MHD Effects on Ciliary-Induced Peristaltic Flow Coatings with Rheological Hybrid Nanofluid
<p>Geometry of the physical problem.</p> "> Figure 2
<p><span class="html-italic">h</span>-curve for the <math display="inline"><semantics> <mi mathvariant="sans-serif">ψ</mi> </semantics></math> at <math display="inline"><semantics> <mrow> <mi mathvariant="normal">z</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi mathvariant="normal">Q</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi mathvariant="sans-serif">ε</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>.</p> "> Figure 3
<p><span class="html-italic">h</span>-curve for the <math display="inline"><semantics> <mi mathvariant="sans-serif">ϕ</mi> </semantics></math> at <math display="inline"><semantics> <mrow> <mi mathvariant="normal">z</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi mathvariant="normal">Q</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi mathvariant="sans-serif">ε</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>.</p> "> Figure 4
<p><span class="html-italic">h</span>-curve for the <math display="inline"><semantics> <mi mathvariant="sans-serif">θ</mi> </semantics></math> at <math display="inline"><semantics> <mrow> <mi mathvariant="normal">z</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi mathvariant="normal">Q</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi mathvariant="sans-serif">ε</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>.</p> "> Figure 5
<p>Variation in <span class="html-italic">w</span>(<span class="html-italic">r</span>) towards <span class="html-italic">R</span>.</p> "> Figure 6
<p>Variation in <span class="html-italic">w</span>(<span class="html-italic">r</span>) towards <span class="html-italic">Gr</span>.</p> "> Figure 7
<p>Variation in <span class="html-italic">w</span>(<span class="html-italic">r</span>) towards <span class="html-italic">M</span>.</p> "> Figure 8
<p>Variation in <span class="html-italic">H<sub>z</sub></span>(<span class="html-italic">r</span>) towards <span class="html-italic">R</span>.</p> "> Figure 9
<p>Variation in <math display="inline"><semantics> <mrow> <mi>θ</mi> <mrow> <mo>(</mo> <mi mathvariant="normal">r</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> towards <span class="html-italic">G<sub>r</sub></span>.</p> "> Figure 10
<p>Variation in <math display="inline"><semantics> <mrow> <mi>θ</mi> <mrow> <mo>(</mo> <mi mathvariant="normal">r</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> towards <span class="html-italic">R</span>.</p> "> Figure 11
<p>Three-dimensional (3-D) velocity profile towards <span class="html-italic">R</span>.</p> "> Figure 12
<p>Three-dimensional (3-D) velocity profile towards <span class="html-italic">Gr</span>.</p> "> Figure 13
<p>Three-dimensional (3-D) velocity profile towards <span class="html-italic">M</span>.</p> "> Figure 14
<p>Pressure rise versus <span class="html-italic">Q</span> for <span class="html-italic">Ω</span>.</p> "> Figure 15
<p>Pressure rise versus <span class="html-italic">Q</span> for <span class="html-italic">M</span>.</p> "> Figure 16
<p>Pressure rise versus <span class="html-italic">Q</span> for <span class="html-italic">Gr</span>.</p> "> Figure 17
<p>Pressure rise versus <span class="html-italic">Q</span> for <span class="html-italic">R</span>.</p> "> Figure 18
<p>Pressure gradient for Ω.</p> "> Figure 19
<p>Pressure gradient for <span class="html-italic">M</span>.</p> "> Figure 20
<p>Pressure gradient for <span class="html-italic">R</span>.</p> "> Figure 21
<p>Pressure gradient for <span class="html-italic">Gr</span>.</p> "> Figure 22
<p>Behavior of streamlines for different values of magnetic Reynold number (<b>a</b>–<b>d</b>).</p> "> Figure 22 Cont.
<p>Behavior of streamlines for different values of magnetic Reynold number (<b>a</b>–<b>d</b>).</p> "> Figure 23
<p>Behavior of streamlines for different values of Grashof number (<b>a</b>–<b>d</b>).</p> "> Figure 23 Cont.
<p>Behavior of streamlines for different values of Grashof number (<b>a</b>–<b>d</b>).</p> "> Figure 24
<p>Behavior of streamlines for different values of Hartmann number (<b>a</b>–<b>d</b>).</p> "> Figure 24 Cont.
