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1.
The Darcy Model with the Boussinesq approximation is used to study natural convection in a shallow porous layer, with variable
permeability, filled with a binary fluid. The permeability of the medium is assumed to vary exponentially with the depth of
the layer. The two horizontal walls of the cavity are subject to constant fluxes of heat and solute while the two vertical
ones are impermeable and adiabatic. The governing parameters for the problem are the thermal Rayleigh number, R
T, the Lewis number, Le, the buoyancy ratio, φ, the aspect ratio of the cavity, A, the normalized porosity, ε, the variable permeability constant, c, and parameter a defining double-diffusive convection (a = 0) or Soret induced convection (a = 1). For convection in an infinite layer, an analytical solution of the steady form of the governing equations is obtained
on the basis of the parallel flow approximation. The onset of supercritical convection, or subcritical, convection are predicted by the present theory. A linear stability analysis of the parallel flow model is conducted and the
critical Rayleigh number for the onset of Hopf’s bifurcation is predicted numerically. Numerical solutions of the full governing
equations are found to be in excellent agreement with the analytical predictions. 相似文献
2.
We prove a regularity result for the anisotropic linear elasticity equation ${P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f}We prove a regularity result for the anisotropic linear elasticity equationP u : = div ( C ·?u) = f{P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f} , with mixed (displacement and traction) boundary conditions on a curved polyhedral domain
W ì \mathbbR3{\Omega \subset \mathbb{R}^3} in weighted Sobolev spaces Km+1a+1(W){\mathcal {K}^{m+1}_{a+1}(\Omega)} , for which the weight is given by the distance to the set of edges. In particular, we show that there is no loss of Kma{\mathcal {K}^{m}_{a}} -regularity. Our curved polyhedral domains are allowed to have cracks. We establish a well-posedness result when there are
no neighboring traction boundary conditions and |a| < η, for some small η > 0 that depends on P, on the boundary conditions, and on the domain Ω. Our results extend to other strongly elliptic systems and higher dimensions. 相似文献
3.
This paper presents an analytical and numerical study of natural convection of nanofluids contained in a rectangular enclosure
subject to uniform heat flux along the vertical sides. Governing parameters of the problem under study are the thermal Rayleigh
number Ra, the Prandtl number Pr, the aspect ratio of the cavity A and the solid volume fraction of nanoparticles, Φ. Three types of nanoparticles are taken into consideration: Cu, Al2O3 and TiO2. Various models are used for calculating the effective viscosity and thermal conductivity of nanofluids. In the first part
of the analytical study, a scale analysis is made for the boundary layer regime situation. In the second part, an analytical
solution based on the parallel flow approximation is reported for tall enclosures (A ≫ 1). In the boundary layer regime a good agreement is obtained between the predictions of the scale analysis and those of
the analytical solution. Solutions for the flow fields, temperature distributions and Nusselt numbers are obtained explicitly
in terms of the governing parameters of the problem. A numerical study of the same phenomenon, obtained by solving the complete
system of the governing equations, is also conducted. A good agreement is found between the analytical predictions and the
numerical simulations. 相似文献
4.
Ding Zhou Weiqing Liu O. G. McGeeIII 《Archive of Applied Mechanics (Ingenieur Archiv)》2011,81(4):473-487
This paper offers a three-dimensional elasticity-based variational Ritz procedure to examine the natural vibrations of an
elastic hollow torus with annular cross-section. The associated energy functional minimized in the Ritz procedure is formulated
using toroidal coordinates (r,q, j)({r,\theta , \varphi}) comprised of the usual polar coordinates (r, θ) originating at each circular cross-sectional center and a circumferential coordinate j{\varphi} around the torus originating at the torus center. As an enhancement to conventional use of algebraic–trigonometric polynomials
trial series in related solid body vibration studies in the associated literature, the assumed torus displacement, u, v and w in the r, θ and j{\varphi } toroidal directions, respectively, are approximated in this work as a triplicate product of Chebyshev polynomials in r and the periodic trigonometric functions in the θ and j{\varphi} directions along with a set of generalized coefficients. Upon invoking the stationary condition of the Lagrangian energy
functional for the elastic torus with respected to these generalized coefficients, the usual characteristic frequency equations
of natural vibrations of the elastic torus are derived. Upper bound convergence of the first seven non-dimensional frequency
parameters accurate to at least five significant figures is achieved by using only ten terms of the trial torus displacement
functions. Non-dimensional frequencies of elastic hollow tori are examined showing the effects of varying torus radius ratio
and cross-sectional radius ratio. 相似文献
5.
