Abstract
A method is proposed for constructing variational models of continuous media for reversible and irreversible processes based on the generalized Hamilton–Ostrogradskii principle, which reduces to the principle of stationarity for a non-integrable variational form of a spatial-temporal continuum. For dissipative processes, the corresponding linear variational form is constructed as a sum of variation of the Lagrangian of the reversible part and a linear combination of dissipation channels of physically nonlinear processes. Examples of using the variational approach to the description of hydrodynamic models are considered. The corresponding variational models of the Darcy hydrodynamics, linear Navier–Stokes hydrodynamics, Brinkman hydrodynamics, gradient hydrodynamics, and some generalization of the classical nonlinear Navier–Stokes hydrodynamics are constructed. For modeling irreversible processes of hydrodynamics with allowance for coupling of deformation with the associated physical processes of heat transfer, it is proposed to use variational formalism for the spatial-temporal continuum, where the spatial and temporal processes are considered simultaneously and consistently because the normalized time is a coordinate.
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REFERENCES
V. N. Matveenko and E. A. Kirsanov, “Viscosity and Structure of Disperse Systems," Vestn. Mosk. Univ., Ser. 2, Khimiya 66 (4), 243–276 (2011).
Al. A. Kovtun, “On Equations of the Biot Model and their Modification," Vopr. Geofiz. 44 (4), 3–25 (2011).
M. A. Biot, “Mechanics of Deformation and Acoustic Propagation in Porous Media," J. Appl. Phys. 33, 1482–1498 (1962).
T. Kaya and J. Goldak, “Numerical Analysis of Heat and Mass Transfer in the Capillary Structure of a Loop Heat Pipe," Intern. J. Heat Mass Transfer 49, 3211–3220 (2006).
R. Sonar, S. Hardman, J. Pell, et al., “Transient Thermal and Hydrodynamic Model of Flat Heat Pipe for the Cooling of Electronics Components," Intern. J. Heat Mass Transfer 51, 6006–6017 (2008).
S. Storring, “Design and Manufacturing of Loop Heat Pipes for Electronics Cooling," Diss. Ottawa: Carleton Univ., 2006.
B. D. Annin and S. N. Korobeinikov, “Admissible Forms of Elastic Laws of Deformation in the Governing Relations of Elastoplasticity," Sib. Zh. Industr. Mat. 1 (1), 21–34 (1998).
B. R. Babin, G. P. Peterson, and D. Wu, “Steady-State Modeling and Testing of Micro Heat Pipe," J. Heat Transfer 112, 595–601 (1990).
H. Darcy, Les Fontaines Publiques de la Ville de Dijon (Dalmont, Paris, 1856).
F. Lefevre and M. Lallemand, “Coupled Thermal and Hydrodynamic Models of Flat Micro Heat Pipes for the Cooling of Multiple Electronic Components," Intern. J. Heat Mass Transfer 49, 1375–1383 (2006).
B. Sieder, V. Sartre, and F. Lefevre, “Literature Review: Steady-State Modelling of Loop Heat Pipe," Appl. Thermal Engng. 75, 709–723 (2015).
V. V. Pukhnachev, “Viscous Flows in Domains with a Multiply Connected Boundary," in New Directions in Mathematical Fluid Mechanics (Birkhauser Verlag, Basel, 2010). P. 333–348.
V. V. Pukhnachev, “Symmetries in the Navier–Stokes equations," Usp. Mekh., No. 6, 3–76 (2006).
P. A. Belov and S. A. Lurie, “On Variation Models of the Irreversible Processes in Mechanics of Solids and Generalized Hydrodynamics Lobachevskii," J. Math. 40 (7), 896–910 (2019).
P. A. Belov, H. Altenbach, S. A. Lurie, et al., “Generalized Brinkman-Type Fluid Model and Coupled Heat Conductivity Problem," Lobachevskii J. Math. 42 (8), 1786–1799 (2021).
D. D. Joseph and L. Preziosi, “Heat Waves," Rev. Modern Phys. 61, 41–73 (1989).
S. Lepri, R. Livi, and A. Politi, “Thermal Conduction in Classical Low-Dimensional Lattices," Phys. Rep. 377, 1–80 (2003).
A. Dhar, “Heat Transport in Low-Dimensional Systems," Adv. Phys. 57, 457 (2008).
Moran Wang, Bin-Yang Cao, and Zeng-Yuan Guo, “General Heat Conduction Equations Based on the Thermomass Theory," Frontiers Heat Mass Transfer 1, 013004 (2010).
S. L. Sobolev, “Transport Processes and Travelling Waves in Locally Nonequilibrium Systems," Usp. Fiz. Nauk 161 (3), 5–29 (1991).
S. L. Sobolev, “Rapid Phase Transformation under Local Non-Equilibrium Diffusion Conditions," Materials Sci. Technol. 31 (13), 1607–1617 (2015).
S. Lurie and P. Belov, “From Generalized Theories of Media with Fields of Defects to Closed Variational Models of the Coupled Gradient Thermoelasticity and Thermal Conductivity," Adv. Structured Materials 120, 135–154 (2019).
S. A. Lurie and P. A. Belov, “Theory of Space-Time Dissipative Elasticity and Scale Effects," NanoMMTA 2, 166–178 (2013).
S. A. Lurie and P. A. Belov, “On the Nature of the Relaxation Time, the Maxwell — Cattaneo and Fourier Law in the Thermodynamics of a Continuous Medium, and the Scale Effects in Thermal Conductivity," Continuum Mech. Thermodynamics 32, 709–728 (2020).
P. A. Belov, A. G. Gorshkov, and S. A. Lurie, “Variational Model of Nonholonomic 4D Media," Izd. Ross. Akad. Nauk, Mekh. Tverd. Tela 6, 29–46 (2006).
P. A. Belov and S. A. Lurie, “Ideal Asymmetric 4D Medium as a Model of Reversible Dynamic Thermoelasticity," Izd. Ross. Akad. Nauk, Mekh. Tverd. Tela 52, 243–276 (2011).
L. I. Sedov, “Mathematical Methods of Constructing New Models of Continuous Media," Usp. Mat. Nauk 20 (5), 121–180 (1965).
C. Lanczos, The Variational Principles of Mechanics (Toronto, 1962).
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Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, 2021, Vol. 62, No. 5, pp. 145-160. https://doi.org/10.15372/PMTF20210515.
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Lurie, S.A., Belov, P.A. VARIATIONAL FORMULATION OF COUPLED HYDRODYNAMIC PROBLEMS. J Appl Mech Tech Phy 62, 828–841 (2021). https://doi.org/10.1134/S0021894421050151
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DOI: https://doi.org/10.1134/S0021894421050151