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1.
We use a mathematical technique, the self-similar functional renormalization, to construct formulas for the average conductivity
that apply for large heterogeneity, based on perturbative expansions in powers of a small parameter, usually the log-variance
of the local conductivity. Using perturbation expansions up to third order and fourth order in obtained from the moment equation approach, we construct the general functional dependence of the scalar hydraulic conductivity
in the regime where is of order 1 and larger than 1. Comparison with available numerical simulations show that the proposed method provides reasonable
improvements over available expansions. 相似文献
2.
Two-Phase Inertial Flow in Homogeneous Porous Media: A Theoretical Derivation of a Macroscopic Model
The purpose of this article is to derive a macroscopic model for a certain class of inertial two-phase, incompressible, Newtonian
fluid flow through homogenous porous media. Starting from the continuity and Navier–Stokes equations in each phase β and γ, the method of volume averaging is employed subjected to constraints that are explicitly provided to obtain the macroscopic
mass and momentum balance equations. These constraints are on the length- and time-scales, as well as, on some quantities
involving capillary, Weber and Reynolds numbers that define the class of two-phase flow under consideration. The resulting
macroscopic momentum equation relates the phase-averaged pressure gradient to the filtration or Darcy velocity in a coupled nonlinear form explicitly given by
or equivalently
In these equations, and are the inertial and coupling inertial correction tensors that are functions of flow-rates. The dominant and coupling permeability tensors and and the permeability and viscous drag tensors and are intrinsic and are those defined in the conventional manner as in (Whitaker, Chem Eng Sci 49:765–780, 1994) and (Lasseux
et al., Transport Porous Media 24(1):107–137, 1996). All these tensors can be determined from closure problems that are to
be solved using a spatially periodic model of a porous medium. The practical procedure to compute these tensors is provided. 相似文献
3.
When a porous melt layer saturated by liquid is solidified from above, convection often sets in due to buoyancy forces. In
this study, the onset of buoyancy-driven convection during time-dependent solidification is investigated by using the similarly
transformed disturbance equations. The thermal disturbance distribution of the solid phase is approximated by the WKB method
and effects of various parameters on the stability condition of the melt phase are analyzed theoretically. For the limiting
case of λ → 0 and finite k
r, the critical conditions approach asymptotically and . This study presenting a constant-temperature cooling model predicts greater instability and gives more unstable results
than those obtained from the constant solidification rate model. 相似文献
4.
Let be the set of m × m matrices A(λ) depending analytically on a parameter λ in a closed interval . Consider one-parameter families of quasi-periodic linear differential equations: , where is analytic and sufficiently small. We prove that there is an open and dense set in , such that for each the equation can be reduced to an equation with constant coefficients by a quasi-periodic linear transformation for almost
all in Lebesgue measure sense provided that g is sufficiently small. The result gives an affirmative answer to a conjecture of Eliasson (In: Proceeding of Symposia in
Pure Mathematics).
Dedicated to Professor Zhifen Zhang on the occasion of her 80th birthday 相似文献
5.
We study the limit of the hyperbolic–parabolic approximation
The function is defined in such a way as to guarantee that the initial boundary value problem is well posed even if is not invertible. The data and are constant. When is invertible, the previous problem takes the simpler form
Again, the data and are constant. The conservative case is included in the previous formulations. Convergence of the , smallness of the total variation and other technical hypotheses are assumed, and a complete characterization of the limit
is provided. The most interesting points are the following: First, the boundary characteristic case is considered, that is,
one eigenvalue of can be 0. Second, as pointed out before, we take into account the possibility that is not invertible. To deal with this case, we take as hypotheses conditions that were introduced by Kawashima and Shizuta
relying on physically meaningful examples. We also introduce a new condition of block linear degeneracy. We prove that, if
this condition is not satisfied, then pathological behaviors may occur. 相似文献
6.
