Basing on the rheological mesoscopic Pokrovskii–Vinogradov model, the equations ofnonrelativistic magneto-hydrodynamics, and the heat conduction equation with dissipative terms,we obtain a closed coupled system of nonlinear partial differential equations that describes theflow of solutions and melts of linear polymers. We take into account the rheology and inducedanisotropy of a polymeric fluid flow as well as mechanical, thermal, and electromagnetic impacts.The parameters of the equations are determined by mechanical tests with up-to-date materialsand devices used in additive technologies (as 3D printing). Thestatement is given of the problems concerning stationary polymeric fluid flows in channels withcircular and elliptical cross-sections with thin inclusions (some heating elements). We show that,for certain values of parameters, the equations can have three stationary solutions of high order ofsmoothness. Just these smooth solutions provide the defect-free additive manufacturing.To search for them, some algorithm is used that bases on the approximations without saturation,the collocation method, and the relaxation method. Under study are the dependences of thedistributions of the saturation fluid velocity and temperature on the pressure gradient in thechannel.