A class of two-level explicit difference schemes for solving three dimensional heat conduction equation

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｉｓｐａｐｅｒｄｅａｌｓｗｉｔｈｔｈｅｉｎｉｔｉａｌ＿ｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｏｆｔｈｒｅｅ＿ｄｉｍｅｎｓｉｏｎａｌｈｅａｔｃｏｎｄｕｃｔｉｏｎｅｑｕａｔｉｏｎｉｎｔｈｅｒｅｇｉｏｎＤ :0≤ｘ,ｙ ,ｚ≤Ｌ ,0 ≤ｔ≤Ｔ ｕ ｔ= 2 ｕ ｘ2 2 ｕ ｙ2 2 ｕ ｚ2 ,ｕ｜ｘ=0 =ｆ1(ｙ,ｚ,ｔ) ,　ｕ｜ｘ=Ｌ =ｆ2 (ｙ ,ｚ,ｔ) ,ｕ｜ｙ=0 =ｇ1(ｚ,ｘ,ｔ) ,　ｕ｜ｙ=Ｌ =ｇ2 (ｚ,ｘ,ｔ) ,ｕ｜ｚ=0 =ｈ1(ｘ ,ｙ ,ｔ) ,　ｕ｜ｚ=Ｌ =ｈ2 (ｘ ,ｙ ,ｔ) ,ｕ｜ｔ=0 =φ(ｘ ,ｙ,ｚ) .(1 )(2 )…     

A family of high-order accuracy explicit difference schemes with branching stability for solving 3-D parabolic partial differential equation

ＩｎｔｒｏｄｕｃｔｉｏｎＷｅｏｆｔｅｎｍｅｅｔｔｈｅｐｒｏｂｌｅｍｏｆｓｏｌｖｉｎｇｅｑｕａｔｉｏｎｏｆｐａｒａｂｏｌｉｃｔｙｐｅｉｎｍａｎｙｆｉｅｌｄｓｓｕｃｈａｓｓｅｅｐａｇｅ ,ｄｉｆｆｕｓｉｏｎ ,ｈｅａｔｃｏｎｄｕｃｔｉｏｎａｎｄｓｏｏｎ .Ｉｎｔｈｅｃａｓｅｏｆ3＿ｄｉｍｅｎｓｉｏｎ ,ｔｈｅｍｏｄｅｌｉｓａｎｉｎｉｔｉａｌａｎｄｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍａｓｆｏｌｌｏｗｓ: ｕ ｔ = 2 ｕ ｘ2 2 ｕ ｙ2 2 ｕ ｚ2 　　　　 (0 <ｘ,ｙ,ｚ<1 ;ｔ>0 ) ,ｕ(ｘ ,ｙ,ｚ,0 ) =φ(ｘ ,ｙ ,ｚ)…     

A new high-order accuracy explict difference scheme for solving three-dimensional parabolic equations

In this paper, a new three-level explicit difference scheme with high-orderaccuracy is proposed for solving three-dimensional parabolic equations. The stabilitycondition is r＝△t/△x² ＝△t/△y²＝△t/△z²≤1/4, and the trumcation error is O(△t²+△x⁴).     

Existence of the minimal positive solution of some nonlinear elliptic systems when the nonlinearity is the sum of a sublinear and a superlinear term

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒ,ｗｅｃｏｎｓｉｄｅｒｔｈｅｅｌｌｉｐｔｉｃｓｙｓｔｅｍ(1λ)　-Δｕ=ｆ(λ,ｘ,ｕ)-ｖ　　(ｉｎΩ),-Δｖ=δｕ-γｖ(ｉｎΩ),ｕ=ｖ=0(ｏｎΩ),ｗｈｅｒｅΩｉｓａｓｍｏｏｔｈｂｏｕｎｄｅｄｄｏｍａｉｎｉｎＲＮ(Ｎ≥2)ａｎｄλｉｓａｒｅａｌｐａｒａｍｅｔｅｒ.Ｔｈｅｓｏｌｕｔｉｏｎｓ(ｕ,ｖ)ｏｆｔｈｉｓｓｙｓｔｅｍｒｅｐｒｅｓｅｎｔｓｔｅａｄｙｓｔａｔｅｓｏｌｕｔｉｏｎｓｏｆｒｅａｃｔｉｏｎｄｉｆｆｕｓｉｏｎｓｙｓｔｅｍｓｄｅｒｉｖｅｄｆｒｏｍｓｅｖｅｒａｌａｐ…     

