Shear Thinning in the Prandtl Model and Its Relation to Generalized Newtonian Fluids
The Prandtl model is certainly the simplest and most generic microscopic model describing solid friction. It consists of a single, thermalized atom attached to a spring, which is dragged past a sinusoidal potential representing the surface energy corrugation of a counterface. While it was primarily...
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MDPI AG
2020-03-01
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| Online Access: | https://www.mdpi.com/2075-4442/8/4/38 |
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| author | Martin H. Müser |
| author_facet | Martin H. Müser |
| author_sort | Martin H. Müser |
| collection | DOAJ |
| description | The Prandtl model is certainly the simplest and most generic microscopic model describing solid friction. It consists of a single, thermalized atom attached to a spring, which is dragged past a sinusoidal potential representing the surface energy corrugation of a counterface. While it was primarily introduced to rationalize how Coulomb’s friction law can arise from small-scale instabilities, Prandtl argued that his model also describes the shear thinning of liquids. Given its success regarding the interpretation of atomic-force-microscopy experiments, surprisingly little attention has been paid to the question how the Prandtl model relates to fluid rheology. Analyzing its Langevin and Brownian dynamics, we show that the Prandtl model produces friction−velocity relationships, which, converted to a dependence of effective (excess) viscosity on shear rate <inline-formula> <math display="inline"> <semantics> <mrow> <mi>η</mi> <mo>(</mo> <mover accent="true"> <mi>γ</mi> <mo>˙</mo> </mover> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, is strikingly similar to the Carreau−Yasuda (CY) relation, which is obeyed by many non-Newtonian liquids. The two dimensionless parameters in the CY relation are found to span a broad range of values. When thermal energy is small compared to the corrugation of the sinusoidal potential, the leading-order <inline-formula> <math display="inline"> <semantics> <msup> <mover accent="true"> <mi>γ</mi> <mo>˙</mo> </mover> <mn>2</mn> </msup> </semantics> </math> </inline-formula> corrections to the equilibrium viscosity only matter in the initial part of the cross-over from Stokes friction to the regime, where <inline-formula> <math display="inline"> <semantics> <mi>η</mi> </semantics> </math> </inline-formula> obeys approximately a sublinear power law of <inline-formula> <math display="inline"> <semantics> <mrow> <mn>1</mn> <mo>/</mo> <mover accent="true"> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </semantics> </math> </inline-formula>. |
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| language | English |
| last_indexed | 2024-04-11T14:07:12Z |
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| spelling | doaj.art-990d9a466625488c96a68f15644bc6a72022-12-22T04:19:51ZengMDPI AGLubricants2075-44422020-03-01843810.3390/lubricants8040038lubricants8040038Shear Thinning in the Prandtl Model and Its Relation to Generalized Newtonian FluidsMartin H. Müser0Department of Materials Science and Engineering, Universität des Saarlandes, 66123 Saarbrücken, GermanyThe Prandtl model is certainly the simplest and most generic microscopic model describing solid friction. It consists of a single, thermalized atom attached to a spring, which is dragged past a sinusoidal potential representing the surface energy corrugation of a counterface. While it was primarily introduced to rationalize how Coulomb’s friction law can arise from small-scale instabilities, Prandtl argued that his model also describes the shear thinning of liquids. Given its success regarding the interpretation of atomic-force-microscopy experiments, surprisingly little attention has been paid to the question how the Prandtl model relates to fluid rheology. Analyzing its Langevin and Brownian dynamics, we show that the Prandtl model produces friction−velocity relationships, which, converted to a dependence of effective (excess) viscosity on shear rate <inline-formula> <math display="inline"> <semantics> <mrow> <mi>η</mi> <mo>(</mo> <mover accent="true"> <mi>γ</mi> <mo>˙</mo> </mover> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, is strikingly similar to the Carreau−Yasuda (CY) relation, which is obeyed by many non-Newtonian liquids. The two dimensionless parameters in the CY relation are found to span a broad range of values. When thermal energy is small compared to the corrugation of the sinusoidal potential, the leading-order <inline-formula> <math display="inline"> <semantics> <msup> <mover accent="true"> <mi>γ</mi> <mo>˙</mo> </mover> <mn>2</mn> </msup> </semantics> </math> </inline-formula> corrections to the equilibrium viscosity only matter in the initial part of the cross-over from Stokes friction to the regime, where <inline-formula> <math display="inline"> <semantics> <mi>η</mi> </semantics> </math> </inline-formula> obeys approximately a sublinear power law of <inline-formula> <math display="inline"> <semantics> <mrow> <mn>1</mn> <mo>/</mo> <mover accent="true"> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </semantics> </math> </inline-formula>.https://www.mdpi.com/2075-4442/8/4/38frictionviscosityshear thinningtheorymolecular dynamicsfokker planck equationblood flow |
| spellingShingle | Martin H. Müser Shear Thinning in the Prandtl Model and Its Relation to Generalized Newtonian Fluids Lubricants friction viscosity shear thinning theory molecular dynamics fokker planck equation blood flow |
| title | Shear Thinning in the Prandtl Model and Its Relation to Generalized Newtonian Fluids |
| title_full | Shear Thinning in the Prandtl Model and Its Relation to Generalized Newtonian Fluids |
| title_fullStr | Shear Thinning in the Prandtl Model and Its Relation to Generalized Newtonian Fluids |
| title_full_unstemmed | Shear Thinning in the Prandtl Model and Its Relation to Generalized Newtonian Fluids |
| title_short | Shear Thinning in the Prandtl Model and Its Relation to Generalized Newtonian Fluids |
| title_sort | shear thinning in the prandtl model and its relation to generalized newtonian fluids |
| topic | friction viscosity shear thinning theory molecular dynamics fokker planck equation blood flow |
| url | https://www.mdpi.com/2075-4442/8/4/38 |
| work_keys_str_mv | AT martinhmuser shearthinningintheprandtlmodelanditsrelationtogeneralizednewtonianfluids |