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1.
General nonlocal diffusive and dispersive transport theories are derived from molecular hydrodynamics and associated theories of statistical mechanical correlation functions, using the memory function formalism and the projection operator method. Expansion approximations of a spatially and temporally nonlocal convective-dispersive equation are introduced to derive linearized inverse solutions for transport coefficients. The development is focused on deriving relations between the frequency-and wave-vector-dependent dispersion tensor and measurable quantities. The resulting theory is applicable to porous media of fractal character.Nomenclature
C
v
(t)
particle velocity correlation function
-
C
v
,(t)
particle fluctuation velocity correlation function
-
C
j
(x,t)
current correlation function
-
D(x,t)
dispersion tensor
-
D(x,t)
fluctuation dispersion tensor
-
f
0(x,p)
equilibrium phase probability distribution function
-
f(x, p;t)
nonequilibrium phase probability distribution function
-
G(x,t)
conditional probability per unit volume of finding a particle at (x,t) given it was located elsewhere initially
-
(k,t)
Fourier transform ofG(x,t)
-
G(x,t)
fluctuation conditional probability per unit volume of finding a particle at (x,t) given it was located elsewhere initially
-
k
wave vector
-
K(t)
memory function
-
L
Liouville operator
-
m
mass
-
p(t)
particle momentum coordinate
-
P
= (0)( , (0))
projection operator
-
Q
=I-P
projection operator
-
s
real Laplace space variable
-
S(k, )
time-Fourier transform of(k,t)
-
t
time
-
v(t)
particle velocity vector
-
v(t)
particle fluctuation velocity vector
-
V
phase space velocity
-
time-Fourier variable
-
(itn)(k)
frequency moment of(k,t)
-
x(t)
particle displacement coordinate
-
x(t)
particle displacement fluctuation coordinate
-
friction coefficient
- (t)
normalized correlation function
General Functions
()
Dirac delta function
- ()
Gamma function
Averages 0
Equilibrium phase-space average
-
Nonequilibrium phase-space average
- (,)
L
2 inner product with respect tof
0 相似文献
2.
The Rouse model is a well established model for nonentangled polymer chains and its dynamic behavior under step strain has been fully analyzed in the literature. However, to the knowledge of the authors, no analysis has been made for the orientational anisotropy for the Rouse eigenmodes during the creep and creep recovery processes. For completeness of the analysis of the Rouse model, this anisotropy is calculated from the Rouse equation of motion. The calculation is simple and straightforward, but the result is intriguing in a sense that respective Rouse eigenmodes do not exhibit the single Voigt-type retardation. Instead, each Rouse eigenmode has a distribution in the retardation time. This behavior, reflecting the interplay among the Rouse eigenmodes of different orders under the constant stress condition, is quite different from the behavior under rate-controlled flow (where each eigenmode exhibits retardation/relaxation associated with a single characteristic time).List of abbreviations and symbols a Average segment size at equilibrium - Ap(t) Normalized orientational anisotropy for the p-th Rouse eigenmode defined by Eq. (14) -
p-th Fourier component of the Brownian force (=x, y) - FB(n,t) Brownian force acting on n-th segment at time t - G(t) Relaxation modulus - J(t) Creep compliance - JR(t) Recoverable creep compliance - kB Boltzmann constant - N Segment number per Rouse chain - Qj(t) Orientational anisotropy of chain sections defined by Eq. (21) - r(n,t) Position of n-th segment of the chain at time t - S(n,t) Shear orientation function (S(n,t)=a–2<ux(n,t)uy(n,t)>) - T Absolute temperature - u(n,t) Tangential vector of n-th segment at time t (u = r/n) - V(r(n,t)) Flow velocity of the frictional medium at the position r(n,t) - Xp(t), Yp(t), and Zp(t) x-, y-, and z-components of the amplitudes of p-th Rouse eigenmode at time t -
Strain rate being uniform throughout the system - Segmental friction coefficient - 0 Zero-shear viscosity - p Numerical coefficients determined from Eq. (25) - Gaussian spring constant ( = 3kBT/a2) - Number of Rouse chains per unit volume - (t) Shear stress of the system at time t - steady Shear stress in the steadily flowing state - R Longest viscoelastic relaxation time of the Rouse chain 相似文献
3.