The Decay Rate and Higher Approximation of Mild Solutions to the Navier�CStokes Equations

In this paper, we consider v(t) = u(t) − e ^tΔ u ₀, where u(t) is the mild solution of the Navier–Stokes equations with the initial data u₀ ? L²(\mathbb Rⁿ)?Lⁿ(\mathbb Rⁿ){u_0\in L^2({\mathbb R}^n)\cap L^n({\mathbb R}^n)} . We shall show that the L ² norm of D ^β v(t) decays like t^{-\frac |b|-1 2-\frac n4}{t^{-\frac {|\beta|-1} {2}-\frac n4}} for |β| ≥ 0. Moreover, we will find the asymptotic profile u ₁(t) such that the L ² norm of D ^β (v(t) − u ₁(t)) decays faster for 3 ≤ n ≤ 5 and |β| ≥ 0. Besides, higher-order asymptotics of v(t) are deduced under some assumptions.     

The high accuracy explicit difference scheme for solving parabolic equations 3-dimension

ＩｎｔｒｏｄｕｃｔｉｏｎＦｒｏｍｐｒａｃｔｉｃａｌｐｒｏｂｌｅｍ,ｗｅｃｏｕｌｄｃｏｎｃｌｕｄｅｍａｎｙｐｒｏｂｌｅｍｓａｂｏｕｔｓｏｌｖｉｎｇｐａｒａｂｏｌｉｃｐａｒｔｉａｌｄｉｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎ．Ｎｏｗｔｈｅｒｅａｒｅｍａｎｙｎｕ...     

A further assessment of interpolation schemes for window deformation in PIV

We have evaluated the performances of the following seven interpolation schemes used for window deformation in particle image velocimetry (PIV): the linear, quadratic, B-spline, cubic, sinc, Lagrange, and Gaussian interpolations. Artificially generated images comprised particles of diameter in a range 1.1 ≤ d _p ≤ 10.0 pixel were investigated. Three particle diameters were selected for detailed evaluation: d _p = 2.2, 3.3, and 4.4 pixel with a constant particle concentration 0.02 particle/pixel². Two flow patterns were considered: uniform and shear flow. The mean and random errors, and the computation times of the interpolation schemes were determined and compared.     

Convection in a Lid-Driven Heat-Generating Porous Cavity with Alternative Thermal Boundary Conditions

Mixed convection flow in a two-sided lid-driven cavity filled with heat-generating porous medium is numerically investigated. The top and bottom walls are moving in opposite directions at different temperatures, while the side vertical walls are considered adiabatic. The governing equations are solved using the finite-volume method with the SIMPLE algorithm. The numerical procedure adopted in this study yields a consistent performance over a wide range of parameters that were 10⁻⁴ ≤ Da ≤ 10⁻¹ and 0 ≤ Ra _I ≤ 10⁴. The effects of the parameters involved on the heat transfer characteristics are studied in detail. It is found that the variation of the average Nusselt number is non-linear for increasing values of the Darcy number with uniform or non-uniform heating condition.     

