Resonant Penetrative Convection with an Internal Heat Source/Sink

We develop a detailed linear instability and nonlinear stability analysis for the situation of convection in a horizontal plane layer of fluid when there is a heat sink/source which is linear in the vertical coordinate which is in the opposite direction to gravity. This can give rise to a scenario where the layer effectively splits into three sublayers. In the lowest one the fluid has a tendency to be convectively unstable while in the intermediate layer it will be gravitationally stable. In the top layer there is again the possibility for the layer to be unstable. This results in a problem where convection may initiate in either the lowest layer, the upmost layer, or perhaps in both sublayers simultaneously. In the last case there is the possibility of resonance between the upmost and lowest layers. In all cases penetrative convection may occur where convective movement in one layer induces motion in an adjacent sublayer. In certain cases the critical Rayleigh number for thermal convection may display a very rapid increase which is much greater than normal. Such behaviour may have application in energy research such as in thermal insulation.     

流经有热源多孔平板并伴有化学反应的传热传质混合对流MHD流动

对流经无限竖直多孔平板的不可压缩粘性导电流体,稳定的传热传质混合对流MHD流动问题,给出了精确解和数值解.假定均匀磁场横向作用于流动方向,考虑了感应磁场及其能量的粘性和磁性损耗.多孔平板有恒定的吸入速度并均匀地混入流动速度.用摄动技术和数值方法求解控制方程.得到了平板上速度场、温度场、感应磁场、表面摩擦力和传热率的分析表达式.相关参数取不同数值时,用图形表示出问题的数值结果.讨论了从平板到流体的Hartmann数、化学反应参数、磁场的Prandtl数,以及包括速度场、温度场、浓度场和感应磁场等其它参数的影响.可以发现,热源/汇或Eckert数的增大,极大地提高了流体的速度值.x-方向的感应磁场随着Hartmann数、磁场的Prandtl数、热源/汇和粘性耗散的增大而增大.但是,研究表明,随着破坏性化学反应(K0)的增大,流动速度、流体温度和感应磁场将减小.对色谱分析系统和材料加工的磁场控制,该研究在热离子反应堆模型、电磁感应、磁流体动力学传输现象中得到了应用.     

饱和的多孔介质中含夹杂时的磁流体动力学自然对流

在一个多孔介质中含倾斜矩形夹杂物时,数值地研究磁流体动力学自然对流热交换问题,矩形夹杂的边界条件为两个绝热壁和两个等温壁,使用适当的变量集,将控制方程:连续性条件、推广的Darcy定理--Forchheimer律、能量方程变换为无量纲形式,然后使用有限差分格式求解,磁效应数、Darcy-Rayleigh数、矩形夹杂物的倾斜角以及纵横比作为控制参数,得到多孔介质中含矩形夹杂物时,磁效应数和倾斜角参数影响流体流动和热交换的规律.     

非Darcy多孔介质中的非线性对流

利用温度-浓度-密度关系,研究非Darcy多孔介质中的自由对流问题.对于不同的惯性参数、传递参数、Rayleigh数、Lewis数、Soret数和Dufour数,分析了非线性温度参数和浓度参数对非线性对流的影响.浮力对对流起着辅助的附加作用,当惯性作用不计时,切向速度随着非线性温度和浓度的增加而急剧地增加.然而,当惯性效应不为0时,非线性温度和浓度对切向速度的影响是有限的.对两个传递参数、惯性影响参数以及控制非线性温度和浓度的其他参数,取不同的数值时,浓度分布有点儿变化,并在不同的范围内传播.随着非线性温度和浓度的增加,传热/传质在很大的范围内变化,这取决于是Dacry多孔介质,还是非Darcy多孔介质.当所有的影响(惯性的影响、两个传递系数的影响、Soret和Dufour的影响)同时为0/不为0,在非线性温度/浓度参数以及浮力的共同作用下,分析了传热/传质的变化.发现在Darcy多孔介质中,温度和浓度以及它们的交叉扩散,对传热/传质的影响,要比非Darcy多孔介质要大.发现了浮力的负面作用,随着非线性温度系数的增加,传热/传质率是提高的,而随着非线性浓度系数的增加,传热/传质率是下降的.     

多孔介质中的具有热源的Brinkman方程解的收敛性

研究多孔介质中的一类含有热源的Brinkman方程的结构稳定性,得到解对Brinkman系数的收敛.即当Brinkman系数趋于零时,Brinkman方程组的解收敛于相应的Darcy方程组的解.     

Pattern Formations in Heat Convection Problems

After Bénard＇s experiment in 1900, Rayleigh formulated heat convection problems by the Oberbeck-Boussinesq approximation in the horizontal strip domain in 1916. The pattern formations have been investigated by the bifurcation theory, weakly nonlinear theories and computational approaches. The boundary conditions for the velocity on the upper and lower boundaries are usually assumed as stress-free or no-slip. In the first part of this paper, some bifurcation pictures for the case of the stress-free on the upper boundary and the no-slip on the lower boundary are obtained. In the second part of this paper, the bifurcation pictures for the case of the stress-free on both boundaries by a computer assisted proof are verified. At last., Bénard-Marangoni heat convections for the ease of the free surface of the upper boundary are considered.     

在有热源/热汇的竖直通道中作混合对流时的第3类边界条件

在竖向平行板通道中,研究粘性耗散和热源/热汇对充分发展的混合层状对流的影响.平板与外部流体进行了热交换,考虑了两种边界条件:与外部流体的参考温度相同的和不相同的.首先,数值地求解Brinkman数或Grashof数可以忽略的简单情况,然后,运用摄动级数法分析有热源/热汇存在时,浮力和粘性耗散对流动的组合影响,并证明摄动参数不大时是有效的.为了放宽摄动参数的使用条件,运用4阶Runge-Kutta打靶法,求解速度和温度场,对平板处的速度、温度、表面摩擦和Nusselt数进行数值讨论并绘出图形.     

