Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2015 / Article
Special Issue

Modeling and Analysis in Thermodynamics and Heat Transfer

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Research Article | Open Access

Volume 2015 |Article ID 151707 | 15 pages | https://doi.org/10.1155/2015/151707

Coupled Thermal Field of the Rotor of Liquid Floated Gyroscope

Academic Editor: Di Zhang
Received24 May 2014
Revised27 Aug 2014
Accepted27 Aug 2014
Published12 Aug 2015

Abstract

Inertial navigation devices include star sensor, GPS, and gyroscope. Optical fiber and laser gyroscopes provide high accuracy, and their manufacturing costs are also high. Magnetic suspension rotor gyroscope improves the accuracy and reduces the production cost of the device because of the influence of thermodynamic coupling. Therefore, the precision of the gyroscope is reduced and drift rate is increased. In this study, the rotor of liquid floated gyroscope, particularly the dished rotor gyroscope, was placed under a thermal field, which improved the measurement accuracy of the gyroscope. A dynamic theory of the rotor of liquid floated gyroscope was proposed, and the thermal field of the rotor was simulated. The maximum stress was in x, 1.4; y, 8.43; min 97.23; and max 154.34. This stress occurred at the border of the dished rotor at a high-speed rotation. The secondary flow reached 5549 r/min, and the generated heat increased. Meanwhile, the high-speed rotation of the rotor was volatile, and the dished rotor movement was unstable. Thus, nanomaterials must be added to reduce the thermal coupling fluctuations in the dished rotor and improve the accuracy of the measurement error and drift rate.

1. Introduction

MEMS (microelectromechanical systems) gyroscopes, such as micromachined vibrating gyroscope [1, 2] and piezoelectric vibrating gyroscope [3, 4], are inertial navigation devices that measure angular velocity and altitude angle. They are widely used in numerous fields because of their small volume, light quality, affordability, and high accuracy. Piezoelectric vibrating gyroscope is generally used in electronic products and personal navigation systems [3, 4] and its bias stability reaches 0.1°/h [5]. However, the mechanical coupling of this kind of vibrating gyroscope causes various problems. Tuning fork gyroscope, whose accuracy reaches the desired theoretical level, was proposed to address such problems [6]. The Shanghai Institute of Microsystem and Information Technology and Shanghai Jiaotong University presented this kind of gyroscope. Beijing University conducted a research on tuning fork gyroscope in 2009 [7, 8]. According to a study conducted by the Georgia Institute of Technology, tuning fork gyroscope exhibits high transition and has a bias drift as low as 0.15°/h and ARW of 0.003°/h, which are the lowest rates among the silicon MEMS gyroscopes. The largest scale factor of a gyroscope is 88 mV; the bandwidth of the microsystem can be configured between 1 and 10 Hertz [9]. Vibrating ring gyroscope is another kind of gyroscope that has excellent pattern matching, high resolution, low ZRO, and long-term stability. These property parameters are applied in a large number of navigation fields [1012].

The angular vibration of the MEMS gyroscope is mainly used to measure angle and velocity. The University of Minnesota presented a highly accurate angular vibratory gyroscope [13]. Meanwhile, the University of Michigan presented the integrating rate of a gyroscope based on a 3D gyroscope using high-Q material manufacturing process [14]. The piezoelectric vibrating gyroscope has the following advantages: robustness, wide measurement range, and high resistance to external shocks and shaking. Thus, it can operate in atmospheric environment and vacuum packed environment. In 2009, Hydragog University of Japan proposed a piezoelectric micromachined modal gyroscope [15].

As shown in Figure 1, it was suspended by the gyroscope’s rotor that eliminates the machinery’s friction and increases its accuracy. In 2003, Murakoshi et al. proposed the micromachined electrostatic suspension spinning gyroscope, Figure 1 [16]. Tsinghua University suggested a ring rotor gyroscope, Figure 2. The electric bearing of this gyroscope provides a noncontact suspension rotor, which can be slowly rotated in two orthogonal input shaft axes. It has an input range of 100 ± 0.5° s−1, scaling factor of 39.8 mV Hz, and noise floor of 0.015° s−1Hz−1/2. And the bias stability is 50.95°/h [17].

