Open flow boundaries are often present in viscoelastic flow calculations; their presence is not dictated by the physics of the problem, but rather by the need of truncating the computational domain. Viscoelastic liquids flowing in complex two- and three-dimensional domains are normally modeled by hyperbolic transport equations for the viscoelastic stress or conformation tensor, , where is the stress or conformation tensor, is the velocity, and ∇ denotes gradient in space. In steady flows, the streamlines are the characteristics of these hyperbolic equations and boundary conditions on are necessary where the liquid enters the flow domain.Open flow boundaries are almost always located in regions of fully-developed, rectilinear flow. Traditionally, several methods have been used to prescribe inflow conditions; each of them has one or more drawbacks in terms of applicability to general models, computational expense and complexity, and inability to deal with unknown flowrates, inflow-outflow boundaries, or unstructured meshes.Here, we propose a new, general way of imposing inflow boundary conditions based on solving the coupled algebraic equations of fully developed flow at the inflow, i.e., solving the equation at the inflow coupled with the flow inside the domain. The equation holds because in fully developed rectilinear flow. Imposing the inflow boundary condition in this fashion is fully general and does not require additional programming in solvers based on finite elements, spectral methods, and finite differences, whether or not Newton’s method is used for solving the nonlinear algebraic equations arising from the discretized partial differential equation.We test this method and find excellent agreement with analytical results in combined Poiseuille and Couette flow of Oldroyd-B and Giesekus liquids in 2-D and 3-D channels and annuli, for which analytical expressions of the velocity and conformation (elastic stress) fields are available. We demonstrate that the new method yields shorter or equal upstream lengths than traditional ones.