On the perturbational global attractivity of nonautomous delay differential equations

Consider the perturbed nonautonomous linear delay differential equation x(t) = - a(t)x(t-τ) + F(t, x₁, t ⩾ 0 where x₁(s)=x(t+s) for −δ≤s≤0. Suppose that a(t) ∈ C([0,∞), (0,∞)), τ≥0,F:[0, ∞) x C[−δ,0] → R is a continuous functions and F(t,0) ≡ 0. Here C[−δ,0] is the space of continuous functions Φ: [−δ,0] → R with ∥Φ∥<H for the norm | Φ |, where |·| is any norm in R and 0<H≤+∞. Most of the known papers [1–5,7] have been concerned with the local or global asymptotic behavior of the zero solution of Eq. (*) when a(t) is independent of t i. e., a(t) is autonomous. The aim in this paper is to derive the sufficient conditions for the global attractivity of the zero solution of of Eq. (*) When a(t) is nonautomous. Our results, which extend and improve the known results, are even “sharp”. At the same time, the method used in this paper can be applicable to the perturbed nonlinear equation. Project supported by the Natural Science Foundation of Hunan     

Numerical investigation of natural convection of Al<Subscript>2</Subscript>O<Subscript>3</Subscript> nanofluid in vertical annuli

This paper presents the numerical study of internal free convection of Al₂O₃ water nanofluid in vertical annuli. Vertical walls are maintained at constant temperatures and horizontal walls are adiabatic. Results are validated by experimental data. Effect of nanofluids on natural convection is investigated as a function of geometrical and physical parameters and particle fractions for aspect ratio of 1 ≤ H/L ≤ 5, Grashof number of 10³ ≤ Gr ≤ 10⁵ and concentration of 0 ≤ ϕ ≤ 0.06. More than 330 different numerical cases are investigated to develop a new correlation for the Nusselt number. This correlation is presented as a function of Nusselt number of base fluid and particle fraction which is a linear decreasing function of particle fraction. The developed correlation for annuli is also valid for the natural convection of Al₂O₃ water nanofluid in a square cavity. Furthermore, the effect of the viscosity and conductivity models on the Nusselt number of nanofluids in cylindrical cavities are discussed.     

Secondary flow patterns and mixing in laminar pulsating flow through a curved pipe

Mixing by secondary flow is studied by particle image velocimetry (PIV) in a developing laminar pulsating flow through a circular curved pipe. The pipe curvature ratio is η = r ₀/r _c  = 0.09, and the curvature angle is 90°. Different secondary flow patterns are formed during an oscillation period due to competition among the centrifugal, inertial, and viscous forces. These different secondary-flow structures lead to different transverse-mixing schemes in the flow. Here, transverse mixing enhancement is investigated by imposing different pulsating conditions (Dean number, velocity ratio, and frequency parameter); favorable pulsating conditions for mixing are introduced. To obviate light-refraction effects during PIV measurements, a T-shaped structure is installed downstream of the curved pipe. Experiments are carried out for the Reynolds numbers range 420 ≤ Re_st ≤ 1,000 (Dean numbers 126.6 ≤ Dn ≤ 301.5) corresponding to non-oscillating flow, velocity component ratios 1 ≤ (β = U _max,osc/U _m,st) ≤ 4 (the ratio of velocity amplitude of oscillations to the mean velocity without oscillations), and frequency parameters 8.37 < (α = r ₀(ω/ν)^0.5) < 24.5, where α² is the ratio of viscous diffusion time over the pipe radius to the characteristic oscillation time. The variations in cross-sectional average values of absolute axial vorticity (|ζ|) and transverse strain rate (|ε|) are analyzed in order to quantify mixing. The effects of each parameter (Re_st, β, and α) on transverse mixing are discussed by comparing the dimensionless vorticities (|ζ _P |/|ζ _S |) and dimensionless transverse strain rates (|ε _P |/|ε _S |) during a complete oscillation period.     