In recent years, the researchers from the University of Electronic Science and Technology in China investigated the LC tuning of magnetic suspension rotor gyroscope. This gyroscope mainly consists of a stator and a rotor. Its suspended rotor is driven, as shown in Figure 3. Magnetic suspension rotor gyroscope is electromagnetically suspended, with its rotor located at the center [18, 19].

In 2012, Tsinghua University and Harbin Industrial University investigated the rotor of liquid floated gyroscope. This gyroscope has high accuracy and mainly consists of a stator, rotor, coil, and test version. The edge of its structure is shown in Figure 4.

2. Theory of Dished Rotor Gyroscope for Heat Coupling

The heat coupling of gyroscope’s rotor is produced by a magnetic field heat, heat flow field, and the composition of heat flow field and magnetic field coupling. The principle of these fields is shown in Figure 5. Dishing rotor gyroscope is produced by the stator coil in the magnetic dipole and is driven in a sealed cavity during high-speed rotation. The gyroscope is filled with #3 industrial white oil in the rotating magnetic heated. The oil convection in the dished rotor is hot and it affects the precision of gyroscope measurements. The concrete principle is shown in Figure 5.

2.1. Dynamic Torque Model of the Dishing Rotor Gyroscope

This is the moment of gyroscope equation:

is the driving moment of the stator coil magnetic field, is the dishing rotor by heavy torque, is the float on the torque surface, is the float on the down surface, is the pressure strain torque rotor of dishing #3 industrial white oil, is the torque produced by the rotation of #3 industrial white oil, is the circulating heat of the disc rotor (enthalpy) generated by #3 industrial white oil, is the stator magnetic field (entropy) for the heat generated, and is the turbulence on the dished rotor torque.

The dynamic torque equation of the dishing rotor gyroscope in the stator coordinate system relative to the frame coordinate system is as follows:

The dynamic torque equation of the dishing rotor gyroscope in the rotor coordinate system relative to the stator coordinate system is as follows:

The dynamic torque equation of the dishing rotor gyroscope in the stator coordinate system relative to the inertial coordinate system is as follows:

The dynamic torque equation of the dishing rotor gyroscope in the rotor coordinate system relative to the inertial coordinate system is as follows:

2.2. Establishing the Dynamic Equations of the Dished Rotor Gyroscope

In order to establish an equation about gyroscope torque, firstly, the hypothesis is as follows. The quality of the dishing rotor is represented by and its radius by . The , , and axes in the coordinate system at the moment of the momentum are represented by , , , respectively. The dishing rotor moment of the momentum , , represents the rotor moment of the momentum in the , , and axes of the direction projection. The setup , represents the dishing rotor at the polar moment of the inertia moment and the equator. The moment of each coordinate system and the other parameters were previously mentioned. Through the moment of the momentum theorem is to the list of the dished rotor around 0, and the Euler mechanics equation is as follows:

The and axes are the polar and equatorial axes of the dishing rotor, respectively. Given that , , the rotor’s origin was set, and its name was . It is , and its cosine is Meanwhile, it is relative to the inertial coordinate system as follows: , , , and the moment of inertia of the dishing rotor is as follows:Therefore, there are

The front dished rotor about the equation of mechanical characteristics is calculated as follows:

The dished rotor moves at a high-speed in a closed chamber filled with #3 industrial white oil. The rotor is in the - and - axes with deflection angles , . Thus, it can be approximated.

The dynamic equation is simplified. Consider

Solving (10), the following equations are obtained:

2.3. Dynamic Equation of Generation of the Dished Rotor Gyroscope

The dynamic equation (6) is substituted to the relevant torque as follows: At the same time,

Substituting (18) into (11), (13), and (16), the following equations are obtained.

