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.

Stability of viscosity stratified flows down an incline: Role of miscibility and wall slip

Hydroelastic response of a floating plate on the falling film: A stability analysis

The influence of externally imposed shear on a surfactant-laden gravity-driven fluid flow over an inclined porous substrate is studied using the linear perturbation theory. The hydrodynamic instability of the flow system corresponding to infinitesimal disturbances is examined in the framework of the Orr–Sommerfeld (OS) boundary value problem. Furthermore, the generalized OS model is obtained by including the Marangoni stress and external shear on the flow dynamics. The formulated stability problem is solved as an eigenvalue problem by the Chebyshev spectral collocation technique. The analysis encounters the existence of different classes of unstable modes, namely, the surface, surfactant, and shear modes. The surface mode instability occurs in the low range of Reynolds number and is the dominant mode of instability in particular parameter ranges. The imposed shear at the top surface along and opposite to the flow direction induces possible destabilization and stabilization of the flow, respectively. The permeability and porosity of the porous medium have a mixed impact on the surface mode instability. The temporal growth rate of the surface mode enhances for a thicker porous medium. The surface mode of the flow contaminated by an insoluble surfactant is less unstable than that of the clean free surface flow. This is due to the co-existence of the damped surfactant mode together with the unstable surface mode. On the other hand, the shear mode instability is identified at higher Reynolds numbers for a very small inclination angle, and the shear mode propagates faster for stronger imposed shear in the downstream direction. This trend is reversed for the upstream imposed shear. Moreover, the Marangoni effects exhibit the stabilizing influence on the shear mode. Conclusively, the external shear force would be helpful in regulating the instability of the surfactant-laden film flow down a porous medium.

Linear instability of a surfactant-laden shear imposed falling film over an inclined porous bed

The stability of a Newtonian thin film flow over a porous slippery wall approximated by Darcy's law is investigated. The modified Orr-Sommerfeld system is derived for the frequency-dependent linear stability analysis and energy-budget analysis. Moreover, in the longwave regime, both linear and weakly nonlinear stability analysis are conducted for small aspect ratios. In addition, the multiple scale approach is performed directly in the nonlinear deformation equation of the free surface to predict the extraordinary behaviour of the amplitude and speed of the nonlinear disturbance in the subcritical and supercritical regimes. The study finds that the larger slip-velocity and externally imposed shear on the thin film increase the total kinetic energy of the infinitesimal perturbations. In a longwaveregime, the critical conditions of the primary instability are described as a function of imposed shear stress that destabilizes the film flow for low critical Reynolds number. Further, in the supercritical stable zone, both the nonlinear wave amplitude and phase speed increase with an increase of induced shear in the flow direction and velocity slip, and a reverse trend is observed in applying the imposed shear in the opposite flow direction. On the other hand, the nonlinear wave amplitude in the subcritical unstable zone increases and decreases, corresponding to the larger values of imposed shear and slip parameters, respectively.

https://pubs.aip.org/aip/pof/article-pdf/doi/10.1063/5.0120882/16592770/114124_1_online.pdf

Shear-imposed falling thin Newtonian film over a porous slippery surface

The linear stability characteristics of pressure-driven miscible two-fluid flow with same density and varying viscosities in a channel with velocity slip at the wall are examined. A prominent feature of the instability is that only a band of wave numbers is unstable whatever the Reynolds number is, whereas shorter wavelengths and smaller wave numbers are observed to be stable. The stability characteristics are different from both the limiting cases of interface dominated flows and continuously stratified flows in a channel with velocity slip at the wall. The flow system is destabilizing when a more viscous fluid occupies the region closer to the wall with slip. For this configuration a new mode of instability, namely the overlap mode, appears for high mass diffusivity of the two fluids. This mode arises due to the overlap of critical layer of dominant instability with the mixed layer of varying viscosity. The critical layer contains a location in the flow domain at which the base flow velocity equals the phase speed of the most unstable disturbance. Such a mode also occurs in the corresponding flow in a rigid channel, but absent in either of the above limiting cases of flow in a channel with slip. The flow is unstable at low Reynolds numbers for a wide range of wave numbers for low mass diffusivity, mimicking the interfacial instability of the immiscible flows. A configuration with less viscous fluid adjacent to the wall is more stable at moderate miscibility and this is also in contrast with the result for the limiting case of interface dominated flows in a channel with slip, where the above configuration is more unstable. It is possible to achieve stabilization or destabilization of miscible two-fluid flow in a channel with wall slip by appropriately choosing the viscosity of the fluid layer adjacent to the wall. In addition, the velocity slip at the wall has a dual role in the stability of flow system and the trend is influenced by the location of the mixed layer, the location of more viscous fluid and the mass diffusivity of the two fluids. It is well known that creating a viscosity contrast in a particular way in a rigid channel delays the occurrence of turbulence in a rigid channel. The results of the present study show that the flow system can be either stabilized or destabilized by designing the walls of the channel as hydrophobic surfaces, modeled by velocity slip at the walls. The study provides another effective strategy to control the flow system.

