## Abstract

Cu–water nanofluid with carbon nanotubes is considered for the peristaltic flow in an endoscope. The peristaltic flow for nanofluid is modelled considering that the peristaltic rush wave is a sinusoidal wave that propagates along the walls of the tube. The governing equations for the proposed model are simplified by using the assumptions of long-wavelength and low Reynolds number. Exact solutions have been evaluated for velocity, temperature, and pressure gradient. Graphical results for the numerical values of the flow parameters, i.e. Hartmann number M, the solid volume fraction *ϕ* of the nanoparticles, Grashof number Gr, heat absorption parameter *β*, and radius of the inner tube *ε*, have been presented for the pressure difference, frictional forces, velocity profile, and temperature profile, and trapping phenomena have been discussed at the end of the article.

- Nomenclature
*u*̅ and*w*̅Velocity components in wave frame

*p*̅Pressure

*T*̅Temperature

*B*_{o}Applied magnetic field

*Q*_{0}Constant heat addition/absorption

- Gr
Local temperature Grashof number

*g*Gravitation due to acceleration

- M
Hartmann number

*a*_{1}Radius of the inner tube

*a*_{2}Radius of the outer tube

*ϕ*Solid volume fraction of the nanoparticles

*δ*Long wave length parameter

*μ*_{f}Fluid viscosity

*c*_{1}Wave speed

*b*Wave amplitude

*T*̅_{0}and*T*̅_{1}Wall temperatures

*λ*Wavelength

*β*Heat absorption parameter

*ε*Radius of the inner tube

*α*_{nf}Effective thermal diffusivity

*ρ*_{nf}Effective density

*μ*_{nf}Effective dynamic viscosity

*k*_{nf}Effective thermal conductivity

*θ*Dimensionless temperature

- Pr
Prandtl number

*ϖ*Amplitude ratio

## 1 Introduction

Endoscopy means observing secret and characteristically a privilege to looking inside the body for therapeutic motives. It is a tool used to inspect the inner part of a tissue or cavity of the body. Distinct from most other homeopathic imaging procedures, endoscopes are injected straight into the body. Endoscopy is also used to periscope in methodological conditions where straight line-of-sight observation is not possible. Latham [1] was the first person who discussed the peristaltic flow. Later on, researchers and scientists emphasised their considerations to analyse the peristaltic flows with different geometries [2–10]. This mechanism occurs in urine transport from the kidney to the bladder through the ureter, movement of chyme in small intestine, the locomotion of some warms, etc. Mekheimer and Abd Elmaboud [11] discussed the influence of heat transfer and magnetic field on the peristaltic flow of Newtonian fluid in a vertical annulus under a zero Reynolds number and long-wavelength approximation. Peristaltic flow of a couple stress fluid with the influence of endoscope is presented by Mekheimer and Abd Elmaboud [12]. The influence of an inserted endoscope and fluid with variable viscosity on the peristaltic motion has been investigated under zero Reynolds number by Hakeem et al. [13]. A new numerical technique for the magneto hydrodynamic (MHD) peristaltic flow of a bio-fluid with variable viscosity in a circular cylindrical tube is presented by Ebaid [14]. He made a comparison between the results obtained by the Adomian series and those obtained previously by perturbation technique and found good accuracy. Akbar and Nadeem [15] discussed the application of the Rabinowitsch fluid model in peristalsis. In another article, Akbar and Butt [16] carried out a heat transfer analysis for the peristaltic flow of Herschel–Bulkley fluid in a non-uniform inclined channel.