At the same time, (19) was calculated by the torque, and the angular velocity is as follows:

These three equations are angular velocity calculated by coupling torque about this kind of liquid floated gyroscope. At the same time, (20) were substituted into the moment. To calculate the reduction of the angular velocity, the following are assumed: Simplifying the equation yields

Equation (23) is substituted into the above equation:

2.4. Theory of Hot Fluid about the Gyroscope

For the theory coupled thermal field of dished rotor, these are as follows. First is the coupling thermodynamics calculation; second is the establishment of boundary conditions.

2.4.1. Coupled Thermodynamic Theory of the Gyroscope’s Rotor

The volume of the dishing rotor is assumed to consist of two parts: spherical and cylindrical. The volume of the sphere is calculated as follows:

The whole dishing rotor is in the thermal field, and the temperature is simulated. Consider

The equation above can be reduced. The following was established and the dished rotor was in coupled thermal field about the calculation of volume and temperature. Consider

The thermodynamic analysis of the dishing rotor gyroscope shows a continuous connection between the well-posed equation of differential thermodynamics and the continuum mechanics of the problems, because the dishing rotor gyroscope is filled with #3 industrial white oil; thus, a Godunov-type equation can be applied [20]:

This gyroscope operates under high-speed rotation: a form of nonequilibrium thermodynamics and state variables can be established as follows:Among them,

The Boltzmann equation of thermodynamics of the dishing rotor heat entropy in the growth equation is as follows [20]:

The medium oil film of the dishing rotor gyroscope is produced by a model of the turbulent flow of inhomogeneous superfluid thermodynamics. The specific liquid properties of #3 industrial white oil are given. The average vortex line length is measured in unit volume and heat flux restructuring expression. The second principle of thermodynamics is combined with the Lagrange equation using the Legendre transformation through the constitutive theory. The complete expression of nonequilibrium and its entropy flux has been established [21], and the constitutive relation is as follows:

This is a part of the calculated heat about dished rotor, where represents the average velocity, is density, is velocity, is energy density, is an internal variable, and is the density of the vortex line [22].

The turbulence theory of nonlinear superfluid involves entropy flux with different temperatures and heat flux of the dishing rotor [21]. The complete expression of the nonequilibrium entropy flux is as follows:The thermodynamic analysis for the oil film is performed on the isothermal surface [23].

In thermodynamic analysis, the lattice Boltzmann method is successfully applied in various problems related to isothermal fluid dynamics [24]. However, the application of the method in nonisothermal problems is limited; the heat model causes numerical instability [25]. The thermal lattice Boltzmann model can generally be divided into two types [26]. The first type is the multispeed model, and the second type is the passive scalar. The main advantage of the passive scalar model over the multispeed model is that the former enhances numerical stability; thus, the stability of the gyroscope must be ensured to enhance the hybrid finite thermal model in the difference method [27, 28]. Applying the thermal lattice Boltzmann equation in the gyroscope has been proposed [29]. The model of uniform lattice BGK collision must be adopted; it can be expressed as follows: where and are the particle density and the function energy distribution, particularly of the particle velocity direction . and represent the dimensionless relaxation time, whose control interest rates are close to the state of balance. Consider

Balance the function of density distribution for these two models. ConsiderThese are the coupled thermodynamic theory of gyroscope’s rotor.

2.4.2. Boundary Conditions of the Dished Rotor Gyroscope in Fluid Dynamics

The boundary of the dished rotor gyroscope was assumed to be in the #3 industrial white oil. The function of the particle density distribution is expressed as follows:

A pressure oil body exists but is applicable only along the edge of the density distribution function. As proposed by Fermi [30], the boundary conditions of #3 industrial white oil velocities can be determined as follows:

The common thermodynamic framework extensive key depends on the assumption of entropy [30]. Some of the common thermodynamic relations, including Claudius thermodynamic entropy, heat, and entropy, are as shown in Figure 6. The theoretical formula is also suitable for dished rotor about the liquid floated gyroscope [31]:

3. Thermodynamic Simulation of the Dished Rotor

The thermodynamic theory of simulation for dished rotor, we demonstrate respectively from two aspects of 2D and 3D simulation.

