The Effect of Variable Magnetic Field on Viscous Fluid between 3-D Rotatory Vertical Squeezing Plates: A Computational Investigation
<p>Geometry of the flow problem.</p> "> Figure 2
<p>Total residual error, with <math display="inline"><semantics> <mrow> <msub> <mi>S</mi> <mi>z</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>x</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>y</mi> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> "> Figure 3
<p>Error profile for <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>η</mi> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>(</mo> <mi>η</mi> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>(</mo> <mi>η</mi> <mo>)</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>(</mo> <mi>η</mi> <mo>)</mo> </mrow> </semantics></math>, with fixed <math display="inline"><semantics> <mrow> <msub> <mi>S</mi> <mi>z</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>x</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>y</mi> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> "> Figure 4
<p>3D graph for <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>η</mi> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>(</mo> <mi>η</mi> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>(</mo> <mi>η</mi> <mo>)</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>(</mo> <mi>η</mi> <mo>)</mo> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>S</mi> <mi>z</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>x</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>y</mi> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> "> Figure 5
<p>Impact of squeeze reynolds number <math display="inline"><semantics> <msub> <mi>S</mi> <mi>z</mi> </msub> </semantics></math> on the velocity component <span class="html-italic">f</span> and <math display="inline"><semantics> <msup> <mi>f</mi> <mo>′</mo> </msup> </semantics></math>, keeping <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>x</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>3.25</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>y</mi> </msub> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>0.75</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> "> Figure 6
<p>Impact of squeeze reynolds number <math display="inline"><semantics> <msub> <mi>S</mi> <mi>z</mi> </msub> </semantics></math> on velocity component <span class="html-italic">g</span> and <span class="html-italic">n</span>, keeping <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>x</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>y</mi> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>0.75</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> "> Figure 7
<p>Impact of squeeze Reynolds number <math display="inline"><semantics> <msub> <mi>S</mi> <mi>z</mi> </msub> </semantics></math> on the magnetic field component <span class="html-italic">m</span>, keeping <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>x</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>1.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>y</mi> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> "> Figure 8
<p>Impact of magnetic field strength <math display="inline"><semantics> <msub> <mi>M</mi> <mi>x</mi> </msub> </semantics></math> on the velocity component <span class="html-italic">f</span>, keeping <math display="inline"><semantics> <mrow> <msub> <mi>S</mi> <mi>z</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>y</mi> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> "> Figure 9
<p>Impact of magnetic field strength <math display="inline"><semantics> <msub> <mi>M</mi> <mi>x</mi> </msub> </semantics></math> on velocity component <math display="inline"><semantics> <msup> <mi>f</mi> <mo>′</mo> </msup> </semantics></math>, keeping <math display="inline"><semantics> <mrow> <msub> <mi>S</mi> <mi>z</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>y</mi> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> "> Figure 10
<p>Impact of Magnetic field strength <math display="inline"><semantics> <msub> <mi>M</mi> <mi>x</mi> </msub> </semantics></math> on velocity component <span class="html-italic">g</span> and magnetic field component <span class="html-italic">n</span>, Keeping <math display="inline"><semantics> <mrow> <msub> <mi>S</mi> <mi>z</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>0.25</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>y</mi> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> "> Figure 11
