Entropy Generation in MHD Radiative Flow of CNTs Casson Nanofluid in Rotating Channels with Heat Source/Sink

Kumam, Poom; Shah, Zahir; Dawar, Abdullah; Rasheed, Haroon Ur; Islam, Saeed

doi:https://doi.org/10.1155/2019/9158093

Mathematical Problems in Engineering

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Abstract Introduction Results and Discussion Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 9158093 | https://doi.org/10.1155/2019/9158093

Entropy Generation in MHD Radiative Flow of CNTs Casson Nanofluid in Rotating Channels with Heat Source/Sink

Poom Kumam,^1,2,3Zahir Shah,⁴Abdullah Dawar,⁵Haroon Ur Rasheed,⁶and Saeed Islam²

Academic Editor: Chaudry M. Khalique

Received06 Oct 2018

Revised28 Nov 2018

Accepted31 Jan 2019

Published26 Mar 2019

Abstract

We presented the applications of entropy generation for SWCNTs and MWCNTs based on kerosene oil for Casson nanofluid flow by rotating channels. Kerosene oil has advanced thermal conductivity and exclusive features and has a lot of practical uses due to its unique behavior. That is why we have used kerosene oil as a based fluid. For the entropy generation second law of thermodynamics is applied and implemented for the nanofluid transport mechanism. In the presence of magnetic field, the effects of thermal radiations and heat source/sink on the temperature profiles are studied. The fluid flow is supposed in steady state. With the help of suitable similitude parameters, the leading equations have been transformed to a set of differential equations. The solution of the modeled problem has been carried out with the homotopic approach. The physical properties of carbon nanotubes are shown through tables. The effects of the imbedded physical parameters on the velocities, temperature, entropy generation rate, and Bejan number profiles are investigated and presented through graphs. Moreover, the impact of significant parameters on surface drag force and heat transfer rate is tabulated.

1. Introduction

Nanofluid is the potential heat transfer fluid which is one of the fundamental fragments of nanotechnology. Engine transmission oil, boiler exhaust flue gas recovery, cooling of electronics, nuclear systems cooling, nanofluids in drilling, defense, high-power lasers, space, biomedical applications, microwave tubes, drilling, thermal storage, lubrications, drag reductions, solar water heating, in diesel electric generator as jacket water coolant, heating and cooling of buildings, engine cooling, nanofluids in transformer cooling oil, cooling of welding, refrigeration (chillers, domestic refrigerator), etc. are the applications of carbon nanotubes (CNTs). These applications got the researchers attention to work on CNTs. Choi [1] has presented the idea of nanofluids for the first time by dipping the nanometer sized particle into the base fluid. The single-walled carbon nanotubes (SWCNTs) have higher heat transfer rate and surface drag force than multiwalled carbon nanotubes (MWCNTs) reported by Haq et al. [2]. Liu et al. [3] studied the synthetic engine oil and ethylene glycol in the presence of MWCNTs. They claimed that ethylene glycol with CNTs has higher thermal conductivity. Nadeem and Lee [4] presented the concept of nanofluid past over an exponentially stretching sheet. The boundary layer flow of nanoparticles over a stretching/shrinking surface is studied by Nadeem and Lee [5–8]. The mathematical model for the examination of nanofluids flow was established by Boungiorno [9]. Sheikholeslami [10–15] experimentally and theoretically studied the properties, applications, and impacts of nanofluids. Using mesoscopic approach, Sheikholeslami and Rokni [16–19] examined simulation of nanofluids and CuOeH2O nanofluid in a curved porous enclosure. Elias et al. [20] have inspected the impact of shape on heat transmission.

