The internal energy as a function of state parameters in steady and unsteady Poiseuille flows
-
Konrad Giżyński
Abstract
We studied planar compressible Poiseuille flows of an ideal gas, both in steady and unsteady states, to identify the minimal number of state parameters required to describe changes in internal energy. In previous work (Phys. Rev. E 104, 055107 (2021)), five parameters were needed for steady flows. Here, using global non-equilibrium thermodynamics, we reduce this number to three: non-equilibrium entropy S *, volume V, and number of particles N. The internal energy U(S *, V, N) of such systems in stationary and non-stationary states is the function of non-equilibrium entropy S *, volume V and number of particles N in the system irrespective of any processes, number of boundary conditions or imposed constraints. We tested this by placing a cylinder inside the channel, finding that U depends on the cylinder’s location y c only via the state parameters S *(y c ) and N(y c ) for V = const. Moreover, in cases where the flow becomes unstable and parameters such as velocity and pressure oscillate, U depends on time t only through S *(t) and N(t) for V = const. These results demonstrate that this formulation of internal energy remains robust and consistent, even in unsteady flows with varying boundary conditions.
-
Research ethics: Not applicable.
-
Informed consent: Not applicable.
-
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
-
Use of Large Language Models, AI and Machine Learning Tools: Used for language improvements only.
-
Conflict of interest: The authors state no conflict of interest.
-
Research funding: This research was founded by NCN within Sonata grant 2019/35/D/ST5/03613.
-
Data availability: The data that support the findings of this study are available from the corresponding author, RH, upon reasonable request.
1 Solver tolerance tests
A solver relative tolerance of 10−5 was selected based on a convergence test of internal energy U (see Figures A1a and A1c) and number of moles n (see Figures A1b and A1d). In case (i) reducing the tolerance to 10−5 or 10−6 changed the result of U by less than 6·10−9 J, corresponding to a relative error below 3·10−12. This is at least two orders of magnitude smaller compared to mesh discretization errors. In case (ii) the results for U changed by 4·10−9 J and the associated relative error is 1·10−12. This is at least one order of magnitude smaller than mesh discretization errors.
Sensitivity of the computed U and n to solver tolerance settings. Panels (a) and (b) show the variation of U and n for case (i), while panels (c) and (d) present the corresponding results for case (ii). The results demonstrate that both U and n become stable for solver relative tolerances of 10−5 and lower. The vertical axis in each plot uses an offset notation to highlight small differences between the tested solver tolerances.
2 Calculation and propagation of numerical errors
The numerical error for each computed quantity was estimated based on the observed convergence with respect to mesh refinement. For a given observable Q, calculations were performed for three increasingly refined meshes. The error was then estimated as the average of the absolute differences between successive mesh results, according to the following formula:
where Q 1, Q 2, Q 3 denote the values of the observable obtained for the coarsest, intermediate, and finest mesh, respectively. The procedure was applied to all key observables considered in the study namely: ΔU, ∫μ *dn, ∫T*dS and ∫p *dV. For quantities that are added or subtracted, like ΔU-∫μ *dn-∫T *dS*+∫p *dV, the total error was calculated as:
where each σ is the error for a single term.
References
[1] G. V. Demirel Yaşar, Nonequilibrium Thermodynamics, Amsterdam, Elsevier, 2019.10.1016/B978-0-444-64112-0.00014-9Search in Google Scholar
[2] D. Jou, G. Lebon, and J. Casas-Vázquez, Extended Irreversible Thermodynamics, Netherlands, Springer, 2010.10.1007/978-90-481-3074-0_2Search in Google Scholar
[3] S. R. De Groot and P. Mazur, Non-Equilibrium Thermodynamics, New York, Dover Publications, 2013.Search in Google Scholar
[4] I. Prigogine and R. Lefever, “Symmetry breaking instabilities in dissipative systems. II,” J. Chem. Phys., vol. 48, no. 4, pp. 1695–1700, 1968, https://doi.org/10.1063/1.1668896.Search in Google Scholar