<p>Impact of magnetic field strength <math display="inline"><semantics> <msub> <mi>M</mi> <mi>x</mi> </msub> </semantics></math> on magnetic field component <span class="html-italic">m</span>, keeping <math display="inline"><semantics> <mrow> <msub> <mi>S</mi> <mi>z</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>1.75</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>y</mi> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> "> Figure 12
<p>Impact of magnetic field strength <math display="inline"><semantics> <msub> <mi>M</mi> <mi>y</mi> </msub> </semantics></math> on velocity component <span class="html-italic">g</span> and magnetic field component <span class="html-italic">n</span>, keeping <math display="inline"><semantics> <mrow> <msub> <mi>S</mi> <mi>z</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>x</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>0.5</mn> </mrow> </semantics></math> (for g), <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>x</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>1.5</mn> </mrow> </semantics></math> (for <span class="html-italic">n</span>), <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> </mrow> </semantics></math> (for <span class="html-italic">g</span>), <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mi>m</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math> (for <span class="html-italic">n</span>), and <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> "> Figure 13
<p>Impact of magnetic Reynolds number <math display="inline"><semantics> <msub> <mi>R</mi> <mi>m</mi> </msub> </semantics></math> on velocity component <span class="html-italic">f</span> and <math display="inline"><semantics> <msup> <mi>f</mi> <mo>′</mo> </msup> </semantics></math>, keeping <math display="inline"><semantics> <mrow> <msub> <mi>S</mi> <mi>z</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>1.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>x</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>1.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>y</mi> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> "> Figure 14
<p>Impact of magnetic Reynolds number <math display="inline"><semantics> <msub> <mi>R</mi> <mi>m</mi> </msub> </semantics></math> on velocity component <span class="html-italic">g</span>, keeping <math display="inline"><semantics> <mrow> <msub> <mi>S</mi> <mi>z</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>1.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>x</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>0.75</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>y</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> "> Figure 15
<p>Impact of magnetic Reynolds number <math display="inline"><semantics> <msub> <mi>R</mi> <mi>m</mi> </msub> </semantics></math> on magnetic field components <span class="html-italic">m</span> and <span class="html-italic">n</span>, keeping <math display="inline"><semantics> <mrow> <msub> <mi>S</mi> <mi>z</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>1.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>x</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>0.75</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>y</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> ">
Abstract
:1. Introduction
2. Modeling and Formulation of the Physical Problem
3. Boundary Conditions
4. Method of Solution
5. Optimal Convergence Control Parameters
6. Error Analysis
7. Results and Discussion
8. Conclusions
- It was observed that increasing the squeeze effect on the upper plate causes an increase in the flow velocity along the y- and z-direction, while along the x-direction, the velocity increase initially, but a decrease in the velocity has been observed in the upper domain ().
- It was also investigated and concluded that, by increasing the squeeze Reynolds number, the magnetic field component decreased the effect of the magnetic field along the z-component, whereas the effect increased along the y-component.
- Furthermore, from the above problem, it was observed that increasing the magnetic field strength parameter , which is the strength of the magnetic field along the x-axis, increases the fluid velocity along the z-axis; however, velocity along the y-axis showed a gradual decrease by increasing . Moreover, along the x-axis, first an increase in the velocity component was observed, but as the velocity started decreasing.
- An inverse relation was observed between the magnetic field strength parameter and the magnetic field component along the z-axis, i.e., increasing the value of showed a decreasing effect in the value of the magnetic field along the z-component, and a direct relation could be seen along the y-axis.