<p>Behavior of streamlines for different values of Hartmann number (<b>a</b>–<b>d</b>).</p> "> Figure 25
<p>Behaviorof streamlines for different values of amplitude ratio (<b>a</b>–<b>d</b>).</p> "> Figure 25 Cont.
<p>Behaviorof streamlines for different values of amplitude ratio (<b>a</b>–<b>d</b>).</p> ">
Abstract
:1. Introduction
2. Problem Description
3. Methodology and Convergence of HAM Solutions
3.1. Methodology
3.2. Convergence of HAM Solutions
4. Quantitative Analysis
5. Conclusions
- ❖
- Velocity of the hybrid nanofluid reduces for M and Gr near the endoscope but it increases near the external peristaltic tube having a ciliary surface due to a decreasing pressure gradient, even in creeping flow condition as observed from 2-D and 3-D plots. These results show that buoyancy effects are more prominent near the peristaltic tube and magnetic induction enhances peristalsis in presence of Ag-TiO2 nanohybrids with 0.2% concentration.
- ❖
- The magnetic induction profile displays a similar behavior as that of velocity towards the magnetic Reynolds number. An increase in values of R upgrades the flow rate and hence it is concluded that velocity and induced magnetic field relatively generate elastic oscillations. Consequently, fluid having hybrid nanoparticles can deeply interact with tumors and efficiently deliver drugs to specified section.
- ❖
- The temperature of hybrid nanofluid depicts a decreasing behavior for Gr while a conflicting trend is seen for R. This trend of temperature increase of the fluid will be helpful in the removal of a cancer tumor and abnormal cells without affecting healthy parts within the body during an endoscopy.
- ❖
- The behavior of the pressure rise for different parameters show that the pressure rise declines in the co-pumping region whereas it is enhanced in the pumping and free-pumping regions. Pumping rate increases for increment in radius ratio parameter which is favorable for accurate endoscopic imaging.
- ❖
- The pressure gradient decreases throughout the length of the endoscope close to the ciliated peristaltic tube for r2 = 1.
- ❖
- Numerical values of velocity and temperature against embedding parameters explore a similar behavior as noticed in graphs. Flow profiles towards variation in radial distance are examined which satisfy the conditions of the problem.
- ❖
- The presence of cilia shows a dominant effect on the behavior of the flow variables. In most cases, the sensitive interior surface of organs may be protected due to cilia as they assist velocity near the peristaltic tube.
- ❖
- The peristaltic flow pattern due to ciliary activity for different parameters is displayed via streamline configuration and a reduction in the size of the trapped bolus is observed towards R, M and Gr but conversely enlarges for ε.
- ❖
- The present work appears to be the first attempt in the literature dealing with the effects of electromagnetic induction on peristaltic transport and the heat transfer of non-Newtonian hybrid nanofluid through a ciliated tube inserted by an endoscope. Additional developments and characteristics of the problem can be examined.
- ❖
- Results for Newtonian nanofluid [50] can be obtained in a limiting case.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Nomenclature
b | Wave Amplitude in Fixed Frame Wall (m) |
c | Wave speed (ms−1) |
n | Power law index |
k | Consistency index (Pa sn) |
g | Gravitational acceleration (ms−2) |
s | Nanoparticles shape factor |
E | Electric field strength (N/C) |
Pressure (Pa) | |
Temperature (K) | |
Temperature of inner tube (K) | |
Temperature of outer tube (K) | |
Grashof number (dimensionless) | |
Heat sink/source parameter (Wm−2K−1) | |