[5] H. J. M. Hanley and D. J. Evans, “A thermodynamics for a system under shear,” J. Chem. Phys., vol. 76, no. 6, pp. 3225–3232, 1982, https://doi.org/10.1063/1.443315.Search in Google Scholar
[6] D. J. Evans, “Computer “experiment” for nonlinear thermodynamics of Couette flow,” J. Chem. Phys., vol. 78, no. 6, pp. 3297–3302, 1983, https://doi.org/10.1063/1.445195.Search in Google Scholar
[7] D. J. Evans, “Rheology and thermodynamics from nonequilibrium molecular dynamics,” Int. J. Thermophys., vol. 7, no. 3, pp. 573–584, 1986, https://doi.org/10.1007/BF00502391.Search in Google Scholar
[8] T. Taniguchi and G. P. Morriss, “Steady shear flow thermodynamics based on a canonical distribution approach,” Phys. Rev. E, vol. 70, no. 5, pp. 56124, 2004, https://doi.org/10.1103/PhysRevE.70.056124.Search in Google Scholar PubMed
[9] P. J. Daivis and M. L. Matin, “Steady-state thermodynamics of shearing linear viscoelastic fluids,” J. Chem. Phys., vol. 118, no. 24, pp. 11111–11119, 2003, https://doi.org/10.1063/1.1574776.Search in Google Scholar
[10] P. J. Daivis, “Thermodynamic relationships for shearing linear viscoelastic fluids,” J. Nonnewton. Fluid Mech., vol. 152, no. 1, pp. 120–128, 2008. https://doi.org/10.1016/j.jnnfm.2007.02.004.Search in Google Scholar
[11] A. Baranyai, “Thermodynamic integration for the determination of nonequilibrium entropy,” J. Mol. Liq., vol. 266, pp. 472–477, 2018. https://doi.org/10.1016/j.molliq.2018.05.113.Search in Google Scholar
[12] R. K. Niven, “Invariance properties of the entropy production, and the entropic pairing of inertial frames of reference by shear-flow systems,” Entropy, vol. 23, no. 11, p. 1515, 2021, https://doi.org/10.3390/e23111515.Search in Google Scholar PubMed PubMed Central
[13] M. Song, et al.., “Inspection of Couette and pressure-driven Poiseuille entropy-optimized dissipated flow in a suction/injection horizontal channel,” Anal. Solut.,” vol. 21, no. 1, p. 20230109, 2023, https://doi.org/10.1515/phys-2023-0109.Search in Google Scholar
[14] D. Hochberg and I. Herreros, “Extended thermodynamic and mechanical evolution criterion for fluids,” Commun. Nonlinear Sci. Numer. Simul., vol. 146, p. 108775, 2025. https://doi.org/10.1016/j.cnsns.2025.108775.Search in Google Scholar
[15] Y. Zhang, K. Gizyński, A. Maciołek, and R. Hołyst, “Storage of energy in constrained non-equilibrium systems,” Entropy, vol. 22, no. 5, p. 557, 2020. https://doi.org/10.3390/E22050557.Search in Google Scholar PubMed PubMed Central
[16] K. Gizynski, K. Makuch, J. Paczesny, Y. Zhang, A. Maciołek, and R. Holyst, “Internal energy in compressible Poiseuille flow,” Phys. Rev. E, vol. 104, no. 5, pp. 1–8, 2021, https://doi.org/10.1103/PhysRevE.104.055107.Search in Google Scholar PubMed
[17] P. J. Zuk, K. Makuch, R. Hołyst, and A. Maciołek, “Transient dynamics in the outflow of energy from a system in a nonequilibrium stationary state,” Phys. Rev. E, vol. 105, no. 5, pp. 1–12, 2022, https://doi.org/10.1103/PhysRevE.105.054133.Search in Google Scholar PubMed
[18] R. Hołyst, K. Makuch, A. Maciołek, and P. J. Żuk, “Thermodynamics of stationary states of the ideal gas in a heat flow,” J. Chem. Phys., vol. 157, no. 19, p. 194108, 2022, https://doi.org/10.1063/5.0128074.Search in Google Scholar PubMed
[19] R. Hołyst, P. J. Żuk, K. Makuch, A. Maciołek, and K. Giżyński, “Fundamental relation for the ideal gas in the gravitational field and heat flow,” Entropy, vol. 25, no. 11, p. 1483, 2023, https://doi.org/10.3390/e25111483.Search in Google Scholar PubMed PubMed Central
[20] R. Hołyst, K. Makuch, K. Giżyński, A. Maciołek, and P. J. Żuk, “Fundamental relation for gas of interacting particles in a heat flow,” Entropy, vol. 25, no. 9, p. 1295, 2023, https://doi.org/10.3390/e25091295.Search in Google Scholar PubMed PubMed Central
[21] A. Maciołek, R. Hołyst, K. Makuch, K. Giżyński, and P. J. Żuk, “Parameters of state in the global thermodynamics of binary ideal gas mixtures in a stationary heat flow,” Entropy, vol. 25, no. 11, p. 1505, 2023, https://doi.org/10.3390/e25111505.Search in Google Scholar PubMed PubMed Central
[22] K. Makuch, R. Hołyst, K. Giżyński, A. Maciołek, and P. J. Żuk, “Steady-state thermodynamics of a system with heat and mass flow coupling,” J. Chem. Phys., vol. 159, no. 19, p. 194113, 2023, https://doi.org/10.1063/5.0170079.Search in Google Scholar PubMed
[23] R. Hołyst, P. J. Żuk, A. Maciołek, K. Makuch, and K. Giżyński, “Direction of spontaneous processes in non-equilibrium systems with movable/permeable internal walls,” Entropy, vol. 26, no. 8, p. 713, 2024, https://doi.org/10.3390/e26080713.Search in Google Scholar PubMed PubMed Central