- Furthermore, it was seen that an increasing value of the magnetic field strength parameter along the y-component caused a decrease in the velocity of the fluid along the y-axis and the effect of the magnetic field along the y-axis.
- It is concluded that for the magnetic Reynolds number , a decrease in the flow velocity along the z-axis was observed with increasing . On the other hand, velocity along the y-axis showed an increasing effect by increasing the flow velocity along the x-axis; this showed a decreasing pattern initially, but as , an increasing effect was observed.
- It was also observed that for the magnetic field, increasing the magnetic Reynolds number showed a decrease in the value of the magnetic field along both the y- and z-axes.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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m | CPU Time | ||||
---|---|---|---|---|---|
1 | 5.56818 | 1.50838 | 2.75325 | 3.42075 | 0.42 s |
5 | 1.04968 | 4.31742 | 7.60909 | 1.89839 | 4.19 s |
10 | 1.48326 | 9.10580 | 1.64090 | 4.81645 | 18.5 s |
15 | 1.47350 | 9.86847 | 1.86358 | 1.38805 | 33.5 s |
20 | 1.48626 | 9.78372 | 1.79738 | 1.50240 | 61.6 s |
25 | 1.47036 | 9.78372 | 1.79738 | 1.48434 | 106.08 s |
30 | 1.47036 | 9.78372 | 1.79738 | 1.48434 | 163.13 s |
35 | 1.47036 | 9.78372 | 1.79738 | 1.48434 | 240.95 s |
40 | 1.47036 | 9.78372 | 1.79738 | 1.48434 | 456.57 s |
Order | |||||
---|---|---|---|---|---|
2 | −1.00242 | −1.02522 | −0.99578 | −1.01863 | 7.07983 |
3 | −1.00597 | −1.02361 | −0.98895 | −1.00881 | 3.11944 |
4 | −1.00854 | −1.02152 | −0.98596 | −1.01663 | 1.17465 |
5 | −1.02616 | −1.03854 | −0.96740 | −0.97698 | −3.79267 |
6 | −0.93502 | −1.07429 | −1.07428 | −0.94309 | 2.53023 |
7 | −0.89878 | −0.95454 | −1.09525 | −0.90992 | 3.93815 |
0. | 0. | 1. | 0. | 0. |
0.1001 | 0.013999 | 1.014380 | 0.099721 | 0.099865 |
0.2002 | 0.052032 | 1.026218 | 0.199465 | 0.199696 |
0.3003 | 0.108119 | 1.034940 | 0.299257 | 0.299529 |
0.4004 | 0.176245 | 1.040324 | 0.399126 | 0.399401 |
0.5005 | 0.250378 | 1.042131 | 0.499097 | 0.499346 |
0.6006 | 0.324481 | 1.040296 | 0.599192 | 0.599396 |
0.7007 | 0.392515 | 1.034878 | 0.699428 | 0.699577 |
0.8008 | 0.448449 | 1.026084 | 0.799814 | 0.799904 |
0.9009 | 0.486271 | 1.014256 | 0.900345 | 0.900383 |
1. | 0.5 | 1. | 1. | 1. |
m | ||||
---|---|---|---|---|
1 | 2.990612172696 | −0.153255933196 | −0.996111208570 | −1.011230294203 |
5 | 2.990713652636 | −0.154752127452 | −0.996176676870 | −1.011568276329 |
10 | 2.990713652641 | −0.154752127484 | −0.996176676875 | −1.011568276329 |
15 | 2.990713652641 | −0.154752127484 | −0.996176676875 | −1.011568276329 |
20 | 2.990713652641 | −0.154752127484 | −0.996176676875 | −1.011568276329 |
25 | 2.990713652641 | −0.154752127484 | −0.996176676875 | −1.011568276329 |
30 | 2.990713652641 | −0.154752127484 | −0.996176676875 | −1.011568276329 |
35 | 2.990713652641 | −0.154752127484 | −0.996176676875 | −1.011568276329 |
40 | 2.990713652641 | −0.154752127484 | −0.996176676875 | −1.011568276329 |
−0.1 | 2.990713 | −0.154752 | −0.996176 | −1.011568 |
−0.5 | 2.953293 | −0.888098 | −0.996174 | −1.013783 |
−0.75 | 2.929678 | −1.472508 | −0.996172 | −1.015593 |
−1.1 | 2.896317 | −2.546940 | −0.996170 | −1.018992 |
0.1 | 2.990701 | −0.154715 | −0.996177 | −0.997930 |
0.5 | 2.990713 | −0.154752 | −0.996177 | −1.011568 |
1 | 2.990753 | −0.154882 | −0.996177 | −1.028616 |
1.5 | 2.990818 | −0.155101 | −0.996177 | −1.045666 |
1 | 2.990713 | −0.154740 | −0.996177 | −0.999635 |
3 | 2.990713 | −0.154705 | −0.996177 | −0.996226 |
5 | 2.990713 | −0.154670 | −0.996177 | −0.995544 |
7 | 2.990713 | −0.154635 | −0.996177 | −0.995252 |
0.1 | 2.990819 | −0.155092 | −0.962698 | −1.110618 |
0.5 | 2.991071 | −0.156280 | −0.831865 | −1.456702 |
1 | 2.991040 | −0.157259 | −0.701194 | −1.729091 |
1.5 | 2.990777 | −0.157899 | −0.597468 | −1.883300 |
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Alam, M.K.; Bibi, K.; Khan, A.; Fernandez-Gamiz, U.; Noeiaghdam, S. The Effect of Variable Magnetic Field on Viscous Fluid between 3-D Rotatory Vertical Squeezing Plates: A Computational Investigation. Energies 2022, 15, 2473. https://doi.org/10.3390/en15072473
Alam MK, Bibi K, Khan A, Fernandez-Gamiz U, Noeiaghdam S. The Effect of Variable Magnetic Field on Viscous Fluid between 3-D Rotatory Vertical Squeezing Plates: A Computational Investigation. Energies. 2022; 15(7):2473. https://doi.org/10.3390/en15072473
Chicago/Turabian StyleAlam, Muhammad Kamran, Khadija Bibi, Aamir Khan, Unai Fernandez-Gamiz, and Samad Noeiaghdam. 2022. "The Effect of Variable Magnetic Field on Viscous Fluid between 3-D Rotatory Vertical Squeezing Plates: A Computational Investigation" Energies 15, no. 7: 2473. https://doi.org/10.3390/en15072473