Exact solutions for quasi-one-dimensional compressible viscous flows in conical nozzles

New analytical solutions for the one-dimensional steady-state compressible viscous adiabatic flow of an ideal gas through a conical nozzle or diffuser have been obtained. In order to analytically solve the problem, it is essential to determine the correct transformations of the variables and to identify the kinetic energy per unit of mass as the physical variable that appears in the final Abel ordinary differential equation.A dimensionless representation is given of the new solution, which points out the fundamental role exerted by some dimensionless groups in problems where viscous power dissipation and variable flow areas are present simultaneously as driving factors of flow changes. Furthermore, a steady-state fluid dynamics analysis of the compressible viscous flows in conical nozzles and diffusers has been carried out to improve the physical interpretation of the solutions.Finally, the thus determined analytical solutions have been validated for both subsonic and supersonic flows through a comparison with numerical solutions pertaining to the same ordinary differential equation. However, when the exact solution includes shocks, the time-asymptotic numerical solutions of the Euler equations for the quasi-one-dimensional unsteady-state gas dynamics are used for validation and the discretisation is performed by applying a finite volume technique.The proposed analytical solutions are complementary to the Fanno and nozzle models that refer to a viscous adiabatic constant cross-section pipe flow and an inviscid adiabatic variable cross-section pipe flow, respectively, and extend the collection of the exact solutions of gas dynamics.