[24] R. Holyst, K. Giżyński, A. Maciołek, K. Makuch, J. Wróbel, and P. J. Żuk, “Global non-equilibrium thermodynamics for steady states like never before,” Europhys. Lett., vol. 149, no. 3, p. 30001, 2025, https://doi.org/10.1209/0295-5075/ada182.Search in Google Scholar
[25] R. Hołyst, J. Wróbel, T. Andryszewski, K. Giżyński, A. Maciołek, and P. J. Żuk, “Global stability of steady states in chemical reactors,” Chem. Eng. J., vol. 518, p. 164076, 2025, https://doi.org/10.1016/j.cej.2025.164076.Search in Google Scholar
[26] F. Durst, Fluid Mechanics: An Introduction to the Theory of Fluid Flows, Berlin Heidelberg, Springer, 2008.Search in Google Scholar
[27] R. Aris, Vectors, Tensors and the Basic Equations of Fluid Mechanics, New York, Dover Publications, 1990.Search in Google Scholar
[28] “CFD Module User’s Guide. COMSOL Multiphysics® v. 6.1. COMSOL AB, Stockholm, Sweden. 2022.”.Search in Google Scholar
[29] “Heat Transfer Module User’s Guide. COMSOL Multiphysics® v. 6.1. COMSOL AB, Stockholm, Sweden. 2022.”.Search in Google Scholar
[30] M. Schäfer, S. Turek, F. Durst, E. Krause, and R. Rannacher, “Benchmark Computations of Laminar Flow Around a Cylinder BT - Flow Simulation with High-Performance Computers II: DFG Priority Research Programme Results 1993–1995,” E. H. Hirschel, Ed. Wiesbaden: Vieweg+Teubner Verlag, 1996, pp. 547–566.10.1007/978-3-322-89849-4_39Search in Google Scholar
[31] “Reference Manual. COMSOL Multiphysics® v. 6.1. COMSOL AB, Stockholm, Sweden. 2022.”.Search in Google Scholar
[32] R. Holyst, “Dream chemistry award lecture,” 2024. https://www.youtube.com/watch?v=a78VB3FsL6k.Search in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Original Research Articles
- Numerical and optimization analysis of natural convection and entropy-generation in wavy triangular cavity with Casson fluid under magnetohydrodynamics and radiation
- Anisotropic turbulent flow of water through converging wavy-aluminum-circular pipe with five half-cycles: insight into the significance of four-branch minor-inlet angle
- A Hydrogen-fueled hybrid system based on HT-PEMFCs for simultaneous electrical power generation and high-value heat storage
- Optimization of two-cold one-hot reservoir low-dissipation heat engines
- Numerical and experimental heat transfer analysis of two-phase flow through microchannel for development of heat dissipation correlation
- Applying irreversible thermodynamics to the paradigmatic secondary transporter: lactose permease (LacY)
- Single-, two-, three-, and four-objective optimizations for an irreversible vacuum thermionic generator via finite-time thermodynamics, NSGA-II and three decision-making techniques
- A variational principle for extended irreversible thermodynamics: heat conducting viscous fluids
- Heat transfer analysis of plate versus pin fin heat sinks with GnP-MWCNT/water hybrid nanofluid
- The internal energy as a function of state parameters in steady and unsteady Poiseuille flows
Articles in the same Issue
- Frontmatter
- Original Research Articles
- Numerical and optimization analysis of natural convection and entropy-generation in wavy triangular cavity with Casson fluid under magnetohydrodynamics and radiation
- Anisotropic turbulent flow of water through converging wavy-aluminum-circular pipe with five half-cycles: insight into the significance of four-branch minor-inlet angle
- A Hydrogen-fueled hybrid system based on HT-PEMFCs for simultaneous electrical power generation and high-value heat storage
- Optimization of two-cold one-hot reservoir low-dissipation heat engines
- Numerical and experimental heat transfer analysis of two-phase flow through microchannel for development of heat dissipation correlation
- Applying irreversible thermodynamics to the paradigmatic secondary transporter: lactose permease (LacY)
- Single-, two-, three-, and four-objective optimizations for an irreversible vacuum thermionic generator via finite-time thermodynamics, NSGA-II and three decision-making techniques
- A variational principle for extended irreversible thermodynamics: heat conducting viscous fluids
- Heat transfer analysis of plate versus pin fin heat sinks with GnP-MWCNT/water hybrid nanofluid
- The internal energy as a function of state parameters in steady and unsteady Poiseuille flows