Hans Vollmayr 《Transport in Porous Media》1996,22(2):137-159
A lattice gas algorithm is proposed for the simulation of water flow in the unsaturated zone. Microscopic dynamics of a two-dimensional model system are defined. Up to four fluid particles occupy the sites of a square lattice. At each time step, the particles are sent to neighbouring sites according to probabilistic rules which depend on the permeability and the potential but not on the input velocities of the particles. On the macroscopic scale, the flow is described by a diffusion term and a Darcy term. Several extensions including higher dimension are discussed.List of Symbols
c
(n)
constant in the definition of the rejection probabilityP forn = 1,2,3 particles at a site 0 c
(n)
1
-
D
diffusion constant
- D
vertical extent of the system, measured in cells
- E
i
vector connecting a site to its neighbour in directioni
-
i
direction of a nearest neighbour site,i = 1,..., 4
-
j
direction of a nearest neighbour site,j = 1,..., 4
-
j
mass transport (fluid flow),j =
v
-
j
x
x-component of the flowj
-
k(x)
spatial dependence of the permeability, user defined under the constraint 0 k 1
-
k
()
the part of the permeability which depends on the degree of saturation (seek)
-
k
(n)
(x)
effective permeability at a sitex that holdsn particles
- L
horizontal extent of the system, measured in cells
-
l
mac
macroscopic length scale, e.g. one meter
-
l
mic
microscopic length scale (one lattice constant)
-
m
integer number of time steps
-
n (x)
number of particles at the lattice sitex
-
N
A
total number of particles on all A-sites
-
P
probability for rejection of a randomly selected direction or set of directions
-
p
arithmetic mean of the probability for a site to receive a particle from a particular neighbour (the average is taken over the four neighbours)
-
p
i
(n)
probability that one out ofn particles at a site is sent in directioni
-
p
ij
(2)
probability that the two particles at a site are sent in directionsi andj
-
t
time
-
t
mac
macroscopic time scale, e.g. one day
-
t
mic
microscopic time scale (one time step)
-
v
fluid velocity
-
x
space vector, mostly two-dimensional:x = (x, y)
-
x
horizontal component ofx
-
y
vertical component ofx
-
quotient of microscopic and macroscopic time scales,t
mic
/t
mac
-
quotient of microscopic and macroscopic length scales,l
mic
/l
mac
- i
p + i is the probability that a particle is received from the neighbour atx +E
i
-
K(X, )
effective permeability,k =k(x)k
()
-
correlation length
-
degree of saturation, used synonymously with density (homogeneous porosity)
-
0
value of a homogeneous particle density
- ø(x)
external potential (user defined), ø = gr + mat
- ø(x)
arithmetic mean of the external potential at the four sites surroundingx
- ø
i
external potential at the sitex +E
i
-
total potential, = ø + den
- gr(x)
gravitational potential
- mat(x)
matrix potential
- den()
density-dependent potential
-
n
potential depending on the occupation number
-
(n)
(x)
probability that sitex is occupied byn particles
-
0
(n)
(n)
in a system with homogeneous particle density
- mac
macroscopic
- mic
microscopic 相似文献
4.
Roger Young 《Transport in Porous Media》1993,11(2):179-185
Two-phase mixtures of hot brine and steam are important in geothermal reservoirs under exploitation. In a simple model, the flows are described by a parabolic equation for the pressure with a derivative coupling to a pair of wave equations for saturation and salt concentration. We show that the wave speed matrix for the hyperbolic part of the coupled system is formally identical to the corresponding matrix in the polymer flood model for oil recovery. For the class ofstrongly diffusive hot brine models, the identification is more than formal, so that the wave phenomena predicted for the polymer flood model will also be observed in geothermal reservoirs.Roman Symbols
A,B
coefficient matrices (5)
-
c(x,t)
salt concentration (primary dependent variable)
-
C(p, s, c,
q
t)
wave speed matrix (6)
-
f
source term (5)
-
g
acceleration due to gravity (constant)
-
h
b(p, c)
brine specific enthalpy
-
h
v(p)
vapour specific enthalpy
-
j
conservation flux (1)
-
k
absolute permeability (constant)
-
k
b(s), kv(s)
relative permeabilities of the brine and vapour phases
-
K
conductivity
-
p(x,t)
pressure (primary dependent variable)
-
q
volume flux (Darcy velocity) (3)
-
s(x,t)
brine saturation (primary dependent variable)
-
t
time (primary independent variable)
-
T=T
sat(p)
saturation temperature
-
u
b(p, c)
brine specific internal energy
-
u
m T
rock matrix specific internal energy
-
u
v(p)
vapour specific internal energy
-
U(x, t)
shock velocity
-
x
space (primary independent variable)
Greek Symbols
porosity (constant)
-
b(p, c)
brine dynamic viscosity
-
v(p)
vapour dynamic viscosity
-
(p, s, c)
conservation density (1)
-
b(p, c)
brine density
-
v(p)
vapour density
Suffixes
b
brine
-
m
rock matrix
-
t
total
-
v
vapour
-
S
salt
-
M
mass
-
E
energy 相似文献
5.
