Linear Stability Analysis of Viscoelastic Fluids in a Plane Channel Flow

In this paper, the linear hydrodynamic stability analysis of a viscoelastic fluid in a plane channel flow driven by a constant pressure gradient is investigated numerically. A nonlinear eigenvalue problem for the perturbed state is obtained from a generalized Orr-Sommerfeld equation. The Chebyshev spectral collocation method with expansions in Lagrange’s polynomials are used for the numerical calculation. The complex wave speed, the critical Reynolds and wavenumber numbers are calculated for different parameters. We mainly examine the combined effects of relaxation time (via the Deborah number) and delay time (via the elasticity number) on the onset of instability. Results show that the relaxation time has a destabilizing effect and the retardation time has a stabilizing effect, with a displacement towards the long-wave region. Moreover, an important feature occurs when the Reynolds number is equal to the ratio of both characteristic times, the Jeffrey’s fluid model behaves like the Newtonian model.