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The hot spots conjecture is only known to be true for special geometries. This paper shows numerically that the hot spots conjecture can fail to be true for easy to construct bounded domains with one hole. The underlying eigenvalue problem for the Laplace equation with Neumann boundary condition is solved with boundary integral equations yielding a non-linear eigenvalue problem. Its discretization via the boundary element collocation method in combination with the algorithm by Beyn yields highly accurate results both for the first non-zero eigenvalue and its corresponding eigenfunction which is due to superconvergence. Additionally, it can be shown numerically that the ratio between the maximal/minimal value inside the domain and its maximal/minimal value on the boundary can be larger than 1 + 10− 3. Finally, numerical examples for easy to construct domains with up to five holes are provided which fail the hot spots conjecture as well.
The hybrid K+/Ca2+ sensor based on laser scanned silicon transducer for multi-component analysis
(2002)
Rocket engine test facilities and launch pads are typically equipped with a guide tube. Its purpose is to ensure the controlled and safe routing of the hot exhaust gases. In addition, the guide tube induces a suction that effects the nozzle flow, namely the flow separation during transient start-up and shut-down of the engine. A cold flow subscale nozzle in combination with a set of guide tubes was studied experimentally
to determine the main influencing parameters.