Thermophysical properties of different base fluid and CNTs.

## Abstract

This chapter discusses the magnetic field effects on the flow of Cattanneo-Christov heat flux model for water-based CNT suspended nanofluid over a stretching sheet. According to the authors, knowledge idea of Cattanneo-Christov heat flux model for water-based CNT suspended nanofluid is not explored so far for stretching sheet. The flow equations are modeled for the first time in the literature transformed into ordinary differential equations using similarity transformations. The numerical solutions are computed using shooting technique and compared with the literature for the special case of pure fluid flow and found to be in good agreement. Graphical results are presented to illustrate the effects of various fluid flow parameters on velocity, heat transfer, Nusselt number, Sherwood number, and skin friction coefficient for different types of nanoparticles.

### Keywords

- boundary layer flow
- nanofluid
- stretching sheet
- Cattanneo-Christov heat flux model
- numerical solution

## 1. Introduction

From recent few decades, heat transfer enhancement of the nanofluid has turned out to be a topic of main interest for the researchers and scientists. The word “nanofluid” was derived by Choi [1]. He defines a liquid suspension comprising ultrafine particles whose diameter is less than 50 nm. Xuan and Roetzel [2] investigated the mechanism of heat transfer enhancement of the nanofluid. According to them, the nanofluid is a solid-liquid mixture in which metallic or nonmetallic nanoparticles are suspended. The suspended ultrafine particles change transport properties and heat transfer performance of the nanofluid, which exhibits a great potential in enhancing heat transfer. They found that the reduced Nusselt number is a decreasing function of each nanofluid parameters. Khanafer et al. [3] discussed buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. The natural convective boundary-layer flow of a nanofluid over a vertical plate is studied analytically by Kuznetsov and Nield [4]. Boundary layer laminar nanofluid flow over the stretching flat surface has been investigated numerically by Khan and Pop [5]. They show that the reduced Nusselt number is a decreasing function of each dimensionless number, while the reduced Sherwood number is an increasing function of higher Prandtl number. Ebaid and his co-authors [6–10] present boundary-layer flow of a nanofluid past a stretching sheet with different flow geometries and with different conditions. Wang [11] discussed free convection on a vertical stretching surface. Scaling group transformation for MHD(Magneto hydrodynamic) boundary-layer flow of a nanofluid past a vertical stretching surface in the presence of suction/injection was discussed by Kandasamy et al. [12].

Fourier [13] was the first who discussed the heat transfer phenomenon in 1822. The equation presented by him was parabolic in nature and has draw back that in initial disturbance is felt instantly throughout the whole medium. Cattaneo [14] modifies the “Fourier law of heat conduction in which he added the thermal relaxation term. The addition of thermal relaxation time causes heat transportation in the form of thermal waves with finite speed.” Christov [15] in this contest discussed Oldroyd upper-convected derivative as an alternative of time plagiaristic to complete the material-in variant formulation. This model is known as Cattaneo-Christov heat flux model. Tibullo et al. [16] described the uniqueness of Cattaneo-Christov heat flux model for incompressible fluids. Mustafa [17] presented the Cattaneo-Christov heat flux model for Maxwell fluid over a stretching sheet. According to him, velocity is inversely proportional to the viscoelastic fluid parameter. Further, fluid temperature has inverse relationship with the relaxation time for heat flux and with the Prandtl number. Very recently, Salahuddin et al. [18] discussed MHD flow of Cattanneo-Christov heat flux model for Williamson fluid over a stretching sheet with variable thickness. They solved nonlinear problem numerically by using implicit finite difference scheme known as Keller box method. They observed that large values of wall thickness parameter and Weissenberg number are suitable for reduction in velocity profile. For further details, see Refs. [11, 12, 19–32].

