The model can be used to analyse the thermal processes due to a frictional block or disc braking or to analyse the temperature conditions close to the ril/wheel contact patch, when macroscopic sliding occurs. It is especially useful in the evaluation of rail chill, head check and other contact tribology-based local heat and stress effects. The contact seizure theory of gear drives developed by Blok is based on a simplified approach to the determination of the temperature due a moving heat source generated by sliding friction. In a simplifying step, the heat flux parallel to the surface of the contacting bodies is disregarded. One-dimensional temperature patterns that vary with time are arranged along the line of motion, based on the displacement of the source. This one-dimensional model is applicable if the velocity (i.e. Péclet number) is 'high enough'. The influence of the width of the source has not been examined. The literature contains analytical results for various heat flux models, and one can make use of them to overcome the restrictions inherent in the one-dimensional theory. However, their rather complex structure makes them unsuited for further analysis, and only the numerical analysis method appears to be fruitful. One approach is to describe the temperature conditions created in a plate, which extends on a half-plane, by heating a part of its edge. The obtained time-dependent temperature results can be arranged into a sequence in a similar manner as in the approach of Blok. This shows the spatial temperature distribution that can be obtained by enhancing the results obtained using the one-dimensional theory by considering the effect of the crosswise dispersed heat. For heat sources of limited width, the cross-corrected approach better fits the reference values given by a full three-dimensional model. In railway traction, the analysis can be most directly applied to gear drives with a localized load pattern.
Special streamlines in the flow with circulation around a cylinder cross themselves, maybe even three times. The simple crossing happens orthogonally, while the threefold one shows up π/3 angles among the branches. There are no discontinuous changes as the pattern develops with growing circulation. These observations yield a general statement. Here we show that if a z = F (x, y) is a solution of the Δz = 0 Laplace equation and a z = const curve intersects itself (once or several times), then the branches running to that crossing point shall form equal angles.
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