Lorenzo Brandolese 《Archive for Rational Mechanics and Analysis》2009,192(3):375-401
We study the solutions of the nonstationary incompressible Navier–Stokes equations in , of self-similar form , obtained from small and homogeneous initial data a(x). We construct an explicit asymptotic formula relating the self-similar profile U(x) of the velocity field to its corresponding initial datum a(x). 相似文献
7.
Michael Winkler 《Journal of Dynamics and Differential Equations》2008,20(1):87-113
The paper deals with positive solutions of the initial-boundary value problem for with zero Dirichlet data in a smoothly bounded domain . Here is positive on (0,∞) with f(0) = 0, and λ1 is exactly the first Dirichlet eigenvalue of −Δ in Ω. In this setting, (*) may possess oscillating solutions in presence
of a sufficiently strong degeneracy. More precisely, writing , it is shown that if then there exist global classical solutions of (*) satisfying and . Under the additional structural assumption , s > 0, this result can be sharpened: If then (*) has a global solution with its ω-limit set being the ordered arc that consists of all nonnegative multiples of the
principal Laplacian eigenfunction. On the other hand, under the above additional assumption the opposite condition ensures that all solutions of (*) will stabilize to a single equilibrium.
相似文献
8.
Numerical simulations are used to study laminar vortex ring formation under the influence of background flow. The numerical
setup includes a round-headed axisymmetric body with an opening at the posterior end from which a column of fluid is pushed
out by a piston. The piston motion is explicitly included into the simulations by using a deforming mesh. A well-developed
wake flow behind the body together with a finite-thickness boundary layer outside the opening is taken as the initial flow
condition. As the jet is initiated, different vortex evolution behavior is observed depending on the combination of background
flow velocity to mean piston velocity () ratio and piston stroke to opening diameter () ratio. For low background flow () with a short jet (), a leading vortex ring pinches off from the generating jet, with an increased formation number. For intermediate background
flow () with a short jet (), a leading vortex ring also pinches off but with a reduced formation number. For intermediate background flow () with a long jet (), no vortex ring pinch-off is observed. For high background flow () with both a short () and a long () jet, the leading vortex structure is highly deformed with no single central axis of fluid rotation (when viewed in cross-section)
as would be expected for a roll-up vortex ring. For , the vortex structure becomes isolated as the trailing jet is destroyed by the opposite-signed vorticity of the background
flow. For , the vortex structure never pinches off from the trailing jet. The underlying mechanism is the interaction between the vorticity
layer of the jet and the opposite-signed vorticity layer from the initial wake. This interaction depends on both and . A comparison is also made between the thrust generated by long, continuous jets and jet events constructed from a periodic
series of short pulses having the same total mass flux. Force calculations suggest that long, continuous jets maximize thrust
generation for a given amount of energy expended in creating the jet flow. The implications of the numerical results are discussed
as they pertain to adult squid propulsion, which have been observed to generate long jets without a prominent leading vortex
ring.
PACS 02.60.Cb, 47.32.cf, 47.32.cb, 47.20.Ft, 47.63.M- 相似文献
9.
We prove the existence of multiscale Young measures associated with almost periodic homogenization. We give applications of
this tool in the homogenization of nonlinear partial differential equations with an almost periodic structure, such as scalar
conservation laws, nonlinear transport equations, Hamilton–Jacobi equations and fully nonlinear elliptic equations. Motivated
by the application in nonlinear transport equations, we also prove basic results on flows generated by Lipschitz almost periodic
vector fields, which are of interest in their own. In our analysis, an important role is played by the so-called Bohr compactification
of ; this is a natural parameter space for the Young measures. Our homogenization results provide also the asymptotic behavior
for the whole set of -translates of the solutions, which is in the spirit of recent studies on the homogenization of stationary ergodic processes. 相似文献
10.
Jean Dolbeault Peter Markowich Dietmar Oelz Christian Schmeiser 《Archive for Rational Mechanics and Analysis》2007,186(1):133-158
Kinetic transport equations with a given confining potential and non-linear relaxation type collision operators are considered.
General (monotone) energy dependent equilibrium distributions are allowed with a chemical potential ensuring mass conservation.
