Ordered Rate Constitutive Theories for Non-Classical Thermofluids Based on Convected Time Derivatives of the Strain and Higher Order Rotation Rate Tensors Using Entropy Inequality
This paper considers non-classical continuum theory for thermoviscous fluids without memory incorporating internal rotation rates resulting from the antisymmetric part of the velocity gradient tensor to derive ordered rate constitutive theories for the Cauchy stress and the Cauchy moment tensor base...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-04-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/22/4/443 |
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| author | Karan S. Surana Stephen W. Long |
| author_facet | Karan S. Surana Stephen W. Long |
| author_sort | Karan S. Surana |
| collection | DOAJ |
| description | This paper considers non-classical continuum theory for thermoviscous fluids without memory incorporating internal rotation rates resulting from the antisymmetric part of the velocity gradient tensor to derive ordered rate constitutive theories for the Cauchy stress and the Cauchy moment tensor based on entropy inequality and representation theorem. Using the generalization of the conjugate pairs in the entropy inequality, the ordered rate constitutive theory for Cauchy stress tensor considers convected time derivatives of the Green’s strain tensor (or Almansi strain tensor) of up to orders <inline-formula> <math display="inline"> <semantics> <msub> <mi>n</mi> <mi>ε</mi> </msub> </semantics> </math> </inline-formula> as its argument tensors and the ordered rate constitutive theory for the Cauchy moment tensor considers convected time derivatives of the symmetric part of the rotation gradient tensor up to orders <inline-formula> <math display="inline"> <semantics> <msub> <mi>n</mi> <mi mathvariant="sans-serif">Θ</mi> </msub> </semantics> </math> </inline-formula>. While the convected time derivatives of the strain tensors are well known the convected time derivatives of higher orders of the symmetric part of the rotation gradient tensor need to be derived and are presented in this paper. Complete and general constitutive theories based on integrity using conjugate pairs in the entropy inequality and the generalization of the argument tensors of the constitutive variables and the representation theorem are derived and the material coefficients are established. It is shown that for the type of non-classical thermofluids considered in this paper the dissipation mechanism is an ordered rate mechanism due to convected time derivatives of the strain tensor as well as the convected time derivatives of the symmetric part of the rotation gradient tensor. The derivations of the constitutive theories presented in the paper is basis independent but can be made basis specific depending upon the choice of the specific basis for the constitutive variables and the argument tensors. Simplified linear theories are also presented as subset of the general constitutive theories and are compared with published works. |
| first_indexed | 2024-03-10T20:29:27Z |
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| id | doaj.art-dd57e4f2ffd24bb7800e821c6a549ae8 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T20:29:27Z |
| publishDate | 2020-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-dd57e4f2ffd24bb7800e821c6a549ae82023-11-19T21:34:41ZengMDPI AGEntropy1099-43002020-04-0122444310.3390/e22040443Ordered Rate Constitutive Theories for Non-Classical Thermofluids Based on Convected Time Derivatives of the Strain and Higher Order Rotation Rate Tensors Using Entropy InequalityKaran S. Surana0Stephen W. Long1Department of Mechanical Engineering, University of Kansas, Lawrence, KS 66048, USADepartment of Mechanical Engineering, University of Kansas, Lawrence, KS 66048, USAThis paper considers non-classical continuum theory for thermoviscous fluids without memory incorporating internal rotation rates resulting from the antisymmetric part of the velocity gradient tensor to derive ordered rate constitutive theories for the Cauchy stress and the Cauchy moment tensor based on entropy inequality and representation theorem. Using the generalization of the conjugate pairs in the entropy inequality, the ordered rate constitutive theory for Cauchy stress tensor considers convected time derivatives of the Green’s strain tensor (or Almansi strain tensor) of up to orders <inline-formula> <math display="inline"> <semantics> <msub> <mi>n</mi> <mi>ε</mi> </msub> </semantics> </math> </inline-formula> as its argument tensors and the ordered rate constitutive theory for the Cauchy moment tensor considers convected time derivatives of the symmetric part of the rotation gradient tensor up to orders <inline-formula> <math display="inline"> <semantics> <msub> <mi>n</mi> <mi mathvariant="sans-serif">Θ</mi> </msub> </semantics> </math> </inline-formula>. While the convected time derivatives of the strain tensors are well known the convected time derivatives of higher orders of the symmetric part of the rotation gradient tensor need to be derived and are presented in this paper. Complete and general constitutive theories based on integrity using conjugate pairs in the entropy inequality and the generalization of the argument tensors of the constitutive variables and the representation theorem are derived and the material coefficients are established. It is shown that for the type of non-classical thermofluids considered in this paper the dissipation mechanism is an ordered rate mechanism due to convected time derivatives of the strain tensor as well as the convected time derivatives of the symmetric part of the rotation gradient tensor. The derivations of the constitutive theories presented in the paper is basis independent but can be made basis specific depending upon the choice of the specific basis for the constitutive variables and the argument tensors. Simplified linear theories are also presented as subset of the general constitutive theories and are compared with published works.https://www.mdpi.com/1099-4300/22/4/443ordered rateconstitutive theoriesconvected time derivativesstrain rategradients of rotation ratesrepresentation theorem |
| spellingShingle | Karan S. Surana Stephen W. Long Ordered Rate Constitutive Theories for Non-Classical Thermofluids Based on Convected Time Derivatives of the Strain and Higher Order Rotation Rate Tensors Using Entropy Inequality Entropy ordered rate constitutive theories convected time derivatives strain rate gradients of rotation rates representation theorem |
| title | Ordered Rate Constitutive Theories for Non-Classical Thermofluids Based on Convected Time Derivatives of the Strain and Higher Order Rotation Rate Tensors Using Entropy Inequality |
| title_full | Ordered Rate Constitutive Theories for Non-Classical Thermofluids Based on Convected Time Derivatives of the Strain and Higher Order Rotation Rate Tensors Using Entropy Inequality |
| title_fullStr | Ordered Rate Constitutive Theories for Non-Classical Thermofluids Based on Convected Time Derivatives of the Strain and Higher Order Rotation Rate Tensors Using Entropy Inequality |
| title_full_unstemmed | Ordered Rate Constitutive Theories for Non-Classical Thermofluids Based on Convected Time Derivatives of the Strain and Higher Order Rotation Rate Tensors Using Entropy Inequality |
| title_short | Ordered Rate Constitutive Theories for Non-Classical Thermofluids Based on Convected Time Derivatives of the Strain and Higher Order Rotation Rate Tensors Using Entropy Inequality |
| title_sort | ordered rate constitutive theories for non classical thermofluids based on convected time derivatives of the strain and higher order rotation rate tensors using entropy inequality |
| topic | ordered rate constitutive theories convected time derivatives strain rate gradients of rotation rates representation theorem |
| url | https://www.mdpi.com/1099-4300/22/4/443 |
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