scholarly journals Ordered Rate Constitutive Theories for Non-Classical Thermoviscoelastic Fluids with Internal Rotation Rates

2018 ◽  
Vol 09 (08) ◽  
pp. 907-939 ◽  
Author(s):  
K. S. Surana ◽  
S. W. Long ◽  
J. N. Reddy
Keyword(s):  
Internal Rotation ◽  
Constitutive Theories ◽  
Rotation Rates ◽  
Ordered Rate Constitutive Theories

Ordered Rate Constitutive Theories for Non-Classical Thermoviscoelastic Fluids Incorporating Internal and Cosserat Rotation Rates

2018 ◽  
Vol 10 (02) ◽  
pp. 1850012 ◽  
Author(s):  
K. S. Surana ◽  
A. D. Joy ◽  
J. N. Reddy
Keyword(s):  
Internal Rotation ◽  
Velocity Gradient ◽  
Cauchy Stress ◽  
Balance Laws ◽  
Gradient Tensor ◽  
Moment Tensors ◽  
Velocity Gradient Tensor ◽  
Constitutive Theories ◽  
Rotation Rates ◽  
Derivatives Of

This paper considers conservation and balance laws for non-classical fluent continua in the presence of internal rotation rates due to the velocity gradient tensor and the rotation rates due to Cosserat rotations. In these balance laws, the internal rotation rates are completely defined as functions of the velocity gradient tensor, but the Cosserat rotation rates are additional three degrees of freedom at each material point. When these rotation rates are resisted by the deforming continua, conjugate moments are created. For thermoviscoelastic fluent continua, these result in additional dissipation mechanism as well as rheology. This paper presents a thermodynamically consistent derivation of constitutive theories for such fluids based on the entropy inequality in conjunction with the representation theorem using integrity, i.e., complete basis. Material coefficients are derived and discussed. The constitutive theories are presented in the absence as well as presence of the balance of moments of moments as a balance law and are compared with the classical continuum theories. Retardation moduli corresponding to the Cauchy stress tensor, both symmetric and antisymmetric, as well as the symmetric and antisymmetric Cauchy moment tensors are derived. The constitutive theories presented in this paper are ordered rate theories. These incorporate convected time derivatives of the strain tensor up to order [Formula: see text] and the convected time derivatives of the stress and moment tensors up to certain orders. Simplified forms of the constitutive theories are also derived. It is shown and concluded that the constitutive theories for thermoviscoelastic fluids based on classical continuum theory such as Maxwell model, Oldroyd-B model, Giesekus model, etc., are all a subset of the constitutive theories presented in this paper for non-classical physics. Both compressible and incompressible thermoviscoelastic fluids are considered.

scholarly journals 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 ◽  
2020 ◽  
Vol 22 (4) ◽  
pp. 443
Author(s):  
Karan S. Surana ◽  
Stephen W. Long
Keyword(s):  
Representation Theorem ◽  
Moment Tensor ◽  
Strain Tensor ◽  
Entropy Inequality ◽  
Cauchy Stress ◽  
Symmetric Part ◽  
Gradient Tensor ◽  
Constitutive Theories ◽  
Derivatives Of ◽  
Ordered Rate Constitutive Theories

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 n ε 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 n Θ . 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.

scholarly journals Internal Rotation, Mixing and Lithium Abundances

1998 ◽  
Vol 185 ◽  
pp. 25-36
Author(s):  
Brian Chaboyer
Keyword(s):  
Internal Rotation ◽  
Transport Process ◽  
Main Sequence ◽  
Convection Zone ◽  
Nuclear Reactions ◽  
Open Clusters ◽  
Current Status ◽  
The Sun ◽  
Rotation Rates ◽  
Destruction Zone

Lithium is an excellent tracer of mixing in stars as it is destroyed (by nuclear reactions) at a temperature around ~ 2.5 × 106 K. The lithium destruction zone is typically located in the radiative region of a star. If the radiative regions are stable, the observed surface value of lithium should remain constant with time. However, comparison of the meteoritic and photospheric Li abundances in the Sun indicate that the surface abundance of Li in the Sun has been depleted by more than two orders of magnitude. This is not predicted by solar models and is a long standing problem. Observations of Li in open clusters indicate that Li depletion is occurring on the main sequence. Furthermore, there is now compelling observational evidence that a spread of lithium abundances is present in nearly identical stars. This suggests that some transport process is occurring in stellar radiative regions. Helioseismic inversions support this conclusion, for they suggest that standard solar models need to be modified below the base of the convection zone. There are a number of possible theoretical explanations for this transport process. The relation between Li abundances, rotation rates and the presence of a tidally locked companion along with the observed internal rotation in the Sun indicate that the mixing is most likely induced by rotation. The current status of non-standard (particularly rotational) stellar models which attempt to account for the lithium observations are reviewed.

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