×

zbMATH — the first resource for mathematics

A Navier-Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients. (English) Zbl 1167.76316
Summary: We consider a complete thermodynamic model for unsteady flows of incompressible homogeneous Newtonian fluids in a fixed bounded three-dimensional domain. The model comprises evolutionary equations for the velocity, pressure and temperature fields that satisfy the balance of linear momentum and the balance of energy on any (measurable) subset of the domain, and is completed by the incompressibility constraint. Finding a solution in such a framework is tantamount to looking for a weak solution to the relevant equations of continuum physics. If in addition the entropy inequality is required to hold on any subset of the domain, the solution that fulfills all these requirements is called the suitable weak solution. In our setting, both the viscosity and the coefficient of the thermal conductivity are functions of the temperature. We deal with Navier’s slip boundary conditions for the velocity that yield a globally integrable pressure, and we consider zero heat flux across the boundary. For such a problem, we establish the large-data and long-time existence of weak as well as suitable weak solutions, extending thus J. Leray’s [Acta Math. 63, 193–248 (1934; JFM 60.0726.05)] and L. Caffarelli, R. Kohn and L. Nirenberg’s [Commun. Pure Appl. Math. 35, 771–831 (1982; Zbl 0509.35067)] results, that deal with the problem in a purely mechanical context, to the problem formulated in a fully thermodynamic setting.

MSC:
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q35 PDEs in connection with fluid mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] D. Bucur, E. Feireisl, Š Nečasová, J. Wolf, On the asymptotic limit of the Navier-Stokes system on domains with rough boundaries, J. Differential Equations, 2006 (submitted for publication)
[2] Bucur, D.; Feireisl, E.; Nečasová, Š., On the asymptotic limit of flows past a ribbed boundary, J. math. fluid mech., (2007)
[3] Bulíček, M.; Málek, J.; Rajagopal, K.R., Navier’s slip and evolutionary navier – stokes-like systems with pressure and shear-rate dependent viscosity, Indiana univ. math. J., 56, 1, 51-85, (2007) · Zbl 1129.35055
[4] Caffarelli, L.; Kohn, R.; Nirenberg, L., Partial regularity of suitable weak solutions of the navier – stokes equations, Comm. pure appl. math., 35, 6, 771-831, (1982) · Zbl 0509.35067
[5] Callen, H., Thermodynamics and an introduction to thermostatics, (1985), Wiley London
[6] Clopeau, T.; Mikelić, A., Nonstationary flows with viscous heating effects, (), 55-63, (electronic) · Zbl 0896.35108
[7] Consiglieri, L., Weak solutions for a class of non-Newtonian fluids with energy transfer, J. math. fluid mech., 2, 3, 267-293, (2000) · Zbl 0974.35090
[8] Evans, L.C.; Gariepy, R.F., Measure theory and fine properties of functions, () · Zbl 0804.28001
[9] Feireisl, E., Dynamics of viscous compressible fluids, () · Zbl 1080.76001
[10] Feireisl, E.; Málek, J., On the navier – stokes equations with temperature-dependent transport coefficients, Differ. equ. nonlinear mech., (2006), 14 pp. (electronic) Art. ID 90616 · Zbl 1133.35419
[11] Frehse, J.; Málek, J., Problems due to the no-slip boundary in incompressible fluid dynamics, (), 559-571 · Zbl 1080.35070
[12] Jäger, W.; Mikelić, A., On the roughness-induced effective boundary conditions for an incompressible viscous flow, J. differential equations, 170, 1, 96-122, (2001) · Zbl 1009.76017
[13] Jäger, W.; Mikelić, A., Couette flows over a rough boundary and drag reduction, Comm. math. phys., 232, 3, 429-455, (2003) · Zbl 1062.76012
[14] Koch, H.; Solonnikov, V.A., \(L_p\)-estimates for a solution to the nonstationary Stokes equations, J. math. sci. (New York), 106, 3, 3042-3072, (2001), Function theory and phase transitions
[15] C. Le Roux, K.R. Rajagopal, A review of boundary conditions for the flow of fluids past solid surfaces, 2007 (in preparation) (A Review article)
[16] Leray, J., Sur le mouvement d’un liquide visquex emplissant l’espace, Acta math., 63, 193-248, (1934) · JFM 60.0726.05
[17] Lions, P.L., (), Incompressible models, Oxford Science Publications
[18] Málek, J.; Rajagopal, K.R., Mathematical issues concerning the navier – stokes equations and some of its generalizations, (), 371-459 · Zbl 1095.35027
[19] Málek, J.; Nečas, J.; Rokyta, M.; Růžička, M., Weak and measure-valued solutions to evolutionary pdes, (1996), Chapman & Hall London · Zbl 0851.35002
[20] Naumann, J., On the existence of weak solutions to the equations of non-stationary motion of heat-conducting incompressible viscous fluids, Math. methods appl. sci., 29, 16, 1883-1906, (2006) · Zbl 1106.76016
[21] C.W. Oseen, Hydrodynamik, Akademische Verlagsgesellschaft, Leipzig, 1927 (in German)
[22] Priezjev, N.V.; Troian, S.M., Influence of periodic wall roughness on the slip behaviour at liquid/solid interfaces: molecular versus continuum predictions, J. fluid mech., 554, 25-46, (2006) · Zbl 1091.76016
[23] Rajagopal, K.R.; Srinivasa, A.R., On thermomechanical restrictions of continua, Proc. R. soc. lond. ser. A math. phys. eng. sci., 460, 631-651, (2004) · Zbl 1041.74002
[24] Simon, J., Compact sets in the spaces \(L^p(0, t; B)\), Ann. mat. pura appl., 146, 65-96, (1987) · Zbl 0629.46031
[25] Solonnikov, V.A., \(L_p\)-estimates for solutions to the initial boundary-value problem for the generalized Stokes system in a bounded domain, J. math. sci. (New York), 105, 5, 2448-2484, (2001), Function theory and partial differential equations
[26] Solonnikov, V.A., Estimates for solutions of a non-stationary linearized system of navier – stokes equations, Trudy mat. inst. Steklov., 70, 213-317, (1964) · Zbl 0163.33803
[27] Walter, W., ()
[28] Wolf, J., Existence of weak solutions to the equations of nonstationary motion of non-Newtonian fluids with shear-dependent viscosity, J. math. fluid mech., 9, 104-138, (2007) · Zbl 1151.76426
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.