Magnetic Reynolds number | |
Magnetic field strength (Am−1) | |
Radial magnetic induction componentComponent (Am−1) | |
Axial magnetic induction component (Am−1) | |
Specific heat (Jkg−1K−1) | |
Hartman number | |
U | Radial velocity component (ms−1) |
W | Axial velocity component (ms−1) |
Greek Symbol | |
Density (kgm−3) | |
Electric conductivity (S/m) | |
Magnetic force function (A2m−1) | |
Stream function (m2s−1) | |
Thermal expansion coefficient (K−1) | |
Wavelength (m) | |
Wave number (dimensionless) | |
Magnetic Diffusivity (m2s−1) | |
Dynamic viscosity (kgm-1s−1) | |
Wave amplitude in moving frame (m) | |
Thermal conductivity (Wm−1K−1) | |
Dimensionless heat source/sink parameter | |
Rate of deformation tensor (s−1) | |
Shear stress (Pa) | |
Measure of eccentricity | |
Subscript | |
Hybrid Nanofluid | |
Base fluid | |
Solid nano particles of Ag | |
Solid nano particles of TiO2 |
Appendix A
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Properties\Constituents | H2O | Ag | TiO2 |
---|---|---|---|
Density, ρ (kg/m3) | 997 | 10,500 | 4250 |
Specific heat, Cp (J/kg K) | 4179 | 235 | 686.2 |
Thermal conductivity, κ (W/m K) | 0.613 | 429 | 8.95 |
Thermal expansion coefficient, (10−5 m/(mK)) | 21 | 1.89 | 0.9 |
Properties | Hybrid Nanofluid |
---|---|
Density | |
Heat Capacity | |
Viscosity | |
Thermal Conductivity | |
Thermal Expansion Coefficient |
0.1 | 1.000000 | |||||
0.3 | ||||||
0.6 | ||||||
0.9 | ||||||
1.2 |
0.1 | 1.000000 | |||||
0.3 | ||||||
0.6 | ||||||
0.9 | ||||||
1.2 |
r | Velocity | Temperature | Induced Magnetic Field |
---|---|---|---|
0.1 | |||
0.15 | |||
0.2 | |||
0.25 | |||
0.3 | |||
0.35 | |||
0.4 | |||
0.45 | |||
0.5 | |||
0.55 | |||
0.6 | |||
0.65 | |||
0.7 | |||
0.75 | |||
0.8 | |||
0.85 | |||
0.9 | |||
0.95 | |||
0.1 | |||
1.05 | |||
1.1 | |||
1.15 | |||
1.2 | 1.000013 |
Existing | [46] | Existing | [46] | Existing | [46] | |
---|---|---|---|---|---|---|
0.1 | −1.000000 | − | − | − | − | −1.000000 |
0.17 | − | − | − | − | − | − |
0.24 | − | − | − | − | − | − |
0.31 | − | − | − | − | − | − |
0.38 | − | − | − | − | − | − |
0.45 | −0.396309 | −0.396313 | −0.384359 | −0.384359 | − | − |
0.52 | −0.403142 | −0.403151 | −0.396393 | −0.396393 | −0.389636 | −0.389636 |
0.59 | −0.431481 | −0.431483 | −0.429839 | −0.429839 | −0.428195 | −0.428195 |
0.66 | −0.478539 | −0.478546 | −0.481408 | −0.481408 | −0.48427 | −0.48427 |
0.73 | −0.542336 | −0.542343 | −0.54867 | −0.54867 | −0.554997 | −0.554997 |
0.80 | −0.621365 | −0.621371 | −0.629747 | −0.629747 | −0.638123 | −0.638123 |
0.87 | −0.714447 | −0.714452 | −0.723131 | −0.723131 | −0.73181 | −0.73181 |
0.94 | −0.820564 | −0.820638 | −0.827574 | −0.827574 | −0.834509 | −0.834509 |
1.01 | −0.939132 | −0.939145 | −0.942016 | −0.942016 | −0.944887 | −0.944887 |
1.04382 | −1.00061 | −1.00061 | −1.00061 | −1.00061 | −1.00061 | −1.00061 |
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Awais, M.; Shah, Z.; Perveen, N.; Ali, A.; Kumam, P.; Rehman, H.u.; Thounthong, P. MHD Effects on Ciliary-Induced Peristaltic Flow Coatings with Rheological Hybrid Nanofluid. Coatings 2020, 10, 186. https://doi.org/10.3390/coatings10020186
Awais M, Shah Z, Perveen N, Ali A, Kumam P, Rehman Hu, Thounthong P. MHD Effects on Ciliary-Induced Peristaltic Flow Coatings with Rheological Hybrid Nanofluid. Coatings. 2020; 10(2):186. https://doi.org/10.3390/coatings10020186
Chicago/Turabian StyleAwais, M., Zahir Shah, N. Perveen, Aamir Ali, Poom Kumam, Habib ur Rehman, and Phatiphat Thounthong. 2020. "MHD Effects on Ciliary-Induced Peristaltic Flow Coatings with Rheological Hybrid Nanofluid" Coatings 10, no. 2: 186. https://doi.org/10.3390/coatings10020186