The aim of this chapter is to discuss the magnetic field effects on the flow of Cattanneo-Christov heat flux model for water-based CNT suspended nanofluid over a stretching sheet. Because according to the authors, knowledge idea of Cattanneo-Christov heat flux model for water-based CNT suspended nanofluid is not explored so far for stretching sheet. The flow equations are modeled for the first time in the literature transformed into ordinary differential equations using similarity transformations. The numerical solutions are computed using shooting technique and compared with the literature for the special case of pure fluid flow and found to be in good agreement. Graphical results are presented to illustrate the effects of various fluid flow parameters on velocity, heat transfer, Nusselt number, Sherwood number, and skin friction coefficient for different types of nanoparticles.

## 2. Formatting mathematical model

We discuss the two-dimensional nanofluid flow over a stretching sheet with water as based fluids surrounding single- and multi-wall CNTs. The flow is supposed to be laminar, steady, and incompressible. The base fluid and the CNTs are usual to be in updraft stability. Sheet is whispered to be stretched with the dissimilar velocity *U*_{w}, *V*_{w} along the *x*-axis and *y*-axis, correspondingly. We have taken the invariable ambient temperature *T*_{∞}. Supplementary new heat model named as Cattanneo-Christov heat flux model is considered to analyze heat transfer phenomena. The *x*-axis is taken along the sheet, and *y*-axis is chosen normal to it. Magnetic field of strength *B*_{0} is applied normal to the sheet (as shown in Figure 1).

With the above analysis, the boundary layer equations for the proposed model, i.e., continuity, momentum, and energy equations, can be written as follows:

where *u* and *v* are the velocity components along *x* and *y* directions, respectively, *T* is the temperature of the fluid, *B*_{0} is the magnitude of magnetic field, and *q* is the heat flux. Equation (3) is the Cattaneo-Christov flux model and has the following form:

where *λ*_{2} is the thermal relaxation time. Eliminating *q* from Eqs. (3) and (4) gives as follows:

Further, *ρ*_{nf} is the effective density, *μ*_{nf} is the effective dynamic viscosity, (*ρc*_{p})_{nf} is the heat capacitance, *α*_{nf} is the effective thermal diffusibility, and *k*_{nf} is the effective thermal conductivity of the nanofluid, which are defined as follows:

where *μ*_{f} is the viscosity of base fluid, *φ* is the nanoparticles fraction, (*ρC*_{p})_{f} is the effective heat capacity of a fluid, (*ρC*_{p})_{CNT} is the effective heat capacity of a carbon nanotubes, *k*_{f} and *k*_{CNT} are the thermal conductivities of the base fluid and carbon nanotubes, respectively, *ρ*_{f} and *ρ*_{CNT} are the thermal conductivities of the base fluid and carbon nanotubes, respectively.

Corresponding boundary conditions are as follows:

where *T*, *T*_{w} and *N* are the ambient, wall fluid temperature, and slip parameter, respectively.

Introducing the following similarity transformations, we have

Making use of Eqs. (6, 8) in Eqs. (1–5), we have

where *γ* = *aλ*_{2} is the non-dimensional thermal relaxation time, and

The quantity of practical interest, in this study, is the skin friction coefficient *c*_{f} and Nusselt number *Nu*_{x}, which is defined as follows:

where *q*_{w} is the heat flux and *K*_{nf} is the effective thermal conductivity. Using variables (8), we obtain:

## 3. Numerical scheme

The nonlinear ordinary differential equations (9)–(10) subject to the boundary conditions (11) have been solved numerically using an efficient Runge-Kutta fourth-order method along with shooting technique. The asymptotic boundary conditions given by Eq. (11) were replaced by using a value of 15 for the similarity variable *η*_{max}. The choice of *η*_{max} = 15 and the step size *Δη* = 0.001 ensured that all numerical solutions approached the asymptotic values correctly. For validating of the proposed scheme, a comparison for the Nusselt number with the literature [4, 8, 9] has been shown in Table 2 for both active and passive control of *φ* in the special case when. Therefore, we are confident that the applied numerical scheme is very accurate.

## 4. Results and discussion

In this section, the graphical explanation of the numerical results for velocity, temperature, skin friction coefficients, Nusselt number, and stream lines is expressed with respect to certain changes in the physical parameters through illustrations (Figures 2–7). A comparative study for pure water, SWCNT and MWCNT, is also depicted through Tables 1–5.

