Branching T-channel is a very common element of a piping system for the transportation of liquids and gases. The brachesbranches of the T-channel can be inclined at different angles which affects the flow dynamics and heat transfer characteristics significantly. Thus, the present work focuses on the flow and thermal characteristics of the laminar forced convection of power-law fluids in a rectangular branching channel which have been numerically investigated over a wide range of parameters such as Reynolds number, 50 <= Re <= 300, Prandtl number, 10 <= Pr <= 50, inclination angle, 30 degrees <= alpha <= 90 degrees and power-law index, 0.2 <= n <= 1.4 (includes shear-thinning, n < 1, shear-thickening, n > 1 and Newtonian, n = 1 fluids). This is perhaps the first systematic study which examines the role of power-law fluid behaviour and of branch inclination on momentum and heat transfer characteristics. New extensive results for the flow and temperature fields are presented in terms of streamline contours and separated-flow zones, pressure coefficient, recirculation length, critical Reynolds number, isotherm contours, temperature profiles and local Nusselt number. The pressure coefficient is found to be higher for shear-thickening fluids than that for the Newtonian and shear-thinning fluids while the inclination angle has only a weak effect. The recirculation length bears a positive dependence on the Reynolds number and inclination angle while an inverse relationship is observed with power-law index in both branches. The critical Reynolds number, at which the onset of flow recirculation is observed, is found to be lower for higher inclination angles and a strong influence of the power-law index is also seen on the critical Reynolds number in both branches. Also, the local Nusselt number is seen to be higher at lower Prandtl numbers, low power law index values and high inclination angles for both branches. Overall, the inclination angle plays a significant role in determining the heat transfer characteristics. (C) 2019 Elsevier Ltd. All rights reserved.