# Dynamics and stability of a power-law film flowing down a slippery slope

TL;DR: In this article, a low-dimensional two-equation model is formulated using a weighted-residual approach in terms of two coupled evolution equations for the film thickness h and a local velocity amplitude or the flow rate q within the framework of lubrication theory.

Abstract: A power-law fluid flowing down a slippery inclined plane under the action of gravity is deliberated in this research work. A Newtonian layer at a small strain rate is introduced to take care of the divergence of the viscosity at a zero strain rate. A low-dimensional two-equation model is formulated using a weighted-residual approach in terms of two coupled evolution equations for the film thickness h and a local velocity amplitude or the flow rate q within the framework of lubrication theory. Moreover, a long-wave instability is shown in detail. Linear stability analysis of the proposed two-equation model reveals good agreement with the spatial Orr-Sommerfeld analysis. The influence of a wall-slip on the primary instability has been found to be non-trivial. It has the stabilizing effect at larger values of the Reynolds number, whereas at the onset of the instability, the role is destabilizing which may be because of the increase in dynamic wave speed by the wall slip. Competing impressions of shear-thinning/shear-thickening and wall slip velocity on the primary instability are captured. The impact of slip velocity on the traveling-wave solutions is discussed using the bifurcation diagram. An increasing value of the slip shows a significant effect on the traveling wave and free surface amplitude. Slip velocity controls both the kinematic and dynamic waves of the system, and thus, it has the profound passive impact on the instability.A power-law fluid flowing down a slippery inclined plane under the action of gravity is deliberated in this research work. A Newtonian layer at a small strain rate is introduced to take care of the divergence of the viscosity at a zero strain rate. A low-dimensional two-equation model is formulated using a weighted-residual approach in terms of two coupled evolution equations for the film thickness h and a local velocity amplitude or the flow rate q within the framework of lubrication theory. Moreover, a long-wave instability is shown in detail. Linear stability analysis of the proposed two-equation model reveals good agreement with the spatial Orr-Sommerfeld analysis. The influence of a wall-slip on the primary instability has been found to be non-trivial. It has the stabilizing effect at larger values of the Reynolds number, whereas at the onset of the instability, the role is destabilizing which may be because of the increase in dynamic wave speed by the wall slip. Competing impressions of shear-thinni...

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TL;DR: Lauga et al. as mentioned in this paper studied the stability of channel flow with streamwise and spanwise slip separately as two limiting cases of anisotropic slip and explore a broader range of slip length than previous studies did.

Abstract: In this work, we revisit the temporal stability of slip channel flow. Lauga & Cossu (Phys. Fluids 17, 088106 (2005)) and Min & Kim (Phys. Fluids 17, 108106 (2005)) have investigated both modal stability and non-normality of slip channel flow and concluded that the velocity slip greatly suppresses linear instability and only modestly affects the non-normality. Here we study the stability of channel flow with streamwise and spanwise slip separately as two limiting cases of anisotropic slip and explore a broader range of slip length than previous studies did. We find that, with sufficiently large slip, both streamwise and spanwise slip trigger three-dimensional leading instabilities. Overall, the critical Reynolds number is only slightly increased by streamwise slip, whereas it can be greatly decreased by spanwise slip. Streamwise slip suppresses the non-modal transient growth, whereas spanwise slip enlarges the non-modal growth although it does not affect the base flow. Interestingly, as the spanwise slip length increases, the optimal perturbations exhibit flow structures different from the well-known streamwise rolls. However, in the presence of equal slip in both directions, the three-dimensional leading instabilities disappear and the flow is greatly stabilized. The results suggest that earlier instability and larger transient growth can be triggered by introducing anisotropy in the velocity slip.

21 citations

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TL;DR: In this article, the effects of wall velocity slip on the linear stability of a gravity-driven two-fluid flow down an incline are examined, and the results show that the presence of slip exhibits a promise for stabilizing the miscible flow system by raising the critical Reynolds number at the onset and decreasing the bandwidth of unstable wave numbers beyond the threshold of the dominant instability.

Abstract: The effects of wall velocity slip on the linear stability of a gravity-driven miscible two-fluid flow down an incline are examined. The fluids have the matched density but different viscosity. A smooth viscosity stratification is achieved due to the presence of a thin mixed layer between the fluids. The results show that the presence of slip exhibits a promise for stabilizing the miscible flow system by raising the critical Reynolds number at the onset and decreasing the bandwidth of unstable wave numbers beyond the threshold of the dominant instability. This is different from its role in the case of a single fluid down a slippery substrate where slip destabilizes the flow system at the onset. Though the stability properties are analogous to the same flow system down a rigid substrate, slip is shown to delay the surface mode instability for any viscosity contrast. It has a damping/promoting effect on the overlap modes (which exist due to the overlap of critical layer of dominant disturbance with the mixed layer) when the mixed layer is away/close from/to the slippery inclined wall. The trend of slip effect is influenced by the location of the mixed layer, the location of more viscous fluid and the mass diffusivity of the two fluids. The stabilizing characteristics of slip can be favourably used to suppress the non-linear breakdown which may happen due to the coexistence of the unstable modes in a flow over a substrate with no slip. The results of the present study suggest that it is desirable to design a slippery surface with appropriate slip sensitivity in order to meet a particular need for a specific application.