(Figure 2a and b) represents the changes in the fluid velocity profiles with respect to different values of solid nanoparticle volume fraction. Figure 2(a) shows the variation in solid volume fraction of nanoparticles with respect to Hartmann number M. As Hartmann number is the ratio of electromagnetic forces to the viscous forces. It is observed that when Hartmann number increases, electromagnetic forces will be dominant to the viscous forces that give declines in the velocity field (see Figure 2(a)). Figure 2(b) shows the variation in solid nanoparticle volume fraction with slip parameter β on velocity profile. It is analyzed that with an increase in slip parameter, velocity profile decreases. Further with an increase in solid nanoparticle volume fraction, velocity profile increases, and boundary layer thickness also increases with the increase in Hartmann number M, slip parameter, and solid nanoparticle volume fraction.

Temperature profile for different values of solid nanoparticle volume fraction with the variation in thermal relaxation time γ, Hartmann number M, and slip parameter β is presented in Figure 3(a–c). Temperature profile decreases with the rise in thermal relaxation time, but thermal boundary layer increases with an increase in thermal relaxation time (see Figure 3(a)). Figure 3(b) depicts that with an increase in electromagnetic forces as compared to the viscous forces, temperature profile and thermal boundary layer increase rapidly. Temperature profile and thermal boundary layer also increase rapidly with the rise in slip parameter (see Figure 3(c)). Moreover, temperature profile and thermal boundary layer increase with an increase in solid nanoparticle volume fraction.

Physical properties | Base fluid | Nanoparticles | |
---|---|---|---|

Water | SWCNT | MWCNT | |

ρ (kg/m^{3}) | 997 | 2600 | 1600 |

c_{p} (J/kg K) | 4179 | 425 | 796 |

k (W/m K) | 0.613 | 6600 | 3000 |

φ | Cf | ||||
---|---|---|---|---|---|

M = 0 | M = 0.5 | ||||

β = 0 | β = 1 | β = 0 | β = 1 | ||

SWCNT | 0.0 0.1 0.2 | 1 1.22904 1.51940 | 0.43016 0.54480 0.70245 | 1.11803 1.35495 1.65690 | 0.46912 0.58833 0.75286 |

MWCNT | 0.0 0.1 0.2 | 1 1.17475 1.39935 | 0.43016 0.53298 0.67377 | 1.11803 1.30590 1.54755 | 0.46912 0.57942 0.73047 |

φ | Nusselt number | ||||||||
---|---|---|---|---|---|---|---|---|---|

γ = 0 | γ = 0 | γ = 0.1 | γ = 0.1 | ||||||

M = 0 | M = 0.5 | M = 1 | M = 2 | ||||||

β = 0 | β = 1 | β = 0 | β = 1 | β = 0 | β = 1 | β = 0 | β = 1 | ||

SWCNT | 0.0 0.1 0.2 | 1.77095 2.66543 3.22397 | 1.33685 2.03231 2.49289 | 1.74513 2.60424 3.13585 | 1.25641 1.89346 2.31928 | 1.71287 2.48958 2.95535 | 1.08479 1.58280 1.92489 | 1.52688 2.07111 2.37985 | 0.70076 0.92785 1.11466 |

MWCNT | 0.0 0.1 | 1.77095 2.63339 | 1.33685 2.02350 | 1.74513 2.57329 | 1.25641 1.88386 | 1.71287 2.46327 | 1.08479 1.57551 | 1.52688 2.05686 | 0.70076 0.92736 |

Variation in skin friction coefficient for SWCNT and MWCNT with slip parameter β and Hartmann number M is presented in Figure 4(a and b). It is seen that with the augment in M, electromagnetic strength is elevated in contrast to thick strength, skin friction coefficient rises for SWCNT as well as for MWCNT, but with the increase in slip parameter, skin friction coefficient decreases for both SWCNT and MWCNT. It is also seen that density and thermal conductivity of SWCNT are greater as compared to the MWCNT; therefore, the skin friction coefficient for SWCNT is greater as compared to the MWCNT.