An Efficient Derivation of the Aronsson Equation

For 1<p<∞, the equation which characterizes minima of the functional u↦∫ _U |Du| ^p ,dx subject to fixed values of u on ∂U is −Δ _p u=0. Here −Δ _p is the well-known ``p-Laplacian'. When p=∞ the corresponding functional is u↦|| |Du|<sup>2</sup>|| <sub>L∞(U)</sub> . A new feature arises in that minima are no longer unique unless U is allowed to vary, leading to the idea of ``absolute minimizers'. Aronsson showed that then the appropriate equation is −Δ_∞ u=0, that is, u is ``infinity harmonic' as explained below. Jensen showed that infinity harmonic functions, understood in the viscosity sense, are precisely the absolute minimizers. Here we advance results of Barron, Jensen and Wang concerning more general functionals u↦||f(x,u,Du)|| _L∞(U) by giving a simplified derivation of the corresponding necessary condition under weaker hypotheses. (Accepted September 6, 2002) Published online April 14, 2003 Communicated by S. Muller     

Maximum density effects on convective instability of horizontal plane poiseuille flows in the thermal entrance region

A linear stability analysis is used to study the conditions marking the onset of secondary flow in the form of longitudinal vortices for plane Poiseuille flow of water in the thermal entrance region of a horizontal parallel-plate channel by a numerical method. The water temperature range under consideration is 0∼30°C and the maximum density effect at 4°C is of primary interest. The basic flow solution for temperature includes axial heat conduction effect and the entrance temperature is taken to be uniform at far upstream location jackie=−∞ to allow for the upstream heat penetration through thermal entrance jackie=0. Numerical results for critical Rayleigh number are obtained for Peclet numbers 1, 10, 50 and thermal condition parameters (λ ₁, λ ₂) in the range of −2.0≤λ ₁≤−0.5 and −1.0≤λ ₂≤1.4. The analysis is motivated by a desire to determine the free convection effect on freezing or thawing in channel flow of water.     

On {SL(2, \mathbb R)} Valued Smooth Proximal Cocycles and Cocycles with Positive Lyapunov Exponents Over Irrational Rotation Flows

Consider the class of C ^r -smooth SL(2, \mathbb R){SL(2, \mathbb R)} valued cocycles, based on the rotation flow on the two torus with irrational rotation number α. We show that in this class, (i) cocycles with positive Lyapunov exponents are dense and (ii) cocycles that are either uniformly hyperbolic or proximal are generic, if α satisfies the following Liouville type condition: |a-\fracp_nq_n| ￡ C exp (-q^r+1+k_n)\left|\alpha-\frac{p_n}{q_n}\right| \leq C {\rm exp} (-q^{r+1+\kappa}_{n}), where C >  0 and 0 < k < 1{0 < \kappa <1 } are some constants and \fracP_nq_n{\frac{P_n}{q_n}} is some sequence of irreducible fractions.     

A class of variational difference schemes for a singular perturbation problem

In this paper, a singularly perturbed boundary value problem for second order self-adjoint ordinary differential equation is discussed. A class of variational difference schemes is constructed by the finite element method. Uniform convergence about small parameter is proved under a weaker smooth condition with respect to the coefficients of the equation. The schemes studied in refs. [1], [3], [4] and [5] belong to the class.     

A Variational Approach to the Fredholm Alternative for the p-Laplacian near the First Eigenvalue

We are concerned with the existence of a weak solution to the degenerate quasi-linear Dirichlet boundary value problem

Existence,uniqueness and asymptotics of steady jets

We consider the three-dimensional flow through an aperture in a plane either with a prescribed flux or pressure drop condition. We discuss the existence and uniqueness of solutions for small data in weighted spaces and derive their complete asymptotic behaviour at infinity. Moreover, we show that each solution with a bounded Dirichlet integral, which has a certain weak additional decay, behaves like O(r ⁻²) as r=|x|→∞ and admits a wide jet region. These investigations are based on the solvability properties of the linear Stokes system in a half space ℝ ₊ ³ . To investigate the Stokes problem in ℝ ₊ ³ , we apply the Mellin transform technique and reduce the Stokes problem to the determination of the spectrum of the corresponding invariant Stokes-Beltrami operator on the hemisphere.     