In this article, a similarity solution of the steady boundary layer flow near the stagnation-point flow on a permeable stretching
sheet in a porous medium saturated with a nanofluid and in the presence of internal heat generation/absorption is theoretically
studied. The governing partial differential equations with the corresponding boundary conditions are reduced to a set of ordinary
differential equations with the appropriate boundary conditions via Lie-group analysis. Copper (Cu) with water as its base
fluid has been considered and representative results have been obtained for the nanoparticle volume fraction parameter f{\phi} in the range 0 £ f £ 0.2{0\leq \phi \leq 0.2} with the Prandtl number of Pr = 6.8 for the water working fluid. Velocity and temperature profiles as well as the skin friction coefficient and the local
Nusselt number are determined numerically. The influence of pertinent parameters such as nanofluid volume fraction parameter,
the ratio of free stream velocity and stretching velocity parameter, the permeability parameter, suction/blowing parameter,
and heat source/sink parameter on the flow and heat transfer characteristics is discussed. Comparisons with published results
are also presented. It is shown that the inclusion of a nanoparticle into the base fluid of this problem is capable to change
the flow pattern. 相似文献
6.
Fluid flow and heat and mass transfer induced by double-diffusive natural convection in a horizontal porous layer subjected
to vertical gradients of temperature and concentration are studied analytically and numerically using the Brinkman-extended
Darcy model. Both cases of rigid and free horizontal boundaries are examined in the present study. The parameters governing
the problem are the Rayleigh number RT, the Lewis number Le, the buoyancy ratio N, the Darcy number Da and the aspect ratio Ar. The analytical solution is based on the parallel flow approximation. The critical
Rayleigh number corresponding to the onset of the parallel flow in this system is determined analytically as a function of
Le, N and Da. For sufficiently small Da, both free and rigid boundaries yield results which are identical to those predicted by
the Darcy model. The present investigation shows that there exists a region in the plane (N, Le) where the convective flow is not possible in the layer regardless of the Rayleigh and Darcy numbers considered.
Received on 21 December 1998 相似文献
7.
Tobias Hansel 《Journal of Mathematical Fluid Mechanics》2011,13(3):405-419
We consider the equations of Navier–Stokes modeling viscous fluid flow past a moving or rotating obstacle in
\mathbb Rd{\mathbb R^d} subject to a prescribed velocity condition at infinity. In contrast to previously known results, where the prescribed velocity
vector is assumed to be parallel to the axis of rotation, in this paper we are interested in a general outflow velocity. In
order to use L
p
-techniques we introduce a new coordinate system, in which we obtain a non-autonomous partial differential equation with an
unbounded drift term. We prove that the linearized problem in
\mathbb Rd{\mathbb R^d} is solved by an evolution system on
Lps(\mathbb Rd){L^p_{\sigma}(\mathbb R^d)} for 1 < p < ∞. For this we use results about time-dependent Ornstein–Uhlenbeck operators. Finally, we prove, for p ≥ d and initial data
u0 ? Lps(\mathbb Rd){u_0\in L^p_{\sigma}(\mathbb R^d)}, the existence of a unique mild solution to the full Navier–Stokes system. 相似文献
8.
Positivity and Almost Positivity of Biharmonic Green’s Functions under Dirichlet Boundary Conditions
Hans-Christoph Grunau Frédéric Robert 《Archive for Rational Mechanics and Analysis》2010,195(3):865-898
In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped
plate boundary value problem, neither a maximum principle nor a comparison principle or—equivalently—a positivity preserving
property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem
from being reasonably written as a system of second order boundary value problems. It is shown that, on the other hand, for
bounded smooth domains
W ì \mathbbRn{\Omega \subset\mathbb{R}^n} , the negative part of the corresponding Green’s function is “small” when compared with its singular positive part, provided
n\geqq 3{n\geqq 3} . Moreover, the biharmonic Green’s function in balls
B ì \mathbbRn{B\subset\mathbb{R}^n} under Dirichlet (that is, clamped) boundary conditions is known explicitly and is positive. It has been known for some time
that positivity is preserved under small regular perturbations of the domain, if n = 2. In the present paper, such a stability result is proved for
n\geqq 3{n\geqq 3} . 相似文献
9.
Matthias Geissert Matthias Hess Matthias Hieber Céline Schwarz Kyriakos Stavrakidis 《Journal of Mathematical Fluid Mechanics》2010,12(1):47-60
Introducing a new localization method involving Bogovskiĭ's operator we give a short and new proof for maximal Lp – Lq-estimates for the solution of the Stokes equation. Moreover, it is shown that, up to constants, the Stokes operator is an
R{\mathcal{R}}-sectorial operator in Lps(W)L^{p}_{\sigma}(\Omega), 1 < p < ¥1 < p < \infty, of R{\mathcal{R}}-angle 0, for bounded or exterior domains of Ω. 相似文献
10.
