Endoscopy Analysis for the Peristaltic Flow of Nanofluids Containing Carbon Nanotubes with Heat Transfer

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 c₁ along the walls of the tube. The r-coordinate is measured along the endoscope where z-coordinate transverse to it. The magnetic field B₀ 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 T̅₀ and T̅₁ 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.

Figure 1:

Geometry of the problem.

(1)R¯1 = a1,R¯2 = a2 + bsin2πλ(Z¯ − c1t¯). (1)

The transformations between the two frames are

(2)r¯ = R¯,  z¯ = Z¯ − c1t¯,u¯ = U¯,  w¯ = W¯ − c1. (2)

The governing equations for an incompressible nanofluid can be written as

(3)1r¯∂(r¯ u¯)∂r¯ + ∂w¯∂z¯ = 0, (3)

(4)[u¯∂u¯∂r¯ + w¯∂u¯∂r¯] = −1ρnf∂P¯∂r¯ + μnfρnf∂∂r¯[2∂u¯∂r¯] + μnfρnf2r¯(∂u¯∂r¯ − u¯r¯) + μnfρnf∂∂z¯[(∂u¯∂r¯ + ∂w¯∂z¯)], (4)

(5)[u¯∂w¯∂r¯ + w¯∂w¯∂z¯] = −1ρnf∂P¯∂z¯ + μnfρnf∂∂z¯[2∂w¯∂z¯]  + μnfρnf1r¯∂∂r¯[r¯(∂u¯∂z¯ + ∂w¯∂r¯)] − σρnfBo2(w¯+c1), (5)

(6)[u¯∂T¯∂r¯ + w¯∂T¯∂z¯] = αnf[∂2T¯∂r¯2 + 1r¯∂T¯∂r¯ + ∂2T¯∂z¯2] + μnf(ρcp)nfQ0. (6)

The corresponding boundary conditions at the walls are

(7a)w¯ = −c1, T¯ = T¯0,  at  r¯ = r¯1, (7a)

(7b)w¯ = −c1, T¯ = T¯1,  at  r¯ = r¯2. (7b)

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

(8)ρnf = (1 − ϕ) ρf + ϕρf,  μnf = μf(1 − ϕ)2.5,(ρcp)nf = (1 − ϕ)(ρcp)f + ϕ(ρcp)s, αnf = knf(ρcp)nf,K=knfkf = (1 − ϕ) + 2ϕkCNTkCNT − kflnkCNT + kf2kf(1 − ϕ) + 2ϕkfkCNT − kflnkCNT + kf2kf. (8)

We introduce the non-dimensional variables:

(9)R = R¯a2,  r = r¯a2, Z = Z¯λ, z = z¯λ, W = W¯c1, w = w¯c1, U = λU¯a2c1, u = λu¯a2c1,P = a22P¯c1λμf, θ = (T¯ − T¯0)(T¯1 − T¯0), t = c1t¯λ, δ = a2λ, Re = ρca2μf, Pr = να, r2 = r¯2a2, α = k(ρc)f,  β = Q0a22(T¯1 − T¯0)kf, M2 = σB02a22μf, Gr = gαa22(T¯1 − T¯0)νc1. (9)

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

(10)∂P∂r = 0, (10)

(11)∂P∂z = 1(1 − ϕ)2.51r∂∂r[r(∂w∂r)] − M2(w + 1) + Grθ, (11)

(12)0 = 1r∂∂r(r∂θ∂r) + ((1−ϕ) + 2ϕkfkCNT − kflnkCNT + kf2kf(1−ϕ)+2ϕkCNTkCNT − kflnkCNT + kf2kf)β. (12)

The corresponding boundary conditions are

(12a)w = −1,  θ = 0,  at  r = r1, (12a)

(12b)w = −1,  θ = 1,  at r = r2 = 1 + ωsin2πz. (12b)

2.2 Exact Solutions

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

(13)w(r, z) = −1M2dPdz + I0(Mr(1 − ϕ)54)(A5dPdz + A6) + K0(Mr(1 − ϕ)54)(A 7dPdz + A8) −A13r2M2(1 − ϕ)52 − A12M2(1 − ϕ)52− 4A13M4(1 − ϕ)5 − A14log(r)M4(1 − ϕ)52. (13)

(14)dPdz = (F − A9(r22 − r12) − A10)A11, (14)

(15)θ(r, z) = −14(1 − ϕ) + 2ϕkfkCNT − kflnkCNT + kf2kf(1 − ϕ) + 2ϕkCNTkCNT − kflnkCNT + kf2kfr2β + A1log(r) + A2, (15)

where A₁–A₁₄ 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

(16)ΔP = ∫01dPdzdz, (16)

(17)F1λ = ∫01 − (r12dPdz) dz, (17)

(18)F2λ = ∫01 − (r22dPdz) dz. (18)

Expression for stream function is given as follows:

(19)w = 1r∂ψ∂r. (19)

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 (ΔP<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 (ΔP>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 (ΔP>0), in co pumping region (ΔP<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 (ΔP<0), the pressure rise gives the opposite behaviour for all the parameters as compared to the pumping region (ΔP>0). Free pumping region exists when (ΔP=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.