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1.
2.
Existence of Weak Solutions to the Equations of Non-Stationary Motion of Non-Newtonian Fluids with Shear Rate Dependent Viscosity 总被引:1,自引:0,他引:1
Jörg Wolf 《Journal of Mathematical Fluid Mechanics》2007,9(1):104-138
In the present paper we prove the existence of weak solutions
to the equations of non-stationary motion of an incompressible fluid with shear rate dependent viscosity in a cylinder Q = Ω × (0,T), where
denotes an open set. For the power-low model with
we are able to construct a weak solution
with ∇ · u = 0. 相似文献
3.
In this paper, a digital photoelastic system was employed to observe the stress distribution generated by an ultrasonic wave
impinged at the edge of a strip. According to the classical photoelastic theory, gray level distribution of the photoelastic
fringe patterns was formulated and expressed in the form of
, where J0 is the zeroth-order Bessel function. This new technique is called the time-averaged photoelastic method. To verify the proposed
method, the distribution of
was superimposed onto the distribution of gray level of the experimentally obtained photoelastic fringe pattern caused by
standing wave only. Except regions near the center of fringes, well-matched results were found. 相似文献
4.
We show two examples of systems
in
with
such that |Zt| is strictly decreasing in time for any n but
as
. 相似文献
5.
A connection between the symmetries of manifolds and differential equations is sought through the geodesic equations of maximally symmetric spaces, which have zero, constant positive or constant negative curvature. It is proved that for a space admitting so(n+1) or so(n,1) as the maximal isometry algebra, the symmetry of the geodesic equations of the space is given by so( or (where d
2 is the two-dimensional dilation algebra), while for those admitting (where represents semidirect sum) the algebra is sl(n+2). A corresponding result holds on replacing so(n) by so(p,q) with p+q = n. It is conjectured that if the isometry algebra of any underlying space of non-zero curvature is h, then the Lie symmetry algebra of the geodesic equations is given by , provided that there is no cross-section of zero curvature at the point under consideration. If there is a flat subspace of dimension m, then the symmetry group becomes ). 相似文献
6.
We consider the motion of a generalized Newtonian fluid, where the extra stress tensor is induced by a potential with p-structure (p = 2 corresponds to the Newtonian case). We focus on the three dimensional case with periodic boundary conditions and extend the existence result for strong solutions for small times from
\tfrac{5}{3}$$
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(see [16]) to
\tfrac{7}{5}.$$
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Moreover, for
we improve the regularity of the velocity field and show that
for all
0.$$
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Within this class of regularity, we prove uniqueness for all
\tfrac{7}{5}.$$
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We generalize these results to the case when p is space and time dependent and to the system governing the flow of electrorheological fluids as long as
相似文献
7.
We consider the Cauchy problem for incompressible Navier–Stokes equations
with initial data in
, and study in some detail the smoothing effect of the equation. We prove that for T < ∞ and for any positive integers n and m we have
, as long as
stays finite. 相似文献
8.
This paper presents an analytical and numerical study of transient free convection from a horizontal surface that is embedded in a fluid-saturated porous medium. It is assumed that for time
steady state velocity and temperature fields are obtained in the boundary-layer which occurs due to a uniform flux dissipation rate q
1 on the surface. Then, at
the heat flux on the surface is suddenly changed to q
2 and maintained at this value for
. Firstly, solutions which are valid for small and large
are obtained. The full boundary-layer equations are then integrated step-by-step for the transient regime from the initial unsteady state (
) until such times at which this forward marching approach is no longer well posed. Beyond this time no valid solutions could be obtained which matched the final solution from the forward integration to the steady state profiles at large times
. 相似文献
9.
A nonlinear nonlocal model arising from synaptically coupled neuronal networks with two integral terms is considered. The existence and stability of several traveling wave solutions are established by using ideas in differential equations and functional analysis. Steady-state solutions of some inhomogeneous integral–differential equations are also investigated. We consider several types of kernel functions: (I) positive functions, such as
and
, where ρ>0 is a constant; (II) nonnegative kernels with compact supports, for examples, (i)
1$$" align="middle" border="0">
, and (ii)
{\pi\over 2}$$" align="middle" border="0">
; (III) Mexican hat type kernel functions, such as
and
, where A>B>0 and a>b>0 are constants.Dedicated to Professor Yulin Zhou and Professor Boling Guo on the Occassions of their birthdays. 相似文献
10.
This paper addresses quasi-static loading of the shear compression specimen (SCS), that has been especially developed to investigate the shear response of
materials at various strain rates. Previous work [4, 5] addressed bi-linear hardening materials, whereas the present work
concerns parabolic hardening materials. The investigation is done numerically using three-dimensional elastoplastic finite
element simulations. The analyses show that the averaged von Mises stress (
) and strain (
) on a mid-section of the gauge reflect accurately the prescribed parabolic hardening model. A method for finding the parabolic
hardening coefficients and reducing the measured load, P, and displacement, d, into equivalent stress
and strain
is introduced and tested. A very good agreement is observed, thus confirming the potential of the technique for large strain
testing of parabolic hardening materials. 相似文献
11.
