Nonlinear elastic instability of gravity-driven flow of a thin viscoelastic film down an inclined plane

doi:10.1016/0377-0257(94)01333-D

Journal of Non-Newtonian Fluid Mechanics

Volume 57, Issues 2–3, May 1995, Pages 243-252

https://doi.org/10.1016/0377-0257(94)01333-D Get rights and content

Abstract

A nonlinear evolution equation for the thickness of a thin viscoelastic film flowing down an inclined plane is derived for an Oldroyd-B fluid, using a long wave approximation. The evolution equation is valid to the second order in a small parameter which measures the relative thickness of the film to a typical wavelength. For a very thin film, viscoelasticity dominates the stability of the film and it can cause a purely elastic instability. The weakly nonlinear development of a monochromatic wave resulting from this elastic instability is studied using the second-order evolution equation which allows us to investigate the effects of inclination angle on the bifurcation. It is found that although extremely long waves bifurcate subcritically, the linearly most amplified wave does bifurcate supercritically when the surface tension parameter J > 13.8035. It is demonstrated that for a fixed bifurcation parameter δ = Wi − Wi_c, increasing the inclination angle can reduce the equilibrium amplitude for a supercritical bifurcating wave.

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Cited by (30)

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This paper investigates the linear stability of viscoelastic liquid films flowing down an inclined porous substrate analytically and numerically. It focuses on the Stokes flow of viscoelastic films and uncovers two unstable modes triggered by elasticity. The elastic surface mode with a long wave number is solved analytically and numerically. Our results also indicate elasticity can trigger an elasto-porous mode at small incline angle and ratio of film thickness to substrate thickness. The Oldroyd-B model is used for the constitutive relation between the strain and polymer stress. The classical Beavers–Joseph condition is applied to describe the boundary conditions at the fluid-porous interface (Beavers and Joseph, 1967). This condition represents the linear relationship between velocity gradient of fluid layer and velocity difference between two layers, $\partial_{z} u = \frac{α}{\sqrt{κ}} (u - u_{m})$ , where $α$ is the Beavers–Joseph coefficient, representing slip flow at the interface; $κ$ is the permeability of the porous medium. Effects of porous medium properties, including permeability and depth ratio, as well as the impact of slip flow at the interface on the unstable modes are examined.
Wave dynamics of a viscoelastic liquid
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A falling film of a weakly viscoelastic liquid (Walters’ liquid $B^{''}$ ) is studied in detail for low to moderate values of the Reynolds number. A linear stability analysis within the framework of the Orr–Sommerfeld-type boundary value problem reveals a destabilizing effect of the viscoelastic parameter near the threshold of instability but a stabilizing effect of the viscoelastic parameter on the primary instability far away from the threshold of instability. The surface equation is derived to decipher the existence of nonlinear waves based on the centre manifold approach. However, it renders an unbounded solution in the short-wave regime. In order to avoid singular behaviour found in the surface equation, a regularized equation, a simplified second-order model, and a full second-order model are proposed. In the linear regime, both the regularized equation and the model equations show excellent agreement with the results of the Orr–Sommerfeld-type boundary value problem, i.e., the viscoelastic parameter demonstrates a dual role in the primary instability. In the nonlinear regime, the results of the regularized equation are verified with those of the simplified second-order model. Both the regularized equation and the simplified second-order model show qualitatively similar characteristics of the nonlinear waves. The speed and maximum amplitude corresponding to the steady-state travelling waves attain fixed values in the drag-inertia regime. Moreover, the speed and maximum amplitude acquired from the regularized equation underestimate the results of the simplified second-order model when the Reynolds number lies in the drag-inertia regime. The time-dependent simulation of the regularized equation displays a train of solitary waves downstream to a monochromatic time periodic forcing at the inlet. The separation distance between the first two solitary pulses is computed, and it seems to approach an asymptotic value when the time is very large. This fact assures the possibility of a bound-state formation among the solitary pulses downstream for the weakly viscoelastic liquid.
Effect of imposed shear on the stability of a viscoelastic liquid flowing down a slippery plane
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We investigate a linear stability analysis for the Oldroyd-B liquid draining down a slippery inclined plane where a constant shear stress is applied at the liquid surface in the streamwise direction. The aim is to expand the previous study (Shaqfeh et al., 1989) in the presence of imposed shear stress when the bounding plane is slippery. The stability analysis is executed analytically as well as numerically for low to high values of the Reynolds number, which detects the existence of surface and shear modes. In the low Reynolds number regime, we use the analytical long-wave asymptotic expansion to find the onset of primary instability for the surface mode. On the other hand, we implement the numerical Chebyshev spectral collocation technique to determine the short-wave instability when the Reynolds number changes from a moderate to a high value. It is found that the critical Reynolds number for the surface mode decreases with the increasing values of the slip length and imposed shear stress, and thereby, the surface instability