As the time passes the geometry of the problem translated to more complex nature. One factor of the complexity of the problem geometry is the magnetic effect. Fluids coming under this umbrella are called magneto fluids, for example, liquid metals, plasmas, salt water, and electrolytes. Crystal growth, cooling of devices, magnetic drug targeting, and electromagnetic casting are the real-world uses of magnetic hydrodynamics (MHD) in the field of engineering and technology. Magnetic hydrodynamics (MHD) was proposed by Hannes Alfven [21]. Hayat et al. [22] studied the Magneto hydrodynamic flow of Casson nanofluid effects past over a stretched surface. Ali et al. [23] examined the blood flow of Casson fluid under the impact of Magneto hydrodynamics (MHD) in axisymmetric cylindrical tube. Shah et al. [24–26] examined the MHD nanofluid in rotating systems. Dawar et al. [27] examined the MHD carbon nanotube Casson nanofluid and radiative heat transfer in rotating channels. Kumar et al. [28] examined the MHD Williamson fluid over a curved sheet under the impact of nonthermal heat source/sink. Kumaran and Sandeep [29] studied the parabolic flow of MHD Casson and Williamson fluids under the impact of thermophoresis and Brownian motion parameters. The impact of Brownian motion and thermophoresis on bioconvective flow of nanoliquids has been examined by Kumar et al. [30]. The other related studies of Kumar et al. can be seen in [31–36].

Casson fluid model characterizes a shear retreating fluid which is presumed to obligate infinite viscosity at zero rate of shear stress. This famous model after the initiation becomes the main point of interest for researchers. Development has been made in model by adding extra agents to the geometry of the modeled problem. Recently, Mehmood et al. [37, 38] numerically inspected Casson micropolar fluid over a stretched sheet with internal heat transmission. Singh Megahe et al. [39] examined the film flow of Casson fluid in the existence of varied heat flux using slip velocity. Abolbashari et al. [40] have studied the Casson nanofluid with entropy generation. The study about the fundamental models of CNTs can be read in [41–45].

The goal of this research work is to examine the nanoparticles of SWCNTs and MWCNTs Casson fluid between two rotating parallel plate. The kerosene oil is taken as base fluid. The solution of the modeled problem has been carried out with the homotopic approach. Homotopy analysis method (HAM) is used due to better results. Many researchers [46–55] used HAM. The influence of thermal radiations, magnetic parameter, and heat source/sink is taken into account. The effect of all embedded parameters on velocities and temperature profiles has been studied graphically and discussed in detail.

2. Some Basic Thermal Conductivities and Fluid Models

Model of thermal conductivity for CNTs proposed by Maxwell [41] defined aswhere is known as nanoparticle volume fraction.

Model of thermal conductivity for CNTs proposed by Jeffery [42] is the form:Davis [43] modified Jeffery’s model asHamilton and Crosser [44] proposed a new model defined asModels (1)-(3) did not depend on shape of particles and deliver the approximations of thermal conductivity for small . On the other hand, model (4) depends on the shape of particles. The symbol in (4) represents shape factor of the particles. Recently, Xue [45] proposed a model of a very large axial ratio recompensing the CNTs space distribution.The rheological model for Casson fluid is defined asIn (1)-(6), . , , represent the nanoparticle volume fraction, base fluid, the nanofluid, Cauchy stress tensor, dynamic viscosity, components of strain rate, the yield stress of fluid and the critical value of .

3. Mathematical Modeling

Assume the flows of CNTs nanofluid based on kerosene liquid between two parallel (lower and upper) plates which are kept at the distance . The nanoscale materials (SWCNTs and MWCNTs) are used in this model. The mechanism is examined through the thermal radiation and heat transportation. The constant angular velocity rotates around y-axis. The three possible cases which specify that both upper and lower plates rotated in the same direction specify that both the plates rotated in opposite directions and for the static case. The lower plate is moving quickly with a velocity than the upper plate. A three dimensional coordinate system is chosen such that x is parallel to the plate, y is normal to the plate, and z is normal to xy-plane. The upper and lower plates are kept at and respectively. With the help of two opposite forces with same magnitude, the lower plate is being kept stretchable so the position cannot change. The fluid flow and heat transfer is assumed in steady state which is incompressible, laminar, and stable. The magnetic field is applied in y-direction with which the fluid is rotating as shown in Figure 1 [46].

The governing equations for the state of problem are as follows [46]:where , and represent the modified pressure, electrical conductivity, and dynamic viscosity, respectively. The absenteeism of signifies the mesh cross flow beside z-axis and is the Casson fluid parameter.

The heat transfer phenomenon is specified as [46]:Here is the fluid temperature, is the thermal diffusivity, and is the radiative heat flux, respectively.

The radiative heat flux is defined aswhere and indicate the absorption coefficient and Stefan Boltzmann constant, respectively.

