Laser welding is commonly used in the automotive-, steel- and aerospace industry. It is a highly non-linear and coupled process where the weld geometry is strongly affected by the flow pattern in the melt pool. Experimental observations are challenging since the melt pool and melt flow below the surface are not yet accessible during welding. Improved process control would allow maintaining, or improving, product quality with less material and contribute further to sustainability by reducing production errors. Numerical modelling with Computational Fluid Dynamics, CFD, provides complementary understanding with access to process properties that are not yet reachable with experimental observation. However, the existing numerical models lack predictability when considering the weld shape. The work presented here is the development of a model for conduction laser welding. The solver upon which the model is based is first described in detail. Then different validation cases are applied in order to test specific parts of the physics implemented. Two cases focus on thermocapillary convection in two-phase and three-phase flows with surface deformation. Finally, a third case considers the melt pool flow during conduction mode welding.It is concluded that the convection of fusion enthalpy, which was neglected in former studies, should be included in the model. The implementation of the thermo capillary force is recommended to be consistent with the other surface forces to avoid unphysical solution. Free surface oscillations, known from experimental observations, are also computed numerically. However, further investigation is needed to check that these oscillations are not disturbed b ynumerical oscillations.