occurs at a lower Reynolds number than that when the slip length and the imposed shear stress are excluded. More specifically, the imposed shear stress has a destabilizing effect on the surface mode when acting in the co-flow direction but a stabilizing effect when acting in the counter-flow direction. Furthermore, the viscosity ratio exhibits a stabilizing effect on the surface mode near the onset of instability but exhibits a destabilizing effect far away from the onset of instability. In the high Reynolds number regime, the primary instability induced by the shear mode can be stabilized by the viscosity ratio and slip length, but can be destabilized by applying a constant shear stress in the co-flow direction. In a special case, the result for the surface mode corresponding to Walters’ liquid $B^{''}$ can be recovered from that of the Oldroyd-B liquid if the viscoelastic parameter is kept at a very low value.
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Weakly-nonlinear interfacial instabilities in two-fluid planar Couette flow are investigated for the case where one layer is thin. Taking this thin-layer thickness as a small parameter, asymptotic analysis is used to derive a nonlinear evolution equation for the interface height valid for wavelengths that scale with the channel height. Consequently, the influence of the thick layer is felt through a non-local coupling term which is obtained by solving a system of linear equations which are a simplified viscoelastic analogue to the Orr–Sommerfeld equation. The evolution equation allows for the clear identification of the influence of normal stresses at the interface on both the initial instability and the subsequent nonlinear dynamics. Results from numerical simulations illustrate: (1) an array of non-stationary states including traveling waves and chaos, (2) competition between elastic instability and instability due to viscosity stratification, and (3) the accuracy of a simplified ’localized’ evolution equation (derived using a long-wave approximation to the coupling term) when either the elasticity of the thick-layer fluid is sufficiently weak or the elasticities of the two fluids are sufficiently well-matched.
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Citation Excerpt :
Nonlinear computations of the instability of a thin viscoelastic film falling down an inclined wall are done by Joo [18]. Kang and Chen [19] find in the linear limit a purely elastic instability. This flow is investigated by Dávalos-Orozco [20] when the wall is smoothly deformed.
In this paper the non axisymmetric longwave instability of a thin viscoelastic liquid film flowing down a vertical heated cylinder is investigated. The stability of the film coating a cylinder in the absence of gravity is also investigated. In a previous paper it is found that viscoelasticity stimulates the appearance of azimuthal modes but the axial mode is the most unstable one. Other calculations in a former paper show that for flow outside a heated cylinder azimuthal modes can be the more unstable when the Marangoni number is large and, in particular, when the Reynolds number and wavenumber are small. Therefore, the small wavenumber and large cylinder radius approximation is assumed with the simultaneous action of viscoelasticity and thermocapillarity on the stability of azimuthal modes. In the presence and in the absence of gravity, it is found that, in comparison with the Newtonian case, it is easier to excite the azimuthal modes when viscoelasticity and thermocapillarity destabilize at the same time. Moreover, it is shown that, despite the axial mode is the most unstable one, there are wide wavenumber ranges where higher modes are the more unstable and they can show up by means of a periodic time dependent perturbation.
Computational analysis of gravity driven flow of a variable viscosity viscoelastic fluid down an inclined plane
2013, Computers and Fluids
Citation Excerpt :
Typical applications can be found say in the petroleum industry during fractional distillation processes, industrial coating applications and fabrication of a variety of plastic and polymeric substances in chemical engineering [1,2]. A number of investigations have been conducted for the gravity driven flow of both Newtonian and non-Newtonian fluids, [1–8]. A common thread in practically all the literature on such viscoelastic flows is that they are all conducted under isothermal conditions.
We investigate the unsteady non-isothermal gravity driven flow of a variable viscosity viscoelastic liquid along an inclined plane with convective cooling at the free surface. It is assumed that the convective heat exchange with the ambient at the free surface follows Newton’s law of cooling and that the viscoelastic fluid behaves according to the generalized Oldroyd-B model with the necessary adjustments for thermodynamics. The unsteady and coupled nonlinear partial differential equations governing the problem are derived and solved numerically using a semi-implicit finite difference scheme. Graphical results are presented and discussed quantitatively and qualitatively with respect to various parameters embedded in the problem. In particular, the effects of shear-rate dependent viscosity and viscoelasticity are explored in depth as they represent the main findings of our investigation and indeed represent the novel contributions of our investigation under the non-isothermal framework.

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Nonlinear elastic instability of gravity-driven flow of a thin viscoelastic film down an inclined plane

Abstract

J. Non-Newtonian Fluid Mech.

J. Non-Newtonian Fluid Mech.

Modeling wavy film flows

Instabilities of three-dimensional viscous falling films

J. Fluid Mech.

Measurement of the primary instabilities of film flows

J. Fluid Mech.

Wave evolution on a falling film

Annu. Rev. Fluid Mech.

Stability of a visco-elastic liquid film flowing down an inclined plane

J. Fluid Mech.

Stability of an elastico-viscous liquid film flowing down an inclined plane