids, thermal conductivity of nanofluid and radiative heat flux.The corresponding boundary conditions are: The external pressure gradient in the flow direction is taken as − ∂p ∂x = λεexp(iωt) , where λ is constant and ω is the oscillation frequency.As in the following, Xue model 40 is used in this problem for thermal conductivity and dynamic viscosity.
The density ρ nf , thermal expansion coefficient (ρ β) nf , heat capacitance ρc p nf , thermal conductivity σ nf , are derived by using the relations given by Refs.20 and 37.
where φ is the nanoparticles volume fraction, ρ f and ρ CNT is the density of the base fluid and CNTs, the volumetric coefficient of thermal expansions of carbon nanotubes and base fluids are denoted by β CNT and β f respectively, (c p ) CNT and (c p ) f is the specific heat capacities of CNTs and base fluids at constant pressure.The radiative heat flux is given by: where α is the radiation absorption coefficient.Substituting Eq. ( 13) into Eq.( 7) yields: Introducing the following dimensionless quantities into Eqs.(6, 7, 8, 9 and 14) and in view of the above relations, we get ( * Symbol is dropped for convenience) where Here Re, M, Gr, Pe, N, Ri are the Reynolds number, the magnetic parameter, the thermal Grashoff number, Peclet number and the radiation parameter and Richardson number respectively.For the sake of simplification other constants are used as in Eq. (20).

V. GRAPHICAL RESULTS AND DISCUSSION
A detailed analysis is made on the effect of MHD mixed convection Poiseuille flow of carbon nanotubes with Casson fluids inside a vertical channel.Both the flow and heat transfer analysis of single and multiple walls CNTs suspended in three different types of base fluids (Water, Kerosene and engine Oil) have been investigated.The governing partial differential equations with associated boundary conditions in non-dimensional form are analyzed for analytic solutions using perturbation method.Solutions for temperature and velocity are obtained on the basis of Xue model. 40For different parameters, the physics of the problem is discussed graphically and a detailed analysis is performed.The shapes of single and multiple walls carbon nanotubes are given in Fig. 1.Thermophysical properties of base fluids and CNTs are given in Tables II and III provides variation of thermophysical properties of nanofluids with solid volume fraction of CNTs.

A. Temperature field
The temperature of the flow suffers a notable change with the variation of volume fraction φ and radiation parameter N. The variation in temperature profile with volume fraction and radiation parameter is shown in Figs.2-4.An increase in temperature is observed with the increase in radiation parameter N for all the three types of base fluids because when N increases it increases conduction from channel to the fluid hence an increase occurs in temperature of the fluid.Temperature increases with the increase in volume fraction.It is due to the fact that thermal conductivity increases with the increase of the addition of solid CNTs which in turn enhances temperature.The temperature of single wall CNTs is greater than multiple wall CNTs.Engine oil/kerosene oil-based nanotubes have greater temperature as compared to water-based nanotubes because of higher thermal conductivity and greater density of water.

B. Velocity field
The velocity of carbon nanotubes nanofluid is detected to vary more or less with the variation of flow parameters shown graphically.The major factors influencing the velocity are Casson parameter, magnetic and radiation parameter, volume fraction of nanoparticles, Reynolds number and Richardson number.Velocity of MWCNTs is observed to be less than that of SWCNTs.It is due to the fact that SWCNTs has greater density as well as greater thermal conductivity than MWCNTs which is the factor of enhancement in heat absorption and hence an increases occurs in velocity.
Figures 5-7 highlight the effect of magnetic and Casson parameter simultaneously on velocity profile.Almost identical pattern is observed for single and multiple wall CNTs for the same values of M. The behavior of velocity near the channel walls and center is not similar.It shows that for all the three base fluids (Water/Engine oil/kerosene oil), in the absence of magnetic field (that is we considered M=0.000001 very small, negligible value) velocity is higher and it decreases with increases magnetic field.Although, magnetic field can induce current in the conductive fluid, but it create a resistive force namely, Lorentz force which reduces the fluid flow.The velocity is reduced under influence of the exterior magnetic field.It is interesting that velocity is maximum along the centerline and minimum at the walls.A similar behavior of M on velocity is observed by Nadeem et al. 16 and Aaiza et al. 49  density as compared to MWCNTs but they have greater thermal conductivity as shown in Table II, which leads to enhancement of heat absorption, hence their velocity is greater than MWCNTs.The diameter and length of CNTs also influence their thermal conductivities as Jiang et al. 41 found in their work.They found that the thermal conductivities of nano-refrigerant whose diameter is 15nm are much higher than those whose diameter is 80nm.Smaller diameter means larger specific surface of CNTs and larger specific area stands for more obvious Brownian movement which is an important factor of increasing thermal conductivity of nanofluid.Larger specific area means there are more liquid molecules close to the surface of CNT if volume fraction is same.These liquid molecules can make a layer structure, called Interfacial layer, which can increase their thermal conductivity.Smaller diameter means thicker interfacial layer and greater thermal conductivity enhancement.Similarly for CNTs with same diameters, the larger the aspect ratio of CNT, the higher their thermal conductivity is.But the influence of diameter is more as compared to aspect ratio.Figure 8-10 show the effect of radiation parameter and volume fraction simultaneously on fluid velocity.In the absence and presence of radiation parameter, velocity decreases with increasing values of volume fraction because the fluid becomes more viscous by adding CNTs but velocity is same for both single and multiple wall carbon nanotubes.Velocity exhibits a decrease by increasing radiation parameter.Velocity for multiple CNTs is less than that of single wall CNTs.At the boundaries velocity is zero for both types of CNTs.Greater Reynolds number stands for greater inertial forces and lesser viscous force making the fluid less viscous thus fluid flows with a greater velocity.Greater Richardson number stands for greater Grashoff number which increase temperature gradiant and buoyancy force (convection), hence leads to an increase in velocity.Velocity of multiple wall CNTs is slightly less than single wall CNTs.