Since , (11) becomes

The density, heat capacity, dynamic viscosity, and thermal conductivity of the nanofluid are mathematically represented as follows [47–50]:where signifies the carbon nanotubes.

is the temperature at lower wall and is the temperature at upper wall which is taken as .

The above system of equations has the following boundary conditions.In (15), at the upper wall of the channel, is the uniform suction/injection velocity. If , it is called uniform suction velocity and if , it is called uniform injection velocity.

For the proposed problem we have defined the following similarities variables:In view of (8)-(10) and (13), the nondimensional system of equations iswhere the relevant boundary conditions areIn (17)-(20), is the suction and injection parameter. is for the case suction parameter and if is for the case of injection. signifies the Reynolds number, represents the rotation parameter, signifies the Prandtl number, indicates the thermal radiation parameter, represents the magnetic parameter, and signifies the heat generation or absorption coefficient.

The equations for the surface drag force and heat transfer rate are defined as follows [51]:

4. Expression for Entropy Analysis and Bejan Number

For the stated problem of nanofluid, the local entropy rate per unit volume is [52]where and is related to viscous dissipation.

In our caseThe characteristic entropy generation rate is defined asThe entropy generation is defined asTo calculate the entropy generation , from (25), (26) with (16), implementing in (27), we obtainwhere , and are the Reynolds number, Brinkman number, dimensionless temperature, and Hartmann number, respectively, whose expressions are given byThe Bejan number is defined asor

5. Solution by HAM

We use homotopy analysis method (HAM) to solve (17)-(19) with (20) by the succeeding process.

The primary assumptions are as follows:The linear operators , and are chosen such thatwhich have the propertieswhere .

The consequences of nonlinear operators , and are indicated as follows:-order problems areThe equivalent boundary conditions arewhere is the emerging parameter. In case of and , we haveExpanding and by Taylor’s serieswhereAs the series (44) converges at , using in (44), we getThe -order problem gratifies the following:The equivalent boundary conditions therefore becomeHerewhere

5.1. HAM Solution Convergence

The assisting parameters , and appear when we calculate the series solution of velocities and temperature functions by using HAM. These parameters lead the convergence of the modeled problem. Figure 2 is schemed to see the convergence of the problem. The velocity function converges in the region , the velocity function converges in the region , and the temperature function converges in the region , respectively.

6. Results and Discussion

To discuss the impact of embedded parameters on velocities , temperature , entropy generation rate , and Bejan numbers , Figures 3–21 are plotted. These parameters are Reynolds number (), Rotation parameter (), magnetic parameter , Casson parameter , Prandtl number , thermal radiation parameter , and heat source/sink . Figure 1 shows physical shape of the nanofluid flow. Figure 2 is plotted to see the convergence region of modeled problem.

6.1. Velocities and Temperature Profiles

Figures 3–5 are plotted to observe the effect of on and . Reynolds is a dimensionless number used to calculate the flow pattern in different situation of the fluid flow. At low Reynolds numbers, the fluid flow tends to laminar while at low Reynolds numbers, the fluid flow tends to turbulent flow. From here, we observed that with the escalating values of the velocities and temperature profiles decrease. This is due to the stretching of the lower plate. The effect of () on velocities profiles is shown in Figures 6-7. From Figure 6, we observed the increasing behavior with the increasing rotation parameter while the opposite result was observed in Figure 7. It is interesting to mention that in the absence of rotation parameter the problem reduces to 2-dimensional flow in a channel. Figures 8-9 are plotted to observe the impact of parameter on velocities profiles. According to Lorentz force theory, the magnetic field parameter has an inverse effect on profiles, that is, the upsurges in magnetic field parameter the profiles reduce. Figures 10-11 depict the impact of on profiles. In practice, the increasing results in the rise in dynamic viscosity that creates a resistance in the flow and a decline in fluid velocity thereof. Therefore, the increasing in Casson fluid parameter reduces the velocities profiles. Figure 12 is schemed to observe the influence of on temperature profile . Physically, nanofluids have large thermal diffusivity with small and vice versa. Therefore, the liquid temperature decreases. From here we observed that the growing shows decline in . Figure 13 is schemed to observe the impact of on . Thermal radiations have leading rules in heat transmission, when the coefficient of convection heat transmission is small. From here we observed that the increasing shows a rapid increase in . Figure 14 specifies the impact of on . The porosity parameter has commanding character in fluid motion. Physically, the higher values of porosity parameter improve the porous surface which resists to the flow motion and as a result the motion of the fluid reduces. From here we see that decreases with increasing .

