A long-wave evolution equation is derived using an asymptotic analysis, and the linear stability of a viscoelastic film flowing along the direction of parallel grooves over a uniformly heated topography is studied. A numerical approach adopting spectral collocation technique is used to demarcate the stable and unstable flow regimes. The combined influence of thermocapillarity and viscoelasticity on the films stability is analyzed. By accounting the bottom topography comprising longitudinal gutters, the possibilities of regulating the film dynamics under iso- and nonisothermal conditions and/or optimizing design structure of an apparatus for desirable flow outcomes have been focussed.

Falling films are relevant to a broad class of interfacial instability problems over a wide range of length and time scales in various technological setups since they offer small thermal resistance and large contact area at small specific flow rates. This serves as an advantage in many processes involving cooling, condensation, absorption, and evaporation, thereby regulating the transport of mass, momentum, and heat across the liquid-gas and liquid-solid interfaces. For a detailed review of falling film problems, refer to Nepomnyashchy et al. [

Although many studies available in literature have focused attention on Newtonian fluids, many fluids used in industrial applications are rheologically complex and non-Newtonian in nature. This has led to the generalization of Navier-Stokes model to satisfy highly nonlinear constitutive laws to arrive at complex partial differential equations, which are one order higher than the Navier-Stokes equations [

Unlike Newtonian fluids, which respond virtually instantaneously to an imposed deformation rate, viscoelastic fluids respond on a macroscopically large time scale, known as the relaxation time. The viscoelastic fluids, a subclass of microstructure flows, display both elastic (for deformation rates larger than the inverse relaxation time) and viscous (for deformation rates smaller than the magnitude of the inverse relaxation time) characteristics. The stress in this liquid is neither directly proportional to the strain nor to the rate of strain but displays a complex relationship [

There are various constitutive models which address the elasticoviscous aspect of a viscoelastic fluid. The Walters-B liquid model [

Chang [

Tackling the complete set of mass, momentum, and energy equations poses a formidable challenge. Therefore, with lubrication approximation at the forefront, dimension reduction techniques that take advantage of thin film geometry have been successfully developed in the majority of theoretical work that has appeared to date. This assumption neglects relatively unimportant terms while scaling complex equations by considering the depth-length ratio to be small and is an essential analytical tool in modeling thin film low-Reynolds number flows (a subregime of

Fluid actuation and transport are core functionalities and challenges in many fluidic systems. The shape and property of the substrate (its curvature and/or flexibility, patterns and/or obstacles) play an important role in the development of flow instabilities and affect the film dynamics and heat transfer. In particular, the interplay between the physical surface and the local fluid flow affects the entire system, where the viscosity, surface tension, thermocapillarity, and the local interface, for instance decides the flow mechanism. A better understanding of these properties and how they can be harnessed and controlled is extremely important. Their impact in relation to optimizing resources and manufacturing cost in the industrial sector have yet to be fully realized, with their use in relation to forming novel functional coatings for applications in many engineering disciplines still at an early stage.

There are several investigations on the flow of a thin film

Literature on non-Newtonian fluids, their stability, and dynamics on substrates with topography is not abundant. For a power-law based non-Newtonian model, the effect of substrate topography, inertia, and non-Newtonian rheology was examined for a two-dimensional thin Carreau-Bird model based liquid which has direct relevance in problems connected to the early stages of coating process [

Shaqfeh et al. [

Figure

Schematic representation of the flow of a viscoelastic film over a topography comprising furrows and ridges.

Constitutive equations which result from a retarded-motion expansion represent the correct constitutive equations for flows in which the rate-of-strain tensor and its time derivatives are small [

Dandapat and Samanta [

The components of the symmetric Cauchy stress tensor,

Since the model proposed by Dandapat and Samanta [

Such an idealized model represents an approximation to first order in elasticity [

The equations governing the flow are those for an incompressible fluid such that the compressibility effects and viscous heat generation are neglected in the heat equation:

In (

In order to nondimensionalize systems (

After nondimensionalizing, the asterisks are dropped from the scaled variables (cf. Appendix

The physical variables

It is assumed that the primary flow,

For

It should be remarked that contrary to the observations in Gambaryan-Roisman and Stephan [