放/吸热流体在两个充满多孔材料的竖直平行板之间作不稳定的自然对流

在一个由两块无限竖直平行板组成的管道中,充满着多孔的介质材料,使用Darcy模型(Brinkman模型的推广)的动量方程,连同能量方程,计算不可压缩、粘性、放/吸热流体在该管道中的不稳定自然对流,即Couette流动.流动是由于边界平板有不对称的加热,以及作加速运动所引起.选用合理的无量纲参数,对控制方程进行简化,通过Laplace变换进行解析求解,得到闭式的速度和温度分布曲线解,随后导出表面摩擦力和传热率.发现在竖直管道中的不同剖面,流体的流动及温度分布曲线随着时间而增加,且在运动平板附近更高.特别是,流体的速度和温度随着平板间距的增加而增加,但是,表面摩擦力和热传导率随着平板间距的增加而减小.     

On the Continuous Dependence of the Exterior Dirichlet Problem on the Boundary

We consider a sequence of exterior domains D_j,j∈ℕ₀, and assume that the boundaries ∂D_j converge to ∂D₀ with respect to the Hausdorff distance. We investigate solutions to the exterior Dirichlet problem for the Laplace equation and for the Helmholtz equation in these domains. Assuming convergence of the boundary data and D_j⊂D₀, j∈ℕ, then, by essentially using the method of Perron, we show that the solutions in the domains D_j converge to the solution in the domain D₀ with respect to the maximum norm. We prove the same result in case that the requirement D_j⊂D₀,j∈ℕ, is replaced by an equicontinuity property of all barrier functions to all boundary points. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd. Math. Meth. Appl. Sci., Vol. 20, 707–716 (1997)     

Double-Diffusive Convection in a Porous Medium,Nonlinear Stability,and the Brinkman Effect

This paper establishes a nonlinear energy stability theory for the double-diffusive convection in a porous medium when both viscosity and thermal expansion coefficient are allowed to vary with temperature. After presenting a nonlinear stability theorem, a variational problem is formulated and the numerical solutions via the compoound matrix method are carried out. It is noted that higher values of the thermal coefficient and higher values of the viscosity ratio have the effect of delaying the onset of convection.     

Emarks on Differential Inclusions Without Existence or Continuous Dependence

We study differential inclusions with existence which are natural approximations of differential inclusions without existence. In a Banach space setting, we also establish a multivalued version of a recent result of Colombo and Garay [3].     

Thermal Convection in a Darcy Porous Medium with Anisotropic Spatially Varying Permeability

We consider the problem of thermal convection in a saturated porous material of Darcy type. We allow a non-zero inertia term and also allow the permeability to be an anisotropic tensor. For a particular numerical result we consider the case of permeability tensor \(\mathbf{K}=\operatorname{diag} \{ {K_{\amalg},K_{\amalg},K_{\perp}(z) } \}\) . Linear instability results are calculated numerically and it is proved that the nonlinear energy stability bound is the same as the linear one.     

Convection Problems in Anisotropic Porous Media with Nonhomogeneous Porosity and Thermal Diffusivity

Convection problem in anisotropic and inhomogeneous porous media has been analysed. In particular, the effect of variable permeability and thermal diffusivity with respect to the vertical direction, has been studied. Linear and nonlinear stability analysis of the conduction solution have been performed.     

On the Existence and Parameter Dependence of Similarity Solutions for Free Convection on a Vertical Plate

The free convection on a heated vertical plate is considered, both for the case of prescribed plate temperature and the case of prescribed heat flux. The parameter dependence of the similarity solutions are investigated, and the existence proofs given for a range of the parameter values.     

伴有化学反应的磁流体传热传质混合对流流经吸入/喷出多孔楔形体时的局部非相似解

研究伴有化学反应的流经多孔楔形体的,传热传质磁流体的自由、受迫和混合对流.使用结合打靶法的Runge-Kutta-Gill方法,和直到3阶截断误差的局部非相似法,将偏微分的控制方程简化为9个常微分方程.通过Falkner-Skan变换,将边界层控制方程表示为无量纲形式.由于楔形体壁面的吸入/喷出,以及可变的壁面温度和浮力的影响,使得流场呈局部非相似性.就一些特定的无量纲参变数,给出具有3阶截断误差的数值计算.图形显示可变壁面温度和浓度条件下,伴有化学反应时磁场强度对无量纲速度、温度和浓度分布的影响.     

不可压缩Navier-Stokes方程解的非线性连续性

研究了不可压缩的Navier-Stokes方程弱解的粘性的连续依赖性.首先利用Moser迭代得到当T> 0时,在Ω×(0,T)上速度■的L^∞范数估计.其次讨论了对粘度μ的连续依赖性,并给出了精确的估计.     

Separately Continuous Functions on Products and Their Dependence on ℵ Coordinates

We investigate necessary and sufficient conditions on topological products X = ∏_{s ∈ s} X _s and Y = ∏_{t ∈ T} Y _t for every separately continuous function f: X × Y → ℝ to be dependent on at most ℵ coordinates with respect to a certain coordinate.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 10, pp. 1357 – 1368, October, 2004.     

Limits on the Transport of Heat and Momentum by Turbulent Convection with Large-Scale Flow

Under certain conditions, high-Rayleigh-number convection in a horizontal layer exhibits an organized pattern of large-scale flow superposed on an irregular structure of turbulent plumes. Bounds on the heat and momentum fluxes in such a flow, which provide upper estimates of the possible level of the large-scale flow, are presented.