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This study presents a two-dimensional (2D) finite-element (FE) linear model for predicting the thermal and fluid dynamics of an oil-filled distribution transformer. The model is based on the solution of the Navier-Stokes and Maxwell equations which are weakly coupled. Two electromagnetic FE models are used to calculate the load and no load losses of the transformer. The thermal-fluid equations are solved using first-order 2D FE and the characteristic-based-split (CBS) scheme. The CBS is used for the first time in the thermal fluid modelling of a transformer thereby is the proposed contribution of this study. The CBS algorithm is applicable to all ranges of flow and gives the same results as specialised methods such as the Petrov-Galerkin and Taylor-Galerkin methods. A natural convection analysis for an incompressible Newtonian flow in a distribution transformer is presented. Convection, radiation and symmetry boundary conditions are taken into account in the model. The model predictions are compared against those obtained with a commercial software and good agreement was obtained. The proposed model was programmed in C language.