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1.
We consider the linear heat equation in a domain occupied by a solid material with a network of pipes in which a well-mixed fluid is circulating. The temperature of the fluid in the pipe is uniform and its time variation is determined by the thermal flux on the wall of the pipe, plus a given internal source; continuity of the temperature across the pipe is also assumed. We suppose that we deal with a periodic geometry, with cells of size with inclusions of size rg; we study in detail in the case r, referring to a previous paper for the case r In the limit »0 we get a homogenized equation. The limit depends strongly on the ratio between the time variation of the temperature in the inclusions and the thermal flux through the interface. The homogenized equation has a new specific heat, which depends on the porosity and the constant of proportionality between the time variation of temperature and the flux on the boundary of the pipe. We also have a new thermal conductivity depending on the microstructure, and volume sources appear. The main tool is the energy method and we generalize the classical results for the more standard boundary conditions for parabolic equations. Finally, we consider the network of pipes forming a random ball structure. We prove convergence for this case. The homogenized equation is of the same form as in the periodic case but auxiliary problems are stochastic. 相似文献
2.
Existence for the Cahn-Hilliard phase separation model with a nondifferentiable energy 总被引:3,自引:0,他引:3
Summary The Cahn-Hilliard model for phase separation in a binary alloy leads to the equations (I) ut=w, (II) w= (u)– u with an associated energy functional F(u)=f [(u)+ +¦u¦2/2] dx. In this paper we discuss the existence theory for initial bounday value problems arising from modifications to the Cahn-Hilliard model due to the addition of the non-differentiable term ¦u¦dx to the energy F(u). 相似文献
3.
Andro Mikelic Laetitia Paoli 《Proceedings of the American Mathematical Society》1999,127(7):2019-2028
We consider the non-stationary incompressible Euler equations in a 2D porous medium. We suppose a periodic porous medium, with the period proportional to the characteristic pore size and with connected fluid part. The flow is subject to an external force, corresponding to an inflow. We start from an initial irrotational velocity and prove that the effective filtration velocity satisfies a transient filtration law. It has similarities with Darcy's law, but it now connects the time derivative of the filtration velocity with the pressure gradient. The viscosity does not appear in the filtration law any more and the permeability tensor is determined through auxiliary problems of decomposition type. Using the limit problem, we construct the correction for the fluid velocity and prove that -norm of the error is of order . Similarly, we estimate the difference between the fluid pressure and its correction in as .
4.
Antonio Fasano Andro Mikeli 《NoDEA : Nonlinear Differential Equations and Applications》2000,7(1):91-105
In this paper we consider the incompressible viscous fluid flow through a porous medium whose grains are also permeable and release mass to the flow. In each component porosity and permeability depend on saturation. The flow is modelled with a nonlinear parabolic equation for the pressure, with a degenerate parabolic term, depending not only on the saturations, but also on the space variable and on time averages of the saturation. We generalize the classic approach of Alt and Luckhaus to this situation and establish existence of at least one weak solution and bounds for its absolute value. Received December 1998 相似文献
5.
We consider the Couette flow between two plates. The lower plate is fixed and has periodically placed riblets of the characteristic
size ɛ on it. In the limit ɛ → 0 we find the effective Couette-Navier flow as an O(ɛ
2
) approximation for the effective mass flow and an O(ɛ
2
)L
1
-approximation for the velocity. In the effective solution the effect of roughness enters through the Navier slip condition
with the matrix coefficient in front of the effective shear stress, calculated using a boundary layer problem. Furthermore,
an O(ɛ
2
) approximation for the tangential drag force is found. In all estimates explicit dependence on the kinematic viscosity ν,
the velocity of the upper plate and the distance between the plates L
3
is kept. Also the uniqueness of the solution is expressed through a non-linear algebraic condition linking and L
3
. Then the result is applied to the viscous sub-layers around immersed bodies, strictly containing the surface riblets. It
is found that for the riblets of the characteristic size ɛ, being of the order smaller or equal to , the approximation obtained for the tangential drag could be applied. We compare ɛ and for realistic data and our results lead to the conclusion that the riblets reduce significantly tangential drag, which may
explain their presence on the skin of Nektons.
Received: 14 December 2001 / Accepted: 1 August 2002 Published online: 7 November 2002 相似文献
6.
Dina Aronzon Eric P. Levy Peter J. Collings Andro Chanishvili Guram Chilaya Gia Petriashvili 《Liquid crystals》2007,34(6):707-718
Trans-cis isomerization was investigated in a room temperature liquid crystal mixture of two azoxybenzene compounds. Experiments were performed on isolated molecules in dilute solutions and on the liquid crystal phase composed of the pure compounds. The absorption spectra of the trans and cis isomers were found to be similar to those of azobenzene compounds, as were the birefringence and order parameter of the nematic liquid crystal phase. The photo-optic properties were also similar in that irradiation by ultraviolet light caused the conversion from trans to cis isomers, while short wavelength visible light incident on these compounds resulted in the conversion from cis to trans isomers. The activation energy for thermal relaxation from the cis to trans isomer in the liquid crystal phase was determined to be (66±7) kJ/mole, which is less than for azobenzene in solution. While a photostationary state in a dilute solution with approximately equal numbers of trans and cis isomers was achieved, the nematic-isotropic transition of the mixture of the pure compounds decreased from 70°C to room temperature with a cis concentration of only about 12%. One unusual finding was that the photostationary concentration of trans and cis isomers due to irradiation with light of a specific visible wavelength depended on the starting concentrations of the two isomers, indicating that there may be a molecular conformation that is not photo-responsive and relaxes only thermally. 相似文献
7.
Andro Mikelić Mario Primicerio 《NoDEA : Nonlinear Differential Equations and Applications》2006,13(3):349-367
We study a mathematical model describing the nonlinear diffusion of oxygen in a living tissue, in presence of consumption
due to metabolism. The tissue is perfused by a system of parallel capillaries in which oxygen is carried by the blood both
in the form of gas freely diffusing in plasma and bound to hemoglobin. We prove global existence of a unique smooth solution
to the resulting parabolic-hyperbolic system. 相似文献
8.
9.
We study the flow of a viscous incompressible fluid through a long and narrow elastic tube whose walls are modeled by the Navier equations for a curved, linearly elastic membrane. The flow is governed by a given small time dependent pressure drop between the inlet and the outlet boundary, giving rise to creeping flow modeled by the Stokes equations. By employing asymptotic analysis in thin, elastic, domains we obtain the reduced equations which correspond to a Biot type viscoelastic equation for the effective pressure and the effective displacement. The approximation is rigorously justified by obtaining the error estimates for the velocity, pressure and displacement. Applications of the model problem include blood flow in small arteries. We recover the well-known Law of Laplace and provide a new, improved model when shear modulus of the vessel wall is not negligible. To cite this article: S. ?ani?, A. Mikeli?, C. R. Mecanique 330 (2002) 661–666. 相似文献
10.