20 citations

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TL;DR: In this article, the stability of a thin viscous Newtonian fluid with broken time-reversal symmetry draining down a slippery inclined plane is examined, and a weakly nonlinear stability analysis is performed using the normal mode approach and a critical Reynolds number is obtained.

Abstract: The stability of a thin viscous Newtonian fluid with broken time-reversal-symmetry draining down a slippery inclined plane is examined. The presence of the odd part of the Cauchy stress tensor with an odd viscosity coefficient brings new characteristics in fluid flow as it gives rise to new terms in the pressure gradient of the flow. By odd viscosity, it is meant that apart from the well-known coefficient of shear viscosity, a classical liquid with broken time-reversal symmetry is endowed with a second viscosity coefficient. The model implements a Navier slip condition at the solid–liquid interface with the slip length being the parameter that measures the deviation from the no-slip condition. The classical long-wave expansion technique is performed and a nonlinear evolution equation of Benney-type is derived in terms of film thickness h(x, t), which is significantly modified due to the presence of odd viscosity in the liquid. The parameters governing the film flow system and the slippery substrate strongly influence the waveforms and their amplitudes and hence the stability of the fluid. The linear stability analysis is performed using the normal mode approach and a critical Reynolds number is obtained. The results of the linear stability analysis reveal that larger odd viscosity leads to the higher critical Reynolds number while the higher slip length makes the critical Reynolds number lower. In other words, odd viscosity has a stabilizing effect while the slip length promotes instability. Based on the method of multiple scales, a weakly nonlinear stability analysis is carried out, which shows that there is a range of wave numbers with a supercritical bifurcation and a range of larger wave numbers with a subcritical bifurcation. Different instability zones are also demarcated. The weakly nonlinear study shows that with an increase in the odd viscosity, the supercritical stable region and the explosion area shrink, whereas the unconditional stable and the subcritical unstable regions increase. It has also been shown that the spatial uniform solution corresponding to the sideband disturbance may be stable in the unstable region. The spatiotemporal evolution of the model has been analyzed numerically by employing the Crank–Nicolson method in a periodic domain for different values of the odd viscosity and slip length. The nonlinear simulations are found to be in good agreement with the linear and weakly nonlinear stability analysis.

13 citations

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TL;DR: In this article, the authors considered the stability problem for wide, uniform stationary open flows down a slope with constant inclination under gravity and showed that under certain conditions, oblique perturbations can grow even when the perturbation propagating along the flow are damped.

Abstract: We consider the stability problem for wide, uniform stationary open flows down a slope with constant inclination under gravity. Depth-averaged equations are used with arbitrary bottom friction as a function of the flow depth and depth-averaged velocity. The stability conditions for perturbations propagating along the flow are widely known. In this paper, we focus on the effect of oblique perturbations that propagate at an arbitrary angle to the velocity of the undisturbed flow. We show that under certain conditions, oblique perturbations can grow even when the perturbations propagating along the flow are damped. This means that if oblique perturbations exist, the stability conditions found in the investigation of the one-dimensional problem are insufficient for the stability of the flow. New stability criteria are formulated as explicit relations between the slope and the flow parameters. The ranges of the growing disturbances propagation angles are indicated for unstable flows.

11 citations

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TL;DR: In this article, the stability of weakly viscoelastic film (Walter's B″) flowing down under gravity along a slippery inclined plane was investigated, where the interaction of the bottom slip with the viscous parameter was investigated.