Carbon nanotubes (CNTs) are allotropes of carbon with a cylindrical nanostructure. Nanotubes have been constructed with a length-to-diameter ratio of up to 132,000,000:1 [17], significantly larger than for any other material. These cylindrical carbon molecules have unusual properties, which are valuable for nanotechnology, electronics, optics, and other fields of materials science and technology. In particular, owing to their extraordinary thermal conductivity and mechanical and electrical properties, CNTs find applications as additives to various structural materials. Nanotubes form a tiny portion of the material(s) i.e. carbon fiber etc. [18]. Applications of CNTs have instigated researchers and scientists to reflect on CNTs and use them in numerous fields. CNTs have impending applications in fields such as nanotechnology, electronics, optics, materials science, and architecture. Recently, new applications have taken advantage of their unique electrical stuffs, surprising strength, and efficiency in heat conduction [19]. CNTs in connection with peristalsis have their application in the Kanzius cancer therapy, where single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs) are inserted around cancerous cells, and then excited with radio waves, which causes them to heat up and kill the surrounding cells. CNTs and their polymer nanocomposites are suitable scaffold materials for bone cell proliferation [20, 21] and bone formation. Very recently, peristaltic flow with nanofluids containing CNTs in a vertical circular tube with thermal and velocity slip effects was investigated by Akbar et al. [22]. According to their observation, the pressure rise for SWCNT is dominant as compared to the MWCNT. In another article, the combined effects of slip and convective boundary conditions on stagnation point flow of CNT-suspended nanofluid over a stretching sheet were also studied by Akbar et al. [23]. Recently, Tripathi and Bég [24, 25] discussed mathematical modelling of peristaltic pumping of nanofluids and its application in drug delivery systems. Heat transfer and carbon nanotubes analysis for the peristaltic flow in a diverging tube was studied by Akbar [26]. Exact solutions have been evaluated for the simplified governing flow equations. Oblique stagnation point flow of CNT-based fluid over a convective surface has been presented by Nadeem et al. [27]. They evaluated the numerical solutions and discussed the physics of the flow parameters through graphs and their interpretation.

In the present article, CNT Cu nanoparticles for peristaltic flow of an incompressible viscous fluid in an endoscope with water as base fluid are discussed. The CNT Cu nanoparticles for peristaltic movement with water as base fluid in an endoscope are not explored thus far. The problem formulation for the CNT Cu nanoparticles for the peristaltic flow is done first time in the literature and simplified under the long-wavelength and low Reynolds number approximation. Exact solutions have been calculated for velocity, temperature, and pressure gradient. Graphical results for the numerical values of the flow parameters, i.e. Hartmann number M, the solid volume fraction *ϕ* of the nanoparticles, Grashof number Gr, heat absorption parameter *β*, and radius of the inner tube *ε*, have been presented for the pressure difference, frictional forces, velocity profile, and temperature profile.

## 2 Mathematical Development

### 2.1 Governing Equations and Boundary Conditions

We have discussed the endoscopic analysis on the peristaltic flow of an incompressible CNT nanofluid with water as base fluid. The flow is due to sinusoidal wave oscillation with constant speed *c*_{1} along the walls of the tube. The *r*-coordinate is measured along the endoscope where *z*-coordinate transverse to it. The magnetic field *B*_{0} is imposed on the flow in the *z*-direction normal to the endoscope. It is assumed that the walls of the tubes of the endoscope are heated uniformly at constant temperatures *T*̅_{0} and *T*̅_{1} to the inner and outer walls, respectively.

Flow is incompressible, two directional, and two dimensional.

The inner tube is rigid, while the outer is flexible.

Sinusoidal wave is propagating along the wall of the outer tube.

The effect of magnetic field induced by the Reynolds number is sufficiently small to be neglected as compared to the external magnetic field.

Due to long-wavelength and low Reynolds number approximation, the velocity component in the

*z*-direction is more important than the one in the*r*-direction.

The geometry of the wall explosives are defined as in Figure 1.

The transformations between the two frames are

The governing equations for an incompressible nanofluid can be written as

The corresponding boundary conditions at the walls are

Properties of CNT are defined as (see Refs. [16–22])

We introduce the non-dimensional variables:

With (8) and (9), under the assumptions of long-wavelength and low Reynold number approximation, (2)–(7) take the form

The corresponding boundary conditions are

### 2.2 Exact Solutions

Exact solutions for the velocity, temperature, and pressure gradient have been evaluated from (10) to (12):

where *A*_{1}–*A*_{14} have been calculated from Mathematica 8 presented in the appendix.

The dimensionless pressure rise and friction force on outer and inner tube are defined, respectively, by

Expression for stream function is given as follows:

## 3 Results and Discussion

Here, we present the pressure rise, pressure gradient, velocity, temperature profile, and stream lines graphically. For that purpose, numerical values are plotted in Figures 2–10. Both the SWCNT and MWCNT are discussed graphically, and their properties can be seen through Table 1. Tables 2 and 3 give the numerical values for pressure rise, velocity, and temperature different flow parameters.