Existence and uniqueness of solutions is proved for initial data bounded by equilibrium distributions. The diffusive macroscopic
limit is carried out using compensated compactness theory. The results are drift-diffusion equations with non linear diffusion.
The most notable examples are of the form , ranging from porous medium equations to fast diffusion, with the exponent satisfying in . 相似文献
11.
S. M. Zhuk 《Nonlinear Oscillations》2007,10(4)
For a linear operator generated by the differential equation
we prove that its graph is closed and determine the adjoint operator . For elements of the linear manifolds and , we propose an analog of the formula of integration by parts. We establish a criterion for the existence of a pseudosolution
of the operator equation and formulate sufficient conditions for the normal solvability of the operator in terms of relations for blocks of the matrix C(t). The results obtained are illustrated by examples.
__________
Translated from Neliniini Kolyvannya, Vol. 10, No. 4, pp. 464–480, October–December, 2007. 相似文献
12.
A Jordan Curve Spanned by a Complete Minimal Surface 总被引:1,自引:0,他引:1
Francisco Martín Nikolai Nadirashvili 《Archive for Rational Mechanics and Analysis》2007,184(2):285-301
In this paper we construct complete (conformal) minimal immersions
which admit continuous extensions to the closed disk,
. Moreover,
is a homeomorphism and
is a (non-rectifiable) Jordan curve with Hausdorff dimension 1.
It turns out that the set of Jordan curves
constructed by the above procedure is dense in the space of Jordan curves of
with the Hausdorff metric. 相似文献
13.
Kalabin et al. (Numer. Heat Transfer A 47, 621-631, 2005) studied the unsteady natural convection for the sinusoidal oscillating wall temperature on one side wall
and constant average temperature on the opposing side wall. The present article is on the unsteady natural convective heat
transfer in an inclined porous cavity with similar temperature boundary conditions as those of Kalabin et al. The inclined
angle of the cavity is varied from 0° to 80°. The flow field is modeled with the Brinkman-extended Darcy model. The combined effects
of inclination angle of the enclosure and oscillation frequency of wall temperature are studied for Ra* = 103, Da = 10−3, , and Pr=1. Some results are also obtained with the Darcy–Brinkman–Forchheimer model and Darcy’s law and are compared with the present
Brinkman-extended Darcy model. The maximal heat transfer rate is attained at the oscillating frequency f = 46.7π and the inclined angle . 相似文献
14.
Manuel Del Pino Michał Kowalczyk Juncheng Wei 《Archive for Rational Mechanics and Analysis》2008,190(1):141-187
We consider the Allen–Cahn equation in a bounded, smooth domain Ω in , under zero Neumann boundary conditions, where is a small parameter. Let Γ0 be a segment contained in Ω, connecting orthogonally the boundary. Under certain nondegeneracy and nonminimality assumptions
for Γ0, satisfied for instance by the short axis in an ellipse, we construct, for any given N ≥ 1, a solution exhibiting N transition layers whose mutual distances are and which collapse onto Γ0 as . Asymptotic location of these interfaces is governed by a Toda-type system and yields in the limit broken lines with an
angle at a common height and at main order cutting orthogonally the boundary. 相似文献
15.
Hildebrando M. Rodrigues J. Solà-Morales 《Journal of Dynamics and Differential Equations》2006,18(4):961-974
We present an example of a contraction diffeomorphism in infinite dimensions that is not
-linearizable, and we construct a regular ordinary differential equation in a Hilbert space whose time-one map is that diffeomorphism. With this we have an example of an asymptotically stable ODE that is not
-conjugate to its linear part. 相似文献
16.
