Existence and Uniqueness of the Viscous Burgers’ Equation with the p-Laplace Operator
In this paper, we investigate the existence and uniqueness of solutions for the viscous Burgers’ equation for the isothermal flow of power-law non-Newtonian fluids <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><m...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-02-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/5/708 |
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| Summary: | In this paper, we investigate the existence and uniqueness of solutions for the viscous Burgers’ equation for the isothermal flow of power-law non-Newtonian fluids <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ρ</mi><mrow><mo>(</mo><msub><mo>∂</mo><mi>t</mi></msub><mi>u</mi><mo>+</mo><mi>u</mi><msub><mo>∂</mo><mi>x</mi></msub><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mi>μ</mi><msub><mo>∂</mo><mi>x</mi></msub><mfenced separators="" open="(" close=")"><msup><mfenced separators="" open="|" close="|"><msub><mo>∂</mo><mi>x</mi></msub><mi>u</mi></mfenced><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><msub><mo>∂</mo><mi>x</mi></msub><mi>u</mi></mfenced><mo>,</mo></mrow></semantics></math></inline-formula> augmented with the initial condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msub><mi>u</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>x</mi><mo><</mo><mi>L</mi></mrow></semantics></math></inline-formula>, and the boundary condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>L</mi><mo>)</mo><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula> is the density, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> the viscosity, <i>u</i> the velocity of the fluid, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. We show that this initial boundary problem has an unique solution in the Buchner space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mn>2</mn></msup><mfenced separators="" open="(" close=")"><mn>0</mn><mo>,</mo><mi>T</mi><mo>;</mo><msup><mrow><msub><mi>W</mi><mn>0</mn></msub></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msup><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mfenced></mrow></semantics></math></inline-formula> for the given set of conditions. Moreover, numerical solutions to the problem are constructed by applying the modeling and simulation package COMSOL Multiphysics 6.0 at small and large Reynolds numbers to show the images of the solutions. |
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| ISSN: | 2227-7390 |