# Linear Stability of a Viscoelastic Liquid Film on an Oscillating Plane

## TL;DR

This paper studies how viscoelastic liquid films on oscillating plates become unstable, showing how parameters like elasticity and surface tension affect stability.

## Contribution

The study provides new analytical solutions and insights into the linear stability of viscoelastic films under oscillation.

## Key findings

- Long-wave instability occurs only within specific bandwidths related to the Ohnesorge number.
- Increasing the relaxation-to-delay time ratio or elasticity parameter reduces unstable bandwidths and enhances instability intensity.
- Higher surface-tension-related parameters suppress wave growth and stabilize the system.

## Abstract

This paper investigates the linear stability of the liquid film of Oldroyd-B fluid on an oscillating plate. The time-dependent Orr–Sommerfeld boundary-value problem is formulated through the assumption of a normal modal solution and the introduction of the stream function, which is further transformed into the Floquet system. A long-wavelength expansion analysis is performed to derive the analytical solution of the Orr–Sommerfeld equation. The results indicate that long-wave instability occurs only within specific bandwidths related to the Ohnesorge number (Oh). Fixing the elasticity parameter (El) and increasing the relaxation-to-delay time ratio (λ˜) from 2 to 4 or fixing (λ˜) and increasing (El) from 0.001 to 0.01 decreases the number of unstable bandwidths while enhancing the intensity of unstable modes. Increasing the surface-tension-related parameter (ζ∗) from 0 to 100 suppresses the wave growth rate, stabilizing the system. Additionally, increasing (λ˜) from 2 to 4 reduces the maximum values of the coupling of viscoelastic, gravitational, and surface-tension forces, as well as the maximum value of the Floquet exponent, further stabilizing the system. These findings provide supplements to the theoretical research on the stability of viscoelastic fluids and also offer a scientific basis for engineering applications in multiple fields.