C. Skin-friction
Table IV

D. Rate of heat transfer
Different values of Nusselt numbers which measures the rate of heat transfer are shown in Table IV.Heat transfer rate increases with volume fraction φ and radiation parameter N, its different values are shown in Table IV.The same pattern is followed for both types of CNTs and all the three base fluids (water/kerosene-oil/Engine-oil).Nusselt number is the ratio of convection to conduction so here it means that convection increases with increase in φ and radiation parameter.As by adding more CNTs in the base fluid, increases its thermal conductivity and hence heat transfer get enhanced.

VI. CONCLUSIONS
In this attempt, the effect of single and multiple wall CNTs on the thermal conductivity of Casson nanofluids is observed in the presence of MHD where water/kerosene-oil/Engine-oil was used as base fluids.The governing equations are formulated, non-dimensionalized and then solved by perturbation technique.We came up with the following outcomes: Increase in volume fraction causes a decrease in velocity of the fluid as the nanofluid becomes more viscous by adding CNTs.Velocity increases with increase in Reynolds number and Richardson number because of greater Grashoff number (thermal bouncy force) and less viscous forces.Velocity increase with increase in radiation parameter due to increase in heat transfer to the fluid.Temperature increases with increase in radiation parameter and volume fraction because of enhancement in thermal conductivity.The temperature of single wall CNTs is greater than multiple wall CNTs for all the three base fluids.Engine oil and kerosene oil-based nanotubes have greater temperature as compared to waterbased nanotubes because water has a greater density.

015036- 2 Aman
et al.AIP Advances 7, 015036 (2017) Jaffery, Walters' B, Maxwell, Oldroyd-B, Burgers and generalized Burgers models.However, there is another popular model namely, Casson model.Casson 1 introduced this model for the first time to predict flow behavior of pigment oil suspensions of the printing ink type.Casson fluid is a shear thinning liquid which is assumed to have infinite viscosity at zero rate of shear, a yield stress below which no flow occurs, and a zero viscosity at an infinite rate of shear. 2 Examples of Casson fluids are jelly, tomato sauce, honey, soup and human blood.Eldabe and Salwa 3 have studied the Casson fluid for the flow between two rotating cylinders.Boyed et al. 4 investigated the Casson fluid flow for the steady and oscillatory blood flow.Swati et al. 5 surveyed numerically Casson fluid flow over an unsteady stretching surface.Pramanik 6 studied Casson fluid flow and heat transfer past an exponentially porous stretching surface in the presence of thermal radiation.Noreen 7 analyzed the influence of magnetic field on peristaltic flow of a Casson fluid in an asymmetric channel.Farhad et al. 8 studied closed form solutions for unsteady free convection flow of a second grade fluid over an oscillating vertical plate.Hussanan et al. 9 investigated the unsteady boundary layer flow and heat transfer of Casson fluid over an oscillating plate with Newtonian heating.Asma et al. 10 studied unsteady magnetohydrodynamic (MHD) free convection flow of Casson fluid past an oscillating vertical plate embedded in a porous medium.Imran et al. 11 studied the effects of slip on free convection flow of Casson fluid over an oscillating vertical plate.Heat and mass transfer in Casson fluid over an exponentially permeable stretching surface was studied by Raju et al.

FIG. 3 .
FIG. 3. Temperature profile for different values of N and φ in engine oil-based nanofluid for SWCNT and MWCNT when N = 1.5, 2.2.

Figure 11 -
13, illustrates the effect of Reynolds number and Richardson number on velocity profile.It is observed that velocity increases with increase in Reynolds and Richardson number.

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et al.AIP Advances 7, 015036 (2017)Rate of heat transfer increases with increase in radiation parameter and volume fraction because of enhancement in thermal conductivity.Skin friction decreases with increasing volume fraction for small values of β while increases with volume fraction for β = 1, 2 .Skin friction values for water-based nanotubes is greater as compared to kerosene oil and engine oil-based nanotubes.The thermal conductivity of CNTs nanofluids increases with volume fraction of CNTs as well as with their diameter and length. )

TABLE I .
Conventional models for effective thermal conductivities of solid/liquid suspensions. ) ) ) )

TABLE II .
Thermophysical properties of different base fluids and CNTs.(Hone et al. 25 ; Antar et al. 26 )

TABLE III .
Variation of thermophysical properties of nanofluids with solid volume fraction of CNTs.

TABLE IV .
Variation of and Skin friction with φ for different values of Casson parameter.