3.1. 2D Simulation of the Thermodynamics of the Dished Rotor Gyroscope in the COMSOL

When the dishing rotor gyroscope is in a confined space, the pressure of the fluid in the COMSOL is simulated, particularly the temperature field of the gyroscope’s rotor. It can be considered by the rotor flow field; these are for the gyroscope that can produce the Eddy current. The temperature of this gyroscope’s rotor is high. In the COMSOL simulation, the inner cavity joins the laminar flow in the four corners of the stator. The maximum temperature is shown in Figure 7(a). When the rotor is operating at 5549 r/min, the turbulent flow velocity of the simplified version of the gyroscope increases the turbulence, particularly the inner cavity (Figure 17). At this point, the rotor reaches the highest temperature, indicating the occurrence of the phenomenon of chaos (Figure 7(b)). Thus, the rotor must be processed under the same conditions shown in Figure 7(b), where the temperature and the phenomenon of laminar flow decrease, particularly the temperature of the plate above the board.

3.2. 3D Simulation of the Thermodynamics of the Dished Rotor Gyroscope in the COMSOL

The coupled thermal simulation of the dishing rotor and heat coupling were performed in the 3D simulation of COMSOL. They were conducted using a grid, as shown in Figure 8.

The rotor’s coupling heat was simulated with the gyroscope’s rotor speeds of 5000 rad/min and 5549 rad/min, and the results are shown in Figures 9(a) and 9(b).

3.2.1. Simulation Measurement of the Dished Rotor Gyroscope

The coupling force field of the dishing rotor gyroscope and the rotor’s coupling heat were simulated with the gyroscope’s rotor speeds of 5000 and 5549 rad/min, as shown in Figures 10(a) and 10(b).

The rotor is driven by a magnetic field, and the gyroscope’s rotor speeds are 5000 and 5549 rad/min. The analysis of the thermal coupling characteristic curve is shown in Figures 11(a) and 11(b). One edge of the rotor is in the largest thermal field force and has a maximum temperature of . Thus, the size of the rotor thermal coupling must be improved to precisely enhance the detection of the gyroscope. This improvement can be achieved by two methods. First, the rotor boundary is processed into 0.005 mm radian; second, the rotor is processed using nanomaterials

3.3. Moment of the Dished Rotor Gyroscope in ADAMS Simulation

ADAMS has established a model of the liquid floated gyroscope. A grid should be constructed for the gyroscope during the simulation of the dishing rotor. The gyroscope’s rotor is placed in the grid, as shown in Figure 12.

The simulation model is established by measuring the rotation of the dishing rotor, particularly its angular velocity and the force and speed of the gyroscope’s rotor. The results are shown in Figure 13. Therefore, we need to process the dished rotor.

4. The Heat Thermal Experiment about the Thermal Fluid of the Dished Rotor

This experiment adopts the phase field equation that can be derived from the extremum principle of thermodynamics, which describes the state of the system. The quantity can be obtained by applying the extremum principle model for thermodynamics and the evolution of the constitutive equation for the thermodynamic system. The phase field method is a potential simulation tool. The microstructure evolution of a system is complex, and several parameters are introduced as the standards in thermodynamics. The relationship between thermodynamics and phase field parameters is analyzed, simulated, and verified [32].

The total Gibbs energy of the GS mechanochemical system is given by Let the Gibbs energy of the pure aphasiac volume density be . is the pure phase flow, is the interface phase, and is the thermodynamics of the phase transformation driving force. This kind of GS for the volume density Gibbs energy system can be described as follows at any point in time [32]:Function relation is suitable to describe the thermodynamics of interface properties. They are , , and . The continuous function of weight has the following properties, which correspond to the total Gibbs energy as follows [32]: The partial derivative of , , and is determined as follows:For the heat dissipation system, the total dissipation is in the framework of linear nonequilibrium thermodynamics, as given above. Considerwhere is independent of energy, and is the thickness of the interface.

