Impact of Addition of a Newtonian Solvent to a Giesekus Fluid: Analytical Determination of Flow Rate in Plane Laminar Motion
In this study, the influence of the presence of a Newtonian solvent on the flow of a Giesekus fluid in a plane channel or fracture is investigated with a focus on the determination of the flow rate for an assigned external pressure gradient. The pressure field is nonlinear due to the presence of the...
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MDPI AG
2024-12-01
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| Series: | Fluids |
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| author | Irene Daprà Giambattista Scarpi Vittorio Di Federico |
| author_facet | Irene Daprà Giambattista Scarpi Vittorio Di Federico |
| author_sort | Irene Daprà |
| collection | DOAJ |
| description | In this study, the influence of the presence of a Newtonian solvent on the flow of a Giesekus fluid in a plane channel or fracture is investigated with a focus on the determination of the flow rate for an assigned external pressure gradient. The pressure field is nonlinear due to the presence of the normal transverse stress component. As expected, the flow rate per unit width <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></semantics></math></inline-formula> is larger than for a Newtonian fluid and decreases as the solvent increases. It is strongly dependent on the viscosity ratio <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi></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>ε</mi><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula>), the dimensionless mobility parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo> </mo></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>β</mi><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula>) and the Deborah number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mi>e</mi></mrow></semantics></math></inline-formula>, the dimensionless driving pressure gradient. The degree of dependency is notably strong in the low range of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi></mrow></semantics></math></inline-formula>. Furthermore, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></semantics></math></inline-formula> increases with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mi>e</mi></mrow></semantics></math></inline-formula> and tends to a constant asymptotic value for large <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mi>e</mi></mrow></semantics></math></inline-formula>, subject to the limitation of laminar flow. When the mobility factor <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi></mrow></semantics></math></inline-formula> is in the range <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced open="[" close="]" separators="|"><mrow><mn>0.5</mn><mo>÷</mo><mn>1</mn></mrow></mfenced></mrow></semantics></math></inline-formula>, there is a minimum value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><mo> </mo></mrow></semantics></math></inline-formula> to obtain an assigned value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mi>e</mi></mrow></semantics></math></inline-formula>. The ratio <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>U</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>/</mo><mi>U</mi></mrow></semantics></math></inline-formula> between Newtonian and actual mean velocity depends only on the product <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msqrt><mi>β</mi></msqrt><mi>D</mi><mi>e</mi></mrow></semantics></math></inline-formula>, as for other non-Newtonian fluids. |
| format | Article |
| id | doaj-art-f14b6e179975493e87d7c7a7adbf778e |
| institution | Kabale University |
| issn | 2311-5521 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fluids |
| spelling | doaj-art-f14b6e179975493e87d7c7a7adbf778e2025-01-24T13:32:32ZengMDPI AGFluids2311-55212024-12-01101110.3390/fluids10010001Impact of Addition of a Newtonian Solvent to a Giesekus Fluid: Analytical Determination of Flow Rate in Plane Laminar MotionIrene Daprà0Giambattista Scarpi1Vittorio Di Federico2Department of Civil, Chemical, Environmental, and Materials Engineering, University of Bologna, 40136 Bologna, ItalyDepartment of Civil, Chemical, Environmental, and Materials Engineering, University of Bologna, 40136 Bologna, ItalyDepartment of Civil, Chemical, Environmental, and Materials Engineering, University of Bologna, 40136 Bologna, ItalyIn this study, the influence of the presence of a Newtonian solvent on the flow of a Giesekus fluid in a plane channel or fracture is investigated with a focus on the determination of the flow rate for an assigned external pressure gradient. The pressure field is nonlinear due to the presence of the normal transverse stress component. As expected, the flow rate per unit width <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></semantics></math></inline-formula> is larger than for a Newtonian fluid and decreases as the solvent increases. It is strongly dependent on the viscosity ratio <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi></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>ε</mi><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula>), the dimensionless mobility parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo> </mo></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>β</mi><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula>) and the Deborah number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mi>e</mi></mrow></semantics></math></inline-formula>, the dimensionless driving pressure gradient. The degree of dependency is notably strong in the low range of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi></mrow></semantics></math></inline-formula>. Furthermore, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></semantics></math></inline-formula> increases with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mi>e</mi></mrow></semantics></math></inline-formula> and tends to a constant asymptotic value for large <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mi>e</mi></mrow></semantics></math></inline-formula>, subject to the limitation of laminar flow. When the mobility factor <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi></mrow></semantics></math></inline-formula> is in the range <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced open="[" close="]" separators="|"><mrow><mn>0.5</mn><mo>÷</mo><mn>1</mn></mrow></mfenced></mrow></semantics></math></inline-formula>, there is a minimum value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><mo> </mo></mrow></semantics></math></inline-formula> to obtain an assigned value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mi>e</mi></mrow></semantics></math></inline-formula>. The ratio <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>U</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>/</mo><mi>U</mi></mrow></semantics></math></inline-formula> between Newtonian and actual mean velocity depends only on the product <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msqrt><mi>β</mi></msqrt><mi>D</mi><mi>e</mi></mrow></semantics></math></inline-formula>, as for other non-Newtonian fluids.https://www.mdpi.com/2311-5521/10/1/1Giesekus modelflow rateviscoelastic fluidNewtonian solventanalytical solutionplane flow |
| spellingShingle | Irene Daprà Giambattista Scarpi Vittorio Di Federico Impact of Addition of a Newtonian Solvent to a Giesekus Fluid: Analytical Determination of Flow Rate in Plane Laminar Motion Fluids Giesekus model flow rate viscoelastic fluid Newtonian solvent analytical solution plane flow |
| title | Impact of Addition of a Newtonian Solvent to a Giesekus Fluid: Analytical Determination of Flow Rate in Plane Laminar Motion |
| title_full | Impact of Addition of a Newtonian Solvent to a Giesekus Fluid: Analytical Determination of Flow Rate in Plane Laminar Motion |
| title_fullStr | Impact of Addition of a Newtonian Solvent to a Giesekus Fluid: Analytical Determination of Flow Rate in Plane Laminar Motion |
| title_full_unstemmed | Impact of Addition of a Newtonian Solvent to a Giesekus Fluid: Analytical Determination of Flow Rate in Plane Laminar Motion |
| title_short | Impact of Addition of a Newtonian Solvent to a Giesekus Fluid: Analytical Determination of Flow Rate in Plane Laminar Motion |
| title_sort | impact of addition of a newtonian solvent to a giesekus fluid analytical determination of flow rate in plane laminar motion |
| topic | Giesekus model flow rate viscoelastic fluid Newtonian solvent analytical solution plane flow |
| url | https://www.mdpi.com/2311-5521/10/1/1 |
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