Physical properties | Fluid phase (water) | SWCNT | MWCNT |
---|---|---|---|

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

ρ (kg/m^{3}) | 997.1 | 2600 | 1600 |

k (W/mk) | 0.613 | 6600 | 3000 |

↓Q | ΔP (SWCNT) | ↓Q | ΔP (SWCNT) | ↓Q | ΔP (SWCNT) |
---|---|---|---|---|---|

–3.0 | 48.9617 | –1.0 | 15.4632 | 1.0 | –21.5613 |

–2.5 | 40.14631 | –0.5 | 6.6478 | 1.5 | –30.3767 |

–2.0 | 31.3309 | 0.0 | –2.1675 | 2.5 | –48.0075 |

–1.5 | 22.51554 | 0.5 | –12.7459 | 3.0 | –56.8229 |

↓Q | ΔP (MWCNT) | ↓Q | ΔP (MWCNT) | ↓Q | ΔP (MWCNT) |

–3.0 | 49.3004 | –1.0 | 15.8019 | 1.0 | –21.2226 |

–2.5 | 40.1463 | –0.5 | 6.9865 | 1.5 | –30.0380 |

–2.0 | 31.6696 | –0.0 | –1.8287 | 2.5 | –38.8533 |

–1.5 | 22.8543 | –0.5 | –12.4072 | 3.0 | –47.6687 |

↓r | w(r,z) (SWCNT) | w(r,z) (MWCNT) | θ(r,z) (SWCNT) | θ(r,z) (MWCNT) |
---|---|---|---|---|

0.3 | –1.0000 | –1.0000 | 1.0000 | 1.0000 |

0.4 | –0.2734 | –0.2809 | 0.7056 | 0.7000 |

0.5 | 0.1366 | 0.1248 | 0.8055 | 0.6470 |

0.6 | 0.3255 | 0.3117 | 0.5180 | 0.5146 |

0.7 | 0.3432 | 0.3291 | 0.4016 | 0.3981 |

0.8 | 0.2359 | 0.2046 | 0.2957 | 0.2926 |

0.9 | –0.0364 | –0.0464 | 0.2959 | 0.1949 |

1.0 | –0.4101 | –0.4163 | 0.1644 | 0.1029 |

1.1 | –0.9015 | –0.9025 | 0.0252 | 0.0151 |

1.2 | –1.5120 | –1.5068 | 0.0000 | 0.0000 |

### 3.1 Medical Interpretation

From the biomedical point of view, CNTs with peristalsis have important applications. These CNTs with copper water inserted inside the human body can be used in the diagnosis of diseases, because copper water makes the arteries flexible. CNTs in connection with peristaltic pumping have their application in the Kanzius cancer therapy, where SWCNTs and MWCNTs are inserted around cancerous cells, and then excited with radio waves, which causes them to heat up and kill the surrounding cells. CNTs and their polymer nanocomposites are suitable scaffold materials for bone cell proliferation and bone formation.

### 3.2 Pumping Characteristics

Numerical integration is carried out for the pressure rise per wavelength. The pressure rise against volume flow rate for the solid volume fraction of the nanoparticles *ϕ*, Hartmann number M, Grashof number Gr, and radius of the inner tube *ε* is plotted in Figures 2a–5a. It is observed that the pressure rise and volume flow rate have the opposite behaviour. It is seen that when the electromagnetic forces and buoyancy forces are high as compared to the viscous forces, it declines the pressure rise in the pumping region; but it gives the opposite behaviour in the augmented pumping region, i.e. when electromagnetic forces and buoyancy forces are dominant as compared to the viscous forces, then the pressure rise increases in the augmented pumping region (Δ*P*<0), see Figures 2a and 3a, and it is also seen that Hartmann number M and Grashof number Gr have the same influence on pressure rise. From Figure 4a, it is depicts that in pumping region (Δ*P*>0), with the increase in solid volume fraction of the nanoparticles *ϕ*, means the increase in the copper rate when copper rate increases the stomach becomes relaxed so the pressure rise decreases in the pumping region (Δ*P*>0), in co pumping region (Δ*P*<0) the pressure rise decreases. It is depicted that when we increase the radius of the inner tube, flow can easily move in the stomach, which gives rise in the pressure in the pumping region, see (Fig. 5a), but the pressure declines in the augmented pumping region. Figures 2a–5a also show that, in the augmented pumping region for (Δ*P*<0), the pressure rise gives the opposite behaviour for all the parameters as compared to the pumping region (Δ*P*>0). Free pumping region exists when (Δ*P*=0). It is also analysed that the pressure rise for SWCNT is more dominant as compared to the MWCNT with the variation of *ε*, while with the variation of solid volume fraction of the nanoparticles *ϕ*, Hartmann number M, and Grashof number Gr, pressure rise for MWCNT is greater as compared to the SWCNT. Frictional forces for inner and outer tubes are plotted in Figures 2b and c–5b and c. It is seen that, for the frictional forces, results are quite opposite for all the parameters as compared to the pressure rise.