The value of the flow of is given by

5. Measurements of the Experiment

5.1. Rough Hydrophobic Surface

The rotor’s coupling heat is given in the theoretical calculation and the simulation above. The coupled heat is related to an experiment using PIV measurements, and the results are shown in Figures 14(a) and 14(b). The rotor operates at speeds of 5000 rad/min and  5549 rad/min and the rotor’s flow field velocity distribution of heated.

Consider , as measured from the center, and the oil of the measurement experiment is for the rotor.

For the rotor operates at speeds of 5000 rad/min and 5549 rad/min, and the rotor’s flow field velocity distribution of heated. Now, it is marked for analysis, and the thermal maximum edge away from the speed of rotor is included . The results are shown in Figures 15(a) and 15(b). Theoretical analysis and simulation of the thermal fields, it is that the rotor’s maximum coupling thermal boundary, the maximum temperature of gyroscope rotor is that 88.13°C.

5.2. Smooth Surface

Consider , as measured from the center. The measurements for the experiment of the rotor oil body are as follows. The rotor speed is 5549 rad/min given that it is located 2 mm away from the center. The results are shown in Figure 16.

The experimental results of the coupled thermal simulation and theoretical results are as follows. The maximum rotor coupled border coupling heat for the coupled thermal simulation results is (degree centigrade) and that for the rotor coupled heat results in the experiment is (degree centigrade). One of the biggest coupled thermal locations can be obtained by calculating the maximum stress, which is in , 1.4; , 8.43; min 97.23; and max 154.34. Two methods can be employed to improve the precision of the gyroscope, considering the rotor coupled thermal and dynamic characteristics. In the first method, the rotor boundary is processed into a 0.005 mm radian. In the second method, the rotor is processed with nanomaterials.

6. Conclusions

The dishing rotor gyroscope was coupled and was established in different coordinates of the system for the thermodynamic analysis. Dynamic equations were solved, and the dynamic equation of the disc rotor angular rate was obtained. The model theory was solved in the COMSOL through 2D and 3D simulations coupled with thermal simulation, and it was in the mechanics of the dynamic system simulation in the ADAMS components. Maximum stress was found in , 1.4; , 8.43; min 97.23; and max 154.34. The maximum stress occurred in the dishing rotor border at a high-speed rotation. The secondary flow reached 5549 r/min, and the heat generated increased. Meanwhile, the high-speed rotation of the rotor was volatile, and the dishing rotor movement was unstable. Adding nonmaterial processed at high temperature at the top and the bottom surfaces of the dished rotor is required.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the Ministry of Science and Technology of China (Grant nos. 2012CB934103 and 2011ZX02403).

