Abstract <p>This study examines the flow patterns of viscoelastic-plastic filler in a molded volume during the manufacturing of repair parts for mining machinery and equipment. Promising technologies for the rapid manufacturing of metal-composite mining machine parts, such as metal-composite technology, require a mathematical description and functional dependencies describing the flow of metal-polymer composite materials (MPCMs) in complex channels formed by mold voids. The objectives of this study are to mathematically describe the MPCM flow process in mold channels and to derive mathematical dependencies describing the optimal channel shape. The paper analyzes a number of studies related to the search for the optimal surface shape ensuring minimal descent time when solving various engineering problems. It is revealed that all solutions are based on the properties of brachistochrone. However, in the process under consideration, additional conditions are imposed when formulating the problem, such as resistance, viscous or dry friction, the movement of solid or viscoelastic-plastic bodies, and others, resulting in a brachistochrone shape other than a cycloid. An equation was obtained for the generatrix that ensures the least work done by the friction forces of the wall-adjacent layers of the MPCM on the mold walls. The resulting mathematical relationship can be applied to the design of internal mold channels using specialized computer programs.</p>