Derivation of Two-Fluid Model Based on Onsager Principle
Abstract
:1. Introduction
2. Onsager Principle
2.1. State Variables
- Volume fraction of the polymer .The corresponding “velocity” variable is , the polymer velocity. The polymer volume fraction and the polymer velocity are related by the conservation lawHere, and .
- Conformation tensor .is a non-dimensional tensor to characterize the microscopic state of the polymer chain. The -tensor is equal to unit tensor when the polymer is at equilibrium, and deviates from when the polymer is deformed. Later, we will introduce the dumbbell model, which presents the polymer chain as a dumbbell consisting of two beads at positions and . These two beads are connected by an elastic spring that has an end-to-end vector and a spring constant k. The conformation of the dumbbell is then specified by the -tensor defined by .The corresponding “velocity” variable is the material time derivative of defined byNotice that, here, we are using the polymer velocity to define the material time derivative, not the medium velocity , which will be introduced next.
- In order to discuss the phenomena of the diffusion or migration of polymers, we need to introduce another “velocity” variable representing the velocity of the surroundings. This can be represented by the solvent velocity , or the medium velocity (volume-average velocity) defined byHere, we will use following rheology convention.
2.2. Free Energy
2.3. Dissipation Function
2.4. Time Evolution Equations
3. Doi–Onuki Derivation
3.1. Polymer Solution
- This derivation is different from the standard derivation based on the Onsager principle. We normally start with a free energy as a function of the state variables and then calculate the change rate by performing the time derivative.
- Here, the polymer stress is input by hand; therefore, we still need a constitutive equation to relate the stress to the state variables.
- Here, the polymer velocity is used. In Ref. [10], it was noted that “it is the deformation of the polymer which causes the change of the free energy”. It turns out that it is very important to specify which velocity is coupled to the polymer stress.
3.2. Polymer Blends
4. Dumbbell Model
4.1. Free Energy
4.2. Dissipation Function
- Using the polymer velocity, . This will give the same results of Doi–Onuki.
- Using the solvent velocity, . This will lead to a different dynamics.
- Using a combination of the polymer and solvent velocities, , withThis includes a special case of volume-average velocity , with and .
4.3. Time Evolution Equations
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Zhou, J.; Doi, M. Derivation of Two-Fluid Model Based on Onsager Principle. Entropy 2022, 24, 716. https://doi.org/10.3390/e24050716
Zhou J, Doi M. Derivation of Two-Fluid Model Based on Onsager Principle. Entropy. 2022; 24(5):716. https://doi.org/10.3390/e24050716
Chicago/Turabian StyleZhou, Jiajia, and Masao Doi. 2022. "Derivation of Two-Fluid Model Based on Onsager Principle" Entropy 24, no. 5: 716. https://doi.org/10.3390/e24050716
APA StyleZhou, J., & Doi, M. (2022). Derivation of Two-Fluid Model Based on Onsager Principle. Entropy, 24(5), 716. https://doi.org/10.3390/e24050716