Stability of a thin viscous Newtonian fluid draining down a uniformly heated porous inclined plane is examined. The long-wave linear stability analysis is performed within the generic Orr-Sommerfeld framework both theoretically and numerically. An evolution equation for the local film thickness for two-dimensional disturbances is derived to analyze the effect of long-wave instabilities. The parameters governing the film flow system and the porous substrate strongly influence the wave forms and their amplitudes and hence the stability of the fluid. The long-time wave forms are either time-independent wave forms that propagate or time-dependent modes that oscillate slightly in the amplitude. The role of permeability and Marangoni number is to increase the amplitude of the disturbance leading to the destabilization state of the film flow system. The permeability of the porous medium promotes the oscillatory behavior. © 2010 Elsevier Ltd. All rights reserved.

Usha R

Department of Mathematics

Evolution equations

FILM FLOWS

Inclined planes

LIQUID-FILM FLOW

LOCAL FILM THICKNESS

LONGWAVE INSTABILITY

MARANGONI NUMBERS

Oscillatory behaviors

Porous media

porous medium

POROUS SUBSTRATES

Time-dependent

VISCOUS NEWTONIAN FLUIDS

Wave forms

Contacts (fluid mechanics)

FLUID MECHANICS

FLUIDS

Heat exchangers

Heat transfer

Linear stability analysis

Liquid films

Machinery

Porous materials

Waves

System stability

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IIT Madras

Chemical Engineering Science

Steady solutions of an inverse problem relevant to gravity-driven film flows over an undulated slippery bottom are considered. Given a target free surface shape, the goal is to obtain the corresponding bottom topography of a slippery substrate which causes the specified free surface shape for a film flowing over it. The approaches followed by Sellier (Phys Fluids 20:062106, 2008) for creeping films and by Heining and Aksel (Phys Fluids 21:083605, 2009) for inertial films for reconstructing a rigid bottom topography for a target free surface profile are extended to the reconstruction of a slippery bottom topography. The model equations for film thickness above the bottom topography are derived for creeping flows under lubrication approximation and for inertial films using the weighted-residual integral boundary layer method and are solved numerically. The influence of inertia, slip parameter and surface tension on the shape of the reconstructed bottom topography is analyzed for different prescribed free surface shapes (sinusoidal, trench and bell-shaped). It is observed that the nonlinearities that appear in the reconstructed rigid bottom substrate with no slip at the substrate are suppressed by seeking the bottom substrate to be reconstructed as a slippery substrate. A spatial linear stability analysis of the corresponding direct problem is examined using Floquet theory, and the results reveal that the slip parameter and surface tension have a high influence on the critical Reynolds number. The results provide a strategy for controlling surface defects in a gravity-driven film over a substrate; namely, in order to achieve a target free surface profile, one can design the bottom substrate to be an undulated rough/textured/grooved or a superhydrophobic surface which can be modelled as an undulated smooth substrate with velocity slip at the substrate. © 2016, Springer-Verlag Wien.

Acta Mechanica

Steady solution and spatial stability of gravity-driven thin-film flow: reconstruction of an uneven slippery bottom substrate

This manuscript provides a theoretical stability analysis for the configuration of two-layer immiscible flow down an inclined plane with velocity slip along the incline in the limit of zero Reynolds number. Surfactants may be present at the air–liquid interface, liquid–liquid interface, or both. In addition to an Orr–Sommerfeld analysis (again at zero Reynolds number), a long wavelength stability analysis is performed and the results are shown to be consistent. The interface mode, namely the mode of instability that arises because of viscosity stratification, is examined. Stability results (growth rates as a function of slip parameter and neutral stability boundaries) for various configurations of viscosity, surfactant placement, and layer thickness are compared with those of the previous literature and found to agree. It is found that velocity slip along the inclined plane reduces the maximum growth rate of instabilities in configurations where they occur, and the range of unstable wave numbers shrinks as well, indicating that slip has a promise for stabilization. This suggests that there is a possibility of using this favourably as a control option for two-layer flows in the absence or presence of surfactants, in relevant applications by designing the substrate to be a porous substrate with small permeability or a slippery substrate or a rough substrate or a hydrophobic substrate which can be modelled as substrates with velocity slip. © 2015, Springer-Verlag Wien.

Effects of velocity slip on the inertialess instability of a contaminated two-layer film flow

Instabilities of a pressure driven two-layer Poiseuille flow confined between a rigid wall and a Darcy-Brinkman porous layer are explored. A linear stability analysis of the conservation laws leads to an Orr-Sommerfeld system, which is solved numerically with appropriate boundary conditions to identify the time and length scales of the instabilities.fde The study uncovers the coexistence of twin instability modes, (i) long-wave interfacial mode-engendered by the viscosity stratification across the interface and (ii) finite wave number shear mode-originating from the inertial stresses, for almost all combinations of the viscosity (μr) and thickness (h r) ratios of the liquid layers. The presence of the porous layer reduces the frictional influence on the films, which significantly alters the length and time scales of the shear mode while the interfacial mode remains dormant to this effect. This is in stark contrast to the two-layer flow confined between non-porous plates where an unstable interfacial (shear) mode is observed when μr&gt;hr2(μr&lt;hr2). The study reveals that strength of the shear mode, (a) increases with porosity, (b) initially increases and then becomes constant with porous layer thickness, (c) initially increases then reduces with increase in permeability, and (d) reduce with increase in the stress jump coefficient at the porous-liquid interface. Moreover, the gravity expedites the destabilization of both the modes in the inclined channels as compared to the similar non-inclined channels. The parametric study presented can find important applications in enhancing heat and mass transfer, mixing, and emulsification especially in the microscale flows. © 2013 Elsevier Ltd.

