Dynamics of gravity-driven viscoelastic films on wavy walls

Abstract

The linear stability and nonlinear dynamics of viscoelastic liquid films flowing down inclined surfaces with sinusoidal topography are investigated. The Oldroyd-B constitutive model is used, and numerical solutions of a long-wave nonlinear evolution equation for the film thickness (introduced by Davalos-Orozco[1]) provide insight into the influence of elasticity and wall topography on the nonlinear film dynamics, while Floquet analysis of the linearized evolution equation is used to study the onset of linear instability. Focusing initially on inertialess films (with zero Reynolds number), linear stability results are organized into three regimes based on the wall wavelength. For su ciently short and su ciently long wall wavelengths, the onset of instability is not tangibly a↵ected by the topography. There is, however, an intermediate range of wavelengths where, as the wall wavelength is increased, the critical Deborah number for the onset of instability first decreases (topography is destabilizing) and then increases su ciently for topography to be stabilizing (relative to the flat wall). Solutions to a perturbation amplitude equation indicate that the character of the instability changes substantially within this intermediate range; topography induces streamwise variations in the base-state velocity at the free surface which couple with perturbations and substantially influence the instability growth rate. Very similar trends are observed for Newtonian films and variations in the critical Reynolds number. Simulations of the full nonlinear evolution equation produce a broad range of nonlinear states including traveling waves, time-periodic waves, and chaos. Perturbations to the film generally saturate at higher amplitudes for cases with larger linear growth rates (e.g. with increasing Deborah number or for a destabilizing wall wavelength), and topography introduces finer temporal scales in the dynamics. The qualitative influences of inclination and inertia on the nonlinear dynamics are shown to be simply related to the influence of elasticity using analytical linear stability results for the flat-wall case.

Stability analysis of viscoelastic film flows over an inclined substrate with rectangular trenches

“…As the length increases, linear stability calculations show a much higher sensitivity to wall deformation. The latter is in qualitative agreement with the results of Sharma, Ray & Papageorgiou (2019), while for L = 10 a stable isle appears at supercritical conditions in the range 6.66 < Re < 7.40. Increasing further the size of the unit cell, the impact of the topography on the free surface is more prominent.…”

“…Although it is already known that fluid elasticity has an overall stabilizing effect on the film flow over structured surfaces (Pettas et al . 2019 b ; Sharma, Ray & Papageorgiou 2019), we demonstrate that the shape of the substrate topography may also affect the intensity of the elastic phenomena considerably and in turn have an impact on the stability of the viscoelastic film. Moreover, when the depth of the trenches exceeds a critical threshold, the overall stability of the flow remains unaffected.…”

“…Next, in figure 12( a – d ), we present the stability diagrams for four different sizes of the periodic unit cell, and 20, respectively, while keeping constant the scales of the other geometrical characteristics. For short wall wavelengths, see figure 12( a ), the film is relatively thick compared to the substrate variations resulting in an almost planar film shape (Sharma, Ray & Papageorgiou 2019). Hence, the neutral curve resembles that of the Nusselt flow, see the black dashed line in figure 12( a ).…”

“…The predicted values from (5.2) appear in figure 8(a) as points ( ) and are in perfect agreement with our calculations. As discussed by Sharma, Ray & Papageorgiou (2019) and Huang & Khomami (2001) for viscoelastic film flow over flat substrates, due to the effect of normal stresses in the disturbed flow, elasticity 'pushes' the fluid in the streamwise direction towards local maxima of the film thickness and away from local minima.…”

“…According to this study, the deformation of the free surface due to the presence of the substrate undulations may lead to stabilization by fluid elasticity. Very recently, Sharma, Ray & Papageorgiou (2019), also employing lubrication theory, showed that the impact of elasticity on the stability of the inertialess flow is profoundly affected by the geometrical characteristics of the substrate. They found that as the wavelength of the periodic wall increases, the topography initially exerts a destabilizing influence, while for shorter wall-wavelengths, the topography may stabilize the flow.…”