Strong Solutions to Non-stationary Channel Flows of Heat-Conducting Viscous Incompressible Fluids with Dissipative Heati
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Strong Solutions to Non-stationary Channel Flows of Heat-Conducting Viscous Incompressible Fluids with Dissipative Heating Michal Beneš
Received: 19 February 2011 / Accepted: 30 August 2011 / Published online: 27 September 2011 © Springer Science+Business Media B.V. 2011
Abstract We study an initial-boundary-value problem for time-dependent flows of heatconducting viscous incompressible fluids in channel-like domains on a time interval (0, T ). For the parabolic system with strong nonlinearities and including the artificial (the so called “do nothing”) boundary conditions, we prove the local in time existence, global uniqueness and smoothness of the solution on a time interval (0, T ∗ ), where 0 < T ∗ ≤ T . Keywords Navier-Stokes equations · Heat equation · Heat-conducting fluid · Qualitative properties · Mixed boundary conditions Mathematics Subject Classification (2000) 35Q30 · 35K05 · 76D03
1 Introduction 1.1 Preliminaries Let ∈ C 0,1 be a two-dimensional bounded domain with the boundary ∂. Let ∂ = D ∪ N be such that D and N are open, not necessarily connected, the one-dimensional (i) (i) (j ) measure of D ∩ N is zero and D = ∅ (N = m i N , N ∩ N = ∅ for i = j ). In a physical sense, represents a “truncated” region of an unbounded channel system occupied by a moving heat-conducting viscous incompressible fluid. D will denote the “lateral” surface and N represents the open parts of the region . We assume that in/outflow channel segments extend as straight pipes. All portions of N are taken to be flat and the boundary N and rigid boundary D form a right angle at each point where the boundary conditions change. Moreover, we assume that D is smooth (of class C ∞ ). M. Beneš () Department of Mathematics and Centre for Integrated Design of Advanced Structures, Faculty of Civil Engineering, Czech Technical University in Prague, Thákurova 7, 166 29 Prague 6, Czech Republic e-mail: [email protected]
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The flow of a viscous incompressible heat-conducting constant-property fluid is governed by balance equations for linear momentum, mass and internal energy [5] (ut + (u · ∇)u) − νu + ∇π = (1 − αθ )f , div u = 0, cp (θt + u · ∇θ ) − κθ − νe(u) : e(u) = αθ f · u + h.
(1) (2) (3)
Here u = (u1 , u2 ), π and θ denote the unknown velocity, pressure and temperature, respectively. Tensor e(u) denotes the symmetric part of the velocity gradient. Data of the problem are as follows: f is a body force and h a heat source term. Positive constant material coefficients represent the kinematic viscosity ν, density , heat conductivity κ, specific heat at constant pressure cp and thermal expansion coefficient of the fluid α. The energy balance equation (3) takes into account the phenomena of the viscous energy dissipation and adiabatic heat effects. For rigorous derivation of the model like (1)–(3) we refer the readers to [11]. Concerning the boundary conditions of the flow, it is a standard situation to prescribe the non-homogeneous Dirichlet boundary condition for temperature
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