Pattern formations in fluids and computer assisted analysis

Pattern formations from the equilibrium state in fluid motions may be treated by the bifurcation theories. Bifurcation theorems can be applied to explain Taylor vortices of Taylor problems and hexagonal cells of heat convection problems as the first bifurcation. Computer assisted analysis becomes necessary to see global bifurcation structures. In this lecture examples of computer assisted proofs and analysis are explained for some heat convection problems. 1 Examples of pattern formations in fluids 1.1 Bénard observed that fluid in a plane heated from below develops a regular pattern of convection cells, above a certain threshold of temperature. The cell pattern is roughly classified into the roll-type, rectangle-type and hexagon-type. Rayleigh first analyzed the pattern formation theoretically. Taylor showed theoretically that fluid in the gap between two rotating cylinders on the same axis develops axisymmetric toroidal vortices, above a certain threshold in angular velocity of the cylinders. Kármán studied that a flow of fluid past a rigid body develops swirling vortices, above a certain threshold in velocity of the flow. 1.2 We focus our attention on the heat convection problems. Let us consider the general equation of motion of compressible fluid with viscosity and thermal conductivity in terms of Bénard’s experiment. For any difference ∆T between the temperature of the lower and upper boundaries, we have the (trivial) heat conduction solutions which correspond to the motionless state of the fluid. It is expected that, above a certain threshold of ∆T , the heat conduction solutions get unstable and nontrivial solutions appear which may be stable and correspond to the cell patterns mentioned above. The main tool for ∗Department of Pure and applied Mathematics, Waseda University, Tokyo 169-8555, Japan. Supported in part by JSPS Kakenhi No. 20540141 and by Japan-Germany joint project on Mathematical Fluid Dynamics.