Existence and nonuniqueness of rectangular solutions of the Bénard problem
Existence and nonuniqueness of rectangular solutions of the Bénard problem
复制标题
DOI：
10.1007/bf00256457
复制
发表时间：
1968
影响因子：
2.5
通讯作者：
P. Rabinowitz
中科院分区：
文献类型：
--
作者：
P. Rabinowitz
本文将研究B6 nard问题这一相对简单的对流问题。我们需要对以下理想化的实验进行数学描述：在两个刚性的、完全导电的壁之间，有一个最初静止的无限大的水平粘性流体层。在壁之间保持恒定的温度梯度，下壁较热。如果温度梯度很小，则流体保持静止，并且热量仅通过传导通过流体传输。然而，当温度梯度增加超过某一临界值时，流体经历称为对流的与时间无关的运动。热量现在通过对流和传导在流体中传递。在实际实验中，流体以规则的细胞模式排列，并且运动仅在细胞内发生[1，2]。细胞的形状似乎强烈依赖于容器的形状[2]。人们可以对上述现象作一个简单的定性解释。流体的底部部分由于加热而膨胀并且变得不那么致密。因此，它趋于上升。然而，粘性的流体抵抗这种浮力。如果温度梯度很小，粘性力占主导地位，流体保持静止，热量仅通过传导传递。在超过临界温度梯度时，浮力变得足够大以克服流体的粘性，并且对流开始。B~ NARD进行了这方面的原始实验。他的流体有一个自由的上表面，他发现细胞呈六边形[2]。然而，这种效应后来被证明[3]主要是由于表面张力，它在我们这里考虑的问题中起着微不足道的作用。我们试图表明，上述对流现象可以从流体的运动方程数学上得到。传导解很容易得到，并且对所有温度梯度值都存在。因此，这个问题在数学上就变成了一个非唯一性问题。我们必须证明，在超过临界温度梯度时，方程的新解，对应于对流的分支或从传导解分叉。实际上，我们会给出一个更完整的，这些方程的结构图。相应的线性化方程的检查表明，温度梯度出现的特征值。我们的主要结果将是存在的“对流”解决方案的温度梯度附近的线性理论的特征值。当然，主要的物理利益的解决方案是那些，
In this paper we will study the B6nard problem which is a relatively simple convection problem. We want a mathematical description of the following idealized experiment: An infinite horizontal layer of viscous fluid initially at rest lies between two rigid perfectly conducting walls. A constant temperature gradient is maintained between the walls, the lower wall being warmer. If the temperature gradient is small, the fluid remains at rest and heat is transported through the fluid only by conduction. However when the temperature gradient is increased beyond a certain critical value, the fluid undergoes time independent motions called convection corrents. Heat is now transported through the fluid by convection as well as conduction. In actual experiments, the fluid arranges itself in a regular cellular pattern, and motions take place only within the cells [1, 2]. The shape of the cells seems to depend strongly on the shape of the container [2]. One can give a simple qualitative explanation of the above phenomena. The bottom portion of the fluid expands because of the heating and becomes less dense. It therefore tends to rise. However the fluid, being viscous, resists this buoyancy force. If the temperature gradient is small the viscous forces are dominant and the fluid remains at rest, heat being transported only by conduction. On exceeding the critical temperature gradient, the buoyancy force becomes large enough to overcome the viscosity of the fluid and convection begins. B~ NARD conducted the original experiments in this area. His fluid had a free upper surface and he found the cells to be in the shape of hexagons [2]. However this effect was later shown [3] to be due primarily to surface tension, which plays a negligible role in the problem we consider here. We seek to show that the above convective phenomena can be obtained mathematically from the equations of motion of the fluid. The conduction solution is easily obtained and exists for all values of the temperature gradient. Thus the problem mathematically becomes one of nonuniqueness. We must show that on exceeding a critical temperature gradient, new solutions of the equations, corresponding to convection branch or bifurcate from the conduction solution. Actually we will give a much more complete picture of the structure of these equations. An examination of the corresponding linearized equations shows that the temperature gradient appears as an eigenvalue. Our main result will be the existence of" convective" solutions for temperature gradients near the eigenvalues of linear theory. Of course the solutions of major physical interest are those which