Abstract: We study the stability of weakly viscoelastic film (Walter's B″) flowing down under gravity along a slippery inclined plane. The focus is to investigate the interaction of the bottom slip with the viscoelastic parameter as well as the influence of the other flow parameters on the stability of the flow. For the slippery substrate, we use the Navier-slip boundary condition at the solid–liquid interface. The dimensionless slip length β is first assumed to be small and its order is considered same as the order of the film aspect ratio ϵ=H/L, where H is the mean film thickness and L is a typical wavelength. To discuss the coupled effect of slip length β and viscoelastic parameter γ, we have used the classical Benney equation model (BEM) as well as the weighted residual method (WRM). For linear stability, the normal mode analysis is carried out to capture the instability threshold. The critical Reynolds numbers (Rec) are obtained for BEM and WRM separately for the system. We found that the first-order WRM is a better choice to capture the instability threshold in comparison with a first-order BEM when β is small. Another noteworthy result we obtain is that, in the absence of β, WRM and BEM yield the same expression for the critical Reynolds number. Further, we have extended the study for moderate values of β, that is, β of order unity and it is found that both BEM and WRM are able to capture the effects of β and γ. We derive the Orr–Sommerfeld (OS) type eigenvalue problem and an asymptotic analysis is performed for small perturbation wavenumbers, which gives an expression for the critical Reynolds number for the instability of very long perturbations. The critical Reynolds number obtained by the OS eigenvalue problem exactly matches with that obtained by BEM. Finally, we validate our analytical predictions by performing a direct numerical simulation of the WRM and good agreement between the results of the linear stability analysis, weighted residual model, and the numerical simulations is found.

11 citations

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TL;DR: In this article, a simple theory based on replacing the effect of the boundary layer with a slip velocity proportional to the exterior velocity gradient is proposed and shown to be in reasonable agreement with experimental results.

Abstract: Experiments giving the mass efflux of a Poiseuille flow over a naturally permeable block are reported. The efflux is greatly enhanced over the value it would have if the block were impermeable, indicating the presence of a boundary layer in the block. The velocity presumably changes across this layer from its (statistically average) Darcy value to some slip value immediately outside the permeable block. A simple theory based on replacing the effect of the boundary layer with a slip velocity proportional to the exterior velocity gradient is proposed and shown to be in reasonable agreement with experimental results.

2,898 citations

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TL;DR: In this article, a unified mathematical theory is presented that takes advantage of the disparity of the length scales and is based on the asymptotic procedure of reduction of the full set of governing equations and boundary conditions to a simplified, highly nonlinear, evolution equation or to a set of equations.

Abstract: Macroscopic thin liquid films are entities that are important in biophysics, physics, and engineering, as well as in natural settings. They can be composed of common liquids such as water or oil, rheologically complex materials such as polymers solutions or melts, or complex mixtures of phases or components. When the films are subjected to the action of various mechanical, thermal, or structural factors, they display interesting dynamic phenomena such as wave propagation, wave steepening, and development of chaotic responses. Such films can display rupture phenomena creating holes, spreading of fronts, and the development of fingers. In this review a unified mathematical theory is presented that takes advantage of the disparity of the length scales and is based on the asymptotic procedure of reduction of the full set of governing equations and boundary conditions to a simplified, highly nonlinear, evolution equation or to a set of equations. As a result of this long-wave theory, a mathematical system is obtained that does not have the mathematical complexity of the original free-boundary problem but does preserve many of the important features of its physics. The basics of the long-wave theory are explained. If, in addition, the Reynolds number of the flow is not too large, the analogy with Reynolds's theory of lubrication can be drawn. A general nonlinear evolution equation or equations are then derived and various particular cases are considered. Each case contains a discussion of the linear stability properties of the base-state solutions and of the nonlinear spatiotemporal evolution of the interface (and other scalar variables, such as temperature or solute concentration). The cases reducing to a single highly nonlinear evolution equation are first examined. These include: (a) films with constant interfacial shear stress and constant surface tension, (b) films with constant surface tension and gravity only, (c) films with van der Waals (long-range molecular) forces and constant surface tension only, (d) films with thermocapillarity, surface tension, and body force only, (e) films with temperature-dependent physical properties, (f) evaporating/condensing films, (g) films on a thick substrate, (h) films on a horizontal cylinder, and (i) films on a rotating disc. The dynamics of the films with a spatial dependence of the base-state solution are then studied. These include the examples of nonuniform temperature or heat flux at liquid-solid boundaries. Problems which reduce to a set of nonlinear evolution equations are considered next. Those include (a) the dynamics of free liquid films, (b) bounded films with interfacial viscosity, and (c) dynamics of soluble and insoluble surfactants in bounded and free films. The spreading of drops on a solid surface and moving contact lines, including effects of heat and mass transport and van der Waals attractions, are then addressed. Several related topics such as falling films and sheets and Hele-Shaw flows are also briefly discussed. The results discussed give motivation for the development of careful experiments which can be used to test the theories and exhibit new phenomena.

2,689 citations

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TL;DR: The dynamics and stability of thin liquid films have fascinated scientists over many decades: the observations of regular wave patterns in film flows along a windowpane or along guttering, the patterning of dewetting droplets, and the fingering of viscous flows down a slope are all examples that are familiar in daily life.