6.2. Entropy Generation and Bejan Number

Figures 15–21 are plotted to see the influences of emerging parameters for both SWCNTs and MWCNTs based on kerosene nanoliquids on entropy generation and Bejan number . Generally, the entropy generation rate is more for SWCNTs as compared to MWCNTs. Figure 15 shows the impact of on . Physically, due to magnetic parameter Lorentz force generates and decreases velocity of the flow and thus entropy generation declines. Figure 16 expresses the impact of on . With high Reynolds number, hectic motion arises for the reason that as escalates, the fluid moves more upsettingly and hence contribution of the fluid fraction and heat transfer on entropy result tends to escalation in entropy generation rate. Figure 17 shows the influence of on . The entropy generation rate escalates with the escalation in . In addition, an increase in created by fluid friction and joule dissipation happens with the escalating values of . Figure 18 shows the influence of on . From the figure we see that at and the escalating values of show increasing behavior in . But at the mean position (i-e ), the radiation parameter has no impact on . It is due to the stretching of the lower plate. Figure 19 shows the influence of on . It is clear from the figures that the accelerating values of shows decline in . Figure 20 depicts the influence of on . The increasing values of escalate the . Figure 21 indicates the influence of on . From the figure we see that at and the increasing values of show increasing behavior in , while at the radiation parameter does not show increasing or decreasing behavior. It is due to the stretching of the lower plate.

6.3. Tables Discussion

Table is schemed to observe the impact of , and on skin fraction coefficient . From Table 1, we observed that the surface drag force increases with the escalation in and while it reduces with the escalation in and . Table 2 is schemed to observe the impact of , and on local Nusselt number . From here it is observed that the heat transfer rate increases with the escalation in and while it reduces with the escalation in and . In Tables 3–5, some physical and thermal properties of CNTs with thermal conductivity are presented.

7. Conclusion

Entropy generation examination in two-dimensional mixed convection flow of nanofluid over an inclined stretching sheet has been analyzed. The kerosene liquid is taken as base fluid. The impact of magnetohydrodynamics, thermal radiation, and heat/source has been examined. The transformed system of equations has been solved by homotopy analysis method. On the achieved study, the key remarks are listed below.(i)The velocity function escalates with the escalation in rotation parameter and reduces with rise in Reynolds number, magnetic field parameter, and Casson fluid parameter.(ii)The velocity function is reduced with escalation in Reynolds number, rotation parameter, magnetic field, and Casson fluid parameter.(iii)The temperature function is increased with the escalation in thermal radiation parameter while reduced with the escalation in Reynolds number, Prandtl number, and heat source/sink.(iv)The entropy generation is increased with the escalation in Reynolds number, Brinkman number, and thermal radiation parameter while reduced with the escalation in magnetic field parameter.(v)The Bejan number is increased with the increase in Brinkman number and thermal radiation parameter while reduced with the upsurge in magnetic field parameter.(vi)The surface drag force increases with the escalation in Reynolds number and Casson fluid parameter while reduces with the escalation in rotation and magnetic parameter.(vii)The heat transfer rate increases with the escalation in Reynolds number and heat source/sink while reduces with the escalation in thermal radiation parameter and Prandtl number.

Nomenclature

:	Constants
:	Suction/injection
:	Magnetic field
:	Specific heat
:	Skin friction coefficient
	Distance between the plates
	Thermal conductivity ()
:	Stefan Boltzmann constant
	Rotation parameter
	Nusselt number
	Origen
:	Fluid pressure
:	Mesh cross flow
	Prandtl number
:	Heat flux
:	Radioactive heat flux
	Radiation parameter
	Fluid temperature
:	Temperature at upper wall
:	Temperature at lower wall
	Velocity components
:	Stretching velocity
	Coordinates

Greek Letters

	Casson parameter
	Angular velocity
:	Thermal diffusivity
	Similarity variable
:	Dynamic viscosity
:	Base fluid density
:	Electrical conductivity of nanofluid
	Nanoparticle volume fraction
:	Absorption coefficient
	Assisting parameter.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2019 Poom Kumam et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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