Nusselt number for SWCNT and MWCNT shows the variation in Hartmann number M, thermal relaxation time γ, and slip parameter β. It is observed that the higher values of thermal relaxation time γ raise the Nusselt number for SWCNT as well as for MWCNT, and it is also analyzed that Nusselt number gives the larger values for SWCNT than MWCNT (see Figure 5(a–c)). Increasing values of Hartmann number M and slip parameter β decrease the Nusselt number for both SWCNT and MWCNT. But due to high density and thermal conductivity of SWCNT, Nusselt number for SWCNT is higher than MWCNT.

Streamlines and Isotherms are presented in Figures 6(a–c) and 7(a–c), respectively. It is analyzed from Figures 6 and 7 that for increasing Hartmann number M and thermal relaxation time γ, streamlines and Isotherms are going close to origin.

## 5. Conclusions

This chapter discussed the magnetic field effects on the flow of Cattanneo-Christov heat flux model for water-based CNT suspended nanofluid over a stretching sheet. Key points of the performed analysis are as follows:

It is observed that when Hartmann number increases, electromagnetic forces will be dominant to the viscous forces that give declines in the velocity.

It is analyzed that with an increase in slip parameter, velocity profile decreases. Further, with an increase in solid nanoparticle volume fraction, velocity profile increases.

Boundary layer thickness also increases with the increase in Hartmann number M, slip parameter β, and solid nanoparticle volume fraction ϕ.

Temperature profile decreases with the rise in thermal relaxation time, but thermal boundary layer increases with an increase in thermal relaxation time.

It depicts that with an increase in electromagnetic forces as compared to the viscous forces, temperature profile and thermal boundary layer increase rapidly.

Temperature profile and thermal boundary layer also increase rapidly with the rise in slip parameter.

Temperature profile and thermal boundary layer increase with an increase in solid nanoparticle volume fraction.

Skin friction coefficient increases for SWCNT as well for MWCNT, but with the increase in slip parameter, skin friction coefficient decreases for both SWCNT and MWCNT.

It is also seen that thickness and thermal conductivity of SWCNT are better as compared to the MWCNT; consequently, the skin friction coefficient for SWCNT is better as compared to MWCNT.

It is pragmatic that the superior values of thermal relaxation time γ hoist the Nusselt number for SWCNT as well as for MWCNT, and it is also analyzed that Nusselt number gives the well-built principles for SWCNT as evaluated to MWCNT.

It is analyzed from Figures 6 and 7 that for rising Hartmann number M and thermal relaxation time γ, streamlines and isotherms are departing shut to source.

**Nomenclature**

μ_{f}: Viscosity of base fluid | q: Heat flux |

(ρC_{p})_{f}: Effective heat capacity of a fluid | U_{w}, V_{w} along the x-axis and y-axis: Thermal Grashof number |

(ρC_{p})_{CNT}: Effective heat capacity of a carbon nanotubes | g: Acceleration due to gravity |

k_{f}: Thermal conductivities of the base fluid | λ_{2}: Thermal relaxation time |

k_{CNT}: Thermal conductivities of the carbon nanotubes | B_{C}: Solutal Grashof number |

B_{0}: Magnitude of magnetic field strength | ν: Kinematic viscosity of the fluid |

T: Local fluid temperature | P_{r}: Prandtl number |

T_{∞}: Ambient temperature | M: Hartmann number |

u, v: Velocity components along x and y directions | (ρc_{p})_{nf}: Heat capacitance |

P: Pressure | ρ_{nf}: Effective density |

φ: Nanoparticle volume fraction | Re_{x}: Local Reynolds number |

η: Similarity variable (transformed coordinate) | α_{nf}: Effective thermal diffusibility |

Nu_{x}: Local Nusselt number | x, y: Coordinate along and normal to the sheet |

(ρc)_{p}: Effective heat capacity of the nanoparticle material | (ρc)_{f}: Heat capacity of the fluid |

θ: Dimensionless temperature | μ_{nf}: Effective dynamic viscosity |

γ: Non-dimensional thermal relaxation time | k_{nf}: Effective thermal conductivity of the nanofluid |