For the flow on a planar wall, the terms contributing to the real and imaginary parts of the complex eigenvalue

When

Since the coefficients of (

In order to solve the linear stability problem (

Instead of choosing a particular fluid and analyzing its stability (which has only a limited scope), the parameters are changed by varying the components of

In the numerical calculations performed, the maximal values of

Free surface profiles corresponding to the primary flow on an uneven substrate with

Stability characteristics of Newtonian and non-Newtonian isothermal layers are examined first. It is known that the flow of an isothermal liquid film with a smooth free surface down a vertical plane is unstable at all Reynolds numbers [

It is ascertained from the numerical behavior of the linear amplification curves that the inertial force increases the instability threshold for Newtonian films falling along parallel longitudinal grooves (Figure

Variation of growth rate with wavenumber for a vertically falling Newtonian isothermal film: (a)

When the non-Newtonian effects are included, the infinitesimal disturbance to the primary flow leads the viscoelastic film to destabilize in relation to the Newtonian film. The sketch in Figure

Dispersion of linear waves for isothermal Newtonian and viscoelastic films, temporal growth rate at

Variation of growth rate with wavenumber for a vertically falling viscoelastic isothermal film with

To substantiate the information availed from Figures

Stability chart in

Neutral stability curves of an isothermal film at

In Figure

Phase velocity curves:

For the case of a topographical substrate subjected to uniform heating, D’Alessio et al. [

The growth rate of the amplitudes at

Influence of Marangoni stress on the neutral stability curves at

Growth rate as a function of wavenumber at

Figure

Linear amplification rate:

Curves corresponding to the growth rate on a vertical topography with

The present investigation, although valid only at the region where the viscous forces dominate inertia due to long-wave modeling, has revealed interesting results associated with the linear stability. The results showed the dramatic impact of the bottom topography on the films stability in conjunction with the force of surface-tension and Marangoni stresses. The analytical expansion of the streamwise velocity at the free surface averaged over a typical bottom undulation does reveal only its increasing tendency contributing towards flow destabilization, but the numerical investigation on the films stability exposes the additional effects involved in the problem. A close-up theoretical investigation can be performed by referring to Figure

The linear stability of a thin Walters-B fluid due to the linear variation of the surface-tension along a liquid interface induced by temperature gradients along the liquid layer was examined on a topography comprising sinusoidal longitudinal grooves. The nontrivial eigenvalue problem was numerically solved using a Chebyshev spectral collocation technique and the results were compared with those occurring from a central difference scheme of second order accuracy to ensure numerical correctness.

Numerical investigation of the eigenvalue problem corresponding to a falling film on a vertically held substrate revealed interesting results. In principle, when the viscous effects dominate the inertial forces, the flow tends to stabilize the system on a longitudinal groove topography, provided the surface-tension effects are weak. However, when the magnitude of the surface-tension force increases, interesting trends were exhibited by the stability curves. Some of the interesting key observations in the study can be enlisted as follows: (i) increased surface-tension effects contribute towards flow destabilization at low-Reynolds number as the amplitude of the bottom undulations increase (ii) Marangoni stresses can lead the system towards flow stabilization for certain increasing range over

Calculations were performed through numerical approach to analyze the effect of perturbation on the undisturbed state on a longitudinal grooved topography. Theoretically, it was also found that the average flow rate increases when the surface-tension force increases at small groove amplitudes. Although the above facts are findings based on a numerical linear stability study, such a behavior is yet to be confirmed through laboratory investigations. However, the latest research article by D’Alessio et al. [

It is not only with a fundamental research point of interest but also due to benefits gained through peer research that the thin film community intensively and actively strives to shed light on understanding intricate and complex mechanisms involved in falling film problems and fill up unknown gaps. It should be remarked that Mazouchi and Homsy [

Walters-B fluid finds its usage in medicinal diagnosis and clinical applications. For instance, Nadeem et al. [

Mathematical models obtained by combining a systematic gradient expansion with weighted residual techniques using polynomials as test functions may possibly shed more light on the stability thresholds beyond the low-Reynolds number regime [

The expressions for the components of the stress tensor

The nondimensional equations accurate up to

Equations and BC’s at

Equations and BC’s at