Fix a strictly increasing right continuous with left limits function ${W: \mathbb{R} \to \mathbb{R}}Fix a strictly increasing right continuous with left limits function
W: \mathbbR ? \mathbbR{W: \mathbb{R} \to \mathbb{R}} and a smooth function
F: [l,r] ? \mathbb R{\Phi : [l,r] \to \mathbb R}, defined on some interval [l, r] of
\mathbb R{\mathbb R}, such that
0 < b\leqq F¢\leqq b-1{0 < b\leqq \Phi'\leqq b^{-1}}. On the diffusive time scale, the evolution of the empirical density of exclusion processes with conductances given by W is described by the unique weak solution of the non-linear differential equation ?t r = (d/dx)(d/dW) F(r){\partial_t \rho = ({\rm d}/{\rm d}x)({\rm d}/{\rm d}W) \Phi(\rho)}. We also present some properties of the operator (d/dx)(d/dW). 相似文献
11.
Non-Newtonian fluid flow through porous media is of considerable interest in several fields, ranging from environmental sciences
to chemical and petroleum engineering. In this article, we consider an infinite porous domain of uniform permeability k and porosity f{\phi} , saturated by a weakly compressible non-Newtonian fluid, and analyze the dynamics of the pressure variation generated within
the domain by an instantaneous mass injection in its origin. The pressure is taken initially to be constant in the porous
domain. The fluid is described by a rheological power-law model of given consistency index H and flow behavior index n; n, < 1 describes shear-thinning behavior, n > 1 shear-thickening behavior; for n = 1, the Newtonian case is recovered. The law of motion for the fluid is a modified Darcy’s law based on the effective viscosity μ
ef
, in turn a function of f, H, n{\phi, H, n} . Coupling the flow law with the mass balance equation yields the nonlinear partial differential equation governing the pressure
field; an analytical solution is then derived as a function of a self-similar variable η = rt
β
(the exponent β being a suitable function of n), combining spatial coordinate r and time t. We revisit and expand the work in previous papers by providing a dimensionless general formulation and solution to the problem
depending on a geometrical parameter d, valid for plane (d = 1), cylindrical (d = 2), and semi-spherical (d = 3) geometry. When a shear-thinning fluid is considered, the analytical solution exhibits traveling wave characteristics,
in variance with Newtonian fluids; the front velocity is proportional to t
(n-2)/2 in plane geometry, t
(2n-3)/(3−n) in cylindrical geometry, and t
(3n-4)/[2(2−n)] in semi-spherical geometry. To reflect the uncertainty inherent in the value of the problem parameters, we consider selected
properties of fluid and matrix as independent random variables with an associated probability distribution. The influence
of the uncertain parameters on the front position and the pressure field is investigated via a global sensitivity analysis
evaluating the associated Sobol’ indices. The analysis reveals that compressibility coefficient and flow behavior index are
the most influential variables affecting the front position; when the excess pressure is considered, compressibility and permeability
coefficients contribute most to the total response variance. For both output variables the influence of the uncertainty in
the porosity is decidedly lower. 相似文献
12.
Changyou Wang 《Archive for Rational Mechanics and Analysis》2011,200(1):1-19
We investigate the well-posedness of (1) the heat flow of harmonic maps from
\mathbb Rn{\mathbb R^n} to a compact Riemannian manifold N without boundary for initial data in BMO; and (2) the hydrodynamic flow (u, d) of nematic liquid crystals on
\mathbb Rn{\mathbb R^n} for initial data in BMO−1 × BMO. 相似文献
13.
We prove that, if ${u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N}We prove that, if
u : W ì \mathbbRn ? \mathbbRN{u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N} is a solution to the Dirichlet variational problem
minwòW F(x, w, Dw) dx subject to w o u0 on ?W,\mathop {\rm min}\limits_{w}\int_{\Omega} F(x, w, Dw)\,{\rm d}x \quad {\rm subject \, to} \quad w \equiv u_0\; {\rm on}\;\partial \Omega, 相似文献
14.
In the regime of weakly transverse long waves, given long-wave initial data, we prove that a non-dimensionalized water wave
system in an infinite strip under the influence of gravity and surface tension on the upper free interface has a unique solution
on [0, T/ e{0, T/ \varepsilon} ] for some e{\varepsilon} independent of constant T. We shall prove in the subsequent paper (Ming et al., The long wave approximation to the three-dimensional capillary gravity
waves, 2011) that on the same time interval, these solutions can be accurately approximated by sums of solutions of two decoupled Kadomtsev–Petviashvili
(KP) equations. 相似文献
15.
We classify new classes of centers and of isochronous centers for polynomial differential systems in
\mathbb R2{\mathbb R^2} of arbitrary odd degree d ≥ 7 that in complex notation z = x + i
y can be written as
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