<p>A semi-analytical solution is proposed for laminar pipe flow of a viscoelastic fluid, in which the polymer contribution is described by the Giesekus model and the solvent behaves as a Newtonian fluid. The velocity profiles, stress tensor components, and mean velocity are derived as functions of three key parameters: the Deborah number, the mobility factor, and the ratio of solvent viscosity to total viscosity. The analysis reveals that, when the mobility factor exceeds 0.5 and the solvent viscosity is negligible or very low, a maximum Deborah number exists beyond which no valid solution can be found. The results of our analysis suggest that considering the viscosity of the solvent in addition to that of the polymer leads to an effect analogous to an increase in flow resistance. Furthermore, a global sensitivity analysis, facilitated by an efficient model reduction technique based on Polynomial Chaos Expansion, is performed to assess the influence of the three parameters on velocity and stress characteristics.</p>