An analysis is made of the transient free convection from a vertical flat plate which is embedded in a fluid-saturated porous medium. It is assumed that for time
a steady state temperature or velocity has been obtained in the boundary-layer which occurs due to a uniform flux dissipation rate
. Then at time
the heat flux on the plate is suddenly changed to
and maintained at this value for
0$$
" align="middle" border="0">
. An analytical solution has been obtained for the temperature/velocity field for small times in which the transport effects are confined within an inner layer adjacent to the plate. These effects cause a new steady boundary layer. A numerical solution of the full boundary-layer equations is then obtained for the whole transient from
to the steady state, firstly by means of a step-by-step method and then by a matching technique. The transition between the two distinct solution methods is always observed to occur very near to the turning point of the plate surface temperature, a time at which the fluid temperature is close to its steady state profile. The solution obtained using the step-by-step method shows excellent agreement with the small time analytical solution. Results are presented to illustrate the occurrence of transients from both small and large increases and decreases in the levels of existing energy inputs. 相似文献
12.
A regular perturbation technique is employed to approximate the solution for fluid infiltration from a circular opening into an unsaturated medium. Introducing two empirical constitutive relations
and
relating the permeability k and water content with pore fluid pressure p, a nonlinear diffusion equation in terms of pore pressure is established. After rearranging the nonlinear diffusion equation, a parameter perturbation on
is performed and an approximate solution with an error of
is obtained, which correlate to a condition in which = . This approximate solution is verified by a finite difference solution and compared also with a linear solution in which the diffusivity is constant. It is shown that the perturbation solution with terms up to and including first-order can give a reasonably accurate solution for the parameter range for p
0 selected in this paper. The solution procedure provided in this paper also avoids the numerical problem normally encountered for a small time solution. The solution may also be used to overcome difficulties arising in solution procedure by the similarity transformation (Boltzmann), commonly conducted on diffusion equation, which cannot be applied for a finite wellbore problem. 相似文献
13.
Some Results on the <Emphasis Type="BoldItalic">m</Emphasis>-Laplace Equations with Two Growth Terms
We prove the existence of positive radial solutions of the following equation:
and give sufficient conditions on the positive functions K1(r) and K2 (r) for the existence and nonexistence of ground states (G.S.) and Singular ground states (S.G.S.), when
or
. We also give sufficient conditions for the existence of radial S.G.S. and G.S. of equation
when
and
, respectively. We are also able to classify all the S.G.S. of this equation. The proofs use a new Emden–Fowler transform which allow us to use techniques taken from dynamical system theory, in particular the ones developed in Johnson et al. (Nonlinear Anal, T.M.A. 20, 1279–1302 (1993)) for the problems obtained by substituting the ordinary Laplacian Δ for the m-Laplacian Δm in the preceding equations.MSC: 37B55, 35H30, 35J70 相似文献
14.
Flow in a porous medium with a random hydraulic conductivity tensor K(x) is analyzed when the mean conductivity tensor
(x) is a non-constant function of position x. The results are a non-local expression for the mean flux vector
(x) in terms of the gradient of the mean hydraulic head
(x), an integrodifferential equation for
(x), and expressions for the two point covariance functions of q(x) and (x). When K(x) is a Gaussian random function, the joint probability distribution of the functions q(x) and (x) is determined. 相似文献
15.
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
16.
We investigate the dynamics of the semiflow φ induced on H01(Ω) by the Cauchy problem of the semilinear parabolic equation
on Ω. Here
is a bounded smooth domain, and
has subcritical growth in u and satisfies
. In particular we are interested in the case when f is definite superlinear in u. The set
of attraction of 0 contains a decreasing family of invariant sets
distinguished by the rate of convergence
. We prove that the Wk’s are global submanifolds of
, and we find equilibria in the boundaries
. We also obtain results on nodal and comparison properties of these equilibria. In addition the paper contains a detailed exposition of the semigroup approach for semilinear equations, improving earlier results on stable manifolds and asymptotic behavior near an equilibrium.Supported by DFG Grant BA 1009/15-1. 相似文献
17.
Xinyu He 《Journal of Mathematical Fluid Mechanics》2007,9(3):398-410
Let
be the exterior of the closed unit ball. Consider the self-similar Euler system
Setting α = β = 1/2 gives the limiting case of Leray’s self-similar Navier–Stokes equations. Assuming smoothness and smallness of the boundary
data on ∂Ω, we prove that this system has a unique solution
, vanishing at infinity, precisely
The self-similarity transformation is v(x, t) = u(y)/(t* − t)α, y = x/(t* − t)β, where v(x, t) is a solution to the Euler equations. The existence of smooth function u(y) implies that the solution v(x, t) blows up at (x*, t*), x* = 0, t* < + ∞. This isolated singularity has bounded energy with unbounded L
2 − norm of curl v. 相似文献
18.
This paper deals with connected branches of nonstationary periodic trajectories of Hamilton equations
emanating from the degenerate stationary point
for H being the generalized Hénon-Heiles (HH) Hamiltonian:
or the generalized Yang-Mills (YM) Hamiltonian:
The existence of such branches has been proved. Minimal periods of searched trajectories near x0 have been described. 相似文献
19.
R. Danchin 《Journal of Mathematical Fluid Mechanics》2006,8(3):333-381
This paper is devoted to the study of the initial value problem for density dependent incompressible viscous fluids in a bounded
domain of
with
boundary. Homogeneous Dirichlet boundary conditions are prescribed on the velocity. Initial data are almost critical in term
of regularity: the initial density is in W1,q for some q > N, and the initial velocity has
fractional derivatives in Lr for some r > N and
arbitrarily small. Assuming in addition that the initial density is bounded away from 0, we prove existence and uniqueness
on a short time interval. This result is shown to be global in dimension N = 2 regardless of the size of the data, or in dimension N ≥ 3 if the initial velocity is small.
Similar qualitative results were obtained earlier in dimension N = 2, 3 by O. Ladyzhenskaya and V. Solonnikov in [18] for initial densities in W1,∞ and initial velocities in
with q > N. 相似文献
20.
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