Stationary solutions to a hybrid viscous hydrodynamic model with classical boundaries

In this paper we present a quantum-classical hybrid model based on the hydrodynamic equations in steady state form. The approach presented here, which has already been proposed in previous works, consists in considering an intrinsically hybrid version of the Bohm potential, which acts only in the region of the domain where quantum effects play an important role, while it disappears where the quantum contribution is essentially negligible and the operation of the device can be well described by using a classical model. Compared to previous results from the same line of research, here we assume that the device at the boundaries of the domain behaves classically, while quantum effects are localised in the central part of it. This is the case of greatest scientific interest, since, in real devices, quantum effects are generally localized in a small area within the device itself. The well posedness of the problem is ensured by adding a viscous term necessary for the convergence of the hybrid limit to an appropriate weak solution. Some numerical tests are also performed for different values of the viscous coefficient, in order to evaluate the effects of the viscosity, especially on the boundaries of the device.