### 3.3 Flow Characteristics

Variations of Hartmann number M, Grashof number Gr, flow rate *Q*, and solid volume fraction of the nanoparticles *ϕ* on the velocity profile are presented in Figure 5a–d. It is seen that the behaviour of velocity field is not the same in view of the Hartmann number M, flow rate *Q*, and Grashof number Gr as compared to the solid volume fraction of the nanoparticles *ϕ*. The velocity field rises due to high electromagnetic forces and buoyancy forces as compared to viscous forces, because in that case flow can easily move in the stomach; so velocity field rises rapidly for large values of Hartmann number M and Grashof number Gr, see Figure 5a and c, and it is also shown in Figure 5d that when flow rate *Q* increases, the speed of fluid in stomach increases and gives rise to the velocity, but when copper rate increases, the stomach becomes relax and the velocity declines with an increase in the solid volume fraction of the nanoparticles *ϕ*, see Figure 5b. It is also observed that the velocity field for SWCNT is greater as compared to the MWCNT for the variation of volume fraction of the nanoparticles *ϕ.* Heat absorption through a substance gives more rise in temperature, so the temperature increases rapidly with the increase in heat absorption parameter *β* and solid volume fraction of the nanoparticles *ϕ*, see Figure 6a and b. The temperature for SWCNT is higher as compared to the MWCNT.

### 3.4 Pressure Gradient

The pressure gradient for different values of M, Gr, *ϕ*, and *ε* are plotted in Figure 7a–d. Magnitude of pressure gradient increases when the electromagnetic forces are high as compared to the viscous forces and when we increase the radius of the inner tube; however, when buoyancy forces are dominant as compared to the viscous forces and when we rise the copper rate, the pressure gradient decreases. It is also observed that the maximum pressure gradient occurs when *z*=0.55, and near the tube walls, the pressure gradient is small. This leads to the fact that flow can easily pass in the middle of the stomach. Pressure gradient decreases with the increase in *ϕ* and Gr. It is also noticed that the pressure gradient for MWCNT is greater when we compared it to SWCNT.

### 3.5 Trapping Phenomena

Trapping phenomena have been discussed through Figures 8 and 9. These phenomena occur when streamlines are going to be enclosed. Figure 8 shows the streamlines for different values of SWCNT and MWCNT. It is noticed that, for MWCNT, the size of the trapping bolus increases, while the number of the trapping bolus decreases as compared to SWCNT. Streamlines for different values of Hartmann number M are presented in Figure 9. It is analysed that when there will be electromagnetic forces in the fluid flow, the size of the trapping bolus decreases, while the number of the trapping bolus increases rapidly.

## 4 Summary

Heat transfer and endoscopy analysis of nanofluids containing CNTs is presented. Summary of the conducted work is as follows:

It is observed that the pressure rise and volume flow rate have opposite behaviour.

It is depicted that in pumping region, the pressure rise decreases with the increase in solid volume fraction for the nanoparticles

*ϕ*, Hartmann number M, and Grashof number Gr, while pressure rise increases with the increase in the value of radius of inner tube*ε*.Free pumping region exists when (Δ

*P*=0).It is also analysed that the pressure rise for SWCNT is more dominant as compared to the MWCNT with the variation of

*ε*.It is seen that for frictional forces, results are quite opposite for all the parameters as compared to the pressure rise.

The velocity field increases due to increase in Hartmann number M, flow rate

*Q*, and Grashof number Gr, while decreases with an increase in solid volume fraction of the nanoparticles*ϕ*.The temperature for SWCNT is higher as compared to the MWCNT.

It is also noticed that the pressure gradient for MWCNT is greater when compared it to SWCNT.

## Acknowledgment:

The author is thankful to the Higher Education Commission Pakistan for the financial support to complete this work.

## Appendix

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**Received:**2015-3-27

**Accepted:**2015-6-9

**Published Online:**2015-8-1

**Published in Print:**2015-9-1

©2015 by De Gruyter