1 / 24

Weakly Nonlocal Heat Conduction: Modeling Memory and Structure with Nonequilibrium Thermodynamics

This research paper explores the concepts of memory and structure in weakly nonlocal heat conduction using nonequilibrium thermodynamics. It examines different heat equations, such as the Cattaneo-Veronese and Guyer-Krumhansl equations, and discusses the hierarchy of heat equations and the importance of nonequilibrium thermodynamics. The paper also delves into the application of Liu's procedure in solving the Liu equations and the implications of extended thermodynamics in heat conduction.

melindas
Download Presentation

Weakly Nonlocal Heat Conduction: Modeling Memory and Structure with Nonequilibrium Thermodynamics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Weakly nonlocal heat conduction – modeling memory and structure with nonequilibrium thermodynamicsPeterVán HAS, RIPNP, Department of Theoretical Physics • Introduction • memory and structure? • different heat equations • Memory – Cattaneo-Vernote • Structure – Guyer-Krumhansl • Hierarchy of heat equations - balances • Discussion

  2. Why nonequilibrium thermodynamics? science of temperature Thermodynamics science of macroscopic energy changes Thermodynamics • general framework of any • Thermodynamics (?) macroscopic (?) • continuum (?) • theories • General framework: • Second Law • fundamental balances • objectivity - frame indifference reversibility – special limit

  3. Nonlocalities: Restrictions from the Second Law. change of the entropy current change of the entropy Change of the constitutive space

  4. T0 T Q Weakly nonlocal memory: inertia

  5. Non-homogeneous equilibrium: structure s a Single well

  6. s a Double well: two phases

  7. Stable mixed sructure: twinning in shape memory alloys Initial and boundary conditions!

  8. Second Law: basic balances • basic state: • constitutive state: • constitutive functions: Second law: (universality) Constitutive theory Method: Liu procedure + …- solving the Liu equations

  9. Ginzburg-Landau (thermodynamic, relocalized) constitutive state space constitutive functions local state Liu procedure (Farkas’s lemma) ?

  10. Basic state, constitutive state and constitutive functions: Heat conduction – Extended Thermodynamics • basic state: • constitutive state: • constitutive functions: Heat conduction: Fourier Cattaneo-Vernote Guyer-Krumhansl

  11. Weakly nonlocalextended thermodynamics state space constitutive space constitutive functions local state: Liu procedure (Farkas’s lemma): It is not solvable! Currents and forces? solution?

  12. extended (Gyarmati) entropy concave entropy (stability) entropy current (Nyíri) (B – current multiplier) plausible general (dE=TdS ~q=TJ)

  13. Solution, conditions (e.g. L11 p.d.) gradients Guyer-Krumhansl equation + new terms applications? Liu?

  14. Weakly nonlocalextended thermodynamics (again) state space constitutive space constitutive functions Specific questions: Why? balance form Why? locality Closure? hierarchy

  15. Weakly nonlocalextended thermodynamics (again) state space constitutive space constitutive functions First order nonlocality Liu procedure (Farkas’s lemma): Liu equations

  16. Liu’s theorem: Conditions: Usage: Consequences:

  17. Balance form evolution + local s, H: Liu equations: Dissipation inequality:

  18. “New” independent variables! potential structure no dissipation

  19. Weakly nonlocalextended thermodynamics (again) state space constitutive space constitutive functions Second order nonlocality + local state: s(e,q) Liu procedure (Farkas’s lemma): Liu equations

  20. Once more: extended entropy current multiplier Closed (trivial) Almost balance:

  21. Discussion: • Kinetic – phenomenological • Universality – independent on the micro-modell • Constructivity – Liu + force-current systems • Origin of balances? • Closure • C=(weakly nonlocal in time)? • Second Law

  22. References: General: Gyarmati, I., The wave approach of thermodynamics and some problems of non-linear theories, Journal of Non-Equilibrium Thermodynamics, 1977, 2, p233-260. Müller, I. and Ruggeri, T.: Rational Extended Thermodynamics, Springer Verlag, 1998, Springer Tracts in Natural Philosophy V 37, New York-etc. Jou, D. , Casas-Vázquez, J. and Lebon, G., Extended Irreversible Thermodynamics, Springer Verlag, 2001, Berlin-etc., 3rd, revised edition. Heat conduction: Cimmelli, V. A. and Ván, P., The effects of nonlocality on the evolution of higher order fluxes in non-equilibrium thermodynamics, Journal of Mathematical Physics, 2005, 46, p112901, (cond-mat/0409254). Ciancio, V. , Cimmelli, V. A. and Ván, P., On the evolution of higher order fluxes in non-equilibrium thermodynamics, 2006, (cond-mat/0407530).

  23. Thank you for your attention!

More Related