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Linear stability analysis of miscible two-fluid flow in a channel with velocity slip at the walls

The linear instability of a surfactant-laden two-layer falling film over an inclined slippery wall is analyzed under the influence of external shear, which is imposed on the top surface of the flow. The free surface of the flow and the interface among the fluids are contaminated by insoluble surfactants. Dynamics of the fluid layers are governed by the Navier–Stokes equations, and the surfactant transport equations regulate the motion of the insoluble surfactants at the interface and free surface. Instability mechanisms are compared by imposing the external shear along and opposite to the flow direction. A coupled Orr–Sommerfeld system of equations is derived using the perturbation technique and normal mode analysis. The eigenmodes corresponding to the Orr–Sommerfeld eigenvalue problem are obtained by employing the spectral collocation method. The numerical results imply that the stronger external shear destabilizes the interface mode instability. However, a stabilizing impact of the external shear on the surface mode is noticed if the shear is imposed in the flow direction, which is in contrast to the role of imposed external shear on the surface mode for a surfactant-laden single layer falling film. Furthermore, in the presence of strong imposed shear, the overall stabilization of the surface mode by wall velocity slip for the stratified two-fluid flow is also contrary to that of the single fluid case. The interface mode behaves differently in the two zones at the moderate Reynolds numbers, and higher external shear magnifies the interfacial instability in both zones. An opposite trend is observed in the case of surface instability. Moreover, the impression of shear mode on the primary instability is analyzed in the high Reynolds number regime with sufficiently low inclination angle. Under such configuration, dominance of the shear mode over the surface mode is observed due to the weaker impact of the gravitational force on the surface instability. The shear mode can also be stabilized by applying the external shear in the counter direction of the streamwise flow. Conclusively, the extra imposed shear on the stratified two-layer falling film plays an active role in the control of the attitude of the instabilities.

Effect of imposed shear on the dynamics of a contaminated two-layer film flow down a slippery incline

The linear instability of a surfactant-laden two-layer falling film over an inclined slippery wall is analyzed under an influence of external shear which is imposed on the top surface of the flow. The free surface of the flow as well as the interface among the fluids are contaminated by insoluble surfactants. Dynamics of both the layers are governed by the Navier--Stokes equations, and the surfactant transport equation regulates the motion of the insoluble surfactants at the interface and free surface. Instability mechanisms are compared by imposing the external shear along and opposite to the flow direction. A coupled Orr--Sommerfeld system of equations for the considered problem is derived using the perturbation technique and normal mode analysis. The eigenmodes corresponding to the Orr--Sommerfeld eigenvalue problem are obtained by employing the spectral collocation method. The numerical results imply that the stronger external shear destabilizes the interface mode instability. However, a stabilizing impact of the external shear on the surface mode is noticed if the shear is imposed in the flow direction, which is in contrast to the role of imposed external shear on the surface mode for a surfactant laden single layer falling film. Moreover, the impression of shear mode on the primary instability is analyzed in the high Reynolds number regime with sufficiently low inclination angle. Under such configuration, dominance of the shear mode over the surface mode is observed due to the weaker impact of the gravitation force on the surface instability. The shear mode can also be stabilized by applying the external shear in the counter direction of the streamwise flow. Conclusively, the extra imposed shear on the stratified two-layer falling film plays an active role to control the attitude of the instabilities.

Stability characteristics of a pressure-driven Poiseuille type flow are explored in a horizontal channel with porous walls. The flow is modified with a thermal gradient, and the temperature slips alongside a uniform cross-flow for three different configurations: (i) the flow having heat influx/outflux and temperature jump across the channel walls with internal heat generation in the fluid, (ii) the flow with constant wall temperatures (not necessarily equal) and no internal heat source, and (iii) the flow having a temperature jump/slip at the upper wall and lower wall with a constant temperature. The Reynolds, Peclet, and Rayleigh numbers govern the behavior of the thermal and velocity profiles along with the physical aspects of the flow. The modified Orr–Sommerfeld and energy equations are derived for the perturbed system by the normal mode analysis and solved using the Chebyshev collocation method, while the energy budget analysis is used to further illustrate the stability characteristics of the system. The resulting eigenvalues and eigenfunctions are used to analyze the growth rate characteristics, neutral stability ranges, velocity isolines, and temperature isotherms for each flow configuration. Further, the energy from various sources is calculated by making use of eigenfunctions corresponding to the unstable eigenmodes. An increase in the Rayleigh and Reynolds numbers promotes the flow instability; whereas, a higher cross-flow Reynolds number suppresses the instability by raising the upward cross-flow. The porous walls, characterized by the velocity slip and wall shear, inhibit the Poiseuille–Rayleigh–Benard instability of the system. The higher temperature difference/lower wall temperature and the stronger thermal slip, respectively, destabilize and stabilize the flow. Conclusively, the flow instability varies depending on the configuration.

Poiseuille–Rayleigh–Bénard instability of a channel flow with uniform cross-flow and thermal slip

The present study deals with the effects of floating flexible membrane on the instability of a gravity-driven flow down an incline. Linear stability of the flow system is explored using normal-mode analysis. Free surface gravity-driven flow is unstable at much lower Reynolds numbers. Instability of such a flow can be controlled either by changing behavioral of the lower wall or by altering the surface waves at the free surface which is done here by including a floating flexible membrane at the top of the liquid layer. Influence of membrane tension is taken into account in terms of stress jump at the free surface. The Orr-Sommerfeld system of the flow is solved numerically using spectral collocation method. The results displays a destabilizing role of membrane tension for a wide range of parameters. The growth rate of the perturbation waves increases with an increase of membrane tension and the critical Reynolds number becomes smaller. Therefore, it is possible to enhance the instability of the flow system with help of membrane properties, which may be useful in Ocean engineering and coating industries.

Muhammad Sani

Papers

Effect of imposed shear on the dynamics of a contaminated two-layer film flow down a slippery incline

Effect of imposed shear on the dynamics of a contaminated two-layer film flow down a slippery incline

Poiseuille–Rayleigh–Bénard instability of a channel flow with uniform cross-flow and thermal slip

Stability Analysis of a Film Flow Down an Incline in the Presence of a Floating Flexible Membrane

Effect of a floating elastic membrane for stabilizing the film flow down a porous inclined plane