References

  1. D. Xia, C. Yu, and L. Kong, “The development of micromachined gyroscope structure and circuitry technology,” Sensors, vol. 14, no. 1, pp. 1394–1473, 2014. View at: Publisher Site | Google Scholar
  2. N. Yezdi, F. Anasazi, and K. Naafi, “Micro-machined inertial sensors,” Proceedings of the IEEE, vol. 86, no. 8, pp. 1640–1659, 1998. View at: Publisher Site | Google Scholar
  3. K. Liu, W. Zhang, W. Chen et al., “The development of micro-gyroscope technology,” Journal of Micromechanics and Microengineering, vol. 19, no. 11, Article ID 113001, 2009. View at: Publisher Site | Google Scholar
  4. L. Yan, Z. Xiao, F. Zhang et al., “Advances of silicon-based integrated photonic devices and applications in optical gyroscope and optical communication,” Chinese Journal of Lasers, vol. 36, no. 3, pp. 547–553, 2009. View at: Publisher Site | Google Scholar
  5. A. Sharma, M. F. Zaman, M. Zucher, and F. Ayazi, “A 0.1°/HR bias drift electronically matched tuning fork microgyroscope,” in Proceedings of the 21st IEEE International Conference on Micro Electro Mechanical Systems (MEMS '08), pp. 6–9, Tucson, AZ, USA, January 2008. View at: Publisher Site | Google Scholar
  6. L. Xie, D. Xiao, and H. Wang, “Sensitivity analysis and structure design for tri-mass structure micromachined gyroscope,” in Proceedings of the 4th IEEE International Conference on Nano/Micro Engineered and Molecular Systems, vol. 5, pp. 126–129, Shenzhen, China, 2009. View at: Google Scholar
  7. F. L. Che, B. Xiong, Y. F. Li, and Y. L. Wang, “A novel electrostatic-driven tuning fork micromachined gyroscope with a bar structure operating at atmospheric pressure,” Journal of Micromechanics and Microengineering, vol. 20, no. 1, pp. 1–6, 2010. View at: Google Scholar
  8. Z. Y. Guo, Z. C. Yang, Q. C. Zhao et al., “A lateral-axis micromachined tuning fork gyroscope with torsional Z-sensing and electrostatic force-balanced driving,” Journal of Micromechanics and Microengineering, vol. 20, no. 2, Article ID 025007, 2010. View at: Publisher Site | Google Scholar
  9. Z. Y. Guo, Z. C. Yang, Q. C. Zhao et al., “A lateral-axis micromachined tuning fork gyroscope with torsional Z-sensing and electrostatic force-balanced driving,” Journal of Micromechanics and Microengineering, vol. 20, no. 2, Article ID 025007, pp. 1–7, 2010. View at: Publisher Site | Google Scholar
  10. F. Ayazi and K. Najafi, “Design and fabrication of high-performance polysilicon vibrating ring gyroscope,” in Proceedings of the 11th IEEE Annual International Workshop on Micro Electro Mechanical Systems (MEMS '98), pp. 621–626, Heidelberg, Germany, January 1998. View at: Publisher Site | Google Scholar
  11. F. Ayazi and K. Najafi, “A HARPSS polysilicon vibrating ring gyroscope,” Journal of Microelectromechanical Systems, vol. 10, no. 2, pp. 169–179, 2001. View at: Publisher Site | Google Scholar
  12. G. He and K. Najafi, “A single-crystal silicon vibrating ring gyroscope,” in Proceedings of the 15th IEEE International Conference on Micro Electro Mechanical Systems (MEMS '02), pp. 718–721, Las Vegas, Nev USA, January 2002. View at: Google Scholar
  13. D. Piyabongkarn, R. Rajamani, and M. Greminger, “The development of a MEMS gyroscope for absolute angle measurement,” IEEE Transactions on Control Systems Technology, vol. 13, no. 2, pp. 185–195, 2005. View at: Publisher Site | Google Scholar
  14. I. P. Prikhodko, S. A. Zotov, A. A. Trusov, and A. M. Shkel, “Foucault pendulum on a chip: rate integrating silicon MEMS gyroscope,” Sensors and Actuators A: Physical, vol. 177, pp. 67–78, 2012. View at: Publisher Site | Google Scholar
  15. X. Wu, W. Chen, Y. Lu et al., “Vibration analysis of a piezoelectric micromachined modal gyroscope (PMMG),” Journal of Micromechanics and Microengineering, vol. 19, Article ID 125008, pp. 1–10, 2009. View at: Publisher Site | Google Scholar
