We show how to formulate two-point boundary value problems to compute laminar channel, tube, and Taylor-Couette flow profiles for some complex viscoelastic fluid models of differential type. The models examined herein are the Pom-Pom Model [McLeish and Larson 42:81-110, (1998)] the Pompon Model [Öttinger 40:317-321, (2001)] and the Two Coupled Maxwell Modes Model (Beris and Edwards 1994). For the two-mode Upper-Convected Maxwell Model, we calculate analytical solutions for the three flow geometries and use the solutions to validate the numerical methodology. We illustrate how to calculate the velocity, pressure, conformation tensor, backbone orientation tensor, backbone stretch, and extra stress profiles for various models. For the Pom-Pom Model, we find that the two-point boundary value problem is numerically unstable, which is due to the aphysical non-monotonic shear stress vs shear rate prediction of the model. For the other two models, we compute laminar flow profiles over a wide range of pressure drops and inner cylinder velocities. The volumetric flow rate and the nonlinear viscoelastic material properties on the boundaries of the flow geometries are determined as functions of the applied pressure drop, allowing easy analysis of experimentally measurable quantitie