Instabilities of a confined two-layer flow on a porous medium: An Orr-Sommerfeld analysis

The instabilities of a free bilayer flowing on an inclined Darcy-Brinkman porous layer have been explored. The bilayer is composed of a pair of immiscible liquid films with a deformable liquid-liquid interface and a liquid-air free surface. An Orr-Sommerfeld analysis of the governing equations and boundary conditions uncovers that this configuration can be unstable by a pair of long-wave interfacial modes at the free surface and at the interface together with a couple of finite wave-number shear modes originating from the inertial influences at the liquid layers. In particular, one of the shear modes originates beyond a threshold flow rate owing to the slippage at the porous-liquid interface and is found to be the dominant one even when the porous medium is moderately thin, porous, and permeable. The strength of the porous media mediated mode (a) grows with increase in porosity, (b) grows and then remains invariant with increase in thickness, and (c) initially grows and then decays with increase in the permeability of the porous layer. Further, the presence of a lower layer with smaller viscosity and a thicker upper layer is found to facilitate the growth of this newly identified porous media mode. Importantly, beyond a threshold upper to lower thickness and viscosity ratios and the angle of inclination the porous media mode dominates over all the other interfacial or shear modes, highlighting its importance in the bilayer flows down an inclined porous medium. The study showcases the importance of a porous layer in destabilizing a free bilayer flow down an inclined plane, which can be of importance to improve mixing, emulsification, and heat and mass transfer characteristics in the microscale devices. © 2013 American Physical Society.

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Instabilities of a free bilayer flowing on an inclined porous medium

The stability of a gravity-driven film flow on a porous inclined substrate is considered, when the film is contaminated by an insoluble surfactant, in the frame work of Orr-Sommerfeld analysis. The classical long-wave asymptotic expansion for small wave numbers reveals the occurrence of two modes, the Yih mode and the Marangoni mode for a clean/a contaminated film over a porous substrate and this is confirmed by the numerical solution of the Orr-Sommerfeld system using the spectral-Tau collocation method. The results show that the Marangoni mode is always stable and dominates the Yih mode for small Reynolds numbers; as the Reynolds number increases, the growth rate of the Yih mode increases, until, an exchange of stability occurs, and after that the Yih mode dominates. The role of the surfactant is to increase the critical Reynolds number, indicating its stabilizing effect. The growth rate increases with an increase in permeability, in the region where the Yih mode dominates the Marangoni mode. Also, the growth rate is more for a film (both clean and contaminated) over a thicker porous layer than over a thinner one. From the neutral stability maps, it is observed that the critical Reynolds number decreases with an increase in permeability in the case of a thicker porous layer, both for a clean and a contaminated film over it. Further, the range of unstable wave number increases with an increase in the thickness of the porous layer. The film flow system is more unstable for a film over a thicker porous layer than over a thinner one. However, for small wave numbers, it is possible to find the range of values of the parameters characterizing the porous medium for which the film flow can be stabilized for both a clean film/a contaminated film as compared to such a film over an impermeable substrate; further, it is possible to enhance the instability of such a film flow system outside of this stability window, for appropriate choices of the porous substrate characteristics. © 2013 American Institute of Physics.

Fulltext

Physics of Fluids

Thin film flow down a porous substrate in the presence of an insoluble surfactant: Stability analysis

The flow of a thin Newtonian fluid layer on a porous inclined plane is considered. Applying the long-wave theory, a nonlinear evolution equation for the thickness of the film is obtained. It is assumed that the flow through the porous medium is governed by Darcy's law. The critical conditions for the onset of instability of a fluid layer flowing down an inclined porous wall, when the characteristic length scale of the pore space is much smaller than the depth of the fluid layer above, are obtained. The results of the linear stability analysis reveal that the film flow system on a porous inclined plane is more unstable than that on a rigid inclined plane and that increasing the permeability of the porous medium enhances the destabilizing effect. A weakly nonlinear stability analysis by the method of multiple scales shows that there is a range of wave numbers with a supercritical bifurcation, and a range of larger wave numbers with a subcritical bifurcation. Numerical solution of the evolution equation in a periodic domain indicates the existence of permanent finite-amplitude waves of different kinds in the supercritical stable region. The long-time waveforms are either time-independent waves of permanent form that propagate or time-dependent modes that oscillate slightly in the amplitude. The presence of the porous substrate promotes this oscillatory behavior. The results show that the shape and amplitude of the waves are influenced by the permeability of the porous medium. © 2008 American Institute of Physics.

Journal	Chemical Engineering Science
ISSN	00092509
Open Access	No