F. Javadpour 《Transport in Porous Media》2009,79(1):87-105
Carbon dioxide (CO2) injections in geological formations are usually performed for enhanced hydrocarbon recovery in oil and gas reservoirs and
storage and sequestration in saline aquifers. Once CO2 is injected into the formation, it propagates in the porous rock by dispersion and convection. Chemical reactions between
brine ions and CO2 molecules and consequent reactions with mineral grains are also important processes. The dynamics of CO2 molecules in random porous media are modeled with a set of differential equations corresponding to pore scale and continuum
macroscale. On the pore scale, convective–dispersive equation is solved considering reactions on the inner boundaries in a
unit cell. A unit cell is the smallest portion of a porous media that can reproduce the porous media by repetition. Inner
boundaries in a unit cell are the surfaces of the mineral grains. Dispersion process at the pore scale is transformed into
continuum macroscale by adopting periodic boundary conditions for contiguous unit cells and applying Taylor-Aris dispersion
theory known as macrotransport theory. Using this theory, the discrete porous system changes into a continuum system within
which the propagation and interaction of CO2 molecules with fluid and solid matrix of the porous media are characterized by three position-independent macroscopic coefficients:
the mean velocity vector , dispersivity dyadic , and mean volumetric CO2 depletion coefficient . 相似文献
17.
We study the dynamics of vortices in solutions of the Gross–Pitaevsky equation in a bounded, simply connected domain with natural boundary conditions on ∂Ω. Previous rigorous results have shown that for sequences of solutions with suitable well-prepared initial data, one can determine limiting vortex trajectories, and moreover that these trajectories
satisfy the classical ODE for point vortices in an ideal incompressible fluid. We prove that the same motion law holds for
a small, but fixed , and we give estimates of the rate of convergence and the time interval for which the result remains valid. The refined
Jacobian estimates mentioned in the title relate the Jacobian J(u) of an arbitrary function to its Ginzburg–Landau energy. In the analysis of the Gross–Pitaevsky equation, they allow us to use the Jacobian to locate
vortices with great precision, and they also provide a sort of dynamic stability of the set of multi-vortex configurations. 相似文献
18.
A basic re-examination of the traditional dimensional analysis of microscopic and macroscopic multiphase flow equations in porous media is presented. We introduce a macroscopic capillary number
which differs from the usual microscopic capillary number Ca in that it depends on length scale, type of porous medium and saturation history. The macroscopic capillary number
is defined as the ratio between the macroscopic viscous pressure drop and the macroscopic capillary pressure.
can be related to the microscopic capillary number Ca and the LeverettJ-function. Previous dimensional analyses contain a tacit assumption which amounts to setting
= 1. This fact has impeded quantitative upscaling in the past. Our definition for
, however, allows for the first time a consistent comparison between macroscopic flow experiments on different length scales. Illustrative sample calculations are presented which show that the breakpoint in capillary desaturation curves for different porous media appears to occur at
1. The length scale related difference between the macroscopic capillary number
for core floods and reservoir floods provides a possible explanation for the systematic difference between residual oil saturations measured in field floods as compared to laboratory experiment. 相似文献
19.
The unsteady dynamics of the Stokes flows, where
, is shown to verify the vector potential–vorticity (
) correlation
, where the field
is the pressure-gradient vector potential defined by
. This correlation is analyzed for the Stokes eigenmodes,
, subjected to no-slip boundary conditions on any two-dimensional (2D) closed contour or three-dimensional (3D) surface. It is established that an asymptotic linear relationship appears, verified in the core part of the domain, between the vector potential and vorticity,
, where
is a constant offset field, possibly zero. 相似文献
20.
Valery M. Levin Juan M. Alvarez-Tostado 《Archive of Applied Mechanics (Ingenieur Archiv)》2006,76(3-4):199-214
An arbitrary anisotropic micro-inhomogeneous (composite) poroelastic medium is considered, containing a random set of ellipsoidal inhomogeneities with different poroelastic characteristics. The properties of these constituents are described by the linear porothermoelastic theory of Biot. One of the self-consistent schemes named effective field method is used to develop explicit expressions for the effective porothermoelastic constants (tensor of the frame elastic compliances , tensor of the generalized Skempton’s coefficients , tensor of thermal expansion coefficients , Biot’s constants , and the heat capacity at constant stress for the static porothermoelastic theory. It is shown that for two components composite porous material these expressions are interconnected and can be expressed only via the components of tensor . Some special cases are considered for the isotropic main material (matrix). 相似文献