Rayleigh Benard: Instability and Boundary Conditions

This lecture covers the Rayleigh Benard convection phenomenon, focusing on the linearization and change of variables in the equations, reduction to two equations, normal mode expansion, and the conditions for instability. It also discusses the Marginal stability curves for Marangoni convection, the stress-free and thermal boundary conditions, and the solution ansatz that satisfies the boundary conditions. The lecture concludes with the dispersion relation, the minimum conditions for instability, and the experimental verification of stress-free boundary conditions. The instructor explains the concepts using mathematical equations and graphical representations.

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In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differential equation, it specifies the values that a solution needs to take along the boundary of the domain. In finite element method (FEM) analysis, essential or Dirichlet boundary condition is defined by weighted-integral form of a differential equation.

Robin boundary condition

In mathematics, the Robin boundary condition (ˈrɒbɪn; properly ʁɔbɛ̃), or third type boundary condition, is a type of boundary condition, named after Victor Gustave Robin (1855–1897). When imposed on an ordinary or a partial differential equation, it is a specification of a linear combination of the values of a function and the values of its derivative on the boundary of the domain. Other equivalent names in use are Fourier-type condition and radiation condition.

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