Yoshihisa Morita Hirokazu Ninomiya 《Journal of Dynamics and Differential Equations》2006,18(4):841-861
We deal with a reaction–diffusion equation u
t
= u
xx
+ f(u) which has two stable constant equilibria, u = 0, 1 and a monotone increasing traveling front solution u = φ(x + ct) (c > 0) connecting those equilibria. Suppose that u = a (0 < a < 1) is an unstable equilibrium and that the equation allows monotone increasing traveling front solutions u = ψ1(x + c
1
t) (c
1 < 0) and ψ2(x + c
2
t) (c
2 > 0) connecting u = 0 with u = a and u = a with u = 1, respectively. We call by an entire solution a classical solution which is defined for all
. We prove that there exists an entire solution such that for t≈ − ∞ it behaves as two fronts ψ1(x + c
1
t) and ψ2(x + c
2
t) on the left and right x-axes, respectively, while it converges to φ(x + ct) as t→∞. In addition, if c > − c
1, we show the existence of an entire solution which behaves as ψ1( − x + c
1
t) in
and φ(x + ct) in
for t≈ − ∞. 相似文献
6.
The present study is concerned with the dynamic anomalous response of an elastic-plastic column struck axially by a massm with an initial velocityv
0. This simple example is considered in order to clarify the influence of the impact characteristics and the material plastic properties on the dynamic buckling phenomenon and particularly on the final vibration amplitudes of the column when it shakes down to a wholly elastic behaviour. The material is assumed to have a linear strain hardening with a plastic with a plastic reloading allowed. These material properties are the reason a number of elastic-plastic cycles to be realized prior to any wholly elastic stable behaviour, which causes different amounts of energy to be absorbed due to the plastic deformations.The column exhibits two types of behaviour over the range of the impact masses — a quasi-periodic and a chaotic response. The chaotic behaviour is caused by the multiple equilibrium states of the column when any small changes in the loading parameters cause small changes in the plastic strains which result in large changes in the response behaviour. The two types of behaviour are represented by displacement-time and phase-plane diagrams. The sensitivity to the load parameters is illustrated by the calculation of a Lyapunov-like exponent. Poincaré maps are shown for three particular cases.Notation
c
elastic wave propagation speed
-
m
impact mass
-
m
c
column mass
-
s
step of the spatial discretization
-
t
time
-
u(x,t)
axial displacement
-
v
0
initial velocity
-
w
0(x)
initial imperfections
-
w(x,t)+w
0(x)
total lateral displacements
-
x
axial axis
-
z
axis along the column thickness
-
A
cross-section areahb
-
E
Young's modulus
-
E
t
hardening modulus (Figure 2)
-
M(x,t)
bending moment
-
N(x,t)
axial force
-
impact mass ratiom/m
c
-
(x,z)
strain
-
Lyapunov-like exponent
-
material density
-
(x,z)
stress 相似文献
7.
As an example of an extended, formally gradient dynamical system, we consider the damped hyperbolic equation u
tt+u
t=u+F(x, u) in R
N
, where F is a locally Lipschitz nonlinearity. Using local energy estimates, we study the semiflow defined by this equation in the uniformly local energy space H1
ul(R
N
)×L2
ul(R
N
). If N2, we show in particular that there exist no periodic orbits, except for equilibria, and we give a lower bound on the time needed for a bounded trajectory to return in a small neighborhood of the initial point. We also prove that any nonequilibrium point has a neighborhood which is never visited on average by the trajectories of the system, and we conclude that any bounded trajectory converges on average to the set of equilibria. Some counter-examples are constructed, which show that these results cannot be extended to higher space dimensions. 相似文献
8.
Recently a third-order existence theorem has been proven to establish the sufficient conditions of periodicity for the most general third-order ordinary differential equation
x+f(t,x,x′,x″)=0