Nonlinear Stability Analysis of Thin Viscoelastic Film Flowing Down on the Inner Surface of a Rotating Vertical Cylinder

Abstract

This paper investigates the stability of thin viscoelastic liquid film flowing down on the inner surface of a rotating vertical cylinder by means of the long wave perturbation. After proving the insufficiency of the linear model in characterization of certain flow behaviors, a generalized nonlinear kinematic model is then derived to represent the physical system. This model is solved through the following procedure. In the first step, the normal mode method is used to characterize the linear behaviors. The amplitude growth rates and the threshold conditions are characterized subsequently and summarized as the by-products of the linear solutions. In the second step, a nonlinear film flow model is solved by using the method of multiple scales to characterize flow behaviors at various states of sub-critical stability, sub-critical instability, supercritical stability, and supercritical explosion. The modeling results indicate that with the increase in the rotation speed and the radius of cylinder R, the film flow system will be more stable.

Nonlinear hydromagnetic stability analysis of a pseudoplastic film flow

“…The solutions obtained by deriving the viscoelastic analog of the Orr-Sommerfeld equation with the requisite boundary conditions revealed that viscoelastic effects tend to destabilize the film flow. A similar finding was reported by Chen et al [4] in their nonlinear stability analysis of a thin viscoelastic liquid film flowing down the internal surface of a rotating vertical cylinder.…”

“…(25)-(30), the governing equations of the thin-film system can be collected and solved on an order-by-order basis. In practice, the non-dimensional surface tension, S n , has a large value, and thus the term α 2 S n can be treated as a zeroth-order quantity [3][4][5]. Furthermore, for y h, the film thickness h is very small, using the power series solution ϕ = ∞ n=0 k n y n about 0, and hence power series approximation solutions can be obtained up to the order of y 5 at the zeroth-and first-orders of the stream function (given in Appendix A).…”

“…The current study applies both linear and nonlinear stability analysis theories to investigate the stability of a thin electrically-conductive pseudoplastic fluid film flowing down the surface of a vertical wall in an applied magnetic field. In contrast to previous related studies [3][4][5], the problem is addressed using a numerical approximation method rather than an analytical solution procedure. The analysis focuses particularly on the respective effects of the Hartmann number and the flow index on the stability of the film flow.…”

“…The parameter ranges of the parameters considered in the analysis are as follows: (1) Reynolds number Re 1 = 0-10, (2) dimensionless perturbation wave number α = 0-0.12, (3) Hartmann number m = 0, 0.1 and 0.2, and (4) flow index n = 0.95, 0.98 and 1.00. Note that in performing the various analyses, a constant dimensionless surfacetension of S 1 = 6173.5 is assumed[3][4][5].…”

Linear Three Dimensional Instability of Viscoelastic Fluid Layers Flowing Down Cylindrical Walls

“…This last fluid model was also used by Shaqfeh et al (1989) including moderate Reynolds numbers and in the non linear problem by Joo (1994) with an extra viscous thinning effect and by Kang and Chen (1995). The nonlinear stability of a Walters B fluid model of a thin film down a flat wall was investigated by Cheng et al (2000) and this was extended for the case of a vertical cylinder only taking into account the axial modes Cheng et al (2001). However, the instability using an Oldroyd B fluid model has not been investigated.…”