This work presents a theoretical and numerical investigation of viscoelastic fluid flow between parallel plates subjected to step strain rate deformation. The study examines the transient stress response following a sudden, nonlinear change in deformation rate that is subsequently maintained at constant value. The governing equations form a system of integro-differential equations incorporating the Caputo derivative to model the fluid’s viscoelastic behaviour. The system is solved sequentially, with the first equation determining the velocity field and the second computing the stress field. Numerical solution of these equations presents significant challenges due to the singular kernel in the integral terms and the rapid timescales of the imposed deformation, occurring within milliseconds and approximated by a logistic-type function. A finite difference scheme is developed to address these computational difficulties, and numerical results show the method’s robustness and stability when modelling complex fluids under demanding initial conditions. The proposed approach provides an effective framework for simulating transient viscoelastic flows in confined geometries subject to abrupt deformations.
Carvalho et al. (Wed,) studied this question.