Finite-wavelength Modes Research Articles

Instability of a viscoelastic film with insoluble surfactants on an oscillating plane

The linear instability of viscoelastic film with insoluble surfactants on an oscillating plane for disturbances with arbitrary wavenumbers is investigated. The combined effects of viscoelastic and insoluble surfactants on the instability are described using Floquet theory. For long-wavelength instability, the solution in the limit of long wave perturbations is obtained by the asymptotic expansion method. The results show that the presence of viscoelastic film shifts the stability boundaries to the low-frequency region in the absence of gravity when the imposed frequency is less than 6. The U-shaped neutral curves with separation bandwidth appear in the presence of gravity. The finite-wavelength instability is solved numerically based on the Chebyshev spectral collocation method. Different from the previous results, a new branch point with special structure of a neutral curve is detected for clean-surface film. Results show that the presence of the surfactants will decrease the unstable frequency bandwidth and increase the critical Reynolds number. Both the travelling-wave mode and standing-wave mode are found due to the existence of surface surfactants. For high-frequency oscillation, the viscoelastic parameter may significantly destabilize the flow and the instability is determined by the finite-wavelength mode over a relatively large frequency range.

Stability of the viscous flow of a polymeric fluid past a flexible surface

The instability in plane Couette flow of a viscoelastic fluid past a deformable surface is examined using the temporal linear stability theory in the zero Reynolds number limit. The polymeric fluid is described using the Oldroyd-B model and the flexible wall is modeled as a linear viscoelastic solid surface. The analysis shows that the wall flexibility tends to reduce the decay rate of the stable discrete modes for the polymeric flow past a rigid wall, and one of the discrete modes becomes unstable when the wall deformability parameter Γ=Vη∕(GR) exceeds a certain critical value Γc. Here, V is the top-plate velocity, η is the zero shear viscosity of the polymeric fluid, G is the shear modulus of the wall, and R is the width of the fluid layer. The analysis reveals the presence of two classes of modes, the first of which becomes unstable for perturbations with wavelength comparable to the channel width (finite wavelength modes), and the second becomes unstable for perturbations with wavelength small compared to the channel width (short wave modes). The latter class of modes are found to be absent for the highly concentrated polymer solutions with β≲0.23, where β is the ratio of solvent-to-solution viscosity. We have mapped out the regions in the parameter space (W¯-H) where the finite wavelength and short wave modes are unstable, where W¯=(λG∕η), and λ is the relaxation time of the viscoelastic fluid. Fluid elasticity is found to have a stabilizing influence on the unstable mode, such that when the shortwave instability is absent for β≲0.23, the flow becomes stable for any Weissenberg number W¯>W¯max. Here, W¯max increases proportional to H for H≫1. However, when the shortwave instability is present, the instability persists for W¯≫1. The behavior of both classes of modes with respect to the parameters, like W¯, H, β, and the ratio of solid-to-fluid viscosity ηr, is examined.

Stability and spectra of the bumpy ϑ pinch

Finite wavelength modes in the bumpy ϑ pinch are examined by performing perturbation expansions in the small bumpiness on the equations of ideal magnetohydrodynamics. The unstable modes are transverse and compressible. The evolution of two unstable classes of modes from a single unstable class is followed numerically as the plasma β is varied.