Impact of Addition of a Newtonian Solvent to a Giesekus Fluid: Analytical Determination of Flow Rate in Plane Laminar Motion

In this study, the influence of the presence of a Newtonian solvent on the flow of a Giesekus fluid in a plane channel or fracture is investigated with a focus on the determination of the flow rate for an assigned external pressure gradient. The pressure field is nonlinear due to the presence of the normal transverse stress component. As expected, the flow rate per unit width Q′ is larger than for a Newtonian fluid and decreases as the solvent increases. It is strongly dependent on the viscosity ratio ε (0 ≤ ε ≤ 1), the dimensionless mobility parameter β (0 ≤ β ≤ 1) and the Deborah number De, the dimensionless driving pressure gradient. The degree of dependency is notably strong in the low range of ε. Furthermore, Q′ increases with De and tends to a constant asymptotic value for large De, subject to the limitation of laminar flow. When the mobility factor β is in the range [0.5 ÷ 1], there is a minimum value of ε to obtain an assigned value of De. The ratio UN/U between Newtonian and actual mean velocity depends only on the rad(βDe), as for other non-Newtonian fluids.