Simulation of the stationary nonisothermal MHD flows of polymeric fluids in channels with interior heating elements

Basing on the rheological mesoscopic Pokrovskii–Vinogradov model, the equations of nonrelativistic magneto-hydrodynamics, and the heat conduction equation with dissipative terms, we obtain a closed coupled system of nonlinear partial differential equations that describes the flow of solutions and melts of linear polymers. We take into account the rheology and induced anisotropy of 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 materials and devices used in additive technologies (as $3D$ printing). The statement is given of the problems concerning stationary polymeric fluid flows in channels with circular 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 of smoothness. 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 some special relaxation method. Under study are the dependencies of the distributions of the saturation fluid velocity and temperature on the pressure gradient in the channel.