Analytical solutions for oscillatory flows of two-layer immiscible viscoelastic fluids

This study theoretically investigates two fundamental oscillatory flows involving two superposed layers of immiscible viscoelastic fluids: Stokes' second problem and oscillatory Couette flow. Both layers are modeled using the linear Oldroyd-B constitutive equation, which captures the essential interplay between viscous and elastic responses in time-dependent flows. Exact analytical solutions for complete velocity and shear stress fields, comprising both transient and steady-periodic components, are derived using the Laplace transform method without invoking simplifying mathematical approximations to the constitutive equations. The analytical framework is developed by systematically enforcing kinematic and dynamic continuity at the fluid–fluid interface. The derived solutions analytically reduce to known limiting cases for single-layer viscoelastic and two-layer Newtonian flows, providing verification of their mathematical consistency and generality. A key physical insight is the explicit formulation of a complex reflection coefficient R and a dynamic impedance ratio Γ that formally quantify the impedance mismatch at the interface, governing shear-wave transmission efficacy and explaining dynamic phenomena, such as stress amplitude non-monotonicity. The solutions elucidate fundamental differences between Newtonian and viscoelastic responses, including frequency-dependent penetration depths, tunable phase lags, and elastic wave interference characteristics. While acknowledging the physical limitations inherent to the constant-viscosity assumption, this work contributes to the analytical study of multi-layer viscoelastic flows by providing exact theoretical baselines that can serve as reliable benchmarks for numerical simulations and validation cases for future interfacial stability analyses in microfluidics, enhanced oil recovery, and biological fluid dynamics.