Natural convection from a downward facing horizontal heated plate was analyzed. An expression for the thickness of the thermal boundary layer was obtained in terms of Rayleigh number. Assuming this thickness as the characteristic length of the problem, the data published by other authors were modified and an equation for Nusselt number is presented. It was observed that this equation correlates the data more precisely than the commonly known equations in the literature that employ the ratio of the area to the perimeter or the shorter side of the plate as the characteristic length. It is concluded that taking the thermal boundary layer as the characteristic length of phenomenon is a proper approach and correlates all the data closely.

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We introduce a numerically tractable formulation of Bayesian joint models for longitudinal and survival data. The longitudinal process is modelled using generalised linear mixed models, while the survival process is modelled using a parametric general hazard structure. The two processes are linked by sharing fixed and random effects, separating the effects that play a role at the time scale from those that affect the hazard scale. This strategy allows for the inclusion of non-linear and time-dependent effects while avoiding the need for numerical integration, which facilitates the implementation of the proposed joint model. We explore the use of flexible parametric distributions for modelling the baseline hazard function which can capture the basic shapes of interest in practice. We discuss prior elicitation based on the interpretation of the parameters. We present an extensive simulation study, where we analyse the inferential properties of the proposed models, and illustrate the trade-off between flexibility, sample size, and censoring. We also apply our proposal to two real data applications in order to demonstrate the adaptability of our formulation both in univariate time-to-event data and in a competing risks framework. The methodology is implemented in rstan.