Two-Fluid Classical and Momentumless Laminar Far Wakes
Two-dimensional two-fluid classical and momentumless laminar far wakes are investigated in the boundary layer approximation. The velocity deficit satisfies a linear diffusion equation and the continuity equation in the upper and lower parts of the wakes. By using the multiplier method, conservation...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-04-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/5/961 |
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| author | Kiara Pillay David Paul Mason |
| author_facet | Kiara Pillay David Paul Mason |
| author_sort | Kiara Pillay |
| collection | DOAJ |
| description | Two-dimensional two-fluid classical and momentumless laminar far wakes are investigated in the boundary layer approximation. The velocity deficit satisfies a linear diffusion equation and the continuity equation in the upper and lower parts of the wakes. By using the multiplier method, conservation laws for the system of partial differential equations (PDEs) in the upper and lower parts of the wake are derived. Lie point symmetries associated with the conserved vectors for the classical and momentumless wakes are obtained. The conserved quantity for the two-fluid classical wake is the total drag on the obstacle, which is rederived. A new conserved quantity for the two-fluid momentumless wake is obtained, which satisfies the condition that the total drag on the obstacle is zero. Using the conserved quantities, it is shown that the equation of the interface is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mo>=</mo><mi>k</mi><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mrow></semantics></math></inline-formula>, where <i>k</i> is a constant and <i>x</i> and <i>y</i> are Cartesian coordinates with origin at the trailing edge of the obstacle. New invariant solutions for the two-fluid classical and momentumless wakes with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> are found. Both solutions depend on the dimensionless parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>χ</mi><mo>=</mo><mrow><mo>(</mo><msub><mi>ρ</mi><mn>1</mn></msub><msub><mi>μ</mi><mn>1</mn></msub><mo>)</mo></mrow><mo>/</mo><mrow><mo>(</mo><msub><mi>ρ</mi><mn>2</mn></msub><msub><mi>μ</mi><mn>2</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> where suffices 1 and 2 refer to the upper and lower parts of the wake. For the special case in which the kinematic viscosity ratio <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ν</mi><mn>2</mn></msub><mo>/</mo><msub><mi>ν</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, two further solutions for the two-fluid momentumless wake are derived with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><mo>±</mo><msqrt><mn>6</mn></msqrt></mrow></semantics></math></inline-formula>. |
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| language | English |
| last_indexed | 2024-03-11T03:16:36Z |
| publishDate | 2023-04-01 |
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| series | Symmetry |
| spelling | doaj.art-ab82c6b913e24045af60e91014dd40902023-11-18T03:28:54ZengMDPI AGSymmetry2073-89942023-04-0115596110.3390/sym15050961Two-Fluid Classical and Momentumless Laminar Far WakesKiara Pillay0David Paul Mason1School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg 2050, South AfricaSchool of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg 2050, South AfricaTwo-dimensional two-fluid classical and momentumless laminar far wakes are investigated in the boundary layer approximation. The velocity deficit satisfies a linear diffusion equation and the continuity equation in the upper and lower parts of the wakes. By using the multiplier method, conservation laws for the system of partial differential equations (PDEs) in the upper and lower parts of the wake are derived. Lie point symmetries associated with the conserved vectors for the classical and momentumless wakes are obtained. The conserved quantity for the two-fluid classical wake is the total drag on the obstacle, which is rederived. A new conserved quantity for the two-fluid momentumless wake is obtained, which satisfies the condition that the total drag on the obstacle is zero. Using the conserved quantities, it is shown that the equation of the interface is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mo>=</mo><mi>k</mi><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mrow></semantics></math></inline-formula>, where <i>k</i> is a constant and <i>x</i> and <i>y</i> are Cartesian coordinates with origin at the trailing edge of the obstacle. New invariant solutions for the two-fluid classical and momentumless wakes with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> are found. Both solutions depend on the dimensionless parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>χ</mi><mo>=</mo><mrow><mo>(</mo><msub><mi>ρ</mi><mn>1</mn></msub><msub><mi>μ</mi><mn>1</mn></msub><mo>)</mo></mrow><mo>/</mo><mrow><mo>(</mo><msub><mi>ρ</mi><mn>2</mn></msub><msub><mi>μ</mi><mn>2</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> where suffices 1 and 2 refer to the upper and lower parts of the wake. For the special case in which the kinematic viscosity ratio <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ν</mi><mn>2</mn></msub><mo>/</mo><msub><mi>ν</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, two further solutions for the two-fluid momentumless wake are derived with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><mo>±</mo><msqrt><mn>6</mn></msqrt></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2073-8994/15/5/961two-fluid classical and momentumless wakesmultiplier methodconservation lawconserved quantityassociated Lie point symmetryinvariant solution |
| spellingShingle | Kiara Pillay David Paul Mason Two-Fluid Classical and Momentumless Laminar Far Wakes Symmetry two-fluid classical and momentumless wakes multiplier method conservation law conserved quantity associated Lie point symmetry invariant solution |
| title | Two-Fluid Classical and Momentumless Laminar Far Wakes |
| title_full | Two-Fluid Classical and Momentumless Laminar Far Wakes |
| title_fullStr | Two-Fluid Classical and Momentumless Laminar Far Wakes |
| title_full_unstemmed | Two-Fluid Classical and Momentumless Laminar Far Wakes |
| title_short | Two-Fluid Classical and Momentumless Laminar Far Wakes |
| title_sort | two fluid classical and momentumless laminar far wakes |
| topic | two-fluid classical and momentumless wakes multiplier method conservation law conserved quantity associated Lie point symmetry invariant solution |
| url | https://www.mdpi.com/2073-8994/15/5/961 |
| work_keys_str_mv | AT kiarapillay twofluidclassicalandmomentumlesslaminarfarwakes AT davidpaulmason twofluidclassicalandmomentumlesslaminarfarwakes |