On perfectly stress field at a mixed-mode crack tip under plane and anti-plane strain

In [1], under the condition that all the perfectly plastic stress components at a crack tip are functions of ϕ only, making use of equilibrium equations, stress-strain rate relations, compatibility equations and yield condition. Lin derived the general analytical expressions of the perfectly plastic stress field at a mixed-mode crack tip under plane and anti-plane strain. But in [1] there were several restrictions on the proportionality factor γ in the stress-strain rate relations, such as supposing that γ is independent of ϕ and supposing that γ=c or cr⁻¹. In this paper, we abolish these restrictions. The cases in [1], γ=cr^d (n=0 or-1) are the special cases of this paper.     

The Critical Dimension for a Fourth Order Elliptic Problem with Singular Nonlinearity

We study the regularity of the extremal solution of the semilinear biharmonic equation ${{\Delta^2} u=\frac{\lambda}{(1-u)^2}}We study the regularity of the extremal solution of the semilinear biharmonic equation D² u=\fracl(1-u)²{{\Delta^2} u=\frac{\lambda}{(1-u)^2}}, which models a simple micro-electromechanical system (MEMS) device on a ball B ì \mathbbR^N{B\subset{\mathbb{R}}^N}, under Dirichlet boundary conditions u=?_n u=0{u=\partial_\nu u=0} on ?B{\partial B}. We complete here the results of Lin and Yang [14] regarding the identification of a “pull-in voltage” λ* > 0 such that a stable classical solution u _λ with 0 < u _λ < 1 exists for l ? (0,l^*){\lambda\in (0,\lambda^*)}, while there is none of any kind when λ > λ*. Our main result asserts that the extremal solution u_l^*{u_{\lambda^*}} is regular (sup_B u_l^* < 1 ){({\rm sup}_B u_{\lambda^*} <1 )} provided N \leqq 8{N \leqq 8} while u_l^*{u_{\lambda^*}} is singular (sup_B u_l^* = 1){({\rm sup}_B u_{\lambda^*} =1)} for N \geqq 9{N \geqq 9}, in which case 1-C₀|x|^4/3 \leqq u_l^* (x) \leqq 1-|x|^4/3{1-C_0|x|^{4/3} \leqq u_{\lambda^*} (x) \leqq 1-|x|^{4/3}} on the unit ball, where C₀:=(\fracl^*[`(l)])<sup>\frac</sup>13{C_0:=\left(\frac{\lambda^*}{\overline{\lambda}}\right)^\frac{1}{3}} and [`(l)]: = \frac89(N-\frac23)(N- \frac83){\bar{\lambda}:= \frac{8}{9}\left(N-\frac{2}{3}\right)\left(N- \frac{8}{3}\right)}.     

On the free energy and variational theorem of elastic problem in thermal shock

In some investigations on variational principle for coupled thermoelastic problems, the free energy Φ(e_ij,θ) ,where the state variables are elastic strain e_ij and temperature increment θ, is expressed as Φ(e_ij,θ)=λ/2e_kke_ij=ue_k1e_k1-γe_kkθ-c/2 p θ²/T₀(0.1) This expression is employed only under the condition of |θ|≤T₀(absolute temperature of reference) But the value of temperature increment is great, even greater than T₀ in thermal shock. And the material properties (λ ,μ ,ν ,c , etc.) will not remain constant, they vary with θ. The expression of free energy for this condition.is derived in this paper. Equation (0.1) is its special case.Euler’s equations will be nonlinear while this expression of free energy has been introduced into variational theorem. In order to linearise, the time interval of thermal shock is divided into a number of time elements Δt_k, (Δt_k=t_k-t_k-1,k=1,2…,n), which are so small that the temperature increment θ_k within it is very small, too. Thus, the material properties may be defined by temperature field T_k-1=T(x₁,x₂,x₃,t_k-1) at instant t_k-1 , and the free energy Φ_k expressed by eg. (0.1) may be employed in element Δt_k.Hence the variational theorem will be expressed partly and approximately.