  16. T. Murakoshi, Y. Endo, K. Fukatsu, and S. Nakamura, “Electrostatically levitated ring-shaped rotational-gyro/accelerometer,” Japanese Journal of Applied Physics, vol. 42, no. 4, pp. 2468–2472, 2003. View at: Publisher Site | Google Scholar
  17. F. T. Han, Y. F. Liu, L. Wang, and G. Y. Ma, “Micromachined electrostatically suspended gyroscope with a spinning ring-shaped rotor,” Journal of Micromechanics and Microengineering, vol. 22, no. 10, Article ID 105032, pp. 1–9, 2012. View at: Publisher Site | Google Scholar
  18. G. Xue, X. Zhang, and H. Zhang, “Electromagnetic design of a magnetically suspended gyroscope prototype,” in Proceedings of the International Conference on Applied Superconductivity and Electromagnetic Devices (ASEMD '09), pp. 369–372, Chengdu, China, September 2009. View at: Publisher Site | Google Scholar
  19. A. N. Gorban and I. V. Karlin, “Method of invariant manifold for chemical kinetics,” Chemical Engineering Science, vol. 58, no. 21, pp. 4751–4768, 2003. View at: Google Scholar
  20. D. V. Tunitsky, “Reducing quasilinear systems to block triangular form,” Sbornik: Mathematics, vol. 204, pp. 438–462, 2013. View at: Publisher Site | Google Scholar
  21. L. Ardizzone, G. Gaeta, and M. S. Mongiovì, “A continuum theory of superfluid turbulence based on extended thermodynamics,” Journal of Non-Equilibrium Thermodynamics, vol. 34, no. 3, pp. 277–297, 2009. View at: Publisher Site | Google Scholar
  22. V. Triani and V. A. Cimmelli, “Entropy principle, non-regular processes, and generalized exploitation procedures,” Journal of Mathematical Physics, vol. 53, no. 6, Article ID 063509, 2012. View at: Publisher Site | Google Scholar | MathSciNet
  23. C.-H. Liu, K.-H. Lin, H.-C. Mai, and C.-A. Lin, “Thermal boundary conditions for thermal lattice Boltzmann simulations,” Computers and Mathematics with Applications, vol. 59, no. 7, pp. 2178–2193, 2010. View at: Publisher Site | Google Scholar
  24. R. Mei, W. Shyy, D. Yu, and L.-S. Luo, “Lattice Boltzmann Method for 3-D Flows with Curved Boundary,” Journal of Computational Physics, vol. 161, no. 2, pp. 680–699, 2000. View at: Publisher Site | Google Scholar
  25. G. R. McNamara, A. L. Garcia, and B. J. Alder, “Stabilization of thermal lattice Boltzmann models,” Journal of Statistical Physics, vol. 81, no. 1-2, pp. 395–408, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  26. X. He, S. Chen, and G. D. Doolen, “A novel thermal model for the lattice Boltzmann method in incompressible limit,” Journal of Computational Physics, vol. 146, no. 1, pp. 282–300, 1998. View at: Publisher Site | Google Scholar | MathSciNet
  27. P. Lallemand and L.-S. Lou, “Hybrid finite-difference thermal lattice Boltzmann equation,” International Journal of Modern Physics B, vol. 17, no. 1-2, pp. 41–47, 2003. View at: Publisher Site | Google Scholar
  28. P. Lallemand and L.-S. Luo, “Theory of the lattice Boltzmann method: acoustic and thermal properties in two and three dimensions,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 68, no. 3, pp. 706–709, 2003. View at: Publisher Site | Google Scholar | MathSciNet
  29. G. R. McNamara, A. L. Garcia, and B. J. Alder, “Stabilization of thermal lattice Boltzmann models,” Journal of Statistical Physics, vol. 81, no. 1-2, pp. 395–408, 1995. View at: Publisher Site | Google Scholar
  30. E. Fermi, Thermodynamics, Dover, New York, NY, USA, 1956.
  31. S. Abe, “Heat and entropy in nonextensive thermodynamics: transmutation from Tsallis theory to Rényi-entropy-based theory,” Physica A: Statistical Mechanics and Its Applications, vol. 300, no. 3-4, pp. 417–423, 2001. View at: Publisher Site | Google Scholar
  32. J. Svoboda, F. D. Fischer, and D. L. McDowell, “Derivation of the phase field equations from the thermodynamic extremal principle,” Acta Materialia, vol. 60, no. 1, pp. 396–406, 2012. View at: Publisher Site | Google Scholar

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