Thermo-capillary convection flows in weld pools may significantly affect both the penetration and the weld bead width. The present work will discuss a mathematical study of the weld pool flows using the Order of Magnitude Scaling (OMS) algorithm. In contrast to finite-element of finite-volume modelling methods, the OMS methodology does not require a meshing of the problem domain; instead, it represents a combination of dimensional and asymptotic analyses. The algorithm has been implemented in prototype form in MATLAB. Since the asymptotic regime of the considered problem may not be known beforehand, the code exhausts all possible iterations of dominant terms, selects self-consistent balances and automatically estimate characteristic values for the asymptotic extremes. In addition to closed-form power-law expressions for characteristic values, the code determines a unique set of dimensionless groups that can be used to form parameter matrices in calibrations against experiments.
Four asymptotic regimes and their scaling laws are identified for the problem of thermo-capillary flows in weld pools. Three of the regimes were already known for small Pr numbers: Regime I for viscous flow with conduction being the dominant heat transfer mechanism, Regime II for conduction-dominated high-Re-number flows, and Regime III for convection-dominated high-Re-number flows. The scaling laws from literature for Regime III are revised. For the first time, the present work introduces Regime IV which corresponds to convection-dominated viscous flows for high Pr-number fluids. The regime represents, for example, the small-scale melt flow of polymers, which is particularly relevant to applications in modern additive manufacturing. The work analyses flows in autogenous fusion zones; possible expansion to wire-based welding processes is discussed.