We study a coupled system of the Navier--Stokes equation and the equation of conservation of mass in a network. The system models the blood circulation in arterial networks. A special feature of the system is that the equations are coupled through boundary conditions at joints of the network. We use a fixed point method to prove the existence and uniqueness of the classic solution to the initial-boundary value problem and discuss the continuity of dependence of the solution and its derivatives on initial, boundary, and forcing functions and their derivatives. We develop a numerical scheme that generates discretized solutions, and we also prove the convergence of the scheme.