The Rayleigh-Bénard Problem

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Mechanics of Fluids and the
Rayleigh-Bénard Problem 1 Introduction

The motion of fluids in certain systems continues to be a relevant to physicists. From the coiling of dribbling fluids to the convection current of a slice of fluid heated from below, patterns forming in natural systems are still unsolved. The best that can be done is to make models of the systems and use approximations.
In this paper I will give an overview of the Rayleigh-Bénard problem as well as describe the derivation of a four-mode lorenz model and a linear stability analysis of the hydrodynamic equations to find the critical Rayleigh number responsible for predicting convection in a system.

2 Basic Fluid Dynamics

2.1 What is a fluid?

A fluid is a substance that will deform (flow) when a shear stress is applied. This is in contrast to a solid, which will compress rather than continuously deform under such a stress. Fluids are a subset of the phases of matter and include liquids (incompressible fluids), and gases (compressible fluids).
Shown in Figure 2.1.1 is of a slice of fluid between two plates undergoing a shear stress. The bottom plate is stationary, while the top one is allowed to move. The fluid adjacent to the top plate will be pulled along at about the same speed as the plate, but the layer of fluid below that resists the motion just a bit and moves slower, same with the layer below that all the way to the layer next to the bottom plate. Figure 2.1.1


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