Abstract: The dynamics and stability of thin liquid films have fascinated scientists over many decades: the observations of regular wave patterns in film flows down a windowpane or along guttering, the patterning of dewetting droplets, and the fingering of viscous flows down a slope are all examples that are familiar in daily life. Thin film flows occur over a wide range of length scales and are central to numerous areas of engineering, geophysics, and biophysics; these include nanofluidics and microfluidics, coating flows, intensive processing, lava flows, dynamics of continental ice sheets, tear-film rupture, and surfactant replacement therapy. These flows have attracted considerable attention in the literature, which have resulted in many significant developments in experimental, analytical, and numerical research in this area. These include advances in understanding dewetting, thermocapillary- and surfactant-driven films, falling films and films flowing over structured, compliant, and rapidly rotating substrates, and evaporating films as well as those manipulated via use of electric fields to produce nanoscale patterns. These developments are reviewed in this paper and open problems and exciting research avenues in this thriving area of fluid mechanics are also highlighted.

1,226 citations

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TL;DR: In this paper, the authors present results from molecular dynamics simulations of newtonian liquids under shear which indicate that there exists a general nonlinear relationship between the amount of slip and the local shear rate at a solid surface.

Abstract: Modelling fluid flows past a surface is a general problem in science and engineering, and requires some assumption about the nature of the fluid motion (the boundary condition) at the solid interface. One of the simplest boundary conditions is the no-slip condition1,2, which dictates that a liquid element adjacent to the surface assumes the velocity of the surface. Although this condition has been remarkably successful in reproducing the characteristics of many types of flow, there exist situations in which it leads to singular or unrealistic behaviour—for example, the spreading of a liquid on a solid substrate3,4,5,6,7,8, corner flow9,10 and the extrusion of polymer melts from a capillary tube11,12,13. Numerous boundary conditions that allow for finite slip at the solid interface have been used to rectify these difficulties4,5,11,13,14. But these phenomenological models fail to provide a universal picture of the momentum transport that occurs at liquid/solid interfaces. Here we present results from molecular dynamics simulations of newtonian liquids under shear which indicate that there exists a general nonlinear relationship between the amount of slip and the local shear rate at a solid surface. The boundary condition is controlled by the extent to which the liquid ‘feels’ corrugations in the surface energy of the solid (owing in the present case to the atomic close-packing). Our generalized boundary condition allows us to relate the degree of slip to the underlying static properties and dynamic interactions of the walls and the fluid.

1,144 citations

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TL;DR: In this article, it was shown that a class of undamped waves exists for all finite values of the Reynolds number R, and that the rates of amplification of unstable waves become very small when R is made fairly small, and their wavelengths to become very large; this provides a satisfactory explanation for the apparent absence of waves in some experimental observations.

Abstract: This paper deals theoretically with a problem of hydrodynamic stability characterized by small values of the Reynolds number R. The primary flow whose stability is examined consists of a uniform laminar stream of viscous liquid running down an inclined plane under the action of gravity, being bounded on one side by a free surface influenced by surface tension. The problem thus has a direct bearing on the properties of thin liquid films such as have important uses in chemical engineering.Numerous experiments in the past have shown that in flow down a wall the stream is noticeably agitated by waves except when R is quite small; on a vertical water film, for instance, waves may be observed until R is reduced to some value rather less than 10. The present treatment is accordingly based on methods of approximation suited to fairly low values of R, and thereby avoids the severe mathematical difficulties usual in stability problems at high R. The formulation of the problem resembles that given by Yih (1954); but the method of solution differs from his, and the respective results are in conflict. In particular, there is dis-agreement over the matter of the stability of a strictly vertical stream at very small R. In contrast with the previous conclusions, it is shown here that the flow is always unstable: that is, a class of undamped waves exists for all finite values of R. However, the rates of amplification of unstable waves are shown to become very small when R is made fairly small, and their wavelengths to become very large; this provides a satisfactory explanation for the apparent absence of waves in some experimental observations, and also for the wide scatter among existing estimates of the ‘quasi-critical’ value of R below which waves are undetectable. In view of the controversial nature of these results, emphasis is given to various points of agreement between the present work and the established theory of roll waves; the latter theory gives a clear picture of the physical mechanism of wave formation on gravitational flows, and in its light the results obtained here appear entirely reasonable.The conditions governing neutral stability are worked out to the third order in a parameter which is shown to be small; but a less accurate approximation is then justified as an adequate basis for an easily workable theory providing a ready check with experiment, This theory is used to predict the value of R at which observable waves should first develop on a vertical water film, and also the length and velocity of the waves. These three predictions are compared with the experimental results found by Binnie (1957), and are substantially confirmed.

904 citations