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This thesis presents a novel multiphysics solver, named gtawFoam, for Gas Tungsten Arc Welding (GTAW) that is applied to simulate orbital GTAW on ultra-thin-walled titanium tubing. In this thesis, ultra-thin-walled tubing refers to tubing where the wall thicknesses are less than 500 $μm$. Orbital welding of tubing with this wall thickness requires both a sufficient heat input to weld the tubing and an internal buttressing gas flow to ensure the tube retains its geometrical integrity. The specific use case is for the commercially pure grade 2 titanium tubing used in the ATLAS ITk cooling system which is 2.275 $mm$ outer diameter and 300 $μm$ wall thickness at the weld. The solver is created using the open source computational fluid dynamics library OpenFOAM and each component of the solver is benchmarked against an appropriate case. With the solver established, it is used to simulate a series of welding procedures that were performed experimentally on the aforementioned titanium tubing. Both the experimental and simulation results show a `goldilocks' region where the weld heat input and inner buttressing gas flow are moderated to a level where a fully penetrating weld is created but the geometric integrity of the tube is not compromised. gtawFoam is then used to simulate hypothetical tubing with larger and smaller wall thicknesses between 250 $μm$ and 350 $μm$. The results suggest that the required buttressing gas pressure once achieved is relatively transferable between wall thickness changes but applying enough heat so as to achieve full penetration is critical. These results are then used to predict effective welding procedures for this hypothetical tubing. gtawFoam is subsequently applied to the welding of turbine blades. This includes the addition of multiple layers of filler metal to mimic additive manufacturing.