Analysis of a Bifurcation and Stability of Equilibrium Points for Jeffrey Fluid Flow through a Non-Uniform Channel
<p>The graphical representation of the geometry of a wall surface.</p> "> Figure 2
<p>A streamline pattern is shown for system (<a href="#FD18-symmetry-16-01144" class="html-disp-formula">18</a>) with different value flow rates <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mn>1</mn> </msub> <mo>∈</mo> <mrow> <mo>(</mo> <mn>0.5</mn> <mo>,</mo> <mn>1.9</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> with constant amplitude <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>. Panel (<b>A<sub>1</sub></b>) shows the backward flow of fluid with <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math> and there are just saddle points. The Panels (<b>A<sub>2</sub></b>,<b>A<sub>3</sub></b>) display two significant changes physically and dynamically with <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.089</mn> <mo>,</mo> <mo> </mo> <mn>1.3</mn> </mrow> </semantics></math> birth red/black critical points refer to center and saddle points, respectively. Also, the appearance of a trapping zone is inside the heteroclinic connection that is created between two different saddle points. The last panel, (<b>A<sub>4</sub></b>), illustrates augmented flow where <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.7</mn> </mrow> </semantics></math> and changes the number of saddle points with a new formation heteroclinic connection.</p> "> Figure 3
<p>(The description is as for <a href="#symmetry-16-01144-f002" class="html-fig">Figure 2</a>, except for the change in the value of the parameter <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>). Streamline patterns for different value flow rates <math display="inline"><semantics> <msub> <mi>Q</mi> <mn>1</mn> </msub> </semantics></math> with constant amplitude <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>. Panel (<b>B<sub>1</sub></b>) shows the backward flow of fluid with <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.85</mn> </mrow> </semantics></math>, and there are just saddle points. Panels (<b>B<sub>2</sub></b>,<b>B<sub>3</sub></b>) display two significant changes physically and dynamically with <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <mn>1.3</mn> </mrow> </semantics></math> birth red/black critical points refer to center and saddle points, respectively. In addition, there is an appearance of a trapping zone inside the heteroclinic connection that is created between two different saddle points. The last panel, (<b>B<sub>4</sub></b>), illustrates augmented flow where <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math> and changes the number of saddle points with a new formation heteroclinic connection.</p> "> Figure 4
<p>This figure shows the bifurcation diagram with <math display="inline"><semantics> <msub> <mi>Q</mi> <mn>1</mn> </msub> </semantics></math> against <math display="inline"><semantics> <msub> <mi>u</mi> <mn>2</mn> </msub> </semantics></math> for Equation (<a href="#FD18-symmetry-16-01144" class="html-disp-formula">18</a>) with the various values of parameters <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mo> </mo> <mn>0.7</mn> <mo>,</mo> <mo> </mo> <mn>0.9</mn> </mrow> </semantics></math> in the panels (<b>A</b>, <b>B</b> and <b>C</b>), respectively. Red/black lines refer to stable/unstable points, green indicates periodic orbit, and a single saddle node is represented by a blue line. At (<b>E</b>), there are two branch lines of saddle-node bifurcation (dash–circle lines) and three regions have a light color (blue, yellow, green) that indicate the existence of three complicated behaviors (backward, trapping, augment), respectively. The symbols I, III, and IV are the number of critical points in every zone.</p> ">
Abstract
:1. Introduction
2. Mathematical Formulations
3. Solution Method
4. Rate of Volume Flow
5. Solution of the Problem
6. The Nonlinear System and Its Bifurcation
- ,
- ,
- ,
- .
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
List of Symbols
geometry of the flow channel | dimensionless stream function | ||
U | velocity field | stream function | |
Cartesian coordinate in the fixed frame | dimensionless volume flow rate | ||
dimensionless cartesian coordinate in the wave frame | volume of flow rate | ||
Cartesian coordinate in the wave frame | extra stress tensor | ||
average radius of the tube | shear rate | ||
d | amplitude of a peristaltic wave | material derivative | |
wavelength | gradient of velocity | ||
wave propagation speed | fluid density | ||
time | pressure in the fixed frame | ||
velocity components in the fixed frame | pressure in the wave frame | ||
velocity components in the wave frame | p | dimensionless pressure in the wave frame | |
elocity components in the wave frame | fluid dynamic viscosity | ||
ratio of relaxation to retardation times | retartation time | ||
instantaneous volume of the flow rate in a wave frame | dimensionless volume of the flow rate | ||
instantaneous volume of the flow rate in a fixed frame | dimensionless wave number |
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Thoubaan, M.G.; Al-Khafajy, D.G.S.; Wanas, A.K.; Breaz, D.; Cotîrlă, L.-I. Analysis of a Bifurcation and Stability of Equilibrium Points for Jeffrey Fluid Flow through a Non-Uniform Channel. Symmetry 2024, 16, 1144. https://doi.org/10.3390/sym16091144
Thoubaan MG, Al-Khafajy DGS, Wanas AK, Breaz D, Cotîrlă L-I. Analysis of a Bifurcation and Stability of Equilibrium Points for Jeffrey Fluid Flow through a Non-Uniform Channel. Symmetry. 2024; 16(9):1144. https://doi.org/10.3390/sym16091144
Chicago/Turabian StyleThoubaan, Mary G., Dheia G. Salih Al-Khafajy, Abbas Kareem Wanas, Daniel Breaz, and Luminiţa-Ioana Cotîrlă. 2024. "Analysis of a Bifurcation and Stability of Equilibrium Points for Jeffrey Fluid Flow through a Non-Uniform Channel" Symmetry 16, no. 9: 1144. https://doi.org/10.3390/sym16091144